Questions: Array Signal Processing and Beamforming

Question 1

A linear array of M equally spaced sensors receives a narrowband signal from angle θ. The phase difference between adjacent sensors is φ = (2π d/λ) sin(θ), where d is spacing and λ is wavelength. In delay-and-sum beamforming, what weights should be applied to each sensor to steer the array toward angle θ₀?

• A. All weights are 1 (simple averaging); delay-and-sum does not need steering
• B. Weights are exp(jm·φ₀) = exp(j·m·(2π d/λ) sin(θ₀)) for sensor m, creating phase shifts that align signals from θ₀
• C. Weights depend on the array shape and sensor spacing; they cannot be specified without this information
• D. Weights are exp(−jm·φ₀); the phase shift is negative to time-reverse the array response

Answer: (B)

A signal arriving from angle θ₀ has phase φ₀ between adjacent sensors. To coherently combine signals (constructive interference), you apply phase shift −φ₀ at each sensor, exactly canceling the arrival phase difference. The weights exp(jm·φ₀) provide this phase shift. Summing weighted sensor outputs gives y = ∑ w_m x_m = ∑ exp(jm·φ₀) x_m, which coherently adds signals from θ₀ and incoherently sums signals from other angles (they have residual phase, causing cancellation). This is the principle of beamforming: steer the 'beam' (high-gain direction) by adjusting phase shifts across the array.

Question 2

Adaptive MVDR (Minimum Variance Distortionless Response) beamforming minimizes output power subject to unit gain in the target direction: minimize w^H R_xx w subject to w^H a(θ₀) = 1, where R_xx is the input covariance matrix and a(θ₀) is the steering vector. Why is this objective sensible, and when can it fail?

• A. It minimizes output power, which suppresses noise; the constraint enforces that the target signal is not distorted. It fails when there are no interferers (output power equals noise power, not signal power)
• B. It explicitly searches for the target direction, maximizing the signal-to-interference-plus-noise ratio. It fails because beamforming cannot explicitly maximize SINR without knowing target signal power
• C. It uses the known covariance matrix, which is always accurate. It never fails
• D. It minimizes total system energy, reducing power consumption. It fails in high-noise environments

Answer: (A)

The insight is elegant: if you minimize total output power (signal + interference + noise) while keeping the target signal power fixed (unit gain in steering direction), you necessarily minimize interference + noise. The constraint w^H a(θ₀) = 1 ensures the target signal passes unchanged through the beamformer (no distortion). The method fails when there are no interferers and only noise — you cannot distinguish signal from noise by power, so the optimizer will try to null the target signal to minimize power, violating the constraint. This is detected when the output power equals the noise-only power; practical MVDR implementations add regularization (diagonal loading) to prevent this.

Question 3

The MUSIC algorithm estimates source locations by (1) computing the sample spatial correlation matrix R_xx from array data, (2) eigendecomposing it to find signal and noise subspaces, (3) searching for angles θ where the steering vector a(θ) is most orthogonal to the noise subspace. Why is orthogonality to the noise subspace a signature of a source direction?

• T. True
• F. False

Answer: True

MUSIC assumes narrowband sources with known (or estimated) number K. The signal subspace is spanned by the K dominant eigenvectors of R_xx, corresponding to signal + noise. The noise subspace is spanned by the remaining eigenvectors, corresponding to noise only. A steering vector a(θ) for the true source direction θ_true lies in the signal subspace, so it is orthogonal to the noise subspace by definition. Search algorithms evaluate g(θ) = 1 / ||proj_{noise}(a(θ))||² — the reciprocal of projection onto noise subspace. At true source directions, the projection is zero (a(θ) is orthogonal to noise), so g(θ) → ∞ (a peak). This gives spectral peaks at source directions without explicitly searching the steering vector — a fundamental subspace method.

Question 4

In adaptive beamforming with imperfect knowledge of the steering vector (e.g., sensor positions are slightly misaligned), the adaptive MVDR beamformer can 'self-null' the target signal, destroying performance. How can this be prevented?

• T. True
• F. False

Answer: True

If the steering vector a(θ₀) used in the constraint is mismatched to the true steering vector due to position errors or miscalibration, the MVDR optimizer will find a weight vector that satisfies w^H a_nominal(θ₀) = 1 but actually nulls the true signal (since the true and nominal steering vectors are different). This self-nulling catastrophically degrades performance. Mitigation strategies: (1) add Quadratic Constraint (QC): relax the constraint to a cone around a_nominal(θ₀), allowing deviations; (2) Diagonal Loading: add regularization R̃_xx = R_xx + λI to stabilize the covariance inverse; (3) Robust Capon beamforming: explicitly model the uncertainty in the steering vector as a bounded region and minimize worst-case power. These trade off adaptation (nulling interferers) for robustness to model error.

Question 5

Explain the computational difference between conventional delay-and-sum beamforming and adaptive MVDR beamforming. Which is more robust to steering vector mismatch, and why?

Model answer: Delay-and-sum beamforming: linear combination of array signals with fixed phase shifts (no optimization), O(M) complexity. MVDR beamforming: solves an optimization problem to find weights minimizing power subject to a constraint, O(M³) complexity (due to covariance matrix inversion). Delay-and-sum is robust: the phase shifts are geometric (determined by array shape and frequency), independent of signal statistics. Even if the arrival angle is slightly wrong, delay-and-sum degrades gracefully (the beam broadens). MVDR is sensitive: the optimal weights depend on the estimated covariance matrix R_xx and steering vector a(θ₀). If either estimate is poor (few data samples, model mismatch), the optimizer can find spurious solutions or self-null the target. Robust variants (Diagonal Loading, QC, Worst-Case Robust Capon) add constraints or regularization to prevent this.

This is the adaptation-robustness tradeoff: adaptive methods (MVDR, LCMV) achieve better interference suppression when the model is correct, but are brittle to deviations. Fixed methods (delay-and-sum, fixed null steering) are suboptimal nominally but fail gracefully. In practice, adaptive beamforming is used when the interference environment is known and relatively stable (e.g., a specific jammer location in radar); fixed beamforming is used when robustness to unknown interference is critical (e.g., sonar in unknown, time-varying ocean).