Tutorials - Digital Signal Processing: An Introduction with MATLAB and Applications - page 239

Tutorial 7

1

Q: Show that d ð t Þ¼ lim a ! 0

2a P 2a ð t Þ [a [ 0]. What is the importance of this

result?
Solution: We should prove that the function g ð t Þ¼ 2a P 2a ð t Þ (shown below)
satisfies the definition and properties of the delta function as a ! 0.

g ( t − t o )

g ( t ) = (1/2 a ) Π 2 a ( t )

1/(2a) 1/(2a)

t t

0 0

t o

− a a − a a

The delta function is defined as R 1

1 x ð t Þ d ð t t o Þ dt ¼ x ð t o Þ for any continuous
function x(t) [see Tables]. Applying the same integral to g(t) we get:

Z

1

Z

t o þ a

x ð t Þ g ð t t o Þ dt ¼ 1
2a

x ð t Þ dt ð 1 Þ

1

t o a

The Mean Value Theorem for integrals states that for any continuous function
s(t) we have:

Z

d

s ð t Þ dt ¼½ d c s ð t m Þ;

c

where t m is a number between c and d (c \ t m \ d). Applying this theorem to Eq. 1
above we get:
Z

1

Z

t o þ a

x ð t Þ g ð t t o Þ dt ¼ 1
2a

x ð t Þ dt ¼ 1

2a ½ð t o þ a Þð t o a Þ x ð t m Þ¼ x ð t m Þ;

1 t o a

where t o - a \ t m \ t o + a.Asa ! 0, we have t m ! t o ; and hence
Z Z

1 1

x ð t Þ lim
a ! 0

f g ð t t o Þg dt ¼ lim
a ! 0

x ð t Þ g ð t t o Þ dt ¼ lim
a ! 0

f x ð t m Þg¼ x ð t o Þ:

1 1