MATLAB Examples

# Figure 11. Illustrating Brennan's Rule: Clutter Eigenspectra eigenspectra for the example radar system with different platform velocities.

## 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 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 = 361; % 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 = -180:180; Lphi = length(phi); f = zeros(1,Lphi); AF = zeros(1,Lphi); % Array Factor vector pre-allocation. 

## Platform Parameters.

beta = [0.6 1 2 2.83 3]; % Beta Parameter Vector. ha = 9e3; % Platform altitude in meters. Rc = 13e4; % (clutter) range of interest 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. Nn = k*Te; % Receiver Noise PSD in Watts/Hz. Pn = Nn*B; % Receiver Noise Power in Watts 

## Clutter Patch Geometry Computations.

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/Rc); % Grazing angle at the clutter patch in rad (flat earth model). gamma = 10^(-3/10); % Terrain-dependent reflectivity factor. theta = psi; 

## 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 and Plot 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))).*cos(phi(k)*pi/180)); end 

## Calculate and Plot the Full Array Transmit Power Gain.

Gtgain = 10^(Gt/10)*abs(AF).^2; % Calculate and Plot the Element Receive Power Gain: grgain = 10^(Gel/10)*10^(Gr/10)*abs(f).^2; 

## Clutter Patch RCS Calculation.

PatchArea = Rc*dphi*dR*sec(psi); sigma0 = gamma*sin(psi); sigma = sigma0*PatchArea; 

## Calculate and Plot the Clutter to Noise Ration (CNR) for each clutter patch:

ksi = Pt*Gtgain.*grgain*lambda^2*sigma/((4*pi)^3*Pn*10^(Ls/10)*Rc^4); 

## Create Spatial Steering Vector:

a = zeros(N,Nc); b = zeros(M,Nc); Vc = zeros(M*N,Nc); Rc = zeros(M*N,M*N); Ksic = diag(ksi); colors = [0 0 1; 0 1 0; 1 0 0 ; 1 1 0; 0 1 1;]; figure('NumberTitle', 'off','Name', ... 'Figure 11. Illustrating Brennan''s Rule: Clutter Eigenspectra for Different Platform Velocities',... 'Position', [50 50 700 550]); for i=1:length(beta) for k=1:Nc fsp = d/lambda*cos(theta)*sin(phi(k)*pi/180); % Spatial frequency of the k-th clutter patch. a(:,k) = exp(1i*2*pi*fsp*(0:N-1)); % Spatial Steering Vector. omegac = beta(i)*fsp; % Normalized Doppler frequency of the k-th clutter patch. b(:,k) = exp(1i*2*pi*omegac*(0:M-1)); % Temporal Steering Vector Vc(:,k) = kron(b(:,k),a(:,k)); % Space-Time Steering Vector. end Rc = Vc*Ksic*Vc'; plot(10*log10(abs(eig(Rc))),'--s','LineWidth',1,'Color', colors(i,:), ... 'MarkerEdgeColor','k','MarkerFaceColor',colors(i,:), 'MarkerSize',5); hold on; end va = round(beta*d*fr/2); legend(['\beta = 0.6, v_a = ',num2str(va(1))], ['\beta = 1, v_a = ',num2str(va(2))], ... ['\beta = 2, v_a = ',num2str(va(3))], ['\beta = 2.83, v_a = ',num2str(va(4))], ... ['\beta = 3, v_a = ',num2str(va(5))]); ylim([-60 80]); xlim([1 100]); grid on; xlabel('Eigenvalue Number'); ylabel('Relative Power (dB)'); for i=1:length(beta) X = [round(N+(M-1)*beta(i)), round(N+(M-1)*beta(i))]; Y = [-60, 80]; line(X,Y,'Color',colors(i,:),'LineWidth',2) end