One important use of inverse systems is in situations in which one wishes to remove dist...

One important use of inverse systems is in situations in which one wishes to remove distortions of some type. A good example of this is the problem of removing echoes from acoustic signals. For example, if an auditorium has a perceptible echo, then an initial acoustic impulse will be followed by attenuated versions of the sound at regularly spaced intervals. Consequently, an often-used model for this phenomenon is an LTI system with an impulse response consisting of a train of impulses, i.e.,

Here the echoes occur T seconds apart, and h_k represents the gain factor on the kth echo resulting from an initial acoustic impulse.

(a) Suppose that x(t) represents the original acoustic signal (the music produced by an orchestra, for example) and that y(t) = x(t)* h(t) is the actual signal that is heard if no processing is done to remove the echoes. In order to remove the distortion introduced by the echoes, assume that a microphone is used to sense y(t) and that the resulting signal is transduced into an electrical signal. We will also use y(t) to denote this signal, as it represents the electrical equivalent of the acoustic signal, and we can go from one to the other via acoustic-electrical conversion systems.

The important point to note is that the system with impulse response given by eq. (P2.64-1) is invertible, Therefore, we can find an LTI system with impulse response g(t) such that

and thus, by processing the electrical signal y(t) in this fashion and then con-verting back to an acoustic signal, we can remove the troublesome echoes.

The required impulse response g(t) is also an impulse train:

Determine the algebraic equations that the successive g_k must satisfy, and solve these equations for g₀, g₁, and g₂ in terms of h_k.

(b) Suppose that What is g(t) in this case?

(c) A good model for the generation of echoes is illustrated in Figure P2.64. Hence, each successive echo represents a fed-back version of y(t), delayed by T seconds and scaled by α. Typically, 0 < a < 1, as successive echoes are attenuated.

(i) What is the impulse response of this system? (Assume initial rest, i.e.,

(ii) Show that the system is stable if 0 < a < 1 and unstable if a > 1.

(iii) What is g(t) in this case? Construct a realization of the inverse system using adders, coefficient multipliers, and T-second delay elements.

(d) Although we have phrased the preceding discussion in terms of continuous-time systems because of the application we have been considering, the same general ideas hold in discrete time. That is, the LTI system with impulse response

is invertible and has as its inverse an LTI system with impulse response

It is not difficult to check that the g_k satisfy the same algebraic equations Part (a).

Consider now the discrete-time LTI system with impulse response

This system is not invertible. Find two inputs that produce the same output.