Tutorials - Digital Signal Processing: An Introduction with MATLAB and Applications - page 242

where we used Euler's formula to get:

a 0 ¼ X 0 ; a k ¼ X k þ X k ; b k ¼ j ð X k X k Þ:

Hence, a o = X 0 = 1/2.
If k is even (k=0), then a k = b k = 0 (see above).
If k is odd, then:

2
ð 2n þ 1 Þ p :

a k ¼ a 2n þ 1 ¼ X ð 2n þ 1 Þ þ X ð 2n þ 1 Þ ¼ 0 and b k ¼ b 2n þ 1 ¼

Therefore, we can write the signal x(t) as follows:

2 þ X

1

x ð t Þ¼ 1 b 2n þ 1 sin ð 2p ð 2n þ 1 Þ f 0 t Þ

n ¼ 0

:

¼ 1

2 þ 2

sin ð x 0 t Þþ 1

3 sin ð 3x 0 t Þþ 1 5 sin ð 5x 0 t Þþ

p

Q:Ifx(t) above is passed through the system shown below, find the output
y(t) and the average power in x(t) and y(t).
Ans. P x = 0.5; P y = 0.47.

x ( t ) LPF, H ( f ) y ( t )

H ( f )

1

f

− 4 / T o 4 / T o

0

Tutorial 10

Q1: Show that x(t)d(t - t o ) = x(t o )d(t - t o ).
Solution: Let g(t) be any function that is continuous at t = t o , and let /

(t) = g(t)x(t). Then using Tables we get:
Z

dt ¼ Z d ð t t o Þ dt ¼ Z

1 1 1

g ð t Þ x ð t Þ d ð t t o Þ ½ ½ g ð t Þ x ð t Þ / ð t Þ d ð t t o Þ dt ¼ / ð t o Þ:

1 1 1