Problem

In this chapter, we have used several properties and ideas that greatly facilitate the a...

In this chapter, we have used several properties and ideas that greatly facilitate the analysis of LTI systems. Among these are two that we wish to examine a bit more closely. As we will see, in certain very special cases one must be careful in using these properties, which otherwise hold without qualification.

(a) One of the basic and most important properties of convolution (in both continuous and discrete time) is associativity. That is, if x(t), h(t), and g(t) are three signals, then

This relationship holds as long as all three expressions are well defined and finite. As that is usually the case in practice, we will in general use the associativity property without comments or assumptions. However, there are some cases in which it does not hold. For example, consider the system depicted in Figure P2.71, with h(t) = u₁ (t) and g(t) = u(t). Compute the response of this system to the input

Do this in the three different ways suggested by eq. (P2.71-1) and by the figure

(i) By first convolving the two impulse responses and then convolving the result with x(t).

(ii) By first convolving x(t) with u₁ (t) and then convolving the result with u(t)

(iii) By first convolving x(t) with u(t) and then convolving the result with u₁(t)

(b) Repeat part (a) for

and

Thus, in general, the associativity property of convolution holds if and only if the three expressions in eq. (P2.71-1) make sense (i.e., if and only if their interpretations in terms of LTI systems are meaningful). For example, in part (a) differentiating a constant and then integrating makes sense, but the process of integrating the constant from t = —∞ and then differentiating does not, and it is only in such cases that associativity breaks down. Closely related to the foregoing discussion is an issue involving inverse systems. Consider the LTI system with impulse response h(t) = u(t). As we saw in part (a), there are inputs—specifically, x(t) = nonzero constant—for which the output of this system is infinite, and thus, it is meaningless to consider the question of inverting such outputs to recover the input. However, if we limit ourselves to inputs that do yield finite outputs, that is, inputs which satisfy

then the system is invertible, and the LTI system with impulse response u₁(t) is its inverse.

(d) Show that the LTI system with impulse response u₁(t) is not invertible. (Hint: Find two different inputs that both yield zero output for all time.) However, show that the system is invertible if we limit ourselves to inputs that satisfy eq. (P2.71-2). [Hint In Problem 1.44, we showed that an LTI system is invertible if no input other than x(t) = 0 yields an output that is zero for all time; are there two inputs x(t) that satisfy eq. (P2.71-2) and that yield identically zero responses when convolved with u₁(t)?]

What we have illustrated in this problem is the following:

(1) If x(t), h(t), and g(t) are three signals, and if x(t) * g(t), x(t) * h(t), and h(t) * g(t) are all well defined and finite, then the associativity property, eq. (P2.71-1), holds.

(2) Let h(t) be the impulse response of an LTI system, and suppose that impulse response g(t) of a second system has the property

Then, from (1), for all inputs x(t) for which x(t) * h(t) and x(t) * g(t) are both well defined and finite, the two cascades of systems depicted in Figure P2.71 act as the identity system, and thus, the two LTI systems can be regarded as inverses of one another. For example, if h(t) = u(t) and g(t) = u₁ (t), then, as long as we restrict ourselves to inputs satisfying (P2.71-2), we can regard these two systems as inverses.

Therefore, we see that the associativity property of eq. (P2.71-1) and the definition of LTI inverses as given in eq. (P2.71-3) are valid, as long as all convolutions that are involved are finite. As this is certainly the case in any realistic problem, we will in general use these properties without comment or qualification. Note that, although we have phrased most of our discussion in terms of continuous-time signals and systems, the same points can also be made in discrete time [as should be evident from part (c)].