Signals and Systems - Electrical Engineering

162 C H A P T E R 2: Continuous-Time Systems

2.14. Steady-state response of averager—MATLAB

An analog averager is given by

y(t)=

1

T

∫t

t−T

x(τ)dτ

(a) Letx(t)=u(t)−u(t− 1 ). Find the average signaly(t)using the above integral. LetT= 1. Carefully

ploty(t). Verify your result by graphically computing the convolution ofx(t)and the impulse response

h(t)of the averager.

(b)To see the effect ofTon the averager, consider the signal to be averaged to bex(t)=cos( 2 πt/T 0 )u(t).

Select the smallest possible value ofTin the averager so that the steady-state response of the system,

y(t)ast→∞, will be 0.

(c)Use MATLAB to compute the output in part (b). Compute the outputy(t)for 0 ≤t≤ 2 at intervals

Ts=0.001. Approximate the convolution integral using the functionconv(usehelpto find aboutconv)

multiplied byTs.

2.15. Echo system modeling

An echo system could be modeled as follows:

(a) Using feedback systems is of great interest in control and in the modeling of many systems. An echo

is created as the sum of one or more delayed and attenuated output signals that are fed back into the

present signal. A possible model for an echo system is

y(t)=x(t)+α 1 y(t−τ)+···+αNy(t−Nτ)

wherex(t)is the present input signal,y(t)is the present output,y(t−kτ)is the previous delayed

outputs, and the|αk|< 1 values are attenuation factors. Carefully draw a block diagram for this system.

(b)Consider the echo model forN= 1 and parametersτ= 1 andα 1 =0.1. Is the resulting echo system

LTI? Explain.

(c)Another possible model is given by a nonrecursive, or without feedback, system,

z(t)=x(t)+β 1 x(t−τ)+···+βMx(t−Mτ)

where several present and past inputs are delayed and attenuated and added up to form the output.

The parameters|βk|< 1 are attenuation factors andτis a delay. Carefully draw a block diagram for the

echo system characterized by the above equation. Does the above equation represent an LTI system?

Explain.

2.16. An ideal low-pass filter—MATLAB

The impulse response of an ideal low-pass filter is

h(t)=

sin(t)

t

or a sinc signal.

(a) Given that the impulse response is the response of the system to an inputx(t)=δ(t)with zero initial

conditions, can an ideal low-pass filter be used for real-time processing? Explain.

(b)Is the ideal low-pass filtering bounded-input bounded-output stable? Use MATLAB to check if the

impulse response satisfies the condition for BIBO stability.