Convex Optimization Approaches for Blind Sensor Calibration using Sparsity

TL;DR

This paper develops convex optimization methods for blind sensor calibration in compressive sensing, enabling joint recovery of unknown gains and sparse signals under various gain conditions.

Contribution

It introduces novel convex formulations for blind calibration with unknown gains, extending basis pursuit and quadratic approaches for different gain scenarios.

Findings

Algorithms successfully recover sparse signals and gains in simulations.

Convex methods are effective with sufficiently many sparse calibration signals.

Proposed approaches are scalable and reduce complexity in certain gain cases.

Abstract

We investigate a compressive sensing framework in which the sensors introduce a distortion to the measurements in the form of unknown gains. We focus on blind calibration, using measures performed on multiple unknown (but sparse) signals and formulate the joint recovery of the gains and the sparse signals as a convex optimization problem. We divide this problem in 3 subproblems with different conditions on the gains, specifially (i) gains with different amplitude and the same phase, (ii) gains with the same amplitude and different phase and (iii) gains with different amplitude and phase. In order to solve the first case, we propose an extension to the basis pursuit optimization which can estimate the unknown gains along with the unknown sparse signals. For the second case, we formulate a quadratic approach that eliminates the unknown phase shifts and retrieves the unknown sparse signals. An alternative form of this approach is also formulated to reduce complexity and memory requirements and provide scalability with respect to the number of input signals. Finally for the third case, we propose a formulation that combines the earlier two approaches to solve the problem. The performance of the proposed algorithms is investigated extensively through numerical simulations, which demonstrates that simultaneous signal recovery and calibration is possible with convex methods when sufficiently many (unknown, but sparse) calibrating signals are provided.