Page 109 - Circuit Analysis II with MATLAB Applications

Sifting Property of the Delta Function

that is, multiplication of any function f t by the delta function G t results in sampling the function
at the time instants where the delta function is not zero. The study of discrete-time systems is based
on this property.

Proof:
Since G t = 0for t 0 andt ! 0 then,

f t G t = 0for t 0 and t ! 0 (3.37)

We rewrite f t as

f t = f0 + > ft – f0 @ (3.38)

Integrating (3.37) over the interval f to t– and using (3.38), we get

t t t
d
d
d
³ f W G W W = ³ f0 G W W + ³ > f W – f0 G @ W (3.39)
W
– f – f – f
The first integral on the right side of (3.39) contains the constant term f 0 ; this can be written out-
side the integral, that is,
t t
d
d
³ f0 G W W = f0 ³ G W W (3.40)
– f – f
The second integral of the right side of (3.39) is always zero because

G t = 0for t 0 andt ! 0
and
> f W – f0 @ = f0 – f0 = 0
W = 0
Therefore, (3.39) reduces to

t t
d
d
³ f W G W W = f0 ³ G W W (3.41)
– f – f
Differentiating both sides of (3.41), and replacing with , we get
t
W
ft G t = f0 G t
(3.42)
Sampling Property of G t

3.6 Sifting Property of the Delta Function G t

The sifting property of the delta function states that

Circuit Analysis II with MATLAB Applications 3-13
Orchard Publications