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Inverse Systems

Module by: Mark A. Davenport. E-mail the author

Inverse systems

Many signal processing problems can be interpreted as trying to undo the action of some system. For example, echo cancellation, channel obvolution, etc. The problem is illustrated below.

Figure 1
flow chart showing movement form x[n] to H to y[n] to H sub I, to x-hat[n].

If our goal is to design a system HIHI that reverses the action of HH, then we clearly need H(z)HI(z)=1H(z)HI(z)=1. In the case where

H ( z ) = P ( z ) Q ( z ) H ( z ) = P ( z ) Q ( z )
(1)

then this can be achieved via

H I ( z ) = Q ( z ) P ( z ) . H I ( z ) = Q ( z ) P ( z ) .
(2)

Thus, the zeros of H(z)H(z) become poles of HI(z)HI(z), and the poles of H(z)H(z) become zeros of HI(z)HI(z). Recall that H(z)H(z) being stable and causal implies that all poles are inside the unit circle. If we want H(z)H(z) to have a stable, causal inverse HI(z)HI(z), then we must have all zeros inside the unit circle, (since they become the poles of HI(z)HI(z).) Combining these, H(z)H(z) is stable and causal with a stable and causal inverse if and only if all poles and zeros of H(z)H(z) are inside the unit circle. This type of system is called a minimum phase system.

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