## Inverse Filters

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

*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 1The 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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