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Matched Filtering

Reasons for Using Matched Filtering

You can see from the results in Receiver Operating Characteristic (ROC) Curves that the probability of detection increases with increasing SNR. For a deterministic signal in white Gaussian noise, you can maximize the SNR at the receiver by using a filter matched to the signal. The matched filter is a time-reversed and conjugated version of the signal. The matched filter is shifted to be causal.

Support for Matched Filtering

Use phased.MatchedFilter to implement a matched filter.

When you use phased.MatchedFilter, you can customize characteristics of the matched filter such as the matched filter coefficients and window for spectrum weighting. If you apply spectrum weighting, you can specify the coverage region and coefficient sample rate; Taylor, Chebyshev, and Kaiser windows have additional properties you can specify.

Matched Filtering of Linear FM Waveform

This example shows how to compare the results of matched filtering with and without spectrum weighting. Spectrum weighting is often used with linear FM waveforms to reduce the sidelobes in the time domain.

Create a linear FM waveform with a duration of 0.1 milliseconds, a sweep bandwidth of 100 kHz, and a pulse repetition frequency of 5 kHz. Add noise to the linear FM pulse and filter the noisy signal using a matched filter. This example applies a matched filter with and without spectrum weighting.

% Specify the waveform.
hwav = phased.LinearFMWaveform('PulseWidth',1e-4,'PRF',5e3,...
    'SampleRate',1e6,'OutputFormat','Pulses','NumPulses',1,...
    'SweepBandwidth',1e5);
w = getMatchedFilter(hwav);

% Create a matched filter with no spectrum weighting, and a
% matched filter that uses a Taylor window for spectrum
% weighting.
hmf = phased.MatchedFilter('Coefficients',w);
hmf_taylor = phased.MatchedFilter('Coefficients',w,...
    'SpectrumWindow','Taylor');

% Create the signal and add noise.
sig = step(hwav);
rng(17)
x = sig+0.5*(randn(length(sig),1)+1j*randn(length(sig),1));

% Filter the noisy signal separately with each of the filters.
y = step(hmf,x);
y_taylor = step(hmf_taylor,x);

% Plot the real parts of the waveform and noisy signal.
t = linspace(0,numel(sig)/hwav.SampleRate,...
    hwav.SampleRate/hwav.PRF);
subplot(2,1,1);
plot(t,real(sig)); title('Input Signal');
xlim([0 max(t)]); grid on
ylabel('Amplitude');
subplot(2,1,2);
plot(t,real(x)); title('Input Signal + Noise');
xlim([0 max(t)]); grid on
xlabel('Seconds'); ylabel('Amplitude');

% Plot the magnitudes of the two matched filter outputs.
figure;
plot(t,abs(y),'b--');
title('Matched Filter Output');
xlim([0 max(t)]); grid on
hold on;
plot(t,abs(y_taylor),'r-');
ylabel('Magnitude'); xlabel('Seconds');
legend('No Spectrum Weighting','Taylor Window');
hold off;

Matched Filtering to Improve SNR for Target Detection

This example shows how to improve the SNR by performing matched filtering.

Place an isotropic antenna element at the global origin [0;0;0]. Then, place a target with a nonfluctuating RCS of 1 square meter at [5000;5000;10], which is approximately 7 km from the transmitter. Set the operating (carrier) frequency to 10 GHz. To simulate a monostatic radar, set the InUseOutputPort property on the transmitter to true. Calculate the range and angle from the transmitter to the target.

hsensor = phased.IsotropicAntennaElement(...
   'FrequencyRange',[5e9 15e9]);
htx = phased.Transmitter('Gain',20,'InUseOutputPort',true);
fc = 10e9;
htgt = phased.RadarTarget('Model','Nonfluctuating',...
   'MeanRCS',1,'OperatingFrequency',fc);
txloc = [0;0;0];
tgtloc = [5000;5000;10];
htxloc = phased.Platform('InitialPosition',txloc);
htgtloc = phased.Platform('InitialPosition',tgtloc);
[tgtrng,tgtang] = rangeangle(htgtloc.InitialPosition,...
   htxloc.InitialPosition);

Create a rectangular pulse waveform 25 μs in duration with a PRF of 10 kHz. Use a single pulse for this example. Determine the maximum unambiguous range for the given PRF. Use the radareqpow function to determine the peak power required to detect a target. This target has an RCS of 1 square meter at the maximum unambiguous range for the transmitter operating frequency and gain. The SNR is based on a desired false-alarm rate of 1e-6 for a noncoherent detector.

hwav = phased.RectangularWaveform('PulseWidth',25e-6,...
   'OutputFormat','Pulses','PRF',1e4,'NumPulses',1);
c = physconst('LightSpeed');
maxrange = c/(2*hwav.PRF);
SNR = npwgnthresh(1e-6,1,'noncoherent');
Pt = radareqpow(c/fc,maxrange,SNR,...
   hwav.PulseWidth,'RCS',htgt.MeanRCS,'Gain',htx.Gain);

Set the peak transmit power to the output value from radareqpow.

htx.PeakPower = Pt;

Create radiator and collector objects that operate at 10 GHz. Create a free space path for the propagation of the pulse to and from the target. Then, create a receiver and a matched filter for the rectangular waveform.

hrad = phased.Radiator('PropagationSpeed',c,...
   'OperatingFrequency',fc,'Sensor',hsensor);
hspace = phased.FreeSpace('PropagationSpeed',c,...
   'OperatingFrequency',fc,'TwoWayPropagation',false);
hcol = phased.Collector('PropagationSpeed',c,...
   'OperatingFrequency',fc,'Sensor',hsensor);
hrec = phased.ReceiverPreamp('NoiseFigure',0,...
   'EnableInputPort',true,'SeedSource','Property','Seed',2e3);
hmf = phased.MatchedFilter(...
   'Coefficients',getMatchedFilter(hwav),...
   'GainOutputPort',true);

After you create all the objects that define your model, you can propagate the pulse to and from the target. Collect the echo at the receiver, and implement the matched filter to improve the SNR.

% Generate waveform
wf = step(hwav);
% Transmit waveform
[wf,txstatus] = step(htx,wf);
% Radiate pulse toward the target
wf = step(hrad,wf,tgtang);
% Propagate pulse toward the target
wf = step(hspace,wf,txloc,tgtloc,[0;0;0],[0;0;0]);
% Reflect it off the target
wf = step(htgt,wf);
% Propagate the pulse back to transmitter
wf = step(hspace,wf,tgtloc,txloc,[0;0;0],[0;0;0]);
% Collect the echo
wf = step(hcol,wf,tgtang);

% Receive target echo
rx_puls = step(hrec,wf,~txstatus);
[mf_puls,mfgain] = step(hmf,rx_puls);
% Get group delay of matched filter
Gd = length(hmf.Coefficients)-1;
% The group delay is constant
% Shift the matched filter output
mf_puls=[mf_puls(Gd+1:end); mf_puls(1:Gd)];
subplot(2,1,1);
t = unigrid(0,1e-6,1e-4,'[)');
rangegates = c.*t;
rangegates = rangegates/2;
plot(rangegates,abs(rx_puls)); title('Received Pulse');
ylabel('Amplitude'); hold on;
plot([tgtrng, tgtrng], [0 max(abs(rx_puls))],'r');
subplot(2,1,2)
plot(rangegates,abs(mf_puls)); title('With Matched Filtering');
xlabel('Meters'); ylabel('Amplitude'); hold on;
plot([tgtrng, tgtrng], [0 max(abs(mf_puls))],'r');

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