We consider the problem of recovering two unknown vectors, w and x, of length L from their circular convolution. We make the structural assumption that the two vectors are members known subspaces, one with dimension N and the other with dimension K. Although the observed convolution is nonlinear in both w and x, it is linear in the rank-1 matrix formed by their outer product wx * . 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 w and x exactly when the maximum of N and K is almost on the order of L. That is, we show that if x is drawn from a random subspace of dimension N , and 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 wx * 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 × 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 N + K, to within constant and log factors.

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We propose several sampling architectures for the efficient acquisition of an ensemble of correlated signals. We show that without prior knowledge of the correlation structure, each of our architectures (under different sets of assumptions) can acquire the ensemble at a sub-Nyquist rate. Prior to sampling, the analog signals are diversified using simple, implementable components. The diversification is achieved by injecting types of "structured randomness" into the ensemble, the result of which is subsampled. For reconstruction, the ensemble is modeled as a low-rank matrix that we have observed through an (undetermined) set of linear equations. Our main results show that this matrix can be recovered using a convex program when the total number of samples is on the order of the intrinsic degree of freedom of the ensemble -the more heavily correlated the ensemble, the fewer samples are needed.To motivate this study, we discuss how such ensembles arise in the context of array processing.

This paper considers recovering L-dimensional vectors w, and x 1 , x 2 , . . . , x N from their circular convolutions y n = w * x n , n = 1, 2, 3, . . . , N . The vector w is assumed to be S-sparse in a known basis that is spread out in the Fourier domain, and each input x n is a member of a known K-dimensional random subspace.We prove that whenever K + S log 2 S L/ log 4 (LN ), the problem can be solved effectively by using only the nuclear-norm minimization as the convex relaxation, as long as the inputs are sufficiently diverse and obey N log 2 (LN ). By "diverse inputs", we mean that the x n 's belong to different, generic subspaces. To our knowledge, this is the first theoretical result on blind deconvolution where the subspace to which w belongs is not fixed, but needs to be determined.We discuss the result in the context of multipath channel estimation in wireless communications. Both the fading coefficients, and the delays in the channel impulse response w are unknown. The encoder codes the K-dimensional message vectors randomly and then transmits coded messages x n 's over a fixed channel one after the other. The decoder then discovers all of the messages and the channel response when the number of samples taken for each received message are roughly greater than (K + S log 2 S) log 4 (LN ), and the number of messages is roughly at least log 2 (LN ).