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A great many applications in signal processing, image processing, and beyond involve determining the presence and location of a target signal within some other signal. A radar system, for example, searches for copies of a transmitted radar pulse in order to determine the presence of and distance to reflective objects such as buildings or aircraft. A communication system searches for copies of waveforms representing digital 0s and 1s in order to receive a message.
Two key mathematical tools that contribute to these applications are inner products and the Cauchy-Schwarz inequality . As is shown in the module on the Cauchy-Schwarz inequality, the expression $\left|\u3008\frac{x}{\left|\right|x\left|\right|},,,\frac{y}{\left|\right|y\left|\right|}\u3009\right|$ attains its upper bound, which is 1, when $y=ax$ for some scalar $a$ in a real or complex field. The lower bound, which is 0, is attained when $x$ and $y$ are orthogonal. In informal intuition, this means that the expression is maximized when the vectors $x$ and $y$ have the same shape or pattern and minimized when $x$ and $y$ are very different. A pair of vectors with similar but unequal shapes or patterns will produce relatively large value of the expression less than 1, and a pair of vectors with very different but not orthogonal shapes or patterns will produce relatively small values of the expression greater than 0. Thus, the above expression carries with it a notion of the degree to which two signals are “alike”, the magnitude of the normalized correlation between the signals in the case of the standard inner products.
This concept can be extremely useful. For instance consider a situation in which we wish to determine which signal, if any, from a set $X$ of signals most resembles a particular signal $y$ . In order to accomplish this, we might evaluate the above expression for every signal $x\in X$ , choosing the one that results in maxima provided that those maxima are above some threshold of “likeness”. This is the idea behind the matched filter detector, which compares a set of signals against a target signal using the above expression in order to determine which is most like the target signal.
The simplest variant of the matched filter detector scheme would be to find the member signal in a set $X$ of signals that most closely matches a target signal $y$ . Thus, for every $x\in X$ we wish to evaluate
in order to compare every member of $X$ to the target signal $y$ . Since the member of $X$ which most closely matches the target signal $y$ is desired, ultimately we wish to evaluate
Note that the target signal does not technically need to be normalized to produce a maximum, but gives the desirable property that $m(x,y)$ is bounded to $[0,1]$ .
The element ${x}_{m}\in X$ that produces the maximum value of $m(x,y)$ is not necessarily unique, so there may be more than one matching signal in $X$ . Additionally, the signal ${x}_{m}\in X$ producing the maximum value of $m(x,y)$ may not produce a very large value of $m(x,y)$ and thus not be very much like the target signal $y$ . Hence, another matched filter scheme might identify the argument producing the maximum but only above a certain threshold, returning no matching signals in $X$ if the maximum is below the threshold. There also may be a signal $x\in X$ that produces a large value of $m(x,y)$ and thus has a high degree of “likeness” to $y$ but does not produce the maximum value of $m(x,y)$ . Thus, yet another matched filter scheme might identify all signals in $X$ producing local maxima that are above a certain threshold.
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