How antenna array factor is related to FFT of antenna weights

I have an uniform linear antenna array of N elements with unknown (uniform) element spacing. Each element is having phase weights phi1,phi2,... ,phiN. Each element is fed with unity amplitude.

I will take the M point Fast Fourier Transform - FFT (M >> N) of this array to get a frequency response R(fr). How R(fr) is related to the array factor of the array. What is the value of inter element spacing R(fr) assumes.

>> N = 12;M = 64; 
>> phis = randi([0,360],1,N)*pi/180;
>> ComplexWgts = exp(2*pi*1j*phis);
>> R_fr = abs(fftshift(fft(ComplexWgts,M)));

Where as the Array factor can be coded as

theta = (-90:90)*pi/180;
d = 0.5;% How to interpret this d from the FFT
AF = zeros(size(theta));
for n = 1:N

      AF = AF + a(n) * ComplexWgts(n) * exp(1j * n * 2 * pi * d * sin(theta));

end

How to correlate between AF and R_fr?

[HAMDI Bilel]

Nelements =17;

Npoints = 1024;

Ntheta = Npoints;

thetalimit = 90;

theta_deg = linspace(-thetalimit,thetalimit,Ntheta);

theta= pi*theta_deg/180;

% Uniform linear broadside equispaced array

d = 0.5; % in wavelengths

Start = ones(1,Nelements); %uniform excitation, 0 phase

AF = zeros(size(theta));

a = abs(Start);

phi = angle(Start);

ComplexWgts = exp(1j*phi);

%Classical Array Factor

for n = 1:Nelements

AF = AF + a(n) * ComplexWgts(n) * exp(1j * n * 2 * pi * d * sin(theta));

end

%AF = AF / max(abs(AF));

%--------------------------------------------------------------------------

% FFT Array Factor

AF_FFT = fftshift(fft(Start,Npoints));

figure

hold on;

plot(theta,20*log10(abs(AF)),'Color','r','LineWidth',2);

plot(theta,20*log10(abs(AF_FFT)),'Color','b','LineWidth',1);

% polar(theta,abs(AF),'r')

% polar(theta,abs(AF),'b')

[ak]

what happens if d is lambda, or 2*lambda? FFT only seems to work when element spacing is lambda/2. how do i change the code so that it can work for other element spacings as well?

Answer 1

[Honglei Chen]

FFT is simply a computation tool, you can interpret the result based on your application. Using your ULA as an example, the phase shift between each element can be considered as exp(1i*2*pi*f*d*sin(theta)/c) where c is the propagation speed. On the other hand, for FFT, the phase shift between each sample is exp(1i*2*pi*f*t) . Therefore, when computing the array factor, the array can be considered as a spatially sampled sequence so instead of sampling interval t , you now use d*sin(theta)/c instead.

HTH.