Signals and Systems - Electrical Engineering

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

conditions under which the series converges, we need to classify signals with respect to their

smoothness.

A signalx(t)is said to bepiecewise smoothif it has a finite number of discontinuities, while asmooth

signal has a derivative that changes continuously. Thus, smooth signals can be considered special

cases of piecewise smooth signals.

The Fourier series of a piecewise smooth (continuous or discontinuous) periodic signalx(t)converges for all

values oft. The mathematician Dirichlet showed that for the Fourier series to converge to the periodic signal

x(t), the signal should satisfy the following sufficient (not necessary) conditions over a period:

n Be absolutely integrable.

n Have a finite number of maxima, minima, and discontinuities.

The infinite series equalsx(t)at every continuity point and equals the average

0.5[x(t+ 0 +)+x(t+ 0 −)]

of the right limitx(t+ 0 +)and the left limitx(t+ 0 −)at every discontinuity point. Ifx(t)is continuous

everywhere, then the series converges absolutely and uniformly.

Although the Fourier series converges to the arithmetic average at discontinuities, it can be observed

that there is some ringing before and after the discontinuity points. This is called theGibb’s phe-

nomenon.To understand this phenomenon it is necessary to explain how the Fourier series can be

seen as an approximation to the actual signal, and how when a signal has discontinuities the conver-

gence is not uniform around them. It will become clear that the smoother the signalx(t)is, the easier

it is to approximate it with a Fourier series with a finite number of terms.

When the signal is continuous everywhere, the convergence is such that at each pointtthe series

approximates the actual valuex(t)as we increase the number of terms in the approximation. How-

ever, that is not the case when discontinuities occur in the signal. This is despite the fact that a

minimum mean-square approximation seems to indicate that the approximation could give a zero

error. Let

xN(t)=

∑N

k=−N

Xkejk^0 t (4.24)

be theNth-order approximation of a periodic signal x(t), of fundamental frequency 0 , that

minimizes the average quadratic error over a period

EN=

1

T 0

∫

T 0

|x(t)−xN(t)|^2 dt (4.25)