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

2π

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(