Signals and Systems - Electrical Engineering

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

For a periodic signalx(t), of periodT 0 , if we know or can easily compute the Laplace transform of a period

ofx(t),

x 1 (t)=x(t)[u(t 0 )−u(t−t 0 −T 0 )] for anyt 0

Then the Fourier coefficients ofx(t)are given by

Xk=

1

T 0

L[x 1 (t)]s=jk 0  0 =

2 π

T 0

fundamental frequency (4.21)

This can be seen by comparing the equation for theXkcoefficients with the Laplace transform of a

periodx 1 (t)=x(t)[u(t 0 )−u(t−t 0 −T 0 )] ofx(t). Indeed, we have that

Xk=

1

T 0

t (^0) ∫+T 0
t 0
x(t)e−jk^0 tdt

=

1

T 0

t (^0) ∫+T 0
t 0
x(t)e−stdt

∣

∣s=jk 0

=

1

T 0

L[x 1 (t)]s=jk 0

nExample 4.5

Consider the periodic pulse trainx(t), of periodT 0 =1, shown in Figure 4.4. Find its Fourier series.

Solution

Before finding the Fourier coefficients, we see that this signal has a dc component of 1, and that

x(t)−1 (zero-average signal) is well represented by cosines, given its even symmetry, and as such

FIGURE 4.4

Train of rectangular pulses.

· · · · · ·

2

x(t)

−1.25 −0.75 −0.25 0.25 0.75 1.25

t

T 0 = 1