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This module develops the matched filter detector, including its mathematical justification based on the Cauchy-Scwarz inequality, its practical implementation via convolution, and several example applications.

Introduction

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 x | | x | | , y | | y | | attains its upper bound, which is 1, when y = a x 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 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.

Matched filter detector theory

Signal comparison

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 X we wish to evaluate

m ( x , y ) = x | | x | | , y | | y | |

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

x m = argmax x X x | | x | | , y | | y | | .

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 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 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 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.

Questions & Answers

what is variations in raman spectra for nanomaterials
Jyoti Reply
I only see partial conversation and what's the question here!
Crow Reply
what about nanotechnology for water purification
RAW Reply
please someone correct me if I'm wrong but I think one can use nanoparticles, specially silver nanoparticles for water treatment.
Damian
yes that's correct
Professor
I think
Professor
what is the stm
Brian Reply
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Rafiq
industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
Damian
How we are making nano material?
LITNING Reply
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LITNING Reply
What is meant by 'nano scale'?
LITNING Reply
What is STMs full form?
LITNING
scanning tunneling microscope
Sahil
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Santosh
Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
Rafiq
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Mahi
what is simplest way to understand the applications of nano robots used to detect the cancer affected cell of human body.? How this robot is carried to required site of body cell.? what will be the carrier material and how can be detected that correct delivery of drug is done Rafiq
Rafiq
what is Nano technology ?
Bob Reply
write examples of Nano molecule?
Bob
The nanotechnology is as new science, to scale nanometric
brayan
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
Damian
Is there any normative that regulates the use of silver nanoparticles?
Damian Reply
what king of growth are you checking .?
Renato
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
Stoney Reply
why we need to study biomolecules, molecular biology in nanotechnology?
Adin Reply
?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
Adin
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Adin
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Kyle
biomolecules are e building blocks of every organics and inorganic materials.
Joe
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Damian Reply
research.net
kanaga
sciencedirect big data base
Ernesto
Introduction about quantum dots in nanotechnology
Praveena Reply
what does nano mean?
Anassong Reply
nano basically means 10^(-9). nanometer is a unit to measure length.
Bharti
do you think it's worthwhile in the long term to study the effects and possibilities of nanotechnology on viral treatment?
Damian Reply
absolutely yes
Daniel
how did you get the value of 2000N.What calculations are needed to arrive at it
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Source:  OpenStax, Signals and systems. OpenStax CNX. Aug 14, 2014 Download for free at http://legacy.cnx.org/content/col10064/1.15
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