2.

A continuous-time signal

xa (t)=sin2zt+0.1 sin127t

is sampled at a sampling frequency 2, 207 rad/sec. A continuous-time

double differentiator system is defined as

Ye (1)

dt²

The continuous-time signal xe (t) is processed by a discrete-time system as

shown in the following figure.

x(1)

C/D

T

x[n]

Discrete-time

system

y[n]

D/C

T

y,(1) (a)

Compute the Fourier transforms of the continuous-time signal

xe (t) and the continuous-time sampled signal x, (t). Sketch the Fourier

transforms of xe (t) and x, (t). Determine all the frequencies of the

Fourier transform X, (j2) within the bandwidth 2 <.

(b)

Determine the Fourier transform of the discrete-time sampled

signal x [n] for w<. What is the discrete-time sampled signal x [n]?

(c)

Obtain the frequency response of the discrete-time system so

that it is equivalent to the continuous-time system. Then compute the

Fourier transform of the discrete-time output signal y [n].

(d)

The continuous-time output signal y, (t) is reconstructed by

a reconstruction ideal low-pass filter H, (j). Determine the signal

yr (t). Compare y, (t) to ye (t). Are they equal? If not, state the reason

why this is the case.

(e)

If yr (t) is not equal to ye (t), how would you design a re-

construction filter H, (j) to recover the desired reconstructed sig-

nal yr (t)=ye (t). Sketch the reconstruction filter H, (j). Note that

H, (j) needs not be an ideal low-pass filter.