Signals and Systems - Electrical Engineering

Problems 449

7.8. Sampling output of nonlinear system

The input–output relation of a nonlinear system is

y(t)=x^2 (t)

wherex(t)is the input andy(t)is the output.

(a) The signalx(t)is band limited with a maximum frequencyM= 2000 πrad/sec. Determine ify(t)is

also band limited, and if so, what is its maximum frequencymax?

(b) Suppose that the signaly(t)is low-pass filtered. The magnitude of the low-pass filter is unity and the

cut-off frequency isc= 5000 πrad/sec. Determine the value of the sampling periodTsaccording to

the given information.

(c) Is there a different value forTsthat would satisfy the Nyquist sampling rate condition for bothx(t)

andy(t)and that is larger than the one obtained above? Explain.

7.9. Signal reconstruction

You wish to recover the original analog signalx(t)from its sampled formx(nTs).

(a) If the sampling period is chosen to beTs= 1 so that the Nyquist sampling rate condition is satis-

fied, determine the magnitude and cut-off frequency of an ideal low-pass filterH(j)to recover the

original signal and plot them.

(b) What would be a possible maximum frequency of the signal? Consider an ideal and a nonideal low-

pass filter to reconstructx(t). Explain.

7.10. CD player versus record player
Explain why a CD player cannot produce the same fidelity of music signals as a conventional record player.
(If you do not know what these are, ignore this problem, or get one to find out what they do or ask your
grandparents about LPs and record players!)

7.11. Two-bit analog-to-digital converter—MATLAB
Letx(t)=0.8 cos( 2 πt)+0.15, 0 ≤t≤ 1 , and zero otherwise, be the input to a 2-bit analog-to-digital
converter.
(a) For a sampling periodTs=0.025sec determine and plot using MATLAB the sampled signal,

x(nTs)=x(t)|t=nTS

(b) The four-level quantizer (see Figure 1.2) corresponding to the 2-bit ADC is defined as

k 1 ≤x(nTs) < (k+ 1 )1 → ˆx(nTs)=k 1 k=−2,−1, 0, 1 (7.26)

wherex(nTs), found above, is the input andˆx(nTs)is the output of the quantizer. Let the quantization

step be 1 =0.5. Plot the input–output characterization of the quantizer, and find the quantized output

for each of the sample values of the sampled signalx(nTs).

(c) To transform the quantized values into unique binary 2-bit values, consider the following code:

xˆ(nTs)=− 21 → 10

ˆx(nTs)=− 1 → 11

ˆx(nTs)= 01 → 00

xˆ(nTs)= 1 → 01

Obtain the digital signal corresponding tox(t).