Signals and Systems - Electrical Engineering

604 C H A P T E R 10: Fourier Analysis of Discrete-Time Signals and Systems

FIGURE 10.9

Computation of

Fourier series

coefficients of different

periodic signals: the

corresponding

magnitude line

spectrum for each

signal is shown on

the right.

0 10 20 30

0102030

0102030

− 1

0

1

x[

n]

x^1

[n

]

x^2

[n

]

− 1 −0.5 0 0.5

− 1 −0.5 0 0.5

− 1 −0.5 0 0.5

0

0.2

0.4

− 1

0

1

0

0.2

0.4

0.6

− 1

0

1

n

0

0.1

0.2

ω/π

|X

(e

jω

)|

|X

( 1

e

jω

)|

|X

( 2

e

jω

)|

that of a period or of multiples of a period. Notice that the MATLAB functionsignis used to generate

a periodic train of pulses from the cosine function. The need to divide by the number of periods used

will be discussed later in section 10.4.3.

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

% Fourier series using FFT

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

N = 10; M = 10; N1 = M∗N;n = 0:N1 - 1;

x = cos(2∗pi∗n/N); % sinusoid

x1 = sign(x); % train of pulses

x2 = x - sign(x); % sinusoid minus train of pulses

X = fft(x)/M;X1 = fft(x1)/M;X2 = fft(x2)/M; % ffts of signals

X = X/N;X1 = X1/N;X2 = X2/N; % FS coefficients

10.3.4 Response of LTI Systems to Periodic Signals

Letx[n], a periodic signal of periodN, be the input of an LTI system with transfer functionH(z). If the Fourier

series ofx[n]is

x[n]=

N∑− 1

k= 0

X[k]ej(kω^0 )n ω 0 =

2 π

N

fundamental frequency