In this letter, we consider a multiple-input single-output (MISO) linear time-varying system whose output is a superposition of scaled and time-frequency shifted versions of inputs. The goal of this letter is to determine system characteristics and input signals from the single output signal. More precisely, we want to recover the continuous time-frequency shift pairs, the corresponding (complex-valued) amplitudes and the input signals from only one output vector. This problem arises in a variety of applications such as radar imaging, microscopy, channel estimation and localization problems. While this problem is naturally ill-posed, by constraining the unknown input waveforms to lie in separate known low-dimensional subspaces, it becomes tractable. More explicitly, we propose a semidefinite program which exactly recovers time-frequency shift pairs and input signals. We prove uniqueness and optimality of the solution to this program. Moreover, we provide a grid-based approach which can significantly reduce computational complexity in exchange for adding a small gridding error. Numerical results confirm the ability of our proposed method to exactly recover the unknowns.

We investigate eigenvectors of rank-one deformations of random matrices$\boldsymbol B = \boldsymbol A + \theta \boldsymbol {uu}^{*}$in which$\boldsymbol A \in \mathbb R^{N \times N}$is a Wigner real symmetric random matrix,$\theta \in \mathbb R^{+}$, and$\boldsymbol u$is uniformly distributed on the unit sphere. It is well known that for$\theta > 1$the eigenvector associated with the largest eigenvalue of$\boldsymbol B$closely estimates$\boldsymbol u$asymptotically, while for$\theta < 1$the eigenvectors of$\boldsymbol B$are uninformative about$\boldsymbol u$. We examine$\mathcal O({1}/{N})$correlation of eigenvectors with$\boldsymbol u$before phase transition and show that eigenvectors with larger eigenvalue exhibit stronger alignment with deforming vector through an explicit inverse law${1}/{\theta ^{*} - x}$with$\theta ^{*}:= \theta + ({1}/{\theta })$. This distribution function will be shown to be the ordinary generating function of Chebyshev polynomials of the second kind. These polynomials form an orthogonal set with respect to the semicircle weighting function. This law is an increasing function in the support of semicircle law for eigenvalues$(-2\:,+2)$. Therefore, most of energy of the unknown deforming vector is concentrated in a$cN$-dimensional ($c < 1$) known subspace of$\boldsymbol B$. We use a combinatorial approach to prove the result. We also extend the result to constant rank-$r$deformations.

This work is about the total variation (TV) minimization which is used for recovering gradient-sparse signals from compressed measurements. Recent studies indicate that TV minimization exhibits a phase transition behavior from failure to success as the number of measurements increases. In fact, in large dimensions, TV minimization succeeds in recovering the gradient-sparse signal with high probability when the number of measurements exceeds a certain threshold; otherwise, it fails almost certainly. Obtaining a closed-form expression that approximates this threshold is a major challenge in this field and has not been appropriately addressed yet. In this work, we derive a tight lower-bound on this threshold in case of any random measurement matrix whose null space is distributed uniformly with respect to the Haar measure. In contrast to the conventional TV phase transition results that depend on the simple gradient-sparsity level, our bound is highly affected by generalized notions of gradient-sparsity. Our proposed bound is very close to the true phase transition of TV minimization confirmed by simulation results.

In this letter, we consider the problem of recovering analysis-sparse signals from under-sampled measurements when some prior information about the support is available. We incorporate such information in the recovery stage by suitably tuning the weights in a weighted L1-analysis optimization problem. Indeed, we try to set the weights such that the method succeeds with minimum number of measurements. For this purpose, we exploit the upper-bound on the statistical dimension of a certain cone to determine the weights. Our numerical simulations confirm that the introduced method with tuned weights outperforms the standard L1-analysis technique.

We study the problem of reconstructing a block-sparse signal from compressively sampled measurements. In certain applications, in addition to the inherent block-sparse structure of the signal, some prior information about the block support, i.e., blocks containing non-zero elements, might be available. Although many block-sparse recovery algorithms have been investigated in the Bayesian framework, it is still unclear how to incorporate the information about the probability of occurrence into regularization-based block-sparse recovery in an optimal sense. In this letter, we bridge between these fields by the aid of a new concept in conic integral geometry. Specifically, we solve a weighted optimization problem when the prior distribution about the block support is available. Moreover, we obtain the unique weights that minimize the expected required number of measurements. Our simulations on both synthetic and real data confirm that these weights considerably decrease the required sample complexity.

Evaluating the statistical dimension is a common tool to determine the asymptotic phase transition in compressed sensing problems with Gaussian ensemble. Unfortunately, the exact evaluation of the statistical dimension is very difficult and it has become standard to replace it with an upper-bound. To ensure that this technique is suitable, [1] has introduced an upper-bound on the gap between the statistical dimension and its approximation. In this work, we first show that the error bound in [1] in some low-dimensional models such as total variation and ℓ1analysis minimization becomes poorly large. Next, we develop a new error bound which significantly improves the estimation gap compared to [1]. In particular, unlike the bound in [1] that fails in some settings with overcomplete dictionaries, our bound exhibits a decaying behavior in such cases.

This letter considers the use of total variation (TV) minimization in the recovery of a given gradient sparse vector from Gaussian linear measurements. It has been shown in recent studies that there exists a sharp phase transition behavior in TV minimization for the number of measurements necessary to recover the signal in asymptotic regimes. The phase-transition curve specifies the boundary of success and failure of TV minimization for large number of measurements. It is a challenging task to obtain a theoretical bound that reflects this curve. In this letter, we present a novel upper bound that suitably approximates this curve and is asymptotically sharp. Numerical results show that our bound is closer to the empirical TV phase-transition curve than the previously known bound obtained by Kabanava.

Interference alignment (IA) techniques mostly attain their degrees of freedom benefits as the number of channel extensions tends to infinity. Intuitively, the more interfering signals that need to be aligned, the larger the number of dimensions needed to align them. This requirement poses a major challenge for IA in practical systems. This study evaluates the necessary and sufficient conditions on channel structure of a fully connected interference network with time‐varying fading to make perfect IA feasible within limited number of channel extensions. The authors propose a method based on the obtained conditions on the channel structure to achieve perfect IA. For the case of 3 user interference channel, it is shown that only one condition on the channel coefficients is required to make perfect IA feasible at all receivers. IA feasibility literatures have mainly focused on network topology so far, in contrast, derived channel aiding conditions in this study can be considered as the perfect IA feasibility conditions on the channel structure.

This letter presents the sparse vector signal detection from one bit compressed sensing measurements, in contrast to the previous works that deal with scalar signal detection. Available results are extended to the vector case and the generalized likelihood ratio test (GLRT) detector and the optimal quantizer design are obtained. A double-detector scheme is introduced, in which a sensor level threshold detector is integrated into network level GLRT to improve the performance. The detection criteria of oracle and clairvoyant detectors are also derived. Simulation results show that with careful design of the threshold detector, the overall detection performance of double-detector scheme would be better than the sign-GLRT proposed in [J. Fang et al., “One-bit quantizer design for multisensor GLRT fusion,” IEEE Signal Process. Lett., vol. 20, no. 3, pp. 257-260, Mar. 2013] and close to oracle and clairvoyant detectors. The proposed detector is applied to spectrum sensing and the results are near the well-known energy detector, which uses the real valued data, while the proposed detector only uses the sign of the data.

Polarity coincidence correlator (PCC), when used to estimate the covariance matrix on an element-by-element basis, may not yield a positive semi-definite (PSD) estimate. Devlin et al. (Devlin, 1975) claimed that element-wise PCC is not guaranteed to be PSD in dimensions p > 3 for real signals. However, no justification or proof was available on this issue. In this letter, it is proved that for real signals with p les 3 and for complex signals with p les 2, a PSD estimate is guaranteed. Counterexamples are presented for higher dimensions which yield invalid covariance estimates.

In this letter, we address the theoretical limitations in estimating the mixing matrix in noisy sparse component analysis (SCA) for the two-sensor case. We obtain the Cramer-Rao lower bound (CRLB) error estimation of the mixing matrix. Using the Bernouli-Gaussian (BG) sparse distribution, and some simple assumptions, an approximation of the Fisher information matrix (FIM) is calculated. Moreover, this CRLB is compared to some of the main methods of mixing matrix estimation in the literature.