Most discrete-time filters aim to modify an input signal according to desired performance in the frequency domain; for example, a low-pass filter attenuates the high frequency components of an input signal. However, there may be instances in which the goals of the filter are best understood in the time domain. One example of such a filter is the matched filter .
The objective of a matched filter is very straightforward: find the location a particular signal occurs in another (larger) signal. While this is a task often employed in children's games (I Spy) or puzzles (Where's Waldo), it of course has more "grown up" uses, as well: for example, sonar/radar work by transmitting a signal and then listening (searching) for an echoed version of it.
We have already considered a signal processing technique that works very well at finding one signal in another. Recall what the inner product operation does. The inner product of two signals $x$ and $y$, $/lt x,y/rt=\sum_n x[n]y*[n]$, measures the their correspondence: the more alike the two signals are, the greater the inner product will be. The more unlike they are, the smaller it will be, with a minimum absolute value of zero (which means the signals are orthogonal).
So it makes sense, then, that the process of matched filtering will use inner products. But it will take more than a single inner product, for we are looking for where a small signal occurs in a larger one. What we would like to do is perform an inner product between the small signal at every possible time location in the larger signal, and then examine were in the large signal this inner product is at its greatest. If we would like to find where the signal $x[n]$ occurs in signal $y[n]$, then we would simply calculate $\lt y[n],x[n-m]\gt$ at every time instance $m$ and note where the maximum value occurs.
If the process of taking an inner product at every time location in a signal sounds familiar, it is because that is essentially what discrete-time convolution is! Noting that the usual definition switches the $m$ and $n$ variables, recall the convolution sum:
$y\ast x=\sum_n y[n]x[m-n]$At every time location in the signal, the convolution sum is performing an inner-product like operation between $y[n]$ and $x[-n]$. The only thing that is missing for it to be an inner product is complex conjugation, so technically the convolution sum is performing an inner product between $y[n]$ and $x*[m-n]$; it is determining how similar $y[n]$ and $x*[-n]$ are at time $m$. So if we would like to find where $x[n]$ occurs in $y[n]$, then we would simply need to carry out the convolution $y[n]\ast x*[-n]$ and find its maximum value.