UNIVERSITY OF ILLINOIS AT URBANACHAMPAIGN
Department of Electrical and Computer Engineering
ECE 410
Digital Signal Processing
Homework 3
Wednesday, September 10, 2008
Prof. Bresler, Prof. Jones
Due by September 17, 2008
Problem 1
(10 points)
Let
X
[
k
] denotes the Npoint DFT of the discretetime signal
x
[
n
] (0
≤
n
≤
N

1)
(a) Show that
X
[0] = 0 if
x
[
n
] =

x
[
N

1

n
].
(b) Given N is an even number, show that
X
[
N/
2] = 0 if
x
[
n
] =
x
[
N

1

n
].
Problem 2
(25 points)
Let
x
[
n
] be a discretetime sequence:
x
[
n
] =
‰
(

1)
n
,
0
≤
n
≤
3
0
,
otherwise
(a) Show that the analytical expression for DTFT of
x
[
n
] is
X
d
(
ω
) =
(
e

j
(
3
ω
2

π
2
)
.
sin(2
ω
)
cos(
ω/
2)
ω
∈
[0
,
2
π
]
, ω
6
=
π
4
ω
=
π
Hint: You can consider
X
d
(
ω
)
as the sum of a finite geometric series.
Plot its magnitude and phase for 0
≤
ω <
2
π
using the following matlab function and com
mands:
w = [0:1281]/128*2*pi;
Xdtft = zeros(1,128);
Xdtft(65) = 4;
Xdtft([1:64,66:end]) = ...
exp(i*(3/2)*w([1:64,66:end])+i*pi/2).*sin(2*w([1:64,66:end])) ...
./cos(w([1:64,66:end])/2);
mag_Xdtft = abs(Xdtft);
ang_Xdtft = angle(Xdtft);
figure;
subplot(2, 1, 1);
plot(w/(2*pi), mag_Xdtft);
grid on;
1
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xlabel(’\omega / (2\pi)’); ylabel(’X_d(\omega)’);
title(’DTFT of x[n]’);
subplot(2, 1, 2);
plot(w/(2*pi), ang_Xdtft);
grid on;
xlabel(’\omega/ (2\pi)’); ylabel(’phase of X_d(\omega) in radians’);
(b) Compute the 4point DFT of
x
[
n
]
,
0
≤
n <
4 and stem plot its magnitude and phase. You can
use the following commands:
x = [1, 1, 1, 1];
M = 4;
Xdft_4 = fft(x, M);
m = 0:(M1);
mag_Xdft_4 = abs(Xdft_4);
ang_Xdft_4 = angle(Xdft_4);
figure;
subplot(2, 1, 1);
stem(m, mag_Xdft_4);
grid on;
title(’magnitude of 4point DFT of x’);
xlim([0.2 4]);
xlabel(’m’); ylabel(’X[m]’);
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