[Paper Review] A Compressive Sensing Based Approach to Sparse Wideband Array Design

Matthew Hawes, Wei Liu|arXiv (Cornell University)|Mar 19, 2014

Antenna Design and Optimization19 references3 citations

TL;DR

This paper proposes a compressive sensing-based method for designing sparse wideband arrays by formulating a modified l₁ norm minimization problem that simultaneously suppresses all tap coefficients associated with each sensor, ensuring sparsity across the entire tapped delay-line. The approach uses iterative reweighting to enhance sparsity and achieves performance comparable to genetic algorithm (GA)-based methods with significantly reduced computation time—130 minutes vs. 436 minutes—while maintaining low sidelobe levels and good frequency-invariant beamforming response.

ABSTRACT

Sparse wideband sensor array design for sensor location optimisation is highly nonlinear and it is traditionally solved by genetic algorithms, simulated annealing or other similar optimization methods. However, this is an extremely time-consuming process and more efficient solutions are needed. In this work, this problem is studied from the viewpoint of compressive sensing and a formulation based on a modified $l_1$ norm is derived. As there are multiple coefficients associated with each sensor, the key is to make sure that these coefficients are simultaneously minimized in order to discard the corresponding sensor locations. Design examples are provided to verify the effectiveness of the proposed methods.

Motivation & Objective

• To address the computational inefficiency of traditional optimization methods like genetic algorithms (GAs) and simulated annealing in sparse wideband array design.
• To overcome the limitation of standard l₁ norm minimization in wideband arrays, where individual coefficients may be small but the entire tap delay-line (TDL) for a sensor must be zeroed out for sparsity.
• To develop a method that simultaneously minimizes all coefficients of a TDL for each sensor, ensuring true sparsity across the array.
• To improve sparsity and beamforming performance through an iterative reweighting scheme that approximates l₀ norm minimization.
• To provide a computationally efficient alternative to GA-based methods while achieving comparable beam pattern performance.

Proposed method

• Formulates the sparse array design as a modified l₁ norm minimization problem where the l₁ norm of the TDL coefficients for each sensor is minimized collectively.
• Introduces a constraint that groups all coefficients of a sensor's TDL together using the l₂ norm, ensuring that if the norm is small, all coefficients are suppressed.
• Employs an iterative reweighting scheme where weights are updated based on the inverse of the l₂ norm of the previous iteration's coefficients to enhance sparsity.
• Uses a convex optimization framework (CVX) to solve the reweighted problem at each iteration, ensuring convergence to a local minimum.
• Applies frequency-invariant beamforming constraints by enforcing the desired response across a range of normalized frequencies.
• Integrates the TDL structure into the array model, where the steering vector includes phase shifts dependent on sensor position and frequency.

Experimental results

Research questions

• RQ1 Can compressive sensing be effectively adapted to wideband array design where sparsity must be enforced across entire tapped delay-line vectors?
• RQ2 How can the l₁ norm minimization be modified to ensure that all coefficients of a sensor's TDL are simultaneously suppressed, rather than individual coefficients?
• RQ3 Does an iterative reweighting scheme improve the sparsity of the solution compared to standard l₁ minimization in wideband array design?
• RQ4 How does the proposed method compare to GA-based optimization in terms of beam pattern performance and computational efficiency?
• RQ5 Can the proposed method achieve frequency-invariant beamforming while maintaining low sidelobe levels and high mainlobe directivity?

Key findings

• The proposed reweighted compressive sensing method achieves a sparse array with 11 active sensors over a 10λ aperture, matching the performance of a GA-based design.
• The beam response of the proposed method shows a mainlobe at 90° and sufficient attenuation in sidelobe regions (0°–80° and 100°–180°), with good frequency-invariant properties.
• Computation time for the proposed method is 130 minutes, significantly lower than the 436 minutes required by the GA-based method.
• The JCLS metric (a measure of beam pattern performance) is 0.0372 for the proposed method and 0.0376 for the GA method, indicating slightly better performance with the proposed approach.
• The mean adjacent sensor spacing is 0.62λ in both methods, confirming that both achieve similar levels of sparsity, but the proposed method does so more efficiently.
• The reweighted l₁ minimization approach successfully promotes sparsity across entire TDLs, avoiding the issue of partial suppression where only some coefficients are small.