Detecting Signals with Known Form: Matched Filters

Abstract

The focus of this chapter is on the detection of signals that have a known form or structure when they are occluded by additive noise. The most common example of signals with known form is from active remote sensing where a known source signal is projected into the underwater environment and the sounds measured by a sensor are then analyzed to achieve one of the inferential objectives. Matched-filter detectors (also known as replica correlators or pulse compressors) are derived under a variety of assumptions about the signal amplitude and phase. Waveform autocorrelation and ambiguity functions are introduced to represent the response of the matched filter to mismatch in time and Doppler and presented for the standard sonar pulses (continuous-wave, linear- and hyperbolic-frequency-modulated pulses). Conventional beamforming of an array of sensors is shown to be a spatial matched filter detector. The differences between resolution, accuracy, and ambiguity with respect to parameter estimation are articulated. Lower bounds on the variance are derived for estimating signal strength, phase, time of arrival, and Doppler scale, including examples using the standard sonar pulses. Many practical aspects of sonar signal processing are covered including the effect of oversampling on false alarm rate, normalization of a time-varying background noise and reverberation power (cell-averaging and order-statistic normalizers), Doppler filter banks (with a fast-Fourier-transform implementation for continuous-wave pulses), and using incoherent integration to surmount temporal spreading losses.

Appendix 8.A: Example MATLAB® Code for a Doppler Filter Bank

The following MATLAB^® code implements a Doppler filter bank using fast Fourier transforms (FFTs) as described in Sect. 8.7.1.4. It is assumed that the data have been basebanded and decimated to a band appropriate to the radial velocities of interest. In producing the filter bank, a 50% overlap in time has been implemented and zero-padding of the FFT has been used to interpolate in frequency, which produces correlated Doppler channels. The response of the filter bank is converted to radial velocity assuming a monostatic active sonar geometry. The sample signal generated includes reverberation and an echo near 30 s with approximately 10-kn radial velocity.

% Setup for Doppler filter bank Ts=60;     % Pulse repetition period (s) Tp=1;      % Pulse duration (s) fc=1500;   % Frequency of pulse fs=40;     % Sampling frequency (Hz) cwkn=2916; % Speed of sound (kn) %--------------------------------------------------------- % Generate sample complex-envelope data Nx=round(Ts∗fs); tx=(0:Nx-1)'/fs; x=randn(Nx,2)∗[1;1j]/sqrt(2)+5./sqrt(1+tx)...   +2∗exp(1j∗2∗pi∗10∗tx).∗(abs(tx-Tp/2-30)<Tp/2); %--------------------------------------------------------- % Number of samples in pulse duration N=round(Tp∗fs); % FFT size (increase the '1' to interpolate more in frequency) Nfft=2ˆ(nextpow2(N)+1); % Number of FFTs without overlapping Nt=floor((length(x)-N/2)/N); % Pulse shading vector & matrix a=tukeywin(N,0.1); A=a∗ones(1,Nt); % FFTs for non-overlapped windows Y1=fftshift(fft(A.∗reshape(x(1:Nt∗N),N,Nt),Nfft),1); % FFTs for 50% overlapping Y2=fftshift(fft(A.∗reshape(x(round(N/2)...    +(1:Nt∗N)),N,Nt),Nfft),1); % Combine into one matrix Y=10∗log10(abs(reshape([Y1;Y2],Nfft,2∗Nt)).ˆ2)'; % Velocity and time vectors for plotting vkn=[((Nfft/2):(Nfft-1))-Nfft 0:(Nfft/2-1)]'∗...      fs∗cwkn/(2∗fc∗Nfft); ts=(0:size(Y,2)-1)∗(N/2)/fs; %--------------------------------------------------------- % Plot normalized intensity in decibels figure(2); clf; imagesc(vkn,ts,Y-median(Y(:))); set(gca,'YDir','normal'); set(gca,'TickDir','out'); shading flat xlabel('Radial velocity (kn)'); ylabel('Time (s)'); caxis([-5 25]); colorbar;