Signal model and doa estimation for Moving Targets

Question

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?

Answer 1

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.