MATLAB Examples

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

Contents

Coded by Ilias Konsoulas, 16 Dec. 2014. Code provided for educational purposes only. All rights reserved.

clc; clear; close all;

Radar System Operational Parameters

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)

Adapted Patterns

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

Plot the Adapted Pattern

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)');

Plot the Principal Cut at Doppler = 100 Hz:

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;