L1 Median

### L1-SVD Algorithm for DOA Estimation Using Uniform Circular Arrays In the context of Direction-of-Arrival (DOA) estimation using uniform circular arrays (UCA), L1-SVD is an effective method that leverages sparse signal representation to enhance resolution and robustness against noise. The approach combines singular value decomposition with sparsity-inducing norms such as L1-norms. #### Overview of L1-SVD Methodology The core idea behind L1-SVD involves decomposing the received data matrix into its constituent components while promoting sparsity through regularization techniques. This allows for more accurate identification of sources even under challenging conditions where traditional methods may fail due to closely spaced signals or high levels of interference[^1]. For implementing this technique specifically within MATLAB: ```matlab % Define parameters N = 8; % Number of sensors in UCA d = lambda / 2; % Sensor spacing half wavelength theta_true = [-30 45]; % True angles of incoming waves snr_db = 10; % Signal-to-noise ratio in dB % Generate synthetic dataset phi = linspace(-pi, pi, N); X = exp(1i * 2*pi*d*[cos(phi); sin(phi)]*sin(theta_true*pi/180)); Y = X + awgn(zeros(size(X)), snr_db); % Perform SVD on covariance matrix Rxx [U,S,V] = svd(cov(Y)); % Apply soft thresholding operator T_λ(|u|) lambda = median(diag(S))/sqrt(log(N)); % Threshold parameter λ selection rule soft_thresholded_U = sign(U).*max(abs(U)-lambda, 0); % Estimate directions via MUSIC spectrum search over grid points θ ∈ (-90°,+90°) angles = -90:0.1:+90; for k=1:length(angles), steer_vec(:,k)=exp(-1j*2*pi*d*(cosd(angles(k))*ones(N,1)+... sind(angles(k)).*(-floor((N-1)/2):ceil((N-1)/2))')); end spec_music=-sum(abs(steer_vec'*diag(eye(N)-soft_thresholded_U*soft_thresholded_U')*steer_vec).^(-2)); figure(); plot(angles, spec_music/max(spec_music)) xlabel('Angle (\circ)') ylabel('Normalized Power') title(['L1-SVD Based DOA Spectrum at SNR=' num2str(snr_db)]) grid on ``` This code snippet demonstrates how one might implement L1-SVD-based DOA estimation for UCAs in MATLAB by generating a simulated scenario involving two incident plane waves arriving from specified angles. After constructing the measurement model based on these assumptions, singular values are extracted after performing SVD operations followed by applying appropriate shrinkage rules before finally searching across possible angular positions to identify peaks corresponding to actual source locations.
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