  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.
