  We consider the problem of recovering two unknown vectors, $\boldsymbol{w}$
and $\boldsymbol{x}$, of length $L$ from their circular convolution. We make
the structural assumption that the two vectors are members of known subspaces,
one with dimension $N$ and the other with dimension $K$. Although the observed
convolution is nonlinear in both $\boldsymbol{w}$ and $\boldsymbol{x}$, it is
linear in the rank-1 matrix formed by their outer product
$\boldsymbol{w}\boldsymbol{x}^*$. This observation allows us to recast the
deconvolution problem as low-rank matrix recovery problem from linear
measurements, whose natural convex relaxation is a nuclear norm minimization
program.
  We prove the effectiveness of this relaxation by showing that for "generic"
signals, the program can deconvolve $\boldsymbol{w}$ and $\boldsymbol{x}$
exactly when the maximum of $N$ and $K$ is almost on the order of $L$. That is,
we show that if $\boldsymbol{x}$ is drawn from a random subspace of dimension
$N$, and $\boldsymbol{w}$ is a vector in a subspace of dimension $K$ whose
basis vectors are "spread out" in the frequency domain, then nuclear norm
minimization recovers $\boldsymbol{w}\boldsymbol{x}^*$ without error.
  We discuss this result in the context of blind channel estimation in
communications. If we have a message of length $N$ which we code using a random
$L\times N$ coding matrix, and the encoded message travels through an unknown
linear time-invariant channel of maximum length $K$, then the receiver can
recover both the channel response and the message when $L\gtrsim N+K$, to
within constant and log factors.
