How is angle of arrival estimated?

Question

I have simulated a phased array antenna. I have also simulated a point source. I don't get how to calculate pseudo spectrum vector without using the position of the source. Algorithms such as Capon and Bartlett use the position of the source for calculate the auto- correlation matrix which is further used to calculate pseudo spectrum. How is the angle of arrival being estimated if the source position is required to run the algorithm in the first place?

Here is the code:

M=6;

sig2=.1;

th1=-3*pi/180;

th2=3*pi/180;

a1=[1];

a2=[1];

a=[1];

for i=2:M

a1=[a1 exp(-1j*i*pi*sin(th1))]; 

a2=[a2 exp(-1j*i*pi*sin(th2))]; 

end

A=[a1' a2'];

Rss=[1 0;0 1];

Rrr=ARssA'+sig2*eye(6);

for k=1:180;

th(k)=-pi/6+pi*k/(3*180);

clear a

a=[1];

for jj=2:M

  a = [a exp(-1j*jj*pi*sin(th(k)))]; 

end

P(k)=real(1./(conj(a)*inv(Rrr)*a.'));

end

figure;

plot(th*180/pi,10*log10(P/max(P)),'k')

grid on

xlabel('Angle')

ylabel('|P(\theta)| (dB)')

Answer 1

How is angle of arrival estimated?

The estimation of angle in linear array antenna is based on the relative phase variation between the target location and antenna elements. This phase variation in spatial dimension between antenna elements will provide you with the location of the target (in terms of angle).

Thus, the basic approach to obtain an angle through array antenna is to perform Fast Fourier Transform (FFT) over the antenna elements. This conventional method with provide you with the angle of target but with very coarse angular resolution (around 2/N radians) (angular resolution using FFT over linear array)

For obtaining the angle with higher resolution, we can use CAPON beamformer or Super resolution algorithms such as MUSIC, ESPIRIT etc.

In order to answer your question,

Algorithms such as Capon and Bartlett use the position of the source for calculate the auto- correlation matrix which is further used to calculate pseudo spectrum

This is not entirely true, as the auto correlation matrix in spatial dimension is the input to the CAPON beamformer and till that point the algorithm does not know the location of the target.

The angle of arrival of the target is determined by taking possible set of direction of arrivals and then suppress the other direction which is not under consideration. One has to scan through all possible direction of arrivals (using array manifold matrix) and then with the spectrum estimate the direction of arrival (normally choosing the highest peak).

For your kind reference:

Algorithm for CAPON is mentioned below:

1. Take auto correlation (A) (among antenna elements)
2. Take inverse of auto correlation matrix (Ainv)
3. Create array manifold matrix (M)
4. Obtain capon spectrum using = 1/ ((hermitian(M) * Ainv * M), where hermitian operation is the conjugate transpose.
5. Finding the peak spectrum location (provides angle of arrival)

How is the angle of arrival being estimated if the source position is required to run the algorithm in the first place?

So we do not need to know the source dimension, but we have to find it among the possible source position/direction (obtaining the maximum likelihood of the actual direction with possible directions)