I am reading a journal on optimal signal detection by mobile sensor. It is mentioned:

How should we plan to robots’ motion in order to maximize the detection probability? Assuming the source emits a known signal, the optimal detection algorithm is a matched filter (i.e., convolve the known waveform with the received signal and threshold).

Question: Why should we perform convolution of the known signal with the observed signal instead of cross-correlating the two, which shall detect the presence of similar patterns?

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    $\begingroup$ The "journal" is using incorrect language. The matched filter's impulse response is the time-reverse of the known signal, and so when the received signal is passed through the matched filter (which gives the convolution of the impulse response and the received signal), the result is the same as the crosscorrelation of the received signal with the known signal. See the latter part of this answer of mine for pictures etc. $\endgroup$ – Dilip Sarwate May 3 '20 at 21:44
  • $\begingroup$ yes, understood. Plus your answer was quite insightfull as well. +1 $\endgroup$ – GENIVI-LEARNER May 4 '20 at 12:44

Convolving with Matched Filter is same as cross-correlation.

enter image description here Suppose say your known signal is $x[n]$ for $0 \le n \le N-1$.

The matched filter is $$ h[n] = x^*[N-n] $$ where $*$ denotes conjugate (considering a generic complex signal). You can drop the conjugate for real signals.

The convolution with matched filtering operation is $$ y[p] = \sum x[l]h[p-l] $$ $$ h[p-l]=x^*[N-(p-l)]=x^*[l-p+N] $$ Therefore $$ y[p]=\sum x[l]x^*[l-p+N]=\sum x[l]x^*[l-\tau] $$ where $\tau = p-N$ is the difference between index $p$ and $N$.

You can already see the matched filter output $y[p]=\sum x[l]x^*[l-\tau]$ is kind of cross correlation operation.

The Matched Filter output at $p=N$ is $$ y[N]=\sum x[l]x^*[l-N+N]=\sum x[l]x^*[l-\tau]\Big|_{\tau=0} $$ So as you can see here, Matched filtering output at $p=N$ which maximizes the SNR is the same as cross correlation output at zero lag, that is $\tau=0$.

  • $\begingroup$ So match filter reverses the signal and when we do convolution it reverses the signal again and thats how we get cross correlation? $\endgroup$ – GENIVI-LEARNER May 3 '20 at 16:43
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    $\begingroup$ @GENIVI-LEARNER Yes. That is correct. $\endgroup$ – jithin May 3 '20 at 16:47

Matched filtering means projecting the received signal on a known signal direction. This sense of direction can be in a vector/eucledian space or the more general Hilbert space of functions and other quantities. Projection by itslef is a correlation measure of the two signals.

If you look at convolution it is nothing but correlation of the seuqences with one of the sequences time reversed. So usually a matched filter is implemented such that the convolution is performed with the time reversed version of the sequence. Which is equivalent to correlating with the signal. Convolution comes into play when we are viewing the operation as an LTI system. It can be efficiently implemented using FFTs.

  • $\begingroup$ Allright so here the wording was missused. The author meant to say cross correlation right? As I also understood convolution is same as correlation with the one of the sequence reversed. But here, nothing is reversed. right? $\endgroup$ – GENIVI-LEARNER May 3 '20 at 16:41
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    $\begingroup$ Yes, it would be more clear if he wrote an equation, where he explicitly mentions that the convolution is performed with the time reversed signal, many authors do. I remember reading about it in Simon Haykin's communication systems book, he clearly mentions it. $\endgroup$ – Dsp guy sam May 3 '20 at 16:48

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