hmwk9.pdf - ECE 513 DIGITAL SIGNAL PROCESSING Cranos M Williams Assigned HOMEWORK 9 DUE 1 Frequency Representation of Up-sampling and Down-sampling(15

hmwk9.pdf - ECE 513 DIGITAL SIGNAL PROCESSING Cranos M...

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ECE 513 - DIGITAL SIGNAL PROCESSING Cranos M. Williams Assigned: November 11, 2019 HOMEWORK 9, DUE NOVEMBER 20, 2019 1.Frequency Representation of Up-sampling and Down-sampling: (15 Pts)The discretetime Fourier transform (DTFT) of a signalx[n] is shown in Figure 1a. The responseis zero forB≤ |ω| ≤π.Mx[n]y1[n]Lx[n]y2[n](a)(c)(b)ωB-B|X(ω)|1Figure 1: a.)DTFT ofx[n], b.)Down-sampling block diagram, c.)Up-sampling blockdiagram(a) Referring to Figure 1b, draw the DTFT|Y1(ω)|of the output signal for a down-sampling factor of M = 2 for the following cases. Your plots should range from-πωπ. Label all relevant frequencies. i. B = π 5 ii. B = π 2 iii. B = 3 π 4 iv. B = π (b) Referring to Figure 1c, draw the DTFT|Y2(ω)|of the output signal for the followcases. Your plots should range from-πωπ. Label all relevant frequencies. 1
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ECE 513 - Homework 9 Solution 2 (c) Use the MATLAB functionfir2to produce a 101 sample sequencex[n] whosemagnitude response matches the response in Figure 1a.Usingyour ownup-sampling and down-sampling routines, test the results above. Did your drawingsmatch the results MATLAB gave? If they didn’t, explain why not. Turn in yourdrawings and the MATLAB plots.2.Equivalence of Polyphase Implementation: (30 Pts)Given thatxd[n] is a sampling ratereduced (down-sampling) version ofx[n],xd[n] =x[nM] =xc(nMT),(1)whereTis the original sampling interval,Mis an integer greater than 0, andxc(t) isa continuous-time signal.(a) Prove the down-sampling identityM-1Xi=0X(ej(ω/M-2πi/M)),(2)whereX(e) andXd(e) are the DTFT ofx[n] andxd[n], respectively.
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