A filter in signal processing is an electronic circuit that allows certain frequencies while filtering out other frequencies. Therefore, a filter can remove the intended frequencies from different signals that also include irrelevant or undesirable frequencies. In the electronics field, a filter is used in different practical applications like audio electronics, radio communications, DC power supplies, ADC, etc. There are different kinds of filters available like an active filter, bandpass filter, passive filter, low pass filter, matched filter, high pass filter, bandstop filter. Each filter has its own function and application. So this article discusses an overview of one of the types of filter namely matched filter – working & its applications.
The characteristics of matched filter include the following.
The block diagram of the matched filter is shown below. Consider the following diagram where g(t) is the input signal & w(t) is the white noise. These two signals are fed to the h(t) filter, which maximizes the signal-to-noise ratio(SNR) of the y(t) output.
The input of filter ‘x(t)’ includes a pulse signal ‘g(t) that is corrupted through additive channel noise ‘w(t)’ which is shown in the following.
X(t) = g(t) + w(t)
Where ‘t’ is an arbitrary observation interval. g(t) is the pulse signal that may signify a with a binary symbol like 0 or 1 within a digital communication system. The ‘w(t)’ is the sample function of a white noise procedure of zero mean & power spectral density is No/2. The source of uncertainty mainly lies in the noise.
The receiver shown in the diagram should be able to receive the pulse signal g(t) with a good SNR, so the output can be given as y(t) that is sampled at t =T.
So this requirement can be satisfied by optimizing the filter design to reduce the noise effects at the output of the filter in some numerical sense, and thus improve the pulse detection of the signal g(t). Since this filter is linear, then the output ‘y(t)’ may be simply expressed as
y(t) = g0(t) + n(t)
From the above equation, go(t) is the linear output corresponding to the pulse signal g(t) & n(t) is filtered noise. Both the go(t) and n(t) are generated by the signal & the noise components of the input x(t) correspondingly. The purpose of the matched filter is to maximise the output signal-to-noise ratio.
Here, the matched filter is a type of linear filter having an impulse response h(t) in the time domain, and in frequency response, is denoted by H(f) and when signal go(t) sampled at t= T >> the average power of the filter noise. Then the maximum signal to noise ratio denoted by ‘Ƞ’ would be
Ƞ = |go(T) |2 / E[n^2 (t)]
Where,
The working of a matched filter in signal processing is done by comparing a recognized template or delayed signal with an unidentified signal to notice the existence of the template within the unknown signal. So this is analogous to convolving the unidentified signal throughout a conjugated time-reversed template version.
The derivations of matched filter mainly include frequency response and an impulse response which are discussed below.
The Matched filter’s frequency response is proportional to the complexity of the spectrum of input signals. So the expression for matched filer frequency response can be mathematically written as
H(f) = GaS* (f) e^-j2πft1
Where,
Generally, the value of ‘Ga’ is considered as ‘1’, so the equation will become as
H(f) = S* (f) e^-j2πft1
The matched filter frequency response equation will have the magnitude of S∗(f) & phase angle of e^-j2πft1, which changes through frequency consistently.
In the time domain, the output of filter receiver ‘h(t)’ can be obtained through applying the IFT or inverse Fourier transform of the frequency response function, H(f).
We know the equation of H(f) = GaS* (f) e^-j2πft1, substitute this in the above equation.
We know the relation S*(f) = S*(-f)
Substitute the above equation in the ‘h(t)’ equation.
h(t) = GaS (t1-t)
Generally, the ‘Ga’ value can be considered as ‘1’ then the equation will become as
h(t) = S (t1-t)
The equation above proves that the Matched filter’s impulse response ‘h(t)’ is the received signal’s ‘s(t)’ mirror image about a time instant ‘ t1’.
Matched filters are classified into two types two dimensional (2D) and three dimensional (3D) matched filters which are discussed below.
The advantages of matched filter are shown below.
The disadvantages of matched filter are shown below.
The applications of matched filters are shown below.
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