Contents

%
%   k-space 2D imaging
%   Thomas Fromenteze
%


clc
clear all

Scene definition

% Parameters
nbf = 201;                  % number of frequency samples
f = linspace(10e9,20e9,nbf); % frequency vector
c = 3e8;                     % speed of light
k = 2*pi*f/c;                % wavenumber
df = f(2)-f(1);              % frequency resolution
t = linspace(0,1/df,nbf);    % time vector

% Transmit antenna
xT = 0;
yT = -0.05;

% Receive array
dxR = c/f(end)/2;            % receive antenna array sampling
nbR = 51;                   % number of antennas
xR = (1:nbR)*dxR;            % receive array coordinates
xR = xR - mean(xR);          % centered on x = 0;
yR = xR .*0;                 % receive array on the plane y = 0;

% Target
xC = 0.1;
yC = 0.4;
SigC = 1; % Reflectivity

fig2 = figure(2);clf
hold on
plot(xT,yT,'kv')
plot(xR,yR,'rv')
plot(xC,yC,'bo')
hold off
legend('Transmit antennas','Receive antennas','Target(s)')
grid on
xlabel('x (m)')
ylabel('y (m)')
daspect([1 1 1])
xlim([-0.4 0.4])
ylim([-0.1 1])

fig1 = figure(1);
fig1.Position = [680 542 871 420];

set(fig1,'color','w')

Propagation

% for loops are used to facilitate understanding. It's a very bad idea if you try to optimize computation times

S = zeros(nbR,numel(f));

rT = sqrt((xT-xC).^2 + (yT-yC).^2); % distance from transmit antennas to target(s)
rR = sqrt((xR-xC).^2 + (yR-yC).^2); % distance from targets to the receive antenna

for mf = 1:numel(f)
    S(:,mf) = S(:,mf) + (exp(-1j*k(mf)*rT)./sqrt(rT) .* SigC .* exp(-1j*k(mf)*rR)./sqrt(rR)).'; % interaction of the transmit waves with the target(s) and measurement
end

Skx = fftshift(fft(S,[],1),1);
dx = mean(diff(xR));
kx = linspace(-pi/dx,pi/dx,nbR);

[Kx,K] = ndgrid(kx,k);
KyR = sqrt(K.^2 - Kx.^2);
Ky = K + KyR;

y = linspace(0.05,0.75,101);

Skx_y = zeros(numel(kx),numel(y));
for my = 1:numel(y)
%         Skx_y(:,my) = sum(Skx .* exp(1j*Ky*y(my)),2);
    Skx_y(:,my) = sum(Skx .* exp(1j*(KyR*y(my)+K*(y(my)-yT))),2);
end

dkx = mean(diff(kx));
x = linspace(-pi/dkx,pi/dkx,numel(kx));
Im = (ifft(Skx_y,[],1));

figure(1), clf()
subplot(2,3,[1 4])
hold on
plot(xT,yT,'kv')
plot(xR,yR,'rv')
plot(xC,yC,'bo')
hold off
legend('Transmit antennas','Receive antennas','Target(s)')
grid on
xlabel('x (m)')
ylabel('y (m)')
daspect([1 1 1])
xlim([-0.4 0.4])
ylim([-0.1 1])

subplot(2,3,[2 5])
scatter(Kx(:)/k(end),real(KyR(:))/k(end),50,abs(Skx(:)),'filled')
title('Magnitude of S(k_x,k_y)')
axis equal tight
xlabel('k_x/k (rad/m)')
ylabel('k_y/k (rad/m)')

subplot(2,3,[3 6])
pcolor(x,y,abs(Im).')
shading flat
hold on
plot(xT,yT,'kv')
plot(xR,yR,'rv')
hold off
title('Reconstructed image')
axis equal tight
xlabel('x (m)')
ylabel('y (m)')
xlim([-0.4 0.4])
ylim([-0.1 1])

Published with MATLAB® R2017b