Page 108 - Circuit Analysis II with MATLAB Applications

Chapter 3 Elementary Signals

3.4 The Delta Function G t

The unit impulse or delta function, denoted as G t , is the derivative of the unit step u t . It is also
0
defined as

t
d
³ GW W = u t (3.33)
0
– f
and

G t = 0 for all t z 0 (3.34)

To better understand the delta function G t , let us represent the unit step u t as shown in Figure
0
3.20 (a).

Figure (a)
0 t
H H
1
Area =1 2H Figure (b)

0
H H t

Figure 3.20. Representation of the unit step as a limit.
The function of Figure 3.20 (a) becomes the unit step as H o 0 . Figure 3.20 (b) is the derivative of
Figure 3.20 (a), where we see that as H o 0 , 12eH becomes unbounded, but the area of the rectangle
1
remains . Therefore, in the limit, we can think of G t as approaching a very large spike or impulse
at the origin, with unbounded amplitude, zero width, and area equal to . 1

Two useful properties of the delta function are the sampling property and the sifting property.

3.5 Sampling Property of the Delta Function G t

The sampling property of the delta function states that

ft G t – a = fa G t (3.35)

or, when a = , 0

ft G t = f0 G t (3.36)

3-12 Circuit Analysis II with MATLAB Applications

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