Blind Deconvolution Demixing using Modulated Inputs

TL;DR

This paper shows 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 can be estimated from the observed mixture using gradient descent algorithm.

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.

Figures (4)

• Figure 1: Analog implementation of uplink NOMA using single carrier OFDM for real time protection against channel interference. Different user equipment's (UE) are transmitting continuous time signal $s_n(t)$ which is bandlimited to $B$ Hz. At each transmitter, random binary waveform $r_n(t)$, alternating at a rate $Q$, is used to modulate $s_n(t)$. The modulated signals are passed through different unknown LTI channels having an $M$-tap impulse response $h_n(t)$ and received at the common base station (BS). The received signal, $y(t)$, is a mixture of the convolution of the transmitted signals and the channels through which the signals passed. At a sampling rate $Q$, the received signal is sampled by ADC where $Q \gtrsim N^2 (B+M)$ (scale with coherences), and recover unknown signals $\{s_n(t)\}_{n=1}^N$, and channels $\{h_n(t)\}_{n=1}^N$ using algorithm \ref{['algo:gradient-descent']}.
• Figure 2: For fixed $L$, and $N$, phase diagrams of $K$ vs. $M$ for different $Q$. Phase transitions show that larger modulated inputs, larger $Q$, allow recovery with larger values of $K$, and $M$.
• Figure 3: Number of transmitters vs. the number of observations.
• Figure 4: SNR (dB) vs. average relative error for $N=2$.

Theorems & Definitions (19)

• Theorem 1
• Theorem 2
• Lemma 1
• Lemma 2: In li2018rapid:Lemma 5.9
• proof
• Lemma 3: Lemma 6.12 in li2018rapid
• proof
• Lemma 4
• Lemma 5
• Lemma 6