Signals and Systems - Electrical Engineering

4.7 Convergence of the Fourier Series 267

with respect to the Fourier coefficientsXk. To minimizeENwith respect to the coefficientsXkwe set
its derivative with respect toXkto zero. Letε(t)=x(t)−xN(t), so that

dEN

dXk

=

1

T 0

∫

T 0

2 ε(t)

dε∗(t)

dXk

dt

=−

1

T 0

∫

T 0

2[x(t)−xN(t)]e−jk^0 tdt

= 0

which after replacingxN(t)and using the orthogonality of the Fourier exponentials gives

Xk=

1

T 0

∫

T 0

x(t)e−j^0 ktdt (4.26)

corresponding to the Fourier coefficients ofx(t)for−N≤k≤N. AsN→∞the average error
EN→0.

The only issue left is howxN(t)converges tox(t). As indicated before, ifx(t)is smoothxN(t)approxi-
matesx(t)at every point, but if there are discontinuities the approximation is in an average fashion.
The Gibb’s phenomenon indicates that around discontinuities there will be ringing, regardless of the
orderNof the approximation, even though the average quadratic errorENgoes to zero asNincreases.
This phenomenon will be explained in Chapter 5 as the effect of using a rectangular window to obtain
a finite-frequency representation of a periodic signal.

nExample 4.9

To illustrate the Gibb’s phenomenon consider the approximation of a train of pulsesx(t)with

zero mean and periodT 0 =1 (see the dashed signal in Figure 4.12) with a Fourier seriesxN(t)

withN=1,..., 20.

Solution

We compute analytically the Fourier coefficients ofx(t)and use them to obtain an approxima-

tionxN(t)ofx(t)having a zero DC component and up to 20 harmonics. The dashed-line plot in

Figure 4.12 isx(t)and the solid–line plot isxN(t)whenN=20. The discontinuities of the pulse

train cause the Gibb’s phenomenon. Even if we increase the number of harmonics there is an

overshoot in the approximation around the discontinuities.