Identifiability Conditions for Compressive Multichannel Blind Deconvolution

TL;DR

This paper establishes conditions under which sparse multichannel blind deconvolution can be uniquely identified from compressive Fourier measurements, reducing the number of measurements and channels needed compared to prior work.

Contribution

It derives sharp identifiability conditions for sparse-MBD from compressive Fourier measurements and proposes a kernel-based sampling scheme, improving measurement efficiency.

Findings

L-sparse filters identifiable from 2L^2 Fourier measurements across two channels

At least 2L measurements per channel are necessary for identifiability

As the number of channels increases, the required Fourier samples per channel decrease asymptotically

Abstract

In applications such as multi-receiver radars and ultrasound array systems, the observed signals can often be modeled as a linear convolution of an unknown signal which represents the transmit pulse and sparse filters which describe the sparse target scenario. The problem of identifying the unknown signal and the sparse filters is a sparse multichannel blind deconvolution (MBD) problem and is in general ill-posed. In this paper, we consider the identifiability problem of sparse-MBD and show that, similar to compressive sensing, it is possible to identify the sparse filters from compressive measurements of the output sequences. Specifically, we consider compressible measurements in the Fourier domain and derive identifiability conditions in a deterministic setup. Our main results demonstrate that $L$-sparse filters can be identified from $2L^2$ Fourier measurements from only two coprime channels. We also show that $2L$ measurements per channel are necessary. The sufficient condition sharpens as the number of channels increases asymptotically in the number of channels, it suffices to acquire on the order of $L$ Fourier samples per channel. We also propose a kernel-based sampling scheme that acquires Fourier measurements from a commensurate number of time samples. We discuss the gap between the sufficient and necessary conditions through numerical experiments including comparing practical reconstruction algorithms. The proposed compressive MBD results require fewer measurements and fewer channels for identifiability compared to previous results, which aids in building cost-effective receivers.