Problems 233

(a) Find and plot the poles and zeros ofH(s). Is this BIBO stable?

(b)Find the frequency 0 of the inputx(t)=2 cos( 0 t)u(t)such that the output of the given system is

zero in the steady state. Why do you think this happens?

(c)If the input is a sine instead of a cosine, would you get the same result as above? Explain why or why

not.

3.26. Zero steady-state response of analog averager—MATLAB
The analog averager can be represented by the differential equation
dy(t)
dt

1
T
[x(t)−x(t−T)]

wherey(t)is the output andx(t)is the input.

(a) If the input–output equation of the averager is

y(t)=

1

T

∫t

t−T

x(τ)dτ

show how to obtain the above differential equation and thaty(t)is the solution of the differential

equation.

(b) Ifx(t)=cos(πt)u(t), choose the value ofTin the averager so that the output isy(t)= 0 in the steady

state. Graphically show how this is possible for your choice ofT. Is there a unique value forTthat

makes this possible? How does it relate to the frequency 0 =πof the sinusoid?

(c) Use the impulse responseh(t)of the averager found before, to show using Laplace that the steady

state is zero whenx(t)=cos(πt)u(t)andTis the above chosen value. Use MATLAB to solve the

differential equation and to plot the response for the value ofTyou chose. (Hint:Considerx(t)/Tthe

input and use superposition and time invariance to findy(t)due to(x(t)−x(t−T))/T.)

3.27. Partial fraction expansion—MATLAB
Consider the following functionsYi(s)=L[yi(t)],i=1, 2and 3 :

Y 1 (s)=

s+ 1

s(s^2 + 2 s+ 4 )

Y 2 (s)=

1

(s+ 2 )^2

Y 3 (s)=

s− 1

s^2 ((s+ 1 )^2 + 9 )

where{yi(t),i=1, 2, 3}are the complete responses of differential equations with zero initial conditions.

(a) For each of these functions, determine the corresponding differential equation, if all of them have as

inputx(t)=u(t).

(b) Find the general form of the complete response{yi(t),i=1, 2, 3}for each of the{Yi(s)i=1, 2, 3}. Use

MATLAB to plot the poles and zeros for each of the{Yi(s)}, to find their partial fraction expansions,

and the complete responses.

3.28. Iterative convolution integral—MATLAB
Consider the convolution of a pulsex(t)=u(t+0.5)−u(t−0.5)with itself many times. Use MATLAB for
the calculations and the plotting.
(a) Consider the result forN= 2 of these convolutions—that is,

y 2 (t)=(x∗x)(t)