I have seen using singular value decomposition (SVD) to solve deconvolution problem for example truncated SVD (TSVD) . It appears there is also a connection between Tikhonov regularization and SVD. My question is: can we use this methods say TSVD for single channel deconvolution?

  • $\begingroup$ Can you define what you mean by single channel deconvolution and how it's different from "deconvolution problem" where you've seen SVD being used? $\endgroup$ – Atul Ingle Aug 22 '16 at 14:22
  • $\begingroup$ @AtulIngle I think this is what OP means; The deonvolution problem is shown as a solution to find x for Ax=s, where A is known, But given a desired vector s and received signal x how do you find A for SVD? $\endgroup$ – Creator Aug 22 '16 at 18:17
  • $\begingroup$ If that's what the OP means, then a standard deconvolution algorithm can still be applied. The goal of deconvolution is to recover the original signal $f(t)$ given measured signal $g(t) = \int h(t-\tau) f(\tau) d\tau$ where $h(t)$ is the "channel". However, note that due to symmetry, one can easily swap the definitions of "original signal" and "channel" and instead write $ g(t) = \int h(\tau) f(t - \tau) d\tau$ and recover $h(t)$ using a deconvolution algorithm. The form $Ax=s$ can be obtained by discretizing the convolution integrals. $\endgroup$ – Atul Ingle Aug 22 '16 at 18:23
  • $\begingroup$ @AtulIngle Exactly that OP is asking (I think) how to get A (discretize x to get A) $\endgroup$ – Creator Aug 22 '16 at 19:58
  • $\begingroup$ How about approximating the convolution integral as a Riemann sum $g(t_i) = \sum_k h( u_k ) f( t_i - u_k)$ and then using the known values of $g$ and $f$ set up a system of linear equations to solve for the values of $h$? (The system of linear equations can be expressed in matrix form, if you will.) This might be too handwavy, but if this is indeed what the OP asked for, I can try to elaborate in a real answer. $\endgroup$ – Atul Ingle Aug 23 '16 at 3:11

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