This paper considers recovering @math -dimensional vectors @math , and @math from their circular convolutions @math . The vector @math is assumed to be @math -sparse in a known basis that is spread out in the Fourier domain, and each input @math is a member of a known @math -dimensional random subspace. We prove that whenever @math , 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 @math . By "diverse inputs", we mean that the @math 's belong to different, generic subspaces. To our knowledge, this is the first theoretical result on blind deconvolution where the subspace to which @math 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 @math are unknown. The encoder codes the @math -dimensional message vectors randomly and then transmits coded messages @math '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 @math , and the number of messages is roughly at least @math .
We introduce a new algorithm for multichannel blind deconvolution. Given the outputs of K linear time- invariant channels driven by a common source, we wish to recover their impulse responses without knowledge of the source signal. Abstractly, this problem amounts to finding a solution to an overdetermined system of quadratic equations. We show how we can recast the problem as solving a system of underdetermined linear equations with a rank constraint. Recent results in the area of low rank recovery have shown that there are effective convex relaxations to problems of this type that are also scalable computationally, allowing us to recover 100s of channel responses after a moderate observation time. We illustrate the effectiveness of our methodology with a numerical simulation of a passive noise imaging" experiment. Abstract Since (1991) demonstrated the feasibility of identifying possibly nonminimum phase channels using second-order statistics, considerable research activity, both in algorithm development and fundamental analysis, has been seen in the area of blind identification of multiple FIR channels. Many of the recently developed approaches invoke, either explicitly or implicitly, the algebraic structure of the data model, while some others resort to the use of cyclic correlation spectral fitting techniques. The objective of this paper is to establish insightful connections among these studies and present recent developments of blind channel equalization. We also unify various representative algorithms into a common theoretical framework.  Blind deconvolution is the recovery of a sharp version of a blurred image when the blur kernel is unknown. Recent algorithms have afforded dramatic progress, yet many aspects of the problem remain challenging and hard to understand. The goal of this paper is to analyze and evaluate recent blind deconvolution algorithms both theoretically and experimentally. We explain the previously reported failure of the naive MAP approach by demonstrating that it mostly favors no-blur explanations. We show that, using reasonable image priors, a naive simulations MAP estimation of both latent image and blur kernel is guaranteed to fail even with infinitely large images sampled from the prior. On the other hand, we show that since the kernel size is often smaller than the image size, a MAP estimation of the kernel alone is well constrained and is guaranteed to succeed to recover the true blur. The plethora of recent deconvolution techniques makes an experimental evaluation on ground-truth data important. As a first step toward this experimental evaluation, we have collected blur data with ground truth and compared recent algorithms under equal settings. Additionally, our data demonstrate that the shift-invariant blur assumption made by most algorithms is often violated.  A new algorithm is proposed for the deconvolution of an unknown, possibly colored, Gaussian or nonstationary signal that is observed through two or more unknown channels described by rational system transfer functions. More specifically, not only the root (pole and zero) locations but also the orders of the channel transfer functions are unknown. It is assumed that the channel orders may be overestimated. The proposed algorithm estimates the orders and root locations of the channel transfer functions, therefore it can also be used in multichannel system identification problems. The input signal is allowed to be nonstationary and the channel transfer functions may be a nonminimum phase as well as noncausal, hence the proposed algorithm is particularly suitable for applications such as dereverberation of speech signals recorded through multiple microphones. Several experimental results indicate improvement compared to the existing methods in the literature. > A new blind channel identification and equalization method is proposed that exploits the cyclostationarity of oversampled communication signals to achieve identification and equalization of possibly nonminimum phase (multipath) channels without using training signals. Unlike most adaptive blind equalization methods for which the convergence properties are often problematic, the channel estimation algorithm proposed here is asymptotically ex-set. Moreover, since it is based on second-order statistics, the new approach may achieve equalization with fewer symbols than most techniques based only on higher-order statistics. Simulations have demonstrated promising performance of the proposed algorithm for the blind equalization of a three-ray multipath channel. >