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.
The setup in is reminiscent of a single-antenna multi-user spread spectrum communication scenario @cite_6 . There, the matrix @math represents the spreading matrix assigned to the @math -th user and @math models the associated multipath channel. There are numerous papers on blind channel estimation in connection with CDMA, including the previously cited articles @cite_19 @cite_24 @cite_23 . Our work differs from the existing literature on this topic in several ways: As mentioned before, we do not require that we have multiple received signals, we allow all multipath channels @math to differ from each other, and do not impose a particular channel model. Moreover, we provide a rigorous mathematical theory, instead of just empirical observations.
@cite_24 @cite_19 @cite_23 @cite_6
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We show that very large bandwidths on fading multipath channels cannot be effectively utilized by spread-spectrum systems that (in a particular sense) spread the available power uniformly over both time and frequency. The approach is to express the input process as an expansion in an orthonormal set of functions each localized in time and frequency. The fourth moment of each coefficient in this expansion is then uniformly constrained. We show that such a constraint forces the mutual information to 0 inversely with increasing bandwidth. Simply constraining the second moment of these coefficients does not achieve this effect. The results suggest strongly that conventional direct-sequence code-division multiple-access (CDMA) systems do not scale well to extremely large bandwidths. To illustrate how the interplay between channel estimation and symbol detection affects capacity, we present results for a specific channel and CDMA signaling scheme. 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. 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 . The problems of channel estimation and multiuser detection for direct sequence code division multiple access (DS CDMA) systems employing long spreading codes are considered. With regard to channel estimation, several procedures are proposed based on the least-squares approach, relying on the transmission of known training symbols but not requiring any timing synchronization. In particular, algorithms suited for the forward and reverse links of a single-rate DS CDMA cellular system are developed, and the case of a multirate multicode system, wherein high-rate users are split into multiple virtual low-rate users, is also considered. All of the proposed procedures are recursively implementable with a computational complexity that is quadratic in the processing gain, with regard to the issue of multiuser detection, an adaptive serial interference cancellation (SIC) receiver is considered, where the adaptivity stems from the fact that it is built upon the channel estimates provided by the estimation algorithm. Simulation results show that coupling the proposed estimation algorithms with a SIC receiver may yield, with a much lower computational complexity, performance levels close to those of the ideal linear minimum mean square error (MMSE) receiver, which assumes perfect knowledge of the channels for all of the users and which (in a long-code scenario) has a computational complexity per symbol interval proportional to the third power of the processing gain.