Signals and Systems - Electrical Engineering

506 C H A P T E R 8: Discrete-Time Signals and Systems

8.14. LTI of ADCs

An ADC can be thought of as composed of three subsystems: a sampler, a quantizer, and a coder.

(a) The sampler, as a system, has as input an analog signalx(t)and as output a discrete-time signal

x(nTs)=x(t)|t=nTswhereTsis the sampling period. Determine whether the sampler is a linear system

or not.

(b) Samplex(t)=cos(0.5πt)u(t)andx(t−0.5)usingTs= 1 to gety(nTs)andz(nTs), respectively. Plot

x(t),x(t−0.5), andy(nTs)andz(nTs). Isz(nTs)a shifted version ofy(nTs)so that you can say the

sampler is time invariant? Explain.

8.15. LTI of ADCs (part 2)

A two-bit quantizer of an ADC has as inputx(nTs)and as outputxˆ(nTs), such that if

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

(a) Is this system time invariant? Explain.

(b) Suppose that the value of 1 in the quantizer is0.25, and the sampled signal isx(nTs)=nTs,Ts=0.1

and− 5 ≤n≤ 5. Use the sampled signal to determine whether the quantizer is a linear system or not.

Explain.

(c) From the results in this and the previous problem, would you say that the ADC is an LTI system?

Explain.

8.16. Rectangular windowing system—MATLAB

A window is a signal w[n]that is used to highlight part of another signal. The windowing process consists

in multiplying an input signalx[n]by the window signal w[n], so that the output is

y[n]=x[n]w[n]

There are different types of windows used in signal processing. One of them is the so-calledrectangular

window, which is given by

w[n]=u[n]−u[n−N]

(a) Determine whether the rectangular windowing system is linear. Explain.

(b) Supposex[n]=nu[n]. Plot the outputy[n]of the windowing system (withN= 6 ).

(c) Let the input bex[n−6]. Plot the corresponding output of the rectangular windowing system, and

indicate whether the rectangular windowing system is time invariant.

8.17. Impulse response of an IIR system—MATLAB

A discrete-time IIR system is represented by the difference equation

y[n]=0.15y[n−2]+x[n] n≥ 0

wherex[n]is the input andy[n]is the output.

(a) To find the impulse responseh[n]of the system, letx[n]=δ[n],y[n]=h[n], and the initial conditions

be zero,y[n]=h[n]=0, n< 0. Find recursively the values ofh[n]for values ofn≥ 0.

(b) As a second way to do it, replace the relation between the input and the output given by the difference

equation to obtain a convolution sum representation that will give the impulse responseh[n]. What

ish[n]?

(c) Use the MATLAB functionfilterto get the impulse responseh[n](usehelpto learn about the function

filter).