In this problem, we examine a few of the properties of the unit impulse function. (a)...

In this problem, we examine a few of the properties of the unit impulse function.

(a) Show that

(b) In Section 1.4, we defined the continuous-time unit impulse as the limit of the signal . More precisely, we defined several of the properties of δ(t) by examining the corresponding properties of . For example, since the signal

or by viewing δ(t) as the formal derivative of u(t).

This type of discussion is important, as we are in effect trying to define δ(t) through its properties rather than by specifying its value for each t, which is not possible. In Chapter 2, we provide a very simple characterization of the behaviour of the unit impulse that is extremely useful in the study of linear time invariant systems. For the present, however, we concentrate on demonstrating that the important concept in using the unit impulse is to understand how it behaves. To do this, consider the six signals depicted in Figure P1.38. Show

Therefore, it is not enough to define or to think of δ(t) as being zero for t ≠ 0 and infinite for t = 0. Rather, it is properties such as eq. (P1 -38-1.) that define the impulse. In Section 2.5 we will define a whole class of signals known as singularity functions, which are related to the unit impulse and which are also defined in terms of their properties rather than their values.