MVDR in MATLAB

 

MATLAB Code

clc; clear; close all;

%% Step 1: Define Parameters

M = 8; % Number of array sensors

d = 0.5; % Sensor spacing (lambda/2)

K = 2; % Number of sources

N = 200; % Number of snapshots

theta = [-20 30]; % True signal angles (degrees)

SNR = 10; % Signal-to-noise ratio (dB)

fprintf('Step 1: Parameters Initialized\n');

%% Step 2: Generate Source Signals

t = 1:N;

s1 = exp(1j*2*pi*0.05*t); % Source 1

s2 = exp(1j*2*pi*0.1*t); % Source 2

S = [s1; s2];

figure;

plot(real(S(1,:))); title('Signal 1 (Real Part)'); xlabel('Samples'); ylabel('Amplitude');

figure;

plot(real(S(2,:))); title('Signal 2 (Real Part)'); xlabel('Samples'); ylabel('Amplitude');

fprintf('Step 2: Source Signals Generated\n');

%% Step 3: Construct Steering Matrix

A = zeros(M,K);

for k = 1:K

A(:,k) = exp(-1j*2*pi*d*(0:M-1)'*sin(theta(k)*pi/180));

end

fprintf('Step 3: Steering Matrix Created\n');

%% Step 4: Generate Received Signals with Noise

X = A*S;

noise = (randn(M,N) + 1j*randn(M,N))/sqrt(2);

noise = noise * 10^(-SNR/20);

X = X + noise;

figure;

plot(real(X(1,:))); title('Received Signal at Sensor 1'); xlabel('Samples'); ylabel('Amplitude');

fprintf('Step 4: Received Signals with Noise Generated\n');

%% Step 5: Estimate Covariance Matrix

R = (X*X')/N;

figure;

imagesc(abs(R)); colorbar; title('Covariance Matrix Magnitude'); xlabel('Sensors'); ylabel('Sensors');

fprintf('Step 5: Covariance Matrix Computed\n');

%% Step 6: Eigenvalue Decomposition

[Evec, Eval] = eig(R);

eigenvalues = diag(Eval);

figure;

stem(sort(eigenvalues,'descend')); title('Eigenvalues of Covariance Matrix'); xlabel('Index'); ylabel('Eigenvalue');

fprintf('Step 6: Eigen Decomposition Done\n');

%% Step 7: Sort Eigenvalues and Eigenvectors

[eigenvalues_sorted, idx] = sort(eigenvalues,'descend');

Evec_sorted = Evec(:,idx);

fprintf('Step 7: Eigenvalues Sorted\n');

%% Step 8: Separate Signal and Noise Subspace

Es = Evec_sorted(:,1:K); % Signal subspace

En = Evec_sorted(:,K+1:end); % Noise subspace

fprintf('Step 8: Signal and Noise Subspaces Separated\n');

%% Step 9: MUSIC Spectrum Calculation

angles = -90:0.1:90;

Pmusic = zeros(size(angles));

for i = 1:length(angles)

steering = exp(-1j*2*pi*d*(0:M-1)'*sin(angles(i)*pi/180));

Pmusic(i) = 1/(steering'*(En*En')*steering); % MUSIC denominator

end

Pmusic = abs(Pmusic);

Pmusic = 10*log10(Pmusic/max(Pmusic));

figure;

plot(angles,Pmusic,'LineWidth',2); grid on;

xlabel('Angle (degrees)'); ylabel('Spectrum (dB)'); title('MUSIC Spatial Spectrum');

fprintf('Step 9: MUSIC Spectrum Computed\n');

%% Step 10: MVDR Beamforming to Reconstruct Sources

S_beamformed = zeros(K,N); % Preallocate

for k = 1:K

a_k = A(:,k); % Steering vector for k-th source

% MVDR weights

w_mvdr = (R\ a_k) / (a_k' * (R\ a_k));

% Apply beamforming

S_beamformed(k,:) = w_mvdr' * X;

end

% Plot MVDR beamformed signals

figure;

for k=1:K

subplot(K,1,k);

plot(real(S_beamformed(k,:)));

title(['MVDR Beamformed Signal ', num2str(k)]);

xlabel('Sample'); ylabel('Amplitude');

end

fprintf('Step 10: MVDR Beamforming Completed\n');

Output

 

 

 

 

 

 

 

 

Workflow of the Multi-Antenna Signal Processing and MUSIC/MVDR Code

This section describes the step-by-step workflow of the MATLAB code that simulates multiple sources, separates them using subspace methods, and reconstructs signals via MVDR beamforming.

1. Step 1: Parameter Initialization

   Define key parameters including:

   • Number of array sensors (M)
   • Sensor spacing (d)
   • Number of sources (K)
   • Number of snapshots (N)
   • True source angles (theta)
   • Signal-to-noise ratio (SNR)

   These parameters form the foundation for the simulation.

2. Step 2: Generate Source Signals

   Create narrowband complex exponential signals representing the sources:

   s1 = exp(1j*2*pi*0.05*t);

   s2 = exp(1j*2*pi*0.1*t);

   These signals form the source matrix S.

3. Step 3: Construct Steering Matrix

   Compute the steering vectors for each source direction and form the matrix A. Each column represents the array’s response to a source angle.

4. Step 4: Generate Received Signals

   The array receives a combination of source signals and additive noise:

   X = A*S + noise;

   This models the real-world scenario of multiple antennas capturing overlapping signals with noise.

5. Step 5: Covariance Matrix Estimation

   Compute the sample covariance matrix of the received signals:

   R = (X*X')/N;

   This matrix captures correlations between sensors and is essential for subspace separation.

6. Step 6: Eigenvalue Decomposition

   Decompose the covariance matrix into eigenvalues and eigenvectors:

   [Evec, Eval] = eig(R);

   Eigenvalues indicate the power in each subspace, separating signal and noise components.

7. Step 7: Sort Eigenvalues and Eigenvectors

   Sort eigenvalues in descending order to identify the signal subspace (largest eigenvalues) and noise subspace (smallest eigenvalues).

8. Step 8: Subspace Separation

   Define:

   • Signal subspace: Es = Evec_sorted(:,1:K)
   • Noise subspace: En = Evec_sorted(:,K+1:end)

   Noise and signal subspaces are orthogonal, which is the foundation for MUSIC.

9. Step 9: MUSIC Spectrum Calculation

   Scan angles using steering vectors and compute the pseudo-spectrum:

   Pmusic(i) = 1/(steering'*(En*En')*steering);

   Peaks in the spectrum indicate the directions of arrival (DOAs) of the sources.

10. Step 10: MVDR Beamforming

    For each source, apply MVDR weights to reconstruct the signal while minimizing noise:

    w_mvdr = (R\ a_k) / (a_k' * (R\ a_k));

    S_beamformed(k,:) = w_mvdr' * X;

    This produces clean estimates of each source signal from the mixed array observations.

Summary: The workflow models a practical multi-antenna system: simulate sources, capture them with an array, compute correlations, separate signal/noise subspaces, find DOAs (MUSIC), and reconstruct signals (MVDR beamforming). This closely mimics real-world array signal processing in radar, wireless communications, and MIMO systems.