MVDR in MATLAB (Minimum Variance Distortionless Response)

 

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.

Try Interactive Online Simulators

1. Direction of Arrival (DoA) Online Simulator

Further Reading

1. Minimum Variance Distortionless Response (MVDR)
2. Hydrophones vs Vector Sensors
3. MUSIC Algorithm
4. MATLAB Code for MUSIC
5. Direction of Arrival (DOA) Estimation