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

> 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
![$ h=[1,2]$](http://www.dsprelated.com/josimages_new/filters/img2003.png)




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