Talk:Matched filter

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[edit] Derivations of matched filter

On July 29, 2006, I made significant changes to the (formerly) "Problem statement" section of this article. The derivation that I found there was of great help. I added in a few missing lines of derivation noted by a previous contributor and clarified some points, such as the covariance matrix. I also added another section with an alternate derivation of the matched filter. I believe that it is best to derive the matched filter in the context of the inner conjugate product of the filter and the observed signal, since this requires only the use of vectors, matrices, and their conjugate transposes -- without a need to use transposes and complex conjugation alone. I believe that this significantly reduces the mathematical complexity. I have edited the Lagrangian derivation accordingly. --Rabbanis 04:46, 30 July 2006 (UTC)

[edit] Notational Problem in definition of y, the conjugate inner product of the filter and the observed signal

The definition for y, the conjugate inner product of the filter and the observed signal, is given as a discrete convolution

y = \sum_{k=-\infty}^{+\infty} h^{*}[k] x[h] = h^{H}x = ... .

This looks wrong to me: how can h, *the filter*, be the index parameter to x? Shouldn't it actually be something like

y = \sum_{k=-\infty}^{+\infty} h^{*}[k] x[n-k] = h^{H}x = ... ?

69.121.100.249 22:48, 27 October 2006 (UTC)

You are absolutely right. This is a typo. I intended to use the index k. I did not use the discrete convolution, however, because I find that the derivation of the matched filter is more intuitive with the conjugate inner product. I realize that this is slightly confusing, as it is inconsistent with the use of h earlier, where it is the impulse response of an LTI system. Thanks for pointing this out. I have changed the index to k. Rabbanis 15:31, 31 October 2006 (UTC)
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