Signals and Systems - Electrical Engineering

Problems 229

3.11. Convolution integral
Consider the following problems related to the convolution integral:
(a) The impulse response of an LTI system ish(t)=e−^2 tu(t)and the system input is a pulsex(t)=u(t)−
u(t− 3 ). Find the output of the systemy(t)by means of the convolution integral graphically and by
means of the Laplace transform.
(b) It is known that the impulse response of an analog averager ish(t)=u(t)−u(t− 1 ). Consider the
input to the averagerx(t)=u(t)−u(t− 1 ), and determine graphically as well as by means of the
Laplace transform the corresponding output of the averagery(t)=h∗x. Isy(t)smoother than
the input signalx(t)? Provide an argument for your answer.
(c) Suppose we cascade three analog averagers each with the same impulse responseh(t)=u(t)−u(t−
1 ). Determine the transfer function of this system. If the duration of the support of the input to the first
averager isMsec, what would be the duration of the support of the output of the third averager?

3.12. Deconvolution
In convolution problems the impulse responseh(t)of the system and the inputx(t)are given and one
is interested in finding the output of the systemy(t). The so-called ”deconvolution” problem consists in
giving two ofx(t),h(t), andy(t)to find the other. For instance, given the outputy(t)and the impulse
responseh(t)of the system, one wants to find the input. Consider the following cases:
(a) Suppose the impulse response of the system ish(t)=e−tcos(t)u(t)and the output has a Laplace
transform

Y(s)=

4

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

What is the inputx(t)?

(b) The output of an LTI system isy(t)=r(t)− 2 r(t− 1 )+r(t− 2 ), wherer(t)is the ramp signal.

Determine the impulse response of the system if it is known that the input isx(t)=u(t)−u(t− 1 ).

3.13. Application of superposition
One of the advantages of LTI systems is the superposition property. Suppose that the transfer function of
a LTI system is

H(s)=

s

s^2 +s+ 1

Find the unit-step responses(t)of the system, and then use it to find the response due to the following

inputs:

x 1 (t)=u(t)−u(t− 1 )

x 2 (t)=δ(t)−δ(t− 1 )

x 3 (t)=r(t)

x 4 (t)=r(t)− 2 r(t− 1 )+r(t− 2 )

Express the responsesyi(t)due toxi(t)fori=1,..., 4in terms of the unit-step responses(t).

3.14. Properties of the Laplace transform
Consider computing the Laplace transform of a pulse

p(t)=

{

1 0 ≤t≤ 1

0 otherwise