Signals and Systems - Electrical Engineering

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

FIGURE 4.12

Approximate Fourier seriesxN(t)of the pulse

trainx(t)(discontinuous) using the DC component

and 20 harmonics. The approximatexN(t)

displays the Gibb’s phenomenon around the

discontinuities.

0 0.2 0.4 0.6 0.8

− 1

−0.5

0

0.5

1

x(

t),

xN

(t

)

t(sec)

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

% Example 4.9---Simulation of Gibb’s phenomenon

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

clf; clear all

w0 = 2∗pi; DC = 0; N = 20; % parameters of periodic signal

% computation of Fourier series coefficients

for k = 1:N,

X(k) = sin(k∗pi/2)/(k∗pi/2);

end

X = [DC X]; % Fourier series coefficients

% computation of periodic signal

Ts = 0.001; t = 0:Ts:1−Ts;

L = length(t); x = [ones(1, L/4) zeros(1, L/2) ones(1, L/4)]; x = x−0.5;

% computation of approximate

xN = X(1)∗ones(1,length(t));

for k = 2:N,

xN = xN + 2∗X(k)∗cos(2∗pi∗(k−1).∗t); % approximate signal

plot(t, xN); axis([0 max(t) 1.1∗min(xN) 1.1∗max(xN)])

hold on; plot(t, x, ’r’)

ylabel(’x(t), xN(t)’); xlabel(’t (sec)’);grid

hold off

pause(0.1)

end

When you execute the above script, it pauses to display the approximation for an increasing num-

ber of terms in the approximation. At each of these values ringing around the discontinuities the

Gibb’s phenomenon is displayed. n