Signals and Systems - Electrical Engineering

4.9 Response of LTI Systems to Periodic Signals 273

FIGURE 4.13
(a) Modulated square-wave
x 1 (t)cos( 20 πt)and (b) cosine
x 2 (t)cos( 20 πt)and their
respective magnitude line
spectra. (a) (b)

0 2 4

− 1

−0.5

0

0.5

1

t

x^1

(t)

0 50 100 150

0

0.1

0.2

Ω

|X

1 k

|

024

− 1

−0.5

0

0.5

1

t

x^2

(t

)

0

0 50 100 150

0.1

0.2

Ω

|X

2 k

|

n

4.9 Response of LTI Systems to Periodic Signals...........................................

The most important property of LTI systems is the eigenfunction property.

Eigenfunction property: In steady state, the response to a complex exponential (or a sinusoid) of a certain

frequency is the same complex exponential (or sinusoid), but its amplitude and phase are affected by the

frequency response of the system at that frequency.

Suppose that the impulse response of an LTI system ish(t)and that H(s)=L[h(t)] is the
corresponding transfer function. If the input to this system is a periodic signalx(t), of periodT 0 ,
with Fourier series

x(t)=

∑∞

k=−∞

Xkejk^0 t  0 =

2 π

T 0

(4.29)

then according to the eigenfunction property the output in the steady state is

yss(t)=

∑∞

k=−∞

[XkH(jk 0 )]ejk^0 t (4.30)