Blind calibration for compressed sensing: State evolution and an online algorithm

TL;DR

This paper introduces an online algorithm for blind calibration in compressed sensing, extending previous offline methods, and provides a theoretical analysis of its performance using State Evolution, validated by numerical simulations.

Contribution

It extends the cal-AMP algorithm from offline to online calibration and offers a theoretical performance analysis via State Evolution formalism.

Findings

The online cal-AMP algorithm effectively refines calibration with new measurements.

Theoretical predictions of performance match numerical simulations.

Both offline and online algorithms perform efficiently in compressed sensing tasks.

Abstract

Compressed sensing, allows to acquire compressible signals with a small number of measurements. In applications, a hardware implementation often requires a calibration as the sensing process is not perfectly known. Blind calibration, that is performing at the same time calibration and compressed sensing is thus particularly appealing. A potential approach was suggested by Sch\"ulke and collaborators in Sch\"ulke et al. 2013 and 2015, using approximate message passing (AMP) for blind calibration (cal-AMP). Here, the algorithm is extended from the already proposed offline case to the online case, where the calibration is refined step by step as new measured samples are received. Furthermore, we show that the performance of both the offline and the online algorithms can be theoretically studied via the State Evolution (SE) formalism. Through numerical simulations, the efficiency of cal-AMP and the consistency of the theoretical predictions are confirmed.