Signals and Systems - Electrical Engineering

290 C H A P T E R 4: Frequency Analysis: The Fourier Series

systems will be introduced and exemplified by its application in filtering. Modulation is the basic

tool in communications and can be easily explained in the frequency domain.

Problems............................................................................................

4.1. Eigenfunctions and LTI systems

The eigenfunction property is only valid for LTI systems. Consider the cases of nonlinear and of time-

varying systems.

(a) A system represented by the following input–output equation is nonlinear:

y(t)=x^2 (t)

Letx(t)=ejπt/^4. Find the corresponding system outputy(t). Does the eigenfunction property hold?

Explain.

(b) Consider a time-varying system

y(t)=x(t)[u(t)−u(t− 1 )]

Letx(t)=ejπt/^4. Find the corresponding system outputy(t). Does the eigenfunction property hold?

Explain.

4.2. Eigenfunctions and LTI systems

The output of an LTI system is

y(t)=

∫t

0

h(τ)x(t−τ)dτ

where the inputx(t)and the impulse responseh(t)of the system are assumed to be causal. Letx(t)=

2 cos( 2 πt)u(t). Compute the outputy(t)in the steady state and determine if the eigenfunction property

holds.

4.3. Eigenfunctions and frequency response of LTI systems

The input–output equation for an analog averager is

y(t)=

1

T

∫t

t−T

x(τ)dτ

Letx(t)=ej^0 t. Since the system is LTI, then the output should be

y(t)=ej^0 tH(j 0 )

(a) Findy(t)for the given input and then compare it with the above equation to findH(j 0 ), the response

of the averager at frequency 0.

(b) FindH(s)and verify the frequency response valueH(j 0 )obtained above.

4.4. Generality of eigenfunctions

The eigenfunction property holds for any input signal, periodic or not, that can be expressed in sinusoidal

form.

(a) Consider the inputx(t)=cos(t)+cos( 2 πt),−∞<t<∞, into an LTI system. Isx(t)periodic? If so,

indicate its period.