Signals and Systems - Electrical Engineering

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

If the inputx(t)of a causal and stable LTI system, with impulse responseh(t), is periodic of periodT 0 and has

the Fourier series

x(t)=X 0 + 2

∑∞

k= 1

|Xk|cos(k 0 t+∠Xk)  0 =

2 π

T 0

(4.34)

the steady-state response of the system is

y(t)=X 0 |H(j 0 )|cos(∠H(j 0 ))+ 2

∑∞

k= 1

|Xk||H(jk 0 )|cos(k 0 t+∠Xk+∠H(jk 0 )) (4.35)

where

H(jk 0 )=

∫∞

0

h(τ)e−jk^0 τdτ (4.36)

is thefrequency response of the system atk 0.

Remarks

n If the input signal x(t)is a combination of sinusoids of frequencies that are not harmonically related, the

signal is not periodic, but the eigenfunction property still holds. For instance, if

x(t)=

∑

k

Akcos(kt+θk)

and the frequency response of the LTI system is H(j), the steady-state response is

y(t)=

∑

k

Ak|H(jk)|cos(kt+θk+∠H(jk))

n It is important to realize that if the LTI system is represented by a differential equation and the input

is a sinusoid, or combination of sinusoids, it is not necessary to use the Laplace transform to obtain the

complete response and then let t→∞to find the sinusoidal steady-state response. The Laplace transform

is only needed to find the transfer function of the system, which can then be used in Equation (4.35) to

find the sinusoidal steady state.

4.9.2 Filtering of Periodic Signals

According to Equation (4.35) if we know the frequency response of the system (Eq. 4.36), at the

harmonic frequencies of the periodic input,H(jk 0 ), we have that in the steady state the output of

the systemy(t)is as follows:

n Periodic of the same period as the input.

n Its Fourier coefficients are those of the inputXkmultiplied by the frequency response at the

harmonic frequencies,H(jk 0 ).