Page 110 - Circuit Analysis II with MATLAB Applications

Chapter 3 Elementary Signals

f
d
³ ft G t – D t = f D (3.43)

– f

that is, if we multiply any function f t by G t – D and integrate from f – to +f , we will obtain the
value of f t evaluated at t = D .
Proof:
Let us consider the integral

b
d
³ ft G t – D t where a D b (3.44)

a
We will use integration by parts to evaluate this integral. We recall from the derivative of products
that

dxy = xdy + ydx or xdy = dxy – ydx (3.45)

and integrating both sides we get
d
d
³ xy = xy – ³ y x (3.46)
=
Now, we let x = f t ; then, dx = f t . We also let dy = G t – D ; then, y u t – D . By substitu-

c
0
tion into (3.46), we get
b b b
–
d
d
³ ft G t – D t = ft u t – D ³ u t – D ft t (3.47)
c

0
0
a a a

We have assumed that a D b ; therefore, u t – 0 D = 0 for D a , and thus the first term of the
right side of (3.47) reduces to f b . Also, the integral on the right side is zero for D a , and there-
fore, we can replace the lower limit of integration by . We can now rewrite (3.47) as
a
D
b b
d
d
c

³ ft G t – D t = fb – ³ ft t = fb – fb + f D
a D
and letting a o – f and b o f for any D f , we get
f
d
³ ft G t – D t = f D

– f (3.48)
Sifting Property of G t
3-14 Circuit Analysis II with MATLAB Applications
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