Signals and Systems - Electrical Engineering

4.6 Fourier Coefficients from Laplace 261

0 0.2 0.4 0.6 0.8 1

0

0.2

0.4

0.6

0.8

1

t

Period

0 50 100

0

0.1

0.2

0.3

0.4

0.5

0.6

Magnitude line spectrum

|X

|k

Ω (rad/sec)

0 50 100

0

1

2

3

4

Phase line spectrum

<X

k

Ω (rad/sec)

FIGURE 4.7
Period of full-wave rectified signalx(t)and its magnitude and phase line spectra.

and using the Laplace transform we have

X 1 (s)e−0.5s=

π

s^2 +π^2

[1+e−s]

so that

X 1 (s)=

π

s^2 +π^2

[e0.5s+e−0.5s]

The Fourier coefficients are then

Xk=

1

T 0

X 1 (s)|s=j 0 k

whereT 0 =1 and 0 = 2 π, giving

Xk=

π

(j 2 πk)^2 +π^2

2 cos( 2 πk/ 2 )

=

2 (− 1 )k

π( 1 − 4 k^2 )