Signals and Systems - Electrical Engineering

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

FIGURE 4.2

(a) Magnitude (top left) and

phase (bottom left) line spectra

of raised cosine and (b)

magnitude (top right) and

phase (bottom right) line

spectra of raised sine.

B

−Ω 0 Ω 0

kΩ 0

|Xk|

−Ω 0 Ω 0

kΩ 0

∠Xk

A

2

A

2

(a)(b)

kΩ 0

−Ω 0 Ω 0

|Yk|

B

A

2

A

2

2

−Ω 0

kΩ 0

Ω 0

∠Yk

−π

π

2

Using symbolic MATLAB integration we can easily find the Fourier series coefficients, and the

corresponding magnitude and phase are then plotted usingstemto obtain the line spectra. Using

our MATLAB functionfourierseriesthe magnitude and phase of the line spectrum corresponding to

the periodic raised siney(t)= 1 +sin( 100 t)is shown in Figure 4.3.

function [X, w] = fourierseries(x, T0, N)

%%%%%

% symbolic Fourier Series computation

% x: periodic signal

% T0: period

% N: number of harmonics

% X,w: Fourier series coefficients at harmonic frequencies

%%%%%

syms t

% computation of N Fourier series coefficients

for k = 1:N,

X1(k) = int(x∗exp(−j∗ 2 ∗pi∗(k - 1)∗t/T0), t, 0, T0)/T0;

X(k) = subs(X1(k));

w(k) = (k−1)∗ 2 ∗pi/T0; % harmonic frequencies

end