FIR Filter Deconvolution 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 $m &gt; n$ 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.
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 $X(f) = Y(f)/H(f)$.