Channel Protection Using Random Modulation

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February 1, 2022
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This paper shows that modulation protects a bandlimited signal against convolutive interference. A signal? $s(t)$ , bandlimited to BHz, is modulated (pointwise multiplied) with a known random sign sequence? $r(t)$ , alternating at a rate? $Q$ , and the resultant 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? $Q \geq B + M$ (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.