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Inverse Filters

Note that the filter matrix $ \mathbf{h}$ is often invertible [58]. In that case, we can effectively run the filter backwards:

$\displaystyle {\underline{x}}= \mathbf{h}^{-1} \underline{y} $

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 $ h=[1,2]$ is in fact unstable, having impulse response $ h_i(n)=(-2)^n$, $ n=0,1,2,\ldots\,$, which grows to $ \infty$ with $ n$. 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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Cyclic Convolution Matrix