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An architecture for parallel processing of "sparse" data streams

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An architecture for parallel processing of "sparse" data streams

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Content AN A R C H IT E C T U R E FO R P A R A L L E L PROCESSING O F "S P A R S E " DATA ST RE A M S by T h o m as T rillin g A D is s e r ta tio n P r e s e n t e d to the F A C U L T Y O F TH E G R A D U A T E SCHOOL UNIVERSITY O F SO U TH ERN CALIFORNIA In P a r t i a l F u lfillm e n t of the R e q u ir e m e n ts fo r the D e g re e D OCTOR O F P H IL O SO PH Y (C o m p u te r Science) A ugust 1978 UMI Number: DP22731 All rights reserved INFORMATION TO ALL USERS The quality of this reproduction is dependent upon the quality of the copy submitted. In the unlikely event that the author did not send a complete manuscript and there are missing pages, these will be noted. Also, if material had to be removed, a note will indicate the deletion. Dissertation Publishing UMI DP22731 Published by ProQuest LLC (2014). Copyright in the Dissertation held by the Author. Microform Edition © ProQuest LLC. All rights reserved. This work is protected against unauthorized copying under Title 17, United States Code ProQuest LLC. 789 East Eisenhower Parkway P.O. Box 1346 Ann Arbor, Ml 48106- 1346 UNIVERSITY O F S O U T H E R N CALIFORNIA TH E GRADUATE SC H O O L UNIVERSITY PARK LOS A N G ELES, C A LIFO R N IA 9 0 0 0 7 f h . D . eps ' n This dissertation, w ritten by Th 0 M AS T l % I LLfNG under the direction of h t S , ................ Dissertation Committee, and approved by all its members, has been presented to and accepted by the Dean of The Graduate School, in partial fulfillm ent of the re quirements for the degree of < S S Q 9 B O CTO R OF PH ILO SO PH Y Dean DISSERTATION COMMITTEE . ......... Chairman A C K N O W L E D G E M E N T S My long c lim b to the P h . D. s u m m it w ould not have b e e n p o s sib le w ithout the h elp of m a n y p eo p le. It is a p l e a s u r e to h av e this op p o rtu n ity to e x p r e s s m y a p p re c ia tio n fo r a ll th e ir e ffo rts. F i r s t and fo r e m o s t, I w ant to thank m y c o m m itte e c h a ir m a n , D r. Irv in g S. R eed, who p ro v id e d e s s e n tia l te ch n ica l guidance and p e r s o n a l e n c o u ra g e m e n t. It has b e e n a p riv ile g e to w o rk u n d e r so d istin g u ish e d a professor,.- one who h a s m a d e a g r e a t m a n y c o n trib u tio n s to the li t e r a t u r e and the d e v e lo p m e n t of both c o m p u te r s y s te m s and r a d a r s y s te m s . A lso , I a m m o s t g ra te fu l fo r the valuable a s s i s t a n c e of D r. S e y m o u r G in sb u rg a n d D r. J e r r y M. M endel, the o th e r two m e m b e r s of m y c o m m itte e . In addition, I a m ind eb ted to the m e m b e r s of the U. S. G. facu lty who s e r v e d on m y guidance c o m m itte e an d a ls o a s s i s t e d in o th e r w ay s: L ow ell D. A m d ah l, D r. W illiam J. C h a n d le r, D r. David R. F e rg u s o n , and D r. R o b e r t S. K ash ef. I w an t to e x p r e s s th an k s to m y e m p lo y e r, T ech n o lo g y S e rv ic e C o rp o ra tio n (TSC), who p ro v id e d tuition r e i m b u r s e m e n t and o th e r su p p o rt. A lso I w ant to thank m y c o lle ag u e a t TSC and frie n d fo r m a n y y e a r s , Jo h n S. B ailey , w h o se g r e a t m a th e m a ti c a l e x p e r tis e w a s of inv alu ab le a s s i s t a n c e . F in ally , I w an t to ack n o w led g e the s u p p o rt and e n c o u r a g e m e n t of m y fam ily , w h ich w as no le s s im p o r ta n t than the te c h n ic a l su p p o rt I r e c e iv e d . My w ife, E lk e, w as a c o n sta n t s o u r c e of in s p ira tio n , ________________________ii and d e s e r v e s m u c h of the c r e d it f o r the c o m p le tio n of this d i s s e r t a tion. My so n s, M ich a e l, A ndrew , and Steven, w e r e u s u ally u n d e r standing w hen a c a d e m ic a c tiv itie s took p r e c e d e n c e o v e r things they w ould like to have done w ith m e . T hanks a r e due m y p a r e n ts , Viola an d C h a r le s T rillin g fo r m a n y r e a s o n s . In addition, m y fa th e r w as of g r e a t a s s i s t a n c e in the editing and p ro o f re a d in g of this d i s s e r t a tion. A B S T R A C T An im p o r ta n t c la s s of c o m p u te r p r o b le m s involves p a r a lle l p r o c e s s in g of " s p a r s e " d ata s t r e a m s . A " s p a r s e " data s t r e a m is one that co n tain s u sefu l in fo rm a tio n only d u rin g o c c a s io n a l b u r s ts of a ctiv ity , w hen s o m e event o c c u r s . The v a st m a j o r i t y of the tim e su ch a d ata s t r e a m co n tain s only u s e le s s p s e u d o - d a ta . E x a m p le s of " s p a r s e " data s t r e a m s a r e d e s c r ib e d f r o m th e d i v e r s e a r e a s of: m e d ic a l a p p lic a tio n s , r a d a r s y s te m s , and s e is m ic d ata a n a ly s is . " S p a r s e " d ata s t r e a m s a r e p o te n tia lly a light p r o c e s s in g load, p r o vided that the b u r s ts of a c tiv ity can b e re c o g n iz e d quickly. C o n v en tional p a r a lle l p r o c e s s in g a r c h i t e c t u r e s a r e not su ited to the p r o c e s s in g of " s p a r s e " d a ta s t r e a m s b e c a u s e they p r o c e s s e v e r y input of e v e r y d ata s t r e a m a c c o rd in g to a fixed s to r e d p r o g r a m . The c o n ventional m e th o d d o e s not a llo w the p r o c e s s in g s y s te m to take a d vantage of the " s p a r s e n e s s " of the data being p r o c e s s e d . In this d is s e r ta tio n , a new a r c h i t e c t u r e is d ev elo p ed for p a r a l l e l p r o c e s s in g of " s p a r s e " data s t r e a m s . The k ey e le m e n t of this a r c h i t e c t u r e is th at a r e la tiv e ly s m a ll n u m b e r of p r o c e s s o r s a r e s h a r e d am ong a la rg e n u m b e r of d ata s t r e a m s . F o r b re v ity , the t e r m " p r o c e s s o r - s h a r i n g a r c h i t e c t u r e " is u se d to d e s c r ib e this new a r c h i t e c t u r e . A g e n e r a liz e d v e r s io n of the p r o c e s s o r - s h a r i n g - a r c h i - te c t u r e is d e s c r ib e d . Then, s p e c ia liz e d co n fig u ra tio n s a r e developed fo r the th r e e e x a m p le s of " s p a r s e " d a ta s t r e a m s . _______________________________________________________________ iv M ost p r o b le m s '.req u irin g 'p a r a lle l p r o c e s s in g of " s p a r s e " d ata s t r e a m s can be c a te g o riz e d a s "ev e n t re c o g n itio n ." T hat is, the function of the p r o c e s s o r is to re c o g n iz e the u n u su al event of i n t e r e s t that r e s u lts in a b u r s t of a c tiv ity on a data s t r e a m . This p r o b le m c an be c o n s id e r e d a s a ch o ice b e tw ee n a lte r n a t e s ta tis tic a l h y p o th e s e s , b e c a u s e d a ta s t r e a m s a r e g e n e r a lly su b jec t to ra n d o m n o ise o r o th e r u n c e r ta in ty that c an give a fa ls e indication of a ctiv ity . Two s ta t is tic a l d e c is io n m e th o d s a r e c o n sid e re d : the m e th o d of N e y m an and P e a r s o n , that u s e s fixed s iz e s a m p le s , and the seq u e n tia l a n a l y sis m e th o d of W ald that v a r ie s the s a m p le s iz e . A n a d a p ta tio n of s e q u e n tia l a n a ly s is m e th o d s c alled the s eq u e n tia l o b s e r v e r is d e s c r ib e d and a n aly z e d . T his m e th o d is shown to be effective and e f ficien t fo r u se w ith the p r o c e s s o r - s h a r i n g a r c h i t e c t u r e . The vital q u e stio n in the u s e of the p r o c e s s o r - s h a r i n g a r c h i te c t u r e is: "how m a n y p r o c e s s o r s w ill be r e q u ir e d ? " E q u atio n s fo r e s tim a tin g the n u m b e r of p r o c e s s o r s a r e d ev elo p ed . F o r ty p ical c a s e s , the n u m b e r of p r o c e s s o r s r e q u ir e d is found to v a ry b e tw e e n one p e r c e n t and ten p e r c e n t of the n u m b e r of d a ta s t r e a m s . Thus the p r o c e s s o r - s h a r i n g a r c h i t e c t u r e can sav e u p w a rd s of ninety p e r c en t of the n u m b e r of p r o c e s s o r s that w ould be r e q u ir e d by a c o n ventional a r c h i t e c t u r e using one p r o c e s s o r p e r d ata s t r e a m . A c o m p u te r sim u la tio n of the p r o c e s s o r - s h a r i n g a r c h i t e c t u r e ap p lied to the e x a m p le of a r a d a r s y s te m is d e s c r ib e d . T his sim u la tio n obtained r e s u l t s that w e r e in v e ry c lo se a g r e e m e n t w ith p re d ic tio n s m a d e using the eq u atio n s d eveloped. v The sav in g s a c h ie v e d by the p r o c e s s o r - s h a r i n g a r c h i t e c t u r e a r e c r e d ite d to the u se of a s y s te m s a n a ly s is a p p ro a c h , w h ich r e s u lts in d esig n in g the a r c h i t e c t u r e to m e e t the n e ed s of the p r o b le m . C o n v entional g e n e r a l p u r p o s e a r c h i t e c t u r e s often in clu d e m a n y unneeded ex p en siv e c a p a b ilitie s . It is concluded th at m o r e e x te n siv e u se of this s y s te m s a n a ly s is a p p r o a c h to a r c h i t e c t u r a l d e sig n could r e s u lt in c o m p a r a b le sav in g s in m a n y o th e r a r e a s . vi T A B L E O F C O N T E N T S page A C K N O W L E D G E M E N T S ............................... ii A B S T R A C T ................................................................................... iv LIST O F F I G U R E S ............................................................................................... ix LIST O F T A B L E S .................................................................................................. xii C H A P T E R 1 IN T R O D U C T IO N ................................................................... 1 C H A P T E R 2 PRO CESSIN G P R O B L E M S INVOLVING P A R A L L E L "S P A R S E " DATA S T R E A M S ................................................................................. 5 2. 1 D ata S tr e a m s and the P r o c e s s i n g of D ata S t r e a m s ........................................................ 5 2. 2 " S p a r s e " M ed ical P a r a l l e l D ata S t r e a m s .......................................................................... 9 2. 3 " S p a r s e " P a r a l l e l D ata S tr e a m s C ontaining a Signal C o m b in ed w ith N o is e ...................................................................... 14 2. 4 " S p a r s e " S e is m ic D ata S t r e a m s ................ 19 2. 5 P r in c i p le s of P a r a l l e l P r o c e s s i n g of " S p a r s e " D ata S t r e a m s ................................ 20 C H A P T E R 3 A R C H IT E C T U R A L CONFIGURATIONS FO R P A R A L L E L PRO CESSIN G O F "S P A R S E " DATA S T R E A M S ....................................... 24 3. 1 B a sic E le m e n ts of the A r c h ite c tu r e fo r P a r a l l e l P r o c e s s i n g of " S p a r s e " D ata S t r e a m s ................. 24 3. 2 A C o n fig u ratio n fo r M ed ical M o n i t o r i n g ................................................................... 31 3. 3 A C o n fig u ratio n fo r R a d a r S y s t e m s .......................................................................... 34 3. 4 C o n fig u ra tio n fo r S e is m ic D ata S t r e a m s .......................................................................... 39 C H A P T E R 4 E V E N T R E C O G N I T IO N .................................................. 43 _______ vii 4. 1 S ta tis tic a l B a s is of E vent R e c o g n itio n ................................................................... page 43 4. 2 M ethods fo r E vent R e c o g n i t i o n ...................... 44 4. 3 E v en t R eco g n itio n in the P r o c e s s o r S h arin g A r c h ite c tu r e ........................................... 46 4 . 4 A n a ly sis of M out of N R ecognition, A N e y m an and P e a r s o n M e th o d ..................... 48 4. 5 A n a ly s is of the S equential O b s e r v e r . . . . 56 C H A P T E R 5 ESTIM A TIO N O F T H E N U M B ER O F PR O C ESSO R S R E Q U I R E D ........................................... 74 5. 1 A n a ly s is of P r o c e s s o r R e q u i r e m e n t s ................................................................................. 74 5. 2 E s tim a tio n of P ^ .......................................... 77 5. 3 E v en t R eco g n itio n a f te r L o s s of Initial P o r tio n of B u r s t of A c tiv ity ............... 84 C H A P T E R 6 C O M P U T E R SIM ULATION O F P R O C E S S OR SHARING A R C H I T E C T U R E ................................ 86 6. 1 D e s c rip tio n of the C o m p u te r D e s i g n ............................................................................. 86 6. 1. 1 M O P T M o d u l e ........................................... 88 6 . 1 . 2 TRSH M o d u le .............................................. 88 6. 1. 3 BSR M o d u le ................................................. 89 6 . 1 . 4 SN T B L M o d u le ........................................... 89 6 . 1 . 5 G PR O B M odule........................................... 89 6 . 1 . 6 S E Q P M o d u le .............................................. 92 6. 1. 7 SLOSS M o d u l e ..................... .................... 93 6. 2 C o m p u te r S im u latio n R e s u l t s ......................... 93 6. 2. 1 B a s ic C a s e , F iv e L o o k s ..................... 93 6. 2, 2 W id er B e a m C a s e , T en L o o k s ................................................................ 107 CONCLUSIONS.......................................................................................................... 120 R E F E R E N C E S 121 A P P E N D IX A The R atio of M j./N, fo r L a r g e N......................... 123 A P P E N D IX B C alc u la tio n of F(i) in E quation 5 - 1 7 ...................... 127 viii LIST O F F IG U R E S F ig u r e page 2. 1 The N o r m a l EGG .............................................................. 11 2. 2 DetaiLed View of a Single B e a t ................................................ 11 2. 3 T r a c e w ith F r e q u e n t P r e m a t u r e V e n tr ic u la r C o n t r a c t i o n s ......................................................................................... 11 2 .4 R a d a r Scan P a t t e r n w ith R a n g e -A z im u th D ata S t r e a m s ......................................................................................... 16 3. 1 G e n e r a l A r c h i t e c t u r e fo r P r o c e s s i n g of ’’S p a r s e " D ata S t r e a m s ................................................................ 25 3. 2 C o n fig u ra tio n fo r M ed ic a l M o n ito rin g .................................. 32 3. 3 C o n fig u ra tio n f o r R a d a r D ata S t r e a m s .............................. 36 3 .4 C o n fig u ra tio n fo r S e is m ic D a t a ............................................... 41 4. 1 R atio of M , / N ................................................................................... 55 opt 4. 2 State D ia g r a m fo r S eq u en tial O b s e r v e r w ith K = 3 and = 7 . . ...................................... 57 4. 3 S eq u en tial O b s e r v e r E q u iv ale n ts to M out of N D e t e c t o r s ........................................................................... 72 6. 1 O utput of the G P R O B M o d u l e ................................................... 91 6. 2 O utput of the S E Q P M odule. .................................................. 91 6. 3 T ab le of L o s s in Signal to N o ise R atio (db) fo r L im ite d M e m o r y S eq u en tial D etectio n , 5 L o o k s, F a l s e A l a r m P r o b . =0. 10E -06, 1 D e t e c t o r .................................................. 95 6. 4 T able of L o s s in Signal to N o ise R atio (db) fo r L im ite d M e m o r y S eq u en tial D etectio n , 5 L o o k s, F a l s e A l a r m P r o b . = 0. 1 0 E -0 6 , 2 D e t e c t o r s .............................................. 96 F ig u r e page 6. 5 T able of L o s s in Signal to N oise R atio (db) fo r L im ite d M e m o ry S equential D etection, 5 L ooks, F a ls e A l a r m P r o b . = 0. 1 0 E -06, 3 D e t e c t o r s .................................................................................... 6. 6 T able of L o s s in Signal to N oise R atio (db) fo r L im ite d M e m o r y S equential D etectio n 5 L o o k s, F a ls e A l a r m P r o b . = 0. 10E -06, 4 D e t e c t o r s .................................................................................... 6. 7 T ab le of L o s s in Signal to N oise R atio (db) fo r L im ite d M e m o r y S equential D etection, 5 L o o k s, F a l s e A l a r m P r o b . = 0. 10E -06, 5 D e t e c t o r s .................................................................................... 6. 8 S /N L o s s vs N u m b e r of D e te c to rs , 5 L ooks, P D = ° * 5’ P FA= 10" 7 ............................................................... 6. 9 S /N L o s s vs N u m b e r of D e te c to r s , 5 L o o k s, P = 0 7 P = 10-? D * ’ FA ............................................................... 6. 10 S /N L o s s vs N u m b e r of D e te c to r s , 5 L ooks, P D = 0 - 9 - P FA= 10- ? ............................................................... 6. 11 S /N L o s s vs N u m b e r of D e te c to r s , 5 L ooks, P D = 0 ' 5’ P FA~ 10" 5 ............................................................... 6. 12 S /N L o s s vs N u m b e r of D e te c to r s , 5 L ooks, P D = 0 - 7’ P F A = 10" 5............................................................... . . 105 6. 13 S /N L o s s vs N u m b e r of D e te c to r s , 5 L ooks, P D= ° * 9’ P FA= 10' 5............................................................... 6. 14 S /N L o s s vs N u m b e r of D e te c to r s , 5 L ooks, P D = ° - 5' P FA= 10' 8 ............................................................... 6. 15 S /N L o s s vs N u m b e r of D e te c to r s , 5 L ooks, p D = ° . 7 , P f a = I Q " 6 ............................................................... 6. 16 S /N L o s s vs N u m b e r of D e te c to r s , 5 L ooks, P D = ° * 9' P F A = 10" S ............................................................... 6. 17 S /N L o s s vs N u m b e r of D e te c to r s , 5 L ooks, P = 0 5 P = 10-® r D ’ FA ............................................................... 11 L 6. 18 S /N L o s s vs N u m b e r of D e te c to r s , 5 L ooks, P D = °* 7 ’ P F A = 10" 8 ............................................................... 112; X F ig u r e 6. 19 6. 20 6. 21 6. 22 6. 23 page S /N L o s s vs N u m b e r of D e te c to rs , 5 L ooks, P D= 0 .9 , P F A = 1 0 -8 113 . T ab le of S /N L o s s (db) for V ario u s and N u m b e r of D e te c to r s , 10 L o o k s, 1000 B in s, P = 10"® 1 114 FA ............................................................................................ \ U p p er L im it of P j-j vs N u m b e r of D e te c to r s , 10 L ooks, 1000 B ins, P FA = 1 0 -8 ............................................ 116! T able of S /N L o s s (db) fo r V a rio u s P ^ and N u m b e r of D e te c to rs ;' 10 L ooks, 1000 B ins, P F A = 10-?. .....................................................'117 U p p er L im it of P ^ vs N u m b e r of D e te c to r s , lb Looks, 1000 Bins, P _ = 10-?............................................... ITS FA xi LIST OF TABLES T ab le page 5. 1 P r o c e s s o r R e q u ir e m e n ts fo r C a s e w ith P B = 0 . 0 1 ...................................................................................................... 78 5. 2 P r o c e s s o r R e q u ir e m e n ts fo r C ase w ith P b = 0 . 0 5 ..................................................................................................... 79 5. 3 P r o b a b ility of P V C 's fo r a S am p le of 100 B e a ts in w hich the R atio of P V C 's to N o r m a l B ea ts is 0. 1 ........................................................... 83 xii C H A P T E R 1 IN TRO D U C TIO N Since 1946, w hen ENIAC in itia te d the age of the e le c tro n ic d ig i ta l c o m p u te r, th e r e have b een incredibly- ra p id im p r o v e m e n ts in the sp eed , siz e , m em ory- c a p a c ity and o v e r a ll p r o c e s s in g c a p a b ilitie s of d ig ita l s y s te m s . F o r the f i r s t tw enty y e a r s o r so, m o s t of the p r o g r e s s w as due to a d v a n c e s in c o m p o n en t technology, su ch a s vacu u m tubes being re p la c e d by t r a n s i s t o r s , follow ed by in te g ra te d c ir c u its , an d finally by la rg e s c a le in te g ra tio n . H o w ev e r, p r o g r e s s in this a r e a h a s slo w ed down, p a r tia lly b e c a u s e in h e re n t lim ita tio n s, su ch as the s p e e d of e le c t r i c a l p ro p a g a tio n , a r e being a p p ro a c h e d and the e m p h a s is h as b een tu rn in g to im p r o v e m e n ts in s y s te m a r c h i t e c t u r e .^ W hile a d v a n c e s in s y s te m a r c h i t e c t u r e have b e e n o c c u r r in g in m a n y a r e a s , p a r a l l e l p r o c e s s in g has b een re c e iv in g the m o s t a t t e n tion, and has r e s u lte d in the m o s t im p r e s s i v e gains in s y s t e m c a p a b il ity. P a r a l l e l p ro c e ss in g has b een r e a liz e d in a v a r ie ty of c o n f ig u r a tio n s. P r o b a b ly the b e s t know n e x a m p le is the a r r a y p r o c e s s o r o r SIMD (Single I n s tru c tio n M ultiple D ata S tr e a m ), in w hich a n u m b e r of p r o c e s s o r s ex ecu te the s a m e p r o g r a m . This c o n fig u ra tio n is w ell su ite d to p r o b le m s having in h e re n t p a r a l l e l i s m such a s a m a t r i x in v e rs io n , but is difficult to u se e ffe ctiv ely o th e rw is e . The IL LIA C IV 2 c o m p u te r u ses this type of a r c h i t e c t u r e , having 64 p a r a l l e l p r o c e s s o r s . A n o th e r u sefu l a r c h i t e c t u r e , p ip elin e, has the p r o c e s s o r s 1 in a row , like a n a u to m o b ile a s s e m b l y line, and th e input d a ta flows down the line and is o p e r a te d on se q u e n tia lly by e a c h p r o c e s s o r . T his type of a r c h i t e c t u r e is u sefu l in p r o c e s s in g long s tr in g s of data, 3 su ch a s v e c to r s w ith m a n y c o m p o n en ts. The CDC ST A R -100 is one of the la r g e s t e x istin g p ip e lin e c o m p u te r s . A n o th e r in te r e s tin g a r - 4 c h ite c tu r e , G A P P , (C ontent A d d r e s s a b le P a r a l l e l P r o c e s s o r s ) has the p r o c e s s o r s a d d r e s s e d by co n ten ts, not by location o r n u m b e r. 5 The ST A EA N is the only ex istin g c o m p u te r of th is type. It w as d e signed fo r a i r tra ffic c o n tro l, but could be u s e d in su ch ta sk s a s d ata m a n a g e m e n t. R e c e n t a r c h i t e c t u r e s have co m b in ed p a r a l l e l p r o c e s s in g f e a tu r e s w ith s y s te m c a p a b ilitie s . F o r e x am p le, G. L ip o v sk i and A . T rip a th i^ have p r o p o s e d a flexible a r c h i t e c t u r e w ith a n u m b e r of s h o r t w o rd le n g th m i c r o p r o c e s s o r s . T h e s e m i c r o p r o c e s s o r s can be u s e d fo r p a r a l l e l p r o c e s s i n g of s h o r t w o rd le n g th d ata s t r e a m s , o r the m i c r o p r o c e s s o r s can be c o m b in ed fo r s e r i a l p r o c e s s in g of lo n g e r w o rd le n g th data s t r e a m s . The a r c h i t e c t u r e s d e s c r ib e d above w e r e a ll d e sig n e d u n d e r an im p lic it a s s u m p tio n that all the d ata s t r e a m s alw ay s co n tain s ig n ifi cant, valid d a ta. T h e s e a r c h i t e c t u r e s a r e w e ll su ite d to su ch p r o b le m s , and p r o c e s s e v e r y input of e v e r y d a ta s t r e a m in the s a m e m a n n e r , a c c o r d in g to a fixed s to r e d p r o g r a m . H o w ev er, th e r e is an im p o r ta n t c la s s of p r o b le m s involving la r g e n u m b e rs of p a r a l l e l " s p a r s e " d ata s t r e a m s , i. e. , ones that have u sefu l d ata only a t in fre q u e n t in te r v a ls . (T he t e r m " s p a r s e " d ata s t r e a m w ill be d e s c rib e d in m o r e d e ta il in S ection 2. 1, an d e x a m p le s given in S ectio n s 2 .2 , 2 .3 , and 2 . 4 . ) C onventional p a r a l l e l p r o c e s s o r s have no 2 m e a n s of taking a d v a n ta g e of the " s p a r s e " n a tu re of su ch d ata s t r e a m s to re d u c e the p r o c e s s in g Load. T hey w ould r e q u ir e d e d ic a tion of a s e p a r a te p r o c e s s o r to e v e r y d ata s t r e a m even though e v e r y individual p r o c e s s o r h as only a light load. T his is c l e a r l y a m o s t inefficient and o v e rly c o s tly d e sig n a p p ro a c h , one that m a y w ell m a k e d ig ital p r o c e s s in g u n fe a sib le . C e r ta in ly it w ould be m u c h m o r e e f ficien t and e c o n o m ic a l to g r e a tly re d u c e the n u m b e r of p r o c e s s o r s and to have e a c h of the p r o c e s s o r s o p e ra te n e a r c a p a c ity lev el by a s s u m in g the light loads of a n u m b e r of d ata s t r e a m s . The p u rp o s e of this d i s s e r t a t i o n is to supply an a r c h i t e c t u r e to m e e t this need. An a r c h i t e c t u r e w ill be d e s c r i b e d in C h a p te r 3 th at a c h ie v e s sav in g s of an o r d e r of m a g n itu d e in the n u m b e r of p r o c e s s o r s r e q u ir e d , w ith v irtu a lly no lo ss in p e r f o r m a n c e . In so m e p r o b le m s , the new a r c h i te c t u r e m a y be the only fe a sib le m e th o d of p r o c e s s in g the data. The d i s s e r t a t i o n is divided into sev e n c h a p te r s , w ith this i n t r o duction being C h a p te r 1. In C h a p te r 2, the c la s s of p r o b le m s involv ing " s p a r s e " d a ta s t r e a m s is d e s c r ib e d . T h r e e ty p ic a l e x a m p le s a r e c o n s id e r e d in s o m e d e tail. In C h a p te r 3, the a r c h i t e c t u r e fo r p a r a l l e l p r o c e s s in g of " s p a r s e " d a ta s t r e a m s is d e s c r ib e d , and c o n fig u ra tio n s a p p r o p r ia te fo r the s p e c ia l r e q u i r e m e n t s of e a c h of the e x a m p le a p p lic a tio n s a r e given. In C h a p te r 4, m e th o d s fo r r e c ognizing ev en ts of i n t e r e s t on the d ata s t r e a m s a r e d e s c r ib e d , a n a ly ze d , and c o m p a r e d . In C h a p te r 5, e q u atio n s a r e d e riv e d fo r e s tim a tio n of the n u m b e r of p r o c e s s o r s r e q u i r e d by a d ig ital s y s te m . In C h a p te r 6, a c o m p u te r sim u la tio n of the p r o c e s s o r s h a r in g a r c h i 3 te c tu r e is d e s c r ib e d , and s o m e in te r e s tin g r e s u lts a r e shown. F o r ty p ical c a s e s , sav in g s of o v e r 90% in the n u m b e r of p r o c e s s o r s u sed c o m p a r e d to co n v en tio n al a r c h i t e c t u r e s is a c h ie v e d . T h e s e r e s u l t s a g r e e quite w e ll w ith p re d ic tio n s m a d e using the equ atio n s of C h a p te r 5. A s u m m a r y of the d is s e r ta tio n , and the co n clu sio n s a r e given in C h a p te r 7. C H A P T E R 2 PROCESSING P R O B L E M S INVOLVING P A R A L L E L "S P A R S E " DATA ST RE A M S T his c h a p te r d e s c r i b e s the c la s s of p r o b le m s r e f e r r e d to in C h a p te r 1, fo r w hich co n v en tio n al a r c h i t e c t u r e s a r e to tally un su ited . T h e s e p r o b le m s involve p r o c e s s in g of p a r a l l e l " s p a r s e " d ata s t r e a m s . The t e r m " s p a r s e " is d e s c r ib e d , and the c r i t i c a l r e l a t i o n ship of the " s p a r s e n e s s " of a d ata s t r e a m to the s y s t e m p r o c e s s in g r e q u ir e m e n ts d is c u s s e d . T h r e e p r o b le m a r e a s r e p r e s e n t a t i v e of this c l a s s a r e c o n s id e r e d and the p r o c e s s in g r e q u ir e m e n ts of e a c h a r e a d e te r m in e d . A s e t of b a sic p r in c ip le s and a r c h i t e c t u r a l d e sig n o b je c tiv es is d e r iv e d by a b s tr a c tin g the f a c to r s c o m m o n to the th r e e p r o b le m a r e a s . T h e s e p r in c ip le s and o b je c tiv e s a r e in c o r p o r a te d in a g e n e r a l a r c h i t e c t u r e fo r p a r a l l e l p r o c e s s in g of " s p a r s e " d a ta s t r e a m s , w h ich is d e s c r i b e d in C h a p te r 3. 2. 1 D ata S tr e a m s and the P r o c e s s i n g of D ata S tr e a m s The t e r m d a ta s t r e a m is c o m m o n ly u se d to d e s c r ib e a flow of n u m e r ic a l v alu es a n d / o r in s tr u c tio n s into a d ig ital p r o c e s s o r . H ow e v e r , the t e r m can a ls o d e s c r i b e an y continuous seq u e n c e of outputs o r e v en ts f r o m a n in fo r m a tio n s o u r c e , su ch a sto ck /p ric e s com ing f r o m a ti c k e r tape, o r a n n o u n c e m e n ts of a i r lin e ,a r r iv a ls and d e p a r tu r e s co m in g f r o m a n a i r p o r t lo u d s p e a k e r. If a n u m b e r of d ata _____________5 s t r e a m s a r e g e n e r a te d s im u lta n e o u s ly by a single s o u r c e , o r by a se t of s o u r c e s having s o m e im p o r ta n t c o m m o n fe a tu re , then the d a ta s t r e a m s w ill be c a lle d p a r a l l e l d ata s t r e a m s . F o r e x a m p le , an a r r a y of a tm o s p h e r i c s e n s o r s m o n ito r in g s o m e re g io n of i n t e r e s t w ould be g e n e r a tin g p a r a l l e l m e te o r o lo g ic a l d a ta s t r e a m s . In m a n y situ a tio n s w h e r e data s t r e a m s a r e g e n e r a te d , the o u t p u ts of the in fo rm a tio n s o u r c e s m u s t be p r o c e s s e d in s o m e m a n n e r b e fo re they a r e of m u c h u s e to a h u m an o b s e r v e r . F r e q u e n tly the a m o u n t of d a ta is o v e rw h e lm in g , and s c r e e n in g is n e c e s s a r y to id e n tify the sig n ific a n t p o rtio n s of the d ata s t r e a m . In so m e c a s e s , a r i t h m e t i c o r lo g ical o p e ra tio n s a r e n e c e s s a r y to put the d a ta in a s ta n d a rd fo r m . S o m e tim e s d ata inputs m u s t be c o m p a r e d w ith o th e r d ata to d e t e r m i n e the d e g r e e of c o r r e la tio n . D igital p r o c e s s o r s a r e e x tr e m e l y w ell su ite d to a ll th e se ta s k s , and a r e being u se d m o r e and m o r e fre q u e n tly in a r e a s g e n e ra tin g d ata s t r e a m s . If a n o b s e r v e r is in te r e s t e d in only a lim ite d p o rtio n of a d ata s t r e a m , it is p o s s ib le fo r h im to r e c o r d the d ata of i n t e r e s t and p r o c e s s it la te r , "off lin e ." H o w ev e r, if the o b s e r v e r d e s i r e s to c o n tin u o u sly m o n ito r a d ata s t r e a m , it is not u s u a lly p r a c t i c a l to s to r e all the d ata s t r e a m inputs fo r a long p e r io d of tim e . It is g e n e r a lly m o r e effective to d e d ic a te a r e a l - t i m e p r o c e s s o r to the d a ta s t r e a m . T his p r o c e s s o r w ill avoid the p r o b le m of s to r a g e , and g r e a tly re d u c e the a m o u n t of d a ta that the o b s e r v e r has to d e a l w ith. If th e r e a r e a n u m b e r of p a r a l l e l d ata s t r e a m s , say N, then it is n e c e s s a r y to e ith e r have N s e p a r a t e p r o c e s s i n g s y s te m s , o r u se a p a r a l l e l p r o c e s s in g s y s te m w ith N p r o c e s s o r u n its. F o r p a r a l l e l d ata s t r e a m s , 6 it w ould n o r m a lly be the c a s e that the pro cessin g '"o f e a c h data s t r e a m w ould be the s a m e , th at is, the s a m e p r o g r a m w ould be fo l lowed fo r each d a ta s t r e a m , w ith only the n u m e r ic a l output values differin g f r o m s t r e a m to s t r e a m . A ty p ic a l e x a m p le of this type is the c a lc u la tio n of a tm o s p h e r i c r e f r a c tiv ity f r o m the equation: Ng = (F + 4810 p/T) , w h e re N is the r e f r a c tiv ity s T is the a tm o s p h e r i c te m p e r a t u r e in d e g r e e s K elv in F is the total p r e s s u r e of the a t m o s p h e r e in m illi b a r s and p is the p a r t i a l p r e s s u r e of the w a t e r v a p o r co m p o n en t. F o r su ch a p r o b le m , it w ould be e fficien t to u s e a co n v en tio n al a r c h i te c t u r e fo r p a r a l l e l p r o c e s s in g , th e SIMD c o n fig u ra tio n , sin g le input, m u ltip le d ata s t r e a m . A sin g le p a r a lle l p r o c e s s in g s y s te m w ould c l e a r l y be m o r e e c o n o m ic a l than N s e p a r a te p r o c e s s o r s , e a c h w ith its own c o n tro l unit and p r o g r a m . Of c o u r s e th is e x a m p le r e q u ir e d only a m in im a l a m o u n t of p r o c e s s in g , but o th e r c a s e s su ch a s p a t t e r n re c o g n itio n m ig h t involve e x te n siv e c a lc u la tio n s and logical op e r a tio n s . M any d ata s t r e a m s c an be c a te g o riz e d a s " s p a r s e . " T h at is, they have useful, valid outputs only d u rin g o c c a s io n a l b u r s t s of a c tivity, w h ich a r e c a u s e d by s o m e e x te r n a l e v en t o r condition of i n t e r e s t o r c o n c e rn . The v a s t m a jo r ity of the tim e , su ch s t r e a m s h av e no output, o r only a n o is e - lik e p s e u d o - d a ta output. A n e x a m p le w ould be the output of a s e n s o r that is re s p o n s iv e only to v e ry high 7 wind v e lo c itie s . Such a d ata s t r e a m w ould h ave u sefu l d a ta only d u r ing a s t o r m o r h u r r ic a n e . T his u sag e of the t e r m " s p a r s e " is a n a l ogous to that in ‘ the t e r m " s p a r s e m a t r i x , " w h ich in d ic a te s one w h o se e le m e n ts a r e p r i m a r i l y z e r o e s . F o r su ch a m a t r i x , s p e c ia l s to r a g e and p r o c e s s in g m e th o d s a r e u sed to take ad v an tag e of this s p a r s e n e s s . O nly the n o n - z e r o e le m e n ts a r e s to r e d an d p r o c e s s e d . The a b s e n c e of d a ta on a n e le m e n t sig n ifies that the e le m e n t is a z e r o . S im ila r ly , it w ill be shown that it is p o s s ib le to quickly r e c ognize the o c c a s io n a l b u r s t s of a c tiv ity of a " s p a r s e d ata s tr e a m ." P r o c e s s i n g is r e q u ir e d only d u rin g th o se p e r io d s . T hus a " s p a r s e " d a ta s t r e a m is a p o te n tia l light p r o c e s s in g load, on a s ta tis tic a lly a v e r a g e d b a s is . The p o te n tia l saving in p r o c e s s o r load e x is ts only if a " s p a r s e " d a ta s t r e a m r e q u i r e s p r o c e s s in g o v e r a sig n ific a n t p e rio d of tim e . If the b u r s t of a c tiv ity is e x tr e m e l y s h o rt, su ch a s in the e x a m p le of a fire a l a r m , then th e r e is no p o te n tia l fo r sav in g s in p r o c e s s in g . Such d ata s t r e a m s w ill not be c o n s id e r e d " s p a r s e " in this d i s s e r t a tion. This p o in t is d is c u s s e d f u r th e r in S ection 2. 4, in r e g a r d s to s e is m ic d ata s t r e a m s . If a n u m b e r of p a r a l l e l " s p a r s e " d a ta s t r e a m s a r e to be p r o c e s s e d , it is p o s s ib le to tak e a d v an ta g e of the e x p ec te d light load by s h a r in g a r e la tiv e ly s m a ll n u m b e r of p r o c e s s o r s am o n g a la rg e n u m b e r of d a ta s t r e a m s . This is the e s s e n tia l f e a tu r e of the a r c h i te c t u r e d ev elo p ed in this d i s s e r t a t i o n fo r p a r a l l e l p r o c e s s in g of " s p a r s e " d ata s t r e a m s . The co n v en tio n al a r c h i t e c t u r e s d is c u s s e d in C h a p te r 1 p r o c e s s a ll the d a ta s t r e a m s "blindly" in an id en tical 8 m a n n e r and do not h av e any c a p a b ility fo r d e te r m in in g w h en a d ata s t r e a m should be p r o c e s s e d , and w hen it should be d is c a r d e d . In C h a p te r 6, it w ill be show n th at a new p r o c e s s o r - s h a r i n g a r c h i t e c tu r e that u tiliz e s s u c h c a p a b ility can a c h ie v e a sav in g s in p r o c e s s o r s of o v e r 90 p e r c e n t, w ith a m in im a l lo ss in c ap a b ility . An im p o r ta n t type of p r o c e s s in g is the m o n ito r in g of a d a ta s t r e a m o r a s e t of p a r a l l e l d a ta s t r e a m s to d e t e r m i n e if e v ery th in g is n o r m a l, o r if s o m e u n u su al condition o r event of i n t e r e s t is o c c u r rin g . The p r o c e s s o r d o es nothing if the sta tu s is n o r m a l, but is s u e s a n a l e r t fo r the u n u su a l co ndition o r event. M any su ch p r o b le m s in volve ’’s p a r s e " p a r a l l e l d a ta s t r e a m s su ch a s the e x a m p le s given in S ections 2. 2, 2. 3, and 2. 4. 2. 2 " S p a r s e " M ed rcal P a r a l l e l D ata S tr e a m s The field of m e d ic a l c a r e has m a n y e x a m p le s of " s p a r s e " p a r a lle l d a ta s t r e a m s , and th e p o te n tia l fo r u se of a u to m a tic p r o c e s s i n g of this d a ta is j u s t s ta r tin g to be r e a liz e d . O ne a r e a th at s e e m s p a r tic u la r ly a p p r o p r ia te fo r " s p a r s e " d a ta s t r e a m p r o c e s s in g is the m o n ito r in g of p a tie n t vital sig n s su ch a s: h e a r t r a te , r e s p i r a t i o n r a te , blood volum e and p r e s s u r e , an d le v els of v a rio u s e le c tr o l y te s in the blood. In m a n y c a s e s , th e s e sig n s a r e being m o n ito r e d ju s t a s a p r e c a u tio n , to d e te c t the o c c a s io n a l d e v iatio n s f r o m n o r m a l that r e q u i r e s o m e e m e r g e n c y a ctio n . A n e s p e c ia lly in te r e s t in g " s p a r s e " d ata s t r e a m is the output of e l e c t r o c a r d i o g r a m s , E C G 's , w hich a r e u s e d to m o n ito r h e a r t fu n c tion. A n EGG m e a s u r e s e l e c t r i c p o te n tia ls b e tw e e n v a rio u s p a r t s of _____________________ 9 the h e a r t . It is one of the m o s t effective m e th o d s of d e tec tin g d a n g e ro u s h e a r t co n d itio n s. A n o r m a l EGG t r a c e is shown in F ig u r e 7 2. 1. E a c h spike r e p r e s e n t s a h e a r t beat, an d the tim e s e p a r a tio n b e tw ee n b e a ts show s the h e a r t r a te . Sixty to one h u n d re d b e a ts p e r 7 m in u te is c o n s id e r e d the n o r m a l r a n g e . F ig u r e 2 .2 show s a d e tailed view of a sin g le b e a t p a tte r n , w hich c o n tain s th r e e s e g m e n ts , know n a s : the P w ave, the QRS c o m p le x , and the T w ave. A n a ly s is of d e v iatio n s f r o m this n o r m a l p a tte r n is u se d in the d ia g n o sis of v a rio u s h e a r t d i s o r d e r s s u c h a s : m y o c a r d ia l in fa rc tio n s , f lu tte r , and a r - • rtiy th m ia. Such a n a ly s is is, a t p r e s e n t, m a d e p r i m a r i l y by c a r d i o lo g is ts , who h ave u n d e rg o n e e x ten siv e tra in in g and p r a c t i c e . H ow e v e r , s o m e u se is a l r e a d y being m a d e of a u to m a tic d a ta p r o c e s s in g a n a ly s is . One v e ry u sefu l ta s k that h a s been p e r f o r m e d by a u to m a tic d a ta p r o c e s s in g fo r s o m e s p e c ia l p u r p o s e s is the re c o g n itio n of a type of a b n o r m a l b e a t c a lle d a p r e m a t u r e v e n tr ic u la r c o n tra c tio n , 7 P V C . F ig u r e 2. 3 show s an EGG t r a c e containing s o m e n o r m a l b e a ts , and s o m e P V C 's . The two c an be d is tin g u is h e d by the d if f e r e n c e s in a m p litu d e an d sh ap e a s w e ll a s the ch an g e in the i n t e r b e a t tim in g . The r e a s o n th at a u to m a tic m o n ito rin g of P V C 's w ould be u sefu l is th at P V C 's o c c u r c o m m o n ly in p e o p le w ith p e r f e c t ly sound, n o r m a l h e a r t s , a s w e ll a s in p eo p le w ith h e a r t c o n d itio n s. A t p r e s e n t, h o s p ita ls take E C G 's of p a tie n ts s u s p e c te d of having h e a r t p r o b le m s , r e c o rd in g the d a ta on p a p e r r o lls . T h e s e E C G 's a r e e x a m in e d , and any of the following r e s u l t s is c o n s id e r e d sig n ific a n t an d d a n g e r o u s , g re q u ir in g m e d ic a tio n a n d / o r a tte n tio n by the p h y sic ia n . 10 F ig u r e 2. 1. The N o r m a l ECG QRS Complex F ig u r e 2. 2. D e ta ile d V iew of a Single B e a t PVC PVC PVC F ig u r e 2. 3. T r a c e w ith F r e q u e n t P r e m a t u r e V e n tr ic u la r C o n tra c tio n s 1. P V C 's o c c u r r in g m o r e fre q u e n tly than one p e r ten h e a r t b e a ts . 2. P V C 's in g ro u p s of two o r th re e . 3. P V C 's landing n e a r the T w ave (the T w ave is the h e a r t 's r e s t p e rio d , an d this situ a tio n could o v e r s t r a i n the h e a r t) . 4. P V C 's o c c u r r in g in a v a r ie ty of s h a p e s . (T h is in d ic a te s that the P V C 's a r e o rig in a tin g f r o m m o r e than one s o u r c e .) H o sp ital p a tie n ts w ith k now n h e a r t co nditions w ould n o r m a lly be p la c e d in a c o r o n a r y in te n siv e c a r e unit, CIGU, w h e re they would r e c e iv e fr e q u e n t o r c o n sta n t ECG m o n ito r in g . T his d ata s t r e a m m ig h t, o r m ig h t not, be " s p a r s e , " depending on th e co ndition of the p a tie n t. P a tie n ts in the h o s p ita l fo r r e a s o n s o th e r than h e a r t p r o b le m s do not n o r m a lly h av e E C G 's m o n ito r e d , and o c c a s io n a lly s e r io u s h e a r t p r o b le m s d ev elo p w h ich a r e not d e te c te d in tim e , r e s u ltin g in d a m a g e to the h e a r t o r ev en d eath . U n fo rtu n ately , it w ould be p r o h ib itiv ely e x p en siv e to m o n ito r e v e r y p a tie n t co n tin u o u sly w ith an ECG, e ith e r by the c o n v en tio n al p a p e r r e c o r d e r , o r by d ed icatin g a s o p h is tic a te d p r o c e s s o r to h im . H o w ev e r, a d a ta p r o c e s s i n g s y s te m taking a d v a n ta g e of the " s p a r s e " n a tu re of the d a ta s t r e a m could m o n ito r m a n y p a tie n ts w ith r e la tiv e ly few p r o c e s s o r s . It w ould c o n c e n t r a t e its e ffo rts on th o se p a tie n ts who h ave P V C le v e ls n e a r the c r i t i c a l r a tio , and d ev o te r e la tiv e ly little e ffo rt to p a tie n ts having few o r no P V C 's . 12 To give a n id ea how su ch a p r o c e s s i n g m e th o d m ig h t w o rk , a s s u m e th a t the p r o c e s s o r took a one m in u te s a m p le f r o m a p a tie n t e v e r y h o u r a s an in itia l step, and counted the P V G 's . F r o m this s a m p le , it is d e te r m in e d if the p a tie n t a p p e a r s to be having a c r i t i c a l ra tio of P V C 's , one in ten b e a ts . A s s u m in g the s a m p le co n tain ed a h u n d re d b e a ts , a count of ten o r m o r e P V C 's w ould be c a u s e fo r an a l e r t . H o w ev e r, a c o u n t of eight o r nine on this s h o r t s a m p le w ould still be of g r e a t c o n c e r n b e c a u s e the a c tu a l ra tio could w ell be . g r e a t e r than the c r i t i c a l ra tio . A m u c h lo n g e r s a m p le w ould have to be taken, to d e t e r m i n e if the a c tu a l ra tio is in fact below the c r itic a l r a t i o / w ith a su ffic ie n tly high co n fid en ce lev el. Of c o u r s e the d a n g e ro u s conditions lis te d a s n u m b e r s 2, 3, an d 4 w ould be s i m i l a r l y c h ec k e d , and a n a d e q u a te s a m p le siz e u sed . On the o th e r hand, if a p a tie n t h a d only one P V C , o r none, in a s a m p le of one h u n d re d b e a ts, then it w ould be r e a s o n a b ly safe to c o n clu d e th at the p a tie n t's a c tu a l r a tio is below the c r i t i c a l r a tio . T his d is c u s s io n gives a n in d icatio n of the m e th o d b y w hich d e c is io n s w ould be m a d e w ith a given p r o b a b ility . A m o r e d e ta ile d d is c u s s io n of e v e n t're c o g n itio n d e c is io n s is g iven in C h a p te r 4. An a c tu a l m e th o d f o r p r o c e s s i n g " s p a r s e " ECG s t r e a m s would undoubtedly be m o r e e la b o r a te than d e s c r i b e d h e r e , but u s e of the b a s ic p rin c ip le , devoting m o s t of the e ffo rt to the d a ta s t r e a m s s h o w ing e v id en ce of P V C 's , w ould p e r m i t handling of a la rg e n u m b e r of p a tie n ts w ith a r e la tiv e ly s m a l l n u m b e r of p r o c e s s o r s . T his w ill be p o s s ib le , of c o u r s e , only if;m o s t of the p a tie n ts a r e not having h e a r t p r o b l e m s . The a r c h i t e c t u r e of the d a ta p r o c e s s o r to handle such ____________13 ty p e s of p r o b le m s w ill.b e d e s c r ib e d ,in S ection 3. 2. 2 .3 " S p a r s e " P a r a l l e l D ata S tr e a m s C ontaining a Signal C o m b in ed w ith N oise T h e r e a r e a g r e a t v a r ie ty of d a ta p r o c e s s in g a p p lic a tio n s in volving re c o g n itio n of sig n a ls o r signal p a tte r n s in the p r e s e n c e of so m e b a c k g ro u n d d is tu r b a n c e level often called* n o ise. Som e e x a m p le s a r e : r a d a r s y s t e m s , c o m m u n ic a tio n n e tw o rk s , ra d io - a s tr o n o m y , and s o n a r s y s t e m s . A s u r v e illa n c e r a d a r s y s te m is a n e x c e lle n t e x a m p le of p a r a l l e l p r o c e s s in g of " s p a r s e " d a ta s t r e a m s . A r a d a r s y s te m lo c a te s t a r g ets in s p a c e by e m ittin g e n e r g y and s e n s in g the r e fle c tio n s of th at e n e r g y f r o m the t a r g e t s . The e n e r g y re f le c te d by the t a r g e ts and r e tu r n e d to the r a d a r is u s u a lly at a v e ry tlo w p o w e r level, often of the s a m e o r d e r of m a g n itu d e a s the th e r m a l n o ise th at is alw ay s, p r e s e n t . T his m e a n s th at it is often d ifficult to d is tin g u is h b e tw ee n t a r g e t s and n o ise . The ra tio of the re c e iv e d sig n al p o w e r to th a t of the n o ise p o w e r, S /N , d e t e r m i n e s how a c c u r a t e l y this d e c is io n can be m a d e . (S /N is u s u a lly m e a s u r e d in d e c ib e ls , db, w h e re : (2-1) S / N (db) = 10 lo g 1()S /N . It h as b e e n found th a t a r a d a r s y s te m can d e te c t t a r g e ts m o s t e ffe ctiv ely by c o n c e n tra tin g its e n e rg y both in s p a c e and in tim e to m a x im iz e the S /N . T h at is, a t r a n s m i s s i o n of a one m ic r o s e c o n d p u ls e once p e r seco n d , a t a n e n e r g y lev el of one m e g a w a tt, w ill give a f a r b e t t e r d e te c tio n p e r f o r m a n c e than a continuous t r a n s m i s s i o n a t 14 the s a m e a v e r a g e p o w e r level, one w att. The d iff e r e n c e b e tw e e n the m e th o d s is that, in the f i r s t c a s e , the r e t u r n e n e r g y is c o m p a r e d to n o ise r e c e iv e d o v e r one m ic r o s e c o n d . In the s ec o n d c a s e , the s a m e to tal a m o u n t of r e t u r n e n e r g y is c o m p a r e d to the n o ise r e c e iv e d o v e r a full seco n d . S im ila r ly , it is b e n e fic ia l to c o n c e n tr a te the e n e r g y in s p a c e by using an a n te n n a to focus the e n e rg y into a p o w e rfu l n a r r o w b e a m , r a t h e r than a w e a k e r w ide b e a m . A s u r v e illa n c e r a d a r 's ta s k is to c o n tin u o u sly m o n ito r s o m e s p e c ifie d v o lu m e c£ s p a c e a n d to d e te c t a ll ta r g e ts w ith in that volum e. ( T h e r e a r e a ls o s p e c ia l p u r p o s e r a d a r s , su ch a s tra c k in g r a d a r s , w h ich a r e d e d ic a te d to a sin g le t a r g e t . ) F o r the r e a s o n s g iv en above, the s u r v e illa n c e can b e s t be a c c o m p lis h e d by a seq u e n c e of b e a m s , e a c h c o v e r in g a r e la tiv e ly s m a ll p o rtio n of the s u r v e illa n c e v o lu m e. A c o m m o n m e th o d fo r a c c o m p lis h in g the seq u en cin g , o r scan n in g , is to m e c h a n ic a l ly r o ta te the a n te n n a th ro u g h the a n g le (o r a n g les) n e c e s s a r y to c o v e r the s u r v e illa n c e v o lu m e. The scan n in g fo r a tw o -d im e n s io n a l r a d a r is ill u s t r a t e d in F ig. 2. 4. In th is fig u re , the a n g u la r s e c tio n s , n u m b e r e d 1, 2, . . . , N, r e p r e s e n t the c e n t r a l p o rtio n s of s u c c e s s iv e b e a m s t r a n s m i t t e d in s e q u e n ce a s the a n te n n a r o ta te s in a c lo c k w is e d ir e c tio n . (A th r e e d im e n s io n a l r a d a r w ould s c a n in the v e r tic a l, o r e lev a tio n d ir e c tio n a l s o .) The r a d ia l d im e n s io n r e p r e s e n t s ra n g e . T he ra n g e to the t a r g e t is found by m e a s u r i n g the tim e in te rv a l, T^, b e tw ee n the pulse t r a n s m i s s i o n an d the a r r i v a l o f the r e t u r n . This tim e in te rv a l, m u l tip lied by the s p e e d of the e le c tr o m a g n e tic p ro p a g a tio n , c, g iv es the r o u n d - t r i p d is ta n c e f r o m the r a d a r to the ta r g e t, w h ich is tw ice the _____________________ 15 ROTATION DIRECTION F ig u r e 2. 4. R a d a r Scan P a t t e r n w ith R an g e-A z im u th D ata S t r e a m s . 16 t a r g e t ra n g e . This m e a s u r e m e n t of the tim e in te r v a l is m a d e by co n tin u o u sly s a m p lin g the re c e iv e d e n e rg y o v e r in te r v a ls equal to the tr a n s m i t t e d p u lse w idth. In the sen sin g of r e t u r n e n erg y , the r a d a r r e c e i v e r a c ts a s an " in te g r a te and d u m p " c i r c u i t o v e r a tim e equal to the tr a n s m i t t e d p u lse w idth, t* T his p r o c e s s im p o s e s a g r a n u la r ity on the ra n g e m e a s u r e m e n t of c t / 2 , w h ich h a s the effect of dividing the s u r v e i l lan ce ra n g e into w h at a r e c a lle d ra n g e b in s. In F ig . 2. 4, the ra n g e bins a r e n u m b e re d f r o m l . . . j . . . D , w h e r e D r e p r e s e n t s the m a x i m u m ra n g e of the s u r v e illa n c e v olum e. (In the d is c u s s io n , the t e r m p u lse w idth h a s b e en u se d fo r c la r ity , but a c tu a lly the t e r m effectiv e p u ls e w idth w ould have b e e n p r o p e r . In s o m e r a d a r s a long p u ls e of w idth k r is tr a n s m itte d , but is divided into k d is tin g u is h a b le s u b p u ls e s . V a rio u s te c h n iq u e s can be u sed to c o m b in e the e n e r g ie s of th e se k su b p u lse s in a m a n n e r that a c h ie v e s a d e te c tio n p e r f o r m a n c e 9 and ra n g e g r a n u l a r i t y e q u al to that of a p u lse of w id th j . ) G e n e r a lly the s u r v e illa n c e s c a n p a t t e r n o v e r la p s the b e a m s in a z im u th to a v o id le av in g " h o le s " in the c o v e r a g e . A point is thus i l lu m in a te d by a n u m b e r of b e a m s in s u c c e s s io n . T his n u m b e r, L, is c a lle d the n u m b e r of " lo o k s " a t a ta r g e t. L is r e la te d to the o th e r s c a n p a r a m e t e r s by the equation w h e re L. = the n u m b e r of looks B = the b e a m w id th (d e g re e s ) S = the s c a n r a te ( d e g r e e s / s e c ) I = the in te r v a l b e tw ee n p u ls e s (sec). F ig u r e 2. 4 i l l u s t r a t e s the c a s e of L = 5. T hat is, the s c a n p a r a m e t e r s r e s u l t in 5 s u c c e s s i v e b e a m s hitting the ta r g e t. The r e t u r n d ata f r o m a ll looks a t a given point in s p a c e a r e c o m b in e d to d e t e r m i n e w h e th e r o r not a t a r g e t is p r e s e n t . U se of a ll the in fo rm a tio n m a x i m i z e s the c a p a b ility of d is tin g u is h in g betw een a c tu a l t a r g e ts an d p s e u d o - t a r g e t s c a u s e d by n o ise. T his m e a n s that the r e t u r n s f r o m s u c c e s s i v e b e a m s a r e c o r r e l a t e d ra n g e bin by ra n g e bin. T hus, in F ig . 2 .4 , the D ra n g e bins can be c o n s id e r e d a s D p a r a lle l d a ta s t r e a m s , w ith e a c h s u c c e s s i v e a z im u th b e a m p ro v id in g new input s a m p l e s fo r a ll the d a ta s t r e a m s . T h e r e a r e s e v e r a l m e th o d s c o m m o n ly u s e d fo r r a d a r ta r g e t d e te c tio n . O ne m e th o d of p a r t i c u l a r i n t e r e s t is c a lle d M out' of N d e te c tio n . T h at is, e a c h input a m p litu d e is c o m p a r e d w ith a r e f e r ence th re s h o ld , T. If this a m p litu d e e x c e e d s T, it is c o n s id e r e d a hit; but if it does not, the input is c o n s id e r e d a m i s s . If a t le a s t M h its a r e r e c o r d e d in the N looks, the p a tte r n is d e c l a r e d to be a t a r get. T he e v alu a tio n of eac h input can be in e r r o r , e ith e r by a n u n u s u a lly s tro n g n o ise b u r s t ex ceed in g th e th re s h o ld , giving a f a ls e hit; o r by the t a r g e t sig n al fluctuating below the th r e s h o ld , giving a m i s s . H o w e v e r, th e co m b in in g of the r e s u lts of the N looks g r e a tly i n c r e a s e s the p r o b a b ility of m a k in g the c o r r e c t d e c is io n . E a r ly a n a ly s i s of the a p p lic a tio n of this m e th o d to r a d a r s y s t e m s w a s m a d e by J. H a r r in g to n , ^ and by G. U ineen and I. S. R eed . * ^ s’ M out of N d e te c tio n gives good d e te c tio n p e r f o r m a n c e , but is not e ffe ctiv e fo r taking a d v a n ta g e of " s p a r s e " d a ta s t r e a m s , b e c a u s e N inputs a r e r e q u ir e d b e f o r e a d e c is io n is m a d e . A n a lte r n a tiv e 18 m e th o d c a lle d the se q u e n tia l o b s e r v e r , d e s c r i b e d in C h a p te r 4, is m u c h m o r e effectiv e in re d u c in g the p r o c e s s i n g on u s e le s s d ata s t r e a m s . It w ill be shown in C h a p te r 4 th a t this i n c r e a s e in p r o c e s s ing e ffic ie n cy is a tta in e d w ithout any lo s s in d e te c tio n c ap a b ility . An a r c h i t e c t u r a l c o n fig u ra tio n fo r this type of p r o b le m is show n in S e c tion 3. 3. 2 .4 " S p a r s e 1 1 S e is m ic D ata S tr e a m s A n o th e r a r e a in w h ich " s p a r s e " p a r a l l e l d a ta s t r e a m s could be g e n e r a te d is in the m o n ito r in g a n d / o r p r o c e s s i n g of s e is m ic data. O ne p o te n tia l e x a m p le is the C a lte c h E a rth q u a k e D e tec tio n an d R e c o rd in g S y ste m , C ED A R S y s t e m . ^ ’ ^ T he CED A R S y s te m h a s a n e tw o rk of 148 s e i s m o m e t e r s s e n s in g e a r t h m o v e m e n ts a t v a rio u s lo c atio n s in S o u th e rn C a lifo rn ia . A t e a c h location, th e r e a r e o th e r s e n s o r s s u c h a s t r a i n g a u g e s, c r e e p m e t e r s , m a g n e t o m e t e r s , tilt s e n s o r s , an d ra d o n gas m e a s u r e m e n t d e v ic e s . A t p r e s e n t , the C ED A R S y s te m is u s e d only to d e te c t e a r t h q u a k e s , and to s t o r e the d a ta r e c o r d e d f r o m e a c h s e n s o r d u rin g the tim e p e r io d p r e c e d in g the e a r th q u a k e fo r la te r a n a ly s is . A n e a r t h quake la s ts fo r a r e la tiv e ly s h o r t p e r io d of tim e . T his tim e p e r io d is too b r i e f to -c o n s titu te a b u r s t of a c tiv ity . T hus the s e is m ic d ata s t r e a m s u s e d by the C ED A R s y s te m a r e not " s p a r s e " in the s e n s e the t e r m is u s e d in this d i s s e r t a t i o n . The p r o c e s s o r - s h a r i n g a r c h i te c t u r e w ould not be u sefu l fo r e a r th q u a k e d e te c tio n fo r the s a m e r e a s o n — the b u r s t of a c tiv ity is too b rie f. H o w ev e r, the p u r p o s e in re c o r d in g the s e n s o r d a ta is to a tte m p t to d i s c o v e r p a tte r n s that a r e p r e c u r s o r s of e a r th q u a k e s . If su ch p a t t e r n s c a n be identified, then the CED A R S y s te m wouLd m o n ito r a ll th e s e n s o r s su ch a s the s t r a i n g a u g es, an d a tte m p t to p r e d i c t e a r th q u a k e s . The d a ta s t r e a m s f r o m th e s e s e n s o r s w ould be " s p a r s e , " and the a r c h i t e c t u r e f o r p a r a l l e l p r o c e s s i n g of " s p a r s e " d a ta s t r e a m s w ould be of g r e a t u s e . A c o n fig u r a tio n of this type of a r c h i t e c t u r e fo r e a iT h q u a k e 'p re d ic tio n is ' ; i show n in S ectio n 3 .4 . It is in te r e s tin g to note th at the m e th o d u se d by the C ED A R S y s te m to d e te c t e a r th q u a k e s is e s s e n tia l ly a n M out of N d e te c tio n m e th o d s i m i l a r to that d e s c r i b e d fo r th e r a d a r e x a m p le in Section 2. 3. At e a c h s e i s m o m e t e r the a m p litu d e of m o v e m e n t is c o m p a r e d a g a in s t a r e f e r e n c e th re s h o ld . If the th re s h o ld is e x c e e d e d a t one s e i s m o m e t e r , th en the r e s u l t s of the o th e r s e i s m o m e t e r s in the v i cin ity a r e e x a m in e d . If a t le a s t M out of th e s e N s e i s m o m e t e r s have e x c e e d e d the th re s h o ld , then the d e c is io n is m a d e that a n e a r th q u a k e h a s o c c u r r e d . 2. 5 P r i n c i p l e s of P a r a l l e l P r o c e s s i n g of " S p a r s e " D ata S tr e a m s In the t h r e e e x a m p le s of S ections 2. 2, 2 .3 , an d 2 .4 , a s w e ll a s in m a n y o th e r a p p lic a tio n s of in te r e s t , the ta s k of the p r o c e s s o r is to m o n ito r a la r g e n u m b e r of " s p a r s e " p a r a l l e l d a ta s t r e a m s , and to r e c o g n iz e the o c c u r r e n c e of s o m e ev en t of i n t e r e s t . If th e s e d ata s t r e a m s r e p r e s e n t r e a l - t i m e inputs, th en it is n e c e s s a r y to h ave a s y s t e m f a s t enough to p r o c e s s a ll the d a ta in r e a l tim e , to avoid lo ss of d a ta . A s m e n tio n e d , a c o n v en tio n al a r c h i t e c t u r e w ould m e e t this r e q u i r e m e n t by having one p r o c e s s o r p e r d a ta s t r e a m . T he r e s u l t 20 w ould be th at m o s t of the tim e the p r o c e s s o r s w ould be o p e ra tin g on u s e l e s s p se u d o o r fa ls e d a ta. T h e re is thus a p o te n tia l fo r g r e a t sav in g s in p r o c e s s o r s by re d u c in g the n u m b e r of p r o c e s s o r s to a n u m b e r d e te r m in e d by the a v e r a g e load, p lu s s o m e s p a r e s to give a n a d e q u a te sa fe ty f a c to r fo r p e a k s in the load. T his re la tiv e ly s m a ll n u m b e r of p r o c e s s o r s w ould be s h a r e d a m o n g a ll the d a ta s t r e a m s a s n eed ed . Of c o u r s e , this a p p r o a c h in v o lv es s o m e p o s s ib ility of d a ta lo s s , w hen a n e x t r a o r d i n a r y p e a k load o c c u r s . H o w e v e r, this lo ss c a n b e k e p t m i n im a l a s w ill be d is c u s s e d in C h a p te r 5. T h e r e a r e two b a s ic r e q u i r e m e n t s fo r the a c h ie v e m e n t of th e s e sav in g s. The f i r s t r e q u i r e m e n t is a m e th o d to quick ly d e te r m in e w hen a d ata s t r e a m is having a b u r s t of a ctiv ity , an d thus r e q u i r e s p r o c e s s in g . The sec o n d r e q u i r e m e n t is an a r c h i t e c t u r e th a t is c ap a b le of d y n a m ic a lly a s s ig n in g p r o c e s s o r s to d ata s t r e a m s w h en r e q u ir e d , an d of taking th e m aw ay , w hen no lo n g e r n eed ed . The m e a n s of re c o g n iz in g the b u r s t of a c tiv ity of a d ata s t r e a m w ould d ep en d on the type of d a ta being m o n ito r e d . It is a s s u m e d th at the a c tiv ity h a s s o m e c h a r a c t e r i s t i c q u a lity th a t is s ta t is tic a lly s i g nificant, su c h a s a n i n c r e a s e in sig n al a m p litu d e , o r a ch an g e of f r e q uency, o r a n u n u s u a l p a tte r n . If su ch c h a r a c t e r i s t i c e x is ts , it can be u s e d to define an e v alu a tio n function ijr, an d a n o r d e r e d s e t of v alu es V, w h e r e v = tv vi V- T he function is a m a p p in g of an y sin g le input of a d a ta s t r e a m onto the s e t of v a lu es V. Low v alu es of V w ill be a s s ig n e d to inputs 21 th at a r e " n o r m a l ." H ig h e r v a lu e s of V w ill be a s s ig n e d to inputs in d ic a tiv e of the u n u su a l b u r s t of a c tiv ity . F o r e x a m p le , in the m e d ic a l m o n ito r in g , V could be the s e t {0, 1, . . , , 10}. F o r a n o rm a l, o r n e a r n o r m a l h e a r t b e a t, su c h a s show n in F ig u r e 2. 2, w ould a s s i g n the v alu e of 0, o r 1, or. p o s s ib ly 2. F o r a d e fin ite P V C , su ch a s th o se in d ic a ted in F i g u r e 2. 3, f w ould a s s i g n the value of 9 o r 10. F o r in d e te r m in a te c a s e s , w ould a s s i g n a value of 4, 5 o r 6. The e v alu a tio n function, is u sefu l fo r two r e la te d p u r p o s e s . F i r s t , it p r o v id e s a m e a n s of identifying d a ta s t r e a m s th at r e q u i r e p r o c e s s i n g . Second, the s u m of the e v alu a tio n v alu es of a n u m b e r of inputs c an be u s e d to r e c o g n iz e the u n u su al e v en t o r co n d itio n of in t e r e s t . F o r th is p u r p o s e , let \ be defin ed by: b X(a, b,s) = S X. i = a ' w h e r e a, b a r e a n y two in p u ts, n u m b e r e d r e la tiv e to s o m e a r b i t r a r y r e f e r e n c e , s is the n u m b e r of a n y d a ta s t r e a m , X. is th e value a s s ig n e d by $ to the ith 1* s input of the sth d a ta s t r e a m . The p a r a m e t e r \ is a m e a s u r e of the a c tiv ity of a p a r t i c u l a r d ata s t r e a m o v e r a sp e c ifie d p e r io d of tim e . The g r e a t e r the value of X» the m o r e lik ely it is th at the u n u su al event is o c c u r r in g on t h e 'p a r - tic u la r d a ta s t r e a m , a t th a t tim e . F o r the m e d ic a l d a ta s t r e a m e x a m p le , a r e la tiv e ly high v alu e of X f o r a- p a tie n t's d a ta s t r e a m o v e r a one m in u te s a m p le w ould sig n ify th at the p a tie n t's co ndition is of 22 s o m e c o n c e rn , r e q u ir in g f u r th e r in v e stig a tio n . F o r th a t p a tie n t, \ w ould be m e a s u r e d o v e r a lo n g e r p e r io d of tim e , p e r h a p s 10 m in u te s , to m a k e a d e c is io n w ith m u c h h ig h e r c o n fid e n ce . E v e n t re c o g n itio n is d i s c u s s e d in C h a p te r 4. It w ould be d e s i r a b l e fo r V to have a l a r g e n u m b e r of v alu es to m a k e f a s e n s itiv e e v alu a tio n function of e a c h input. H o w ev er, if the e v alu a tio n v alu es of a n u m b e r of inputs f r o m a p a r t i c u l a r d a ta s t r e a m a r e c o m b in e d to g e th e r in a \ ra tin g f o r the d a ta s t r e a m , then it is often a d e q u a te to u se a s e t V containing only two v a lu e s . The sav in g s in p r o c e s s i n g often ju s tify w hat is fre q u e n tly only a slig h t lo s s , a s s u m in g the a v e r a g in g is o v e r .a (reaso n ab le n u m b e r of inputs. F o r the r a d a r e x a m p le , V = {+ 1,0}. The fu nction a s s i g n s +1 to a n input c r o s s i n g the th re s h o ld , a n d 0 to a n input not c r o s s i n g the th re s h o ld . F o r this a p p lic a tio n , \ w ould be tak en o v e r a fixed n u m b e r of inputs, N, s ta r tin g a t input a. Thus: a + N -1 X(a, a+ N -1 , s) = 2 X . i = a l' 8 A n o th e r sim p lifie d v e rs io n , u sefu l fo r m a n y a p p lic a tio n s , is to let V = {+K ,-1}, w h e r e K is a p o s itiv e in te g e r. The u n u su a l c o n d i tion o r s ta tu s of i n t e r e s t is re c o g n iz e d if a n d only if the X ra tin g of the d a ta s t r e a m r e a c h e s s o m e th re s h o ld lev el, T his p r o c e s s in g g iv es a re c o g n itio n m e th o d eq u iv alen t to th at of a s e q u e n tia l o b s e r v e r , a s d e s c r ib e d in S ection 4. 3. 23 C H A P T E R 3 A R C H IT E C T U R A L CO NFIG U R A TIO N S FO R P A R A L L E L PR O C ESSIN G O F "S P A R S E " DATA ST R E A M S In this c h a p te r, a g e n e r a l a r c h i t e c t u r e fo r p a r a l l e l p r o c e s s in g of " s p a r s e " d ata s t r e a m s is d e s c r ib e d . Specific c o n fig u ra tio n s of th is type of a r c h i t e c t u r e a r e d ev elo p e d fo r the th r e e r e p r e s e n t a t i v e p r o b le m s given in C h a p te r 2. 3. 1 B a s ic E l e m e n ts of the A r c h i t e c t u r e fo r P a r a l l e l P r o c e s s i n g of " S p a r s e " D ata S tr e a m s A s p r e v io u s ly in d ic a ted , the k e y f e a tu r e of the a r c h i t e c t u r e fo r p a r a l l e l p r o c e s s i n g of " s p a r s e " d a ta s t r e a m s is that it ta k e s a d v a n tag e of the " s p a r s e " n a tu r e of the d a ta s t r e a m s to a c h ie v e a d r a s t i c re d u c tio n in the n u m b e r of p r o c e s s o r s r e q u ir e d , c o m p a r e d w ith c o n v en tio n al a r c h i t e c t u r e s . T his re d u c tio n is a tta in e d by hav in g a s m a l l e r n u m b e r of p r o c e s s o r s th an the n u m b e r of d a ta s t r e a m s , and s h a r in g th e m a m o n g the a c tiv e d a ta s t r e a m s . F o r this r e a s o n , the t e r m " p r o c e s s o r - s h a r i n g a r c h i t e c t u r e " is a p p r o p r ia te , an d w ill often be u s e d fo r b re v ity , w hen r e f e r r i n g to>the a r c h i t e c t u r e fo r p a r a l l e l p r o c e s s i n g of " s p a r s e " d a ta s t r e a m s . A b lo ck d i a g r a m of the a r c h i t e c t u r e f o r the tim e s h a r in g of p r o c e s s o r s a m o n g p a r a l l e l d ata s t r e a m s is show n in F ig u r e 3. 1. T he m a j o r fu n ctio n al a r e a s in the a r c h i t e c t u r e a r e : the input bus, the input seq u e n c in g c o n tr o l unit, the e v a lu a to r, the e v alu a tio n d ata 24 INPUTS FROM DATA STREAMS t I INPUT BUS I ' INPUT SEQUENTIAL CONTROL UNIT \p EVALUATOR l GOOD DATA FROM NEW DATA FROM EVALUATION ROUTING LOGIC ACTIVE STREAMS ACTIVE STREAMS DISCARD USELESS DATA ! BANK OF PROCESSORS I i AVAILABLE PROCESSOR SECTION ASSIGNED . PROCESSOR SECTION I I f I ASSIGN TO A PROCESSOR PUSHDOWN STACK ASSIGNED PROCESSOR TOP OF STACK AVAILABLE PROCESSOR POINTER TO TOP OF STACK BOTTOM ~ OF STACK DISCARD LOW EST A RATING A RATINGS OVERFLOW AND PRIORITY LOGIC OVERFLOW SIGNAL OUTPUT ALERTS F ig u r e 3. 1. G e n e r a l A r c h i t e c t u r e fo r P r o c e s s ing of " S p a r s e " D ata S t r e a m s . 25 ro u tin g logic, an d , m o s t im p o r ta n t, the b ank of p r o c e s s o r s . To u n d e r s ta n d the o p e ra tio n of this a r c h i t e c t u r e , it is n e c e s s a r y to know b o th the fu n ctio n of e ac h unit, an d the m a j o r p ath w ay s of in fo r m a tio n flow. To s im p lify the d e s c r ip tio n , it w ill be a s s u m e d that the f e v a l u a tio n functionjrriaps’ the inputs of a d a ta s t r e a m onto the 'set +K, ,-1 . The value of +K w ill be a s s ig n e d to inputs e v a lu a te d a s " p o s itiv e ," in d ic a tiv e of the ev en t. The value of -1 w ill be a s s ig n e d to inputs e v a lu a te d a s " n e g a tiv e ," not in d ic a tiv e of the event. A ll d a ta s t r e a m s feed the input b u s. F o r m o s t a p p lic a tio n s , the c u r r e n t input f r o m one of the d a ta s t r e a m s is s e le c te d f r o m the bus a c c o rd in g to a sig n al f r o m th e input seq u e n c in g c o n tro l unit. H o w ev e r, in s o m e c a s e s , s u c h a s d e s c r i b e d in S ection 3. 2, a m u lti- p o r t bus m a y be u s e d . The input seq u e n c in g an d s e le c tio n , an d the r e s u lta n t sa m p lin g r a t e is v e r y m u c h d e p en d e n t on the p r o b le m . F o r the m e d ic a l m o n ito r in g e x a m p le d e s c r i b e d in S ection 3. 2, the s a m p ling r a t e of a d a ta s t r e a m is a function of its X ra tin g , o r a ctiv ity . F o r the r a d a r d e te c tio n e x a m p le d e s c r i b e d in S ection 3 .3 , the s a m p ling r a t e is d e t e r m i n e d by p h y s ic a l dhd te m p o r a l p a r a m e t e r s of the d ata s o u r c e s , a n d is not u n d e r the c o n tro l of the d ig ita l s y s te m . F o r the e a rth q u a k e p r e d ic tio n e x a m p le d e s c r i b e d in S ection 3 .4 , the s a m p lin g r a t e of th e d a ta s t r e a m s could be v a rie d , but only w ith in lim its d e te r m in e d by p h y s ic a l f a c to r s . E v e r y input f r o m an y d a ta s t r e a m is f i r s t s e n t to the e v a lu a to r, w h ic h a s s i g n s to it the value of e ith e r +K, o r -1. 26 The r e s u l t of the e v alu a tio n of a n input f r o m a d ata s t r e a m c an be u tiliz e d in th r e e p o s s ib le w a y s. If the input is f r o m a d ata s t r e a m not p r e v io u s ly d e sig n a te d a s a c tiv e , w h ich is a lm o s t a lw ay s the c a s e fo r " s p a r s e " d a ta s t r e a m s , a n d the e v alu a tio n w a s -1, then the input is s im p ly d ro p p e d . (T hus the \J r e v a lu a to r s c r e e n s ; ; m u c h of the d a ta inputs, a n d k e e p s th e m f r o m e v e r / being a load on the p r o c e s s o r . ) If the input is f r o m a d a ta s t r e a m p r e v io u s ly d e sig n a te d a s a c tive, th en a p r o c e s s o r h as a l r e a d y b e en a s s ig n e d to that d a ta s t r e a m . In th is c a s e , the e v alu a tio n re s u lt, w h e th e r,+ K o r -1, is se n t to the a s s ig n e d p r o c e s s o r . The p r o c e s s o r w ill c o n s id e r the new input in co n ju n ctio n w ith the r e s u l t s of the e v a lu a tio n s of p r e v io u s inputs f r o m the d a ta s t r e a m . The p r o c e s s o r h as t h r e e p o s s ib le o p tio n s. F i r s t , it can d e c l a r e , o r re c o g n iz e , th at the u n u su a l e v en t o r sta tu s of i n t e r e s t h a s o c c u r r e d , and is s u e an a l e r t . T his a c tio n can h a p p en only if the X ra tin g of the a c tiv ity of the d a ta s t r e a m being p r o c e s s e d is q u ite high, and a +K e v alu a tio n of the new input is enough to m a k e the d e c l a r a t i o n d e c is io n w ith su ffic ie n t co n fid en ce. (T h is d e c isio n p r o c e s s is a n a ly z e d in C h a p te r 4. ) Second, it c a n d e c id e th at the d a ta s t r e a m is not r e a l l y a c tiv e and d r o p the d a ta s t r e a m . (In this c a s e , the p r o c e s s o r b e c o m e s u n a s s ig n e d , o r a v a ila b le , and is sw itch e d to the a v a ila b le p r o c e s s o r s e c tio n a s w ill be d e s c r i b e d .) T h is a c tio n c a n o c c u r only if the \ ra tin g of the d a ta s t r e a m w as q u ite low, a n d the f e v a lu a tio n of the new input w as -1 . T h ird , if the p r o c e s s o r d o es not h av e enough in fo r m a tio n to m a k e a d e c is io n w ith a d e q u a te co n fid en ce, th en it w ill w a it fo r m o r e data, re m a in in g a s s ig n e d to the d a ta s t r e a m . _______________________________ 27 If the input is f r o m a d a ta s t r e a m not p r e v io u s ly d e s ig n a te d a s a c tiv e , but the e v alu a tio n is +K, then the d ata s t r e a m b e c o m e s d e s ig n a te d a s a c tiv e . A p r o c e s s o r w ill be a s s ig n e d to this d ata s t r e a m , and the e v alu a tio n r e s u l t w ill be s to r e d by the new ly a s signec p r o c e s s o r . A s w ill be d e s c r ib e d , this a s s ig n e d p r o c e s s o r now b e c o m e s p a r t of the a s s ig n e d s e c tio n of the p r o c e s s o r bank. In the r a r e c a s e th a t th e r e is no p r o c e s s o r a v a ila b le fo r the new a c tiv e d a ta s t r e a m , an o v e rflo w s ig n al is g e n e r a te d , a c tiv a tin g p r i o r i t y logic. This logic d e t e r m i n e s w h ic h of the d a ta is of l e a s t value an d should be d ro p p e d . The c h o ice is b etw ee n the d a ta s t r e a m w ith the lo w e st \ ra tin g a n d the d a ta s t r e a m w ith the c u r r e n t input having a e v a l u atio n of +K. The m o s t im p o r ta n t an d m o s t c o m p le x a r e a is the b ank of p r o c e s s o r s . T he type of p r o c e s s o r u s e d w ill d ep en d on the p r o b le m . F o r a | g r e a t m a n y a p p lic a tio n s , the s e q u e n tia l o b s e r v e r , w h ich w as in tr o d u c e d in S ection 2. 5, w ill be quite e ffe ctiv e. T h is type of p r o c e s s o r , w h ic h is d e s c r i b e d an d a n a ly z e d in C h a p te r 4, is w e ll suited- to the p r o c e s s o r s h a r in g a r c h i t e c t u r e . T he bank of p r o c e s s o r s is d iv id ed into two s e c tio n s . The s e c tio n in d ic a te d on the left c o n tain s p r o c e s s o r s th a t h a v e b e e n a s s ig n e d to a c tiv e d a ta s t r e a m s , i. e. , th o se hav in g a high \ ra tin g . The s e c tio n on the rig h t c o n ta in s u n a s s ig n e d , o r a v a ila b le p r o c e s s o r s . O n ce a s s ig n e d to a d a ta s t r e a m , a p r o c e s s o r w ill n o r m a l l y r e m a i n d e d ic a te d to th at s t r e a m until it e ith e r r e c o g n iz e s the event, o r sta tu s of in te r e s t , an d i s s u e s an a l e r t , o r e l s e m a k e s a n eg ativ e d e c is io n , an d d r o p s the d a ta s t r e a m . W hen one of th e s e d e c is io n s is m a d e , then the p r o c e s s o r is t r a n s - 28 f e r r e d to the a v a ila b le p r o c e s s o r sectio n . O ne of the m a j o r d e sig n p r o b le m s of the p r o c e s s o r s h a r in g a r c h i t e c t u r e is the m e th o d of re c o g n iz in g w h e th e r o r not a d ata s t r e a m h a s b e e n a s s ig n e d to a p r o c e s s o r . If the a n s w e r is " y e s ," th en it is n e c e s s a r y to be a b le to lo c a te the a s s ig n e d p r o c e s s o r a s p r o m p t ly a s p o s s ib le , in o r d e r to p r o p e r l y ro u te the new d ata. T he a s s ig n e d p r o c e s s o r s w ill be o r g a n iz e d to f a c ilita te both the re c o g n itio n of a s s ig n e d d a ta s t r e a m s , and the lo c atio n of the a s s ig n e d p r o c e s s o r s . In a few p r o b l e m s , the inputs f r o m the d a ta s t r e a m s m ig h t o c c u r a t u n p re d ic ta b le in te r v a ls and in r a n d o m s e q u e n c e s . F o r this c a s e , a c o n ten t a d d r e s s a b l e m e th o d could be of g r e a t b en efit. A n o th e r p o s s ib ility w ould be to o r d e r the p r o c e s s o r s by s o m e p a r a m e t e r so that a b in a r y s e a r c h could be u se d . H o w ev e r, r e a r r a n g i n g of the p r o c e s s o r s w ould b e n e c e s s a r y to m a in ta in this o r d e r w h e n e v e r d a ta s t r e a m s c h a n g e d f r o m in a c tiv e and vice v e r s a . In s o m e c a s e s it m ig h t be s i m p l e s t to ju s t put a ll the a c tiv e p r o c e s s o r s in a b lo ck of c o n se c u tiv e lo c a tio n s. This w ould m in im i z e the s e a r c h a r e a . F o r m o s t a p p lic a tio n s , a ll the d a ta s t r e a m s a r e s a m p le d a t the s a m e r a te , in a r e p e titiv e se q u e n c e . F o r su ch a p p lic a tio n s , a 14 double linked c i r c u l a r lis t o rg a n iz a tio n of the p r o c e s s o r s w ould be the lo g ic a l ch o ice, b e c a u s e it p e r m i t s im m e d ia te re c o g n itio n of a s sig n ed d a ta s t r e a m s an d g iv es the lo c atio n of the a s s ig n e d p r o c e s s o r w ith o u t a n y s e a r c h in g . The m e th o d by w h ich this is a c c o m p lis h e d is b e t t e r e x p lain ed w ith a n i llu s tr a tiv e e x a m p le , and h a s t h e r e f o r e been p o stp o n ed to S ectio n 3 .4 . 29 A double linked c i r c u l a r lis t c o n s is ts of a se t of e le m e n ts at v a rio u s lo catio n s in a s to r a g e d e v ic e . E a c h e le m e n t c o n ta in s two " p o i n t e r s ," one giving the lo catio n of the p r e c e d in g e le m e n t, an d one giving the lo c atio n of the follow ing e le m e n t. It should b e n oted th at th e r e a r e o th e r linked lis t m e th o d s that do not r e q u i r e a s m u c h p o in te r s to r a g e . H o w e v e r, th e s e m e th o d s s a c r i f i c e c o n v e n ie n c e a n d fle x ib ility in su c h a r e a s a s in s e r tio n s an d d e le tio n s, w h ich a r e vital to the p r o c e s s o r s h a r in g a r c h i t e c t u r e . If one s t a r t s a n y w h e re on the lis t an d follow s the p o in te r s f r o m e le m e n t to e le m e n t, in e ith e r d ir e c tio n , one e v e n tu a lly r e a c h e s the s ta r tin g point, h e n c e the t e r m " c i r c u l a r . " O ne im p o r ta n t a d v a n ta g e of the double linked lis t o v e r o th e r linked lis ts is th a t in s e r tio n s an d d e le tio n s c an e a s ily be m a d e a t a n y p o in t on the lis t. F o r e x a m p le , s u p p o se it is d e s i r e d 't o d e le te the e le m e n t a t lo c a tio n L ^ , w h o se b a c k w a rd p o in te r is to lo c atio n a n d f o r w a r d p o in te r is to lo c atio n L p . A ll th at is n e c e s s a r y is to go to lo c a tio n a n d ch an g e the f o r w a r d p o in te r f r o m to L^,. S i m ila rly , a t lo c atio n L ^ , the b a c k w a rd p o in te r is c h an g e d f r o m to L g . The e le m e n ts a t L g and h av e now b e e n linked, "cutting out!1 the e le m e n t a t L ^ , a s d e s i r e d . (If only the p o in te r L p Had b e e n a v a ila b le , it w ould h a v e b e e n n e c e s s a r y to go "all a ro u n d the lis t" to find L g . ) The d e le te d e le m e n t w ould n o r m a lly be r e t u r n e d to a lis t of a v a ila b le e le m e n ts . A n in s e r tio n of a n e le m e n t is b a s ic a lly the i n v e r s e of the d e le tio n p r o c e d u r e . 14 The a v a ila b le p r o c e s s o r s a r e o r g a n iz e d in a p u sh d o w n stac k . A pu sh d o w n s ta c k is like a s ta c k of d is h e s . The ite m a t the top is 30 a lw a y s ta k en f i r s t , w h ich th en e x p o se s a new ite m u n d e rn e a th . W hen p r o c e s s o r s b e c o m e a v a ila b le , b e c a u s e of d a ta s t r e a m s b e c o m in g ini- a c tiv e , th e y a r e ad d ed to the top of the s ta c k . If the b o tto m of the p r o c e s s o r is r e a c h e d , then a n o v e rflo w co n d itio n h as o c c u r r e d . This condition w ill a c tu a te p r i o r i t y logic, a s p r e v io u s ly d e s c r ib e d . T his d e s c r ip ti o n h a s b e e n r a t h e r g e n e r a l in s e v e r a l a r e a s to d e m o n s t r a t e the a p p lic a b ility of the p r o c e s s o r s h a r in g a r c h i t e c t u r e to a g r e a t v a r ie ty of p r o b l e m s . In S ections 3. 2, 3. 3, an d 3 .4 , s p e cific c o n fig u ra tio n s w ill be g iv en fo r the t h r e e ty p ic a l p r o b l e m s c o n s id e r e d in C h a p te r 2. 3. 2 A C o n fig u ra tio n fo r M e d ic a l M o n ito rin g A n a r c h i t e c t u r a l c o n fig u ra tio n fo r the p r o c e s s i n g of m e d ic a l d a ta s t r e a m s is show n on F i g u r e 3. 2. This c o n fig u ra tio n follow s the g e n e r a l p r o c e s s o r s h a r in g a r c h i t e c t u r e of F ig . 3. 1, but th e r e a r e a n u m b e r of a r e a s in w h ic h s p e c ific d e s ig n f e a tu r e s h a v e b e en s e le c te d b e c a u s e of the p a r t i c u l a r n a tu r e and r e q u i r e m e n t s of this p r o b le m . A m a j o r f a c to r having a n im p o r ta n t in flu en ce in the d e sig n is th a t the d a ta s t r e a m s c an be s a m p le d a s fr e q u e n tly a s d e s i r e d . T his p e r m i t s a m o r e e ffic ie n t p r o c e s s i n g than is p o s s ib le in th o se c a s e s , su c h a s the r a d a r d e te c tio n e x a m p le , in w h ich s a m p lin g r a t e s a r e c o n tr o lle d by e x te r n a l f a c to r s . B e c a u s e of this c ap a b ility , the m o n i to rin g of p a tie n t's E C G 's fo r sig n s of d a n g e r o u s h e a r t s ta tu s is p e r f o r m e d in two s ta g e s , a s w as d e s c r ib e d in S ection 2. 2. The f i r s t s ta g e is a p e r io d ic c h e c k in w h ic h e a c h p a tie n t's E C G is s a m p le d fo r one m in u te e v e r y h o u r . A n u m b e r of p r o c e s s o r s a r e d e d ic a te d ____________________31 INPUTS FROM DATA STREAMS t I • • INPUT SEQUENTIAL CONTROL UNIT INPUT BUS I DANGER SYMPTOM EVALUATOR r. . . . DATA FROM ACTIVE STREAMS EVALUATION ROUTING LOGIC GOOD DATA FROM NEW ACTIVE STREAMS FI RST LEVEL SE QUENCE DATA X RATINGS OF 2nd LEVEL DATA STREAMS DISCARD VERY LOW DANGER STREAMS BANK OF PROCESSORS ASSIGNED PROCESSOR SECTION AVAILABLE PROCESSOR SECTION A SECTION I XED LOCATIONS ASSIGNED PROCESSOR C PUSHDOWN ~1 STACK J B SECTION DOUBLE LINKED LIST TOP OF STACK AVAILABLE PROCESSOR BOTTOM OF STACK ASSIGN | TO A ! PROCESSOR* POINTER ■ TO TOP : OF STACK i DANGER ALARMS OVERFLOW ALARM F ig u r e 3. 2. C o n fig u ra tio n fo r M ed ic a l M onito rin g . 32 to this p u r p o s e . In F ig u r e 3. 2, the d e d ic a te d f i r s t s ta g e p r o c e s s o r s a r e show n a s the A S ection of the a s s ig n e d p r o c e s s o r s . T h e s e p r o c e s s o r s w ould be in know n fixed lo c a tio n s, and linked lis t m e th o d s w ould not be r e q u ir e d . E a c h p r o c e s s o r could be a s s ig n e d a m a x i m u m of six ty p a tie n ts . The E C G 's f r o m th e s e p a tie n ts w ould be s a m p le d in se q u e n c e . f ’o r the g r e a t m a j o r i t y of p a tie n ts , the f i r s t le v el t e s t w ould be a d e q u a te to r u le out a n y in d ic a tio n of d a n g e r o u s h e a r t s ta tu s . O n r a r e o c c a s io n s , the f i r s t lev el c h e c k w ill in d ic a te a ‘ d a n g e r o u s s ta tu s of a p a tie n t fo r w h o m th e r e h ad b e en no p r e v io u s in d ic a tio n th a t a n y thing w as out of th e o r d in a r y . In this c a s e , a n im m e d ia te a l e r t w ould be is s u e d to s u m m o n h u m a n a tte n tio n a n d a ctio n . F o r s o m e p a tie n ts , the f i r s t lev el t e s t w ill be in c o n c lu siv e w ith r e s p e c t to the c o n fid e n ce le v e ls d e s i r e d . In this c a s e , the p a tie n t’s d a ta h i s t o r y an d d a ta s t r e a m w ould be t r a n s f e r r e d . t o a n a v a ila b le p r o c e s s o r w h ich w ould be p la c e d in the B S ectio n of the a s s ig n e d p r o c e s s o r s . The B S ection p r o c e s s o r s w ill be in double linked c i r c u la r lis t. New input d a ta w ill be se n t to all p r o c e s s o r s in the B S ectio n a s often a s p o s s ib le , w ith p r i o r i t y b ein g g iv en to th o se d ata s t r e a m s w ith \ r a tin g s c lo s e to the d a n g e r level. A B S ection p r o c e s s o r w ill r e m a i n a s s ig n e d to a p a tie n t's ECG d a ta s t r e a m un til the p a tie n t's s ta tu s e ith e r r e t u r n s to n o r m a l, o r r e a c h e s the d a n g e r lev el, re q u ir in g a n a l e r t . If a p a tie n t's s ta tu s d o e s r e t u r n to n o r m a l, th en the p r o c e s s o r w ill be r e tu r n e d to the p u s h down s ta c k of a v a ila b le p r o c e s s o r s a s d e s c r i b e d in S ectio n 3. 1. O f c o u r s e , a p a tie n t r e tu r n in g to n o r m a l s ta tu s w ill c o n tin u e to 33 r e c e iv e f i r s t Level"m onitoring e v e r y h o u r. T he n a tu r e of the d a ta s t r e a m s im p o s e s a h e a v i e r b u r d e n th an n o r m a l on the input b u s. The d a ta o u tp u ts f r o m the EGG a r e in the f o r m of an alo g voltage s ig n a ls , and a r e la tiv e ly long s a m p le is r e q u ir e d fo r a n a ly s is of a n input, s o m e th in g of the o r d e r of one m i n ute. (In c o n tr a s t, the inputs f r o m the r a d a r d a ta s t r e a m s d e s c r ib e d in S ection 2. 3 h a v e a ty p ical d u r a tio n of the o r d e r of one m i c r o s e c o n d . ) F o r this r e a s o n , s o m e type o f;.m u lti-p o rt bus w ill be u se d so th at p a r a l l e l p r o c e s s i n g can b e p e r f o r m e d . The p a tie n t d a ta s t r e a m s w ould be id en tified by tag b its w h ich m ig h t r e p r e s e n t the p a tie n t's n a m e o r s o c ia l s e c u r i t y n u m b e r. The input s eq u e n c in g unit w ould c o m b in e two fu n ctio n s: the f i r s t lev el s a m p lin g , an d th e sec o n d le v el follow up. The f i r s t le v el seq u e n c in g w ould b e a c c o m p lis h e d by a s im p le c o u n te r , w h ic h w ould ta k e the p a tie n ts w h o se s ta tu s is c o n s id e r e d n o r m a l in tu rn , f o r one m in u te of e a c h h o u r. The s a m p lin g of the s e c o n d le v el p a tie n t E C G 's w ould d ep en d on the load in the B Section, and on the n u m b e r of o u t p u t p o r ts on the d a ta b u s. If t h e r e is a conflict, p r i o r i t y is g iv en to the d a ta s t r e a m w ith the h ig h e r X ra tin g . T h is c o n fig u ra tio n is o v e r s im p lif ie d , of c o u r s e , an d is b a s e d on a r b i t r a r y s e le c tio n of s o m e e le m e n ts in the d e sig n . H o w e v e r, it d o e s give an in d ic a tio n of th e m a n y p o s s ib ilitie s fo r u s e of the p r o c e s s o r - s h a r i n g a r c h i t e c t u r e in m e d ic a l a p p lic a tio n s . 3. 3 A C o n fig u ra tio n fo r R a d a r S y s te m s The g e n e r a l a r c h i t e c t u r e d e s c r i b e d in S ection 3. 1 is w ell su ite d to the p r o c e s s i n g of r a d a r d a ta s t r e a m s . In fact, the in h e r e n t 34 o r d e r in g of r a d a r data- s t r e a m s p e r m i t s s o m e sig n ific a n t s a v in g s to be m a d e in the a r c h i t e c t u r e . A c o n fig u ra tio n fo r the p r o c e s s i n g of r a d a r d a ta s t r e a m s is show n in F ig u r e 3. 3. It w ill be noted th a t the input bus and th e input seq u e n c in g c o n tro l unit h a v e b e en e lim in a te d . The sin g le input s t r e a m show n is a c tu a lly a v e ry ra p id r e p e titiv e s e q u e n c e of inputs f r o m e a c h of the r a n g e - a z i m u t h d a ta s t r e a m s . E a c h input s a m p le h a s a d u r a tio n e q u al to the effectiv e p u ls e w idth of the r a d a r , w h ic h is ty p ic a lly of the o r d e r of one m i c r o s e c o n d . The \J f e v a lu a tio n is s im p ly a th r e s h o ld c r o s s i n g d e te c to r . A n input is " p o sitiv e " if it e x c e e d s the th re s h o ld ; o th e r w is e , it is " n e g a tiv e ." The m o s t d ifficu lt p r o b l e m in the d e s ig n of this c o n fig u ra tio n is the m a tc h in g of d a ta s t r e a m in p u ts w ith th e ir a s s ig n e d p r o c e s s o r s . T hat is, w hen a " p o s itiv e " input is r e c e iv e d , it is n e c e s s a r y to q u ick ly d e t e r m i n e w h e th e r this input is f r o m a c u r r e n t l y a c tiv e d a ta s t r e a m a l r e a d y a s s ig n e d to a p r o c e s s o r , o r if it is the f i r s t " p o s i tiv e" input f r o m a p r e v io u s ly in a c tiv e d a ta s t r e a m . The d ata r a t e of the sin g le input s t r e a m th at c o m b in e s a ll the d a ta s t r e a m s is too high fo r a m e m o r y s e a r c h m e th o d to be u sed . H o w e v e r, t h e r e is a m e th o d of o rg a n iz in g the p r o c e s s o r m e m o r y th a t m a k e s 'u s e of the in h e r e n t o r d e r in g of the d a ta s t r e a m s to m a t c h d a ta s t r e a m s w ith th e ir a s s i g n e d p r o c e s s o r s w ithout a m e m o r y s e a r c h . A s d e s c r ib e d in S ectio n 2. 3, the d a ta s t r e a m s in a r a d a r s y s te m r e p r e s e n t s u c c e s s i v e r a n g e b in s, w ith the d a ta s t r e a m inputs a lw a y s being in o r d e r of in c r e a s in g r a n g e . Thus, the ra n g e c a n be u s e d a s a n id e n tific a tio n n u m b e r fo r the d a ta s t r e a m s , a n d a ls o in d i c a te s the input seq u e n c in g o r d e r . T he p r o c e s s o r m e m o r y is o r g a n - 35 V COMBINED DATA STREAM S THRESHOLD CROSSINGS FROM NEW ACTIVE STREAMS DATA FROM ACTIVE STREAMS DISCARD USELESS DATA BANK OF PROCESSORS ASSIGNED PROCESSOR SECTION AVAILABLE PROCESSOR SECTION PUSHDOWN STACK ASSIGNED PROCESSOR ASSIGN — TO A PROCESSOR TOP OF STACK AVAILABLE PROCESSOR POINTER TO’ TOP OF 1 STACK 1 POINTER TO OLDEST PROCESSOR BOTTOM OF STACK 1 DISCARD LOW EST X RATING X RATINGS OVERFLOW SIGNAL OUTPUT IDENTITY GATE RANGE OF CURRENT INPUT OVERFLOW AND PRIORITY LOGIC RANGE OF OLDEST PROCESSOR THRESHOLD CROSSING DETECTOR ALERTS F ig u r e 3. 3. C o n fig u ra tio n fo r R a d a r D ata S tr e a m s . 36 iz ed to tak e a d v a n ta g e of th is o r d e r in g by hav in g a ll p r o c e s s o r s s to r e d in o r d e r of i n c r e a s in g ra n g e of the a s s ig n e d d a ta s t r e a m s . The double linked c i r c u l a r lis t s t r u c t u r e d e s c r i b e d in S ection 3. 1 is v e r y e ffectiv e fo r th is p u r p o s e . The r a n g e o r d e r in g c a n be e a s ily e s ta b lis h e d a n d m a in ta in e d by in s e r tin g an d d e letin g p r o c e s s o r s at the p r o p e r p la c e s in the lis t. T h e r e a r e a few o th e r e s s e n tia l e le m e n ts of this c o n fig u ra tio n th at m u s t be d e s c r i b e d b e f o r e its o p e r a tio n can b e ex p lain ed . The " o ld e st" p r o c e s s o r is the one that h a s b e e n the lo n g e st w ithout an input f r o m its a s s ig n e d d a ta s t r e a m . T h e r e is a p o in te r that a lw a y s c o n tain s the a d d r e s s of the " o ld e s t" p r o c e s s o r . The r a n g e of the d a ta s t r e a m to w h ic h th is o ld e s t p r o c e s s o r is a s s ig n e d is s to r e d in a r e g i s t e r . T h e r e is a n id en tity g ate th at c o n tin u o u sly c o m p a r e s this s to r e d ra n g e w ith th e ra n g e of the d a ta s t r e a m c u r r e n t l y being r e c eiv ed . If the two r a n g e s a r e id e n tic a l, it is c l e a r th a t the new input is f r o m th e d a ta s t r e a m a s s ig n e d to the " o ld e s t" p r o c e s s o r . The s u r p r i s i n g fact is th a t if the r a n g e s a r e not id e n tic a l, then the d ata s t r e a m h a s not b e e n a s s ig n e d to an y p r o c e s s o r . No c h eck in g of the o t h e r . p r o c e s s o r s in the m e m o r y is n e c e s s a r y . To u n d e r s ta n d the r e a s o n th a t only the " o ld e st" p r o c e s s o r n e ed b e te s te d , c o n s id e r the s y s t e m in o p e ra tio n a t s o m e a r b i t r a r y tim e . A s s u m e th at the c i r c u l a r lis t c o n ta in s a n u m b e r of p r o c e s s o r s a s s ig n e d to d a ta s t r e a m s whose: r a n g e s a r e : ^ A 2 ’ * * * ’ "^Aj* T h e s e p r o c e s s o r s w ill h av e b e e n i n s e r t e d su ch th a t ^ A 2 < ’ ** < ^ A j ' taking ^ e p r o c e s s o r w ith the lo w e s t r a n g e d a ta s t r e a m a s a n im a g in a r y s ta r tin g point of the lis t. 37 C o n s id e r the tim e w hen a set of d a ta s t r e a m input r e t u r n s f r o m a b e a m a r e j u s t s ta r tin g to e n te r the th r e s h o ld c r o s s i n g d e t e c to r . The inputs w ill be in the o r d e r of: the d a ta s t r e a m f r o m R j , the d a ta s t r e a m f r o m • • • up to the d a ta s t r e a m f r o m w h e r e R m is the m a x i m u m ra n g e of i n t e r e s t . A t th is in sta n t, the " o ld e st" p r o c e s s o r w ill be the one a s s ig n e d to the a c tiv e d a ta s t r e a m w ith the lo w e st ra n g e , 1 ^ 1 * The p u r p o s e of a ll this e la b o r a te o r g a n iz a tio n can now be se e n . R j is le s s th an o r equal to If R j is not id e n tic a l to then the d a ta s t r e a m f r o m R j h a s hot b e e n a s s i g n e d to an y p r o c e s s o r . It c e r ta in l y w as not a s s ig n e d to the " o ld e s t" p r o c e s s o r , an d a ll the o th e r a s s ig n e d p r o c e s s o r s a r e a s s ig n e d to d a ta s t r e a m s w ith r a n g e s g r e a t e r th an T h u s, t h e r e is no need f o r an y s e a r c h of the p r o c e s s o r list. If R j is not equal to a n d the input f r o m d a ta s t r e a m R j did not c r o s s the th r e s h o ld , no f u r t h e r p r o c e s s i n g is r e q u ir e d . H o w e v e r , if the input did c r o s s the th re s h o ld , th en the d a ta s t r e a m f r o m R j w ill be a s s ig n e d to a p r o c e s s o r ( a s s u m in g one is a v a ila b le ). T his p r o c e s s o r w ill be i n s e r t e d d i r e c t l y a h e a d of the " o ld e s t" p r o c e s s o r , w h ic h w ill k e e p the c i r c u l a r lis t in the c o r r e c t r a n g e o r d e r . If R j e q u als th en the in fo r m a tio n a s to w h e th e r o r not t h e r e w as a th r e s h o ld c r o s s i n g is se n t to the o ld e s t p r o c e s s o r , u sin g the p o in te r to find the lo c atio n . T his p r o c e s s o r now h a s the t h r e e o p tio n s d e s c r i b e d in S ection 3. 1: d e c l a r e a ta r g e t, d r o p the d ata s t r e a m , o r co n tin u e p r o c e s s i n g . If a ta r g e t d e c la r a tio n is m a d e , a n a l e r t is is s u e d . If th e d a ta s t r e a m is d ro p p e d , the p r o c e s s o r is 38 r e t u r n e d to the a v a ila b le lis t. If the d e c is io n is to con tin u e p r o c e s s ing, the " o ld e s t" p r o c e s s o r b e c o m e s the " n e w e s t," a n d the next one, the p r o c e s s o r a s s ig n e d to the d ata s t r e a m f r o m R ^ 2 ’ b e c o m e s the " o ld e s t" p r o c e s s o r . T he p o in te r is ch an g ed to the lo catio n of this p r o c e s s o r in the d ir e c tio n c o r r e s p o n d in g to i n c r e a s e d ra n g e . The ra n g e R ^ 2 *s e n tere,3 i*1 the s to r a g e r e g i s t e r . If R^ d o e s not eq u al R ^ » then R ^ is le s s than, o r equal to R ^ j T his s itu a tio n is th u s e q u iv alen t to th at ju s t d e s c r i b e d fo r R^, and w ill s i m i l a r l y co n tin u e fo r R^» • •» until R ^ ^ is r e a c h e d . A t this point, the p r o c e s s o r a s s ig n e d to the d a ta s t r e a m f r o m R^ 2 w ^ l b e c o m e the " o ld e s t" p r o c e s s o r , a s d e s c r ib e d . The n ex t input w ill be f r o m the d a ta s t r e a m a t a r a n g e of R ^ ^ + l . By a line of re a so n in g s i m i l a r to th a t of th is d is c u s s io n , it c an be co n clu d e d th at the c u r r e n t " o ld e s t" p r o c e s s o r is th e only one th a t could h a v e b e e n a s s ig n e d to this d a ta s t r e a m . T h u s, the double linked c i r c u l a r lis t o rg a n iz a tio n of the p r o c e s s o r s p e r m i t s ra p id re c o g n itio n of a s s ig n e d d a ta s t r e a m s and lo c atio n of th e ir a s s ig n e d p r o c e s s o r s . The o th e r a r e a s of th is c o n fig u ra tio n , s u c h a s the o v erflo w p r o c e s s in g , a r e a s d e s c r i b e d f o r the g e n e r a l a r c h i t e c t u r e in S ection 3. 1. It is s e e n th a t the p r o c e s s o r s h a r in g a r c h i t e c t u r e is v e ry e f fe c tiv e a n d e ffic ie n t fo r r a d a r d a ta s t r e a m p r o c e s s i n g . 3. 4 C o n fig u ra tio n fo r S e is m ic D ata S tr e a m s It is. not yet p o s s ib le to sp e c ify a c o n fig u ra tio n f o r p r o c e s s i n g s e i s m i c d ata to p r e d i c t e a r th q u a k e s , b e c a u s e r e lia b le p r e d ic tio n m e th o d s h a v e not b e e n found. H o w ev e r, a s d i s c u s s e d in S ection 2. 4, ________________________________________________________________________________ 39 s e i s m i c d a ta is being g a th e r e d and a n a ly z e d a t v a rio u s r e s e a r c h i n stitu tio n s su ch a s the C a lifo rn ia In stitu te of T ech n o lo g y . It is hoped th a t p a t t e r n s and r e la tio n s h ip s can b e found in the s e i s m i c d ata s t r e a m s that a r e p r e c u r s o r s of e a r th q u a k e s . In th is e v en tu ality , a c o n fig u ra tio n s u c h a s the one show n in F ig u r e 3. 4 could be u s e d to p r o c e s s th e s e i s m i c d a ta s t r e a m s . The c o n fig u ra tio n of F ig . 3 .4 u s e s a two lev el p r o c e s s i n g a p p r o a c h , co m b in in g c o n v e n tio n a l a r r a y p r o c e s s i n g w ith the p r o c e s s o r s h a r in g a r c h i t e c t u r e . A s s u m e th at lo c atio n s a r e being m o n ito r e d , a n d th a t a t e a c h lo c atio n X^ s e n s o r s a r e g e n e r a tin g d a ta s t r e a m s . T h e s e s e n s o r s w ould be s t r a i n g a u g e s, c r e e p m e t e r s , e t c . , a s d e s c r ib e d in S ection 2. 4. The d a ta r a t e s f r o m a ll th e s e n s o r s a t one lo c a tio n woul'd! p r o b a b ly be low enough th at a ll th e ir d a ta s t r e a m s could be m u ltip le x e d on a sin g le telep h o n e line an d t r a n s m i t t e d to the c e n t r a l d ig ita l p r o c e s s i n g s y s te m . T h e se d ata s t r e a m s w ould not be " s p a r s e . " The p r e d ic tio n m e th o d s w ould p ro b a b ly involve a r i t h m e t i c and lo g ic a l o p e r a tio n s on th e d a ta s t r e a m s f o r su ch p u r p o s e s a s p a tte r n r e c o g n itio n o r d a ta c o r r e l a t i o n . T h is p r o c e s s in g w ould be id e n tic a l fo r the d a ta s t r e a m s a t e a c h lo catio n . T hus, an a r r a y of X^ p r o c e s s o r s in a SIMD (Single I n s tr u c tio n M ultiple D ata S tr e a m ) c o n fig u r a tio n w ould be an e ffic ie n t a r c h i t e c t u r e fo r the f i r s t lev el p r o c e s s ing. The outputs of the f i r s t lev el p r o c e s s o r s w ould be " s p a r s e , " a s s u m i n g that they h a v e s o m e v a lid ity in p re d ic tin g e a r th q u a k e s . 40 LOCATION Y LOCATION LOCATION 2 • • • X1 £ OUTPUTS. MULTIPLEXOR TELEPHONE LINE ) a MULTI PLEXOR t TELEPHONE LINE ) MULTIPLEXOR * > I TELEPHONE LINE < • ARRAY ARRAY ARRAY PROCESSOR PROCESSOR PROCESSOR FIRST LEVEL PROCESSING PROCESSOR SHARING ARCHITECTURE * SECOND LEVEL PROCESSING ALERT F ig u r e 3. 4. C o n fig u ra tio n fo r S e is m ic D ata. 41 T h e s e o utputs could u n d e rg o a s ec o n d le v el p r o c e s s i n g of a m o r e s o p h is tic a te d n a tu r e . The p r o c e s s o r s h a r in g a r c h i t e c t u r e w ould be of g r e a t b en efit in devoting m o s t of the s e c o n d lev el p r o c e s s i n g to the m o s t p r o m i s in g and m o s t a c tiv e d a ta s t r e a m s . A c o n fig u ra tio n along the lin e s of F i g u r e 3. 2 w ould s e e m r e a s o n a b le . A ll of the o utputs of the a r r a y p r o c e s s o r s w ould be s a m p le d a t s o m e r e la tiv e ly low r a te , a n d the a r r a y p r o c e s s o r s w h o se d a ta s t r e a m s in d ic a ted a c tiv ity , o r p o te n tia l e a r th q u a k e s w ould be p r o c e s s e d in m o r e d e ta ile d w a y s. 42 C H A P T E R 4 E V E N T R E C O G N ITIO N The o b je c tiv e of the a r c h i t e c t u r e fo r p a r a l l e l p r o c e s s i n g of " s p a r s e " d a ta s t r e a m s is to r e c o g n iz e e v en ts of i n t e r e s t . In this c h a p te r , m e th o d s of e v en t r e c o g n itio n a r e d e s c r i b e d and a n a ly z e d . T his a n a ly s is s u p p o rts a b a s ic d e s ig n p r in c ip le of the a r c h i t e c t u r e , the u s e of se q u e n tia l a n a ly s is m e th o d s w h e n e v e r p o s s ib le . Such m e th o d s u se a v a ria b le n u m b e r of sam p les, w ith the n u m b e r d e p e n d ing on the d a ta . M o st c o n v en tio n al p r o c e s s o r s u s e a fixed n u m b e r of s a m p l e s . It is show n th a t the se q u e n tia l a n a ly s is m e th o d s r e s u l t in a lig h te r load on the p r o c e s s o r s , th an u sin g a fixed n u m b e r of s a m p l e s , an d th at th is sav in g is a c h ie v e d w ithout an y s ig n ific a n t lo ss in re c o g n itio n p e r f o r m a n c e . 4. 1 S ta tis tic a l B a s is of E v en t R ec o g n itio n E v en t re c o g n itio n can be c o n s id e r e d a s a s t a t i s t i c a l p r o b le m of ch o o sin g b e tw e e n two a l t e r n a t e h y p o th e se s on the b a s is of s o m e n u m b e r of d a ta s a m p l e s . A c o m m o n te r m in o lo g y is to let H q be the h y p o th e sis th at the e v en t h a s o c c u r r e d , and H j be the h y p o th e sis that the e v en t h as not o c c u r r e d . In m o s t r e a l w o rld p r o b l e m s , t h e r e is a n e le m e n t of u n c e r ta in ty in the input d a ta due to n o ise, r a n d o m e r r o r s , o r o th e r u n p re d ic ta b le f a c t o r s . D e c is io n s b a s e d on this d ata m u s t h a v e s o m e p o s s ib ility of being in e r r o r . The p r o b a b ility that a n H q d e c is io n w ill be m a d e , w hen w as a c tu a lly the c a s e is often 43 c a lle d ex. The p ro b a b ility that an d e c is io n w ill be m a d e , w hen w a s a c tu a lly the c a s e is often c a lle d g. In this a n a ly s is , r e p r e s e n ts the p r o b a b ility of a fa ls e a l e r t an d g r e p r e s e n t s the p ro b a b ility of m i s s i n g the event. T he e v en t re c o g n itio n d e c is io n is m a d e by d e te r m in in g if the input d a ta - s a tis fy s o m e s e t of con d itio n s o r c r i t e r i a . T h e s e c r i t e r i a a r e b a s e d on p r o p e r t i e s of the d a ta in d ic a tiv e of the e v en t such a s input m a g n itu d e , o r w a v e fo r m . The p u r p o s e of th e s e c r i t e r i a is to m i n im i z e both # and g, but a c o m p r o m is e is n e c e s s a r y , b e c a u s e th e s e two o b je c tiv e s co n flict. The value of & can be re d u c e d to any d e s i r e d lev el by m a k in g the c r i t e r i a v e ry " s t r i c t , " but th is w ill tend to i n c r e a s e g. S im ila r ly , g can be m a d e a r b i t r a r i l y s m a l l by m a k ing the c r i t e r i a " e a s y ," but this w ill tend to m a k e # la r g e . The n u m b e r of d ata s a m p l e s a v a ila b le , an d th e ir q u ality , d e t e r m i n e the in h e r e n t c a p a b ility of m a k in g the c o r r e c t d e citio n . A s s u m in g th at the q u a lity of the d a ta is not u n d e r the c o n tro l of the d a ta p r o c e s s i n g s y s te m , the d e c is io n p r o c e s s involves a t r a d e - o f f of the t h r e e p a r a m e t e r s : & |3 > and N, the n u m b e r of s a m p l e s . A ny two of th e s e p a r a m e t e r s can be s e t in a d v a n c e a t any d e s i r e d level, w h ich then c o n s t r a i n s the th ir d p a r a m e t e r to s o m e d e te r m in a b le lev el. 4. 2 M ethods fo r E v en t R ec o g n itio n T h e r e a r e two c o m m o n s t a t is tic a l m e th o d s fo r ev en t r e c o g n i tion. O ne m e th o d , b a s e d on a s t a t is tic a l te s t d e v elo p e d by N e y m an 15 a n d P e a r s o n , is c o m m o n ly u sed w hen it is d e s i r e d to fix the n u m b e r of s a m p le s , N, in a d v a n c e . T his N m a y be in h e re n tly d e t e r m in e d by the n a tu r e of the sa m p lin g p r o c e s s , o r it m a y be s e le c te d j u s t a s a m a tte r , of c o n v e n ie n c e . E ith e r a o r g could a ls o be fixed in a d v a n c e , but it is c u s t o m a r y to se t a a t s o m e a c c e p ta b le lev el, a n d then to h av e the p r o c e s s i n g m in im iz e g, s u b je c t to the c o n s t r a i n t s of a an d N. H o w ev e r, if this m e th o d w e r e a p p lie d to the m e d ic a l m o n ito r in g e x a m p le , it w ould s e e m p r e f e r a b l e to fix g, the m i s s p r o b a b ility ,a t s o m e a c c e p ta b le level, su c h a s 0 ,0 0 0 1 , and to m in im i z e <*» s u b je c t to the c o n s tr a in t s of g an d N. A n o th e r a p p r o a c h is c a lle d s e q u e n tia l a n a ly s is , an d w as f i r s t d ev elo p ed by A. W a l d . ^ In this m eth o d , the n u m b e r of s a m p l e s is v a r ia b le , an d the p a r a m e t e r s a an d g a r e fixed in a d v a n c e a t s p e c i fied v alu es d e s ig n a te d a s : A a n d B. The sa m p lin g p r o c e s s is c o n tinued until e ith e r a n Hq o r d e c is io n c an b e m a d e w ith the s p e c i fied co n fid en ce le v e ls . It h a s b e en found th at the a v e r a g e n u m b e r of s a m p l e s r e q u i r e d fo r a d e c is io n using se q u e n tia l a n a ly s is m e th o d s is le s s than the fixed n u m b e r, N, th at w ould be r e q u i r e d by the m e th o d s of N e y m a n an d P e a r s o n to a c h ie v e the s a m e c o n fid e n ce le v e ls of A 17 a n d B. Of c o u r s e , a n y p a r t i c u l a r te s t m ig h t r e q u i r e m o r e th a n N s a m p le s , but the a v e r a g e is le s s than N, g e n e r a lly c o n s id e r a b ly so. F o r p r o b l e m s in w h ich the sa m p lin g p r o c e s s is flex ib le, and a n y d e s i r e d n u m b e r of s a m p l e s m a y be taken, s e q u e n tia l a n a ly s is m e th o d s a r e often u s e d to m in im i z e the a v e r a g e n u m b e r of s a m p l e s , giving a s a v in g s in tim e a n d / o r c o st. A n e x a m p le of this type is the e v alu a tio n of p ro d u c tio n ru n s of e l e c t r i c a l c o m p o n e n ts to s e e if the b a tc h is " b a d ," H^; o r "g o o d ," H^. The d e c is io n is m a d e by te stin g a n u m b e r of c o m p o n e n ts f r o m the b a tc h to se e if th ey m e e t s p e c i f i c a tio n s. U sing s e q u e n tia l a n a ly s is , one c o m p o n en t a t a tim e is te s te d u n til the d ata p e r m i t s a d e c is io n of Hq o r H j to be m a d e w ith the _______ 45 co n fid en ce le v els sp ec ifie d by A and B. T his m e th o d w ill r e q u ir e m u c h le s s te stin g , on the a v e r a g e , than u se of the m e th o d s of N e y m an a n d P e a r s o n w ith a p r e d e t e r m i n e d , fixed n u m b e r of s a m p l e s . 4. 3 E v e n t R ec o g n itio n in the P r o c e s s o r S h arin g A r c h i t e c t u r e S e q u en tial a n a ly s is m e th o d s a r e u s e d w h e n e v e r p o s s ib le in the p r o c e s s o r s h a r in g a r c h i t e c t u r e , b e c a u s e th e ir q u ic k e r d e c is io n m a k ing c a p a b ility r e d u c e s the load on the p r o c e s s o r s . A p p lic a tio n s su ch a s :the m e d ic a l m o n ito r in g e x a m p le fit n a tu r a lly into the se q u e n tia l a n a ly s is m e th o d s , b e c a u s e the p a tie n t's d a ta s t r e a m c a n be s a m p le d fo r a s long a s n e c e s s a r y to m a k e a d e c is io n . H o w ev e r, in a n a p p l i ca tio n s u c h a s r a d a r t a r g e t d e te c tio n , the n u m b e r of s a m p l e s , o r looks, is d e te r m in d by p h y s ic a l f a c t o r s . T his m a k e s it n e c e s s a r y to e ith e r m o d ify the s e q u e n tia l a n a ly s is m e th o d s p r o p o s e d by W ald, o r u s e a N e y m a n and P e a r s o n type m e th o d , b e c a u s e a d e c is io n m u s t be m a d e a f t e r the la s t d ata input in all c a s e s . F o r tu n a te ly , t h e r e is a m o d ific a tio n of s e q u e n tia l a n a ly s is m e th o d s c a lle d the s e q u e n tia l o b s e r v e r that p r e s e r v e s m u c h of the q u ick d e c is io n a d v a n ta g e s of s e q u e n tia l a n a ly s is , and a ls o a c h ie v e s o p tim a l re c o g n itio n c a p a b ility e q u iv alen t to N e y m a n an d P e a r s o n m e th o d s . B e f o re d e s c r ib in g the se q u e n tia l o b s e r v e r , W ald 's a p p r o a c h w ill be b r ie f ly d e s c r ib e d . The b a s ic p a r a m e t e r u s e d is the likelihood ra tio , L q, w h ich is d efin ed by the equation _ fo<>v yn> n ” h ^ i - y z *■ „> ’ 46 w h e r e fg is the jo in t p r o b a b ility d e n s ity function, pdf, of a ll the inputs y^ to y , th a t the ev en t h a s o c c u r r e d , and f^ is the jo in t pdf that the event h a s not o c c u r r e d . It is m o r e c o n v en ie n t to tak e the n a tu r a l lo g a r ith m s of both s id e s , obtaining: n Sn L = £ 2 m . y r y v . n i = 1 fjty ,) T he likelihood ra tio is c o m p u te d fo r n= 1, th at is, a f t e r the f i r s t s a m p le . If > A, w h e r e A is th e sp e c ifie d value fo r a , then Hq is a c c e p te d . If L ^ < B, w h e r e B is the sp e c ifie d value fo r $, then H^ is a c c e p te d . H o w ev e r, if A a § B, then no d e c is io n c an be m a d e , an d a n o th e r s a m p le is tak en . A f te r the s e c o n d s a m p le is tak en , is c o m p u te d an d s i m i l a r l y c o m p a r e d w ith A and B to s e e if a d e c is io n can be m a d e , o r if a th ir d s a m p le is r e q u ir e d . T his p r o c e d u r e c o n tin u e s fo r n= 3, 4, . . . . W ald h as d e m o n s t r a t e d th at e v e n tu ally a d e c is io n can a lw a y s be m a d e w ith the s p e c ifie d c o n fid en ce le v e ls . T h e s e q u e n tia l o b s e r v e r u s e s the g e n e r a l re c o g n itio n m e th o d d e s c r i b e d in S ectio n 2. 5, w ith the \ ra tin g being a m e a s u r e of the event lik elih o o d . T his d e v ic e w as f i r s t d e s c r i b e d and a n a ly z e d by I. S. R eed in r e f e r e n c e 18. A s e q u e n tia l o b s e r v e r can be thought of a s a n a c c u m u la to r an d a s e t of d e c is io n logic. The a c c u m u la to r is s e t to an in itial value of z e r o . On a n input th at is " p o s itiv e ," th at is, in d ic a tiv e of the h y p o th e s is H q (such a s th r e s h o ld c r o s s i n g of a r a d a r input), the a c c u m u l a t o r is i n c r e m e n t e d by a n a m o u n t Z^. On a n in put th at is "n eg ativ e," th a t is, in d ic a tiv e of H j, the a c c u m u la to r is 47 d e c r e m e n t e d by a n a m o u n t Z^* If the a c c u m u l a t o r r e a c h e s s o m e u p p e r th r e s h o ld , T ^ , th en is a c c e p te d . If s o m e lo w e r th re s h o ld , T t , is r e a c h e d , th en H. is a c c e p te d . T h e s e th r e s h o ld s , T tt an d T t , J -j 1 U i-j a r e in effect c o n fid en ce le v e ls , a n alo g o u s to W ald 's A an d B. F ar m o n ito r in g of " s p a r s e " d a ta s t r e a m s , a n output, o r a l e r t , is r e q u i r e d only on a n d e c is io n . No a c tio n is r e q u i r e d fo r a n d e c is io n . In this c a s e , the se q u e n tia l o b s e r v e r c an be s im p lifie d by le ttin g be z e r o . A f u r t h e r s im p lific a tio n c an be m a d e by a s c a l ing ch an g e and a ro u n d -o ff. F i r s t , a ll the n u m b e r s a r e d iv id ed by Z m a k i n g th e i n c r e m e n t value Z^/Z^, th e d e c r e m e n t value 1, and the u p p e r th r e s h o ld T ^ fZ ^ , F o r m o s t c a s e s involving r e c o g n itio n of a n u n u su al ev en t, T y and Z j a r e m u c h g r e a t e r than Z2. T h e r e f o r e , a ro u n d -o ff to in te g r a l v alu es can be m a d e w ith little e r r o r . L e t the ro u n d e d -o ff value of Z^/Z^ be d e sig n a te d a s K, and the ro u n d e d -o ff value of T y / Z j be d e sig n a te d a s T ^ . T hen the r e s u ltin g se q u e n tia l o b s e r v e r is in a c o n v en ie n t f o r m fo r d ig ita l p r o c e s s i n g , in th at it i n c r e m e n t s the a c c u m u la to r by a n in te g e r, K, on a " p o s itiv e " input, d e c r e m e n t s the a c c u m u l a t o r by 1 on a " n e g a tiv e " input, an d r e c o g n iz e s a n e v en t if the a c c u m u l a t o r r e a c h e s the value T ^ . ( F o r the r e s t of this d is s e r t a t i o n , the t e r m " s e q u e n tia l o b s e r v e r " w ill r e f e r to this s im p lifie d v e r s io n of the g e n e r a l s e q u e n tia l o b s e r v e r . ) T he r e c o g n itio n /f a ls e a l e r t c a p a b ilitie s of this d e v ice w ill be a n a ly z e d in S ection 4. 5. 4. 4 A n a ly s is of M out of N R eco g n itio n , A N e y m a n and P e a r s o n M ethod It is u sefu l to c o n s id e r the p e r f o r m a n c e a tta in a b le by the m e th o d s 48 of N e y m a n an d P e a r s o n f o r s e v e r a l r e a s o n s . O ne of th e m o s t i m p o r ta n t r e a s o n s is to e s ta b lis h a r e f e r e n c e by w h ich the p e r f o r m a n c e of the se q u e n tia l o b s e r v e r can be m e a s u r e d , s in c e the N e y m a n and P e a r s o n t e s t is o p tim u m u n d e r the a s s u m e d c o n d itio n s. In addition, s o m e e le m e n ts of this a n a ly s is w ill p r o v e to be of u s e in the a n a ly s is an d d e s ig n of the s e q u e n tia l o b s e r v e r d i s c u s s e d in S ection 4. 5. It is a l s o p o s s ib le th a t fo r s o m e a p p lic a tio n s , the m e th o d s of N e y m a n an d P e a r s o n w ould be u s e d w ith the p r o c e s s o r s h a r in g a r c h i t e c t u r e . The M out of N re c o g n itio n m e th o d w ill be c o n s id e r e d b e c a u s e the a n a ly s is is le s s c o m p lic a te d than fo r o th e r ty p es of N e y m a n an d P e a r s o n m e th o d s , a n d b e c a u s e th e r e is a c o n s id e r a b le body of u sefu l l i t e r a t u r e on th is m e th o d . A s d e s c r ib e d in S ectio n s 2. 3 and 2. 5, M out of N r e c o g n itio n is p e r f o r m e d by e v alu a tin g e a c h input of a 'd a t a s t r e a m a s e ith e r " p o sitiv e " (\jj = l) o r " n e g a tiv e " (\jf = 0). If at le a s t M out of N c o n s e c u tiv e inputs a r e " p o s itiv e ," th en a n d e c is io n is m a d e . O th e r w is e , a n d e c is io n is m a d e . In th e a n a ly s is it is helpful te d efin e P ^ b y the equation: P j-j = 1 - j3, w h e r e P ^ is c a lle d the p r o b a b ility of d e te c tio n . D u rin g the b u r s t of a c tiv ity of a " s p a r s e " d a ta s t r e a m , it w ill be a s s u m e d th at e a c h input is an in d ep en d en t t r i a l , and th at the p r o b a b ility is p g th at a n input w ill be e v alu a te d a s " p o s itiv e ." P ^ is thus the p r o b a b ility of getting a t le a s t M " p o sitiv e " in p u ts, o r s u c c e s s e s , in N t r i a l s , w h ich c a n be found d i r e c t l y f r o m the c u m u la tiv e b in o m ia l d is trib u tio n : 49 N N-.i N { j'( N - j) * * (4-1) F r o m the value of Pj-j’ @ c an e a s ^ y be o b tain ed . The p r o b a b ility of fa lse a le r t, a , c a n be found f r o m th is b a s ic eq u atio n a ls o . It is a s s u m e d that the d a ta s t r e a m inputs wilL o c c a sio n a lly be f a ls e ly e v a lu a te d a s " p o s itiv e ," w hen the ev en t is not o c c u r r in g , due to n o ise o r o th e r d is tu r b a n c e . T h e s e fa ls e inputs w ill be a s s u m e d to be d is tr ib u te d a t r a n d o m , w ith p ro b ab ility p. E q u atio n 4-1 c a n be u s e d , su b stitu tin g p fo r p , yielding s F o r la r g e v a lu es of N, the b in o m ia l d is tr ib u tio n can be a p p r o x im a te d by the G a u s s ia n d is trib u tio n , to obtain: By a ch an g e of v a ria b le , this eq u atio n c a n be put in a s ta n d a r d f o r m . (4-2) N . a = 2 PJ( 1-P) j = M N “j Nt j l ( N - j ) ! * 2 d x , w h e r e , = Np, an d gr = »/Np( 1 -p) . V_ 2 2 a - e dv cp w h e r e cp = M -N p </N p(l-p) 50 T h is d efin ite in te g r a l c a n be e x p r e s s e d a s the s e r i e s a 2 £ _ 2 _ , n ^ , ,vN 6 --------------- 2 T — • ’ -------------2N J T ^N cp cp cp *f2< ncp" w h e r e is a r e m a i n d e r t e r m th a t b e c o m e s s m a l l fo r la r g e N. F o r m a n y c a s e s of i n te r e s t , & is s m a l l enough th at the a p p r o x im a tio n 2 _ C£_ 2 e a '/Zftcp c an be m a d e . T his a p p r o x im a tio n w ill have an e r r o r of only a few _5 p e r c e n t fo r &< 10 . In th e s e c a s e s , cp can be s e t to s o m e c o n sta n t, K p , w h e r e K^, r e p r e s e n t s the n u m b e r of s ta n d a r d d e v ia tio n s by w h ich M, the re c o g n itio n level, e x c e e d s the m e a n , Np. It w ill be a s s u m e d th at this a p p r o x im a tio n can be m a d e , a n d that: (4-3) K f = M - NP---- . Np( 1-p) F o r c o m p a r i s o n p u r p o s e s , it is a ls o of value to find the sig n al to n o ise r a tio r e q u i r e d fo r re c o g n itio n a s a fu n ctio n of the fa ls e a l e r t r a t e . T h is r e s u l t w ill be a p p lic a b le to e x a m p le s involving the r e c o g nition of a w e ak sig n a l in a " s p a r s e " d a ta s t r e a m w ith b a c k g ro u n d n o ise . It w ould be w o rth w h ile to p e r f o r m th is type of a n a ly s is fo r 51 th e m e d ic a l m o n ito r in g e x a m p le an d the e a r th q u a k e p r e d ic tio n e x a m p le a ls o , but m a t h e m a t i c a l m o d e ls a r e not yet a v a ila b le fo r th e s e e x a m p le s . F o r G a u s s ia n n o ise, the p ro b a b ility of a fa ls e " p o s itiv e " e v alu a tio n , p, is g iv en by the p r o b a b ility that the n o ise w ill e x c e e d the th r e s h o ld , T, w h ic h is given by: (4-4) p = e ~ T F o r this a n a ly s is , a sig n a l p ro b a b ility d e n s ity function ty p ical 19 of flu ctu atin g n o is e - lik e sig n als d e v elo p e d by S w e rlin g w ill be u sed . T he s ig n al flu c tu a te s a b o u t its m e a n sig n al to n o is e r a tio , S /N , w ith the d is trib u tio n : i«’ c -v'.T ( 4 - 5> P s = t y T-- <x /1 + x , - T /1 +x T - +k e dx = e w h e re p = the sin g le look p r o b a b ility th a t sig n al p lu s n o ise s e x c e e d s the th r e s h o ld T and x = the m e a n s ig n a l to n o ise ra tio . the p r o b a b ility of getting a t le a s t M " p o s itiv e " e v alu a tio n s in N t r i a l s , c an b e found f r o m the G a u s s ia n a p p r o x im a tio n to the b in o m ia l d is trib u tio n , u sin g the equation: « t,2 - V 7 2 j v e dV VNPg(l-Pg) 52 If M = N pn , then, D a o „ 2 P - — I D e’ dV = 1 /2 0 T his p a r t i c u l a r c a s e is t h e r e f o r e m o r e c o n v en ie n t fo r a n a ly s is , so it w ill be a s s u m e d th at the d e s i r e d P j-j = 1 /2 and M = N pg. L e t E q u atio n 4 -3 c an then be w r itt e n in the fo rm : K f , ____________ R - P = - — v p ( l - p ) . ,/N- Solving this eq u atio n fo r p y ield s: P (2R + K ? ,/ N ) ± y ( 2 R + K 5 ,/ N ) - 4 R 2(l+ K ?,/N ) _£ £ £ _____ 2 ( 1 + K 2 /N) By u se of s e v e r a l a p p r o x im a tio n s an d s im p lific a tio n s by n e glectin g t e r m s of the o r d e r of 1 /N c o m p a r e d to t e r m s of the o r d e r of 1 / */n , the following' eq u atio n is obtained: (4-6) p = R [ l -KFA / N " ]. W ith f u r th e r s im p lific a tio n an d a p p r o x im a tio n , the so lu tio n f o r x is: (4-7) ; = . ^ . </N ^/R^n(R) 53 T h is value of x w ill b e c o m p a r e d w ith that fo r an e q u iv a le n t s e q u e n tia l o b s e r v e r in S ectio n 4. 5. A m a j o r p r o b l e m in the u se of M out of N d e te c tio n is the s e le ctio n of M fo r a given N. F o r s o m e a p p lic a tio n s , p h y s ic a l c o n s id e r a tio n s m ig h t d ic ta te the c h o ice . F o r the m e d ic a l m o n ito r in g e x a m p le , M /N of 0. 1 w as the c r i t i c a l ra tio , an d w ould be u s e d if M out of N d e te c tio n w e r e u s e d in th is e x a m p le . F o r p r o b l e m s involving re c o g n itio n of s ig n a ls in d a ta s t r e a m s s u b je c t to b a c k g ro u n d d i s t u r b a n c e su ch a s t h e r m a l n o ise, the o p tim u m M fo r a g iv e n N h a s b e e n 20-22 found e m p i r i c a l l y fo r m a n y c a s e s . A n u m b e r of a n a ly s ts h a v e m a d e c o m p u te r ru n s to find M thei M th a t g iv e s the b e s t r e c o g n itio n /f a l s e a l e r t p e r f o r m a n c e fo r a given N. F ig u r e 4. 1 show s the 22 r a tio of M ^./N o b tain ed by J. W alk e r fo r two d if f e r e n t d i s t r i b u tion fu n ctio n s of w e ak s ig n a ls in t h e r m a l n o ise . The v alu e of M Qp(. h a s b e en found to be e s s e n tia l ly in d ep en d en t of s ig n a l to n o ise ra tio , p r o b a b ility of d e te c tio n , a n d f a ls e a l e r t r a te . It a p p e a r s th a t this r a tio a p p r o a c h e s a n a s y m p to te of a p p r o x im a te ly 0. 2 f o r la r g e v a lu es of N, a n d in fact, s e v e r a l a n a ly s e s , su ch a s th o se in r e f e r e n c e s 23 an d 24 h a v e found th is to be th e c a s e . In A p p en d ix A, a n o th e r d e r i vation of th is r e s u l t is given, in w hich it is show n that: M . .M , . * ■ - « * = 2 ( - r - o - M O P t T he so lu tio n of th is eq u atio n is: — =0. 203. 54 1000 500 300 150 100 50 \ \ O . \ O * FLUCTUATING TARGET MODEL I 6 i J L PFA= , ° Pd= ° . 9 r-8 \ CONSTANT AMPLITUDE \ t a r g e t m o d el \ \ 9 \ \ \ \ \ \ \ 0.1 0.2 0.3 M . /N opt F ig u r e 4. 1. R atio of M ^ ^ / N . O.k 0.5 55 4. 5 A n a ly s is of the S e q u e n tia l O b s e r v e r F o r a n a ly s is p u r p o s e s , the ev en t re c o g n itio n v e r s io n of the s e q u e n tia l o b s e r v e r d e s c r i b e d in S ection 4. 3 c an be r e p r e s e n t e d a s a finite s ta te m a c h in e w ith s to c h a s tic inputs, w h e r e e a c h s ta te of the m a c h in e c o r r e s p o n d s to a p o s s ib le value of the a c c u m u la to r . T h e r e w ill be, t h e r e f o r e , T ^ t 1 s ta t e s . T h is r e p r e s e n t a t i o n is illu s tr a te d in F ig . 4. 2, fo r a s e q u e n tia l o b s e r v e r w ith the p a r a m e t e r s K = 3 and T = 7. In this d ia g r a m , the a r r o w s in d ic a te the p o s s ib le m o v e m e n ts -K f r o m one s ta te of th e m a c h i n e to a n o th e r . The "p " r e p r e s e n t s the p r o b a b ility of a " p o s itiv e " input, a n d "q " is eq u al to 1-p. The dotted p a th f r o m s ta te 7 to s ta te z e r o is the r e s e t w hich o c c u r s a f t e r r e c o g n ition of a n ev en t. In th is v e r s io n , the count n e v e r goes b elo w z e r o , a s in d ic a te d by the a r r o w f r o m the z e r o s ta te b a c k to itse lf, on a " n e g a tiv e " input. A s d i s c u s s e d in S ectio n 4. 3, th e r e is no b en efit in allow ing the count to go n e g ativ e. H o w ev e r, it w ill be s e e n that a " n e g a tiv e goin g " s e q u e n tia l o b s e r v e r is of th e o r e tic a l i n t e r e s t in f a ls e a l a r m a n a ly s is . The s e q u e n tia l o b s e r v e r a p p e a r s v e r y s i m i l a r to the a u to m a to n 25 o r finite s ta te a c c e p to r , but t h e r e is an im p o r ta n t d istin c tio n . The finite s ta te a c c e p t o r o p e r a t e s on a s trin g of a r b i t r a r y length, and u s e s the c r i t e r i o n th a t an input s tr in g m u s t le av e the m a c h in e in a Z 6 final s ta te , an d thus d e te c ts a c l a s s of s t r i n g s s a id to be " r e g u la r ." The se q u e n tia l o b s e r v e r o p e r a t e s on a co n tin u o u s s t r e a m an d a c c e p tance, o c c u r s if the m a c h in e e v e r r e a c h e s the t e r m i n a l s ta te . It thus a c c e p ts a c e r t a i n d e n s ity of " p o s itiv e " in p u ts. 56 q f \ \ \ \ \ RESET / / F ig u r e 4. 2. State D i a g r a m fo r S e q u en tial O b s e r v e r , w ith K = 3 and TR = 7. 57 To m a k e u se of the se q u e n tia l o b s e r v e r , it is n e c e s s a r y to h a v e a m e a n s of s e le c tin g K an d TR to obtain the r e q u i r e d co n fid en ce le v els fo r & and j-J. F o r this p u r p o s e it is u se fu l to d e r i v e s o m e r e la tio n s h ip s b e tw e e n th e se q u e n tia l o b s e r v e r a n d the M out of N d e t e c to r . By c o n s id e r a tio n of th e se two m e th o d s , it w ill be s e e n th a t fo r a n y M out of N d e te c to r , th e r e is a s e q u e n tia l o b s e r v e r th a t w ill have a n id e n tic a l re c o g n itio n fu n ctio n o v e r an y s e q u e n c e of e x a c tly N in p u ts , p ro v id e d th a t the s e q u e n tia l o b s e r v e r s t a r t s a t the z e r o s ta te . It w ill be s e e n th at the s e q u e n tia l o b s e r v e r w ill be a t the z e r o s ta te a l m o s t a ll of the tim e , if p is s m a ll. The follow ing c o n d itio n s a r e n e c e s s a r y an d s u ffic ie n t to e s t a b lis h this e q u iv alen c e: 1. A ny s e q u e n c e contain in g M " p o s itiv e " in p u ts, and not m o r e than (N -M ) " n e g a tiv e " inputs m u s t r e s u l t in re c o g n itio n . T his co n d itio n c a n be e x p r e s s e d by the in eq u ality : MK - (N-M ) 5 T . X \ 2. ( M - l) c o n s e c u tiv e " p o s itiv e " inputs m u s t not r e s u l t in r e c ognition. T h is co n d itio n c a n be e x p r e s s e d by the in eq u ality : K ( M - l) < Tr . Solving th e s e e q u a tio n s fo r the m in im u m 'v a lu e s of K and T R y ield s: (4-8) K = N -M + l , 58 and (4-9) Tr = K (M -l) + 1 . C o n v e r s e ly , to c o n s t r u c t a n e q u iv alen t M out of N d e te c to r f o r any se q u e n tia l o b s e r v e r : It m u s t be e m p h a s iz e d th at th e s e eq u atio n s p ro v id e a m a tc h in g in re c o g n itio n only o v e r a n N input se q u e n c e , f o r the s e q u e n tia l o b s e r v e r s ta r tin g a t the z e r o s ta te . T his a s s u m p tio n w ill be r e a s o n a b ly a c c u r a t e fo r s m a l l p, and fo r e v en ts w h o se b u r s t of a c tiv ity is of know n length. U n d e r th e s e c o n d itio n s, the r e c o g n itio n p ro b a b ility , P q , fo r the se q u e n tia l o b s e r v e r c an be found by eq u atio n 4. 1, f o r the e q u iv alen t M out of N d e t e c t o r . T his is e q u iv a le n t to finding g. How e v e r the m a tc h in g is of no u s e in finding the f a ls e a l e r t p ro b a b ility , a, w h ich w a s a s s u m e d to be s m a l l f o r the d e r iv a tio n of the equations. F o r this p u r p o s e , a s te a d y s ta te a n a ly s is of the s e q u e n tia l o b s e r v e r is r e q u ir e d , u n d e r th e co n d itio n th at the e v en t is not o c c u r rin g ,, ju s t r a n d o m f a ls e " p o s itiv e " in p u ts. To d e v elo p this a p p r o a c h , c o n s id e r F ig u r e 4. 2, w h e r e p r e p r e s e n t s the p r o b a b ility of a f a ls e " p o s itiv e " input. The s e q u e n tia l o b s e r v e r w ill ch an g e f r o m s ta te to s ta te , follow ing the in d ic a te d p a th s fo r p a n d q, d e p en d in g on w h e th e r the input w as " p o s itiv e " o r " n e g a tiv e ." T his seq u e n c in g of s ta t e s is a M a r k o v c h a in a s Set: and N = K + M - 1 59 d e s c r i b e d a n d a n a ly z e d by F e l l e r in r e f e r e n c e 27. The c h a r a c t e r i s tic p r o p e r t y of M a rk o v c h a in s is th at the p r o b a b ility of tr a n s itio n f r o m s ta te j to s ta te k h as s o m e fixed d e te r m in a b le value, p. , , w h ich is c o m p le te ly in d e p en d e n t of the p r e v io u s h i s t o r y of the m a chine and the input s e q u e n c e . A M a rk o v c h ain is thus sp e c ifie d by a m a t r i x of t r a n s itio n p r o b a b iliti e s . In the e x a m p le of F ig u r e 4. 2, p l, 4 = p a n d p 4, 3 = q ' w hiIe p 4, i = °* In c o n s id e r in g the s e q u e n tia l o b s e r v e r u n d e r a v e r y long s e q u e n ce of r a n d o m inputs, it s e e m s r e a s o n a b le to co n clu d e th at fo r e a c h s ta te t h e r e is s o m e fixed p r o b a b ility th a t the se q u e n tia l o b s e r v e r w ill be in th at s ta te a t an y r a n d o m tim e . C e rta in ly , the s e q u e n tia l o b s e r v e r w ill be a t the z e r o s ta te " m o s t of the t im e " if p is 27 s m a ll. F e l l e r h a s show n th a t su ch p r o b a b iliti e s , c a lle d " s t a t i o n a r y p r o b a b i l i t i e s ," w ill e x is t fo r the s ta t e s of a M a rk o v ch ain u n d e r t h r e e c o n d itio n s. T he M a r k o v c h ain m u s t be a p e r io d ic ; it m u s t be i r r e d u c i b l e , a n d a ll the s ta te s of the M a rk o v ch ain m u s t be e rg o d ic . A c h ain is p e r io d ic if the tim e fo r the m a c h in e to r e t u r n to a n y s ta te is a lw a y s a fixed m u ltip le of s o m e tim e in te r v a l, t, w h e r e t > 1. The s e q u e n tia l o b s e r v e r is c l e a r l y not p e r io d ic , b e c a u s e the r e t u r n tim e to the z e r o s ta te c a n be one in te r v a l, w h ich o c c u r s on a q input. T h e r e f o r e the s e q u e n tia l o b s e r v e r is a p e r io d ic . A M a r k o v c h ain is i r r e d u c i b l e if and only if e v e r y s ta te c a n be r e a c h e d f r o m e v e r y o th er s ta te . T h is is c l e a r l y tr u e fo r the s e q u e n tia l o b s e r v e r . A s ta te is e rg o d ic if: lim p ^ . = U, > 0, w h e r e p!1 . is the p r o b a b ility th at the 1 • x C l£ 1« K n-+ so J m a c h in e r e a c h e s s ta te k, s ta r tin g f r o m s ta te j, in e x a c tly n t r a n s i tio n s. T h is co n d itio n w ill be s a tis f ie d fo r a ll s ta t e s p ro v id e d that 60 p > 0. Since the th r e e co n d itio n s a r e s a tis f ie d fo r the s e q u e n tia l o b s e r v e r , a s ta t io n a r y p ro b a b ility e x is ts fo r e v e r y s ta te of the s e q u e n tia l o b s e r v e r . T h e s e s ta t io n a r y p r o b a b iliti e s c a n be r e la te d by d iff e r e n c e e q u a tio n s. F o r th e e x a m p le shown in F i g u r e 4. 2, let P(i) be the p r o b a b ility th at the s e q u e n tia l o b s e r v e r is in s ta te i. T hen the f o l lowing d if f e r e n c e eq u atio n s m u s t hold b e tw e e n the s ta t io n a r y p r o b a b ilitie s of the s ta t e s , b e c a u s e of the t r a n s itio n p r o b a b ilitie s : P(0) = q:P(0;) + ,qP (l');rP (7) P(D = qp( 2) P(2) = qP(3) P(3) = qP(4) + pP(0) P(4) = q P ( 5 ) + p P ( l ) P(5) = qP(6) + pP(2) P(6) = PP(3) P(7) = PP(4) + p P (5 ) + p P (6 ) . A lso , 7 2 F(j) = 1 • j = l F o r a n e x a m p le w ith r e la tiv e ly few s ta te s , s u c h a s F ig u r e 4. 2, an a p p r o x im a te so lu tio n can be found to the se t of d if f e r e n c e e q u a tio n s by m a k in g a n in itia l e s t i m a t e fo r the s ta tio n a r y p r o b a b ilitie s , an d then u sin g a n ite r a tiv e p r o c e d u r e . S ince p is s m a ll, a good in i tia l e s t i m a t e w ould be th a t P(0) = 1. P(4) w ould be e s ti m a te d as 61 pP(O), o r p. P ( l ) , P(2), an d P(3), wouLd be e s ti m a te d a s p a ls o . The o th e r s ta t io n a r y p r o b a b iliti e s w ould be e s ti m a te d using the d if f e r e n c e e q u atio n s and the in itia l e s t i m a t e s fo r P(0) - P (4 ). The s u m of the in itia l e s t i m a t e p r o b a b iliti e s is c o m p u te d to give, say, S^. T h en a ll of the in itia l e s t i m a t e s a r e divided by S^. A se c o n d s e t of e s ti m a te d v alu es is th en m a d e by using the d if f e r e n c e e q u atio n s in tu rn , s ta r tin g w ith P (4). If p is s m a ll, an a c c u r a t e so lu tio n can be found in a few i t e r a tio n s . H o w e v e r fo r e x a m p le s w ith l a r g e r n u m b e r s of s ta t e s , the p r o c e s s b e c o m e s m u c h m o r e difficult, if not i m p r a c t i c a l . F o r tu n a te ly , a c lo s e d f o r m so lu tio n fo r a s e t of d iff e r e n c e e q u atio n s of th is type h a s b e en found, a s d e s c r i b e d below . The d if f e r e n c e te q u a tio n s f o r the g e n e r a l s e q u e n tia l o b s e r v e r c an be o b ta in e d u sin g the s a m e a p p r o a c h a s show n f o r the e x a m p le of F i g u r e 4. 2. T he a n a ly s is c a n b e c o n s id e r a b ly s im p lifie d if the s e q u e n tia l o b s e r v e r is a s s u m e d to h av e a n infinite n u m b e r of p o s itiv e s ta te s , in s te a d of r e s e tti n g a t T ^ . If the p r o b a b ility of fa ls e a le r t. is s m a ll, this m o d if ic a tio n w ill h av e n e g lig ib le effect on the r e s u l t s . It is c l e a r th a t P(0) is g iv en by the d if f e r e n c e equation: (4-10) P(0) = qP(0) + qP( 1) . The d if f e r e n c e eq u atio n s fo r P ( l) t o 'P ( K - l ) a r e : P ( l) = qP(2) P(2) = qP(3) P ( K - l ) = qP(K) . 62 T hus fo r 2 s i | K, the follow ing d iff e r e n c e e q u atio n is valid: (4-11) P(i) = F ( i - l ) / q . F o r a ll i s K, the follow ing d iff e r e n c e e q u atio n is valid: (4-12) P(i) = q P ( i + l ) + p P ( i - K ) , By solving eq u atio n 4 -1 2 f o r P(i+1), an d then re d u c in g a ll th e in d ic e s by one, the follow ing eq u atio n is o b tain ed , w h ich is valid fo r i n K + l: (4-13) P(i) = P ( i - l ) / q - p / q • P(i-(K +1)) . W ith th e s e d iff e r e n c e e q u atio n s, a c lo se d f o r m e x p r e s s i o n fo r P(0) can be o b tain ed u sin g the m e th o d of c h a r a c t e r i s t i c e q u atio n s, o r g e n e r a tin g fu n ctio n s. L et c(X) = E P (i)x 1. i = 0 T h at is, a s s u m e the e x is te n c e of a fu n ctio n C(X) w h o se p o w e r s e r i e s e x p a n sio n is s u c h th a t th e c o e ffic ie n t of X 1 is P (i). L e t (4-14) G ( x ) = ( l - 2 - £ x K + 1 ] c ( X ) . T he p r o d u c t t e r m XC(X) c a n be e x p r e s s e d by the e q u atio n s: K . . . • XC(X) = E P (i)X + E P (i)X , i = 0 i=K+l or equivalently: 63 K . 00 (4-15) XC(X) = E P ( i - l ) X l + E P ( i - l ) X l . i = 1 i - K + 1 A lso , the p r o d u c t t e r m X ^ ^ C ( X ) c a n be e x p r e s s e d by the equation: (4-16) X ^ + 1C(X) = E P ( i-(K + l) )X I . i = K + l U sing eq u atio n s 4 -1 5 an d 4 -1 6 , eq u atio n 4 - 1 4 c an be e x p r e s s e d as: K , K » (4-17) G(X) = E P (i)X l - - E P ( i - l ) X l + E H(i)Xl , i = 0 q i= 1 i = K +l w h e r e H(i) = P(i) - P ( ^r -') + | P (i-(K + 1)). By eq u atio n 4 -1 3 , H(i) = 0 , fo r i a K + l, T hus, eq u atio n 4 -1 7 r e d u c e s to: K , K G(X) = E P (i)X L - - E P ( i - l ) X l , i = 0 q i= 1 o r K G(X) = P (0) + E [P (i) - 1 ^ J x 1 E q u iv ale n tly , K (4-18) G(X) = P(0) + [ P ( l ) - ^ 1 1 ] X + E [P (i) - P ( i ~ --) i = 2 X 1 By equation 4 -1 1 , the su m m ation term in equation 4 -1 8 is zero . 64 T h e r e f o r e , eq u atio n 4 -1 8 r e d u c e s to: (4-19) G(X) = P(0) + Fp (1) - L * 4 x . Solving eq u atio n 4 -1 0 fo r P ( l ) , and s u b stitu tin g into eq u atio n 4 -1 9 yields: G(X) = P(0) + ] x = P(0) 1 + ^ - x ] . T hus, G(X) = P (0)(1-X ) . S u b stitu tin g th is e x p r e s s i o n f o r G(X) into eq u atio n 4 -1 4 yield s: - P (0)(1-X ) _ q P (0 )(l-X ) C(X) “ ! _ 2£ + £ x K + 1 ' q - X + p X ^ * ’ To o b tain a so lu tio n fo r P(0), note that: C(l) = E P(i) . i = 0 The s u m of a ll the s ta t io n a r y p ro b a b ilitie s m u s t equal one. H ow e v e r , C(X) e v a lu a te d a t X = 1 is in d e te r m in a te . T h e r e f o r e , I 'H o s p ita l's ru le is a p p lie d to obtain: (4-20) C(X) P(0) X=1 - - ( K + l ) £ q q F r o m equation 4 -2 0 , the solution for P(0) is: 65 (4-21) W ith the c lo s e d f o r m e x p r e s s io n fo r P(0), a ll the o th e r s ta t io n a r y p ro b a b ilitie s c an be found f r o m the b a s ic d iff e r e n c e e q u atio n s. T h ey could a ls o be found f r o m C(X), u s in g the equation: H o w e v e r th is m e th o d is only of th e o r e tic a l i n t e r e s t , and not fe a s ib le fo r a c tu a l c a lc u la tio n s of the P ( i) 's . F o r the s e q u e n tia l o b s e r v e r , the f a ls e a l e r t p ro b a b ility , qt, is equal to th e s ta t io n a r y p r o b a b ility of the T ^ t h s ta te . F o r the m o d i fied a n a ly s is v e r s io n , T his value c a n be found fo r an y s p e c ific c a s e by co m p u tin g P(0) an d then a ll th e o th e r s ta t io n a r y p r o b a b iliti e s . H o w ev e r, no c lo s e d f o r m e x p r e s s i o n fo r co m p u tin g & h a s b e en found. A m e th o d h a s b e e n found to o b tain the m e a n and v a r ia n c e of the c u m u la tiv e s u m of a s e q u e n tia l o b s e r v e r , w h ic h g iv es s o m e m e a s u r e of the f a ls e a l e r t p e r f o r m a n c e . 2 L e t u be the m e a n value of the count, an d a be the v a r ia n c e > *c c of the count. Since l_ r a (C(X)) 1 L, r a x l T h is eq u atio n is c l e a r l y c o r r e c t , s in c e C(X) =P(0)+P( 1)X+P(2)X 2 • • • • (4-22) oi = 1 - £ P(i) i = 0 66 o o e © _ C(X) = E P ( i ) x \ C'(X) = E iP(i)Xl . i = 0 i = 0 T h e r e f o r e , C'(X) = E iP(i) =n_. X=1 i = 0 S im ila r ly , G"(X) = E i(i-i)P (i)x 1 -2 , i = 0 and C"(X) = S (1 - i)P(i) ■ X=1 i=(D Thus 2 a c = C'(X) +Uc-Uc- X=1 c c U sing e q u atio n 4 -1 2 , C(X) can be e x p r e s s e d by the equation: (4-23) C(X) _ (q -p K )(l-X ) y K+1 * q-X +pX H o w e v e r w hen this e q u atio n is d iff e r e n tia te d and e v a lu a te d a t X = 1, a n i n d e te r m in a te e x p r e s s i o n r e s u l t s , w h ich r e q u i r e s s e v e r a l lengthy a p p lic a tio n s of I’H o s p i ta l 's ru le to e v a lu a te . A n a l t e r n a t e f o r m of eq u atio n 4 -2 3 is: 67 (4-24) C(X) = . 1 -p E X 1 i = 0 F r o m eq u atio n 4 -2 4 , the follow ing e q u atio n s w e r e d e riv e d , _ pK (K +l) 2ll-p(K+l)J ’ a 2 - pK (K +l)[2(2K + l)-p(K +2)(K + l)1 c 1 2 [ l- p ( K + l) ] 2 o r 2 n c [2(2K + l)-p(K + 2)(K + l)] a c = 6 [l- p (K + l)j * T h e s e e q u atio n s give s o m e in d ic a tio n of the d is tr ib u tio n of the c u m u la tiv e s u m of the s e q u e n tia l o b s e r v e r , but can n o t be u s e d to e s ti m a te the fa ls e a l e r t r a t e by e s t i m a t e s b a s e d on the G a u s s ia n d i s trib u tio n . The r e a s o n fo r this is th at the i m p a s s ib le b a r r i e r a t z e r o h a s the effect of re je c tin g s o m e of the -1 in p u ts, but not a n y of the +K in p u ts. T his r e je c ti o n h as the effect of in tro d u c in g a p o s itiv e b ia s onto w hat w ould o th e r w is e be a G a u s s ia n d is trib u tio n . A n a p p r o a c h th a t g iv e s a m e a n s of c o m p a r in g the re c o g n itio n / f a ls e a l e r t c a p a b ilitie s of the se q u e n tia l o b s e r v e r an d the M out of N d e t e c t o r is the u s e of the " n eg a tiv e goin g " s e q u e n tia l o b s e r v e r . F o r this d e v ice , K w ill be s e le c te d su ch th at pK+q = 0. F o r th e s e p a r a m e t e r s , the c u m u la tiv e s u m of the s e q u e n tia l o b s e r v e r w ill be G a u s s ian d is tr ib u te d a b o u t a m e a n of z e r o . ( T h e r e w ill be a slig h t p e r t u r - 68 b atio n f r o m the G a u s s ia n d is tr ib u tio n b e c a u s e th e r e w ill be a finite n u m b e r of s ta t e s , but if the n u m b e r of s ta t e s is la r g e , this effect w ill be m i n i m a l , ) T he sig n a l to n o is e ra tio w ill b e found f o r the "n e g a tiv e going" s e q u e n tia l o b s e r v e r fo r the s a m e co n d itio n s a s in the e x a m p le in S ection 4. 4. To o b tain the s a m e f a ls e a l e r t p r o b a b ility , T R w ill be s e t to m a k e a re c o g n itio n d e c is io n w hen the n u m b e r of " p o s itiv e " in p u ts is K p s ta n d a r d d e v ia tio n s ab o v e the e x p e c te d n u m b e r of " p o s i tiv e " in p u ts. T his r e la tio n s h ip is e x p r e s s e d by the equation: In o r d e r to h a v e th e js a m e P of 1 /2 a s in the e x a m p le of S ectio n 4 .4 , the following eq u atio n m u s t be satisfie d : (4-25) Tr = [ N p + K F V N p ( l- p ) ] ( K + D - N . (4-26) N p g(K + l)-N = T ■ . C o m b in in g e q u atio n s 4 -2 5 a n d 4 -2 6 y ield s: (4-27) k f ________ Ps = P + — ^P(I-P) • s / N L e t x = the r e q u i r e d S /N fo r the "n e g a tiv e going" s e q u e n - s tia l o b s e r v e r to a tta in P q = 1 /2 and x m = the c o r r e s p o n d in g S /N fo r a n M out of N d e te c to r . F r o m equation 4 -7 , 69 By su b stitu tin g e q u atio n 4 - 4 into eq u atio n 4 -5 , the equ atio n l / d + x ) o p s = p is ob tain ed . F o r s m a l l x , the following a p p r o x im a tio n c a n be m a d e : s U - s ) P s = P = P - X s P0fl (p) . T hus, o r K F P - x P0n(p) = P + — P( 1 -p ), s 7 n k_ f \/p(l -p) (4-29) x = - , .— . s (/N p *"<p ) B y s u b stitu tin g the e x p r e s s i o n fo r p f r o m eq u atio n 4 -6 into eq u atio n 4 -2 9 , and p e r f o r m i n g a n u m b e r of a p p r o x im a tio n s an d s i m p lific a tio n s , the r e s u lt: KF[0n(R) + 2(l-R)] (4 ' 30) x s =" ‘ xm L 1 + 2NR(l-R)0n(R) J w as o b tain ed . In A p p en d ix A, it is show n th a t fo r b e s t d e te c tio n p e r f o r m a n c e : 0n(R) = 2 (R -1 ). E q u a tio n 4 -30 th u s r e d u c e s to: (4-31) x = x ' s m 70 Thus the n eg ativ e going s e q u e n tia l o b s e r v e r h as e s s e n tia l ly the s a m e d e te c tio n c a p a b ility a s the M out of N d e te c to r , fo r the optim um c h o ice of R. O f c o u r s e , the n eg ativ e going se q u e n tia l o b s e r v e r w ould be of a b s o lu te ly no u s e in the p r o c e s s o r - s h a r i n g a r c h i t e c t u r e , b e c a u s e a p r o c e s s o r w ould be r e q u ir e d fo r e v e r y d a ta s t r e a m , j u s t as in a c o n v en tio n al a r c h i t e c t u r e . H o w e v e r, the fu n d a m e n ta l s i m i l a r ity b e tw e e n th e s e two ty p es of se q u e n tia l o b s e r v e r s s tro n g ly in d i c a te s that the m o d ifie d s e q u e n tia l o b s e r v e r u se d in the p r o c e s s o r - s h a r in g a r c h i t e c t u r e is c o m p a r a b le in re c o g n itio n c a p a b ility to an M out of N d e te c to r . It is a ls o of i n t e r e s t th at the u se of the s e q u e n tia l o b s e r v e r in the p r o c e s s o r s h a r in g a r c h i t e c t u r e h a s a n o th e r a d v a n ta g e b e s id e s the q u ick r e je c ti o n c a p a b ility . The s e q u e n tia l o b s e r v e r is a m u c h s i m p l e r p r o c e s s o r to m e c h a n iz e than an M out of N d e te c to r . A s e q u e n tia l o b s e r v e r r e q u i r e s only log^CT^) b its to s t o r e the c u m u la tiv e s u m , w hile an M out of N d e te c to r r e q u i r e s N b its of s to r a g e . In ad d itio n , the s e q u e n tia l o b s e r v e r r e q u i r e s m u c h le s s d e c is io n logic. H o w ev e r, the M out of N d e t e c t o r has an a d v a n ta g e in th a t it r e c o v e r s m o r e q u ic k ly f r o m a s e q u e n c e of f a ls e " p o s itiv e " inputs th at a p p ro a c h , but do not r e a c h the th r e s h o ld level. F ig u r e 4. 3 g iv e s the o p tim u m M fo r v a rio u s v alu es of N f r o m 3 to 500. F o r e a c h M out of N d e te c to r , the e q u iv a le n t se q u e n tia l o b s e r v e r is given, w ith the b its r e q u i r e d p e r d e t e c t o r c ell, an d the p e r c e n ta g e s av in g o v e r the M out of N d e te c to r . A lso , the lo n g e st r e c o v e r y tim e s a r e given. It w ill be noted th at both the p e r c e n ta g e m e m o r y sav in g an d the r e c o v e r y tim e i n c r e a s e w ith N. H o w ev e r, it 71 M out of N D etectors Sequential O b s e rv e r -- N- M , opt K T Bits Needed % Saving M axim um R eco v ery Tim e (listening periods) 3 2 2 3 2 33 1 5 3 3 7 3 40 1.4 10 5 6 25 5 50 2. 5 20 8 13 92 7 65 4. 5 30 12 19 210 8 73 7 50 22 29 610 10 80 12 100 40 61 2, 380 12 88 24 150 50 101 4, 950 13 91 33 300 110 191 20, 820 15 95 70 500 150 351 52, 300 16 97 105 D O F ig u re 4. 3 Sequential O b s e rv e r Equivalents to M out of N D etecto rs. m u s t be noted th a t the value of K is lim ite d by the r e la tio n s h ip : pK 4 q. If this lim ita tio n is e x c e e d e d , then the s e q u e n tia l o b s e r v e r w ill not be a t the z e r o s ta te m o s t of the tim e , wh ich w as a s s u m e d in the a n a ly s is . In s te a d , the c u m u la tiv e s u m w ill tend to i n c r e a s e s te a d ily , an d r e a c h r e la tiv e ly often. T his lim ita tio n p r e v e n ts c o m p le te m a tc h in g of s o m e M out of N d e t e c t o r s , fo r c a s e s of la rg e N a n d / o r high v a lu e s of p. 73 C H A P T E R 5 E S T IM A T IO N O F T H E N U M B E R O F PR O C E S S O R S R E Q U IR E D In this c h a p te r , e q u atio n s a r e d e v elo p e d to e s t i m a t e the n u m b e r of p r o c e s s o r s r e q u i r e d in the p r o c e s s o r - s h a r i n g a r c h i t e c t u r e . T his n u m b e r is p r i m a r i l y a function of the a c tiv ity of the d a ta s t r e a m s , but is a ls o d e p e n d e n t on the s y s te m r e q u i r e m e n t s fo r r e c ognition p e r f o r m a n c e . F o r ty p ic a l c a s e s , the n u m b e r of p r o c e s s o r s r e q u i r e d is only a s m a l l p e r c e n ta g e of the n u m b e r of d a ta s t r e a m s . T hus the p r o c e s s o r - s h a r i n g a r c h i t e c t u r e a c h ie v e s a v e ry s u b s ta n tia l s av in g ' o v e r c o n v en tio n al a r c h i t e c t u r e s th at w ould r e q u ir e one p r o c e s s o r p e r d a ta s t r e a m . 5. 1 A n a ly s is of P r o c e s s o r R e q u i r e m e n ts The u s e of the p r o c e s s o r s h a r in g a r c h i t e c t u r e e n ta ils a c c e p ta n ce of s o m e s a c r i f i c e in re c o g n itio n p e r f o r m a n c e on the u n u su al o c c a s io n s th a t p s e u d o - a c tiv e d a ta s t r e a m s tie up a ll the p r o c e s s o r s . W hen this co n d itio n o c c u r s , a c tiv e d a ta s t r e a m s c an n o t be a s s ig n e d to a p r o c e s s o r . If the c o n d itio n p e r s i s t s fo r the e n tir e b u r s t of a c tivity, th e n the e v en t w ill of c o u r s e be m i s s e d . H o w e v e r, if a p r o c e s s o r should b e c o m e f r e e a f t e r lo ss of s o m e in itia l p o r tio n of the b u r s t of a c tiv ity , th e n r e c o g n itio n m a y s till be p o s s ib le , but is le s s likely th a n if so m e d ata h a d not b een lo st. ( T h e r e is a ls o a slig h t p o s s ib il ity th at a n e v en t c a n be m i s s e d b e c a u s e an a c tiv e d a ta s t r e a m w ith a low X ra tin g is d ro p p e d by the p r i o r i t y Logic in fa v o r of a p s e u d o - a c tiv e d a ta s t r e a m . This p r o b a b ility is q u ite low, a n d w ill not be in c lu d e d in the a n a l y s i s .) It w ill be show n th at w ith p r o p e r se le c tio n -q f the n u m b e r of p r o c e s s o r s , a c c o r d in g to the s y s t e m p a r a m e t e r s and r e q u i r e m e n t s , the Loss of p e r f o r m a n c e c a n be re d u c e d to m i n im a l le v e ls , w hile s till a c h ie v in g a g r e a t s a v in g s in the n u m b e r of p r o c e s s o r s r e q u ir e d . F o r this a n a ly s is , it w ill be a s s u m e d th at the n u m b e r of a c tiv e d a ta s t r e a m s h as a G a u s s ia n d is trib u tio n , w ith a m e a n of ^ j,B , and a s t a n d a r d d e v ia tio n of crB * L e t D be the n u m b e r of d ata s t r e a m s , and P g be the p r o b a b i l ity th at a d a ta s t r e a m is a c tiv e . Then, an d Qg a r e g iv en by: C l e a r l y the n u m b e r of p r o c e s s o r s m u s t e x c e e d the m e a n , jjg, by s o m e s a fe ty fa c to r . If th is sa fe ty f a c to r is e x p r e s s e d a s a p r o b a b ility th a t a p r o c e s s o r w ill be a v a ila b le w hen r e q u i r e d , then the follow ing eq u atio n is o btained: (5-1) (5-2) a B = a/ d p b ( 1 - p b ) . If P B is s m a ll, then e q u atio n 5 -2 c a n be a p p r o x im a te d by: (5-3) (5-4) Np ^ B + K C a B ’ w h e r e N p is the n u m b e r of p r o c e s s o r s r e q u i r e d a n d K^. is the c o n fid e n ce c o n s ta n t e x p r e s s e d in u n its of the s ta n d a r d d e v iatio n . 75 T h u s, = 1 c o r r e s p o n d s to a p r o c e s s o r a v a ila b ility p r o b a b ility of 0. 84, ,= 2 c o r r e s p o n d s to a p r o c e s s o r a v a ila b ility p r o b a b ility of 0. 977, and K^. = 3 c o r r e s p o n d s to a p r o c e s s o r a v a ila b ility p r o b a b ility of 0. 9987. The m e a s u r e of th e e ffe c tiv e n e s s of the p r o c e s s o r - s h a r i n g a r c h ite c tu r e is the ra tio of the n u m b e r of p r o c e s s o r s r e q u i r e d to the n u m b e r of d a ta s t r e a m s . T h is p a r a m e t e r , R p , is g iv en by: NP (5-5) R p = - j f . By su b stitu tin g e q u atio n s 5 -1 , 5-3, and 5 -4 into e q u atio n 5 -5 , the follow ing e q u atio n fo r R p is o btained: K / P T (5-6) R = P In eq u atio n 5 -6 , the P g t e r m is a fu nction only of the " s p a r s e n e s s " of the d a ta s t r e a m , an d is thus not u n d e r the c o n tr o l of the d ig ita l s y s te m . T h u s, P g d e t e r m i n e s the lo w e r lim it on the p e r c e n ta g e of p r o c e s s o r s r e q u ir e d . The K ^ y P g t e r m is the s a fe ty f a c to r, o r c o n fid e n ce level, an d c a n be v a rie d a c c o r d in g to the s y s te m r e q u i r e m e n t s . It w ill be noted th a t this t e r m g e ts s m a l l e r in p r o p o r tio n to P g a s D g e ts l a r g e r . T hus a v e r y high p r o c e s s o r a v a ila b ility p ro b a b ility , su ch a s 0. 9987, c a n be a tta in e d w ith only a m o d e s t p e r c e n ta g e i n c r e a s e in p r o c e s s o r s o v e r the m i n i m u m lev el 76 d e te r m in e d by the value of 3?g* The p a r a m e t e r is thus the c r i t i c a l p a r a m e t e r in the s a v in g s a c h ie v a b le by the p r o c e s s o r - s h a r i n g a r c h i t e c t u r e . T his p a r a m e t e r is a lw a y s " s m a l l " f o r " s p a r s e " d a ta s t r e a m s , but the m e a n in g of " s m a l l " d e p e n d s on the type of p r o b le m . F o r r a d a r t a r g e t d e te c tio n , P g w ould u s u a lly be le s s than 0 .0 1 , often v e r y m u c h l e s s . F o r e a r th q u a k e p r e d ic tio n , w h ich is a m u c h le s s e x a c t s c ie n c e a t p r e s e n t , P B could be p e r h a p s 0. 0 5. S om e e x a m p le s of the s av in g s a c h ie v a b le by the p r o c e s s o r - s h a r in g a r c h i t e c t u r e f o r P g = 0 .0 1 , an d P ^ = 0 .0 5 a r e g iv en in T a b le s 5. 1 a n d 5. 2. In th e se ta b le s , the f i r s t c o lu m n g iv es D, the n u m b e r of d a ta s t r e a m s . The se c o n d a n d th ir d c o lu m n s give |j,g and O jg r e s p e c tiv e ly . The n ex t th r e e c o lu m n s give the n u m b e r of p r o c e s s o r s r e q u i r e d fo r a v a ila b ility p ro b a b ilitie s of: 0. 84, 0. 977, and 0. 9987 r e s p e c tiv e ly . T he next c o lu m n g iv es the ra tio of N p / D fo r an a v a ila b ility p r o b a b ility of 0. 9987. It is s e e n that this p r o b a b ility d o e s a p p r o a c h P g a s D g e ts la rg e , a s p r e d ic te d by eq u atio n 5 -6. The la s t c o lu m n show s the p e r c e n ta g e s a v in g s a c h ie v e d by the p r o c e s s o r - s h a r i n g a r c h i t e c t u r e , fo r the co n d itio n of 0. 9987 a v a ila b ility p ro b a b ility , a s c o m p a r e d w ith a c o n v en tio n al a r c h i t e c t u r e using one p r o c e s s o r p e r d a ta s t r e a m . It is s e e n th at fo r P g = 0 .0 1 , a sav in g s of 96 to 98. 7 p e r c e n t is a c h ie v e d . E v en fo r the r e la tiv e ly high P g = 0. 05, a s a v in g s of 89 to 94. 3 p e r c e n t is s till a c h ie v e d . 5. 2 E s tim a tio n of P g, F o r a p p lic a tio n s in w h ich the s e q u e n tia l o b s e r v e r is u s e d fo r re c o g n itio n , an a p p r o x im a te eq u atio n c an be d e r iv e d fo r the m a j o r 77 TABLE 5.1 P r o c e sso r Requirements for Case with P g = 0.01 D Number of Data Streams Mean aB Standard Deviation Np , Number of P r o c esso rs Np /D Percentage of P r o c e sso r s for 0. 9987 Case Percentage Saving Availability Probability 0. 84 0. 977 0. 9987 100 1 1 2 3 4 4 96 500 5 2 7 9 12 2 .4 97. 6 1,000 10 3 13 16 20 2.0 98.0 5,000 50 7 57 64 71 1 .4 98. 6 10,000 100 10 110 120 130 1.3 98. 7 00 TABLE 5. 2 P r o c e s s o r R eq u irem en ts for C ase with P R = 0.05 D N um ber of Data S tre a m s M b M ean S tandard Deviation Np , N um ber of P r o c e s s o r s Np /D P e rc e n ta g e of P r o c e s s o r s for 0. 9987 C ase P e rc e n ta g e Saving A vailability P robability 0 .8 4 0.977 0.9987 100 5 2 7 9 11 11 89 500 25 5 30 35 40 8 92 1,000 50 7 57 64 71 7 93 5,000 250 16 266 282 298 6 94 10, 000 500 22 522 544 566 5.7 94.3 -j % o c o m p o n e n t of Pg> the p r o b a b ility th at r a n d o m n o ise w ill c a u s e a p r o c e s s o r to be " tie d - u p ." D enote th is p r o b a b ility by P ^ , an d let P-p= 1 - P g . A c c o rd in g to the s y s te m p r i o r i t y r u le s , a d ata s t r e a m w ill h a v e a h ig h e r X ra tin g than a new " p o s itiv e " input, if its a s s ig n e d s e q u e n tia l o b s e r v e r h a s a c u m u la tiv e s u m of K o r g r e a t e r . T h e r e fo r e , P g w ill be ta k en a s the p r o b a b ility th at the c u m u la tiv e s u m of a s e q u e n tia l o b s e r v e r w ill h a v e a count of K o r g r e a t e r u n d e r a s e q u e n ce of r a n d o m inputs not d u rin g a b u r s t of a c tiv ity . The s ta t io n a r y p ro b a b ility e q u atio n s d e v elo p e d in s e c tio n 4. 5 c a n be u s e d fo r this a n a ly s is . T h u s: (5-7) T h e r e f o r e : K - l (5-8) P F = S P(i) . i = 0 E q u atio n 5-8 c a n be r e w r i t t e n a s: (5-9) K -l P = F(0) + S P(i) • F i =l By solving e q u atio n 4 -10 f o r P ( l ) , the equation: P ( l) = ^P (O ) is obtained. U sing equation 4 -1 1 , it is found that: 80 P( 2) = = PF j 0) q q2 S im ila r ly , fo r 1 s i s K , (5-10) P(i) = . l q S u b stitu tio n of eq u atio n 5-10 into eq u atio n 5 -9 yields: (5-11) P f = P (0 ) + KS i = l q E q u atio n 5-11 c an be r e d u c e d to: (5-12) P = it) Ki;1 . q S u b stitu tin g f r o m eq u atio n 4 -21 y ield s: (5-13) P F = . q F r o m eq u atio n 5-1 3 , it follow s that: K (5-14) P g = 1 - . q M a th e m a tic a l m o d e ls fo r m e d ic a l e x a m p le s a r e not a s w e ll d efin ed a s s o m e o th e r a r e a s , but s o m e u se fu l a n a ly s is can be m a d e . To d e r iv e a n e s t i m a t e of P ^ fo r a m e d i c a l e x a m p le , it is n e c e s s a r y 81 to f i r s t c o n s id e r the r e c o g n itio n m e th o d . In the ECG m o n ito r in g e x a m p le of Section 2. 2, the f i r s t lev el p r o c e s s i n g w as a one m in u te s a m p le of a p a tie n t's E C G . If th is s a m p le gave ev en a slig h t in d ic a tion of a d a n g e r o u s h e a r t condition, th en the p a tie n t's d a ta s t r e a m w as a s s ig n e d to a p r o c e s s o r fo r f u r t h e r s a m p lin g . To d e te r m in e the n u m b e r of P V G 's that w ould in d ic a te p o s s ib le d a n g e r , T ab le 5. 3 w ould be u s e d . T h is tab le g iv es the p r o b a b ility of obtain in g a given n u m b e r of P Y C 's in a s a m p le of 100 b e a ts , w h en the p a tie n t w a s a c tu ally having P V G 's a t the c r i t i c a l ra tio of one in ten, d is tr ib u te d a t r a n d o m . T h u s -the p r o b a b ility of g ettin g no P V G 's in one h u n d re d _5 b e a ts is s e e n to be 2. 7 x 10 , an d th e p r o b a b ility of g ettin g one PV C -4 is 2. 9 X 10 . It w ould be r e a s o n a b le to a s s u m e th at a p a tie n t w a s not in d a n g e r if t h e r e w e r e not m o r e th an one PV G in the one h u n d re d b e a t s a m p l e . T h is w ould c o r r e s p o n d to a g, o r m i s s p ro b a b ility , of To d e t e r m i n e a a n d P g , it w ould be n e c e s s a r y to know the p r o b a b ility d is tr ib u tio n of P V G 's fo r p a tie n ts w ithout h e a r t d i s e a s e . U n fo rtu n a te ly , su c h d a ta is not a v a ila b le . A s s u m e th at th e ra tio of P V G 's to n o r m a l b e a ts f f o r s u c h p a tie n ts is s o m e Ppj* d is tr ib u te d a t ra n d o m . T h en the p r o b a b ility th at a n o r m a l p a tie n t w ould give a n in d ic a tio n of d a n g e r , a n d r e q u i r e a s s i g n m e n t of a p r o c e s s o r , i. e. , P g , is g iv en by: 3. 2 X 10 (5-15) 100- i 1001 i !( 100-i) J * The su m m ation in equation 5 -15 is fro m 2, the low er threshold, up 82 T A B L E 5. 3 P r o b a b il ity of P V G 's fo r a s a m p le of 100 B e a ts , in w h ic h the R atio of P V G 's to N o r m a l B e a ts is 0. 1 N u m b e r of P V G 's P r o b a b il ity 0 2. 7 X io" 5 1 2. 9 X i o “4 2 1. 6 X io" 3 3 5. 9 X i o ' 3 4 1. 6 X io" 2 5 3. 4 X i o - 2 6 6. 0 X i o - 2 7 O' 00 X i o - 2 8 1.1 X h -» o 1 H -4 9 1. 3 X I— * o t h -J 10 1. 3 X 1 o 83 to 9. If t h e r e w e r e 10 o r m o r e P V G 's , th e n a n a l e r t w ould be is su e d , w h ich w ould a c tu a lly be a f a ls e a l e r t . The p r o b a b ility of a fa lse a l e r t , is g iv e n by: /k -p - 1v 0 T D i /i ■ £ > a o o - i 1 0 0 ! ( 5 - l6 ) FA H H i ! ( 100-i) 1 * 5. 3 E v en t R ec o g n itio n a f t e r L o s s of In itial P o r t i o n of B u r s t of A ctiv ity The a n a ly s is of S ectio n 5. 1 w a s b a s e d on the p r o b a b ility of h a v ing a p r o c e s s o r a v a ila b le a t the s t a r t of the b u r s t of a c tiv ity in d ic a t ing an ev en t. In s o m e a p p lic a tio n s , e s p e c ia lly th o se involving r e l a tiv e ly long b u r s t s of a c tiv ity , it m a y be n e c e s s a r y to c o n s id e r the in te r m e d ia te c a s e of re c o g n itio n a f t e r lo s s of a n in itia l p o rtio n of the b u r s t of a c tiv ity . F o r s u c h a n e x a m p le , r e c o g n itio n w ould r e s u l t if M " p o s itiv e " e v a lu a tio n s w e r e a tta in e d on the r e m a in in g p o rtio n of the b u r s t of a c tiv ity , w h ic h w ould be le s s th a n N, in p u ts. F o r s u c h a p p lic a tio n s , eq u atio n 4 -1 w ould be m o d ifie d to an eq u atio n of the fo r m : N -M + l (5-17) P = £ F (i)R (i), i = 1 w h e r e F(i) is the p r o b a b ility that a p r o c e s s o r is a v a ila b le fo r the f i r s t tim e on the ith input of the event R(i) is the r e s u ltin g re c o g n itio n p r o b a b ility w hen the p r o c e s s is s t a r t e d on the ith input of the event. F o r the s e q u e n tia l o b s e r v e r , o r for M out of N d e te c tio n , R(i) is 84 given by: (5-18) j ! ( N - i- j) J * E q u atio n 5 -17 w ould r e q u i r e a d a p ta tio n fo r an y p a r tic u la r , a p p lic a tio n . In the c o m p u te r p r o g r a m d e s c r i b e d in C h a p te r 6 , the S E Q P m o d u le w h ich is d i s c u s s e d in S e c tio n 6. 1. 6 , m e c h a n i z e s a n e q u atio n of the type of eq u atio n 5 -1 7 . The d e ta ils of this eq u atio n a r e d e s c r i b e d in A p p en d ix B. 85 C H A P T E R 6 C O M P U T E R SIM U LA TIO N O F P R O C E S S O R SHARING A R C H IT E C T U R E In this c h a p te r , a c o m p u te r p r o g r a m is d e s c r i b e d th at s i m u la te s the p e r f o r m a n c e of the p r o c e s s o r - s h a r i n g a r c h i t e c t u r e fo r the e x a m p le of the d e te c tio n of sig n a ls in d ata s t r e a m s c o n tain in g b a c k g ro u n d n o ise. T h is e x a m p le is u s e d fo r the s im u la tio n b e c a u s e the r e c o g n itio n p r o c e s s is d efin ed by w e ll know n m a t h e m a t i c a l m o d e ls . In fact, the body of u se fu l l i t e r a t u r e in this field is too e x te n s iv e to r e f e r e n c e . In s te ad , the r e a d e r is r e f e r r e d to the R a d a r H andbook 28 e d ite d by Skolnik w h ic h c o n ta in s a w e a lth of in f o r m a tio n a s w e ll a s a c o m p r e h e n s iv e lis t of r e f e r e n c e s in a ll a r e a s of r a d a r . It w ould be in te r e s tin g an d d e s i r a b l e to s im u la te the m e d ic a l m o n ito r in g and the e a r th q u a k e p r e d ic tio n e x a m p le s a ls o , but a t p r e s e n t t h e r e is not enough p a r a m e t e r d ata a v a ila b le to do so. The r e s u l t s of this s i m u latio n a r e in a c c o r d w ith p r e d ic tio n s m a d e u sin g eq u atio n s 5 -1 , 5 -3, an d 5-4. 6. 1 D e s c r ip tio n of the C o m p u te r P r o g r a m T he p u r p o s e of the c o m p u te r p r o g r a m is to c o m p a r e the p e r f o r m a n c e of the p r o c e s s o r - s h a r i n g a r c h i t e c t u r e , u sin g a l i m i t e d n u m b e r of s e q u e n tia l o b s e r v e r s , w ith that of a c o n v e n tio n a l p r o c e s s o r u sin g an M 'o u t of N d e t e c t o r fo r e v e r y d ata s t r e a m . The m e a s u r e 86 of p e r f o r m a n c e is the sig n a l to n o ise r a tio , S /N , r e q u i r e d to a tta in a d e s i r e d P j y a t a s p e c ifie d a, o r f a ls e a l e r t r a te , P ^ . A n i n t e r e stin g r e s u l t found in this s im u la tio n is th at the p r o c e s s o r - s h a r i n g a r c h i t e c t u r e has an in h e r e n t lim ita tio n on the a tta in a b le , ev en fo r an a r b i t r a r i l y high S /N . O n the o th e r hand, a n M out of N d e te c t o r c a n a tta in any d e s i r e d P ^ , fo r a su ffic ie n tly high S /N . T h e r e fo r e the c o m p u te r p r o g r a m w as d e s ig n e d to f i r s t c a lc u la te if the given n u m b e r of s e q u e n tia l o b s e r v e r s , o r " d e te c to r s ," w a s a d e q u a te to a tta in the d e s i r e d P j y u n d e r the sp e c ifie d c o n d itio n s. If the d e s i r e d P q is a tta in a b le , then the d iff e r e n c e b e tw e e n the S /N r e q u i r e d by the p r o c e s s o r - s h a r i n g a r c h i t e c t u r e , an d th a t r e q u i r e d by the c o n v en tio n al a r c h i t e c t u r e is c o m p u te d . The p r o g r a m h a s a m o d u la r o rg a n iz a tio n , th a t is, it has a set of in d e p en d e n t s u b ro u tin e s that c a n be c a lle d in s e q u e n c e . T his o r g a n iz a tio n fa c ilita te s ch ec k o u t, an d a ls o enables, c h a n g e s to be m a d e in one m o d u le w ith o u t a ffe ctin g the r e s t of the p r o g r a m . The input p a r a m e t e r s ! t o the p r o g r a m a r e : P q the d e s i r e d o v e r a ll p r o b a b ility of d e te c tio n P p ^ the s p e c ifie d fa ls e a l e r t r a te N the n u m b e r of inputs e x p e c te d f r o m a ta r g e t D the n u m b e r of d a ta s t r e a m s , o r ra n g e bins Ig the n u m b e r of the t a r g e t m o d e l to be u sed I j-j the n u m b e r of d e t e c t o r s u sed . 87 S om e of the im p o r ta n t p r o g r a m m o d u le s w ill be b r ie f ly d e s c rib e d . 6 . 1 . 1 M O P T M odule The M O P T m o d u le d e t e r m i n e s the o p tim u m M fo r the g iv e n N. F o r tow v alu es of N, tab le lo o k -u p is u sed, but fo r h ig h e r v a lu e s of 22 N, a q u a d r a tic fo r m u la d e v elo p e d by J. W a lk e r is u sed . 6 . 1 . 2 TR SH M odule The TRSH m o d u le p e r f o r m s two r e la te d fu n ctio n s. The f i r s t function is the c o m p u ta tio n of p, the sin g le s c a n p r o b a b ility of n o ise c r o s s i n g the th re s h o ld , th a t c o r r e s p o n d s to the s p e c ifie d f a ls e a l e r t r a te , T he se c o n d fu nction is the d e te r m in a tio n of the t h r e s h old, T, r e q u i r e d to o b tain p. F o r s m a l l p, the t e r m s fo r j > M can be n e g le c te d in e q u atio n 4. 2 re d u c in g the eq u atio n to: F o r s m a l l p (e s p e c ia lly ' i'fi hhis s m a ll) , eq u atio n 6-1 c a n be a p p r o x i m a te d by: F o r the c a s e s c o n s id e r e d in this p r o g r a m , eq u atio n 6 -2 w a s a c c u r a t e enough. If m o r e a c c u r a c y w e r e r e q u ir e d , an ite r a tiv e b in a r y s e a r c h so lu tio n of eq u atio n 6-1 could be m a d e s i m i l a r to th at d e s c r i b e d in ( 6 - 1 ) M !(N -M )! * NJ (6 - 2) P = 88 the BSR m o d u le in S ectio n 6. 1. 3. The th r e s h o ld is c o m p u te d u sin g eq u atio n 4 -4 , an d c o n v e r te d into db, using eq u atio n 2 -1. 6 . 1. 3 BSR M odule The BSR m o d u le finds the p g that s a ti s f i e s eq u atio n 4-1 fo r the d e s i r e d P j y A b in a r y s e a r c h m e th o d is u sed . T h a t is, an in itia l e s t i m a t e of 1 /2 is m a d e fo r p , an d the r e s u ltin g ou tp u t, P_\,of the s O c u m u la tiv e b in o m ia l d is tr ib u tio n is c o m p u te d . If I q > then a p g of 1 /4 is the next t r i a l , w hile if then a p^ of 3 /4 is the next t r i a l . S im ila r ly , the th ir d t r i a l c a n be 1 /8 , 3 /8 , 5 /8 , o r 7 /8 , depending on the f i r s t two t r i a l s . The s e a r c h is stopped, w h en T en to tw enty t r i a l s a r e u s u a lly r e q u ir e d . 6 . 1 . 4 S N T B L M odule The S N T B L m o d u le finds the S /N r e q u i r e d to a tta in the r e q u ir e d w ith the th r e s h o ld of T. F o r the n o n -flu ctu a tin g t a r g e t m o d e l, a table lo o k -u p m e th o d is u sed . F o r the o th e r t a r g e t m o d e ls , the S /N c an be c o m p u te d f r o m the t a r g e t p r o b a b ility d e n s ity function. 6. 1. 5 G P R O B M odule The G P R O B m o d u le c o m p u te s the p a r a m e t e r s fo r a s e q u e n tia l o b s e r v e r e q u iv a le n t to a n M Qp t out N d e te c to r . F i r s t , K and T R a r e c o m p u te d f r o m e q u atio n s 4 -8 an d 4 -9 . T hen, P(0) is c o m p u te d f r o m eq u atio n 4 -2 1 . N ext, the s ta t io n a r y p ro b a b ilitie s a r e c o m p u te d 89 fo r a ll s ta te s e x c e p t u sin g the d if f e r e n c e e q u atio n s 4 -1 0 , 4 -1 1 , an d 4 -1 2 . F in a lly , the c u m u la tiv e s ta t io n a r y p r o b a b iliti e s , C(i), a r e c o m p u te d fro m : The s ta t io n a r y p r o b a b ility of P ( T ^ ) is s e t e q u al to the c o m p u te d by eq u atio n 6 -4. A ty p ic a l output of the G P R O B m o d u le is show n in F ig u r e 6. 1. B elow th is, in F ig u r e 6. 2, the co n d itio n s fo r this e x a m p le a r e p r in t e d out. T h ey a r e : N = 5, M =3 (as d e t e r m i n e d by the M O P T m o d - — 5 ule), Ig= 5, D = 100, P y ,^ = 10 , and 1^= 1. It is s e e n th at the s e q u e n tia l o b s e r v e r h a s a K of 3, and a T ^ of 7, w h ic h is the e x a m p le ill u s t r a t e d in F ig u r e 4. 2. The s ta tio n a r y p r o b a b iliti e s , and the c u m u la tiv e p r o b a b iliti e s a r e lis te d . It is s e e n th a t P(0) is a p p r o x im a te ly 0. 9.7., w h ich is c lo s e to 1, a s e x p e c te d fo r s m a l l P p ^ . It is a ls o _ 5 s e e n th at P(7) is 0. 000010685, w h ich is q u ite c lo s e to 10 , the s p e cified T h is r e s u l t is in a g r e e m e n t w ith the a n a ly s is of S ectio n 4. 5, w h ic h in d ic a te d th at a se q u e n tia l o b s e r v e r m a tc h e d to a n M out of N d e t e c t o r by e q u a tio n s 4 -8 and 4 -9 , w ould a ls o h ave a c o m p a r a b le i (6-3) C(i) = E P(j). j = 0 E q u atio n 4 -2 2 c a n thus be e x p r e s s e d a s: (6-4) P F A ' 90 SEQUENTIAL OBSERVER K = 3 TR = 7 j P ( I ) C ( I ) 0 0 .9 6 9 6 9 6 9 9 3 2 0 .9 6 9 6 9 6 9 9 3 2 1 0 .0 0 9 7 9 4 9 1 1 5 0 .9 7 9 4 9 1 9 0 4 7 2 0 . 0 0 9 89 3 H5 00 0 .9 8 9 3 8 5 7 5 4 7 3 0 .0 0 9 9 9 3 7 8 7 8 0 .9 9 9 3 7 9 5 4 2 4 4 0 .0 0 0 2 9 9 8 2 3 5 .0 .9 9 9 6 7 9 3 6 5 9 5 0 .'0002039136 0 .9 9 9 8 8 3 2 7 9 5 6 0 .0 0 0 1 0 6 0 3 5 5 0 .9 9 9 9 8 9 3 1 5CI 7 0 .0 0 0 0 1 0 6 8 5 0 1.0000 0 0 0 0 0 0 F ig u re 6. 1. Output of the GPROB Module. 1 1 2 3 PS: H : F ( I) 0 .3 44 00 0 .3 6 9 7 6 0 .0 1 7 6 8 R(I) 0 .7 9 48 9 0 .5 9 834 0 .3 0 0 6 3 0 .6699 5 M =- pr - nr-ratEi) 3 PD=0.500 0.5000001 IS= 5 D ITERATIONS:: 18 F A C T 0 R = 0.6302770 100 PF A= 0 . 1 00L-0 4 J.D= 1 F ig u re 6. 2. Output of the SEQ P Module. 6. 1.6 S E Q P M odule The S E Q P m o d u le c o m p u te s p , the r e q u i r e d sin g le s c a n p r o b - s a b ility of d e te c tio n fo r the p r o c e s s o r - s h a r i n g a r c h i t e c t u r e . E q u a tions 5 -17 an d 5 -18 a r e u s e d fo r this p u r p o s e . The c a lc u la tio n of the F(i) t e r m in eq u atio n 5 -17 is quite c o m p lic a te d an d is d e s c r i b e d in A p p en d ix B. The f i r s t o p e r a tio n in the S E Q P m o d u le is to c o m p u te the u p p e r lim it, P ^ , f r o m the equation: N -M + l (6-5) P , = S F(i) . i ~ J . i 1=1 If Pj} £ P j y th en the d e s i r e d Pj^ c an n o t be a c h ie v e d . The S /N is s e t to 100 db to identify this condition. F o r this c a s e , P ^ is p r in t e d out to in d ic a te the u p p e r lim it. H o w ev e r, if P ^ > P j y the d e s i r e d P ^ c an be a tta in e d , and the S E Q P c o m p u te s the r e q u i r e d p . H e r e a lso , s a b in a r y s e a r c h m e th o d is u se d , w ith an in itia l t r i a l of p e q u al to s 1 /2 . The t r i a l value is u s e d in eq u atio n 5-18 to o b tain v alu es of R(i) for: 1 < i < (N -M + l). T h e s e v alu es of R(i) a r e u s e d in eq u atio n 5-17, obtaining s o m e p ro b a b ility , P j . If P ^ > P ^ , then, p g= 1 /4 is the n ex t tr i a l, w h ile if P ^ < P ^ , then p g = 3 /4 is the next t r i a l . The ite r a tio n s c o n tin u e until: (A lim ita tio n of 20 i te r a tio n s is in c lu d e d a s a sa fe ty f a c to r a g a in s t having the p r o g r a m c au g h t in s o m e kind of an infinite loop. ) F i g u r e 6 . 2 sh o w s the output of the S E Q P m o d u le f o r a ty p ic a l e x a m p le . F o r this e x a m p le , N -M + l = 3. T h e r e f o r e the index, i, ^ in eq u atio n 5 -1 7 ru n s f r o m 1 to 3. The v alu es of F(i) a n d R(i) a r e show n fo r the fin al i te r a tio n w ith p = 0. 6699. The re s u ltin g ite r a te d s is show n a s 0. 50000001, a tta in e d in 18 i t e r a tio n s . The t e r m F A C T O R is u s e d in the c a lc u la tio n of F(i), a s d e s c r i b e d in A p p en d ix B. 6 . 1 . 7 S L O S S M odule The SLOSS m o d u le c o m p u te s the d iff e r e n c e b e tw ee n the S /N r e q u i r e d fo r the j p r o c e s s o r s h a r in g a r c h i t e c t u r e to a c h ie v e p , and s th at r e q u i r e d by the M ^ out of N d e te c to r . 6. 2 C o m p u te r S im u la tio n R e s u lts In o r d e r to d e v e lo p s o m e of the fu n d a m e n ta l p a r a m e t e r i n t e r r e la tio n s h ip s , two c a s e s , N = 5 and N =10, w e r e s tu d ie d o v e r a w ide ra n g e of a ll the m a j o r input c o n d itio n s. The o v e r a ll p r o b a b ility of d e te c tio n w a s v a r ie d f r o m 0. 5 to 0. 95 in 0. 05 i n c r e m e n t s . F a l s e — 5 _£) _ y a l a r m p r o b a b iliti e s of 10 , 1 0 , 10 and 10 w e r e u sed . The n u m b e r s of ra n g e bins u s e d w e r e 20, 50, 100, 500 an d 1000. F in ally , the n u m b e r s of d e t e c t o r s u s e d w e r e s e le c te d to c o v e r , in s o f a r a s w a s p o s s ib le , the ra n g e of i n te r e s t ; th at is, f r o m the m i n i m u m of d e t e c t o r s a t w h ich d e te c tio n w as fe a s ib le up to the point w h e r e the lo s s b e c o m e s n eg lig ib le. 6. 2. 1 B a s ic C a s e , F iv e L o o k s O ne ty p ic a l c a s e th at i l l u s t r a t e s the in te r r e l a t i o n s h i p of the _7 p a r a m e t e r s is th at f o r N = 5 a n d P ^ = 10 . F o r this c a s e , M 3. It is found th at the re s u ltin g 3 out of 5 d e te c to r r e q u i r e s a value of 93 _7 p = 0. 0022 to m e e t the sp e c ifie d of 10 . The e q u iv a le n t s e q u e n tia l o b s e r v e r h a s a K of 3 and a of 7. U sing the s a m e value of p = 0 .0 0 2 2 in th e b a s ic s ta t io n a r y e q u atio n s fo r the s e q u e n tia l o b s e r v e r , it is found th at P ^ is a p p r o x im a te ly 0. 005. T he m e a n value, l_Lg, of the n u m b e r of d e te c to r s th at w ill be tie d -u p by n o ise fo r the v a rio u s n u m b e r of d a ta s t r e a m s is a s follow s: 20 d a ta s t r e a m s 0 . 1 d e t e c t o r s tie d - u p 50 d a ta s t r e a m s 0. 25 d e t e c t o r s tie d -u p 100 d a ta s t r e a m s 0. 5 d e t e c t o r s tie d -u p 500 d a ta s t r e a m s 2. 5 d e t e c t o r s tie d -u p 1000 d a ta s t r e a m s 5. 0 d e t e c t o r s tie d - u p . T he n u m b e r of d e t e c t o r s inv o lv ed is too s m a l l fo r eq u atio n 5 -4 to be a p p lic a b le , h o w e v e r it w ould be r e a s o n a b le to p r e d i c t th at one d e te c to r w ould be quite a d e q u a te fo r 20 d a ta s t r e a m s , an d r e a s o n a b ly a d e q u a te fo r 50 d a ta s t r e a m s . F o r 100 d a ta s t r e a m s , one d e t e c t o r sh o u ld be r e a s o n a b ly good, but a little m a r g i n a l . F o r 500 d ata s t r e a m s , p o o r p e r f o r m a n c e w o u ld be e x p e c te d fo r one of two d e t e c t o r s . T h r e e d e t e c t o r s sh ould be r e a s o n a b ly good and fo u r d e te c t o r s q u ite good. F in a lly , f o r 1000 d a ta s t r e a m s , five d e t e c t o r s w ould be r e q u i r e d to give e v en r e a s o n a b le p e r f o r m a n c e . T h e s e p r e d ic tio n s a r e not p r e c i s e , but do give a n in d ic a tio n of w h a t c a n be e x p e c te d . The a c tu a l r e s u l t s f r o m the c o m p u te r p r o g r a m a r e p r e s e n t e d in F i g u r e s 6 -3 to 6-7 w h ich give ta b le s of the d iff e r e n c e , o r lo s s , in S /N fo r the v a rio u s n u m b e r s of d ata s t r e a m s , fo r P D ra n g in g f r o m 0. 5 to 0. 95, fo r one to five d e t e c t o r s . (T he S /N d if f e r e n c e s have 94 TABLE O F LOSS IN SIGNAL T O NOISE RATIO (DB) FO R LIMITED M EM O R Y SEQUENTIAL DETECTION 5 L O O K S F A L S E A L A R M P R O B . = 0 . 1 0 E - 0 6 1 D E T E C T O R S * * B I N S k k 2 0 5 0 . 1 0 0 5 0 0 1 0 0 0 k k * k P R O B . D E T . k k k k k k k k k k 0 . 5 k k 0 . 0 0 . 0 0 . 1 1 . 0 1 0 0 . 0 k k * * 0 . 5 5 k k 0 . 0 0 . 0 0 . 1 1 . 1 1 0 0 . 0 k k k k 0 . 6 k k 0 . 1 0 . 1 0 . 2 1 . 3 1 0 0 . 0 k k k k 0 . 6 5 k k 0 . 0 0 . 1 0 . 1 T . 5 1 0 0 . 0 k k k k 0 . 7 k k 0 . 1 0 . 1 0 . 2 2 . 0 1 0 0 . 0 k k k k 0 . 7 5 k k 0 . 1 0 . 1 0 . 2 1 0 0 . 0 1 0 0 . 0 k k k k 0 . 8 k k 0 . 1 0 . 2 0 . 3 1 0 0 . 0 1 0 0 . 0 k k k k 0 . 8 5 k k 0 . 0 0 . 1 0 . 2 1 0 0 . 0 1 0 0 . 0 k k k k 0 . 9 k k 0 . 1 0 . 1 Q . A 1 0 0 . 0 1 0 0 . 0 k k k k 0 . 9 5 k k Q . 1 0 . 2 0 . 5 1 0 0 . 0 1 0 0 . 0 k k k k i k k k k k k k k k k k ( 1 0 0 I N D I C A T E S T H A T T H E P R O B A B I L I T Y O F D E T E C T I O N C A N N O T B E O B T A I N E D ) Figure 6. 3. Table of Loss in Signal to Noise Ratio (db) for Limited Memory Sequential Detection. . 5 Looks False A larm Prob. = 0. 10E-06 1 Detector TABLE O F LOSS IN SIGNAL T O NOISE RATIO (DB) FO R UNITED M EM O R Y SEQUENTIAL DETECTION 5 L O O K S F A L S E A L A ' R M P R O B . = 0 . 1 0 E - 0 6 2 D E T E C T O R S A A A A A A A A A A A . A A A A * * B I N S k k 2 0 5 0 1 0 0 5 0 0 1 0 0 0 it A * * P R O B . D E T . k k k k k k k k k k * k k i k k A ik * * 0 . 5 k k 0 . 0 0 . 0 0 . 0 0 . 4 1 . 5 A A * * 0 . 5 5 k k 0 . 0 0 . 0 0 . 0 0 . 5 1 . 8 it : A k. k 0 . 6 k k 0 . 0 0 . 0 0 . 1 0 . 4 2 . 7 A it * * 0 . 6 5 k k 0 . 0 0 . 0 0 . 0 0 . 4 1 0 0 . D A A k k 0 . 7 k k 0 . 0 0 . 0 0 . 0 0 . 6 1 0 0 . 0 A A k k 0 . 7 5 k k 0 . 0 0 . 0 0 . 0 0 . 7 1 0 0 . 0 A it k k 0 . 8 k k 0 . 0 0 . 0 0 . 1 0 . 8 1 0 0 . 0 A A k A 0 . 3 5 k k 0 . 0 0 . 0 0 . 0 1 . 1 1 0 0 . 0 A A k k 0 . 9 k k 0 . 0 0 . 0 0 . 1 1 0 0 . 0 1 0 0 . 0 A * k k 0 . 9 5 k k 0 . 0 0 . 0 0 . 1 1 0 0 . 0 1 0 0 . 0 • A * ★ * A A A it A A A A A A A A A A A A \k A A A A A A ( 1 0 0 I N D I C A T E S T H A T T H E . P R O B A B I L I T Y O F D E T E C T I O N C A N N O T B E O B T A I N E D ) Figure 6 .4 . Table of Loss in Signal to Noise Ratio (db) for Limited M emory Sequential Detection. 5 Looks False A larm Prob. = 0. 10E-06 2 Detectors N O TABLE OF L O S S IN SIGNAL T O NOISE RATIO (DB) FO R LIMITED M EM O R Y SEQUENTIAL DETECTION 5 L O O K S F A L S E A L A R M P R O B . = 0 . 1 0 E - 0 6 3 D E T E C T O R S * * B I N S * * + * P R O B . D E T . * * 20 5 0 100 k irk'k'k'kk'k'k-k'kik'kikic 5 0 0 1 0 0 0 * * k k k k 0 . 5 * * 0 . 0 0 . 0 0 . 0 0 . 1 0 . 6 * * * * 0 . 5 5 ★ k 0 . 0 0 . 0 D . O 0 . 1 0 . 6 ■* k k k 0 . 6 k k 0 . 0 0 . 0 0 . 0 0 . 1 0 . 8 •k -k * k k 0 . 6 5 k k 0 . 0 0 . 0 0 . 0 0 . 1 0 . 9 * * k k 0 . 7 k k 0 . 0 0 . 0 0 . 0 0 . 2 1 . 1 * : k k k 0 . 7 5 k k 0 . 0 0 . 0 0 . 0 0 . 2 1 . 6 •k k k k 0 . 8 k k 0 . 0 0 . 0 0 . 0 0 . 2 1 0 0 . 0 -k k k k 0 . 8 5 k k 0 . 0 0 . 0 0 . 0 0 . 2 1 0 0 . 0 * ■ * k k 0 . 9 k k 0 . 0 0 . 0 0.0. 0 . 4 1 0 0 . 0 * * k k 0 . 9 5 k k 0 . 0 0 . 0 0 . 0 0 . 7 1 0 0 . 0 * * < 1 0 0 I N D I C A T E S T H A T T H E P R O B A B I L I T Y O F D E T E C T I O N C A N N O T B E O B T A I N E D ) Figure 6. 5. Table of Loss in Signal to Noise Ratio (db) for Limited M em ory Sequential Detection. 5 Looks False A larm Prob. = 0. 10E-06 3 Detectors TABLE O F LOSS IN SIGNAL T O NOISE RATIO (DB) FO R LIMITED M EM O R Y SEQUENTIAL DETECTION 5 L O O K S F A L S E A L A R M P R 0 8 . = 0 . 1 0 E - 0 6 4 D E T E C T O R S * * BINS * * 20 50 100 500 1000 ** k k P R O B . D E T . ** * * t k k k * * * 0 .5 * * 0 .0 0 .0 0 .0 0.0 0 .2 k k k k 0 .5 5 ** 0.0 0 .0 0 .0 0.0 0.2 k k k * 0 .6 ** 0.0 0 .0 G . O 0.1 0 .3 k k * * 0 .6 5 * * 0 .0 0 .0 0 .0 0 .0 0 .3 k k * * 0 .7 * * 0 .0 0 .0 0 .0 0.1 0 .4 k k * * 0 .7 5 * * 0 .0 0 .0 0 .0 0.1 0 .5 k k k it 0.8 * * 0 .0 0 .0 0 .0 0.1 0 .6 k k k * 0 .8 5 * * 0 .0 0 .0 0 .0 0.0 0 .8 k k k k 0 .9 * * 0 .0 D . O 0 .0 0.1 100-0 k k it k 0 .9 5 ** 0.0 *********; 0 .0 . 0 .0 0.1 100.0 k k ( 1 0 0 I N D I C A T E S T H A T T H E P R O B A B I L I T Y O F D E T E C T I O N C A N N O T B E O B T A I N E D ) Figure 6. 6. Table of L oss in Signal to Noise Ratio (db) for Limited M emory Sequential Detection. 5 Looks F alse A larm Prob. = 0 .1 0 E -0 6 4 Detectors vO oo TABLE O F LOSS IN SIGNAL T O NOISE RATIO (OB) FO R LIMITED M EM O R Y SEQUENTIAL DETECTION 5 L O O K S F A L S E A L A R M P R O B . = 0 . 1 0 E - 0 6 5 D E T E C T O R S * * B I N S * * 2 0 5 0 1 0 0 5 0 0 1 0 0 0 * * * * P R O B . D E T . * * ** * •k 0 . 5 * * 0 . 0 D . O 0 . 0 0 . 0 0 . 0 * * k k 0 . 5 5 * * 0 . 0 0 . 0 0 . 0 0 . 0 0 . 0 * * * * 0 . 6 * * 0 . 0 0 . 0 0 . 0 0 . 0 0 . 1 k k k •k 0 . 6 5 ** 0 . 0 0 . 0 0 . 0 0 . 0 0 . 1 k k k k 0 . 7 * ★ 0 . 0 0 . 0 0 . 0 0 . 0 0 . 2 k k * * 0 . 7 5 * k 0.0 0 .0 0 .0 0.0 0 . 2 k k k k 0 .8 * A 0 .0 0 .0 0 .0 0.0 0 . 2 k k k k 0 .5 5 k k 0 .0 0 .0 0 .0 0.0 0 . 2 k k k k 0 .9 k k 0.0 0 .0 0 .0 0.1 0 .4 k k * * 0 .9 5 * * 0 .0 0 .0 0 .0 0.0 0 .9 k k ■*■************-* + ********** ******* ( 1 0 0 I N D I C A T E S T H A T T H E P R O B A B I L I T Y O F D E T E C T I O N C A N N O T B E O B T A I N E D ) Figure 6. 7. Table of L oss in Signal to Noise Ratio (db) for Limited M em ory Sequential Detection. 5 Looks F alse A larm Prob. = 0. 10E-06 5 Detectors v O v O b e e n ro u n d e d off to 0 . 1 db an d the value of 100 db in d ic a te s a n u n a tta in a b le P j y ) It is s e e n th at fo r 20 d ata s t r e a m s t h e r e is a slig h t d if f e r e n c e f o r one d e te c to r , and e s s e n tia l ly no lo ss f o r m o r e than one d e te c to r . F o r 50 d ata s t r e a m s , the p e r f o r m a n c e is only slig h tly w o r s e th a n fo r 20 d a ta s t r e a m s , w h ic h is in a g r e e m e n t w ith the p r e d ic tio n s . S im ila r ly , fo r 100 d a ta s t r e a m s , the lo s s is g r e a t e r yet fo r only one d e te c to r , but is e s s e n tia l ly e lim in a te d by two o r m o r e d e t e c t o r s . F o r 500 d a ta s t r e a m s , it w as p r e d ic te d th a t d e te c tio n w ould be p o o r fo r one d e t e c t o r , an d in d eed the r e s u l t s show a n u p p e r lim it on of le s s than 0. 75, and w ith a d iff e r e n c e , o r lo s s, in S /N of 1-2 db in the P j-)= 0. 5 to 0. 7 ra n g e w h ic h is a tta in a b le . With two d e t e c t o r s , the u p p e r lim it of Pj-j is a b o v e 0 .8 5 , an d the S /N l o s s e s a r e g r e a t l y re d u c e d . F o r th r e e o r m o r e d e t e c t o r s , the p e r f o r m a n c e im p r o v e s m a r k e d l y a s p r e d ic te d . F o r 1000 d a ta s t r e a m s , the p e r f o r m a n c e is r a t h e r p o o r fo r t h r e e - d e t e c t o r s o r le s s , but b e c o m e s r e a s o n a b le fo r fo u r d e t e c t o r s , an d q u ite good fo r five d e t e c t o r s . T his p e r f o r m a n c e is slig h tly b e t t e r than p r e d ic te d . The effect of the n u m b e r of d e t e c t o r s on S /N lo s s f o r the v a r ious n u m b e r of b in s c a n be b e t t e r o b s e r v e d in F i g u r e s 6 . 8 , 6. 9 and 6. 10, in w h ich the d a ta f r o m the ta b le s has b e e n r e o r g a n iz e d and p lo tte d fo r P D = 0* 5, P ^ = 0 . 7 and P D= 0* 9. T h e s e p lo ts show the effect of the u p p e r lim it of P ^ an d th a t the S /N lo s s i n c r e a s e s w ith P^-j, a s w e ll a s w ith the n u m b e r of b in s. F i g u r e s 6. 11, 6. 12 and 6. 13 a r e the c o r r e s p o n d i n g p lo ts to -5 -5 F i g u r e s 6. 8 , 6 . 9 an d 6. 10 fo r P p ^ = 10 . T he p lo ts f o r P p ^ = 10 _7 follow the s a m e b a s ic p a t t e r n a s th o s e fo r 10 > t>ufc n a tu ra lly 100 lo o t - “ O CO \ 1000 bins 500 bins 00 bins J D etectors 20 & 50 bins ' F ig u r e 6 . 8. S /N L o s s vs N u m b e r of D e te c to r s 5 L o o k s , P D = 0. 5, P pA = 10"7 101 00 ' 1000 b ins .500 b ins 100 b in s 20 £ 50 bins 3 D etectors * F ig u r e 6. 9. S /N L o s s vs N u m b e r of D e te c to r s 5 Looks , P = 0. 7 , P ,.= 10“ 7 102 lOOl oo 1000 bins 100 bins - • 2 3 D etectors W & 50 bins F i g u r e 6. 10. S /N L o s s vs N u m b e r of D e te c to r s 5 Looks* P = 0. 9* P . A = 1 0 - 7 103 100 500 & 1000 Bins 3 - -o CO 20 b in s 2 3 D e te c to rs 4 F ig u r e 6. 11. S /N L o s s vs N u m b e r of D e te c to r s _________________ 5 looks, P p = 0 .5 , P FA= 10~5_______ 104 S/N Loss (db) 100 500 & 1000 b in s 2 . 1. 50 b in s 0. 3 D e te c to r s F i g u r e 6. 12. S /N L o s s vs N u m b e r of D e te c to r s 5 L o o k s , P D = 0 . 7, P P A = 10-5 100 500 S 1000 b in s 2.5 - 100 b in s 50 b in s 0 b in s 2 3 ^ D e te c to rs H F i g u r e 6. 13. S /N L o s s vs N u m b e r of D e te c to r s 5 L o o k s, P e= 0 . 9 , P F A = 10“ 5 106 th e S /N l o s s e s a r e g r e a t e r b e c a u s e of the h ig h e r n o ise level, w h ich fills up the d e t e c t o r s m o r e fr e q u e n tly . S im ila r ly , F i g u r e s 6. 14, - 6 6. 15 and 6. 16 a r e fo r P p ^ = 10 . A s e x p ec te d , the S /N l o s s e s a r e -7 -5 b e tw e e n th o se fo r the c a s e s of 10 an<l ^ * F in a lly , - 8 F i g u r e s 6. 17, 6. 18 and 6. 19 a r e fo r the c a s e of = 1(T . A t this Low value the S /N lo s s e s b e c o m e quite s m a l l f o r a ll c a s e s but the 500 and 1000 d a ta s t r e a m c a s e s 'f o r ju s t one- d e te c to r . E v en fo r 1000 d a ta s t r e a m s , the lo s s b e c o m e s s m a ll with five d e t e c t o r s . T his e x a m p le w ould c o r r e s p o n d to a s av in g s in p r o c e s s o r s of 99. 5 p e r c e n t. 6. 2. 2 W id e r B e a m C a s e , T en L ooks A n u m b e r of c o m p u te r ru n s w e r e m a d e fo r c a s e s w ith 10 looks and 1000 d a ta s t r e a m s to s e e if p r e d ic tio n s m a d e by u sin g e q u atio n 5 -4 could be v e rifie d . F o r 10 looks, M , is 5; thus a 5 out of 10 opt - 8 d e t e c t o r w ould be u s e d . F o r a of 10 , this d e te c to r w ould h a v e p = 0. 0083. The e q u iv a le n t se q u e n tia l o b s e r v e r h as a K = 6 and a T ^ = 25. U sing th e s e p a r a m e t e r s , a of 10, and a g of 3 w e r e c a lc u la te d . U sin g th e s e v alu es in eq u atio n 5 -4 , a n N p of 16 w as found fo r a c o n fid e n ce lev el of two o ^ . F o r a c o n fid e n ce le v el of t h r e e N p w ould be 19, but th is le v e l p r o v e d to be o v e r ly c o n s e r v a t i v e . T hus, it w a s p r e d ic te d th at the p e r f o r m a n c e sh ould be: p o o r, b elo w 10 d e t e c t o r s ; re a s o n a b le , f o r 11 to 13 d e te c to r s ; q u ite good fo r 14 to 16 d e t e c t o r s ; and e x ce lle n t, a b o v e 17 d e t e c t o r s , o r so. The a c tu a l c o m p u te r resuL ts shown in F i g u r e 6. 20 a r e in e x c e lle n t a g r e e m e n t w ith th e s e p r e d ic tio n s . In fact, a t 16 d e t e c t o r s t h e r e is only a m i n im a l lo s s of S /N . T his e x a m p le thus c o r r e s p o n d s to a s a v in g s in p r o c e s s o r s of 9 8 .4 p e r c e n t. 107 100 1000 b in s jd2 T J CO 500 b in s \ 100 b in s 50 \ 2 3 D e te c to rs ** F ig u r e 6. 14. S /N L o s s vs N u m b e r of D e te c to r s 5 L ooks, P.= 0. 5, P - =10~e 108 S/N Loss (db) 1000 bins 500 bins i ns 20 ns Tj N u m ^ r ° o f D e te c to r s 5 L o o k s , P = 0. 7, P F A = 10-* F ig u r e 6. 15. S /N L o s s vs * = > C T vvyOj- « TD - 109; 100 * 1000 b in s 3 " 500 b in s m 1 ;5«>- 100 b in s 0 . 5 ^ 5 0 bi ns 0 b in s ^ D e te c to r s F ig u r e 6. 16. S /N L o s s vs N u m b e r of D e te c to r s 5 L o o k s, P B = 0 .9 » P fa= 10"*5 110 100 -o CO 1000 b in s 500 n b in s 2 3 D etectors 4 F ig u r e 6. 17. S /N L o s s vs N u m b e r of D e te c to r s 100 3 “ 2.5 ” -Q -< 3 t o • 1000 bins 00 bins D etectors 4 F i g u r e 6. 18. S /N L o s s vs N u m b e r of D e te c to r s 5 L o o k s , P D = 0. 7, P F A = 10” ® 20 bins 1000 bins CO '5 0 0 bins 20,50,100 bins 2 3 D etectors k F ig u r e 6. 19. S /N L o s s vs N u m b e r of D e te c to r s 5 L o o k s , P B = 0. 9, P F A = 1 0 -8 \ D P D \ 8 9 10 11 12 13 14 15 16 0. 5 1 . 6 1.0 0. 7 0 / 5 ' 'V 0. 3 0. 2 0 . 1 0 . 1 0 0. 55 2. 1 1 .0 0. 6 0 .4 0. 3 0. 2 0 . 1 0 . 1 0 0. 6 — 1 . 2 0. 8 0. 5 0. 4 0 . 2 0 . 1 0 . 1 0 0. 65 — 1. 6 0. 8 0. 6 0. 3 0. 2 0 . 1 0 . 1 0 0 .7 — — 0. 9 0. 5 0 . 4 0. 2 0 . 1 0 0 0. 75 — — 1 .2 0. 6 0. 3 0. 2 0 . 1 0 0 0. 8 — _ — 0 .9 0 . 4 0. 3 0. 2 0 . 1 0 . 1 0. 85 — — — 1. 7 0 .7 0. 3 0. 2 0 . 1 0 . 1 0. 9 — — — — 1. 1 0. 5 0. 3 0 . 1 0 0. 95 — — — — — — 0. 5 0. 2' 0 . 1 (— in d ic a te s th at the c an n o t be a t t a i n e d . ) F ig u r e 6. 20. T ab le of S /N L o s s (db) f o r V a rio u s and N u m b e r of D e te c to r s . 10 looks, 1000 bins* = 10~^ F ig u r e 6. 21 g iv e s a p lo t of the u p p e r lim it of a s a function of the n u m b e r of d e t e c t o r s . It is s e e n that a t |jg , 10 d e t e c t o r s , the u p p e r lim it of is 0. 782. With 16 d e t e c t o r s , the lim it r i s e s to a P q of 0. 91. T h is s h a r p r i s e in the u p p e r lim it of P ^ , an d a c c o m p a n y in g s h a r p d e c r e a s e in S /N lo s s is q u ite c o n s is te n t w ith the a s s u m e d G a u s s ia n d is tr ib u tio n fo r the n u m b e r of d e te c to r s tie d -u p . A s i m i l a r s e t of p r e d ic tio n s w as m a d e fo r the c a s e of P j , ^ = l(f A m e a n , (j-g, of 17 d e t e c t o r s w a s c a lc u la te d , w ith a of fo u r d e t e c t o r s . T h u s, u sin g eq u atio n 5 -4 , the p r e d i c t e d p e r f o r m a n c e is: p o o r, below 17 d e t e c t o r s ; r e a s o n a b le , 18 to 21 d e t e c t o r s ; q u ite good, 22 to 26 d e t e c t o r s ; a n d e x c e lle n t, 27 d e t e c t o r s and a b o v e . In this c a s e a ls o , the c o m p u te r r e s u l t s a r e in e x c e lle n t a g r e e m e n t w ith the p r e d ic tio n , a s show n by F i g u r e 6. 22. In fact, the lo s s b e c o m e s m in im a l a t 25 d e t e c t o r s , w h ich c o r r e s p o n d s to a s av in g s in p r o c e s s o r s of 97. 5 p e r c e n t . F ig u r e 6. 23 is a p lo t of the u p p e r lim it of P ^ fo r this c a s e , a s a fu n ctio n of the n u m b e r of d e t e c t o r s . The r e s u l t - 8 is s i m i l a r in b a s ic n a tu r e to th at fo r the c a s e of 10 • - 6 F o r a P ^ of 10 , a m e a n , jjg, of 31 d e t e c t o r s w a s c a l c u lated , and a c-g of 6 d e t e c t o r s . T he e x a m p le s of 25 to 29 d e t e c t o r s w e r e ru n , yielding the e x p e c te d r e s u l t s of p o o r p e r f o r m a n c e . H o w e v e r , it w a s found that the r e q u i r e d c o m p u te r tim e r o s e ra p id ly w ith the n u m b e r of d e t e c t o r s . T h e r e f o r e it w a s not fe a s ib le to c o m p le te this c a s e . A c c o rd in g to the p r e d ic tio n , e x c e lle n t p e r f o r m a n c e w ould h ave b e e n a tta in e d a t a b o u t 43 d e t e c t o r s . T h is w ould c o r r e s p o n d to _5 a sav in g s in p r o c e s s o r s of 95. 7 p e r c e n t. F o r of 10 , a m e a n .11 5; 0. 0, 0. 0. n 0. 0. Q . CL 0. D etectors F ig u re 6. 21. Upper L im it of P Q vs N um ber of D etecto rs 10 Looks, 1000 B ins, P pA = 10“ 8 D P D 14 15 16 17 18 19 20 22 25 0. 5 1. 9 1. 3 1 .0 0. 8 0. 6 0. 4 0. 3 0. 1 0 0. 55 2. 9 1 .4 1. 1 0. 8 0. 6 0. 5 0. 3 0. 2 0 0. 6 — 1. 9 1. 2 0. 8 0. 6 0 . 4 0. 3 0. 2 0 0. 65 — — 1. 4 1.0 0. 7 0. 5 0. 3 0. 2 0. 1 0. 7 — — 2. 1 1.0 0. 8 0. 6 0 . 4 0. 1 0 0. 75 — __ _ 1 .6 0. 8 0. 6 0. 4 0. 2 0 0. 8 — — — — 1. 2 0. 7 0. 5 0. 2 0. 1 0. 85 — — — — — 1. 1 0. 7 0. 3 0. 1 0 .9 — — — — — — 1. 1 0 . 4 0 . 1 0. 95 — — — — — — — 0. 7 0 . 1 ( — in d ic a te s th a t the c an n o t be a tta in e d .) F i g u r e 6. 22. T a b le of S /N L o s s (db) f o r V a rio u s a n d N u m b e r of D e te c to r s . 10 lo o k s, 1000 b in s , P F A = 1 0 "? 1.17- 00 12 13 14 15 16 17 18 19 20 21 22 23 2k D etectors F ig u re 6.23. Upper L im it of vs N um ber of D etectors: 10 L ooks, 1000 B ia s, P fA = 10“ 7 of 60 d e t e c t o r s w a s c a lc u la te d , and a of 8 d e t e c t o r s . E x c e lle n t p e r f o r m a n c e w ould be p r e d ic te d to o c c u r a t a p p r o x im a te ly 76 d e t e c t o r s . T his w ould c o r r e s p o n d to a sav in g in p r o c e s s o r s of 92. 4 p e r c e n t. 1 C H A P T E R 7 CONCLUSIONS In this d i s s e r t a t i o n , a new a r c h i t e c t u r e w a s d e v elo p e d fo r a c l a s s of p r o b l e m s involving p a r a l l e l " s p a r s e " d a ta s t r e a m s . T his a r c h i t e c t u r e a c h ie v e s a d r a m a t i c re d u c tio n in the n u m b e r of p r o c e s s o r s r e q u ir e d , a s c o m p a r e d w ith co n v en tio n al a r c h i t e c t u r e s . A n a l y s i s in d ic a te d th a t s a v in g s of o v e r 90 p e r c e n t of the p r o c e s s o r s w ould r e s u l t in ty p ic a l c a s e s . T h e s e p r e d ic tio n s w e r e c o n f ir m e d by a c o m p u te r s im u la tio n fo r a r e p r e s e n t a t i v e s e t of p r o b l e m s . The k e y to th e s e s a v in g s is the u s e of s y s t e m a n a ly s is to d e sig n the a r c h i t e c t u r e to fit the r e q u i r e m e n t s of the p r o b l e m . C o n v e n tio n a l a r c h i t e c t u r e s o ften in clu d e m a n y u n n eed ed e x p e n s iv e c a p a b il itie s fo r c e r t a i n p r o b l e m s . C o n fig u ra tio n s of this new p r o c e s s o r - s h a r in g a r c h i t e c t u r e w e r e d ev elo p e d fo r p r o b l e m s f r o m t h r e e d i v e r s e a r e a s , and r e p r e s e n t a v e r y g r e a t saving o v e r c o n v en tio n al a r c h i t e c t u r e s . It is b e lie v e d that g r e a t e r a p p lic a tio n of the s y s t e m s a n a ly s is a p p r o a c h to a r c h i t e c t u r a l d e s ig n can r e s u l t in c o m p a r a b le s a v in g s in m a n y o th e r a r e a s . 120 R E F E R E N C E S 1. P . E n slo w , J r . (E d ito r): M u l t i p r o c e s s o r s a n d P a r a l l e l P r o c e s s i n g , Jo h n W iley & Sons, N ew Y ork, 1974, pp. 1-25. 2. G. B a r n e s et a l. : " T h e ILLLAC IV C o m p u te r ," I E E E T r a n s . , C -1 7 , A u g u st 1968, pp. 7 4 6 -757. 3. C o n tro l D ata C o rp . : C o n tro l D ata S T A R -1 0 0 C o m p u te r: H a r d - w a r e R e f e r e n c e M an u al, C o n tro l D ata C o r p . , St. P a u l, M inn. , 1975. 4. C. F o s t e r : C o n ten t A d d r e s s a b l e P a r a l l e l P r o c e s s o r s , Van N o s tr a n d R ein h o ld C o m p an y , New Y ork, 1976. 5. L. G a m b in o an d R . B o u lis, S r. : "ST A R A N C o m p le x ," 1975 S a g a m o r e C o m p u te r C o n fe re n c e on P a r a l l e l P r o c e s s i n g , pp. 132-141. 6. G. L ip o v sk i a n d A. T r ip a th i: "A R e c o n fig u r a b le V a r i s t r u c t u r e A r r a y P r o c e s s o r , " P r o c e e d in g s 1977 In te r n a tio n a l C o n fe re n c e on P a r a l l e l P r o c e s s i n g , pp. 165-174. 7. R. M e n a rd : I n tr o d u c tio n to A r r h y t h m i a R e c o g n itio n , C alif. H e a r t A s s o c . , San F r a n c i s c o , C-A. , 1968, pp. 4 -1 6 . 8. L. B r u n n e r an d D. S u d d arth : T extbook of M e d ic a l- S u r g ic a l N u r s in g , J. B. L ip p in c o tt Co. , P h ila d e lp h ia , New Y ork, T o ro n to , 1975, pp 413. 9. A. R ih a c z e k : H igh R e s o lu tio n R a d a r , M c G r a w - H ill B ook Co. , New Y ork, 1969. 10. J. H a r r in g to n : "A n A n a ly s is of the D e te c tio n of R e p e a te d S ignals in N o is e by B in a r y In te g ra tio n ," IR E T r a n s . , IT -1 , M a r c h 1955, pp. 1-9. 11. G. D in een an d I. S. R e e d : "An A n a ly s is of Signal D e te c tio n a n d L o c a tio n by D ig ita l M e th o d s ," IR E T r a n s . , I T -2 , M a r c h 1956, pp. 29-38. 12. D. M ich a lo p o u lo s (E d ito r): " C o m p u te r s A id C a lte c h E a r t h q u ak e R e s e a r c h , " C o m p u t e r , Ju n e 1978, ( P u b lis h e d by I E E E C o m p u te r S ociety, L ong B e a c h , C A .) p. 101. 121' 13. C ED A R S y s te m , u n p u b lish e d d a ta, c o u r te s y of A. B la n c h a r d (staff m e m b e r of the C a lte c h S e is m o lo g y L a b o r a to r y ) , 1978. 14. D. K nuth: F u n d a m e n ta l A l g o r i t h m s , vol. 1, A d d is o n -W e s le y , R ea d in g , M a s s . , 1975, pp. 2 7 0 -2 7 8 . 15. J. N e y m a n an d E. P e a r s o n : "O n the P r o b l e m of the M o st E ffic ien t T e s ts of S ta tis tic a l H y p o t h e s e s ,1 1 P h ilo s . T r a n s . R o y al Soc. , London, S e r ie s A, 23, 1931, pp. 2 8 9 -3 3 7 . 16. A. W ald: S e q u e n tia l A n a l y s i s , John W iley & Sons, New Y ork, 1947. 17. J . B u s s g a n g and D. M iddleton: " O p tim u m S e q u en tial D e tec tio n of S ignals in N o is e ," IR E T r a n s . , IT -1 (3), D e c e m b e r 1955, pp. 5-18. 18. I. S. R eed : A n A n a ly s is of S e q u e n tia l D e te c tio n by the S e q u e n tia l O b s e r v e r , (o r ig in a lly L in c o ln L a b o r a t o r i e s R e p o r t #20, 1953; r e p u b lis h e d a s U. S. D o c u m e n t A D -1 7 0 8 2 ). 19. P . S w erlin g : " P r o b a b ilit y of D e te c tio n fo r F lu c tu a tin g T arg ets', IR E T r a n s . , IT -6 , A p r i l I960, pp. 269-308. 20. M. S c h w a rtz : "A C o in c id e n c e P r o c e d u r e fo r Signal D e t e c tio n ," IR E T r a n s . , I T - 2 (4), D e c e m b e r 1956, pp. 135-139. 21. R. W o rley : " O p tim u m T h r e s h o ld s fo r B in a r y I n te g r a tio n ," I E E E T r a n s . , IT -1 4 , M a r c h 1968, pp. 3 4 9 -353. 22. J. W alk e r: D ouble T h r e s h o ld D e te c tio n , T ech n o lo g y S e rv ic e C o r p o r a tio n D o c u m en t, T S C - P D - 0 0 9 - 14, S anta M onica, CA, Ju ly 1969. 23. P . S w erlin g : T h e "D ouble T h r e s h o ld " M ethod of D e te c tio n , R an d C o r p o r a tio n R e p o r t R M -1 0 0 8 , Santa M o n ica, CA, D e c e m b e r 1952. 24. J. Capon: " O p tim u m C o in c id e n c e P r o c e d u r e s fo r D e te c tin g W eak S ignals in N o is e ," I960 IR E In te rn a tio n a l C onvention R e c o r d , p a r t 4, I960, pp. 154-166. 25. T. B ooth: S eq u en tial M a c h in e s an d A u to m a ta T h e o r y , John W iley & Sons, New Y ork, L ondon, Sydney, 1967, pp. 4 0 9 -4 1 1 . 26. S. G in s b u rg : A n In tro d u c tio n to M a th e m a tic a l M ac h in e T h e o r y , A d d is o n -W e s le y , R ea d in g , MA, 1962, pp. 107-109. 27. W. F e l l e r : A n In tro d u c tio n to P r o b a b il ity T h e o r y an d Its A p p l i c a t i o n s , Jo h n W iley & Sons, New Y ork, 1968, pp. 372-394. 28. M. Skolnik (E d ito r): R a d a r H an d b o o k , M c G r a w - H ill B ook Co. , N ew Y ork, 1970.____________________________________________________ 122 A P P E N D IX A T H E RA TIO O F M t /N , F O R L A R G E N opt A s m e n tio n e d in S ectio n 4. 4, s e v e r a l a n a ly s ts have d e t e r m i n e d th a t M ^/N a p p r o a c h e s a n a s y m p to te of a p p r o x im a te ly 0. 2 fo r la r g e N. In th is a p p en d ix a n in te r e s t in g d e riv a tio n of this r e s u l t is given, s ta r tin g w ith s o m e of the b a s ic equ atio n s d e r iv e d in C h a p te r 4. The follow ing co n d itio n s a r e a s s u m e d : The f a ls e a l e r t p ro b a b ility , P is s m a 11 T he d e s i r e d p r o b a b ility of d e te c tio n , P ^ , is 1 /2 The sig n a l to n o ise r a tio , S /N is s m a ll. It w ould a p p e a r th a t th e s e a s s u m p tio n s w ould lim it the v alid ity of an y r e s u l t s o b ta in e d to only a few c a s e s . H o w e v e r, M ^ h a s b e e n found to be e s s e n t i a l l y in d e p en d e n t of P ^ , an<^ S /N . T hus it is valid to so lv e the p r o b l e m u n d e r the c o n v en ie n t a s s u m p tio n s m a d e , and s till h av e a g e n e r a l, u sefu l r e s u lt. The u s u a l d efin itio n of M , is: the M that r e s u l t s in the low - opt e s t S /N r e q u i r e d to a tta in a given P j y fo r a s p e c ifie d P ^ ^ , fo r the given N. It is e q u iv alen t, an d helpful in the a n a ly s is , to define M ^ a s : the M th at r e s u l t s in the lo w e st P ^ fo r a given S /N , P ^ , and N. Since P ^ is s m a l l, the a p p r o x im a tio n of eq u atio n 4 -3 c an be u sed . In equation 4 -3 , let R = M N ' 123 yielding (A -1) K = V ^ R rE ^- . Vp(l-p) T he m i n i m u m o c c u r s w hen K^, is a m a x im u m , s in c e K^, is the n u m b e r of s ta n d a r d d e v ia tio n s by w h ic h the r e c o g n itio n t h r e s h old e x c e e d s the m e a n . T aking p a r t i a l d e r iv a tiv e s of both s id e s of eq u atio n A - l w ith r e s p e c t to R g iv es 5Kf = */N(i-aP/aR) _ VN(R-p)(i-2p)^p/aR # V p ( l- p ) 2 [ p ( l-p )]3 2 Setting aKp / = 0 gives: 2p (i-P)(i-ap/BR) = (R-p)(i-2p)ap/aR, w h ich y ield s: (A 2) - § £ = 2Fi * - P i C) 3R p+R( 1 -2p) The co n d itio n th at P ^ = 1 /2 is e q u iv a le n t to settin g M = N p g, a s d e s c r i b e d in S ectio n 4. 4. Thus: (A-3) r, = M =n p s N L e t x be the S /N , and (A -4) U = 1 + x . 124 F r o m eq u atio n 4 -5 , p = e S u b stitu tin g eq u atio n 4 - 4 into s eq u atio n 4 -5 y ield s: / a - T / U (A-5) p g= e S ubstituting e q u atio n s A -3 and A -4 into eq u atio n A -5 y ield s: (A-6) R = p 1 /U , or p = R U . T aking p a r i t i a l d e r iv a tiv e s of both s id e s of eq u atio n A -6 w ith r e s p e c t to R y ield s: (A -7) J g = U R U - l = U P . C o m b in in g e q u a tio n s A -2 and A -7 yields: ,A 8) -M lzE L .. = u p p + R (l-2 p ) R * S u b stitu tin g eq u atio n A - 4 into eq u atio n A -8 and sim p lify in g yields: (A -9) Rx (l+ x -2 R x ) = 1 -x . T aking n a tu r a l logs of both s id e s , a n d u s in g the a p p ro x im a tio n : 0n( 1+x) = x (w hich is valid, s in c e x is s m a ll), eq u atio n A -9 b e c o m e s : (A -10) x$n(R)+x( 1 -2R ) = - x . Equation A -1 0 red u ces to: 125 (A -11) R = M S J + i . T he a p p r o x im a te so lu tio n to eq u atio n A - l l is: R = 0. 203 . 126 A P P E N D IX B C A L C U L A T IO N O F F(i) IN E Q U A TIO N 5-17 T he f a c to r F(i) in eq u atio n 5-17 r e p r e s e n t s the p r o b a b ility th at a p r o c e s s o r is f i r s t a v a ila b le fo r a s s ig n m e n t to a n a c tiv e d a ta s t r e a m on the ith input of the b u r s t of a c tiv ity . The n a tu re of th is f a c to r d e p en d s on the type of p r o b l e m to w h ic h the p r o c e s s o r s h a r ing a r c h i t e c t u r e is a p p lie d . In this ap p en d ix , e q u atio n s a r e d e r iv e d fo r a p p lic a tio n s in w h ich s e q u e n tia l o b s e r v e r p r o c e s s o r s a r e u se d fo r r e c o g n itio n of e v e n ts in d a ta s t r e a m s s u b je c t to b a c k g ro u n d n o ise. In th is a n a ly s is it wilL be a s s u m e d th at the e v en ts a r e u n c o m m o n , and th at the p r i m a r y lim ita tio n on p r o c e s s o r a v a ila b ility is the effect of p s e u d o " p o s itiv e " inputs c a u s e d by n o is e. A c c o rd in g to the s y s t e m p r i o r i t y r u le s , a p r o c e s s o r is " tie d - u p " if it h a s a c u m u la tiv e s u m of K o r g r e a t e r . O th e r w is e , it is a v a ila b le to a new a c tiv e d a ta s t r e a m . The p r o b a b ility that a given d a ta s t r e a m w ill tie up a p r o c e s s o r by n o is e inputs c an b e e s ti m a te d by the c u m u la tive s ta t io n a r y p r o b a b iliti e s . The p r o b a b ility , P y , th a t a s e q u e n tia l o b s e r v e r w ould h ave a c u m u la tiv e s u m of K o r g r e a t e r is g iv en by: (B-l) P V = 1 - C ( K - 1 ) . S im ila r ly , the p r o b a b ility that a given d ata s t r e a m w ill not tie up a s e q u e n tia l o b s e r v e r , P-yy> is g iven by: 127 (B -2) P w = C(K-l) F (l) is the p r o b a b ility that a p r o c e s s o r is a v a ila b le on the f i r s t input of a f i r s t a c tiv ity . L e t D = the n u m b e r of d a ta s t r e a m s , E q u atio n B -3 is b a s e d on the c u m u la tiv e b in o m ia l d is trib u tio n . E a c h value of j in the s u m m a tio n r e p r e s e n t s the p r o b a b ility th at t h e r e a r e e x a c tly j d a ta s t r e a m s tiein g up p r o c e s s o r s . F(l) is thus the s u m of the p r o b a b iliti e s f o r w h ich j is le s s th an the n u m b e r of p r o c e s s o r s , ID* F(2) is the p r o b a b ility that t h e r e w a s not a p r o c e s s o r a v a ila b le fo r the f i r s t input of the b u r s t of a c tiv ity , but th at one b e c a m e a v a i l a b le fo r the s e c o n d input. T h is s itu a tio n can o c c u r if and only if a ll the p r o c e s s o r s w e r e tied up on the f i r s t input, but th a t one (or m o r e ) p r o c e s s o r h ad a c u m u la tiv e s u m of e x a c tly K. It is a ls o n e c e s s a r y that t h e r e w a s a n eg ativ eM n p u h to t h i s l p r o c e s s o r ( s ) , re d u c in g th e :• c u m u la tiv e s u m to K - l . In ad d itio n , a n o th e r d a ta s t r e a m m u s t not h a v e tie d up th is p r o c e s s o r in the tim e b e tw e e n the r e l e a s e by the o th e r d a ta s t r e a m , an d the se c o n d input f r o m the tr u l y a c tiv e d a ta s t r e a m . T h e s e co n d itio n s a r e e x p r e s s e d in the follow ing equation: and I p = the n u m b e r of p r o c e s s o r s . Then, F(l) is g iv en by: (B-3) 128, D _______ DJ ( l - p ) F A C . The t e r m F A C in eq u atio n B -4 r e p r e s e n t s the p r o b a b ility th a t the r e l e a s e d d e te c to r is not ta k en by a n o th e r d a ta s t r e a m having a p s eu d o " p o s itiv e " input. If the d a ta s t r e a m s a r e s a m p l e d in s e q u e n c e, FA C is the p r o b a b ility th a t a ll the d a ta s t r e a m s b e tw e e n the one r e le a s in g the p r o c e s s o r an d th e a c tiv e d ata s t r e a m w ill have n eg ativ e in p u ts. T he a v e r a g e value of FA C is g iv en by: The g e n e r a l eq u atio n fo r F(i) is: T h is eq u atio n is an a p p ro x im a tio n , and n e g le c ts t e r m s th at a r e s m a ll, p ro v id in g p is s m a ll. (B-6) w h e r e and P x = P(K + i-2) P y = 1 -C (K + i-?) P z = C(K +i-3) . 129,-

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Creator Trilling, Thomas (author)

Core Title An architecture for parallel processing of "sparse" data streams

Degree Doctor of Philosophy

Degree Program Computer Science

Publisher University of Southern California (original), University of Southern California. Libraries (digital)

Tag Computer Science,OAI-PMH Harvest

Language English

Contributor Digitized by ProQuest (provenance)

Advisor Reed, Irving (committee chair), Ginsburg, Seymour (committee member), Mendel, Jerry M. (committee member)

Permanent Link (DOI) https://doi.org/10.25549/usctheses-c17-68841

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Legacy Identifier DP22731.pdf

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Document Type Dissertation

Rights Trilling, Thomas

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Access Conditions The author retains rights to his/her dissertation, thesis or other graduate work according to U.S. copyright law. Electronic access is being provided by the USC Libraries in agreement with the au...

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