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# Digital Signal Processing Homework 7

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ECE 310 Digital Signal Processing
Homework 7

1. The sequence x[n] = cos
π
3
n

, −∞ < n < ∞ was obtained by sampling the continuous-time signal
xa(t) = cos (Ω0t), −∞ < t < ∞ at a sampling rate of 1000 samples/sec. What are two possible
values of Ω0 that could have resulted in the sequence x[n]?
2. The continuous-time signal xa(t) = sin (10πt) + cos (20πt) is sampled with a sampling period T to
obtain the discrete-time signal x[n] = sin
π
5
n

+ cos

5
n

a) Determine a choice for T consistent with this information.
b) Is your choice for T in part (a) unique? If so, explain why. If not, specify another choice of T
consistent with the information given.
3. The continuous-time signal xa(t) = cos (400πt) is sampled with a sampling period T to obtain a
discrete-time signal x[n] = xa(nT)
a) Compute and sketch the magnitude of the continuous-time Fourier transform of xa(t) and the
discrete-time Fourier Transform of x[n] for T = 1 ms.
b) Repeat part (a) for T = 2 ms.
c) What is the maximum sampling period Tmax such that no aliasing occurs in the sampling
process?
4. The continuous-time signal xa(t) has the continuous-time Fourier transform shown in the figure below.
The signal xa(t) is sampled with sampling interval T to get the discrete-time signal x[n] = xa(nT).
Sketch Xd(ω) (the DTFT of x[n]) for the sampling intervals T = 1/100, 1/200 sec.
5. Let x[n] = xa(nT). Show that the DTFT of x[n] is related to the FT of xa(t) by
Xd(ω) = 1
T
X∞
`=−∞
X

ω + 2`π
T

where Xd(ω) is the DTFT of x[n] and X(Ω) the FT of xa( Digital Signal Processing Homework 7
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