# Compressed Blind De-convolution

## TL;DR

This paper introduces novel L1 minimization techniques for blind deconvolution and compressed sensing, enabling recovery of unknown filters and sparse signals from limited noisy measurements under certain conditions.

## Contribution

It develops new L1 minimization methods and establishes recovery guarantees for blind deconvolution and compressed sensing with unknown stable systems.

## Key findings

- Exact recovery of filter H and sparse signal u from noisy measurements.
- Reconstruction of x, u, and H from O(k log n) measurements under specific conditions.
- Proposed LP optimization algorithm reliably estimates unknown filter and sparse input.

## Abstract

Suppose the signal x is realized by driving a k-sparse signal u through an arbitrary unknown stable discrete-linear time invariant system H. These types of processes arise naturally in Reflection Seismology. In this paper we are interested in several problems: (a) Blind-Deconvolution: Can we recover both the filter $H$ and the sparse signal $u$ from noisy measurements? (b) Compressive Sensing: Is x compressible in the conventional sense of compressed sensing? Namely, can x, u and H be reconstructed from a sparse set of measurements. We develop novel L1 minimization methods to solve both cases and establish sufficient conditions for exact recovery for the case when the unknown system H is auto-regressive (i.e. all pole) of a known order. In the compressed sensing/sampling setting it turns out that both H and x can be reconstructed from O(k log(n)) measurements under certain technical conditions on the support structure of u. Our main idea is to pass x through a linear time invariant system G and collect O(k log(n)) sequential measurements. The filter G is chosen suitably, namely, its associated Toeplitz matrix satisfies the RIP property. We develop a novel LP optimization algorithm and show that both the unknown filter H and the sparse input u can be reliably estimated.