Signals and Systems - Electrical Engineering

4.7 Convergence of the Fourier Series 265

(a) (b)

0 0.5 1 1.5 2

− 1

−0.5

0

0.5

1

t

Period

x(

t)

(^00204060)
0.1
0.2
0.3
0.4
Ω
|X
|k
|X
|k
0 20 40 60
− 2
− 1
0
1
2
Ω
0 0.5 1 1.5 2
0
0.1
0.2
0.3
0.4
0.5
t
Period
y(
t)
|X
|k
0 20 40 60
0
1
2
3
4
Ω
(^00204060)
0.1
0.2
0.3
0.4
Ω
|Y
|k
FIGURE 4.11
(a) Periods of the sawtooth signalx(t)and (b) its integraly(t)and their magnitude and phase line spectra.
%%%%%%%%%%%%%%%%%
% Example 4.8---Saw-tooth signal and its integral
%%%%%%%%%%%%%%%%
syms t
T0 = 2;
m = heaviside(t)−heaviside(t−T0/2);
m1 = heaviside(t−T0/2) - heaviside(t−T0);
x = t∗m + (t−2)∗m1;
y = int(x);
[X, w] = fourierseries(x, T0, 20);
[Y, w] = fourierseries(y, T0, 20);
The signaly(t)is smoother than x(t);y(t)is a continuous function of time, while x(t)is
discontinuous. This is indicated as well by the magnitude line spectra of the two signals. Ignor-
ing the dc components, the{|Yk|}ofy(t)decay a lot faster to zero than the{|Xk|}ofx(t)(See
Figure 4.11). As we will see in Section 4.10, computing the derivative of a periodic signal is equiva-
lent to multiplying its Fourier series coefficients byj 0 k, which emphasizes the higher harmonics.
If the periodic signal is zero-mean so that its integral exists, the Fourier coefficients of the integral
can be found by dividing them byj 0 kso that now the low harmonics are emphasized. n

4.7 Convergence of the Fourier Series

It can be said, without overstating it, that any periodic signal of practical interest has a Fourier series.
Only very strange signals would not have a converging Fourier series. Establishing convergence is
necessary because the Fourier series has an infinite number of terms. To establish some general