Beamforming Algorithms

The Bartlett beamformer scans the steering vector across angles:

$$P_{\text{Bartlett}}(\theta) = \frac{\mathbf{a}^H(\theta)\mathbf{R}_{xx}\mathbf{a}(\theta)} {\|\mathbf{a}(\theta)\|^4}$$

where $\mathbf{R}_{xx} = E[\mathbf{x}\mathbf{x}^H]$ is the array covariance.

def bartlett(Rxx, N, d_lambda, theta_scan):
    P = np.zeros(len(theta_scan))
    for i, th in enumerate(theta_scan):
        a = steering_vector(N, d_lambda, th)
        P[i] = np.real(a.conj() @ Rxx @ a) / (np.abs(a @ a.conj())**2)
    return P

The Capon (Minimum Variance Distortionless Response) beamformer minimizes output power while maintaining unit gain at the look direction:

$$P_{\text{Capon}}(\theta) = \frac{1}{\mathbf{a}^H(\theta)\mathbf{R}_{xx}^{-1}\mathbf{a}(\theta)}$$

This provides much better resolution than Bartlett by placing nulls toward interferers.

def capon(Rxx, N, d_lambda, theta_scan):
    Rinv = np.linalg.inv(Rxx)
    P = np.zeros(len(theta_scan))
    for i, th in enumerate(theta_scan):
        a = steering_vector(N, d_lambda, th)
        P[i] = 1 / np.real(a.conj() @ Rinv @ a)
    return P

MUSIC (MUltiple SIgnal Classification) achieves super-resolution by exploiting the noise subspace:

1. Estimate $\mathbf{R}_{xx}$ and compute its eigendecomposition
2. Partition eigenvectors into signal ($\mathbf{U}_s$) and noise ($\mathbf{U}_n$)
3. Compute the pseudospectrum:

$$P_{\text{MUSIC}}(\theta) = \frac{1}{\mathbf{a}^H(\theta)\mathbf{U}_n\mathbf{U}_n^H\mathbf{a}(\theta)}$$

def music(Rxx, N, d_lambda, theta_scan, n_sources):
    eigvals, eigvecs = np.linalg.eigh(Rxx)
    Un = eigvecs[:, :N-n_sources]  # noise subspace
    P = np.zeros(len(theta_scan))
    for i, th in enumerate(theta_scan):
        a = steering_vector(N, d_lambda, th)
        P[i] = 1 / np.real(a.conj() @ Un @ Un.conj().T @ a + 1e-10)
    return P

ESPRIT (Estimation of Signal Parameters via Rotational Invariance) estimates DOAs without spectral scanning by exploiting the shift invariance of ULAs. It computes angles directly from the eigenvalues of a rotation matrix derived from two subarrays.

The sample covariance from $L$ snapshots:

$$\hat{\mathbf{R}} = \frac{1}{L}\sum_{l=1}^{L} \mathbf{x}[l]\mathbf{x}^H[l]$$

More snapshots improve the estimate. Diagonal loading ($\hat{\mathbf{R}} + \delta\mathbf{I}$) improves robustness for Capon.