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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.

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  • $\begingroup$ what's the problem with having more equations than unknowns if our equations were consistent? $\endgroup$ – Mohammad M Oct 5 '19 at 21:42
  • $\begingroup$ 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) $\endgroup$ – Mohammad M Oct 5 '19 at 21:56
  • $\begingroup$ By existence theorem, the rows of A must be linearly independent. $\endgroup$ – eet Oct 5 '19 at 22:00
  • $\begingroup$ Please see imgur.com/a/ECHXosB @MohammadM $\endgroup$ – eet Oct 5 '19 at 22:08
  • $\begingroup$ 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. $\endgroup$ – Mohammad M Oct 5 '19 at 22:20
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A few comments:

  • An overdetermined system (with more equations than unknowns) can have exact solutions.

  • 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 $X(f) = Y(f)/H(f)$.

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