Optimal SRP Beamformer

• The paper presents an SDP-relaxed QCQP formulation that robustly optimizes the steering vector to maximize output SINR under interference constraints.
• It employs semidefinite programming to convert a nonconvex beamforming problem into a tractable convex optimization, ensuring reliable rank-one recovery.
• Empirical simulations show superior performance compared to conventional methods, especially under steering vector mismatches and limited training data.

The Optimal Steered Response Power (SRP) Beamformer refers to a class of beamforming algorithms that provide robust, high-performance spatial filtering and localization via explicit optimization of the steering vector under power maximization, subject to normalization and interference suppression constraints. The most widely cited realization in this context is presented in "Robust Adaptive Beamforming Based on Steering Vector Estimation via Semidefinite Programming Relaxation" (Khabbazibasmenj et al., 2010), which demonstrates not only the theoretical formulation but also practical algorithms exhibiting superior output signal-to-interference-plus-noise ratio (SINR) compared to traditional SRP and subspace-projection-based beamformers, especially under steering vector uncertainties and limited data regimes.

1. Objective and Constraint Structure

The design of the optimal SRP beamformer is based on maximizing the output power of the beamformer, equivalently minimizing a quadratic form involving the inverse of the estimated sample covariance matrix $\hat{R}$ and the estimated steering vector $\hat{a}$. The foundational criterion can be formulated as:

$\min_{\hat{a}} \ \hat{a}^H \hat{R}^{-1} \hat{a}$

subject to:

• Steering vector normalization: $\|\hat{a}\|^2 = M$ (where $M$ is the number of array elements).
• Interference avoidance via quadratic constraint: $\hat{a}^H \tilde{C} \hat{a} \leq \Delta_0$

Here, $\tilde{C}$ encodes spatial directions outside the sector of the desired signal, constructed via directional integration, and $\Delta_0$ is chosen—e.g., as the sector-wise maximum $d^H(\theta)\tilde{C}d(\theta)$ so that the optimal estimate does not collapse to an interfering source vector. These constraints are explicitly free of tunable or heuristic ambiguity, in contrast to many robust beamforming methods.

2. Nonconvex QCQP Formulation

The above optimization is a homogeneous quadratically constrained quadratic program (QCQP) with a quadratic (pseudo-power) objective and two quadratic constraints. The nonconvexity arises due to the equality in the norm constraint, rendering the global optimization NP hard in general. The nominal QCQP is given by:

$$\begin{array}{ll} \text{minimize} & \hat{a}^H \hat{R}^{-1} \hat{a}\ \text{subject to} & \|\hat{a}\|^2 = M, \ & \hat{a}^H \tilde{C} \hat{a} \leq \Delta_0 \end{array}$$

Both the objective and all constraints are quadratic in $\hat{a}$.

3. Semidefinite Programming Relaxation

To render the problem tractable, a standard semidefinite "lifting" relaxation is employed. Introducing $A = \hat{a}\hat{a}^H$ (so $A \succeq 0$ and $\text{rank}(A) = 1$), the problem can be recast as:

$$\begin{array}{ll} \text{minimize} & \mathrm{Tr}(\hat{R}^{-1}A) \ \text{subject to} & \mathrm{Tr}(A) = M, \ & \mathrm{Tr}(\tilde{C}A) \leq \Delta_0, \ & A \succeq 0, \ & \text{rank}(A) = 1 \end{array}$$

Relaxation is achieved by omitting the nonconvex rank-one constraint, yielding a convex semi-definite program (SDP) amenable to interior-point optimization. The original nonconvexity is thus isolated to the rank constraint.

The relaxation is shown to be tight: if the original (vector) problem's solution is unique (up to a phase rotation), the relaxed optimum is guaranteed to be rank-one. In other (rare) cases, the authors describe a constructive method for extracting a rank-one solution directly from the SDP result, allowing the optimal steering vector to be efficiently reconstructed.

4. Duality, Algorithmic Efficiency, and Closed-Form Solutions

Strong duality holds for this relaxation, offering insight and additional computational efficiency. The Lagrange dual involves only two variables (corresponding to the norm and interference constraints), and is

$$\begin{array}{ll} \text{maximize} & \gamma_1 M - \gamma_2 \Delta_0 \ \text{subject to} & \hat{R}^{-1} - \gamma_1 I + \gamma_2 \tilde{C} \succeq 0 \end{array}$$

Solving the dual yields the optimal objective and can, in many scenarios, directly produce the optimal steering vector via the structure of the maximizer.

In special cases, such as when the interference constraint is inactive (i.e., high SINR limit) or certain forms of projection constraint are set, closed-form analytical solutions for the optimal $\hat{a}$ can be written explicitly.

The net effect is a highly efficient algorithmic framework for beamformer design—exploiting SDP and duality—which avoids computational expenses associated with sequential quadratic programming (SQP) or iterative subspace-projection methods traditionally used for robust beamforming.

5. Empirical Performance and Robustness

Comprehensive simulations using the SDP-based optimal SRP beamformer demonstrate:

• Higher output SINR than worst-case norm-bound robust beamformers, SQP/projection beamformers, and conventional eigenspace-projection approaches.
• Robustness to multiple classes of steering vector errors, including unstructured phase perturbations (modeling incoherent local scattering), coherent multipath (coherent local scattering: multiple steering vectors), and limited or small-sample training regimes.
• In scenarios where the presumed steering vector is exact, the optimal SRP approach achieves equality with conventional MVDR and subspace beamformers. When mismatch errors are present, the proposed method exhibits pronounced superiority, particularly as the number of training samples decreases.

6. Distinction from Conventional SRP/Eigenspace Beamformers

The optimal SRP beamformer is distinguished from traditional (projection-based) SRP and related subspace beamformers (e.g., eigenspace-based matched filter or pseudo-inverse beamformers) in several important ways:

• No requirement to estimate (often unreliably) the rank or dimension of the signal-plus-interference subspace, circumventing the problem of subspace swaps at low SNR.
• Avoidance of ambiguous design parameters (e.g., arbitrary loading factors or norm bounds required by regularization-based approaches).
• Explicit incorporation of angular sector information via quadratic constraint $\hat{a}^H \tilde{C} \hat{a} \leq \Delta_0$, preventing convergence of the solution towards directions associated with strong interference.
• Direct maximization of output power subject to robust, scenario-grounded constraints, rather than indirect correction of presumed steering vectors via projection.

This framework provides a mathematically principled, "parameter-free" means to achieve robust output maximization and interference suppression, supported by the duality-theoretic underpinnings of the SDP relaxation.

7. Practical Implementation Considerations

The optimal SRP beamformer can be implemented with standard convex optimization toolkits supporting SDP (e.g., MOSEK, CVX), and the computational complexity is dominated by the size of the array ($M$). Since the dual problem has only two variables, efficient line-search or low-dimensional optimization methods can be employed to further accelerate computation.

Large problem instances may benefit from warm-start strategies or leveraging problem structure (e.g., Hermitian Toeplitz covariance matrices for ULAs).

The method does not require heuristically chosen user parameters (unlike diagonal loading, etc.), making it particularly suitable for automated or real-time beamforming applications where tuning opportunities are limited or not feasible.

Summary Table: Comparison with Conventional SRP Beamformers

Feature	Optimal SRP (SDP-relaxed)	Conventional SRP/Projection
Steering Vector Estimation	Power-maximizing, QCQP+SDP	Subspace projection/correction
Constraint Type	Norm+quadratic (sector/interf.)	Typically subspace, heuristic
Parameter Dependence	Parameter-free (no user choices)	Norm bounds, loading, subspace
Interference Suppression	Explicit via $\tilde{C}$ constraint	Implicit by subspace proj.
Performance @ Steering Mismatch	Superior, especially at low SNR	Degraded due to subspace swap
Solution Approach	Convex SDP + tightness guarantee	SQP or heuristic projection

The optimal SRP beamformer introduces a rigorous, convex optimization-based methodology to robust beamforming, achieving efficient and robust steering vector estimation that exhibits significant empirical gains over previously established approaches—especially in the presence of uncertainty and limited training data (Khabbazibasmenj et al., 2010).