Signals and Systems - Electrical Engineering

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

n The Fourier coefficients{Xk}are easily obtained using the orthonormality of the Fourier functions: First,

we multiply the expression for x(t)in Equation (4.11) by e−j`^0 tand then integrate over a period to get

∫

T 0

x(t)e−j`^0 tdt=

∑

k

Xk

∫

T 0

ej(k−`)^0 tdt

=

∑

k

XkT 0 δ(k−`)

=X`T 0

given that when k=`, then

∫

T 0 e

j(k−`) 0 tdt=T 0 ; otherwise it is zero according to the orthogonality of the

Fourier exponentials. This then gives us the expression for the Fourier coefficients{X`}in Equation (4.12).

You need to recognize that the k and`are dummy variables in the Fourier series, and as such the expression

for the coefficients is the same regardless of whether we use`or k.

n It is important to realize from the given Fourier series equations that for a periodic signal x(t), of period

T 0 , any period

x(t), t 0 ≤t≤t 0 +T 0

provides all the necessary information in the time-domain characterizing x(t). In an equivalent way the

coefficients and their corresponding frequencies{Xk,k 0 }provide all the necessary information about x(t)

in the frequency domain.

4.4 Line Spectra

The Fourier series provides a way to determine the frequency components of a periodic signal and the

significance of these frequency components. Such information is provided by the power spectrum of

the signal. For periodic signals, the power spectrum provides information as to how the power of the

signal is distributed over the different frequencies present in the signal. We thus learn not only what

frequency components are present in the signal but also the strength of these frequency components.

In practice, the power spectrum can be computed and displayed using a spectrum analyzer, which

will be described in Chapter 5.

4.4.1 Parseval’s Theorem—Power Distribution over Frequency

Although periodic signals are infinite-energy signals, they have finite power. The Fourier series

provides a way to find how much of the signal power is in a certain band of frequencies.

The powerPxof a periodic signalx(t), of periodT 0 , can be equivalently calculated in either the time or the

frequency domain:

Px=

1

T 0

∫

T 0

|x(t)|^2 dt=

∑

k

|Xk|^2 (4.14)