Array toolbox array gain calculation for tapered planar array seems inaccurate.

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Jonathan Williams
Jonathan Williams 2022-3-18
编辑： Jonathan Williams 2022-3-18
According to Van Trees, H.L., "Optimum Array Processing," section 2.6.2., the array gain for any arbitrary array geometry should be equal to under the assumption that the weight vector, , has been normalized to satisfy the distortionless constraint that , where is the array manifold vector. If the weight vector is not normalized, then the array gain can instead be calculated as .
In MATLAB, create a URA as such:
% set up the array
fc = 3e9; % can be anything
c = physconst('LightSpeed');
dy = c/fc/2;
dz = dy;
N = 48; % can be any integer
M = 48; % can be any integer
% create a 2-D Taylor taper array
nbar = 4; % can be any integer. Set to '1' for untapered
SLL = -40; % can be any negative value
wn = taylorwin(N,nbar,SLL);
wm = taylorwin(M,nbar,SLL);
W = wn * wm.';
% create the array object
array = phased.URA([N M],[dz dy],'Taper',W);
Now calculate the array gain using MATLAB's toolbox and Van Trees's method:
% calculate the array gain using MATLAB
gain = phased.ArrayGain("SensorArray",array);
g_MATLAB = gain(fc,[0;0]);
% note: MATLAB seems to expect that the taper vector is NOT normalized.
% In other words, sum(W(:)) = N*M
% calculate the array gain using Van Trees's method:
% the manifold vector is presumed to be ones(length(W),1) for a signal at broadside
g_tapered = 10*log10(abs(sum(W(:)))^2/(W(:)'*W(:)));
% or the same answer is obtained if we first normalize the weight vector
W = W/N/M;
g_tapered_normalized = 10*log10(1/(W(:)'*W(:)));
The answer produced by MATLAB is the same as if the array were not tapered, i.e., 10*log10(N*M). This does not seem accurate. Am I doing something wrong? By intuition alone, regardless of calcuation method, the array gain should always go down for a tapered array as compared to untapered.

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