Signals and Systems - Electrical Engineering

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

perpendicularity of vectors: Perpendicular vectors cannot be represented in terms of each other, as

orthogonal functions provide mutually exclusive information. The perpendicularity of two vectors

can be established using thedot or scalarproduct of the vectors, and the orthogonality of functions is

established by theinner product, or the integration of the product of the function and its conjugate.

Consider a set of complex functions{ψk(t)}defined in an interval [a,b], and such that for any pair of

these functions, let’s sayψ`(t)andψm(t),`6=m, their inner product is

∫b

a

ψ`(t)ψm∗(t)dt=

{

0 `6=m

1 `=m

(4.8)

Such a set of functions is calledorthonormal(i.e., orthogonal and normalized).

A finite-energy signalx(t)defined in [a,b] can be approximated by a series

ˆx(t)=

∑

k

akψk(t) (4.9)

according to some error criterion. For instance, we could minimize the energy of the error function

ε(t)=x(t)−ˆx(t)or

∫b

a

|ε(t)|^2 dt=

∫b

a

∣

∣

∣

∣

∣

x(t)−

∑

k

akψk(t)

∣

∣

∣

∣

∣

2

dt (4.10)

The expansion can be finite or infinite, and may not approximate the signal point by point.

Fourier proposed sinusoids as the functions{ψk(t)}to represent periodic signals, and solved the

quadratic minimization posed in Equation (4.10) to obtain the coefficients of the representation.

For most signals, the resulting Fourier series has an infinite number of terms and coincides with the

signal pointwise. We will start with a more general expansion that uses complex exponentials and

from it obtain the sinusoidal form. In Chapter 5 we extend the Fourier series to represent aperiodic

signals—leading to the Fourier transform that is in turn connected with the Laplace transform.

Recall that a periodic signalx(t)is such that

n It is defined for−∞<t<∞(i.e., it has an infinite support).

n For any integerk,x(t+kT 0 )=x(t), whereT 0 is the fundamental period of the signal or the

smallest positive real number that makes this possible.

TheFourier series representationof a periodic signalx(t), of periodT 0 , is given by an infinite sum of weighted

complex exponentials (cosines and sines) with frequencies multiples of the signal’s fundamental frequency

 0 = 2 π/T 0 rad/sec, or

x(t)=

∑∞

k=−∞

Xkejk^0 t  0 =

2 π

T 0

(4.11)