Suppose that we have @math sensors and each one intends to send a function @math (e.g. a signal or an image) to a receiver common to all @math sensors. During transmission, each @math gets convolved with a function @math . The receiver records the function @math , given by the sum of all these convolved signals. When and under which conditions is it possible to recover the individual signals @math and the blurring functions @math from just one received signal @math ? This challenging problem, which intertwines blind deconvolution with blind demixing, appears in a variety of applications, such as audio processing, image processing, neuroscience, spectroscopy, and astronomy. It is also expected to play a central role in connection with the future Internet-of-Things. We will prove that under reasonable and practical assumptions, it is possible to solve this otherwise highly ill-posed problem and recover the @math transmitted functions @math and the impulse responses @math in a robust, reliable, and efficient manner from just one single received function @math by solving a semidefinite program. We derive explicit bounds on the number of measurements needed for successful recovery and prove that our method is robust in the presence of noise. Our theory is actually sub-optimal, since numerical experiments demonstrate that, quite remarkably, recovery is still possible if the number of measurements is close to the number of degrees of freedom.
Problems of the type or are ubiquitous in many applied scientific disciplines and in applications, see e.g @cite_19 @cite_24 @cite_25 @cite_37 @cite_16 @cite_23 @cite_29 @cite_45 @cite_22 @cite_11 @cite_43 . Thus, there is a large body of works to solve different versions of these problems. Most of the existing works however require the availability of multiple received signals @math . And indeed, it is not hard to imagine that for instance an SVD-based approach will succeed if @math (and must fail if @math ). A sparsity-based approach can be found in @cite_8 . However, in this paper we are interested in the case where we have only one single received signal @math -- a single snapshot, in the jargon of array processing. Hence, there is little overlap between these methods heavily relying on multiple snapshots (many of which do not come with any theory) and the work presented here.
@cite_37 @cite_22 @cite_8 @cite_29 @cite_24 @cite_19 @cite_43 @cite_45 @cite_23 @cite_16 @cite_25 @cite_11
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Convolutive mixtures of images are common in photography of semi-reflections. They also occur in microscopy and tomography. Their formation process involves focusing on an object layer, over which defocused layers are superimposed. We seek blind source separation (BSS) of such mixtures. However, achieving this by direct optimization of mutual information is very complex and suffers from local minima. Thus, we devise an efficient approach to solve these problems. While achieving high quality image separation, we take steps that make the problem significantly simpler than a direct formulation of convolutive image mixtures. These steps make the problem practically convex, yielding a unique global solution to which convergence can be fast. The convolutive BSS problem is converted into a set of instantaneous (pointwise) problems, using a short time Fourier transform (STFT). Standard BSS solutions for instantaneous problems suffer, however, from scale and permutation ambiguities. We overcome these ambiguities by exploiting a parametric model of the defocus point spread function. Moreover, we enhance the efficiency of the approach by exploiting the sparsity of the STFT representation as a prior. We apply our algorithm to semi-reflections, and demonstrate it in experiments. Blind source separation is a process in which mixed signals, obtained as a linear combination of various source signals, are decomposed into their original sources. The source signals and their mixture weights are unknown, but a priori information about their statistical behavior and mixing model is available. In this paper, a new algorithm based on generalized cross correlation linear-operator set is proposed. This algorithm significantly improves source-separation quality compared to several other well-known algorithms, such as subband decomposition independent component analysis, block Gaussian likelihood, and convex analysis of mixtures of non-negative sources. The problem of blind demodulation of multiuser information symbols in a high-rate code-division multiple-access (CDMA) network in the presence of both multiple-access interference (MAI) and intersymbol interference (ISI) is considered. The dispersive CDMA channel is first cast into a multiple-input multiple-output (MIMO) signal model framework. By applying the theory of blind MIMO channel identification and equalization, it is then shown that under certain conditions the multiuser information symbols can be recovered without any prior knowledge of the channel or the users' signature waveforms (including the desired user's signature waveform), although the algorithmic complexity of such an approach is prohibitively high. However, in practice, the signature waveform of the user of interest is always available at the receiver. It is shown that by incorporating this knowledge, the impulse response of each user's dispersive channel can be identified using a subspace method. It is further shown that based on the identified signal subspace parameters and the channel response, two linear detectors that are capable of suppressing both MAI and ISI, i.e., a zero-forcing detector and a minimum-mean-square-error (MMSE) detector, can be constructed in closed form, at almost no extra computational cost. Data detection can then be furnished by applying these linear detectors (obtained blindly) to the received signal. The major contribution of this paper is the development of these subspace-based blind techniques for joint suppression of MAI and ISI in the dispersive CDMA channels. This chapter reviews space-time processing methods for CDMA mobile radio applications with emphasis on blind signal detection. We begin with a motivation for the use of blind space-time processing in CDMA. Next, we develop channel and signal models useful for blind processing. We follow this by considering first space-time single user receivers (ST-RAKE) and then review some basic theory of blind ST-RAKE algorithms. The important problem of multi-user detection (MUD) is considered next which leads to a novel technique allowing the estimation of the minimum-mean-square error linear MUD. We consider the blind multiuser detection problem for asynchronous DS-CDMA systems operating in a multipath environment. Using only the spreading code of the desired user, we first estimate the column vector subspace of the channel matrix by multiple linear prediction. Then, zero-forcing detectors and MMSE detectors with arbitrary delay can be obtained without explicit channel estimation. This avoids any channel estimation error, and the resulting methods are therefore more robust and more accurate. Corresponding batch algorithms and adaptive algorithms are developed. The new algorithms are extremely near-far resistant. Simulations demonstrate the effectiveness of these methods. By combining multiple-input multiple-output (MIMO) communication with the orthogonal frequency division multiplexing (OFDM) modulation scheme, MIMO-OFDM systems can achieve high data rates over broadband wireless channels. In this paper, to provide a bandwidth-efficient solution for MIMO-OFDM channel estimation, we establish conditions for channel identifiability and present a blind channel estimation technique based on a subspace approach. The proposed method unifies and generalizes the existing subspace-based methods for blind channel estimation in single-input single-output OFDM systems to blind channel estimation for two different MIMO-OFDM systems distinguished according to the number of transmit and receive antennas. In particular, the proposed method obtains accurate channel estimation and fast convergence with insensitivity to overestimates of the true channel order. If virtual carriers (VCs) are available, the proposed method can work with no or insufficient cyclic prefix (CP), thereby potentially increasing channel utilization. Furthermore, it is shown under specific system conditions that the proposed method can be applied to MIMO-OFDM systems without CPs, regardless of the presence of VCs, and obtains an accurate channel estimate with a small number of OFDM symbols. Thus, this method improves the transmission bandwidth efficiency. Simulation results illustrate the mean-square error performance of the proposed method via numerical experiments 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.