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Microwave Coherence Tomography

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Microwave Coherence Tomography

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Content INFORMATION TO USERS This manuscript has been reproduced from the microfilm master. UMI films the text directly from the original or copy submitted. Thus, some thesis and dissertation copies are in typewriter face, while others may be from any type o f computer printer. T he quality of this reproduction is dependent upon the quality of the copy submitted. Broken or indistinct print, colored or poor quality illustrations and photographs, print blecdthrough, substandard margins, and improper alignment can adversely afreet reproduction. In the unlikely event that the author did not send UMI a complete manuscript and there are missing pages, these will be noted. Also, if unauthorized copyright material had to be removed, a note will indicate the deletion. Oversize materials (e.g., maps, drawings, charts) are reproduced by sectioning the original, beginning at the upper left-hand comer and continuing from left to right in equal sections with small overlaps. Each original is also photographed in one exposure and is included in reduced form at the back of the book. Photographs included in the original manuscript have been reproduced xerographlcally in this copy. Higher quality 6” x 9" black and white photographic prints are available for any photographs or illustrations appearing in this copy for an additional charge. Contact UMI directly to order. UMI A Bell & Howell Information Company 300 North Zeeb Road, Ana Aibor Ml 48106-1346 USA 313/761-4700 800/521-0600 MICROWAVE COHERENCE TOMOGRAPHY by John Catim ir do Sullma-Przyborowskl A T h esis Presentod to tho FACULTY OF THE GRADUATE SCHOOL UNIVERSITY OF SOUTHERN CALIFORNIA In Partial Fulfillment o f tho Requirem ents for the D egree MASTER OF SCIENCE (Applied M athematics) May 1996 UMI Hum bert 1380487 C o p y rig h t 1996 b y d e S u lix n a -P rsy b o ro w e k i, J o h n C a e im ir All rights reserved. UMI Mlcrororro 13804*7 Copyright 1996, by UMI Company. Atl rights reserved. TMs microform edition Is protected against unauthorized copying under Title 17, United States Code. UMI 300 North Zeeb Road Ann Atfcor, MI 48I0J UNIVERSITY OF SOUTHERN CALIFORNIA TH K ORAOUAtK BCHO O L UNIVfMITT PARK LO B AMOKUB, CALIFORNIA B O O O T This thesis, written by J n h a ... Ca b 1 m lx » d a ..S iii.lm a .« tg x z .y .b a c a ic a J c i under the direction of hlA~~.Thesis Committee, and approved by all iti members, has been pre• seated to and accepted by the Dean of The Graduate School, in partial fulfillm ent of the requirements fo r the degree of - - - - - - - - - - - T1 |, HH ! u a Dn i Date M aJ 1 0 1 1996 THESIS COMMITTEE DEDICATION I dedicate this thesis to Casimir and June. ii ACKNOWLEDGEMENTS I first thank Or. Haydn for his enthusiasm and his open-m indedness. I note the support of the staff in the math departm ent at USC. Further, I acknowledge the support of the em ployees at OPCOA, Inc., in particular: Dr. Ken Jam es, for his wide-ranging technical expertise, and Mr. Jon W hinnery, for his excellent preparation of this manuscript. iii TABLE OF CONTENTS Dedication........................,............ ii Acknowledgem ents............................................................................................................... iii A bstract..................................................................................................................................... v Introduction.............................................................................................................................. 1 Part I - System Analysis........................................................................................................7 Part II - Statistical Determination of the P a ram e te r....................................................12 Part III - Applying the Statistical Solution to the System M odel...............................16 Part IV - Conclusion.............................................................................................................18 Bibliography.......................................................................................................................... 20 Appendix I - Fourier Transform ........................................................................................ 21 Appendix II - Hilbert Transform s......................................................................................24 Appendix III - Statem ents from Probability and Statistics..... ............................ 28 iv ABSTRACT The theoretical groundwork is laid for a m odem imaging technique with potential industrial applications. The most obvious application of this work is to imaging over very shod distances through obstructions. It is competitive with recent developm ents in radar theory and w as inspired by Optical C oherence Tomography. In a departure from conventional imaging theory, w e do not use a deterministic function for the transmitted signal; rather, our theory is constructed around the u se of G aussian noise for the transmitted signal. By using a variety of analysis techniques, a simple approach is developed to determ ine the position of target objects. v INTRODUCTION Recently, a new technique of noninvasive biological imaging w as developed. The nam e of the new technique is Optical C oherence Tomography (OCT). W hat m akes OCT different from other imaging technologies is that it u se s an incoherent signal (the term "signal" is synonymous with the term "function") a s a basis of its imaging technology. An incoherent signal is a signal in which the p hase of the signal changes over time. The advantages of using an incoherent signal are: low cost of implementation and high accuracy at short distances. This signal has, to date, been supplied by a pulse diode. OCT is a m odem application of the classical Michelson interferometer (a device that m easures distances on the order of a wavelength of light, that distance being m easured in Angstrom units, which are equal to 10'1 0 meters). OCT is able to determ ine the changes in tissue layers b ecau se a difference in tissue corresponds directly to a difference in reflectivity. OCT, due to its high frequency, h as a short wavelength that results in high resolution; "resolution" being the sm allest distance that can be m easured. The topic of this thesis is to construct an adaptation of the OCT concept for short-range ground imaging. Instead of optical frequencies (i.e., 101 9 Hz) w e will work at microwave frequencies (i.e., 10* Hz) which are better suited to this problem. In particular, we need to isolate the travel time of the electrom agnetic wave from source to first reflection and back to the point of origin. Knowing this time will lead directly to determining the distance at 1 which the first change of medium occurs. W e are not, at present, taking into consideration the change in velocity of the electrom agnetic w ave a s it enters the medium. W e are not considering determination of material characteristics of the new medium. Expanding on the concept of OCT, we will not u se just an incoherent source, but, in fact, will u se a completely incoherent source; that is, G aussian noise. Historically, noise in a physical system is to be avoided, due to the degradation of the information that is being communicated. M ethods for com pensating for the presen ce of noise and ameliorating the corruption of the desired information are concepts upon which m uch of the theory of signals in communications engineering h as been built. Franz Tuteur, in one of his many papers, m ade the following statem ent that exemplifies typical thinking in this a rea of research: The detection of weak signals em bedded in a stronger stationary stochastic process is an old and well-studied problem. Probably the best known exam ple is the detection of radar or sonar signals in zero-m ean G aussian white noise. It is well known that the optimum detector in this problem involves correlation of the observed signal with a replica of the desired signal. Correlation is m ost effective if the sh ap e and time of occurrence of the expected signal is known. It is only marginally effective if this is not the case. Instead of correlation one can also u se a m atched filter, but the effectiveness of a m atched filter detector also dim inishes if the shape, or at least the bandwidth of the expected signal is not reasonably well known. Another approach to the detection of signals in noise is Fourier analysis or spectral estimation. This works best if the expected signal h a s spectral o features that clearly distinguish it from the noise. Its advantage over correlation or m atched filter m ethods is its insensitivity to the shape or time of occurrence of the desired signal, [p.1435] In the context of classical radar system s, in order to evaluate the degree of difference betw een a signal and its tim e-translated replica, it is standard to employ the deterministic autocorrelation function that is equal to the inner product of the two signals [Baskakov, pp. 85-86]. Let u(t) be a signal, then: *.(*) = J r/(r)«(r — r)c/r The absolute value of the autocorrelation function is equal to or greater than its value at any other time w hen the time shift is equal to zero. This fact Is used in long-range radar detection “w hen correlation is performed betw een the emitted and the returned signal. A large peak indicates a resem blance betw een the returned signal and the emitted signal, from which we assum e that a target is present* [Poularikas, p. 105], All of this analysis is contingent on the presence of a highly coherent signal. In M ichelson's interferometer, a light beam is split into two sub-beam s which, after transversing paths of different lengths, are recom bined so that they interfere and form a fringe pattern. By varying the path length of one of the sub-beam s, distances can be accurately expressed in term s of wave lengths of light. [Halliday, p. 735] OCT is an application of the classical Michelson interferometer using m odem incoherent optical technology. It is well-established that wholly incoherent sources, e.g., sun light, will, when applied to a Michelson 3 interferometer, interfere with itself when the difference in the path length is sufficiently small. Thus, it is consistent with established scientific principles to u se a wholly incoherent signal with a Michelson interferometer for short- distance m easurem ents. G eneral electrom agnetic argum ents are applicable, in principle, at all frequency ranges. So, the argum ents used in OCT are equally valid in the microwave frequency range. The only difficulty is that the size of the physical apparatus must be practical and reasonable. W e completely obviate the need for any special technology by using a G aussian noise generator. In typical mathematical analysis, attendant to problem s of the Michelson interferometer and autocorrelation function, what is frequently called time-domatn analysis is employed. It is a central argum ent of this thesis that w e abandon this temporal point of view, and that all of our analysis b e done with consideration of frequency. In light of the fam ous Wiener-Khlnchin theorem, instead of being concerned with the autocorrelation function, our analysis will focus on the power spectral density. Applied physics h a s already developed sound mathem atical m odels to describe the behavior of electrom agnetic w aves a s they interact with their physical surroundings. The amplitude ratio of the electric field to the m agnetic field for the w aves in either direction is called the intrinsic im pedance of the material in which the w ave is traveling . . . [Marshall, p. 320]. A reflection occurs when an electrom agnetic wave traveling through one medium 4 encounters a new medium in which there is an im pedance mismatch betw een the two media. Transm ission occurs when there is no such im pedance mismatch. An electrom agnetic wave, when it interacts with a boundary, will either be reflected by it, or will be transmitted through it, or will exhibit a combination of the two, depending upon the frequency of the electrom agnetic w ave and the physical composition of the boundary. Earth's strata are com posed of a wide range of naturally occurring physical substances. Electrom agnetic w aves at microwave frequencies will experience both transm ission and reflection when interacting with the earth's strata. The boundary we will work with and the strata below it have associated with them a particular reflectivity. Reflectivity is a complex function that is derived from the empirical d ata a s follows: W e require the following param eters: free sp ace permittivity e free sp ace permeability p tho loss tangent that gives the conductivity a the intrinsic permittivity c' From which we obtain the characteristic im pedance of the material by the equation: [Marshall, p. 325] The reflectivity of the material is: 5 p(a>) = ^ ------- (Marshall, p. 337] ' rj(a>) + 377 1 ’ K 1 w here 377 O is the characteristic impedance of air. Our system is basically the frequency domain analog of the OCT device. W e know the spectrum of the transmitted signal - to it, we sum the spectrum of the return signal. W e then determine the power spectral density of this sum. The power spectral density is altered to make the desired information more salient. W e then perform the necessary mathematical analysis of the altered power spectral density to isolate the param eter of interest: the time of travel. This analysis involves standard techniques of complex variables, the application of a novel small theorem from Hilbert transforms, and sorpe classical statistical analysis. 6 PART I - SYSTEM ANALYSIS The boundary that is the location of the first strata change and, correspondingly, the first im pedance mismatch is characterized by the reflectivity function, p e C. The transmitted signal is denoted by: f e R. The received signal, g e C, m ust be derived. It is derived using the standard techniques of Fourier system theory. W e have f{ t) and its Fourier transform, /( a ) ) . W e may split /( to ) . / ( G ) ) = | / ( G ) )| Likewise, we may split p(to): p (g) ) — |p (G ) W e assum e an im pedance match which m eans that a change in the underlying strata, which is the boundary, c au se s the reflection. W e have: £(0)) = |p ( g ) ) | | / < g) w here c,M T represents the p h ase shift due to travel time. Let: h ( 0 ) ) t = ^ ( G ) ) + / ( G ) ) giving: A (o ) = |/(G) )If,# < ">[l+|p (G ) The desired power spectral density (PSD) will be: IMg>)|; = (a ) w here the **" notation indicates the complex conjugate. 7 |/KOI)!1 =|/<£»)l3 -H /(e»)l! Ipto*)!1 +2|/<o»>|’ Ip (o>)KV«<0 (o»)+ 0> n w here the term containing the desired information is the third term. W e need to isolate mT in the argum ent of the cosine so that w e can determ ine 7'. W e now introduce a necessary theorem from Hilbert Transform Theory, a s follows: If y e R and supp/-< (-7',7’ ) and if/(cy) = > (o))f «v/„(u, T and r„ are fixed and or e R then: / / « l / ( t u ) J = ^ c u ) S in /„ft> w here //„ |/(a> )| Is the Hilbert transform of / acting on n>. P ro o f: Define Then, g iv e n / e R an d / is its Fourier transform w e m ay construct: i/(/) = 2 / ( 0 w ( 0 Then, a s follows: + 0 [See Appendix I] Z to So w e may write: f/< 0 «/<*-/„> Further, z, (co) e C, thus: 8 V (w) = /(o> )+ M n {/(CD)} Since r, (c d ) = < )■(«)^?'v,, = j * ( c d )Cos /p>+/r(a))Sin *p) (See Appendix I] Consequently: / / B{ / ( a » } = jt< a)S in tl t w q o .d . Now, since / ( / ) is hermitian, it is fully characterized for t > 0. W e may represent /(« •) in a new form .v(« ) , a s follows: Write i,(/)s= 2 /(0 w (0 a n d f,(r) = — 7*). T h en w eh av e; -,(w ) * = rr (oty"* « [.> ’ (co)+///„{.v(a>)}]<*~ a ( c d ) « Rc{£. (c d )} B^,(^u)Co5tu7'-//w{ ^ |•(a))}SinaJ7, with «(a)) e J v t((o)+I/l{r(w)} TtwO(to)- ^ .Ka») W e may state: a ( oi) » rt(cu) f V«(aj/’+ 0(c d )] . If /»is of duration r , then s is of duration y . W e now apply this theorem and attendant argum ent to the term of interest: Denote the term of interest by H'(w) a 2|/(tt>)f |p (a>)|Cos(0(a>)+flj/’ ) . The supp 41 = ( - r . r ) . Then, let: «(CD)=2|/(CD)p|p(£0)| W e will have a simple system of equations: «(«u) = V i!<«»)+//;L(f(o») *(«) The system is solved a s follows: a( m) = >f r 1 <ftj) + //* (r(w )} Equation #1: «*(«) = i 2(to)+ / / 2 |jr(a))J * (« ) x((i))tanO (o)) = //„{jr(a>)} Equation #2: ( x(o))ttvtO (o ))2 = //„2 {*(«)} Substituting Equation #2 into Equation #1, we obtain: u ! ( g>) * = *2(« ) + (jf(ft>)fclH0 (cu))2 </:(to) = jr2 (a>) + jr* (eu)fcf/r0(w) rt:(a)) = jf:(w)[l+ /uh:0(o))] y*'(W) = , giving 1+ tan'O (to) . «(w) 0 + hm'O (o>) Determining: «V(w) = * jf(w) C o s ( o j 7 ) a s w as desired. It is of interest to note that supp X(t) « (-2r ,2 r ) . 10 W e need to find a way to m easure ,Y(to) in order to determine the value of 7'. This will require that the analysis be statistical in nature, because ,Y(e))is a stochastic process. 11 PART II - STATISTICAL DETERMINATION OF THE PARAMETER The crossing problem, a s presented below, appeared in abbreviated form in the writings of S.t. Baskakov [pp. 201-202]: The C rossing Problem: Definition: The * upward crossing" of the process X(t) at the level x0 is the event consisting in that a realization X(t) cro sses the specified level x0 in the upward direction. Determine: average # of upward crossings . unit time Definition: Stationary: the statistical characteristic of a process rem ains unchanged with time. Definition: M ean Square continuity: if s' 11 -V (t + r ) - . v ( / ) ! ’ I - * 0 a s c -> 0 W e will assu m e X(t) is m ean square continuous and stationary. C hoose At so small that either no or only one upward crossing occurs. For one upward crossing w e require: 12 *{') < & x{/ + A/) > x0 but, x(/ + A f)« x(f) + x'Af So we may write: x (/)+ x ’A/ > x„ = > x(/)> x0-x 'A / we have: x^-x'A r < x { 0 < x „ So, an upward crossing requires that we have both a positive derivative and that we satisfy the inequality. W e assum e the existence of the joint bivariate probability density of the stochastic process and of its derivative at the sam e time instant P (of the e v e n t)« J j />(r, x' }Jx dx' « t »— « A » = J /> (*■ „, x')x'A/t/x' = A/J(x,ltx')x'ci!r* O tt Thus, we may solve the problem as: »(*„> = Af (| A ssum e X(t) G aussian so we have: p(xntx') = /H*o)/H*') [See Appendix III] w here />(x') is a normal density a s well, since taking the derivative is a linear 13 transformation. n •» »(*■„) * 4 1 «(*„) = it The G aussian density function is: /;(* ) c Lr - n , - « o < x < t-» V 2 f f i t Assume the autocorreletion function of X(t) is known. Then, we have: Kx ( r ) = H(r ) so the variance of the derivative is: a I = A ' . i ( O ) = - A*! ( 0 ) = - i r i /4*( 0 ) So we have: »■ i />(*') = ~r~------- 1 ----------- Substituting into the formula: "GO = Pi* ■ « )f "7s ™ --- 7=1==.-v :°i < - * ■ < " » dx' \ " 'j V2ffaJ % p n 0 ) W e integrate by substitution a s follows: ( * ’ )*' it = 2o ; ( - / r ( 0)) 14 — Jx> o l t - i r m n(x \ - tAx \ f x " M - l K r J i j 2noJ _ l n 0 ) ‘ = e M e d p r m V 2 f f From which we obtain; , v - s P m . .1 «(*.,) «■*-=------- f :„l In PART III - APPLYING THE STATISTICAL SOLUTION TO THE SYSTEM MODEL A'(oj) is our stochastic process of interest. W e show ed in Part I that it m ay be represented as: A'(w) = jr(w)Cos(a7) So we may now substitute the actual expressions from Part I a s follows: This stochastic process is one term of a larger expression for the altered PSD. W e have: By observing our expression for the PSD, we can deduce that there are som e requirem ents for u s to apply our statistical solution. It is necessary that the PSD of the original signal b e statistically G aussian. Since our last term ‘ ‘rides" on the other two terms, the variance of the original signal, that is, our G aussian noise, must b e wide enough so a s to b e nearly linear over the portion w e are applying our statistical solution to. The probabilistic autocorrelation in frequency is: C o s ( q j 7 ) W e may now express ,V(a>) in term s of the original signal, / . C osta?) Cos(a>7) 16 A 'j, (M )=A'tA'(o))A'((u+ m * )] To u se the formula from Part II, we need to determine the autocorrelation in frequency of the altered PSD. Denote ]//. (cw)|2 by Z(o>). Then: ^ ( m ) sjfC[Z(o})Z(o)+ w)J M « ) = + t ' H ’ Cos< “ 7> > • (|/<co +tr)]' + |/(a)+H )|'|p(ni + n)|’ + |/(c l + H')|' ^ y Col((M + H ')?))] W e know jrjn the formula because it is the PSD of / . Then, / ’, the desired parameter. So, t*7’ equals the distance to the boundary, where v is the speed of the electromagnetic wave. 17 PART IV - CONCLUSION In contemporary industry, great em phasis is placed on finding readily implemented solutions to problems. The material discussed in this thesis is applicable to ground imaging. W e used the already-active area of OCT as a guide for our work. B ecause of the intended application, we assum e that the electrom agnetic wave will be at the microwave frequency. The physical system motivated the types of m athematical analysis that w e applied to the problem. W e used several branches of formal m athem atics for our analysis. The atypical u se of the Hilbert theorem helped us isolate the desired param eter. The u se of G aussian noise a s the transmitted signal h as not been observed by the author in contemporary literature. This new technique h as the benefit of eliminating the need to conduct noise-immunity studies. A consequence of this new technique w as that a statistical m ethod w as necessary to determ ine the param eter of interest. The crossing problem (se e Pari It) contributed significantly to the solution of this problem. The analysis presented in this thesis, proceeding from m athematical argum ents, yields a method of near- range imaging that is potentially more cost-effective and practicable than other existing techniques. W e have performed here only a first-order analysis. W e have not considered the change in velocity of the electrom agnetic wave a s it enters the new medium. T here is a need to perform further mathematical analysis. In particular, the equation: 18 ) « #;[(|^<wf + |/(<u)|'[PM * + [f( 0 > ) f C o s ( t o 7 ) ) • ( I / ( w +w )]■ + |/ ( t u + M -)]?|p ( to +*■)]* + |/( c o + M)|: 2 |p (co + m )| C o»((cu + m )7 ))] h 1 1 1 V '+ tan ’Gtw+w) m ust be analyzed in greater detail. W e may need to a cc ess techniques from numerical analysis to m ake this expression m ore tractable. It is necessary that the variance of the noise be wide enough so that it will appear nearly linear relative to the term of interest in order to apply the formula from Pari II. There « may be a need to optimize the variance. Through both a statistical and iterative process it should be possible to determ ine the material param eters: conductivity, permeability, and permittivity; i.e., w e also wish to be able to detect the various strata levels and determ ine their physical nature. W e believe that further work motivated by our thesis should strive to identify more than just the location of the first strata change. This thesis h as provided a theoretical model of how this analysis may be performed. The final formulas in this thesis m ust be reevaluated for practicable implementation. O nce this is done, a com puter simulation of one elem ent may be performed. Then, one elem ent may be built and run in a laboratory environm ent to se e if the experimental d ata ag re es with the com puter simulation. Favorable results from this first experiment will motivate a com puter simulation of an imaging array com posed of many elem ents, followed by the actual construction and testing of such an imaging array. 19 BIBLIOGRAPHY Baskakov, S I , Signals and Circuits, (Boris V. Kuznetsov, translator), rev. ed., Moscow, Mir Publishers, 1986. Burden, Richard L., and J, Douglas Faires, Numerical Analysis, 4th. ed., Boston, PW S-Kent Publishing Company, 1989. Halliday, David and Robert Resnick, Fundamentals o f Physics, 2nd. ed., Extended Version, New York, John Wiley & Sons, 1981. (Author unknown], "Optical C oherence Tomography; An Imaging Method with G reat Promise," Biophotonics International, November/December, 1995, pp. 58*59. Marshall, Stanley V., and Gabriel G. Skitek, Electromagnetic C oncepts & Applications, 2nd. ed., Englewood Cliffs, New Jersey, Prentice-Hall, Inc., 1982. Papoulis, Alhanasios, Signal Analysis, New York, McGraw-Hill, Inc., 1977. Poularikas, Alexander D., and Sam uel Seely, Signals and System s, 2nd ed., Malabar, Florida, Krieger Publishing Company, 1994, Tuteur, Franz B., "Wavelet Transform ations in Signal Detection," [Unspecified IEEE publication], c. 1988, pp. 1435-1438. 20 APPENDIX I - FOURIER TRANSFORM Definition: the Fourier transform of f(x) is: A t) = ] f ( * y mdr The inverse is: / ( * ) = “ The following theorem is used in the b o d / of this thesis: If /( w ) * = .Kco)C o s t h e n Proof: Since Cost,.to = 1, - " - " have: 2 2 /( a ) ) R ~ l )’(to)i,,,,/< + j,y(a»)iriw * Then, by the above definition: / ( ! ) = ] [ ^ ( W k “-“ + / ( / ) = ^ J i>(CD )t,* l' ' U ' ,l/C D + ^ j ^■(CD)t*'<,* (,,*</tD assum ing ^(cu)and /(O ar© a Fourier transform pair. q.e.d. Som e basic identities: Let / e R. Then, a s follows: * ■ < « ) = i't (o))+u;(<d). W e may write: f u ) = J (/•;(tu)+//;(o)))<-"-,to f i x ) ~ _ J ( / ; <co)+//;(a»)XCosoir+/ SituJi)f/a) 2JT Necessarily, we require that both: j /;<w)Cosairuto = 0 and J /';(n))SinftmAu = 0. T hus we have: /',((u)is odd and /•; (to) is even, so / j( - w ) « - /.’(to) and /•;(-c i» = / ; « d). /•'(-w) = /•; (*»)-//;( o>)« /**(co) implying, finally: i.e., hermitian. Any function, / ( x ) , may be written as: /< * ) = /,< * )+ /,(* ) w h e r e /,< - x ) = /,( x ) /« ( - * ) * - /,< * ) so / ( - * ) = - /,( * ) + /,( * ) and w e may write: /< * )+ /< -* ) = 2 /.W Assum e the function v(r) is even. Then, a s follows: fe e tm (*(&) = J J’ fJt) CoscoxJx+i j _ > '(jr) Sincurt/r. Since J .v ^ S in o m /r^ 0, it is implied e; e R APPENDIX II - HILBERT TRANSFORMS Hilbert transforms are usually developed in the context of Fourier analysis. The usual definition of the Hilbert transform in the complex plane is: f /fo ra> > 0 for a) < 0 W e may find the Hilbert transform in the real plane with the u se of the following model and the inverse Fourier transform: C hoose c >0. , i> f *~,wfo ra)>0 r for ft) <0 Let /i(/) = — ]n { to W Md(o I 4 1 //(/)*= !im— f (-/H '"V “< /ft) + lini — f ( / ^ 'V V w * ( /) « — lim f </r *"K \/w + — -1 i m (V " **do) 2n I n f_* 1 ii ■ o . <n-r)« - /»(/)*=— lim + — lim-------- 2 ff'-*"£+/'/ -I- 2jrr-*"/V -c A /;(/) = — lim -— + — lim— 2n r- - * < • c +it 2n it - c Below w e give an exam ple of a Hilbert transform. Hilbert Transform of a Square Pulse: A < /) = T T 1 for — < / < — 2 2 0 otherwise Then we have: ".{'< /> }= 1 T ~ r J t -T r ~ * 2 t //,{»<»)}« f — i / r + j — i/r 7 / - r r-* ti //,(»(/)} = lim f </r + } lit - Ji ' - f ,!r'-r i~r — In|r — r11 - l n |r - r | it, {'(/)} = lin||-ln|c| + Ini t + ^ |-ln j/-^ //.{•'</)} = |lnj/ + y - I n j / - ||J r + / - A classical statem ent from Hilbert transform theory is given below. Let us characterize a function by two necessary requirements for real systems: 1. /re R 2. //(r) — o for r< 0 By (1), above, H(at) is hermitian and //, <cu) is even. //, (cu) is odd for //(« ) * = //, (« )+ ///, (ca). W e may write: I /;(/) = f//(c o y ” i/co | “ | //(() = — J//,(oj)Coso)/doj - - ” (w)Sino)/</a) /f(r) e ~ J //, (a)) Cosa) r t/w - —J //, (a>) Sinu> / dco By (2), above, we require: •• •• J //,(w)Coswi d ft) = J / ( (co)Sinwr < /o > for /< 0 . u I ) The choice of If,(to) implies//,(co) and vice-versa. With (2), above, in mind, we construct the model: //(/) = (I+Sgn(f ))/;,(*) Fourier transforming //( « ) * //,(w )+ — * / / (to) where // e R iw Since //(w )« //,(« )+ ///,(& ). we may write: //((o)) = //,( c d ), and 26 2 I/t(o)) = — • //,(w ) or, equivalently: CD » , ( / ) = — ZJ //,(/) = * _ / “ « So, the real and imaginary parts are related by the Hilbert transform. [See Appendix I for identities.) APPENDIX III • STATEMENTS FROM PROBABILITY AND STATISTICS The argum ents below, appeared in abbreviated form in the writings of S. I. Baskakov [pp. 197,199). A’(/) stationary and without a loss of generality we may assum e thatm^ (f) a 0. Let — . d t Then, by the definition of the autocorrelation function, we have: A',(r> = 4 r t f b < r + r ) |» A',(r) = lim + ’ A /-* .. [ A/ A/ J K, ( t) ■ lim — HiixV + Ai) • jr(r + T + A/) * ' - " ( A / ) * 1 - x ( / ) a( / 4 t + A / ) - v ( m r ) x ( / + A O + J r ( O T ( f + r ) J A \(r)« Hm—^ -r[ l*;[x(/+A/)jr(/ + r+ A /)]- /i[x(/)jr(/ + r + A/)] \ * / - ^;|jr(r + r)r(/+ A /)]+ /i’ Jxt/Jxt/ + r)]| A, (r» = Ijm -L rlA T , (n -A -, ( r + A /> - ^ ( r - A /J + A ', <r)] A',(r) = lim -pL .|A 'Jl(r-A /)-2 A 'J ,(r)+ A '1 (r+ A i)], the finile-diffarenco (A/)* representation of the second derivative *7 (f ) = - A'j (r) [Burden, p. 152). 26 Further results are: Normalization: Let A'(0) = a 3 and write /f(r) = , the normalized autocorrelation. Then we a may write: 1\ * A ', (r) = -A 'j(r) = -<ri;/{*(r) for > ’ (/)*= — . Further, for any autocorrelation function: A'(0) = a : , the variance of the stochastic process. S tatistical In d ep en d en ce of G a u ssia n R andom P ro c e s s an d Its Derivative For a G aussian random process, p(x,x')s:p(x)fKx') The argum ent justifying this is a s follows: if wo have a stationary m ean zero random process, X{(), we m ay write with A'A T(r)= i;Ix(/)-.v(/ + r)] A \,( r ) ^ /A [ * < /) * ( f + T ) ] 29 So they are uncorrelated, Thus w e have, for a G aussian random process, independence. 30

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Creator De Sulima-Przyborowski, John Casimir (author)

Core Title Microwave Coherence Tomography

Degree Master of Science

Degree Program Applied Mathematics

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

Tag engineering, electronics and electrical,geophysics,Mathematics,OAI-PMH Harvest

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Advisor Haydn, Nicolai (committee chair), Arratia, Richard (committee member), Weber, Charles L. (committee member)

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