Signals and Systems - Electrical Engineering

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

If we callYk=XkH(jk 0 )we have a Fourier series representation ofyss(t)withYkas its Fourier

coefficients.

4.9.1 Sinusoidal Steady State.............................................................

If the input of a stable and causal LTI system, with impulse responseh(t), isx(t)=Aej^0 t, the

output is

y(t)=

∫∞

0

h(τ)x(t−τ)dτ=Aej^0 t

∫∞

0

h(τ)e−j^0 τdτ

=Aej^0 tH(j 0 )=A|H(j 0 )|ej^0 t+∠H(j^0 ) (4.31)

The limits of the first integral indicate that the system is causal (theh(τ)=0 forτ <0) and that

the inputx(t−τ)is applied from−∞(whenτ=∞) tot(whenτ=0); thusy(t)is the steady-state

response of the system. If the input is a sinusoid—for example,

x 1 (t)=Re[x(t)=Aej^0 t]=Acos( 0 t) (4.32)

then the corresponding steady-state response is

y 1 (t)=Re[A|H(j 0 )|ej^0 t+∠H(j^0 )]

=A|H(j 0 )|cos( 0 t+∠H(j 0 )). (4.33)

As in the eigenfunction property, the frequency of the output coincides with the frequency of the

input, however, the magnitude and the phase of the input signal is changed by the response of the

system at the input frequency.

The following script simulates the convolution of a sinusoidx(t)of frequency= 20 π, amplitude

10, and random phase with the impulse responseh(t)(a modulated decaying exponential) of an LTI

system. The convolution integral is approximated using the MATLAB functionconv.

%%%%%%%%%%%%%%%%%

% Simulation of Convolution

%%%%%%%%%%%%%%%%

clear all; clf

Ts = 0.01; Tend = 2; t = 0:Ts:Tend;

x = 10∗cos(20∗pi∗t + pi∗(rand(1, 1)−0.5)); % input signal

h = 20∗exp(−10.ˆt).∗cos(40∗pi∗t); % impulse response

% approximate convolution integral

y = Ts∗conv(x, h);