Advanced DSP

Deconvolution

If a signal x went through an MA filter f, and we got the output y=x*f. Suppose we know y and f but we do not know x. We can find the AR inverse of f, (call it f ^-1), and find x=y*(f ^-1). There are problems with this method. Suppose, y has noise in it (due to sensor noise, calculational errors and environmental conditions – “backscatter”), then y=x*f+n. Now passing y through f ^-1 will give x'=y*(f ^-1)=x*f*f ^-1+n*f ^-1=x+n*f ^-1. Now f ^-1, being an AR filter, is likely to be unstable, thus causing an oscillatory inverse-noise term n*f ^-1 which will eclipse the deconvolution x.

The AR deconvolver goes “berserk” because it finds further and further “false” explanations to hide its past mistakes – while at the same time being convinced that it is doing a perfect job, there being no errors. What we need to be able to do, is specify that beyond a certain point, the input signal x is known to be zero. This will make it impossible for any algorithm to find a perfect explanation, making a best fit explanation necessary.

When we thus constrain the system, we now have a finitely many inputs x to play with, which cannot match the expected output y perfectly, but only in the least squares sense. If the matrix equivalent of the convolution “*f” is taken to be F, then what we want is to find x such that Fx=y. If this is not achievable, we want to minimize the error ||Fx-y||. This is a least squares matrix inversion problem, which can be solved using the methodology of lecture 1.

LS deconvolution is an application of LS matrix inversion. The LS deconvolution methodology is used in system identification and autocorrelation estimationalgorithms. A fast algorithm for LS deconvolution is Levinson-Durbin.