Tutorials - Digital Signal Processing: An Introduction with MATLAB and Applications - page 244

Tutorial 11

Q: For the periodic pulse train p ð t Þ¼ P k ¼1 d ð t kT Þ find:
(A) Complex Fourier series (FS), (B) trigonometric FS, (C) Fourier Transform
(FT).

p ( t )

1

t

−4 T o −3 T o −2 T o − T o T o 2 T o 3 T o 4 T o

0

Solution:
(A) p ð t Þ¼ P 1

k ¼1 P k e þ j2pkf o t ; where x o ¼ 2pf o ¼ 2 T o is the fundamental
frequency, T o being the fundamental period, and the Fourier coefficients are
given by [see Tables]:

Z

T o = 2 T o = 2

Z

P o ¼ 1
T o

p ð t Þ dt ¼ 1
T o

d ð t Þ dt ¼ 1
T o

T o = 2 T o = 2

T o = 2 T o = 2

Z Z

P k ¼ 1
T o

p ð t Þ e j2pkf 0 t dt ¼ 1
T o

d ð t Þ e j2pkf 0 t dt ¼ 1
T o

T o = 2 T o = 2

X

1

p ð t Þ¼ 1
T o

e j2pkf 0 t )

1

(B) p ð t Þ¼ a o þ P 1

k ¼ 1 ½ a k cos ð kx o t Þþ b k sin ð kx o t Þ , where [see Tables]:
a o ¼ P o ¼ 1 a k ¼ P k þ P k ¼ 2

T o ; T o ;

b k ¼ j ð P k P k Þ¼ 0 :

X

1

p ð t Þ¼ 1
T o

þ 2
T o

) cos ð kx o t Þ:

k ¼ 1

(C) Using part (A) we have [using Tables]:

( )

X X

1 1

Ff p ð t Þg¼F 1
T o

¼ 1
T o

e j2pkf o t F e j2pkf o t

k ¼1 k ¼1

X

d ð f kf o Þ¼ f o X

1 1

¼ 1
T o

d ð f kf o Þ

k 1 k 1

Hence, a pulse train in the time domain is Fourier-transformed to a pulse train in
the frequency domain.