MATLAB Examples

## Contents

```clc; clear; close all; ```

```fo = 450e6; % Operating Frequency in Hz Pt = 200e3; % Peak Transmit Power 200 kW Gt = 22; % Transmit Gain in dB Gr = 10; % Column Receive Gain in dB B = 4e6; % Receiver Instantaneous Bandwidth in Hz Ls = 4; % System Losses in dB fr = 300; % PRF in Hz Tr = 1/fr; % PRI in sec. M = 18; % Number of Pulses per CPI. Tp = 200e-6; % Pulse Width in sec. N = 18; % Number of Array Antenna Elements Gel = 4; % Element Gain in dB be = -30; % Element Backlobe Level in db Nc = 360; % Number of clutter patches uniformly distributed in azimuth. c = 299792458; % Speed of Light in m/sec. lambda = c/fo; % Operating wavelength in meters. d = lambda/2; % Interelement Spacing % Azimuth angle in degrees: phi = -180:179; Lphi = length(phi); f = zeros(1,Lphi); AF = zeros(1,Lphi); % Array Factor pre-allocation. % Platform Parameters: beta = 1; % beta parameter. ha = 9e3; % Platform altitude in meters. ```

## Thermal Noise Power Computations

```k = 1.3806488e-23; % Boltzmann Constant in J/K. To = 290; % Standard room Temperature in Kelvin. F = 3; % Receiver Noise Figure in dB; Te = To*(10^(F/10)-1); % Effective Receiver Temperature in Kelvin. Lr = 2.68; % System Losses on receive in dB. Ts = 10^(Lr/10)*Te; % Reception System Noise Temperature in Kelvin. Nn = k*Ts; % Receiver Noise PSD in Watts/Hz. Pn = Nn*B; % Receiver Noise Power in Watts sigma2 = 1; % Normalized Noise Power in Watts. ```

## Clutter Patch Geometry computations

```Rcik = 130000; % (clutter) range of interest in meters. dphi = 2*pi/Nc; % Azimuth angle increment in rad. dR = c/2/B; % Radar Range Resolution in meters. Re = 6370000; % Earth Radius in meters. ae = 4/3*Re; % Effective Earth Radius in meters. psi = asin(ha/Rcik); % Grazing angle at the clutter patch in rad (flat earth model). theta = psi; % Elevation (look-down angle) in rad. Flat earth assumption. gamma = 10^(-3/10); % Terrain-dependent reflectivity factor. phia = 0; % Velocity Misalignment angle in degrees. ```

## Clutter-to-Noise Ratio (CNR) Calculation

Calculate the Voltage Element Pattern:

```for i =1:Lphi if abs(phi(i))<=90 f(i) = cos(phi(i)*pi/180); else f(i) = 10^(be/10)*cos(phi(i)*pi/180); end end % Calculate the Array Factor (AF) (Voltage): steering_angle = 0; % Angle of beam steering in degrees. for k=1:Lphi AF(k) = sum(exp(-1i*2*pi/lambda*d*(0:N-1)*(sin(phi(k)*pi/180) ... - sin(steering_angle*pi/180)))); end % Calculate the Full Array Transmit Power Gain: Gtgain = 10^(Gt/10)*abs(AF).^2; % Calculate the Element Receive Power Gain: grgain = 10^(Gel/10)*abs(f).^2; % Clutter Patch RCS Calculation: PatchArea = Rcik*dphi*dR*sec(psi); sigma0 = gamma*sin(psi); sigma = sigma0*PatchArea; % Calculate the Clutter to Noise Ratio (CNR) for each clutter patch: ksi = Pt*Gtgain.*grgain*10^(Gr/10)*lambda^2*sigma/((4*pi)^3*Pn*10^(Ls/10)*Rcik^4); Ksic = sigma2*diag(ksi); ```

## Clutter Covariance Matrix Computations

Platform Velocity for beta parameter value:

```va = round(beta*d*fr/2); Ita = d/lambda*cos(theta); % Calculate Spatial and Doppler Frequencies for k-th clutter patch. % Spatial frequency of the k-th clutter patch: fsp = Ita*sin(phi*pi/180); % Normalized Doppler Frequency of the k-th clutter patch: omegac = beta*Ita*sin(phi*pi/180 + phia*pi/180); % Clutter Steering Vector Pre-allocation: a = zeros(N,Nc); b = zeros(M,Nc); Vc = zeros(M*N,Nc); for k=1:Nc a(:,k) = exp(1i*2*pi*fsp(k)*(0:N-1)); % Spatial Steering Vector. b(:,k) = exp(1i*2*pi*omegac(k)*(0:M-1)); % Temporal Steering Vector Vc(:,k) = kron(b(:,k),a(:,k)); % Space-Time Steering Vector. end Rc = Vc*Ksic*Vc'; % Eq. (64) Rn = sigma2*eye(M*N); ```

## Jamming Covariance Matrix Calculation

```J = 2; % Number of Jammers. thetaj = 0; phij = [-40 25]; % Jammer elevation and azimuth angles in degrees. R_j = [370 370]*1e3; Sj = 1e-3; % Jammer ERPD in Watts/Hz. fspj = d/lambda*cos(thetaj*pi/180)*sin(phij*pi/180); % Spatial frequency of the j-th jammer. Lrj = 1.92; % System Losses on Receive in dB. Aj = zeros(N,J); for j=1:J Aj(:,j) = exp(1i*2*pi*fspj(j)*(0:N-1)); % Jammer Spatial Steering Vector. end indices= zeros(1,J); for j=1:J indices(j) = find(phi == phij(j)); end grgn = grgain(indices); ksi_j = (Sj*grgn*lambda^2)./((4*pi)^2.*Nn*10^(Lrj/10).*R_j.^2); Ksi_j = sigma2*diag(ksi_j); Phi_j = Aj*Ksi_j*Aj'; % Eq. (47) % Jamming Covariance Matrix: Rj = kron(eye(M),Phi_j); % Eq. (45) ```

## Total Interference Covariance Matrix

```Ru = Rc + Rj + Rn; % Eq. (98) InvRu = inv(Ru); ```

## Target Space-Time Steering Vector

```phit = 0; thetat = 0; % Target azimuth and elevation angles in degrees. fdt = 100; % Target Doppler Frequency. fspt = d/lambda*cos(thetat*pi/180)*sin(phit*pi/180); omegat = fdt/fr; bt = exp(-1i*2*pi*omegat*(0:M-1)).'; % Target Doppler Steering Vector. at = exp(-1i*2*pi*fspt*(0:N-1)).'; % Target Spatial Steering Vector. ta = chebwin(N,30); % 30 dB Chebychev Spatial Tapper. gt = kron(bt,ta.*at); ```

## Doppler Filter Bank Creation:

```dopplerfilterbank = linspace(-150,150,M+1); omegadopplerbank = dopplerfilterbank/fr; ```

## Doppler Filter Matrix Construction for Adjacent Bin Post-Doppler method:

```K = 2; P = floor(K/2); U2 = zeros(M,M); if mod(K,2) % If K is odd for m=1:M U2(:,m) = 1/sqrt(M)*exp(-1i*2*pi*omegadopplerbank(m)*(0:M-1)); % Doppler Filter Steering Vector end else % if K is even: outomegadoppler = zeros(1,M); for m=1:M outomegadoppler(m) = (omegadopplerbank(m) + omegadopplerbank(m+1))/2; U2(:,m) = 1/sqrt(M)*exp(-1i*2*pi*outomegadoppler(m)*(0:M-1)); % Doppler Filter Steering Vector end end td0ab = ones(M,1); % Uniform Doppler Taper. td30ab = chebwin(M,30); % 30-dB Chebyshev Doppler Taper. td60ab = chebwin(M,60); % 60-dB Chebyshev Doppler Taper. td90ab = chebwin(M,90); % 90-dB Chebyshev Doppler Taper. % Create Doppler Filter Bank in Fab matrix for Adjacent Bin post Doppler method: Fab0 = diag(td0ab)*U2; % Eq. 230 without complex conjugation operator. Fab30 = diag(td30ab)*U2; Fab60 = diag(td60ab)*U2; Fab90 = diag(td90ab)*U2; Fmab0 = zeros(M,K,M); Fmab30 = zeros(M,K,M); Fmab60 = zeros(M,K,M); Fmab90 = zeros(M,K,M); for m=1:M if mod(K,2) % if K is odd and >1. if (m-P>0) && (m+P<=M) % [m m-P:m+P] % omegadopplerbank(m) % omegadopplerbank(m-P:m+P) Fmab0(:,:,m) = Fab0(:,m-P:m+P); % Eq. 231. Fmab30(:,:,m) = Fab30(:,m-P:m+P); Fmab60(:,:,m) = Fab60(:,m-P:m+P); Fmab90(:,:,m) = Fab90(:,m-P:m+P); elseif (m-P<=0) && (m+P<=M) % [m M+(m-P):M 1:m+P] % omegadopplerbank(m) % omegadopplerbank([M+(m-P):M 1:m+P]) Fmab0(:,:,m) = [Fab0(:,M+(m-P):M) Fab0(:,1:m+P)]; % Eq. 231. Fmab30(:,:,m) = [Fab30(:,M+(m-P):M) Fab30(:,1:m+P)]; Fmab60(:,:,m) = [Fab60(:,M+(m-P):M) Fab60(:,1:m+P)]; Fmab90(:,:,m) = [Fab90(:,M+(m-P):M) Fab90(:,1:m+P)]; elseif m+P>M % [m m-P:M 1:m+P-M] % omegadopplerbank(m) % omegadopplerbank([m-P:M 1:m+P-M]) Fmab0(:,:,m) = [Fab0(:,m-P:M) Fab0(:,1:m+P-M)]; % Eq. 231. Fmab30(:,:,m) = [Fab30(:,m-P:M) Fab30(:,1:m+P-M)]; Fmab60(:,:,m) = [Fab60(:,m-P:M) Fab60(:,1:m+P-M)]; Fmab90(:,:,m) = [Fab90(:,m-P:M) Fab90(:,1:m+P-M)]; end else % if K is even. if (m-P>0) && (m+P<=M+1) % [m m-P:m+P-1] % omegadopplerbank(m) % outomegadoppler(m-P:m+P-1) Fmab0(:,:,m) = Fab0(:,m-P:m+P-1); % Eq. 231. Fmab30(:,:,m) = Fab30(:,m-P:m+P-1); Fmab60(:,:,m) = Fab60(:,m-P:m+P-1); Fmab90(:,:,m) = Fab90(:,m-P:m+P-1); elseif (m-P<=0) && (m+P<=M) % [m M+(m-P):M 1:m+P-1] % omegadopplerbank(m) % outomegadoppler([M+(m-P):M 1:m+P-1]) Fmab0(:,:,m) = [Fab0(:,M+(m-P):M) Fab0(:,1:m+P-1)]; % Eq. 231. Fmab30(:,:,m) = [Fab30(:,M+(m-P):M) Fab30(:,1:m+P-1)]; Fmab60(:,:,m) = [Fab60(:,M+(m-P):M) Fab60(:,1:m+P-1)]; Fmab90(:,:,m) = [Fab90(:,M+(m-P):M) Fab90(:,1:m+P-1)]; elseif m+P>M+1 % [m m-P:M 1:m-M+P-1] % omegadopplerbank(m) % outomegadoppler([m-P:M 1:m-M+P-1]) Fmab0(:,:,m) = [Fab0(:,m-P:M) Fab0(:,1:m-M+P-1)]; % Eq. 231. Fmab30(:,:,m) = [Fab30(:,m-P:M) Fab30(:,1:m-M+P-1)]; Fmab60(:,:,m) = [Fab60(:,m-P:M) Fab60(:,1:m-M+P-1)]; Fmab90(:,:,m) = [Fab90(:,m-P:M) Fab90(:,1:m-M+P-1)]; end end end ```

```m = 16; % This is the Target's Doppler Bin. f0abm = Fmab0(:,:,m); % Select a Cluster of K Adjacent Doppler Bins. f30abm = Fmab30(:,:,m); f60abm = Fmab60(:,:,m); f90abm = Fmab90(:,:,m); R0abum = kron(f0abm,eye(N))'*Ru*kron(f0abm,eye(N)); R30abum = kron(f30abm,eye(N))'*Ru*kron(f30abm,eye(N)); R60abum = kron(f60abm,eye(N))'*Ru*kron(f60abm,eye(N)); R90abum = kron(f90abm,eye(N))'*Ru*kron(f90abm,eye(N)); gt0abm = kron(f0abm,eye(N))'*gt; % Eq. 141 gt30abm = kron(f30abm,eye(N))'*gt; gt60abm = kron(f60abm,eye(N))'*gt; gt90abm = kron(f90abm,eye(N))'*gt; % Calculate K*N X 1 Adaptive Weight for m-th Doppler Bin. w0abm = R0abum\gt0abm; % Eq. 205 w30abm = R30abum\gt30abm; w60abm = R60abum\gt60abm; w90abm = R90abum\gt90abm; wab0 = kron(f0abm,eye(N))*w0abm; wab30 = kron(f30abm,eye(N))*w30abm; wab60 = kron(f60abm,eye(N))*w60abm; wab90 = kron(f90abm,eye(N))*w90abm; ```

```phi = -90:90; Lphi = length(phi); fd = -150:150; Lfd = length(fd); fsp = d/lambda*cos(theta*pi/180)*sin(phi*pi/180); omega = fd/fr; Pw0 = zeros(Lfd,Lphi); Pw30 = zeros(Lfd,Lphi); Pw60 = zeros(Lfd,Lphi); Pw90 = zeros(Lfd,Lphi); for m1=1:Lphi for n=1:Lfd a = exp(-1i*2*pi*fsp(m1)*(0:N-1)); % Dummy Spatial Steering Vector. b = exp(-1i*2*pi*omega(n)*(0:M-1)); % Dummy Doppler Steering Vector v = kron(b,a).'; Pw0(n,m1) = abs(wab0'*v)^2; Pw30(n,m1) = abs(wab30'*v)^2; Pw60(n,m1) = abs(wab60'*v)^2; Pw90(n,m1) = abs(wab90'*v)^2; end end ```

## Normalisation:

```max_value0 = max(max(Pw0)); max_value30 = max(max(Pw30)); max_value60 = max(max(Pw60)); max_value90 = max(max(Pw90)); Pw0 = Pw0/max_value0; Pw30 = Pw30/max_value30; Pw60 = Pw60/max_value60; Pw90 = Pw90/max_value90; [rows0 cols0] = find(10*log10(abs(Pw0))<-150); for i=1:length(rows0) Pw0(rows0(i),cols0(i)) = 10^(-150/10); end [rows30 cols30] = find(10*log10(abs(Pw30))<-150); for i=1:length(rows30) Pw30(rows30(i),cols30(i)) = 10^(-150/10); end [rows60 cols60] = find(10*log10(abs(Pw60))<-150); for i=1:length(rows60) Pw60(rows60(i),cols60(i)) = 10^(-150/10); end [rows90 cols90] = find(10*log10(abs(Pw90))<-150); for i=1:length(rows90) Pw90(rows90(i),cols90(i)) = 10^(-150/10); end ```

```figure('NumberTitle', 'off','Name', ... ['Figure 48.2. Adapted Patterns for Adjacent-Bin post Doppler STAP for K = ', ... num2str(K), ' and Doppler bin ' num2str(m)], ... 'Position', [1 1 1000 1000]); subplot(2,2,1); [Az Doppler] = meshgrid(sin(phi*pi/180),fd); colormap jet; pcolor(Az, Doppler, 10*log10(abs(Pw0))); shading interp; xlim([-1 1]) ylim([-150 150]); % xlabel('sin(Azimuth)'); ylabel('Doppler Frequency (Hz)'); h = colorbar; % set(get(h,'YLabel'),'String','Relative Power (dB)'); title('Doppler Filters Untapered'); subplot(2,2,2); pcolor(Az, Doppler, 10*log10(abs(Pw30))); shading interp; xlim([-1 1]) ylim([-150 150]); % xlabel('sin(Azimuth)'); % ylabel('Doppler Frequency (Hz)'); title('30 dB Chebychev Doppler Taper'); h = colorbar; set(get(h,'YLabel'),'String','Relative Power (dB)'); subplot(2,2,3); pcolor(Az, Doppler, 10*log10(abs(Pw60))); shading interp; xlim([-1 1]) ylim([-150 150]); xlabel('sin(Azimuth)'); ylabel('Doppler Frequency (Hz)'); title('60 dB Chebychev Doppler Taper'); h = colorbar; % set(get(h,'YLabel'),'String','Relative Power (dB)'); subplot(2,2,4); pcolor(Az, Doppler, 10*log10(abs(Pw90))); shading interp; xlim([-1 1]) ylim([-150 150]); xlabel('sin(Azimuth)'); % ylabel('Doppler Frequency (Hz)'); title('90 dB Chebychev Doppler Taper'); h = colorbar; set(get(h,'YLabel'),'String','Relative Power (dB)'); tightfig; ```