Problem

The role played by u (t), δ (t) and other singularity functions in the study of linear t...

The role played by u (t), δ (t) and other singularity functions in the study of linear time-invariant systems is that of an idealization of a physical phenomenon, and as we will see, the use of these idealizations allow us to obtain an exceedingly important and very simple representation of such systems. In using singularity functions, we need, however, to be careful. In particular, we must remember that they are idealizations, and thus, whenever we perform a calculation using them, we are implicitly assuming that this calculation represents an accurate description of the behavior of the signals that they are intended to idealize. To illustrate, consider the equation

This equation is based on the observation that

Taking the limit of this relationship then yields the idealized one given by eq. (PI.39-1). However, a more careful examination of our derivation of eq. (P1.39-2) shows that that equation really makes sense only if x(t) is continuous at t = 0. If it is not, then we will not have x(t) ≈ x(0) for t small.

To make this point clearer, consider the unit step signal u(t). Recall from eq. (1.70) that u(t) = 0 for t < 0 and u(t) = 1 for t > 0, but that its value at t = 0 is not defined. [Note, for example, that u_? (0) = 0 for all ?, while (from Problem 1.38(b)).] The fact that u(0) is riot defined is not particularly bothersome, as long as the calculations we perform using u(t) do not rely on a specific choice for u(0). For example, if f (t) is a signal that is continuous at t = 0, then the value of

does not depend upon a choice for u(0). On the other hand, the fact that u (0) is undefined is significant in that it means that certain calculations involving singularity functions are undefined. Consider trying to define a value for the product u(t)δ(t).

To see that this cannot be defined, show that

But

In general, we can define the product of two signals without any difficulty as long as the signals do not contain singularities (discontinuities, impulses, or the other singularities introduced in Section 2.5) whose locations coincide. When the locations do coincide, the product is undefined. As an example, show that the signal