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 .
Another relevant result is blind deconvolution plus demixing @cite_9 , where one observes sum of @math different convolved pairs of @math -dimensional vectors lying in @math , and @math dimensional known subspaces; one of which is generic and the other is incoherent in the Fourier domain. Each generic basis is chosen independently of others. The blind deconvolution plus the demixing problem is again cast as a rank- @math matrix recovery problem. The algorithm is successful when @math .
@cite_9
2140867429 2537174396 2951249062 2011677178
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 of 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 x 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. 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 the best of 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 . Blind deconvolution (BD), the resolution of a signal and a filter given their convolution, arises in many applications. Without further constraints, BD is ill-posed. In practice, subspace or sparsity constraints have been imposed to reduce the search space, and have shown some empirical success. However, existing theoretical analysis on uniqueness in BD is rather limited. As an effort to address the still mysterious question, we derive sufficient conditions under which two vectors can be uniquely identified from their circular convolution, subject to subspace or sparsity constraints. These sufficient conditions provide the first algebraic sample complexities for BD. We first derive a sufficient condition that applies to almost all bases or frames. For blind deconvolution of vectors in @math , with two subspace constraints of dimensions @math and @math , the required sample complexity is @math . Then we impose a sub-band structure on one basis, and derive a sufficient condition that involves a relaxed sample complexity @math , which we show to be optimal. We present the extensions of these results to BD with sparsity constraints or mixed constraints, with the sparsity level replacing the subspace dimension. The cost for the unknown support in this case is an extra factor of 2 in the sample complexity. A time domain blind source separation algorithm of convolutive sound mixtures is studied based on a compact partial inversion formula in closed form. An l1-constrained minimization problem is formulated to find demixing filter coefficients for source separation while capturing scaling invariance and sparseness of solutions. The minimization aims to reduce (lagged) cross correlations of the mixture signals which are modeled stochastically. The problem is non-convex, however it is put in a nonlinear least square form where the robust and convergent Levenberg-Marquardt iterative method is applicable to compute local minimizers. Efficiency is achieved in recovering lower dimensional demixing filter solutions than the physical ones. Computations on recorded and synthetic mixtures show satisfactory performance, and are compared with other iterative methods.