Blind Deconvolution Demixing using Modulated Inputs

TL;DR

This paper introduces a novel method for blind deconvolution demixing of modulated signals using gradient descent, providing theoretical guarantees and demonstrating robustness through extensive simulations.

Contribution

It presents a new approach for blind deconvolution demixing with modulated inputs, establishing sample complexity bounds and demonstrating recovery guarantees.

Findings

Successful recovery of signals and channels under specified conditions

Gradient descent converges to the true solution in simulations

Theoretical phase transition analysis supports practical robustness

Abstract

This paper focuses on solving a challenging problem of blind deconvolution demixing involving modulated inputs. Specifically, multiple input signals $s_n(t)$, each bandlimited to $B$ Hz, are modulated with known random sequences $r_n(t)$ that alter at rate $Q$. Each modulated signal is convolved with a different M tap channel of impulse response $h_n(t)$, and the outputs of each channel are added at a common receiver to give the observed signal $y(t)=\sum_{n=1}^N (r_n(t)\odot s_n(t))\circledast h_n(t)$, where $\odot$ is the point wise multiplication, and $\circledast$ is circular convolution. Given this observed signal $y(t)$, we are concerned with recovering $s_n(t)$ and $h_n(t)$. We employ deterministic subspace assumption for the input signal $s_n(t)$ and keep the channel impulse response $h_n(t)$ arbitrary. We show that if modulating sequence is altered at a rate $Q \geq N^2 (B+M)$ and sample complexity bound is obeyed then all the signals and the channels, $\{s_n(t),h_n(t)\}_{n=1}^N$, can be estimated from the observed mixture $y(t)$ using gradient descent algorithm. We have performed extensive simulations that show the robustness of our algorithm and used phase transitions to numerically investigate the theoretical guarantees provided by our algorithm.