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Higher Order Delta Functions

3.7 Higher Order Delta Functions

An nth-order delta function is defined as the nth derivative of u t , that is,
0

n
G
n
G t = ----- u t > @ (3.49)
dt 0
The function G' t is called doublet, G'' t is called triplet, and so on. By a procedure similar to the
derivation of the sampling property of the delta function, we can show that

ft G' t – a = fa G' t – a – f ' a G t – a (3.50)

Also, the derivation of the sifting property of the delta function can be extended to show that
f n n d n
d

³ ft G t – D t = – 1 -------- ft > n @ (3.51)

– f dt
t = D
Example 3.8
Evaluate the following expressions:

4
a. 3t G t – 1

f
d
b. ³ tG t – 2 t

– f
2
c. t G' t – 3

Solution:
4
a. The sampling property states that f t G t – a = fa G t – a For this example, f t = 3t and

a = 1 . Then,
4 4

3t G t – 1 = ^ 3t ` G t – 1 = 3G t – 1
t = 1
f
b. The sifting property states that ³ ft G t – D t = f D . For this example, f t = t and D = . 2
d

– f
Then,
f
d
³ tG t – 2 t = f2 = t t = 2 = 2

– f
c. The given expression contains the doublet; therefore, we use the relation
Circuit Analysis II with MATLAB Applications 3-15
Orchard Publications