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