Tutorials - Digital Signal Processing: An Introduction with MATLAB and Applications - page 241

Tutorial 9

Q: For the periodic square wave x(t) shown below, find:
(A) The complex Fourier series, (B) the trigonometric Fourier series.

x ( t )

1

t , sec

− T o − T o / 2 T o / 2 T o 3 T o / 2

0

Solution:
(A) Since x(t) is periodic, it has Fourier series expansion [Tables]:

x ð t Þ¼ X

1

X k e þ j2pkf 0 t

k ¼1

From the above figure, the signal fundamental frequency is f o = 1/T o Hz. The
Fourier coefficients (for k=0) are given by [see Tables]:

T o
2

T o
2

Z Z

X k ¼ 1
T o

x ð t Þ e j2pkf o t dt ¼ 1
T o

e j2pkf o t dt

0 0

1
j2pkf o

e j2pkf o t

¼ 1 ð 1 Þ k
j2pk

0 ¼ 1 e jkp

1
T o

T o
2

¼

j2pk

[Note that we used f o T o = 1 and e -jp = cos(p) - jsin (p) =-1].

For k = 0 we have: X o ¼ T o R T x ð t Þ dt ¼ 2 :

0

Now:

1. If k is even, i.e., k = 2n, then X k = X 2n = 0 (if n = 0)
2. If k is odd, i.e., k = 2n + 1, then X k ¼ X 2n þ 1 ¼ 1

jp ð 2n þ 1 Þ :

jp X

1

x ð t Þ¼ 1

2 þ 1

1
2n þ 1 e j ð 2n þ 1 Þ 2pf 0 t

)

n ¼1

(B) From above we have:
x ð t Þ¼ X

1

X k e þ j2pkf 0 t ¼ X 0 þ X

1

X k e j2pkf 0 t þ X k e j2pkf 0 t

k ¼1 k ¼ 1

¼ a 0 þ X

1

f a k cos ð 2pkf 0 t Þþ b k sin ð 2pkf 0 t Þ g

k ¼ 1