Signals and Systems - Electrical Engineering

4.2 Eigenfunctions Revisited 243

the delay. Find the frequency response of the ideal communication system, and use it to determine

the steady-state response when the delay caused by the system isτ=3 sec, and the input isx(t)=

2 cos( 4 t−π/ 4 ).

Solution

The impulse response of the ideal system ish(t)=δ(t−τ)whereτis the delay of the transmission.

In fact, the output according to the convolution integral gives

y(t)=

∫∞

0

δ(ρ−τ)

︸ ︷︷ ︸

h(ρ)

x(t−ρ)dρ=x(t−τ)

as expected. Let us then find the frequency response of the ideal communication system. According

to the eigenvalue property, if the input isx(t)=ej^0 t, then the output is

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

but also

y(t)=x(t−τ)=ej^0 (t−τ)

so that comparing these equations we have that

H(j 0 )=e−jτ^0

For a generic frequency 0≤ <∞, we would get

H(j)=e−jτ

which is a complex function of, with a unity magnitude|H(j)|=1, and a linear phase

∠H(j)=−τ. This system is called anall-pass system, since it allows all frequency components

of the input to go through with a phase change only.

Consider the case whenτ=3, and that we input into this systemx(t)=2 cos( 4 t−π/ 4 ), then

H(j)=1e−j^3 , so that the output in the steady state is

y(t)= 2 |H(j 4 )|cos( 4 t−π/ 4 +∠H(j 4 ))

=2 cos( 4 (t− 3 )−π/ 4 )

=x(t− 3 )

where we usedH(j 4 )= 1 e−j^12 (i.e.,|H(j 4 )|=1 and∠H(j 4 )=12). n

nExample 4.3

Although there are better methods to compute the frequency response of a system represented by

a differential equation, the eigenfunction property can be easily used for that. Consider the RC