Stable Blind Deconvolution over the Reals from Additional Autocorrelations

TL;DR

This paper demonstrates that stable blind deconvolution over the real numbers is achievable from autocorrelations when the signal has sufficient zero separation, with stability bounds depending on signal properties.

Contribution

It extends previous uniqueness results by establishing stability under zero separation conditions and provides explicit stability bounds based on spectral analysis.

Findings

Stable reconstruction is possible with zero separation in the z-domain.

The stability constant depends on signal dimension and boundary coefficients.

An explicit analytical expression for the stability constant is derived.

Abstract

Recently the one-dimensional time-discrete blind deconvolution problem was shown to be solvable uniquely, up to a global phase, by a semi-definite program for almost any signal, provided its autocorrelation is known. We will show in this work that under a sufficient zero separation of the corresponding signal in the $z-$domain, a stable reconstruction against additive noise is possible. Moreover, the stability constant depends on the signal dimension and on the signals magnitude of the first and last coefficients. We give an analytical expression for this constant by using spectral bounds of Vandermonde matrices.