Why Convolve instead of Cross-Correlate for Detection of Known Signal?

Question

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?

Answer 1

Convolving with Matched Filter is same as cross-correlation.

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 $\tau=0$.

Answer 2

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.