# FIR Filter Deconvolution [closed]

Suppose we have a system whose impulse response h has length K and fed with an input x that has length N. Then it is known that the output y has length M = K + N -1. This shows us the convolution matrix which relates x and y has size M x N. By existence theorem, the problems in the format $$A_{m x n}x = b$$ has a solution if and only if $$m \leq n$$. Since $$m > n$$ always holds, it seems to me that there will be always no solution for FIR filter deconvolution problem. Is this correct becase in a somewhere, I read about the possibility of existence.

• what's the problem with having more equations than unknowns if our equations were consistent? – Mohammad M Oct 5 '19 at 21:42
• Considering your notation A must be M by K and not M by N, also how do you calculate the K+N-1 output if you don't assign some predefined value to the x outside of those N values? (the outputs where the kernel completely placed over the input is K-N+1) – Mohammad M Oct 5 '19 at 21:56
• By existence theorem, the rows of A must be linearly independent. – eet Oct 5 '19 at 22:00
• Please see imgur.com/a/ECHXosB @MohammadM – eet Oct 5 '19 at 22:08
• here your equations would be less than your unknowns, but even if you have more equations than unknowns if the equations were consistent you could have at least one answer. – Mohammad M Oct 5 '19 at 22:20

• An overdetermined system can have approximate solutions; for example, in the least-squares sense, where $$\mathbf{x}=(H^TH)^{-1}H^T\mathbf{y}$$ minimizes $$||H\mathbf{x}-\mathbf{y}||$$.
• Deconvolution is usually performed in the frequency domain, where $$X(f) = Y(f)/H(f)$$.