# Matched filter

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vsee on 1 Apr 2011
Commented: MUKESH KUMAR on 3 Jun 2020
How can I write a mtached filter in Matlab? I have the filter design and signal processing tool boxes.

Honglei Chen on 1 Apr 2011
If you have a signal, x, then the matched filter's coefficients is given by time reverse of x, i.e., x(end:-1:1). If your signal is complex, you also need to to use complex conjugate. You can then use it just as an FIR filter. For example,
>> x = ones(10,1);
>> b = x(end:-1:1);
>> y = filter(b,1,x);
Honglei Chen on 12 Jan 2017
It's the same, use the filter function. The only difference is when you come up with the filtering coefficient, you need to add a conjugate, like
b = conj(x(end:-1:1))
HTH

Andrew Newell on 1 Apr 2011
Check out MATLAB Simulations for Radar Systems Design from the File Exchange.

To elaborate on Honglei's answer. If the signal you are trying to design a matched filter for, is x, then --
>> b = x(end-1:1);
>> y = filter(b,1,x);
mmm ssss on 9 Jan 2012
how can implement the same filter on image ?
how i should represent x.

Mathuranathan Viswanathan on 13 Apr 2013
Hi You can try the model given here
Regards Mathuranathan

Nicole Bienert on 7 Jan 2020
The built in xcorr function can be used - you just need to normalize by fs and square the output. See below for an ideallized example:
%Purpose: Demonstrate how to match filter correctly
BW=3.84e6;
fs = 50*BW; %sample rate
T= 1/fs; %sample period
fc = 330e6; %center freq
chirpLen=0.075; %chirp length
A=3; %amplitude of voltage signal (normally unknown)
Ar=2; %amplitude of reference voltage signal (normally unknown)
%create the signal withought noise and zero padded on either side (zero
%padding not necessary because xcorr does that, I'm just demonstrating that
%signals don't need to be the same length.)
sig=[zeros(1,ceil(chirpLen*fs)),A*chirp(t,0,t(end),BW),zeros(1,ceil(chirpLen*fs))];
%create the reference chirp
ref_chirp=Ar*chirp(t,0,t(end),BW);
t=[0:T:(length(ref_chirp)-1)*T];
%normalize reference chirp: The reference chirp needs to have energy of 1
%so that it doesn't bias the output of the match filter. A filter shouldn't
%be applying gain to the signal or changing the units. The signal is in
%volts, so we divide by the square root of the energy to normalize it.
%If you know the signal's amplitude (for CW or FMCW):
energy=Ar^2/2*chirpLen;
%If you don't know the signal's amplitude, integrate to find energy (if it is noiseless):
%energy=trapz(t,ref_chirp.^2)
ref_chirp=ref_chirp/sqrt(energy);
% perform match filtering
[R,lags] = xcorr(sig,ref_chirp); %signals don't need to be the same length
%R is the sum of each data sample as the signals are shifted past
%eachother, so to make the numerical integration correct, you need to
%multiply by dx which is T in this case. Then to get the filtered voltage
%signal in units of energy, you need to square it.
R=(abs(R*T)).^2; %absolute value only necessary if signals are complex
% take only positive side
R = R(lags>=0);
lags=lags(lags>=0);
[matchFiltPeak,index]=max(R);
figure()
plot(lags*T,R)
xlim([index-250 index+250]*T)
display(['Energy in signal was: ',num2str(A.^2/2*chirpLen)])
display(['which is the same as the peak of the match filter: ',num2str(matchFiltPeak)])
MUKESH KUMAR on 3 Jun 2020

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