We consider the problem of recovering two unknown vectors, @math and @math , of length @math from their circular convolution. We make the structural assumption that the two vectors are members of known subspaces, one with dimension @math and the other with dimension @math . Although the observed convolution is nonlinear in both @math and @math , it is linear in the rank-1 matrix formed by their outer product @math . 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 @math and @math exactly when the maximum of @math and @math is almost on the order of @math . That is, we show that if @math is drawn from a random subspace of dimension @math , and @math is a vector in a subspace of dimension @math whose basis vectors are "spread out" in the frequency domain, then nuclear norm minimization recovers @math without error. We discuss this result in the context of blind channel estimation in communications. If we have a message of length @math which we code using a random @math coding matrix, and the encoded message travels through an unknown linear time-invariant channel of maximum length @math , then the receiver can recover both the channel response and the message when @math , to within constant and log factors.
The recently developed compressive sensing (CS) framework enables the design of sub-Nyquist analog-to-digital converters. Several architectures have been proposed for the acquisition of sparse signals in large swaths of bandwidth. In this paper we consider a more flexible multi-channel signal model consisting of several discontiguous channels where the occupancy of the combined bandwidth of the channels is sparse. We introduce a new compressive acquisition architecture, the compressive multiplexer (CMUX), to sample such signals. We demonstrate that our architecture is CS-feasible and suggest a simple implementation with numerous practical advantages.  Abstract Many applications in cellular systems and sensor networks involve a random subset of a large number of users asynchronously reporting activity to a base station. This paper examines the problem of multiuser detection (MUD) in random access channels for such applications. Traditional orthogonal signaling ignores the random nature of user activity in this problem and limits the total number of users to be on the order of the number of signal space dimensions. Contention-based schemes, on the other hand, suffer from delays caused by colliding transmissions and the hidden node problem. In contrast, this paper presents a novel pairing of an asynchronous non-orthogonal code-division random access scheme with a convex optimization-based MUD algorithm that overcomes the issues associated with orthogonal signaling and contention-based methods. Two key distinguishing features of the proposed MUD algorithm are that it does not require knowledge of the delay or channel state information of every user and it has polynomial-time computational complexity. The main analytical contribution of this paper is the relationship between the performance of the proposed MUD algorithm in the presence of arbitrary or random delays and two simple metrics of the set of user codewords. The study of these metrics is then focused on two specific sets of codewords, random binary codewords and specially constructed algebraic codewords, for asynchronous random access. The ensuing analysis confirms that the proposed scheme together with either of these two codeword sets significantly outperforms the orthogonal signaling-based random access in terms of the total number of users in the system.