MATLAB Examples

# Figure 41. Example performance for Doppler bin 6 (100 Hz) with 80-dB Chebyshev Doppler filters.

## 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) ```

## Target Space-Time Steering Vector

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

## Doppler Filter Matrix Construction

```dopplerfilterbank = linspace(0,300,M+1); omegadopplerbank = dopplerfilterbank/fr; U = zeros(M,M); for m=1:M U(:,m) = 1/sqrt(M)*exp(-1i*2*pi*omegadopplerbank(m)*(0:M-1)); % Doppler Filter Steering Vector end td = chebwin(M,80); % 80-dB Chebyshev Doppler Taper. F = diag(td)*conj(U); % Eq. (189) ```

## Solve M Separate N-dimensional Adaptive Problems

```ksicm = zeros(Nc,M+1); W = zeros(N,M); for m=1:M fm = F(:,m); Rum = kron(fm,eye(N))'*Ru*kron(fm,eye(N)); % Eq. (196) wm = Rum\gt; % Eq. (195) W(:,m) = wm; for k=1:Nc ksicm(k,m) = ksi(k)*abs(fm'*b(:,k))^2; % Eq. (200) end end ```

## The Doppler Bin #6 corresponds to the seventh column of the above matrices

```msel = 7; wmsel = W(:,msel); fmsel = F(:,msel); w = kron(fmsel,wmsel); % Eq. (202) ```

```phi1 = -90:90; Lphi = length(phi1); fd = -150:150; Lfd = length(fd); fsp = d/lambda*cos(theta*pi/180)*sin(phi1*pi/180); omega = fd/fr; Pw1 = zeros(Lfd,Lphi); for m=1:Lphi for n=1:Lfd a = exp(1i*2*pi*fsp(m)*(0:N-1)); % Target Spatial Steering Vector. b = exp(1i*2*pi*omega(n)*(0:M-1)); % Dummy Target Doppler Steering Vector v = kron(b,a).'; Pw1(n,m) = abs(w'*v)^2; end end ```

## Normalisation

```    max_value = max(max(Pw1));
Pw = Pw1/max_value;```
```[rows cols] = find(10*log10(abs(Pw1))<-100); for i=1:length(rows) Pw1(rows(i),cols(i)) = 10^(-100/10); end ```

```figure('NumberTitle', 'off','Name', ... ' Figure 41. Example performance for Doppler bin 6 (100 Hz) with 80-dB Chebyshev Doppler filters.',... 'Position', [1 1 700 600]); [Az Doppler] = meshgrid(phi1,fd); colormap jet; surf(Az, Doppler, 10*log10(abs(Pw1))); shading interp; xlim([-90 90]) ylim([-150 150]); xlabel('sin(Azimuth)'); ylabel('Doppler Frequency (Hz)'); h = colorbar; set(get(h,'YLabel'),'String','Relative Power (dB)'); ```
```figure('NumberTitle', 'off','Name', ... ' Figure 41. Example performance for Doppler bin 6 (100 Hz) with 80-dB Chebyshev Doppler filters.',... 'Position', [1 1 1000 400]); % a. Cut of the Adapted Pattern at Doppler = 100 Hz. subplot(1,2,2); plot( phi1, 10*log10(abs(Pw1(fd == fdt,:))),'LineWidth',1.5); ylim([-80 40]); xlim([-90 90]); ylabel('Magnitude (dB)'); xlabel('Azimuth Angle (deg)'); title('Cut of the Adapted Pattern at Doppler = 100 Hz'); grid on; % Plot the CNR at bin #6. subplot(1,2,1); plot(phi(91:271),10*log10(abs(ksi(91:271))),'--','LineWidth',1.5) hold on; plot(phi(91:271),10*log10(abs(ksicm(91:271,msel))),'r','LineWidth',1.5) ylabel('CNR (dB)'); xlabel('Azimuth Angle (deg)'); title('Clutter Power Spectral Density'); ylim([-80 40]); xlim([-90 90]); grid on; legend('Single Pulse','Doppler Bin #6','Location','NorthWest'); tightfig; ```