EEO 401 Digital Signal Procssing Prof. Mark Fowlr Dtails of th ot St #19 Rading Assignmnt: Sct. 7.1.2, 7.1.3, & 7.2 of Proakis & Manolakis
Dfinition of th So Givn signal data points x[n] for n = 0,, -1 Comput points using: X [ k] 1 n0 x[ n] j2kn / k 0,1,2,..., 1 k 2 k Invrs (I) So Givn points X[k] for k = 0,, -1 Comput signal data points using: x[ n] 1 1 n0 X[ k] j2kn/ n0,1,2,..., 1 2/24
as a Matrix Oprator (Linar Transformation) X [ k] 1 n0 x[ n] j2kn / k 0,1,2,..., 1 1 j2 0 n/ [0] [ ] [0] [1] [ 2] [ 1] X x n x x x x n0 1 X[1] x[ n] x[0] x[1] x[ 2] n0 1 n0 j21 n / j21/ j21( 1)/ X[ 1] x[ n] x[0] x[1] x[ 2] j2 ( 1) n/ j21( 1)/ j2 ( 1)( 1)/ X[0] 1 1 1 1 x[0] j2 11/ j2 2 1/ j2 ( 1) 1/ X[1] 1 x[1] j21( 1)/ j22( 1)/ j2 ( 1)( 1)/ X[ 1] 1 x[ 1] X x Matrix 3/24
It is common to us th W symbol whn discussing th : W j2 / th root of unity : 1 kn Xk [ ] xnw [ ] k0,1,2,..., 1 n0 I: 1 1 kn xn [ ] X[k] W k0 n0,1,2,..., 1 Matrix 1 1 1 1 1 2 1 1 W W W 2 4 2( 1) W 1 W W W 1 2( 1) ( 1)( 1) 1 W W W Easy to Invrt! : X W x I: 1 x W X W X 1 * 4/24
Proprtis of You v larnd th proprtis for CTFT and for DTFT (.g., dlay proprty, modulation proprty, convolution proprty, tc.) and sn that thy ar vry similar (xcpt having to account for th DTFT s priodicity) Sinc th is linkd to th DTFT you d also xpct th proprtis of th to b similar to thos of th DTFT. That is only partially tru! Linarity: x [ n] X [ k] 1 1 x [ n] X [ k] 2 2 ax[ n] ax[ n] ax[ k] ax [ k] 1 1 2 2 1 1 2 2 5/24
Priodicity: 1. points computd using th formula ar priodic Xk [ ] Xk [ ] k 2. Signal sampls computd using th I formula ar priodic xn [ ] xn [ ] n Proof of #2: 1 1 xn [ ] X[ k ] k0 j2 ( n) k/ Proof of #1 is virtually idntical 1 1 k0 Xk [ ] j2 n/ j2 kn/ = 1 xn [ ] #1 is not surprising it coms from th priodicity of th DTFT ot that #2 says th priodicity is for th sampls AFTER doing an I!!! This has a big impact on othr proprtis such as convolution & dlay proprtis!! 6/24
ot that #2 says th priodicity is for th sampls AFTER doing an I!!! x[n] x [n] Extract n RX[k] 30 20 10 0 1 x [ n] X[ k] k0 j2 kn/ n n ImX[k] -10 0 1 2 3 4 5 6 7 n 10 5 0-5 -10 0 1 2 3 4 5 6 7 n Ths ar not rally thr but thy ar mathmatically thr!!! 7/24
This priodic vstig of th signal that ariss in th contxt of th can b capturd this way: Lt this b th L-point signal (sgmnt) with L : xn [ ] Dfin a -priod priodic signal by x [ n] x[ nl] p l = 0 n < 0, n > L -1 Thn th priodicity proprty can b xprssd as: x [ n] I x[ n] p -pt of signal with L points implis zro-padding out to points 8/24
Circular Shift (Dfind) Math otation for Circular Shift x[[ nk]] x[ nk,mod ] n n mod 4-3 1-2 2-1 3 0 0 1 1 2 2 3 3 4 0 5 1 6 2 7 3 8 0 9/24
Circular Shift Proprty of th Rcall th shift proprty of th DTFT (which is virtually th sam as for th CTFT): f jl f yn [ ] xn [ l] Y ( ) X ( ) Rgular tim shift this is for DTFT Imparts additional linar phas trm of intgr slop For w hav a similar proprty but it involvs circular shift rathr than rgular shift!! d kl d j2 kl/ d yn [ ] x[[ nl]] Y [ k] W X [ k] X [ k] This is a dirct rsult of #2 of Priodicity Discrt frquncis @ 2πk/ What this says is: 1. If you circularly shift a signal thn th corrsponding has a linar phas trm addd or altrnativly 2. If you impart a linar phas shift of intgr slop to th, thn th corrsponding I will hav a circular shift impartd to it. #2 is th most common scnario that ariss 10/24
Proof: Th obvious way to s this is to us th priodic xtnsion viw of th I: d x [ ] [ ] p n I X k 1 I X [ k] X [ k] 1 j2 kl/ d j2 kl/ d j2 kn/ 1 k 0 1 j2 k( nl)/ d X k k 0 [ ] x [ nl] x[[ nl]] p I valuatd at n l i.. Shiftd vrsion of priodic xtnsion 11/24
Mor stp-by-stp proof as in th book: 1 x[[ n l]] x[[ n l]] n0 j2 kn/ Whn n l w don t nd th mod opration! Explicit form for mod Combin into singl sum l1 1 j2 kn/ j2 kn/ x[[ n l]] x[ n l] n0 nl x[ mn] l1 1 j2 kn/ j2 kn/ x [ ln ] xn [ l ] n0 nl 1 1l j2 k( ml)/ j2 k( ml)/ xm [ ] xm [ ] ml m0 1 j2 kl/ j2 km/ j2 kl/ d x[ m] X [ k] m0 Chang of variabls in ach sum Split from xp and pull out Final Rsult! 12/24
Circular Modulation Proprty of th yn W xn xn Y k X k m X k m nm j2 nm/ d d d [ ] [ ] [ ] [ ] [[ ]] [( )mod ] What this says is: 1. If you modulat th signal (sgmnt) by a frquncy qual to on of th discrt frquncis, thn th corrsponding will hav a circular shift impartd to it. 2. If you circularly shift a thn its I will hav a modulation impartd at a frquncy qual to a discrt frquncy. Th cyclic natur hr is th sam as for th DTFT du to th fact that th DTFT is priodic with priod of 2π. Whr this diffrs from th DTFT vrsion is that th modulation frquncy must b on of th discrt frquncis of th. 13/24
Circular Convolution Proprty of th For CTFT and DTFT w had th most important proprty of all th convolution proprty: convolution in tim domain givs multiplication in frquncy domain For th this proprty gts changd du to th circular proprtis of & I. Latr w ll s th ramifications of this. d x1[ n] X1[ k] d d x1[ n] x2[ n] X1[ k] X2[ k] d x2[ n] X2[ k] My symbol for circular convolution of two lngth signal sgmnts book uss a diffrnt symbol that I could not mak! Circular convolution itslf is not rally somthing w want rathr w nd up hr bcaus w ask this qustion: Givn that multiplying 2 DTFTs corrsponds to tim-domain convolution Dos th sam thing hold for multiplying two s??? Th answr is: sort of but it givs circular convolution. And sinc LTI systms do rgular convolution this rsult at first sms not that usful. 14/24
Proof: Th priodic xtnsion viw of th I provids som insight that this proprty likly holds but w nd to prov it! You ll notic that th proof follows th lin of th qustion w just askd: What happns in th tim-domain whn I multiply two s??? For m = 1, 2 1 d j2 nk/ m m n0 X [ k] x [ n], k 0,1,2,..., 1 X [ k] X [ k] X [ k], k 0,1,2,..., 1 d d d 3 1 2 1 1 x n X k X k X k 1 1 d j2 nk / d d j2 nk / 3[ ] 3[ ] 1[ ] 2[ ] k0 k0 VERY important to sub in w/ diffrnt summation dummy variabls!!! 1 1 1 1 j2 mk / j2 lk / j2 nk / x1[ m] x2[ l] k0 m0 l0 1 1 1 1 x [ m] x [ l] 1 2 m0 l0 k0 j2 k( nml)/ d to valuat! 15/24
Asid: 1 k0 j2 kn/ n [[ ]], n k ( k is intgr) 0, othrwis Th rsult is obvious for n = k sinc for that cas w ar summing 1s. For othrwis this can b stablishd via th gomtric summation rsult: 1 1 j 1 2 / 2 / k j kn j n 1 k0 k0 2 n/ j2 n/ j2 n/ j2 n/ j2 n/ 1 1 1 0 1 1 This can also b sn graphically: For n = k Im R For n k Im R Lik vnly spacd forcs that cancl out! (Ys holds for odd too!) 16/24
So picking up whr w lft off: 1 1 1 1 x [ n] x [ m] x [ l] 3 1 2 m0 l0 k0 j2 k( nml)/ [[ nml]] 1 1 x1[ m] x2[ l] [[ nml]] m0 l0 Sifting Proprty but w/ mod natur! 1 x [ n] x [ m] x [[ nm]] x [ n] x [ n] 3 1 2 1 2 m0 Hr w finish th proof & dfin th trm circular convolution Flip & Shift but with Mod!!! d to undrstand Circular Tim Rvrsal to s how this works! 17/24
Circular Rvrsal: Rvrs about 0 on th circl x[[ n]] x[ n], 0 n 1 Ys 0!! = 8 x[n] x[ n] x[ n] x[0] x[0] x[1] x[7] x[2] x[6] x[3] x[5] x[4] x[4] x[5] x[3] x[6] x[2] x[7] x[1] Forward Rvrs x[[-n]] x[[ n]] n n 18/24
Circular Convolution Exampl: 1 x [ n] x [ m] x [[ nm]] 3 1 2 m0 Flippd Product Squnc for n = 0 x 3 [0] = sum of ths = 2 + 4 + 6 + 2 = 14 19/24
Circular Convolution Ex. (p. 2): 1 x [ n] x [ m] x [[ nm]] 3 1 2 m0 Flippd for n = 1 Etc. S txtbook for th rst of th xampl Product Squnc for n = 1 x 3 [1] = sum of ths = 4 + 1 + 8 + 3 = 16 20/24
Circular Convolution Ex. (p. 3): Altrnat viw using priodizd signals Original Signals: n = 0 Output Sampl: 1. Flip priodizd vrsion around this point 2. o shift ndd to gt n = 0 Output Valu 3. Sum ovr on cycl 21/24
Circular Convolution Ex. (p. 4): n = 1 Output Sampl: Shift by 1 & Sum ovr on cycl n = 2 Output Sampl: Shift by 2 & Sum ovr on cycl n = 3 Output Sampl: Shift by 3 & Sum ovr on cycl 22/24
of Product of Two Signals d x1[ n] X1[ k] 1 d d x1[ nx ] 2[ n] X1[ k] X2[ k] d x2[ n] X2[ k] This is th dual of th Convolution Proprty of s so th proof is vry similar. Parsval s Thorm for d 1[ ] 1[ ] 1 1 * 1 d d 1 2 1 2 d n0 n0 2[ ] 2[ ] x n X k x n X k x [ nx ] [ n] X [ kx ] [ k] * Spcial Cas: 1 x[ n] X [ k] 1 1 2 d n0 k0 2 23/24
of Complx-Conjugat * * d * d d [ ] [ ] [ ] [[ ]] [ ] x n X k x n X k X k I of Complx-Conjugat Tak conjugat hr * d * * d [ ] [ ] [[ ]] [ ] [ ] x n X k x n x n X k Tak conjugat hr 24/24