Reading about singular value decomposition (SVD) in the context of signal processing applications, one can separate the signal from the noise into orthogonal subspaces. On the surface this sounds like using the decomposed output of the SVD is more optimal than matched filter (or correlation) for detection, demodulation, timing estimation, etc. Is this true or is there more nuance to this problem?


That's not true, it's not better.

The thing is: the matched filter just implements the projection in the signal vector space, onto the signal vector itself (or a multiple thereof). (You'll find correlation is just an inner product in that space.)

The line through that vector is the signal subspace, the plane to which that vector is normal is the noise space. Matched filtering is a method of linear algebra, and not different than using the SVD to find the subspaces. It's just that for matched filtering, the signal space is known.

  • $\begingroup$ Is the idea that with SVD that you are separating out the signal and noise into their own subspaces, but since they are orthogonal, the matched filter operation does not change whether you perform it on the raw data vs the output of the SVD? $\endgroup$ – BigBrownBear00 Jul 21 '20 at 10:07
  • $\begingroup$ as said, the SVD is necessary to find the noise and signal subspaces. The separation, in the end, is just a projection. The matched filter is a projection onto a signal space, which you know beforehand (so you don't need the SVD to determine it). $\endgroup$ – Marcus Müller Jul 21 '20 at 10:42
  • $\begingroup$ Understood. That begs the question, why would you ever use SVD then? $\endgroup$ – BigBrownBear00 Jul 21 '20 at 10:50
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    $\begingroup$ As said, in cases where you don't know the signal and/or noise space. $\endgroup$ – Marcus Müller Jul 21 '20 at 10:51
  • $\begingroup$ Can you stop saying as said, lol? we're not all on your level Marcus ;) $\endgroup$ – BigBrownBear00 Jul 21 '20 at 11:11

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