Channel Protection using Random Modulation

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.