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In this post What features can be used to estimate both azimuth and elevation angle using a deep learning model?, I have shared a signal model:

 SOURCE_angles = doa(ii,:); Ax = zeros(Mx, SOURCE_K); Ay = zeros(My - 1, SOURCE_K); for k = 1:2:SOURCE_K Ax(:, k) = exp(-1i*2*pi*fc*dist*(1/cSpeed)*(0:Mx-1)' ... *cosd(SOURCE_angles(k))*sind(SOURCE_angles(k+1))); Ay(:, k) = exp(-1i*2*pi*fc*dist*(1/cSpeed)*(1:My-1)' ... *sind(SOURCE_angles(k))*sind(SOURCE_angles(k+1))); end A = [Ax; Ay]; % Steering matrix of all sensors % The Expected covariance matrix for a angle-pair Ry_the = A*diag(ones(SOURCE_K,1))*A' + noise_power*eye(Mx+My-1); % The Sampled covariance matrix for a angle-pair S = sqrt(sVar)*randn(SOURCE_K, p).*exp(1i*(2*pi*fc*repmat((1:p)/fs, SOURCE_K, 1))); X = A*S; noiseCoeff = 1; Eta = sqrt(noiseCoeff)*randn(Mx + My - 1, p); Y = X + Eta; Ry_sam = Y*Y'/p;

However, here I considered stationary targets but if I have to consider moving target then the signal model should include the doppler returns.

This is the modifications I did, and want to know if I have done it correctly?

n_pulses = 16; M_vec = 1:n_pulses; K = 3; % Number of targets for k = 1:K ndf(k) = -0.5 + (k - 1) / (K - 1); end nn = 1; for current_ndf = ndf %%% ndf = normalized doppler returns for m = M_vec X_tg_m(m, nn) = exp(1i*2*pi*m*current_ndf); end nn = nn + 1; end G = size(doa,1); for ii=1:G SOURCE_angles = doa(ii,:); Ax = zeros(Mx, size(M_vec,2)*K); Ay = zeros(My - 1, size(M_vec,2)*K); %%% NSF idx = 1; idy = 1; for k = 1:2:SOURCE_K Ax(:, idy:1:size(M_vec,2)*idx) = exp(1i*2*pi*fc*dist*(1/cSpeed)*(0:Mx-1)'*cosd(SOURCE_angles(k))*sind(SOURCE_angles(k+1))).*X_tg_m(:,idx)'; Ay(:, idy:1:size(M_vec,2)*idx) = exp(1i*2*pi*fc*dist*(1/cSpeed)*(1:My-1)'*sind(SOURCE_angles(k))*sind(SOURCE_angles(k+1))).*X_tg_m(:,idx)'; idx = idx + 1; idy = idy + size(M_vec,2); end A = [Ax; Ay]; % Steering matrix of all sensors % The Expected covariance matrix for a angle-pair Ry_the = A*diag(ones(size(M_vec,2)*K,1))*A' + noise_power*eye(Mx+My-1); %dim: (Mx+My-1) x (Mx+My-1) S = sqrt(sVar)*randn(size(M_vec,2)*K, p).*exp(1i*(2*pi*fc*repmat((1:p)/fs, size(M_vec,2)*K, 1))); X = A*S; noiseCoeff = 1; Eta = sqrt(noiseCoeff)*randn(Mx + My - 1, p); Y = X + Eta; Ry_sam = Y*Y'/p; end

The calculations of normalized doppler returns are taken from the paper

Liu, Zhe, et al. "Moving target indication using deep convolutional neural network." IEEE Access 6 (2018): 65651-65660.

Mainly I am not sure should I include the doppler returns while calculating the array manifolds? Because doppler returns doesn't affect the phase shifts, so the expected covariance matrix will not be affected, but for the sampled covariance matrix since it is calculated based on the received signal, not including the doppler returns will probably lead to a wrong covariance matrix?

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Shortly after posting this question, I found a paper which explained the signal model for GMT very well. I have attached the link to the paper, if anyone ever needs clarification. End-to-End Moving Target Indication for Airborne Radar Using Deep Learning.

According to this paper the space-time steering vector for a single moving target is formed by the combination of both spatial steering vector (say $A_{s}$) and temporal steering vector (the steering vector formed by the Doppler freq, say $A_{d}$). The final steering vector is $A_{s} \otimes A_{d}$, where $\otimes$ represents the outer product. This answers my first question about including the Doppler frequency in the array manifold calculation. And, including the Doppler frequency in the array manifold calculation, also answers my second question. Hope this helps. Also, correct me if I am wrong.

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    $\begingroup$ At the end of your question post, you ask two questions. Would be cool if you answered them explicitly in your answer, so that future readers don't have to read the full paper you link to here in your answer! $\endgroup$ Commented Nov 19, 2024 at 12:44
  • $\begingroup$ @MarcusMüller, thanks for the suggestion. Feel free to add to my answer, if I have missed out on anything $\endgroup$ Commented Nov 19, 2024 at 15:08
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    $\begingroup$ No, thank you for writing so nicely prepared questions and answers! $\endgroup$ Commented Nov 19, 2024 at 15:30

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