Inverse Filters

Note that the filter matrix is often invertible [58]. In that case, we can effectively run the filter backwards:

However, an invertible filter matrix does not necessarily correspond to a stable inverse-filter when the lengths of the input and output vectors are allowed to grow larger. For example, the inverted filter matrix may contain truncated growing exponentials, as illustrated in the following matlab example:

> h = toeplitz([1,2,0,0,0],[1,0,0,0,0])
h =
  1  0  0  0  0
  2  1  0  0  0
  0  2  1  0  0
  0  0  2  1  0
  0  0  0  2  1
> inv(h)
ans =
    1    0    0    0    0
   -2    1    0    0    0
    4   -2    1    0    0
   -8    4   -2    1    0
   16   -8    4   -2    1

The inverse of the FIR filter is in fact unstable, having impulse response , , which grows to with .

Another point to notice is that the inverse of a banded Toeplitz matrix is not banded (although the inverse of lower-triangular [causal] matrix remains lower triangular). This corresponds to the fact that the inverse of an FIR filter is an IIR filter.

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written by Julius Orion Smith III
Julius Smith's background is in electrical engineering (BS Rice 1975, PhD Stanford 1983). He is presently Professor of Music and Associate Professor (by courtesy) of Electrical Engineering at Stanford's Center for Computer Research in Music and Acoustics (CCRMA), teaching courses and pursuing research related to signal processing applied to music and audio systems. See http://ccrma.stanford.edu/~jos/ for details.