# Channel Protection using Random Modulation

## TL;DR

This paper demonstrates that random modulation, or spread spectrum, can protect bandlimited signals from convolutive interference, enabling provable recovery of both the signal and channel using gradient descent under certain conditions.

## Contribution

The paper introduces a novel modulation-based method for signal protection and provides theoretical guarantees for recovering signals and channels via simple gradient descent algorithms.

## Key findings

- Successful recovery of signals and channels using gradient descent.
- Phase transition analysis for recovery conditions.
- Stable recovery in the presence of noise.

## Abstract

This paper shows that modulation protects a bandlimited signal against convolutive interference. A signal $s(t)$, bandlimited to $B$Hz, is modulated (pointwise multiplied) with a known random sign sequence $r(t)$, alternating at a rate $Q$, and the resultant \textit{spread spectrum} signal $s(t) \odot r(t)$ is convolved against an $M$-tap channel impulse response $h(t)$ to yield the observed signal $y(t)= (s(t)\odot r(t))\circledast h(t),$ where $\odot$ and $\circledast$ denote pointwise multiplication, and circular convolution, respectively. We show that both $s(t)$, and $h(t)$ can be provably recovered using a simple gradient descent scheme by alternating the binary waveform $r(t)$ at a rate $Q \gtrsim B + M$(to within log factors and a signal coherences) and sampling $y(t)$ at a rate $Q$. We also present a comprehensive set of phase transitions to depict the trade-off between $Q$, $M$, and $B$ for successful recovery. Moreover, we show stable recovery results under noise.