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, is...
$$h(t)=\sum_{k=0}^{x} h_{k} \delta(t-k T)$$
Here the echoes occur \(T\) seconds apart, and \(h_{k}\) represents the gain
factor on the \(k\) th 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 \(g(r)\) such that
$$y(t) * g(t)=x(t)$$
and thus, by processing the electrical signal \(y(t)\) in this fashicn and then
converting back to an acoustic signal, we can remove the troublesome echoes.
The required impulse response \(g(t)\) is also an impulse train:
$$g(t)=\sum_{k=0}^{x} g_{k} \delta(t-k T)$$
Determine the algebraic equations that the successive \(g_{k}\) must satisfy,
and solve these equations for \(g_{0}, g_{1},\) and \(g_{2}\) in terms of \(h_{k}\)
(b) Suppose that \(h_{0}=1, h_{1}=1 / 2,\) and \(h_{t}=0\) for all \(i \geq 2\) What
is \(g(t)\) in this case?
(c) A good model for the generation of echocs 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 \(\alpha .\) Typically, \(0<\alpha<1,\) as successive
echoes are attendated.
(i) What is the impulse response of this systern's (Assume initial rest, i.e.,
\(y(t)=0\) for \(t<0\) if \(x(t)=0\) for \(t<0 .\)
(ii) Show that the system is stable if \(0<\alpha<1\) and unstable if \(\alpha>1\)
(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 diserete tome. That is, the CTI systern with impulse
response
$$h[n]=\sum_{k=0}^{\infty} \delta | n-k N$$
is invertible and has as its inverse an LTI sysiern with impulse response
$$g[n]=\sum_{k=0}^{\infty} g_{k} \delta | n-k N$$
It is not difficult to check that the \(g_{r}\) satisfy the same algebrace
equations as in part (a) Consider bow the discrete-time LTI system with
impulse response
$$h[n]=\sum_{k=-\infty}^{\infty} \delta[n-k N]$$
This system is nor invertible. Find two inputs that produce the same output.
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