ECE 513 - DIGITAL SIGNAL PROCESSING
Cranos M. Williams
Assigned: November 11, 2019
HOMEWORK 9 Solution, 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-π≤ω≤π. Label all relevant frequencies.
i. B =
π
5
ii. B =
π
2
iii. B =
3
π
4
iv. B =
π
Figures 2 shows the output of down-sampled signals.
1

ECE 513 - Homework 9 Solution
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(b) Referring to Figure 1c, draw the DTFT|Y2(ω)|of the output signal for the followcases. Your plots should range from-π≤ω≤π. Label all relevant frequencies.

ECE 513 - Homework 9 Solution
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(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.

ECE 513 - Homework 9 Solution
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ECE 513 - Homework 9 Solution
5
-1
-0.5
0
0.5
1
0
0.2
0.4
0.6
0.8
1
Normalized Frequency
|X(
ω
)|
Original Signal: B =
π
/5
-1
-0.5
0
0.5
1
0
0.2
0.4
0.6
0.8
1
Normalized Frequency
|Y
1
(
ω
)|
Original Signal Downsampled by 2
Figure 4: Output with
B
=
π/
5
-1
-0.5
0
0.5
1
0
0.2
0.4
0.6
0.8
1
Normalized Frequency
|X(
ω
)|
Original Signal: B =
π
/2
-1
-0.5