Distributed wavelet compression algorithms for wireless sensor networks

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Z3M#%!e8.D:!’#-]E -m#"0 -8N67#" I.3W3E=43W+;?9b9_3E;4&G#-U3W3E88h@I.I[3E8G/>f5@ dR>3E x(n) /.3E8G -36G3/G#"X#b#"+3E8G - n 86G3 ,.8.0/>9_3E8G0 -87#"%#"X#W;4& *+6 z \’X#"8G!e -9 DZ=43E8 A; X(z) &)#"8G/ x e (n 0 ) = x(2n) #"8G/ x o (n 0 ) = x(2n + 1) ]W -3WI[ -8G/h b‘3E=43E8<#"8G/: A/./I7#"m&.*+6 z \’X#"8Gq!s -9_+DZ=43E8:A; X e (z 2 ) = X(z) + X(z) 2 X o (z 2 ) = X(z) X(z) 2z 1 3E T(z) = a(1 + z) RiU6G3E8~ 2 6 6 4 1 0 T(z) 1 3 7 7 5 2 6 6 4 1 (z) 1 (z) 3 7 7 5 = 2 6 6 4 1 0 a(1 + z) 1 3 7 7 5 2 6 6 4 X e (z) z 1 X o (z) 3 7 7 5 = 2 6 6 4 X e (z) z 1 X o (z) + aX e (z) + azX e (z) 3 7 7 5 s>RIJ 8G A,7#"0 -8 s>RIJ6G K*‘67#"Q6G3_3E=43E8\I7#"Q "!U6G3_/G#"X#3E9M#"8G,.8G]67#"8.D43W/)&*+6.%03b6G3_ ?/./ I7#"‘0+]W -8A=4 -%=43W/:*+6:6G3TPG%3E az 1 + 1 + az R( 8: -6G3E‘*$ -/.K&.*$3]K#"8C]W -9bI.,.3 s>RIJ+#-K~ 3CV5S-0W:n"#7"#5 n r n run!"n x(n) := x(n) r n ruQtKp4p x(n) := ax(n 1) + x(n) + ax(n + 1) 3E x o (n) /.3E8G -3+6G3/G#"X##"6G3‘ ?/./_3E8G - ndo6G3‘/>,7#"%G%!e8.DQ3EIR( ! S(z) = b(1+z) & *a3Q]K#"8C/.3W]E[36G3I.9M#"%%!e8.DM3EIC#-K~ 2 6 6 4 1 S(z) 0 1 3 7 7 5 2 6 6 4 1 (z) 1 (z) 3 7 7 5 = 2 6 6 4 1 b(1 + z) 0 1 3 7 7 5 2 6 6 4 X e (z) z 1 X 0 o (z) 3 7 7 5 = 2 6 6 4 bz 1 X o (z) + X e (z) + b X o (z) z 1 X o (z) 3 7 7 5 s>RIb #"8G/h6G3]W -3WI[ -8G/>8.D5#"%D4 -6.9#"+6G38G ?/.3W+0K~ 3CV5S-0W:n"#7"#5 n r n run!"n x(n) := b x(n 1) + x(n) + b x(n + 1) r n ruQtKp4p x(n) := x(n) 8c iU6G3E3d!s -3Z&A#T*U#W=43E%03EX#"8G!e -9 ]K#"8b[3a!s#-]E -K3W/b8A %!e8.D3EIGK&4#"8G/^9bI.%03E9_3E8F3W/c8 #^3E8G -+8G3E *a -2C#-#_3F,G3E8G]W3 "!/>,7#"%j#"8G/hI.9M#"%j3EIGKRiU6G39bI.%03E9_3E8AX#"0 -8< "!g6G3 *$ 3EIG$0U%%,GX#"3W/58hDZ,.3>RIJ>R3E x(n) /.3E8G -36G3TI. 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"# 2n 2n+1 2n+2 2n+3 2n+4 2n+5 $% &’ () *+ ,- ./ 2n 2n+1 2n+2 2n+3 2n+4 2n+5 (a) (b) Dual Step Primal Step DZ,.3Q>RIb>~)!e8.DM3EIGU!s -‘6G3 s#aPG#"8G/ U3W]W -8G/h%03E=43E% "!g*$#K=43E%03ET/.3W]W -9bI[ Z0 -8R RX 0 & $ -8G0/.3E6G3$PG%3EI7#"g!s -g6G3$ -6G3E8l\ #",.[3W]6.03W \ G3K#",.=43K#",c.0 -6G -D4 -87#"% *$#K=43E%03EIJ .*+6 L=Z#"8.06.8.DO9_ -9_3E8F:!e -h6G3V/>,7#"%Q#"8G/ I.9M#"%*U#W=43E%03EC!,.8G]E0 -8G ’$s>& d&,.I Y# 8G -9M#"%m#"0 -8 !s#-]E -K~ ~ H(z) = 1 8 z 2 + 1 4 z 1 + 3 4 + 1 4 z 1 8 z 2 ~ G(z) = 1 4 z 2 1 2 z 1 + 1 4 iU6G3I[ -%;AI.67#-X39M#"f5!s -+6.0+I7#" "!PG%3E‘0‘DZ=43E8:F; P(z) = 2 6 6 4 1 8 z 1 + 3 4 1 8 z 1 4 + 1 4 z 1 4 z 1 + 1 4 1 2 3 7 7 5 iU6G3;?9b9_3E0]%!e8.D^!s#-]E -m#"0 -8h!s -U6G3#" m=43I -%;?I.67#-3T9M#"f e3W3@I.I[3E8G/>f5@ 0 e -9b8.Dc8G -9M#"%m#"0 -8 P(z) = 2 6 6 4 1 1 4 + 1 4 z 0 1 3 7 7 5 2 6 6 4 1 0 1 2 z 1 1 2 1 3 7 7 5 : s>R#Z iU6G3E3d!s -3Z&[6G3_$Ts>& *$#K=43E%03E]K#"8139bI.%03E9_3E8F3W/VA;:6G3J%!8.D:#"%D4 -6.9 #-T/.3‘\ ]E[3W/8Cij#".%03>RZ~ AI.%8.D e3E=43E8 #"8G/: ?/./ x e (n) x(2n) x o (n) x(2n + 1) ,7#"%%!8.D x o (n) x o (n) 1 2 (x e (n) + x e (n + 1)) D9M#"%%!e8.D x e (n) x e (n) + 1 4 (x o (n 1) + x o (n)) ij#".%03>RZ~!8.D5 -I3EX#"0 -8GU!s -‘6G3Q$s>& U*U#W=43E%03EKR 8J T A/.3^]W -9b9,.8.0]K#"0 -8G8N#53E8G -8G3E *$ -2<]W -9bI.,.8.DC6.0*$#W=43E%03EQ*a -,.%0/1!s -%%0 K* DW\ ,.3c>RIb>&j*+6G3E3b6G3c]W -3WI -8G/>8.DCPG%3EQ*$3EDZ6FJ8NDZ,.3c>RIJ:#"3MDZ=43E8NA; a = 1 2 &#"8G/ b = 1 4 R 7 G Q \ ‘ #^ @T#^9_3K#"8G‘ M K=43E]W -9_36G3]K#",GX#"% ;I. -.%03E9:&[#"8G/h3W/>,G]W36G38F,.9Q[3E "!X#"8G9b0X0 -8GK& 8 -,.cI. -I Z3W/ ;?3E9 3K#-]6 3E8G -_I[3Eq!e -9_MI7#"#"%]W -9bI.,.X#"0 -8G5,G8.D\6G3 #W=Z#"%#".%03 /G#"X#M67#"#"=43WT*+616G3Q8G3E *a -2 7 K*RT3E H(z) = P c k=a h k z k & (a > 0;c 0) & 3^# D43E8G3E0]PG%3ET/.3dPG8G3W/ F;C6G3QX#"8Gq!s -9 M[3#"I.I.%03W/ M6G38G3E *$ -2RiU6G3 z \’X#"8Gq!s -9 "! 6.0UPG%3E]K#"8C3I.%8F b *a _3E9_ H(z) = A(z) + C(z) = 1 X k=a h k z k + c X k=0 h k z k *+6G3E3 A(z) #"8G/ C(z) #"36G3#"8F0]K#",GX#"%g#"8G/:]K#",GX#"%I7#"‘ "!g6G3PG%3EK&.3WI3W]E=43E%;4R ( 8 6G3MPG^3EI "!T6G35I7#"#"%‘]W A3‘_M]E03E8Ac#"I.I. 4#-]6&$3K#-]6O8G ?/.3]W -9bI.,.3Wb6G3]K#",GX#"% I7#" "!6G3PG%3E]W -8F=4 -%,.0 -8&#"8G/<3E8G/.6.08>!s -9M#"0 -8< M6G3Q8G3df>8G ?/.3^#-‘k I7#"#"% ]W ?3‘_M]E03E8Fq|XR1@UJ,.G3F,G3E8A8G ?/.3WK&*+6G3E358G3E*/G#"X#C[3W]W -9_3W^#K=-#"%#".%03Z&g6G3c3E9_^ "!‘6G3 #"8F0]K#",G #"%,.9b9M#"0 -8 #"3#-/./.3W/)&>3dPG8.8.D_6G3]W ?3‘_M]E03E8F,.8F%)+0$!e,.%%;c]W -9bI.,.3W/)R@T‘#"8 3dfG#"9bI.%03Z&4%03E x(n) /.3E8G -3‘6G3UX#K* /G#"X##"8G/ x(n) 6G3UPG87#"%7]W ?3‘_M]E03E8F!e -6G3 n \’6_3E8GX -KRj( 8 -6G3E*$ -/.K&Z![6G3U8.I.,.DZ87#"%.0g [3$PG%3E3W/bF; h(n) &Z6G3E8 x(n) = x(n)h(n) R$ -8G0/.3E 6G3+PG%3E8.D^ -I[3EX#"0 -8 x(n) = x(n1)+x(n)+x(n+1) R( 8M -,.+#"I.I. 4#-]6&>8G]W33E8G - n /. 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f (j) 0 = 0; j = 1 J ) 5-T n 7 )UV5 j = 1 J residue = f (j) 2m %2 j )UV5 s = 1 2m D43E8G3EX#"3%0 ?]K#"%)I7#"#"%7#-3W/h -8 #K=-#"%#".%03I7#"#"%j/G#"X# % ) n%2 j == residue .p n ur ’ .n .ndon Y:7 ) 5 k = 1 a s 7 x (j) p (n 2 j1 [2k 1] f (j) s1 f (j1) 2m )+ := h s k x (j1) (n f (j) s1 f (j1) 2m ) 7)UV5 7 % ) ) 5 ) 5 7 ) V5 H8G 0 & ( 8b6.03W]E0 -8b*$3/.3W]E[3U6G3+I7#"#"% ]W ?3‘_M]E03E8FU#"%D4 -6.9 !s - *$ %03E=43E%0 "!/.3W]W -9bI[ Z0 -8 ,G8.D\6G3CX#"9_3hI7#"M "!*$#K=43E%03EMPG%3EM8Y6G3:3dfG#"9bI.%03: "!?3W]E0 -8 >R KGRZR iU6G3:I -%;?I.67#-3 9M#"f:/.3W]W -9bI Z0 -8:0+3EI. ?/>,G]W3W/:6G3E3!s -‘]W -8F=43E8.03E8G]W3Z~ P(z) = 2 6 6 4 1 1 4 + 1 4 z 0 1 3 7 7 5 2 6 6 4 1 0 1 2 z 1 1 2 1 3 7 7 5 iU6G3%!e8.DM3A,7#"0 -8G+8:9_3/. -9M#"8 #"3 x(2n + 1) = 1 2 x(2n) + x(2n + 1) 1 2 x(2n + 2) x(2n) = 1 4 x(2n 1) + x(2n) + 1 4 x(2n + 1) ( 8h6.0‘]K#-3*$367#W=43 a 1 = 1 a 2 = 1 f 1 = 2 1 1 = 1 f 2 = 1 + (2 1 1) = 2 ( 86G3PGU3EIc!s -U6G3TPGU%03E=43E%) "!/.3W]W -9bI[ Z0 -8&G ?/./5X3E8G - 2n + 1 ]W -9bI.,.36G3 ]K#",GX#"% I7#"$ "!6G3]W A3‘_M]E03E8AKRiU6G3W3I7#"#"%0a*+%%[3+,.I/G#"3W/ f 1 = 1 8G ?/.3‘%#"3E s#"aX3E8G - 2n + 2 dR( 8<6G3X3W]W -8G/<3EI& 6G3JI7#"#"%0!s -6G3^3E=43E8V3E8G - 2n +*+%%g[3Q,.I[/G#"3W/V# !3E f 2 = 2 8G ?/.3WK&G#"+6G3!s -%%0 K*+8.Dc3E=43E8 3E8G -KRiU6G3E3d!s -3Z&.8G A/.3 2n + 2 ,.I[/G#"3W‘I7#"#"%0‘! -9 8G ?/.3W 2n + 1 #"8G/ 2n RiU6G3 -I[3EX#"0 -8G#"3Z&G#-+8:6G3Q3df.#"9bI.%03 "!?3W]E0 -8 >RIG>RZ~ H8c ! #"%$’&)( *+,)’ #")$-"(. xp(2n + 1) := x(2n + 1) 1 2 x(2n) x(2n + 1) := xp(2n + 1) 1 2 x(2n + 2) x(2n) := xp(2n) + 1 4 x(2n + 1) xp(2n + 2) := x(2n + 2) + 1 4 x(2n + 1) ( 86G3TX3W]W -8G/c%03E=43E% "!/.3W]W -9bI Z0 -8&G6G3T #"9_3T -I[3EX#"0 -8G+6G -,.%0/[3T#"I.I.%03W/)&G.,.$8G m* 6G3 8.I.,.5/G#"X#]W -3WI[ -8G/.c 6G3<3E 1 &+6G3 3E=43E8 PGc%03E=43E%T]W ?3‘_M]E03E8F f4n;4n + 2 n = 0;1;2;:::g R ( ! x 0 (n) /.3E8G -3W6G3<3W]W -8G/ %03E=43E%]W A3‘_M]E03E8A!e -3E8G - n &+6G3 %!8.D 3A,7#"0 -8G#"3DZ=43E8CA;~ x 0 (4n + 2) = 1 2 x(4n) + x(4n + 2) 1 2 x(4n + 4) x 0 (4n) = 1 4 x 0 (4n 2) + x(4n) + 1 4 x 0 (4n + 2): E\3b8G3W3W/V hPG8G/#"*+6.0]X6N8G ?/.3W3K#-]6L "!$6G3M3E9_#"[ K=43b[3W]W -9_3c#K=-#"%#".%03Z& h6G3E;1]K#"8 [3J#-/./.3W/< c6G3I7#"#"%/G#"X#>R. -9 s>R#cd&[*a3J67#K=4367#" f (2) 1 = 4 #"8G/ f (2) 2 = 6 &)#"8G/ 6G3 -I[3EX#"0 -8G]K#"8:[3/.3W]E[3W/C#- x 0 (4n + 2) 8 > > > > > > > > > > > > > > < > > > > > > > > > > > > > > : x(4n + 2) [3W]W -9_3W#K=-#"%#".%03J#" t (4n+2)+2 ) #"8G A/.3 (4n + 4) ]W -9bI.,.3 x 0 p (4n + 2) = x(4n + 2) 1 2 x(4n) x(4n + 4) [3W]W -9_3W#K=-#"%#".%03J#" t (4n+4)+2 ) #"8G A/.3 (4n + 6) ,.I[/G#"3 x 0 (4n + 2) = x 0 p (4n + 2) 1 2 x(4n + 4) s>RYX HRX x 0 (4n) 8 > > > > > > > > > > > > > > > > > > > < > > > > > > > > > > > > > > > > > > > : x(4n) 3W]W -9_3W#K=-#"%#".%03J#" t (4n)+2 ; x 0 (4n 2) [3W]W -9_3W#K=-#"%#".%03J#" t (4n2)+4 ) #"+8G A/.3 (4n + 2) ]W -9bI.,.3 x 0 p (4n) = x(4n) + 1 4 x 0 (4n 2) x 0 (4n + 2) [3W]W -9_3W#K=-#"%#".%03J#" t (4n+2)+4 ) #"+8G A/.3 (4n + 6) ,.I[/G#"3 x 0 (4n) = x 0 p (4n) + 1 4 x 0 (4n + 2) s>R#J iU6G3E3d!s -3Z&3W]W -8G/ %03E=43E%‘ ?/./L8G A/.3W 4n + 2 *+%%U,.I[/G#"3I7#"#"%0^!e -9 8G ?/.3W 4n 2 #"8G/ 4n 4 &#"8G/D43E8G3EX#"35I7#"#"%0!e -8G A/.3W 4n R<?3W]W -8G/%03E=43E%+3E=43E88G ?/.3W 4n + 4 *+%% D43E8G3EX#"3I7#"#"%0‘!s -‘8G ?/.3W 4n + 2 R‘BI[3EX#"0 -8G/.3WX]E3W/CF;:3A,7#"0 -8GT>RYXM#"8G/ >R#J_]K#"8 [3,.9b9M#"K3W/ #-K~ ! %$-"( *+,)’ $ (. x 0 (4n 2) := x 0 p (4n 2) 1 2 x(4n) x 0 p (4n + 2) := x(4n + 2) 1 2 x(4n) x 0 (4n 4) := x 0 p (4n 4) + 1 4 x 0 (4n 2) x 0 p (4n) := x(4n) + 1 4 x 0 (4n 2) ‘. ‘ @T3W3E8 8<?3W]E0 -81>R KG&76G3 *a \’*$#K;1#"%D4 -6.9 [3E8G3dPG‘!e -9 6G3Q9bI.%0]E ; #"8G/ ,G]E,.3 "!Q6G3<%!e8.D9bI.%03E9_3E8AX#"0 -8R ST K*a3E=43EK&#"8A0]K#",GX#"% ; 9bDZ6Fh[3W]W -9_3V#\I. -.%03E9:&T8G]W3 #hI[ -3E8F#"%%;N%#"D43M8A,.9Q[3E^ "!3df>X#hX#"8G9b00 -8GJ]K#"8[3b8F ?/>,G]W3W/)&/.3EI[3E8G/>8.D< -86G3 *$#K=43E%03EQPG%3EQ%03E8.DZ6G#"8G/V8A,.9Q[3E "!U%03E=43E%0Q "!‘/.3W]W -9bI Z0 -8RM( 8\ -/.3EQ [3b,G3d!e,.%’&)6G3 *$ \’*$#K;J9bI.%03E9_3E8AX#"0 -8M9,G]W -8G0/.3Eg6G K* 9Q,G]X6_3df?X#T]W -9b9Q,.8.0]K#"0 -8c]K#"8J[3a8F ?/>,G]W3W/ H8J ,G]X667#"jj%%A%03K#-/.j T#"8 m=43EX#"%%?3W/>,G]E0 -8^ "!G6G33E8G3EDZ;]W ZKRiU6G3a0/.3K#+0j ]W -9bI7#"3a6G3 ]W -9bI.3W0 -8I[3Eq!e -9M#"8G]W3 eDZ87#"% \’ \’8G -0X3X#"0 a#"8G/56G3]W Z$ J#-]X6.03E=4367#"UI3Eq!s -9M#"8G]W3 !s -J#C8F,.9Q[3Eb "!‘I ZX.%03c]W ?/>8.D1]6G3E9_3W e3ZR DGR&g*$#K=43E%03Eb=?mRhX#W* /G#"X#CX#"8G9b00 -8 #"8G/ 8F,.93ET "!%03E=43E%0 "!g/.3W]W -9bI[ Z0 -8R ( !Q6G3V/>0X#"8G]W3W[3E *$3W3E8 3E8G -:#"312?8G K*+8&#3K#- -87#".%03\3W9M#"3<!s -c6G3V]W Z "! X#"8G9b8.D^# b \’.I7#-]243E$3E *a3W3E8M *a J3E8G -X3EI7#"X#"3W/_F;_#/>0X#"8G]W3 d ]K#"8M3+DZ=43E8cF; C = bd K8K7&A> &4*+6G3E3 0a#]W -8GX#"8F$67#"a/.3EI[3E8G/.a -8c6G3‘9_3W/>,.9 ;?I.0]K#"%%;cdRA8G]W3 3K#-]6L%!e8.D:!’#-]E -m#"0 -8L3EIL3df?I.%0]E%;\/.3dPG8G3W*+6.0]6O]W ?3‘_M]E03E8FJ8G3W3W/\ C3M3df>]X67#"8.D43W/ [3E *a3W3E8<3K#-]6h8G ?/.3Z&.6G3 -X#"%]W Z‘]K#"8h[3]W -9bI.,.3W/C#-+6G3,.9 "!#"%%)6G38G/>=A0/>,7#"%j]W Z !s -M3K#-]6 /G#"X#\3df.]67#"8.D43ZR (’!X#"8G9b00 -8G5#"3: -9b8.0/>3W]E0 -87#"%’&6G3:]W ZM "!3K#-]X6 I7#"M "! X#"8G9b00 -8G *$ Q8G3EDZ6A -8.Db3E8G -$]K#"8M[3]W -9bI.,.3W/h#-6G3]W Z$ "!6G3‘X#"8G9b0X0 -8 b6G39_ -3/>0X#"8F8G ?/.3ZR E 6.%03^6G3^I7#"#"%]W ?3‘_M]E03E8F#"I.I. 4#-]X6N67#-Q#c9_ -3_]W -9bI.%03df19bI.%03E9_3E8FX#"0 -8L67#"8V6G3 *$ \’*$#K; #"%D4 -6.9 #"8G/ 8G]E3K#-3W^6G3%#"3E8G]E;L8O6G3c8G3E *a -2 e8G]W36G38G ?/.3]E%0 Z3Wb 6G358.2167#-Q C*$#"J!e -J/G#"X#!e -9 6G3_!s#"6G3W d&j^3E%9b87#"3Wb#"8F0]K#",GX#"%UX#"8G9b0X0 -8GK& 3W/>,G]E8.DQ6G3+3E8G3EDZ;b]W -8G,.9bI.0 -8&>#"8G/b 3Ea#X#"DZ6Aq!e -*U#"/M*U#W;b "![]W -9bI.,.8.D6G3U -X#"% ]W ZKRgG -U#8G3E *$ -2_*+6 M 8G A/.3WK&F*+6G3E33K#-]X6M8G ?/.3 n 0#"a#Q/>0X#"8G]W3 d n "!)6G38G3df?8G ?/.3 #"8G/:3E8G]W ?/.3W+‘/G#"X#^,G8.D b n .K&>6G3 -X#"%j]W Z‘]K#"8:[3]W -9bI.,.3W/ #- P M n=1 b n d n R 7‘ ? n J @ 9bI.%03EQ#"%3E87#"=43^ 6G3;EN#K=43E%03EQiX#"8G!e -9H h/.3W]W -3E%#"3_/G#"X#58\#53E8G -8G3E *a -2 ]W -,.%0/5,G3#]W ?/>8.Db]6G3E9_3,G]X6:#-$[Da$LR>[Da$ 0+#I.3W/>0]E=43]6G3E9_3T67#"+]K#"83W/>,G]W3 H8b 3W/>,.8G/G#"8G]E;L8 # X3E03Wb "!6.DZ6.%;]W -3E%#"3W/ /G#"X# 8Z>&gZG&$BK R( 8 6.0b]W ?/>8.D13W]6.8. A,G3 6G3b/> 3E3E8G]W3b3E *a3W3E8\ *a C8G3EDZ6F[ -8.DC/G#"X#hX#"9bI.%03W0]W -9bI.,.3W/#"8G/ F,7#"8AK3W/)RMA8G]W3 6G3J/> 3E3E8G]W3J[3E *a3W3E81 *a ]W -8G3W]E,.=43^]W -3E%#"3W/1X#"9bI.%03WT3E8G/C c39M#"%%’& 6G3E;C]K#"8 [3 3E8G]W ?/.3W/C9_ -3J3‘_M]E03E8F%;4&[3A,.8.DM!s3E*a3ET. _[33EI.3W3E8A3W/)R@ 9bI.%033EI.3W3E8AX#"0 -8 "!#^ D$$ 3E8G]W ?/.3E#"8G/:/.3W]W ?/.3E]K#"8h[33W3E8h8:DZ,.3Q>RZZR Q( ) Predictor (delay) Predictor (delay) Encoder Decoder DZ,.3Q>RZZ~[Da$x3E8G]W ?/.3E#"8G/h/.3W]W ?/.3EKR E 6.%036G39bI.%0]E ; "!T[Da$ ]K#"8 3h#"X#-]E=43h <6G3h3E8G -^8G3E *a -2O]W3E87#"0 \#" PGK&jX#"8Gq!s -9 ]W ?/>8.DC9_3E6G ?/.J#"3M2A8G m*+8N <#-]6.03E=43Z&g8ND43E8G3EX#"%’&g,.I[3E0 -]W -9bI.3WX0 -8 I[3Eq!e -9M#"8G]W3\#-h]W -9bI7#"3W/ LI.3W/>0]E=43\]X6G3E9_3WKR @%0 .&#I.X#-]E0]K#"%9bI.%03E9_3E8FX#"0 -8 "! [Da$H8h#3E8GX -a8G3E *a -2M*$ -,.%0/3E6G3E$3A,.3T0/.3‘8>!s -9M#"0 -8h -a%#"D43T9_3E9_ -;_., 3E #"M6G3 8G ?/.3WKRi 9M#"243C6.0cI[ -8F]E%03K#"3EK&‘]W -8G0/.3E56G3 />0.,.3W/ 9bI.%03E9_3E8FX#"0 -8 "! 6G3C[Da$ ]X6G3E9_3C "!DZ,.3 >RZ<#"%0 -8.DL#"8 #"X#K;Y "! M 8G ?/.3W_I.%,G_6G3 8.2[R G -5#"8F; 8G ?/.3 n &h D$$ ]W ?3‘_M]E03E8Fh0h]W -9bI.,.3W/ #- x D (n) = x(n) ^ x(n 1) &*+6G3E3 x(n) ]W -3WI[ -8G/. 6G3b/G#"X#5=Z#"%,G3_#- ?]E#"3W/ 8G ?/.3 n & hatx(n) 06G3 A,7#"8FK3W/V=43E0 -8\ "! x(n) &#"8G/ x D (1) = ^ x(1) RiU6G3]W A3‘_M]E03E8A_0J6G3E8 F,7#"8AK3W/Y#"8G/N!s -*$#"/.3W/Y 6G38.2[R iU6G3E3#"3‘ *$ Q9M#"85X#"3EDZ03Wa!e -6G3‘8G ?/.3W ]W -9bI.,.36G3E$]W A3‘_M]E03E8AKR( 8_6G3+PG$ -8G3Z& -8.%;Y[Da$ ]W A3‘_M]E03E8Ah#"3:X#"8G9b3W/ #"8G/ " -_!s -*$#"/.3W/ 8 6G3C8G3E *a -2[RiU6G3E3d!s -3Z& 8\ -/.3E :]W -9bI.,.3_]W A3‘_M]E03E8AK&j3K#-]X6N3E8G -03F,.3W/V h -3b6G3b]W A3‘_M]E03E8A!e -9 KZ #"%%I.3E=?0 -,GT8G A/.3WT#"8G/ ]W -9bI.,.36G38F=43E3J -I3EX#"0 -8G c3W]W K=43ET6G3 -DZ87#"%g/G#"X#_=Z#"%,G3 !s -Q6G3cI.3E=A0 -,GJ8G ?/.3c#"8G/L]K#"%0]E,.%#"3c6G3c]E,.3E8Fb]W ?3‘_M]E03E8FKR<G -J3dfG#"9bI.%03Z&g8G ?/.3 3 8G3W3W/. 1 -3 x D (1) = ^ x(1) #"8G/ x D (2) &3W]W -8G,G]E ^ x(2) = x D (2) + ^ x(1) #"8G/6G3E8Y]W -9bI.,.3 x D (3) = x(3)^ x(2) RE 6.%036.09_3E6G ?/Q/. ?3W8G -3A,.3a#"8A;#-/./>0 -87#"%AX#"8G9b0X0 -8 -6G3E 67#"8V6G3b#-]E,7#"%[Da$ ]W ?3‘_M]E03E8Fm&)0]E%03K#"67#"T9_3E9_ -; I. ?]W3W8.Dh3A,.3E9_3E8F!s - %#"D43U=Z#"%,G3W "! 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DPCM WITH SIDE INFO. DZ,.3>R#H>~ Dj3Eq!s -9M#"8G]W3]W -9bI7#"0 -8[3E *a3W3E8:[Da$ *+6:#"8G/c*+6G -,.+0/.38>!s -9M#"0 -8 #"8G/h6G3 Dg#"#"%$ ?3‘_M]E03E8FT@I.I. 4#-]X6h!s -+%0 K* ]W -3E%#"0 -8</G#"X#>R 6G -,.%0/1 3E#"8\#-/>=Z#"8FX#"D43b/>,G3J 56G3J6G3E%9b3W/13EDZ0 -81 "!U,.I.I[ -K& 9_3K#"8.8.Dh67#" -8.%; /G#"X#! -9 #8G3EDZ6F[ -6G A ?/J03F,.3W/J8b -/.3Eg ]W -9bI.,.3‘#X#"8Gq!s -9 ]W ?3‘_M]E03E8FK&F3W/>,G]E8.D 9_3E9_ -;:#"8G/hI. ?]W3W8.Db3A,.3E9_3E8F*+6G3E8C]W -9bI7#"3W/: b(( PG%3EKR [ J U3W]K#",G3 6G3 %!e8.D ]6G3E9_31/. ?3Wc8G -53E%; -8 6G3 7 -,.03EhX#"8Gq!e -9:&‘]K#"8 [3C,G3W/ ]W -8G,G]E*U#W=43E%03Ea*+6G3E3UX#"8G%#"0 -8c#"8G/^/>%#"0 -8 s#"8G/^6F,G6G3+G -,.03EX#"8Gq!s -9 ]K#"8.8G - [3,G3W/)R@ ;?I.0]K#"%’&G.,.9bI -X#"8A3dfG#"9bI.%030+6G33EDZ,.%#"TX#"9bI.%8.Dc]K#-3ZR1#"8F;h3K#"%%!s3 I. -.%03E9_]K#"8C3E8G3dPG+!e -9 7#-0U!e,.8G]E0 -8G#"8G/:X#"8Gq!e -9_#-/G#"I.3W/: ^3EDZ,.%#" #"9bI.%8.DGR KH E 6.%03I.3E=?0 -,G3W3K#"]X6Q67#-#-/./>3W3W/6G3I. -.%03E9 "!G]W -9bI.,.8.D*U#W=43E%03EX#"8Gq!e -9_ -8 3EDZ,.%#"+X#"9bI.%8.D^I7#-]W3W >&?RX &A6G3‘=Z#-9M# - ;M "!j#"%D4 -6.9_+#-,.9_3W6G3T/G#"X#P73E%0/c0 [3E8.D #"9bI.%03W/_#-#‘3EDZ,.%#"K&F*a3E%% \’[3E67#W=43W/_,G]E,.3Z&?#"%9_ Z#"%*$#K;?#T A,7#"3$DZ0/ F,.8G]E,.8>f X#"9bI.%8.D<DZ0/.J67#W=43#"%0 C[3W3E8,G/>03W/ #b.&a#G dRCiU6G35)!e8.D<]X6G3E9_35I. K=?0/.3W^#: A -%$ /.3WDZ8OuKntp Anndo erst !"n 0nu IH8J &’R}3ZR& -8G3WQ67#"#"3b8G -8G3W]W3W #"%; X#"8G%#"3WJ#"8G/ />%#"3W "! -8G3UP.f.3W/^!e,.8G]E0 -8&?#"8G/_67#"]K#"8&F6G3E3d!s -3Z&4[3#-/G#"I.3W/M 3EDZ,.%#"aX#"9bI.%8.DGR( 8 6.03W]E0 -8V*$3^I. -I Z3b#5%DZ6A9_ A/>P7]K#"0 -8N "!$6G3_$Ts>& *U#W=43E%03EQX#"8G!e -9:&[X#"2?8.D #-/>=Z#"8FX#"D43h "!‘6G35%!8.D !’#-]E -m#"0 -8& <9bI. K=4356G35X#"8Gq!e -9x3‘_M]E03E8G]E;N*+6G3E8O/G#"X#C0 3EDZ,.%#"%;OX#"9bI.%03W/)R E 6.%03*a3%9b_6.0^9_ ?/>P7]K#"0 -8 16G3I.3W/>0]E0 -8 3EI&9b%#"%; I7#"#"%%;=Z#";A8.Db,.I[/G#"3]W ?3‘_M]E03E8F]K#"8<#"%0 b[3]W -9bI.,.3W/)R E\33d!s3E‘ KZ)!s -‘/.3EX#"%0KR E\3a]K#"8%%,GX#"3a6G3%!8.D=43EX#"% ;*+66G3$$s>& [*U#W=43E%03EKR1E 6.%036G3$$s>& ]K#"813^/.3E=43W/1,G8.D:X#"8G/G#"/VG -,.03E3W]6.8. A,G3WK&)6G3^8A,.0 -8V3E6.8G/1]K#"8V[3]E%03K#"%; 3W3E8:,G8.DM6G3%!e8.Dc#"I.I. 4#-]X6R‘G -6.0X#"8Gq!s -9:&G6G3I.3W/>0]E - P 0DZ=43E8CF;6G3PG%3E p(z) = 1 2 z 1 +1 1 2 z &A*+6.0]6509bI.%;b6G3T#W=43EX#"D43T "!6G3‘ *$ 3E=43E8c8G3EDZ6F[ -KRiU6G3/.3EX#"% ]W ?3‘_M]E03E8F 1 (n) = x(2n + 1) P(x(2n + 1)) 0+6G3E3d!s -3DZ=43E8CA;~ 1 (n) = x(2n + 1) x(2n) + x(2n + 2) 2 ( !76G3,.8G/.3E%;?8.D/G#"X#0g,l_M]E03E8F%;^]W -3E%#"3W/ s#-6G3a*U#W=43E%03EX#"8Gq!s -9 ,G,7#"%%;J#-X,.9_3W d& #"8G/ 6G31X#"9bI.%03W/DZ87#"%05%0 A]K#"%%; %8G3K#"K&6.0I.3W/>0]E -h0h -I.9M#"%’&T8G]W3 x(2n + 1) = [x(2n) + x(2n + 2)]=2 &#"8G/6G3E3d!s -3 1 (n) = 0 eDZ,.3Q>RBK dR ( !q&j6G K*$3E=43EK&6G3c #"9_3MDZ87#"%a*$#-J :[3_3EDZ,.%#"%;N #"9bI.%03W/)&g6G3MI.3W/>0]E -J*a -,.%0/L8G -Q[3 -I.9M#"%Q#"8F;?9_ -3Z&T8G]W3 x(2n + 1) 6= [x(2n) + x(2n + 2)]=2 R T3E=43E6G3E%03WK&6G3C%!e8.D K8K x(2n+1) x(2n) x(2n+2) DZ,.3Q>RBKG~a$s>& $I.3W/>0]E -*+6C3EDZ,.%#"X#"9bI.%8.Dc#"8G/h%0 ?]K#"%%;5%8G3K#"‘8.I.,.KR #"I.I. 4#-]6N#"%%0 m*‘!s -#"813K#-;C -%,.0 -81 56.0TI. -.%03E9:RT8.D6G3X#"9_3^I7#"#"%#"DZ,.9_3E8AK& #D4 A ?/5I.3W/>0]E -+*a -,.%0/[3 x(2n) + (1 )x(2n + 2) &>*+6G3E3 0+#I7#"X#"9_3E3E‘67#"U=Z#"03W I7#"#"%%;\#"8G/\/.3EI[3E8G/.Q -8\6G3M/>0X#"8G]W3_[3E *$3W3E8X#"9bI.%03W ejDZ,.3c>R#GdRciU6G3bI.3W/>0]E - P *a -,.%0/C6G3E8C]W -8G0+8C#b%8G3K#"‘8F3EI[ -%#" -K&G8G3K#-/C "!#^9bI.%03#W=43EX#"D43ZR x(2n+1) x(2n) x(2n+2) (1 ) DZ,.3+>R#G>~< A/>P73W/b$ s>& I.3W/>0]E -g*+6^3EDZ,.%#"X#"9bI.%8.D#"8G/Q%0 ?]K#"%%;%8G3K#"g8.I.,.KR i;AI.0]K#"%g3E8G -T/.3EI.%0 K;?9_3E8F#"3,G,7#"%%;h3EDZ,.%#"8 87#",.3Z&#"8G/ ]W -,.%0/CDZ3K#"%;h[3E8G3dPG !j#%!e8.D,G]E,.367#"a3 73W]Ea6G3T#-]E,7#"% DZ0/c*a3E3T/.3WDZ8G3W/)RDZ,.3>R#cQ%%,GX#"3Wa6G3 /> 3E3E8G]W3816G3J9M#"DZ8.,G/.3b "!a/.3EX#"%]W ?3‘_M]E03E8F*+6G3E8N#MX#-/>0 -87#"%a$Ts>& X#"8Gq!s -9 #"8G/h+9_ ?/>P73W/h=43E0 -8<#-/G#"I.3W/h b3EDZ,.%#" #"9bI.%8.Dc#"3,G3W/)R iU6G39_ ?/>P73W/=43E0 -8:6G m*‘#JDZ8.P7]K#"8F‘3W/>,G]E0 -8C856G3#K=43EX#"D433E8G3EDZ;c "!6G3/.3EX#"% ]W ?3‘_M]E03E8FK&+#"%%0 K*+8.DO#<9_ -3:3‘_M]E03E8Ac3EI.3W3E8AX#"0 -8&U,G8.DV!s3E*a3E_._!e -M#1 #"9_3hDZ=43E8 />0 -0 -8V%03E=43E%’&[*+6G3E8V]W -9bI7#"3W/1 h#cX#-/>0 -87#"%X#"8Gq!e -9 eDZ,.3b>RYX dRiU6.09_ ?/>P73W/ =43E0 -8C "!g6G3$s>& $0+,G3W/8<#"%%)6G39Q,.%#"0 -8G8h6G3,.I[]W -9b8.DM]67#"I.3EmR KG 0 100 200 300 400 500 600 −0.3 −0.2 −0.1 0 0.1 0.2 0.3 0.4 Irregular Data Sampling Locations Original Data Sampled Data locations 0 5 10 15 20 25 0 0.01 0.02 0.03 0.04 0.05 Magnitude of Wavelet Transform Detail Coefficients irregular transform traditional transform DZ,.3b>R#c>~ -I @ />DZX#"%DZ87#"%j0X#"9bI.%03W/V#" G8Z_X#"8G/. -9%0 ?]K#"0 -8GKR - -9 1#"DZ8. \ ,G/.3 "!/.3EX#"%j]W A3‘_M]E03E8A+!s -+#JX#"8G!e -9 #-/G#"I.3W/ ^3EDZ,.%#" #"9bI.%8.Dc#"8G/:#X#-/>0 -87#"% X#"8Gq!s -9:R Kc 4 6 8 10 12 14 16 20 30 40 50 60 70 80 90 100 110 SNR Vs. 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MI7#"#"%]W ?3‘_M]E03E8FK&7#-3W/< -8C6G3QX#"3 #"%%0 ?]K#"3W/h ^!e,.%%]W A3‘_M]E03E8AKR U A8G]W3J3K#-]613E9#-/./.3W/< c6G3QI7#"#"%g,.9_/.3W]E[3W/C8\?3W]E0 -8V>RIG_67#-T c3F,7#"8AK3W/)& 6G3E3*+%%A3$#"88G]E3K#-3a8Q/>0 -0 -8J8Q6G3PG87#"%FX#"8Gq!s -9 I[3Eq!e -9M#"8G]W3ZR< -3I[3W]EP7]K#"%%;4& %03EJ,GJ#-,.9_3_6G3c*a3h#"3_8A3E3W3W/L86G35,.9 "!‘ *a <3E9_ X 1 #"8G/ X 2 &#"8G/X#K;V67#" 6G3PG87#"%3W,.%‘0‘ b[3 A,7#"8FK3W/:A;:# F,7#"8AK3E Q L R( 8C -6G3E‘*$ -/.K&G6G3PG87#"% F,7#"8AK3W/ X#"8Gq!s -9 ]W ?3‘_M]E03E8F0‘DZ=43E8<F; Q L (X 1 + X 2 ) R+( 8:6G33E8G -8G3E *a -2 ]W3E87#"0 .&6G K*a3E=43EK& 3K#-]6 3E9x6G -,.%0/3c8G/.3EI3E8G/.3E8A%; A,7#"8FK3W/LF;L#"88A3E9_3W/>#"3 F,7#"8AK3EK& Q ` &g3d!s -3 6G3PG87#"% A,7#"8Fm#"0 -8RaiU6G3PG87#"%]W ?3‘_M]E03E8F0+6G3E3d!e -3DZ=43E8:F; Q L (Q ` (X 1 ) + Q ` (X 2 )) R KJ 1 2 3 4 5 6 7 8 9 10 0 10 20 30 40 50 60 70 80 90 100 Partial Quantization Effects B l SNR 1 2 3 4 5 6 7 8 B L Q L (X 1 +X 2 ) Q L (Q ι (X 1 )+Q ι (X 2 )) DZ,.3>H>RZ~C T I3Eq!s -9M#"8G]W3: "!g\’%=A%pG WH1*U#W=43E%03EcX#"8Gq!e -9 *+6 #"8G/Y*+6G -,._I7#"#"% A,7#"8Fm#"0 -8R ( 8F3E9_3W/>#"3 F,7#"8Am#"0 -8 "!,.9b9M#"0 -8Y3E9_ A; Q ` 8A A/>,G]W3W5#-/./> \ 0 -87#"%/>0 -0 -8 DZ,.3^H>R%%,GX#"3Wb6G3h3 3W]Eb "! 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Q L (Q ` (X 1 ) + Q ` (X 2 )) [3 " 0 = Z Z (x 1 + x 2 Q L (Q ` (x 1 ) + Q ` (x 2 ))) 2 p X 1 ;X 2 (x 1 ;x 2 )dx 1 dx 2 : H>R 8Z G8Z E\31*$#"8AC O[3V#".%03V O/.3dPG8G3N# D43E8G3EX#"%Q,.%03V -86G K* O/.3WDZ8 s#"%%0 A]K#"3\.h c6G3 A,7#"8FK3E Q ` ,G]6<67#"6G3J#-/./>0 -87#"%/>0 -0 -8 8A A/>,G]W3W/ 0‘[3E%0 K* #_]W3EX#"8<6.3W6G -%0/)R ( 8h6G33E8G -‘8G3E *a -2h]W3E87#"0 .& Q ` ]W -3WI[ -8G/.‘ A,7#"8Fm#"0 -8 I3Eq!s -9_3W/hA;56G3X3E8G - -8:I7#"#"%/G#"X#>&G*+6.%03 Q L 0+6G3 A,7#"8Fm#"0 -8 "!!e,.%%]W A3‘_M]E03E8AKR ‘ 3E r L k #"8G/ t L k /.3E8G -3Z&3WI3W]E=43E%;4&6G3 k \’63W]W -8G,G]E0 -8 %03E=43E%#"8G//.3W]E00 -8 %03E=43E% #- ?]E#"3W/O V6G3 F,7#"8AK3E Q L R ( 8 -6G3Ec*a -/.K&a!s -c#"8 8.I.,. u & Q L (u) = r L k ! u2 [t L k ;t L k+1 ) RiU6G3E3d!e -3Z&G!6G38.I.,. ^6G3 F,7#"8AK3E0 X 1 + X 2 &>*a367#K=43Z~ Q L (X 1 + X 2 ) = r L k ! t L k X 1 + X 2 t L k+1 H>R > X 1 X 2 t L k r L k t L k+1 t L k t L k+1 DZ,.3;H>R >~+TX#"I.6.0]K#"%8A3EI.3EX#"0 -8\ "! F,7#"0 -8 H>R > dR+(’!g6G33W,.% "! 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p> r YZ:=R0/ R96Cdk> U?YG/=>fOW60/=> rsN=6PO?N , >WUXNoY=/ YMce>W<?gIUXNo> g>,SJMR9RP>W< OfY;gXU 8;6C4k>WT UXNo>w8;<?YZ:=Lj/C6PgXUKJMToOf>wU?Y UXNo> gX69T)d B 5 698;:=<?> ooBCHwgXNoYQr gnUXNo>wUXN=<?>f> ToY=/=> 8;<?YZ:=L 69T >QJZOKN g?Of>WT_JM<X6PY^JMTE/ UXNo>y<?>f]‘:=69<?>1/ RPY‘OQJMR OfY;gXUIg?Y UXN_JMU JMR9R7UXNo>yUX<KJMTog>, 6PgKgX6PYZTog MOfY2, L=:=UKJMUX6PYZTogw<?>f]‘:=69<?>1/ U?YpN_JJ4k> UXNo> OfY >SOW6P>WTVU?g <?>QJ2/ 7 U?Ypt >^\iYZ<Xr JM</=>1/ U?Y UXNo> gX69T)d N_JJ4k> t >f>WTAL >W< \iYZ<>, >1/eB :CJ17 UXN_JMUA, >WUXNoYF/ c OQJMT <?>WL=<?>fg?>WT‘U UXNo> YZ<X698;69T_JMR /oJMUKJp:ogX69T=8 o t=69U?g L >W< g?>WTog?YZ<mh HJKpt=69U?g L >W< @ ToY=/=>fg j 3 JZO?N=6P>54 69T=8 JD/C6PgXU?YZ<XUX6PYZT a rs69UXNAJ RPY OQJMRsOfY;gXU YM\ 18d 2 Bm9 >WUXNoYF/ 6HOQJMT JZOKN=6P>54k> UXNo> gKJp, > /C6PgXU?YZ<XUX6PYZT;a :og?69T=8^@CBE@;@ t=69U?g L >W< g?>WTogKYZ< h HJM t=69U?g L >W< @SToYF/=>fgdj 3 t=:=UI<?>f]‘:=69<X69T=8 JSRPY OQJMR OfY;gXU YM\ 26d 2 B h No>W<?>d\iYZ<?> 3 \iYZ<8, >WUXNoY=/ c 3 UXNo> U?YZUKJMR OfY;gXU U?Y ,SJpdk> JMR9R UXNo> /oJMUKJ 69T UXNo>I8;<?YZ:=L J14;JM69R JMt=RP>yJMU UXNo> gX69T)d 6PgI8;6C4k>WTmtB7 UXNo> RPY‘OQJMR OfY;gXU h 18d 2 jnL=R9:ogIUXNo> OfY;gXUwYM\ \iYZ<Xr JM</C69T=8^UXNo> L=<?Y Of>fg?g?>1/D/oJMUKJ NoYZLog U?Y UXNo> gX69T)d h 12Nd 2 j 3 YZ< C A = 18d 2 + 12Nd 2 B :‘6C, 69R JM<XRC7 3 r > OQJMT OQJMRPOW:=R JMU?>yUXNo> OfY;gXU ‘NK 4 bits/node Total cost: C A = 18d 2 + 12Nd 2 Total Cost: C B = 26d 2 + 10Nd 2 3.33 bits/node sink sink 12 bits over N hops 10 bits over N hops (b) (a) local cost: 18d 2 local cost: 26d 2 57698;:=<?>AooBCH;F hiJkj 9 >WUXNoY=/ c KF J g?6C, L=RP> >WToOfY=/C69T=8 g?O?No>5, > 6Pg :og?>1/ HJK t=69U?g JM<?> g?>WTVU U?Y UXNo> gX69T)d Bshutvj 9 >WUXNoY=/ 6 KFCJ RPY OQJMR9RC7l, YZ<?>I>.- L >WTogX6C4k>f, >WUXNoYF/ 6Pg :og?>1/ 3 t=:=U t >WUXU?>W< OfY2, L=<?>fg?gX6PYZT 6Pg JZO?N=6P>54k>1/ HJMyt=69U?g To>f>1/ U?Y t > \iYZ<Xr JM</=>1/^U?Y UXNo>IgX69T)d Bh No> OfY;g?U YM\ UX<KJMTog>, 69UXUX69T=8 k t=69U?gsY 4k>W<nJ /C6PgXUKJMToOf> d r JZg OfY2, L=:=U?>1/ JZg kd 2 B \iYZ<, >WUXNoYF/m6 JZg C B = 26d 2 + 10Nd 2 B p> OQJMT 3 UXNo>W<?>d\iYZ<?> 3 @oTE/m\iYZ< rsN=6PO?N 4ZJMR9:o>fg YM\ N J , >WUXNoY=/^RP>QJ2/=gsU?Y RPYbr >W<xOfY;g?UsUXN_JMT UXNo> YZUXNo>W<QF C B < C A 26d + 10Nd < 18d + 12Nd 8d < 2Nd N > 4 h No>IJMt Y 4k>n>f]‘:_JMUX6PYZTog g?NoYQr UXN_JMU 6 \ UXNo>xUXN=<?>f>ng?>WTog?YZ<?g JM<?>/C6Pg?UKJMTVU , YZ<?> UXN_JMTAoyNoYZLog \u<?Y2, UXNo> gX69T)d 3 , >WUXNoY=/ 6 wrs69R9R7RP>QJ2/ U?Y J RPYQr >WWTo>W<X8N7 OfY;g?UQB T 8k>WTo>W<KJMR 3 69Ux6Pgn>.-CL >fOWU?>1/ UXN_JMUxgK>WTog?YZ<?g OWRPY;g?>W< U?Y UXNo> gX69T)d gXNoYZ:=R0/ t >WTo>.@oUx\u<?Y2, OfY=/C69T=8 g?OKNo>5, >fgxUXN_JMUIYMce>W< g>,SJMR9RP>W< RPY OQJMR OfY;gXU 3 rsN=69RP> g?>WTog?YZ<?g UXN_JMUSJM<?> \iJM< \q<?Y2, UXNo> gX69T)d gXNoYZ:=R0/ >WToOfY=/=> :ogX69T=8pg?O?No>5, >fgSUXN_JMU <?>f]‘:=69<?> Jmg>,SJMR9RP>W< J14k>W<KJM8k>yTV:), t >W< YM\7t=69U?gsL >W< g?>WTog?YZ< \qYZ<nJ 8;6C4k>WTj/C6Pg?U?YZ<XUX6PYZT 3 JM8;<?>f>W69T=8 rs69UXN CHJM JMTE/LKNI B N=69RP> t YZUXN CHJMxJMTE/ LKNI 69TE/C6POQJMU?>^UXN_JMU UXNo>^:og?> YM\ /C6 c >W<?>WT‘US>WToOfYF/C69T=8 JMR98kYZ<X69UXN), g 69T /C6 ce>W<?>WTVUn<?>W8;6PYZTog YM\ UXNo>wTo>WU r YZ<>d r YZ:=R0/ 6C, L=<?YJ4k> >WTo>W<X8N7^OfYZTogX:), L=UX6PYZT >SOW6P>WToO57^69T UXNo>wTo>WU r YZ<>d 3 UXNo>57m>W69UXNo>W< <?>f]‘:=69<?> UXNo> JZg?gX:), L=UX6PYZTpYM\ g?L >fOW6n@_Og/oJMUKJA, Y=/=>WRPgwU?YG/=>fgX698;T UXNo> >WToOfY=/=>W<?g ‘;@ YZ< UXNo>^OfY2, L=<?>fg?gX6PYZTAJMR98kYZ<X69UXN), gSL >W< \iYZ<>,SJMToOf>^6Pg YZT=RC7 JMT_JMRC7‘aQ>1/ t_JZgK>1/ YZT >WT‘UX<?YZLB7 , Y=/=>WRPg 3 rs69UXN R96C, 69U?>1/ L=<KJZOWUX6POQJMRnJML=L=R96POQJMt=69R969U7kBmh r Y gX698;T=6n@_OQJMT‘U OfYZTVUX<X69t=:=UX6PYZTog YM\ YZ:=< r YZ<>d JM<?> UXN_JMU 6 jn69U ,SJpdk>fgwToYA, YF/=>WR969T=8 JZg?g?:), L=UX6PYZTog JMt YZ:=UIUXNo> :=TE/=>W<XRC7 69T=8P/oJMUKJ g?UKJMUX6PgXUX6POfg hu69TogXU?>QJ2/pJMR9R 6Pg <?>f]‘:=69<?>1/ 6PgsUX<KJM69T=69T=8A/oJMUKJkjsJMTE/ 696 j UXNo> JMR98kYZ<X69UXN), /C69<?>fOWUXRC7 OfY2, L_JM<?>fg UXNo> L >W< \iYZ<>,SJMToOf>yYM\ JZO UX:_JMR OfY=/C69T=8Sg?O?No>5, >fg hu69TogXU?>QJ2/^YM\ OfYZTog?60/=>W<X69T=8S>WTVUX<?YZLF7=jsJMTE/ L=<?YJ4 60/=>fg , YZ<?>w<?>QJMR96PgXUX6POITo>WU r YZ<>d OfY;gXUy>fg?UX6C,SJMU?>fgQBx:=< , >WUXNoY=/p6Pg _>.-C69t=RP>S>WToYZ:=8;N U?Y JZOfOfY2,g, Y=/oJMU?> JMTF7mTo>WU r YZ<>d OfYZT @o8;:=<KJ UX6PYZT huU?YZL YZRPYZ8N7=jdB p>It >WR96P>54k>wUXN_JMUsUXNo>It_JZgX6POIL=<X69ToOW69L=RP>fgxYM\ YZ:=<nJML=L=<?YkJZOKN OfYZ:=R0/ t > JML=L=R96P>1/^U?Y YZUXNo>W< /oJMUKJ <?>WL=<?>fg?>WT‘UKJMUX6PYZTpg?>WRP>fOWUX6PYZT Jp, YZT=8 YZUXNo>W<Ig?>WU?g YM\ OfYF/C69T=8 g?OKNo>5, >fg hu6 B >;B 3 ToYZU R96C, 69U?>1/ U?Y^r J14k>WRP>WU?g jwUXN_JMUyYZL >W<KJMU?>StB7m>.- L=RPYZ69UX69T=8 gXL_JMUX6 JMR <?>1/C:=TE/oJMToO57Ahu69T , >WUXNoY=/=g UXN_JMU/=Y^ToYZU 69T 4kYZRC4k> /oJMUKJ >.-COKN_JMT=8k>fg t >WU r >f>WT ToY=/=>fg769T UXNo> OfY2, L=<?>fg?gX6PYZT L=<?Y Of>fg?g 3 UXNo> YZL=UX6C, 69abJMUX6PYZT t >fOfY2, >fg gXUX<KJM698;N‘U \qYZ<Xr JM</_jdB " k"! " p> OfYZTogX60/=>W<g/oJMUKJ JM8;8;<?>W8VJMUX6PYZT hqOfY2, L=<?>fg?g?6PYZTvj JMRPYZT=8pJ gX69T=8;RP> <?YZ:=U?> L_JMUXN \q<?Y2, JMT >1/C8k> U?Y UXNo> gX69T)d 3 JZg gK>f>WT 69T 57698;:=<?>eooBLK hiJk,y:=R9UX6 <?YZ:=U?>SJML=L=<?YkJZOKN rs69R9R t > J2/)/C<?>fg?g?>1/m69T N_JML=U?>W< DkjdB h N=6Pg L_JMUXN 6Pg JZgKgX:), >1/ U?Y t >jd ToYQrsT 3 rsN=6PO?N 6C, L=R96P>fg UXN_JMU J <?YZ:=UX69T=8AJMR98kYZ<X69UXN), N_JZg t >f>WT JML=L=R96P>1/^U?Y UXNo>wTo>WU r YZ<>dA@o<?gXUQBb< JZO?N g?>WTogKYZ<s6PgnJZg?gX698;To>1/ J T‘:),yt >W< n 3 gXUKJM<XUX69T=8 \q<?Y2,[UXNo>w>1/C8k>;B h No> To>WU r YZ<>d U?YZL YZRPYZ8N7 hiJMTE/ UXN‘:og UXNo> 69T‘U?>W<XToYF/=> /C6PgXUKJMToOf>fg j 6Pgld ToYQrsT 3 JMTE/ >QJZOKNAToY=/=> 69T UXNo>^L_JMUXNAOQJMT YZL >W<KJMU?>^:og?69T=8 JmOfYF/C69T=8 g?O?No>5, > O?NoY;g?>WTA\u<?Y2, JmL=<?>1/=>.@oTo>1/ g?>WU YM\ J14;JM69R JMt=RP> OfY=/C69T=8pg?O?No>5, >fgQB T UXN=6Pg r YZ<>d 3 J14;JM69R JMt=RP> g?OKNo>5, >fgSJM<?>P/C6Pg?OW<?>WU?> r J14k>WRP>WU UX<KJMTog \qYZ<>, gS:ogX69T=8 UXNo> gKJp, >@oR9U?>W<Xt_JMT)d t=:=U rs69UXNk/C6 ce>W<?>WTVU TV:), t >W< YM\eRP>54k>WRPg YM\ /=>fOfY2, L Y;g?69UX6PYZT hqg?>f>f: >fOWUX6PYZT%KCBEDkjdF rsNo>WT UXNo> TV:), t >W< YM\vRP>54k>WRPgb/=>fOfY2, L Y;gX69UX6PYZT 6Pg 69ToOW<?>QJZg?>1/ 3 UXNo> L YZU?>WT‘UX6 JMR_OfY2, L=<?>fg?gX6PYZT >SOW6P>WToO57 ‘po JMRPg?Y 69ToOW<?>QJZg?>fg hu6 \ /oJMUKJ^6PgIN=698;N=RC7mOfYZ<X<?>WR JMU?>1/ JZOW<?Y;g?g g?>WTogKYZ<?g j 3 t=:=U JMUwUXNo> OfY;g?U YM\8, YZ<?> RPY OQJMR 69TC\iYZ<>,SJMUX6PYZTp>.-COKN_JMT=8k> hut >fOQJM:og?>e/oJMUKJS\u<?Y2, , YZ<?> ToY=/=>fgx6Pg To>f>1/=>1/mU?Y OfY2, L=:=U?> g?Y2, > YM\ UXNo> r JJ4k>WRP>WUxOfY >SOW6P>WTVU?g jdB ... ... sink M−1 n 4 2 1 3 M 57698;:=<?> ooBLKCF :‘69T=8;RP> <?YZ:=U?> L_JMUXN rs69UXNj9 ToY=/=>fgsU?Y UXNo> g?69T)d B :‘69ToOf>SUXNo> r JJ4k>WRP>WU UX<KJMTog \iYZ<>, 6Pg OW<X69UX6POQJMR9RC7mg Jp, L=RP>1/ 3 UXNo>STV:),yt >W< YM\sr J14k>WRP>WU OfY‘>SOW6P>WT‘U?g 8k>WTo>W<KJMU?>1/ 6Pg >f]V:_JMR U?Y UXNo>ST‘:),yt >W< YM\nToYF/=>fgbB gX69T=8 UXNo> L_JM<XUX6 JMRsOfY >SOW6P>WTVU JML=L=<?YkJZO?N 3 UXNo> r JJ4k>WRP>WU OfY >SOW6P>WTVU OfYZ<X<?>fgXL YZTE/C69T=8 U?Y ToYF/=> n 6Pg OfY2, L=:=U?>1/A69T gXU?>WLogQF JMU ToY=/=> n J L_JM<XUX6 JMR 4k>W<?gX6PYZT YM\sUXNo> OfY >SOW6P>WTVU 6Pg @o<?gXU 8k>WTo>W<KJMU?>1/ 3 rsN=6POKN t >fOfY2, >fg J \q:=R9R OfY‘>SOW6P>WT‘U JZg 69U 69ToOfYZ< L YZ<KJMU?>fg J2/)/C69UX6PYZT_JMR/oJMUKJ \q<?Y2, \u:=UX:=<?> ToY=/=>fg^hu6 B >;B 3 ToY=/=>fg OWRPY;g?>W< U?YmUXNo> g?69T)d=jdB c g g?>f>WT 69T : >fOWUX6PYZT KCBED 3 UXNo> TV:), t >W< YM\ NoYZLog <?>f]‘:=69<?>1/ :=T‘UX69R J L_JM<XUX6 JMR OfY‘>SOW6P>WT‘UIt >fOfY2, >fgx\q:=R9R&/=>WL >WTE/=g YZT UXNo>wgXL >fOW6n@_OwUX<KJMTog \iYZ<>, @oR9U?>W<?gst >W69T=8 :og?>1/eB !#" T YZ</=>W< U?Yk@oTE/ UXNo> t >fgXUxOfY=/C69T=8^g?O?No>5, > \iYZ<x>QJZO?NpYM\ UXNo> g?>WTogKYZ<?g 3 r > L=<?YZL Y;g?> <?>WL=<?>fg?>WT‘UX69T=8 UXNo>ITo>WU r YZ<>d JZg Jy8;<KJML=N B 5 698;:=<?> ooBE@ 69R9R9:og?UX<KJMU?>fgsUXNo>I8;<KJML=N JZg?g?Y OW6 JMU?>1/ U?Y J g?69T=8;RP> <?YZ:=U?>wL_JMUXN 69ToOWR9:E/C69T=8 M + 1 g?>WTog?YZ<?g huUXNo> R JZgXU t >W69T=8 UXNo>wgX69T)doj 3 rsNo>W<?>w>QJZO?N g?>WTog?YZ<sOQJMT^:og?>wYZTo>wYM\ UXN=<?>f> J14;JM69R JMt=RP> OfYF/C69T=8 g?OKNo>5, >fgQB < JZO?N >1/C8k> 69T UXNo> 8;<KJML=Nm<?> _>fOWU?gwJ L Y;g?gX69t=RP> UX<KJMTogX69UX6PYZT \u<?Y2, YZTo> OfY=/C69T=8 g?O?No>5, >wU?Y JMToYZUXNo>W< 3 JMTE/ N_JZgnJMT JZg?g?Y OW6 JMU?>1/ r >W698;N‘UnUXN_JMUs<?>WL=<?>fg?>WT‘U?g UXNo>IUX<KJMTog>, 6PgKgX6PYZT OfY;gXU U?Y OfYZT‘UX69TV:o>m69T Jm/=>WU?>W<>, 69To>1/ g?OKNo>5, > YZ< UXNo> >.- UX<KJ L=<?Y‘Of>fgKgX69T=8 UX<KJMTog>, 6Pg?gX6PYZT OfY;gXU U?Y ‘;D O?N_JMT=8k> OfY=/C69T=8 g?OKNo>5, >fgQBj< JZO?N L Y;gKgX69t=RP> L_JMUXN 69T UXNo>S8;<KJML=N 6Pg JZg?g?Y OW6 JMU?>1/ U?Y YZTo> OKNoYZ6POf> YM\ OfY=/C69T=8Sg?O?No>5, >w\qYZ< >QJZOKN ToY=/=>I69T UXNo>I<?YZ:=UX69T=8SL_JMUXN^\q<?Y2, JMT >1/C8k>wU?Y UXNo>ygX69T)d B + >WU c n (i;j) t > UXNo>nOfY;g?U YM\ UXNo> UX<KJMTog?69UX6PYZTS\q<?Y2, , >WUXNoY=/ i JMU ToYF/=> n U?Y, >WUXNoY=/ j JMU ToYF/=> n + 1 3 n = 1;::: ;M 3 rsNo>W<?> i;j2 S =fA;B;C;:::g 3 rs69UXN A;B;C;::: <?>WL=<?>fgK>WTVUX69T=8 UXNo> OfY=/C69T=8ng?O?No>5, >fg7OfYZTogX60/=>W<?>1/eB&+ >WU l n (i) /=>WToYZU?> UXNo> RPY OQJMRkL=<?Y‘Of>fgKgX69T=8nOfY;gXUe\iYZ< g?>WTog?YZ< n U?Y >WToOfYF/=> 69U?g /oJMUKJ :ogX69T=8 OfY=/C69T=8 gKO?No>5, > i Bx:=< 8kYkJMR 6Pg U?YP@oTE/pUXNo> g?>f]V:o>WToOf> fi 1 ;i 2 ;::: ;i n ;::: ;i M g YM\ OfYF/C69T=8 g?O?No>5, >fgIJZg?g?Y OW6 JMU?>1/^U?Y >QJZO?N YM\ UXNo>wToY=/=>fg gX:oO?N UXN_JMU P n (c n (i n ;i n+1 ) + l n (i n )) 6Pg , 69T=6C,y:), B sink C B A n M Stage Stage Stage Stage n+1 n+2 states 57698;:=<?> ooBE@CF: >WTog?YZ< To>WU r YZ<>d g?>f>WTSJZg J 8;<KJML=N B< JZO?N g?>WTog?YZ< 6Pg J ToY=/=> UXN_JMU OQJMT t > 69T YZTo> gXUKJMU?> YM\ YZL >W<KJMUX6PYZT 3 JZgKg?Y‘OW6 JMU?>1/ rs69UXN J OfY=/C69T=8 g?O?No>5, > h c 3 6 3 sjIJMTE/pJ L Y;gX69UX6PYZTm69T UXNo> To>WU r YZ<>d B <b/C8k>fg OfYZ<X<?>fgXL YZTE/AU?Y UX<KJMTogX69UX6PYZTog t >WU r >f>WT gXUKJMU?>fg 3 JMTE/ N_J14k> UX<KJMTog>, 6Pg?gX6PYZT JMTE/ L=<?Y Of>fg?gX69T=8 OfY;gXU?gyhur >W698;NVU jxJZg?g?Y 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To>.-CUSToY=/=> rs69R9RxN_J14k> U?YpL >W< \qYZ<>, U?Y >.-CUX<KJZOWU UXNo> RPYbr >W< RP>54k>WRxOfY >SOW6P>WTVU?g \q<?Y2, UXNo> N=698;No>W< RP>54k>WR YZTo>fg 3 YZ<sU?Y gXUKJM<XUsUXNo>IL=<?Y Of>fg?gX69T=8 YM\7N=698;No>W< RP>54k>WR OfY >SOW6P>WTVU?g t_JZgK>1/^YZT^RPYQr >W<nRP>54k>WR YZTo>fg 3 JZg JMR9RPYQr >1/ tF7pUXNo> >5, t >1/)/=>1/ L=<?YZL >W<XU7kB t n (i n ;i n+1 ) 6Pg OfY2, L=:=U?>1/ JZgyUXNo> >WTo>W<X8N7 gXL >WTVU tB7 UXNo>Ig?>WTog?YZ< L=<?Y‘Of>fgKg?YZ< U?Y OfY2, L=:=U?>IUXNo>f,y:=R9UX69L=R96POQJMUX6PYZTogn<?>f]V:=69<?>1/ tF7SUXNo> UX<KJMTogX69UX6PYZT^YZL >W<KJMUX6PYZTog hiJ2/)/C69UX6PYZTog OQJMT t > U 7‘L=6POQJMR9RC7 To>W8;RP>fOWU?>1/ 3 g?69ToOf> UXNo>57SJM<?> YZ</=>W<?g YM\ ,SJM8;T=69UX:E/=>IOfY2, L=:=UKJMUX6PYZT_JMR9RC7 O?No>QJML >W< UXN_JMT ,y:=R9UX69L=R96POQJMUX6PYZTog oVDujdBmln>5,SJM69T=69T=8mYZT UXNo> g Jp, > g?OKNo>5, >^J2/)/=g ToY L=<?Y Of>fg?gX69T=8 YJ4k>W<XNo>QJ2/ 3 t n (j;j) = 0 \qYZ<nJMR9R k B :‘69ToOf> t n (j;j) = 0 69U 6Pg ToYZU/=>WL=6POWU?>1/^69T UXNo> @o8;:=<?>;B n n+1 n+2 n+3 n+2 n+3 sink B+N B B+N B B+N B+N B+N B+N Path 1 Path 2 Level j−1 Level j x(n) x(n) x p (n) x p (n) x p (n) x p (n) 57698;:=<?> ooBLICFbh <KJMTog?69UX6PYZT OfY;gXU?g >.-oJp, L=RP>;B c g JMTS>.-oJp, L=RP> 3 OfYZTog?60/=>W< UXNo>sUX<KJMTog>, 6PgKgX6PYZTSYM\eUXNo> xOfY‘>SOW6P>WT‘U \iYZ< ToY=/=> n JMU RP>54k>WR j 3 YJ4k>W L_JMUXNog 3 69T J To>WU r YZ<>dyYM\ n+3 ToY=/=>fg 3 JZg g?>f>WTy69Ty57698;:=<?>booBLIyhuUXNo> UX<KJMTog>, 6Pg?gX6PYZTog \iYZ< JMR9RwUXNo>pYZUXNo>W< OfY‘>SOW6P>WT‘U?g JM<?>mToYZU OfYZTog?60/=>W<?>1/ No>W<?>bjdB c g?gX:), >mUXN_JMU JMR9RwToY=/=>fg JM<?> JMU J?/C6PgXUKJMToOf> d YM\ >QJZOKN YZUXNo>W< 3 JMTE/ UXN_JMU B t=69U?gSJM<?> JMR9RPY OQJMU?>1/ \qYZ< UXNo> UX<KJMTog \qYZ<>, OfY >SOW6P>WTVU YM\xToY=/=> n 3 x(n) Bp5oYZ< 69R9R9:ogXUX<KJMUX6PYZT L=:=<XL Y;g?>fg 3 JZg?gX:), > UXN_JMU UXNo> OfY‘>SOW6P>WT‘U \qYZ< ToY=/=> n rs69R9R t >fOfY2, > J;@oT_JMR OfY >SOW6P>WTVU JMUSToY=/=> n + 2 3 L=<?YJ4 60/=>1/AUXNo> gKJp, > RP>54k>WRIYM\ /=>fOfY2, L Y;gX69UX6PYZT 6Pg &2K ,SJM69TVUKJM69To>1/ \qYZ<sToY=/=>fg n + 1 JMTE/ n + 2 BnT‘UX69R 69Us<?>QJZO?No>fgnToYF/=> n + 2 3 UXNo>wOfY >SOW6P>WTVUnrs69R9Ret > YZT=RC7^L_JM<XUX6 JMR9RC7 OfY2, L=:=U?>1/ 3 JMTE/ <?>WL=<?>fg?>WT‘U?>1/ tF7 x p (n) 3 rsN=6PO?N 6Pgn]‘:_JMTVUX69aQ>1/ rs69UXN B + N t=69U?gQB h No>s>1/C8k>sr >W698;NVU?g 69T 57698;:=<?> ooBLI /=>WL=6POWU UXNo> T‘:),yt >W< YM\ t=69U?g t >W69T=8wUX<KJMTog, 69UXU?>1/ YJ4k>W< >QJZOKNS>1/C8k> huL=<?Y Of>fg?gX69T=8SOfY;gXU?gxJM<?>IToYZUngXNoYQrsT \qYZ< OWR JM<X69U 7^L=:=<XL Y;g?>fgdjdB n n+1 n+2 n+3 n+2 n+3 sink Level j−1 Level j BD 2 n Nd 2 Nd 2 Nd 2 = 3Nd 2 ND 2 n+1 = 4Bd 2 57698;:=<?> ooBCHJMCF h <KJMTogX69UX6PYZT OfY;gXU?g OfY2, L=:=UKJMUX6PYZT \iYZ< >.-oJp, L=RP> YM\ 57698;:=<?> ooBLICB T7JMUXNmH 3 UXNo>wOfYF/C69T=8SgKO?No>5, >w6Pg ,SJM69T‘UKJM69To>1/ JMTE/^UXNo>wOfY >SOW6P>WTVUn\qYZ<sToY=/=> n 6Pgs<?>.@oTo>1/^:=T‘UX69R 69U t >fOfY2, >fg@oT_JMR 3 JMU ToYF/=> n+2 B c \uU?>W< UXN_JMU L YZ69T‘U 3 69U 6Pg ]‘:_JMTVUX69aQ>1/ YZT=RC7 rs69UXN B t=69U?g JMTE/ x(n) 6Pgx\qYZ<Xr JM</=>1/ U?Y UXNo> gX69T)d B T 7JMUXNmK 3 UXNo> OfY >SOW6P>WTVU OfY2, L=:=UKJMUX6PYZTp6Pg 69T‘U?>W<X<X:=L=U?>1/pgX69ToOf> UXNo> OfY=/C69T=8yg?O?No>5, >n6Pg O?N_JMT=8k>1/ 3 JMTE/ x p (n) rs69R9R_N_J14k> U?Y t > \iYZ<Xr JM</=>1/SU?YwUXNo> gX69T)d JZg J L >W<>,SJMTo>WTVU L_JM<XUX6 JMR 3 ]V:_JMT‘UX69aQ>1/ rs69UXN B + N t=69U?gQB h No> OfY;gXU?gxJ2/)/=>1/ U?Y UXNo>yOW:=<X<?>WTVUIL_JMUXNogIJM<?>yOfY2, L=:=U?>1/ JZg/=>WL=6POWU?>1/ 69T 5 698;:=<?> ooBCHJMCBbh No>wU?YZUKJMR UX<KJMTog, 6Pg?gX6PYZT OfY;gXU \iYZ< JMUXN Hw6Pg C 1 = 4Bd 2 + 2Nd 2 3 JMTE/ \iYZ< 7JMUXN?K 3 C 2 = 4(B + N)d 2 B ! ! !#" h No> L_JM<XUX6 JMR OfY‘>SOW6P>WT‘UyJML=L=<?YkJZO?N 8;:_JM<KJMTVU?>f>fg UXN_JMUwJMTF7 OfY2, L=:=UKJMUX6PYZT JMUwJMTB7 8;6C4k>WTmToYF/=> <?> ]‘:=69<?>fg YZT=RC7D/oJMUKJS\u<?Y2, L=<?>54 6PYZ:ogxToY=/=>fgQB c RPg?Y 3 /C:o> U?Y UXNo> >5, t >1/)/=>1/mL=<?YZL >W<XU7D/C6Pg?OW:og?g?>1/ 69T : >fOWUX6PYZT ooBLKCBE@ 3 JMTB7 ToY=/=> JMR9r JJ7 gSN_JZgSJZOfOf>fg?g U?Y UXNo> L_JZg?USOfY‘>SOW6P>WT‘U?g 69U To>f>1/=g U?YpOfY2, L=:=U?> 69U?g YQrsT L_JM<XUX6 JMR OfY >SOW6P>WTVU 3 <?>W8VJM</CRP>fg?g YM\xrsNo>WUXNo>W< UXNo>%/oJMUKJ 69U 6Pg <?>fOf>W6C4‘69T=8 r JZg 8k>WTo>W<KJMU?>1/ &Z@ \q<?Y2, UXNo> gKJp, > RP>54k>WR j YZ< \u<?Y2, JMTF7 YZUXNo>W<wRP>54k>WR B c gwJ <?>fgX:=R9U 3 UX<KJMTog?69UX6PYZTpOfY;gXU?g /=>WL >WTE/mYZT=RC7 YZT UXNo> L=<?>fg?>WTVU g?UKJMU?>;F UXNo> L=NB7CgX6POQJMR L Y;gX69UX6PYZT YM\nUXNo> ToY=/=> 69T UXNo> To>WU r YZ<>d JMTE/ UXNo> OfY=/C69T=8 g?OKNo>5, > t >W69T=8^:og?>1/ 3 JMTE/ 3 gX69ToOf> UXNo> YZ:=UXL=:=U YM\ J ToY=/=> hu69U?g OfY >SOW6P>WTVU j 6PgwUXNo> g Jp, > <?>W8VJM</ RP>fg?g YM\yUXNo> L=<?>54‘6PYZ:og L_JMUXN 3 OfY;gXU?g :=L U?Y J L_JM<XUX6POW:=R JM< ToYF/=>D/=Y ToYZU 69To:o>WToOf> UXNo> OfY;gXUS\iYZ< J \q:=UX:=<?>^UX<KJMTogX69UX6PYZT B 69UXNoYZ:=U RPY;gX69T=8p8k>WTo>W<KJMR969U 7 3 OfYZTogX60/=>W< UXNo> gXUKJMU?> (n;j) 3 JMTE/ RP>WU S a = fi a 1 ;i a 2 ; ;i a n2 ;j 1g 3 S b =fi b 1 ;i b 2 ; ;i b n2 ;jg JMTE/ S c =fi b 1 ;i b 2 ; ;i b n2 ;j + 1g <?>WL=<?>fg?>WT‘U UXNo>SUXN=<?>f>SYZL=UX6C,y:), h , 69T=6C,y:), OfY;gXU jwg?>f]V:o>WToOf>fg YM\ OfYF/C69T=8 g?O?No>5, >fg :=LpU?Y gXUKJMU?>fg (n1;j1) 3 (n1;j) JMTE/ (n1;j +1) <?>fgXL >fOWUX6C4k>WRC7 huUX<KJMTogX69UX6PYZTog \q<?Y2,HJ 8k>WTo>W<X6POxL=<?>54 6PYZ:og gXUKJMU?> OfYZ:=R0/St >/=>.@oTo>1/ gX6C, 69R JM<XRC7ojdB+ >WU C k (n) <?>WL=<?>fg?>WTVU UXNo> % L_JMUXN OfY;gXU :=L U?YwToYF/=> n hu6 B >;B 3 >.-=OWR9:E/C69T=8 UXNo> YZ:=UXt YZ:=TE/ OfY;gXU?g \iYZ<sToY=/=> n j \qYZ<nJ g?OKNo>5, >yg?>f]‘:o>WToOf> S k Bp>wN_JJ4k>wUXN_JMUQF C k (n) = n X m=1 R m;iam D 2 m + l m (i am ) + NP(m 1;i a m1 ) + t m1 (i a m1 ;i am ): h ooBLKNIkj h No> JM<X<X6C4‘69T=8^L_JMUXN OfY;gXU?g \iYZ<IgXUKJMU?> (n;j) \iYZ<wg?>f]‘:o>WToOf>fg S a 3 S b 3 JMTE/ S c OQJMTmt > gK>f>WT 69T 57698 :=<?> ooBCHNH;B :‘69ToOf>wUXNo>wY 4k>W<XNo>QJ2/ OfY;gXUsU?Y <?>WUX<X6P>54k>wUXNo>ITo>fOf>fg?gKJM<>7%/oJMUKJ \q<?Y2,[UXNo> OfY‘>SOW6P>WT‘U?gs\q<?Y2,[RP>54k>WR j + 1 6Pg J2/)/=>1/ U?Y UXNo> 69ToOfY2, 69T=8mUX<KJMTogX69UX6PYZT 3 UXNo> RPY‘OQJMR L=<?Y‘Of>fg?g?69T=8 OfY;gXU JMTE/ UXNo>STV:), t >W< YM\ YZ:=UXt YZ:=TE/AL_JM<XUX6 JMRPg 3 P(n;j) 3 JMU gXUKJMU?> (n;j) rs69R9RxToYZU O?N_JMT=8k> 3 <?>W8VJM</CRP>fg?g YM\yUXNo> 69ToOfY2, 69T=8 L_JMUXN B 5 698;:=<?> ooBCHJK gXNoYQr g UXNo> 69TVt YZ:=TE/MYZ:=UXt YZ:=TE/ OfY;gXU?gS\iYZ< gXUKJMU?> (n;j) \iYZ<SUXNo> g?YZR9:=UX6PYZT <?>WL=<?>fg?>WT‘U?>1/ tF7 S a 3 S b JMTE/ S c Blh No>W<?>d\iYZ<?> 3 \iYZ< JMTB7pJM<X<X6C4 69T=8 L_JMUXN 3 YZ:=UXt YZ:=TE/ OfY;gXU?gyOQJMTpt > /=>fOfYZ:=L=RP>1/^\q<?Y2,[69TVt YZ:=TE/ OfY;gXU?gbB NoY‘Y;gX69T=8SJ t >fgXUsL_JMUXN JM<X<X6C4 69T=8 U?Y J gXL >fOW6n@_OwgXUKJMU?> /=Y >fg ToYZU >WR96C, 69T_JMU?> YZL=UX6C,SJMR L_JMUXNog 3 JMTE/ g?YSJ L_JMUXN UXN_JMU, 69T=6C, 69aQ>fgxUXNo> L_JMUXNP, >WUX<X6PO huRPYQr >fgXU OfY;g?U j OQJMT t >I\qYZ:=TE/ :ogX69T=8 J gXNoYZ<XU?>fgXU L_JMUXNmJMR98kYZ<X69UXN), B c Un>QJZO?N gXUKJM8k> n YM\7UXNo> /=>fOW6PgX6PYZT 3 UXNo>yJMR98kYZ<X69UXN), & o (n,j) NP(n 1;j + 1)D 2 n1 + t n1 (j + 1;j) R n;j D 2 n + l n (j) NP(n 1;j)d 2 n1 C b (n 1) C c (n 1) NP(n 1;j 1)D 2 n1 + t n1 (j 1;j) C a (n 1) 57698;:=<?> ooBCHNH;F c <X<X6C4‘69T=8 OfY;gXU?gs\qYZ< gXUKJMU?> huT 3 ?jdB @oTE/=gwUXNo> t >fgXUIUX<KJMTogX69UX6PYZTpOfY2, 69T=8 69T‘U?Y^>QJZO?N gXUKJMU?> JMUwUXN_JMU g?UKJM8k> tF7 OfY2, L=:=UX69T=8 UXNo> , >WUX<X6POfg YM\ JMR9R UXNo> L Y;g?g?69t=RP> L_JMUXNog^OfY2, 69T=8 69TVU?Y UXNo>mgXUKJMU?> 3 JMTE/ UXNo>WT g?>WRP>fOWUX69T=8 UXNo> L_JMUXN rs69UXN UXNo> , 69T=6C,y:), , >WUX<X6PO JZg UXNo>mgX:=<>4 6C4kYZ< L_JMUXN OfY2, 69T=8 69TVU?Y UXN_JMU^gXUKJMU?>;B c U UXNo> R JZg?U g?UKJM8k> 3 UXNo> gX:=<>4 6C4kYZ< L_JMUXNArs69UXN UXNo>G, 69T=6C,y:), L_JMUXN , >WUX<X6PO 6Pg g?>WRP>fOWU?>1/ JZg UXNo> YZL=UX6C, :), L_JMUXN B :‘69ToOf> >QJZO?N g?UKJMU?> /=>fg?OW<X69t >fg UXNo>wOfY=/C69T=8 g?OKNo>5, >I:og?>1/ JMUs>QJZO?N YM\7UXNo>xToY=/=>fg 3 UXNo>wYZL=UX6C, :), L_JMUXN JMRPg?Y L=<?YJ4 60/=>fg UXNo> YZL=UX6C,SJMRxgK>WRP>fOWUX6PYZT YM\ OfYF/C69T=8 g?O?No>5, >fgS\qYZ< >QJZO?N g?>WTog?YZ< 69T UXNo> To>WU r YZ<>d 3 gX:oOKN UXN_JMUwUXNo> >WTo>W<X8N7mOfYZTogX:), L=UX6PYZTp6Pg , 69T=6C, 69aQ>1/eB c , YZ<?>g/=>WUKJM69RP>1/ /=>fgKOW<X69L=UX6PYZT JMt YZ:=U/ 7 T_Jp, 6PO L=<?YZ8;<KJp,g, 69T=8 JMTE/ UXNo> gXNoYZ<XU?>fgXU L_JMUXN JMR98kYZ<X69UXN), OQJMT t > \qYZ:=TE/^69TLI B " & * " T UXNo> \qYZR9RPYbrs69T=8 gX6C, :=R JMUX6PYZTog 3 r > OfYZTogX60/=>W<?>1/pU r Y /C6 c >W<?>WT‘U gX6C, L=RP> To>WU r YZ<>dpOfYZT @o8;:=<KJMUX6PYZTogbB p> OfY2, L_JM<?>1/ YZ:=< YZL=UX6C, 69abJMUX6PYZT U?>fO?N=T=6P]‘:o> U?Y^g?YZR9:=UX6PYZTogyrsNo>W<?> UXNo> gKJp, > OfY=/C69T=8 g?OKNo>5, > 6Pg :og?>1/ \iYZ<xJMR9ReToY=/=>fgs69T UXNo>ITo>WU r YZ<>d B &ZD (n,j) (n,j) (n,j) NP(n; j)D 2 n + t n(j; j + 1) NP(n;j)D 2 n + t n(j;j + 1) NP(n;j)D 2 n + t n(j;j + 1) R n;jD 2 n + l n(j) NP(n;j)D 2 n + t n(j; j 1) NP(n;j)d 2 n Ca(n 1) R n;jD 2 n + l n(j) NP(n;j)D 2 n + t n(j;j 1) NP(n;j)d 2 n NP(n 1;j)d 2 n1 R n;jD 2 n + l n(j) NP(n;j)D 2 n + t n(j; j 1) NP(n;j)d 2 n NP(n 1; j + 1)D 2 n1 + t n1(j + 1;j) C b (n 1) Cc(n 1) NP(n 1;j 1)D 2 n1 + t n1(j 1;j) 57698;:=<?> ooBCHJKCF x:=UXt YZ:=TE/ OfY;gXU?g \iYZ< gXUKJMU?> huT 3 Kj \qYZ<s69ToOfY2, 69T=8 g?O?No>5, > gK>f]V:o>WToOf>fgyhiJkj S a 3 hutvj S b JMTE/ hqObj S c B &2‘ h N=6PgIr JZg /=YZTo> rs69UXNmUXNo> UXN=<?>f> J14;JM69R JMt=RP> OfY=/C69T=8 gKO?No>5, >fgQBh No> OfY=/C69T=8^g?O?No>5, >fgyOfYZTogX60/=>W<?>1/ JM<?> <KJfr hq]V:_JMT‘UX69aQ>1/_jA/oJMUKJpUX<KJMTog, 6Pg?gX6PYZT 3 r J14k>WRP>WU?g^rs69UXN YZTo> RP>54k>WR YM\e/=>fOfY2, L Y;gX69UX6PYZT 3 JMTE/ r JJ4k>WRP>WU?gnrs69UXN U r YSRP>54k>WRPg YM\/=>fOfY2, L Y;gX69UX6PYZTogbB sink n M Stage Stage Stage Stage n+1 n+2 states j-1 levels of decomposition j levels of decomposition j+1 levels of decomposition 57698;:=<?> ooBCHQ@CFbawJMUKJ UX<KJMTogX69UX6PYZTog \qYZ<sL_JM<XUX6 JMR7OfY‘>SOW6P>WT‘UxJML=L=<?YkJZO?N B c RPg?Y 3 rs69UXNoYZ:=U RPY;g?69T=8 8k>WTo>W<KJMR969U7 3 JMTE/ \qYZ< gX6C, L=R96n@_OQJMUX6PYZT L=:=<XL Y;gK>fg 3 r > OfYZTogXUX<KJM69To>1/ UXNo> TV:), t >W< YM\ L Y;g?gX69t=RP> UX<KJMTogX69UX6PYZTog 69T UXNo> 8;<KJML=N 3 JZg 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UXNo>S69T=69UX6 JMR JZgKgX:), L=UX6PYZT UXN_JMUnUXNo>57^OfYZ:=R0/ t > >SOW6P>WT‘UXRC7 >WToOfY=/=>1/ rs69UXNoYZ:=UnUXNo>yTo>f>1/ YM\ J RPY OQJMR a 89 B T^\u:=UX:=<?> gX6C, :=R JMUX6PYZTog 69TpUXN=6Pg g?>fOWUX6PYZT JMTE/ 69T : >fOWUX6PYZTog DCBEDCBLK JMTE/ DCBEDCBE@ r > rs69R9R OfYZTogX60/=>W< YZT=RC7 UXNo>S8;RPYZt_JMR ,y:=R9UX69L=RP> OfY >SOW6P>WTVUw>WToOfY=/C69T=8 JML=L=<?YkJZO?N B 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 x 10 7 10 20 30 40 50 60 70 80 Cost/Sensor SNR 1−level 5/3 Wavelet (Selective Local Coding) 2−level 5/3 Wavelet (Selective Local Coding) DPCM (Selective Local Coding) 1−level 5/3 Wavelet 2−level 5/3 Wavelet DPCM 57698;:=<?>yDCBEDCF G RPYZt_JMR 4CgQB g?>WRP>fOWUX6C4k>y>WToOfYF/C69T=8SYM\,y:=R9UX69L=RP> OfY‘>SOW6P>WT‘U?g 3 9 eHJMCB c ToYZUXNo>W< 6C, L YZ<XUKJMT‘USJZgXL >fOWU YM\nUXNo> L=<?YZL Y;g?>1/ , >W<X8;69T=8mg?UX<KJMU?>W8N7 69T UXNo>i@oT_JMRsOfY;gXU 6Pg UXNo> Jp, YZ:=TVU YM\ YJ4k>W<XNo>QJ2/ J2/)/=>1/eBh No> L >W< \iYZ<>,SJMToOf> YM\eUXNo>nJMR98kYZ<X69UXN),HJMRPYZT=8yrs69UXN UXNo>sToY >WToOfY=/C69T=8 gXUX<KJMU?>W8N7phu<KJfr /oJMUKJ UX<KJMTog>, 6Pg?gX6PYZTvj 6Pg gXNoYQrsT 69T 5 698;:=<?>fg DCBL‘ DCB (CB 57698;:=<?>fgxDCBL‘^DCB ( JMRPg?Y gXNoYbr UXN_JMU UXNo> : l gQB Y;gXUnOW:=<>4k>fgs\iYZ</C6 ce>W<?>WTVU JMR98kYZ<X69UXN), g JMTE/ U?YZL YZRPYZ8;6P>fg N_JJ4k>f/C6 ce>W<?>WTVUsgXRPYZL >fgQB&ax6 ce>W<?>WTVU gXRPYZL >fg OQJMT t >xJMUXUX<X69t=:=U?>1/ U?Y O?N_JMT=8k>fg 69TSUXNo>n<?YZ:=U 69T=8 g?UX<KJMU?>W8N7 3 rsN=6PO?N L=<?>fg?>WT‘UIJ UX<KJ2/=> YMc YZT UXNo> JMR98kYZ<X69UXN), g L >W< \qYZ<>,SJMToOf>y69T U?>W<>, gnYM\ >WTo>W<X8N7 Ipo 1.5 2 2.5 3 3.5 4 4.5 x 10 7 10 20 30 40 50 60 70 80 90 100 Cost/Sensor SNR Raw Data Transmission 1−level 5/3 Wavelet 2−level 5/3 Wavelet DPCM 57698;:=<?>yDCBL‘CF c R98kYZ<X69UXN), L >W< \qYZ<>,SJMToOf>fgn:ogX69T=8 8;RPYZt_JMR >WToOfY=/C69T=8SYM\,y:=R9UX69L=RP> OfY‘>SOW6P>WT‘U?g 3 9 eH;B OfYZTogX:), L=UX6PYZT B :‘UX<KJMU?>W8;6P>fg UXN_JMU JMR9RPYbr J N=698;No>W< TV:), t >W< YM\ , >W<X8k>fgxhqgX:oOKN JZg UXNo> : bhnj 69ToOW<?>QJZg?> UXNo>yY 4k>W<KJMR9R >WTo>W<X8N7 OfYZTogX:), L=UX6PYZT tB7^8k>WTo>W<KJMUX69T=8 J R JM<X8k>W< TV:), t >W<wYM\,y:=R9UX69L=RP> OfY >SOW6P>WTVU?g JMU UXNo> , >W<X8k> L YZ69TVU?gQB YZTF4k>W<?g?>WRC7 3 <?YZ:=UX69T=8^UX<?>f>fg UXN_JMU JMUXU?>5, L=U U?Y <?>1/C:oOf> UXNo> , >W<X8k> Y 4k>W<XNo>QJ2/ tB7^R96C, 69UX69T=8 UXNo> T‘:),yt >W<wYM\, >W<X8k> L YZ69TVU?gx69T UXNo> To>WU r YZ<>d 3 JMRPg?YSOfYZT‘UX<X69t=:=U?> U?Y JMT 69ToOW<?>QJZg?> 69T >WTo>W<X8N7 OfYZTogX:), L=UX6PYZT 3 tF7 69ToOW<?>QJZgX69T=8 UXNo> gX:), YM\ UXNo>^g?]V:_JM<?>1/ /C6PgXUKJMToOf>phiJMTE/ UXNo>W<?>d\iYZ<?> UXNo> UX<KJMTog>, 6Pg?g?6PYZT OfY;gXU?g j t >WU r >f>WT ToYF/=>fgbB aI>WL >WTE/C69T=8 YZT UXNo>^To>WU r YZ<>d U?YZL YZRPYZ8N7 3 UXNo>^Y 4k>W<XNo>QJ2/ OfY;gXU/C:o> U?Y UXNo> 69ToOW<?>QJZg?> 69T UXNo>/C6PgXUKJMToOf> , 698;NVUxg?:=<XL_JZg?g UXNo>w8VJM69T rs69UXN UXNo>w<?>1/C:oOWUX6PYZT 69T UXNo> TV:), t >W< YM\ , >W<X8k>fg 3 YZ<4 6POf> 4k>W<?gKJCB c RPgKY 3 , YZ<?> OfY2, L=RP>.- JMR98kYZ<X69UXN), g h K RP>54k>WR r J14k>WRP>WU UX<KJMTog \iYZ<>, jeU?>WTE/ U?YnN_JJ4k> g?R9698;NVUXRC7 RPYbr >W< gXRPYZL >fg UXN_JMT gX6C, L=RP>W< g?O?No>5, >fgsh a 89 j 3 69TE/C6POQJMUX69T=8 JnN=698;No>W< TV:), t >W< YM\ , :=R9UX69L=RP>wOfY‘>SOW6P>WT‘U?g 8k>WTo>W<KJMU?>1/ 3 JZg >.- L >fOWU?>1/eBp>nrs69R9R OfY2, L_JM<?>IUXNo>xL >W< \iYZ<>,SJMToOf> YM\ /C6 c >W<?>WT‘U <?YZ:=UX69T=8 gXUX<KJMU?>W8;6P>fgS69T: >fOWUX6PYZT DCBEDCBE@CB N=69RP> UXNo> JMtog?YZR9:=U?>^L >W< \qYZ<>,SJMToOf> YM\ UXNo> I;D 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 x 10 7 10 20 30 40 50 60 70 80 Cost/Sensor SNR Raw Data Transmission 1−level 5/3 Wavelet 2−level 5/3 Wavelet DPCM 57698;:=<?>yDCB & F c R98kYZ<X69UXN), L >W< \qYZ<>,SJMToOf>fgn:ogX69T=8 8;RPYZt_JMR >WToOfY=/C69T=8SYM\,y:=R9UX69L=RP> OfY‘>SOW6P>WT‘U?g 3 9 KCB JMR98kYZ<X69UXN), g JMU J UKJM<X8k>WUe:xl(,SJ17 O?N_JMT=8k>i/=>WL >WTE/C69T=8 YZT 6Pg?gX:o>fgwR96Cdk> UXNo> To>WU r YZ<>d U?YZL YZRPYZ8N7 3 /oJMUKJ OfYZ<X<?>WR JMUX6PYZT 3 >WU?O;B 3 JMR98kYZ<X69UXN), gIrs69UXN N=698;No>W< :xl gQB Y;gXUxg?RPYZL >fg OQJMT t > UXNoYZ:=8;NVUwYM\ JZg t >W69T=8i, YZ<?>y>SOW6P>WT‘Ux69T UXNo>yg?>WTogK>wUXN_JMUnUXNo>57 OQJMTmJZO?N=6P>54k> J t >WUXU?>W<x6C, L=<?YJ4k>5, >WT‘Uw69T U?>W<>, gxYM\ : l \iYZ<nJ g>,SJMR9RP>W<xJ2/)/C69UX6PYZT_JMR OfY;g?U 3 JZg OfY2, L_JM<?>1/ U?YSJMR98kYZ<X69UXN), gnrs69UXN RPYQr >W<xgXRPYZL >fgQB c gSJmgX:=8;8k>fgXU?>1/ JMR9U?>W<XT_JMUX6C4k> U?YpY 4k>W<?OfY2, > UXNo> >.-CUX<KJ L >WT_JMR9U 7 69ToOW:=<X<?>1/ tB7 UXNo>G, :=R9UX69L=RP> OfY >SOW6P>WTVU 8k>WTo>W<KJMUX6PYZT JMTE/m>WToOfY=/C69T=8 3 r > OQJMT OfYZTogX60/=>W<yJMTpJML=L=<?YkJZO?N rsNo>W<?> ToYk, :=R9UX69L=RP> OfY >SOW6P>WTVU?g r YZ:=R0/ t >y8k>WTo>W<KJMU?>1/ JMU UXNo>e, >W<X8k> L YZ69TVU?gQBh N=6PgwOfYZ:=R0/mt > JZO?N=6P>54k>1/ 3 \qYZ<w>.-oJp, L=RP> 3 tB7 69T‘U?>W<X<X:=L=UX69T=8 UXNo>x>WToOfY=/C69T=8 69T JyL_JMUXNSrsNo>WTo>54k>W< 69U , >W<X8k>fg U?Y JMToYZUXNo>W< L_JMUXN 3 JZg 69R9R9:ogXUX<KJ2/=>1/ 69T 5 698;:=<?>^DCBLI 3 JMTE/ gX6C, L=RC7 \qYZ<Xr JM</C69T=8 UXNo> OfY >SOW6P>WTVU?g \q<?Y2, UXN_JMU L YZ69TVU U?Y UXNo> g?69T)dprs69UXN YZ:=U \u:=<XUXNo>W< <?>.@oT=69T=8oBD:‘:oOKN JP, >WUXNoYF/ r YZ:=R0/ ToYZU gX:Cc >W< UXNo>SL=<?YZt=RP>5, YM\ N_JJ4‘69T=8 U?Y >WToOfY=/=> JMTE/ UX<KJMTog>, 69U ,y:=R9UX69L=RP>SOfY‘>SOW6P>WT‘U?g 3 t=:=Uw69UIr YZ:=R0/ 3 NoYbr >54k>W< 3 Of>W<XUKJM69T=RC7 69ToOW<?>QJZg?> UXNo> T‘:),yt >W< IN‘ 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 x 10 7 10 20 30 40 50 60 70 80 Cost/Sensor SNR Raw Data Transmission 1−level 5/3 Wavelet 2−level 5/3 Wavelet DPCM 57698;:=<?>xDCB (CF c R98kYZ<X69UXN), L >W< \qYZ<>,SJMToOf>fg :ogX69T=8 8;RPYZt_JMRv>WToOfY=/C69T=8 YM\ , :=R9UX69L=RP> OfY >SOW6P>WTVU?g 3 9 eHJMCB YM\ L_JM<XUX6 JMR OfY‘>SOW6P>WT‘U?g UXN_JMU r YZ:=R0/ ToYZU t > \u:=<XUXNo>W< <?>.@oTo>1/ JMTE/ UX<KJMTog>, 69UXU?>1/ JZgygX:oOKN U?Y UXNo> gX69T)d B N=69RP> \u:=<XUXNo>W< 69TB4k>fg?UX698VJMUX6PYZT U?Y ]V:_JMT‘UX69UKJMUX6C4k>WRC7 OfY2, L_JM<?> t YZUXN , >WUXNoYF/=g 6Pg To>fOf>fgKgKJM<>7 3 UXNo> UX<KJMTog>, 6Pg?gX6PYZT YM\7L_JM<XUX6 JMRPg \u<?Y2, ToY=/=>fg \iJM< \q<?Y2, UXNo> g?69T)dSr YZ:=R0/ JMRPgKY 69ToOW:=<s69T J gX698;T=6n@_OQJMT‘U J2/)/C69UX6PYZT_JMR OfY;gXU 3 JMTE/ UXN=6Pg JML=L=<?YkJZOKN r YZ:=R0/ L Y;g?gX69t=RC7 JMRPgKY g?:Cc >W< \q<?Y2,[<?YZ:=UX69T=8 gXUX<KJMU?>W8;6P>fg rs69UXN J R JM<X8k>wT‘:),yt >W<xYM\, >W<X8k>wL YZ69T‘U?gQB ! " ##" ! # 57698;:=<?>fgxDCBCHJM DCBCH5oS69R9R9:ogXUX<KJMU?>wUXNo>yL >W< \iYZ<>,SJMToOf>yYM\ UXNo> /C6 ce>W<?>WTVUIJMR98kYZ<X69UXN), gn69T U?>W<>, gxYM\/oJMUKJ <?>WL=<?>fg?>WT‘UKJMUX6PYZT /C6PgXU?YZ<XUX6PYZT h , >QJZgX:=<?>1/^tB7P:‘698;T_JMR U?Y YZ6Pg?> <KJMUX6PY 3 : l j JZgnJ \q:=ToOWUX6PYZT YM\ U?YZUKJMR >WTo>W<X8N7 OfYZTogX:), L=UX6PYZTp69T UXNo> To>WU r YZ<>d B Tp>QJZO?N;@o8;:=<?> r >l/=>WL=6POWU UXNo> <?YZ:=UX69T=8 U?YZ8k>WUXNo>W< rs69UXN UXNo>/oJMUKJ <?>WL=<?>fgK>WTVUKJMUX6PYZT JMR98kYZ<X69UXN), :og?>1/ \iYZ< >QJZO?N g?>WTog?YZ<sToY=/=>;B h No> >WTo>W<X8N7^OfYZTogX:), L=UX6PYZT 6Pg I’& Path2 Path1 sink Path3 Path1 Path2 Path3 sink (a) (b) 57698;:=<?>yDCBLICFnhiJkj 9 :=R9UX69L=RP> OfY >SOW6P>WTVUwJML=L=<?YkJZO?N Bxhutvj T‘U?>W<X<X:=L=U?>1/ L_JMUXNmJML=L=<?YkJZOKN B J14k>W<KJM8k>1/ YJ4k>W<, :=R9UX69L=RP> <?>QJMR969abJMUX6PYZTog YM\ UXNo>/oJMUKJCBh No> /oJMUKJ <?>WL=<?>fg?>WTVUKJMUX6PYZT 69U?g?>WR \ 6Pg 8k>WTo>W<KJMU?>1/ UKJpd 69T=8 JZg JMT 69T=L=:=UyUX<KJM69T=69T=8?/oJMUKJ^8k>WTo>W<KJMU?>1/ rs69UXN UXNo> c l , YF/=>WR 3 6 B >;B 3 r >k/=Y ToYZUy:ogK> UXNo> , Y=/=>WR_L_JM<KJp, >WU?>W<?g /C69<?>fOWUXRC7SJMTE/SYZ:=< JMR98kYZ<X69UXN),HOQJMT t >nJML=L=R96P>1/ 3 JZg gXNoYQrsT t >WRPYbr 3 U?Y YZL=UX6C, 69aQ> <?>WL=<?>fg?>WT‘UKJMUX6PYZT \qYZ< JMTB7 UX<KJM69T=69T=8 /oJMUKJxg?>WUQB xtog?>W<>4k> UXN_JMU 3 69T JMR9RCOQJZg?>fg 3 UXNo> YZL=UX6C, 69aQ>1/ /oJMUKJn<?>WL <?>fg?>WT‘UKJMUX6PYZT 3 JZg >.-CL >fOWU?>1/ 3 YZ:=UXL >W< \iYZ<>, g OfYZT @o8;:=<KJMUX6PYZTog rsNo>W<?> JIgX69T=8;RP> <?>WL=<?>fg?>WTVUKJMUX6PYZT 6Pg :og?>1/eB YZU?> 3 NoYbr >54k>W< 3 UXN_JMU 69T g?Y2, >nYM\eUXNo>nOQJZgK>fg UXNo> 8VJM69Tog L=<?YJ4 60/=>1/ tB7 UXNo>nYZL=UX6C,SJMR <?>WL=<?>fg?>WT‘UKJMUX6PYZT JM<?>, YF/=>fgXUQB h N=6Pg OQJMTmJMRPg?Y t >yg?>f>WT tB7 ToYZUX69T=8 UXN_JMU , Y;gXUnToY=/=>fg :og?>yYZTo> <?>WL=<?>fg?>WT‘UKJMUX6PYZT 3 g?Y UXNo>/C6 ce>W<?>WToOf>wrs69UXN <?>fgXL >fOWUsU?YSJZg?g?698;T=69T=8 UXN_JMUs<?>WL=<?>fg?>WTVUKJMUX6PYZT U?YSJMR9ReToY=/=>fgs6Pg g>,SJMR9R B T^5 698;:=<?>fg DCBCHJM DCBCHQ@ ToY=/=>fg r >W<?>I:=T=6 \iYZ<>, RC7k/C6Pg?UX<X69t=:=U?>1/eB TogXU?>QJ2/ 3 69T 5 698;:=<?>wDCBCH5o ToY=/=>fg r >W<?> OWR9:ogXU?>W<?>1/ U?Y gXL=R969UnUXNo>/oJMUKJ @_>WR0/ 69T U r YS<?>W8;6PYZTog g?>WL_JM<KJMU?>1/ tF7 J 8VJML 3 JMTE/^UXNo> To>WU r YZ<>d gX69T)d r JZg L=R JZOf>1/ 69T UXNo> <?>W8;6PYZT rsNo>W<?> UXNo> ToY=/=> /=>WTogX69U 7yr JZg RPYQr >W<QB xtog?>W<>4k> UXN_JMU7UXNo> t >WN_J14 6PYZ< 69T 57698;:=<?> DCBCH5o 6Pg 69TVUX:=69UX6C4k>WRC7 <?>QJZg?YZT_JMt=RP>;Fy69T UXNo> OWR9:ogXU?>W< YM\sTo>QJM<XtF7 ToYF/=>fg YZTpUXNo> U?YZL <X698;N‘U <?>W8;6PYZT YM\nUXNo> g?>WTog?YZ<@_>WR0/ J K RP>54k>WR r J14k>WRP>WU 6Pg OKNoY;g?>WT g?Y JZg U?Y , 69T=6C, 69aQ> UXNo> YJ4k>W<KJMR9R <KJMU?> <?>f]‘:=69<?>1/eBh N=6Pgn6Pgnt >fOQJM:ogK> UXN=6Pg g>,SJMR9R OWR9:ogXU?>W<x6Pgs\ JM< \q<?Y2, UXNo> gX69T)d JMTE/ UXN‘:og JMTF7 >.-COf>fg?gn<KJMU?> I( 0 100 200 300 400 500 600 0 100 200 300 400 500 600 1.5 2 2.5 3 3.5 4 4.5 5 x 10 7 10 20 30 40 50 60 70 80 90 Cost/Sensor SNR Optimum Configuration Raw Data Transmission 1−level 5/3 Wavelet 2−level 5/3 Wavelet DPCM 57698;:=<?>yDCBCHJMCF c R98kYZ<X69UXN), L >W< \qYZ<>,SJMToOf> \qYZ<s<KJMTE/=Y2, To>WU r YZ<>d^rs69UXN ToY , >W<X8k>y<?YZ:=UX69T=8 50 100 150 200 250 300 350 400 450 500 550 600 0 100 200 300 400 500 600 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 x 10 7 10 20 30 40 50 60 70 80 90 Cost/Sensor SNR Optimum Configuration Raw Data Transmission 1−level 5/3 Wavelet 2−level 5/3 Wavelet DPCM 57698;:=<?>yDCBCHNH;F c R98kYZ<X69UXN), L >W< \iYZ<>,SJMToOf>w\iYZ<s<KJMTE/=Y2, To>WU r YZ<>d^rs69UXN gXNoYZ<XU?>fg?U L_JMUXN <?YZ:=UX69T=8 69T UXNo>S<?>WL=<?>fg?>WT‘UKJMUX6PYZT h /C:o>SU?Y :ogX69T=8 J RP>fg?g >SOW6P>WTVU >WToOfY=/C69T=8‘jyr YZ:=R0/ <?>fgX:=R9Uy69T gX698;T=6n@_OQJMT‘U >WTo>W<X8N7 <?>f]V:=69<?>1/^\qYZ<sUX<KJMTog>, 6PgKgX6PYZT YM\/oJMUKJ U?Y UXNo> gX69T)d B 57698;:=<?>fg DCBCHQDyJMTE/ DCBCHJ‘y69R9R9:ogXUX<KJMU?>nUXNo> L >W< \qYZ<>,SJMToOf> YM\ YZ:=< , >WUXNoY=/=g rs69UXN <?>QJMR/oJMUKJCB U OQJMT t > g?>f>WT UXN_JMU rsNo>WT ToYj, >W<X8k> 6Pg JMR9RPYQr >1/ hq57698;:=<?> DCBCHQDkj 3 UXNo> YZL=UX6C,y:), JMR98kYZ<X69UXN), L >W< \iYZ<>, g OfYZTogX60/=>W<KJMt=RC7 t >WUXU?>W< UXN_JMT UXNo> YZUXNo>W< gKO?No>5, >fgQB;h N=6Pg OfYZT @o<>, g YZ:=< 69T=69UX6 JMR JZg?gX:), L=UX6PYZT UXN_JMU INI 0 100 200 300 400 500 600 0 100 200 300 400 500 600 0.5 1 1.5 2 2.5 3 3.5 4 4.5 x 10 7 0 10 20 30 40 50 60 70 Cost/Sensor SNR Optimum Configuration Raw Data Transmission 1−level 5/3 Wavelet 2−level 5/3 Wavelet DPCM 5 698;:=<?> DCBCHJKCF c R98kYZ<X69UXN), L >W< \iYZ<>,SJMToOf>w\iYZ< g?]‘:_JM<?>I8;<X60/^To>WU r YZ<>d^rs69UXN ToY , >W<X8k> <?YZ:=UX69T=8oB 0 100 200 300 400 500 600 0 100 200 300 400 500 600 0.5 1 1.5 2 2.5 3 3.5 x 10 7 0 10 20 30 40 50 60 70 Cost/Sensor SNR Optimum Configuration Raw Data Transmission 1−level 5/3 Wavelet 2−level 5/3 Wavelet DPCM 57698;:=<?>yDCBCHQ@CF c R98kYZ<X69UXN), L >W< \iYZ<>,SJMToOf>I\iYZ< g?]V:_JM<?>I8;<X60/ To>WU r YZ<>d^rs69UXN gXNoYZ<XU?>fgXU L_JMUXN <?YZ:=UX69T=8oB , >W<X8;69T=8pL_JMUXNog U?>WTE/ U?Ymt >^OfY;gXUXRC7kB YQr >54k>W< 3 JZg rs69R9R t > g?>f>WT 3 gXNoYZ<XU?>fgXU L_JMUXNA<?YZ:=UX69T=8 gXUX69R9R L=<?YJ4 60/=>fgnY 4k>W<KJMR9R t >WUXU?>W< L >W< \qYZ<>,SJMToOf>;B HJMNM 0 100 200 300 400 500 600 0 100 200 300 400 500 600 4 5 6 7 8 9 10 11 12 13 14 x 10 7 20 30 40 50 60 70 80 90 100 Cost/Sensor SNR Optimum Configuration Raw Data Transmission 1−level 5/3 Wavelet 2−level 5/3 Wavelet DPCM 57698;:=<?>yDCBCH5ooF c R98kYZ<X69UXN), L >W< \iYZ<>,SJMToOf>I\qYZ< OWR9:og?U?>W<?>1/^To>WU r YZ<>d^rs69UXN gXNoYZ<XU?>fgXU L_JMUXN <?YZ:=UX69T=8oB 480 500 520 540 560 580 600 240 250 260 270 280 290 300 310 320 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 x 10 5 0 10 20 30 40 50 60 70 Cost/Sensor SNR Optimum Configuration Raw Data Transmission 1−level 5/3 Wavelet 2−level 5/3 Wavelet DPCM 57698;:=<?>yDCBCHQDCF c R98kYZ<X69UXN), L >W< \iYZ<>,SJMToOf>I\iYZ<s<?>QJMR To>WU r YZ<>dP/oJMUKJ rs69UXN ToY , >W<X8k>w<?YZ:=UX69T=8oB !" # #" ! ! ! " ! !#" YQrAUXN_JMU r >nN_J14k>n>fg?UKJMt=R96PgXNo>1/ UXN_JMU8/oJMUKJw<?>WL=<?>fg?>WT‘UKJMUX6PYZT YZL=UX6C, 69abJMUX6PYZT RP>QJ2/=g U?YwRPYQr >W< Y 4k>W<KJMR9R OfY;gXUy\qYZ< J^8;6C4k>WT To>WU r YZ<>d U?YZL YZRPYZ8N7 3 r > OfY2, L_JM<?>A/C6 ce>W<?>WTVU To>WU r YZ<>d U?YZL YZRPYZ8;6P>fgy69T U?>W<>, g YM\ UXNo>W69<yY 4k>W<KJMR9R OfY;gXUyrs69UXN YZL=UX6C, 69aQ>1/p<?>WL=<?>fg?>WT‘UKJMUX6PYZT hqgK>f> 57698;:=<?>fg DCBCH & mDCBCHJIkjdB T L_JM<XUX6POW:=R JM< r >SOfY2, L_JM<?> UXNo> L >W< \iYZ<>,SJMToOf> YM\ ToY , >W<X8k>SJMTE/pgXNoYZ<XU?>fgXU L_JMUXN <?YZ:=UX69T=8oB c gIrs69R9R t > g?>f>WT 69T HJM H 480 500 520 540 560 580 600 240 250 260 270 280 290 300 310 320 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 x 10 5 −10 0 10 20 30 40 50 60 Cost/Sensor SNR Optimum Configuration Raw Data Transmission 1−level 5/3 Wavelet 2−level 5/3 Wavelet DPCM 57698;:=<?>yDCBCHJ‘CF c R98kYZ<X69UXN), L >W< \iYZ<>,SJMToOf>I\qYZ<s<?>QJMR To>WU r YZ<>dP/oJMUKJ rs69UXN gXNoYZ<XU?>fgXU L_JMUXN <?YZ:=UX69T=8oB UXN=6PgSgK>fOWUX6PYZT 3 UXNo> t_JZgX6PO^69T‘UX:=69UX6PYZTA6Pg UXN_JMU rsN=69RP>^UXNo> T‘:),yt >W< YM\ , >W<X8k>fgi/=Yp6C, L=RC7 69TAg?Y2, > YJ4k>W<XNo>QJ2/ OfY;gXU^h /C:o> U?Y UXNo> >WToOfYF/C69T=8mYM\,y:=R9UX69L=RP> OfY >SOW6P>WTVU?g j 3 U?YZL YZRPYZ8;6P>fg UXN_JMU <?>1/C:oOf> UXNo> TV:), t >W< YM\, >W<X8k>fg 69ToOW<?>QJZgK> UXNo>sJJ4k>W<KJM8k> 69T‘U?>W<XToYF/=> /C6Pg?UKJMToOf> 3 JMRPg?Y 69ToOW<?>QJZgX69T=8IUXNo> UX<KJMTog>, 6PgKgX6PYZT OfY;gXU t >WU r >f>WT^ToY=/=>fgsJMTE/ 3 UXNo>W<?>d\iYZ<?> rsNo>WUXNo>W< N_J14 69T=8 \i>Wr >W< , >W<X8k>fg 6Pg t >WUXU?>W<8/=>WL >WTE/=g YZT NoYbr ,y:oOKN^UXNo>x>.-CUX<KJ/C6PgXUKJMToOf> 6Pg 3 JMTE/ rsN_JMU UXNo> OfY;gXU YM\ UXNo>, >W<X8k>fg r JZg \iYZ< >QJZO?N^gXL >fOW6n@_OnTo>WU r YZ<>d U?YZL YZRPYZ8N7kB 5oYZ< t YZUXN g?6C,y:=R JMU?>1/ <KJMTE/=Y2, To>WU r YZ<>d /oJMUKJ rs69UXN :=T=6 \iYZ<>, gKJp, L=R969T=8 hq5 698;:=<?>^DCBCH &;j JMTE/ <?>QJMR /oJMUKJ hq57698;:=<?> DCBCH (kj 3 r >mOQJMT gK>f> UXN_JMU gXNoYZ<XU?>fgXU L_JMUXN <?YZ:=UX69T=8 RP>QJ2/=g U?Y t >WUXU?>W< Y 4k>W<KJMR9R L >W< \qYZ<>,SJMToOf>;B75oYZ< UXNo> /=>WTog?>W< To>WU r YZ<>dShqgX6C,y:=R JMU?>1//oJMUKJkjeY 4k>W<KJMR9RVL >W< \qYZ<>,SJMToOf> 6Pg 4k>W<>7wg?>WTogX69UX6C4k> U?YIUXNo>sJ14k>W<KJM8k>sRP>WT=8;UXN YM\vUXNo> gX69T=8;RP> <?YZ:=U?>sL_JMUXNog 69T UXNo> To>WU r YZ<>d 3 g?Y UXN_JMU UXNo> 8VJM69Tog \q<?Y2, :ogX69T=8 gXNoYZ<XU?>fgXU L_JMUXNm<?YZ:=UX69T=8^JM<?> g?:=togXUKJMTVUX6 JMR B T UXN=6PgxOQJZgK> 3 \iYZ<xUXNo> gXNoYZ<XU?>fgXU L_JMUXN <?YZ:=UX69T=8 3 KNM YM\ UXNo> ToY=/=>fg 69T UXNo> To>WU r YZ<>d r >W<?>%, >W<X8k>^ToY=/=>fg 3 rsN=69RP> UXNo> ToY , >W<X8k>^<?YZ:=UX69T=8 69ToOW<?>QJZg?>1/ UXNo> J14k>W<KJM8k>/C6PgXUKJMToOf>wt >WU r >f>WT ToY=/=>fgs69T UXNo>ITo>WU r YZ<>d^tB7P‘;@ B HJMNK 1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 x 10 7 20 30 40 50 60 70 80 Optimum Algorithm and Raw Data Transmission Performance Cost SNR Optimum Algorithm, no merges Optimum Algorithm, Shortest−Path Raw Data, Shortest−Path 57698;:=<?> DCBCH & F xL=UX6C, :), JMR98kYZ<X69UXN), L >W< \qYZ<>,SJMToOf> OfY2, L_JM<?>1/AU?Y <KJfr /oJMUKJpUX<KJMTog, 6Pg?gX6PYZT \iYZ< ToY , >W<X8k> JMTE/^g?NoYZ<XU?>fgXU L_JMUXN <?YZ:=UX69T=8 \iYZ<s<KJMTE/=Y2, To>WU r YZ<>d T OfYZTVUX<KJZgXU 3 \iYZ< UXNo> g>,SJMR9RP>W< <?>QJMR/oJMUKJ To>WU r YZ<>d hq57698;:=<?> DCBCH (kj 3 L >W< \iYZ<>,SJMToOf> 6Pg ToYZU JZg gX698;T=6n@_OQJMT‘UXRC7 J ce>fOWU?>1/ tB7 UXNo> <?YZ:=UX69T=8 3 /C:o> U?Y UXNo> <?>1/C:oOf>1/ TV:),yt >W< YM\xToY=/=>fgQB p> t >WR96P>54k> UXN_JMU 69T UXNo>xOQJZgK>nYM\ 57698;:=<?> DCBCH & 3 8;6C4k>WTSUXNo> /=>WTogX69U7 YM\ UXNo>nTo>WU r YZ<>d UXNo>xOfY;g?U YM\ , >W<X8;69T=8 6Pg ToYZU JZg gX698;T=6n@_OQJMTVU hiJZgIJ <?>fgX:=R9U YM\ UXNo> L=<?YZL Y;g?>1/Da 89 OfY=/C69T=8 YM\b, >W<X8k>1/ L_JMUXNog jxJMTE/ UXNo>W<?>d\iYZ<?> UXNo> L >WT_JMR9U7;/C:o> U?Y JMR9RPYQrs69T=8 RPYZT=8k>W< L_JMUXNog hiJZgw69T UXNo> ToY , >W<X8k>SOfYZT @o8;:=<KJMUX6PYZTvj 6Pg U?Y Y N=698;N U?Y t > OfY2, L >WTogKJMU?>1/mtF7 UXNo> <?>1/C:oOWUX6PYZT YM\8, >W<X8k> OfY;gXU?gQB UI6Pg JMRPg?Y 69TVU?>W<?>fg?UX69T=8 U?Y 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Asset Metadata

Creator Ciancio, Alexandre Gomes (author)

Core Title Distributed wavelet compression algorithms for wireless sensor networks

School Andrew and Erna Viterbi School of Engineering

Degree Doctor of Philosophy

Degree Program Electrical Engineering

Publication Date 11/05/2006

Defense Date 07/24/2006

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

Tag distributed compression,distributed wavelets,OAI-PMH Harvest,wireless sensor networks

Language English

Advisor Ortega, Antonio (committee chair), Govindan, Ramesh (committee member), Krishnamachari, Bhaskar (committee member)

Creator Email ciancio@usc.edu

Permanent Link (DOI) https://doi.org/10.25549/usctheses-m122

Unique identifier UC1325127

Identifier etd-Ciancio-20061105 (filename),usctheses-m40 (legacy collection record id),usctheses-c127-30241 (legacy record id),usctheses-m122 (legacy record id)

Legacy Identifier etd-Ciancio-20061105.pdf

Dmrecord 30241

Document Type Dissertation

Rights Ciancio, Alexandre Gomes

Type texts

Source University of Southern California (contributing entity), University of Southern California Dissertations and Theses (collection)

Repository Name Libraries, University of Southern California

Repository Location Los Angeles, California

Repository Email uscdl@usc.edu

Abstract (if available)

Abstract We address the problem of compression for wireless sensor networks from a signal processing point of view. Each of the sensors has limited power, and acquires data that should be sent to a central node. The final goal is to have a reconstructed version of the sampled field at the central node, with the sensors spending as little energy as possible.