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# Dig Sig Proc&Analys EECS 451

UM

GPA 3.8

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This 3 page Class Notes was uploaded by Ophelia Ritchie on Thursday October 29, 2015. The Class Notes belongs to EECS 451 at University of Michigan taught by Andrew Yagle in Fall. Since its upload, it has received 11 views. For similar materials see /class/231528/eecs-451-university-of-michigan in Engineering Computer Science at University of Michigan.

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Date Created: 10/29/15

EECS 451 THE DISCRETE TIME FOURIER SERIES DTFS Cont Time Fourier Series Note Let t t T be periodic with periodT in continuous time Then t can be expanded in the continuous time Fourier series 205 X0 XlejQT X26j477rt X4642 X4644 where Xk ftquotT tej2ktdt for integers k and any time to Conjugate symmetry t real ltgt XLC X for integers k Discrete Time Fourier Series Note Let n N be periodic with periodN in discrete time Then can be expanded in the discrete time Fourier series X0 Xlej2T Xgej4T XN71ej NINI 27v where Xk 2201 ne j2W kquotDFT for k 0 N I Conjugate symmetry real ltgt X N4c X I for integers k EX DTFS DTFS DTFS DTFS Note Note Then Or since 12767 47 67Q7 6747 67 12767 47 6 Periodic period N 4 X0 a0 II Ix2 la3 12646 7 X1 fl0l jl1l 1l2ljl3l f12 6j 46j 2 X2 ash 1m11m2 1m312 64 6 1 X3 gaspja1 1a2 jx3 126j 4 6j 2 X3 X471 Xf 2 2 although both X3 and X1 are real here is a real and even function ltgt X k is a real and even function I 7 263quot 16 2ej37 complex exponential form n 7 4cosn I cos7rn trigonometric form 634 673quot try it and ejm cos7rn 1quot mn nan Power Parseval So Time domain Average powerl22 62 42 62 58 Freq domain Average power72 22 I2 22 58 Compute average power in either time domain or frequency domain So and Then Consider gt LTI gt yn where input I27 67 476 Linear TimeInvariant LTI system is ynf3yn I 3n3an I Then and Then and Frequency response functionHej Huh stay tuned Hej0 31 0 Hej 2 3131 134167j03946 HM 33 7 g Hej3 2 3 1341ej046 yn 07 1341w 0 m2ejm2 315W 1341ej0462ejn3w2 yn 70 41341 cosgn 046 cos7rn which becomes yn 5366 cos n 046 15 cos7rn Note DC term ltered out EXAMPLES OF DTFS PROPERTIES Given DTFS 4 is a discrete time signal with period N n N for all 73 N71 3927rnk N 7 1 N71 7 27rnk N k0 Xkej where Xk i N 2710 ne 3 X0 2201 nDC valuemean value of periodic signal Negative frequencies are second half of Xk Use XEC XNEk llatlab7s fftshift shifts DC to the center from the left end of plot This makes conjugate symmetry XEC XNk X easier to see Matlab fft XN N computes DTFS coef cients if X is one period EX 1 DTFSl 00 0 00 0 0 1 l l l l l l l Impulse in time EX 2 DTFSQ 0 10 00 0 0 51 j 1j 1 j 1j Delayed 6m EX 3 DTFSl l l l l l l 1 1 0 00 0 00 0 Constant in time EX 4 Parseval DTFS11211111 9 j 1j1 j 1j ls linear 181222121212121212 3292 j2 12Ij212 j2 12Ij2 Average power EX 5 Note DTFSces27rn 0 yam M e j96k N M This only works for periodic discretetime sinusoids we 27T EX gt DTFS81216 153 2339 3 3 2339 1 period of and Xk mm 15ej0 3 2jej7 2 03pm 3 2jejlt37r2gtn 1 Huh DTFS16128 153 2339 3 3 2339 Reversal n gt X an 2481216812162481216 gt a n 2416 128 161282416 128 Huh 73392ka DTFSQ 1624 8 15 3 2j3 2j 3 Delay Xn D gt Xke N n 2 12 16 248Q 16 24 8 12 16 248 Huh DTFS 8 12 16 33 2739 15 3 2739 acne N j27rnF gt XkeF 77llodulate77 signal means shift its spectrum by some frequency F Huh DTFS080120160 1532j33 2j1532j33 2j lnterpolate with zeros gtrepeat and halve DFT of lower order Huh DTFS 8 12 16 24 8 12 16 15 0 3 2739 0 3 0 3 270 Repeat in time gtinterpolate with zeros in frequency domain CONCEPTS BEHIND DISCRETE TIME FOURIER SERIES Given DTFS is a discrete time signal with period N n N for all n 5701 XkejgmkN where Xk 2201 nki mkN Huh Fastest oscillating discrete time sinusoid w 7T gt cos7rn 1quot Fourier series of discrete time periodic signal has nite number of terms7 with frequencies 07 2 722 732 N 1e0 2i 2 7 win If N even7 the component with the highest frequency is w 7T If N odd7 the component with the highest frequency is w 7T If is real7 then XNk X conjugate symmetry X0 1 N 1mean value of If N is even7 XIV2 l a2 3 N SIMPLE EXAMPLE WITH N24 Given Goal r59 Then Line Using 247 87 127 1677 87 127 167 247 87 127 16 PeriodN4 Compute DTFSFourier series expansion of discrete time periodic NOTE e j2T 1 je j2T 2 le j2T 3 j XO 124 8 12 16 15 Note this is real X2 24 8 12 16 03 Note this is real X1 g24 8 j 12 1 16j 3 2 X3 124 8j 12 1 16 j 3 2339 Xf n 15ej0 3 mg way 3 2jejquot3n 7139 37139 spectrum 1s perlodlc w1th components at 07 i5 l7r7 i7 i27r o 42 42 3 2 366J33397 ejm cos7rn 63 4 3 e 3 4 7 s1mpl1 es to 15 72 cosgn 3370 3cos7rn Don7t double at w 07 7T PARSEVAL S THEOREM POWER IS CONSERVED Power Time Freq 2201 2531 Xk2average power of periodic 152 l3 2jl2 32 I3 2j2 260 since 3 2j2 13 llt242 82 122 162 260 They are equal

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