The problem of direction of arrival (DOA) estimation has been studied for decades as an essential technology in enabling radar, wireless communications, and array signal processing related applications. In this paper, the DOA estimation problem in the scenario with gain-phase errors is considered, and a sparse model is formulated by exploiting the signal sparsity in the spatial domain. By proposing a new atomic norm, named as GP-ANM, an optimization method is formulated via deriving a dual norm of GP-ANM. Then, the corresponding semidefinite program (SDP) is given to estimate the DOA efficiently, where the SDP is obtained based on the Schur complement. Moreover, a regularization parameter is obtained theoretically in the convex optimization problem. Simulation results show that the proposed method outperforms the existing methods, including the subspace-based and sparse-based methods in the scenario with gain-phase errors.

A New Atomic Norm for DOA Estimation With Gain-Phase Errors

Recently, several promising approximate message passing (AMP) based algorithms have been developed for bilinear recovery with model $\boldsymbol{Y}=\sum _{\boldsymbol{k}=1}^{\boldsymbol{K}} \boldsymbol{b}_{\boldsymbol{k}} \boldsymbol{A}_{\boldsymbol{k}} \boldsymbol{C}+\boldsymbol{W}$ , where $\lbrace \boldsymbol{b}_{\boldsymbol{k}}\rbrace$ and $\boldsymbol{C}$ are jointly recovered with known $\boldsymbol{A}_k$ from the noisy measurements $\boldsymbol{Y}$ . The bilinear recovery problem has many applications such as dictionary learning, self-calibration, compressive sensing with matrix uncertainty, etc. In this work, we propose a new approximate Bayesian inference algorithm for bilinear recovery, where AMP with unitary transformation (UTAMP) is integrated with belief propagation (BP), variational inference (VI) and expectation propagation (EP) to achieve efficient approximate inference. It is shown that, compared to state-of-the-art bilinear recovery algorithms, the proposed algorithm is much more robust and faster, leading to remarkably better performance.

Approximate Message Passing With Unitary Transformation for Robust Bilinear Recovery

This paper considers the generalized bilinear recovery problem, which aims to jointly recover the vector b and the matrix X from componentwise nonlinear measurements ${\text {Y}}\sim p({\text {Y}}|{\text {Z}})=\prod \limits _{i,j}p(Y_{ij}|Z_{ij})$ 
, where ${\text {Z}}={\text {A}}({\text {b}}){\text {X}}$ 
, ${\text {A}}(\cdot)$ is a known affine linear function of b, and $p(Y_{ij}|Z_{ij})$ is a scalar conditional distribution that models the general output transform. A wide range of real-world applications, e.g., quantized compressed sensing with matrix uncertainty, blind self-calibration and dictionary learning from nonlinear measurements, one-bit matrix completion, and joint channel and data decoding, can be cast as the generalized bilinear recovery problem. To address this problem, we propose a novel algorithm called the Bilinear Adaptive Generalized Vector Approximate Message Passing (BAd-GVAMP), which extends the recently proposed Bilinear Adaptive Vector AMP algorithm to incorporate arbitrary distributions on the output transform. The numerical results on various applications demonstrate the effectiveness of the proposed BAd-GVAMP algorithm.

Bilinear Adaptive Generalized Vector Approximate Message Passing

Grid-less variational Bayesian line spectral estimation with multiple measurement vectors

Employing low-resolution analog-to-digital converters (ADCs) coupled with large antenna arrays at the receivers has drawn considerable interests in the millimeter wave (mm-wave) system. Since mm-wave channels are sparse in angular dimensions, exploiting the structure could reduce the number of measurements while achieving acceptable performance at the same time. Motivated by the variational Bayesian line spectral estimation (VALSE) algorithm which treats the angles as random parameters, in contrast to previous works which confine the estimate to the set of grid angle points and induce grid mismatch, this paper proposes the grid-less quantized variational Bayesian channel estimation (GL-QVBCE) algorithm for antenna array systems with low resolution ADCs. Numerical results show the near optimal performance of GL-QVBCE by comparing with the Cramer Rao bound (CRB) and the state-of-art methods.

Grid-Less Variational Bayesian Channel Estimation for Antenna Array Systems With Low Resolution ADCs

In this paper, the line spectral estimation (LSE) problem is studied from one-bit quantized samples where variational line spectral estimation (VALSE) combined expectation propagation (EP) VALSE-EP method is proposed. Since the original measurements are heavily quantized, performing the off-grid frequency estimation is very challenging. Referring to the expectation propagation (EP) principle, this quantized model is decomposed as two modules, one is the componentwise minimum mean square error (MMSE) module, the other is the standard linear model where the variational line spectrum estimation (VALSE) algorithm can be performed. The VALSE-EP algorithm iterates between the two modules in a turbo manner. In addition, this algorithm can be easily extended to solve the LSE with the multiple measurement vectors (MMVs). Finally, numerical results demonstrate the effectiveness of the proposed VALSE-EP method.

Off-grid Variational Bayesian Inference of Line Spectral Estimation from One-bit Samples.

Efficient estimation of line spectral from quantized samples is of significant importance in information theory and signal processing, e.g., channel estimation in energy efficient massive MIMO systems and direction of arrival estimation. The goal of this paper is to recover the line spectral as well as its corresponding parameters including the model order, frequencies and amplitudes from heavily quantized samples. To this end, we propose an efficient grid-less Bayesian algorithm named VALSE-EP, which is a combination of the variational line spectral estimation (VALSE) and expectation propagation (EP). The basic idea of VALSE-EP is to iteratively approximate the challenging quantized model of line spectral estimation as a sequence of simple pseudo unquantized models so that the VALSE can be applied. Note that the noise in the pseudo linear model is heteroscedastic, i.e., different components having different variances, and a variant of the VALSE is re-derived to obtain the final VALSE-EP. Moreover, to obtain a benchmark performance of the proposed algorithm, the Cramer Rao bound (CRB) is derived. Finally, numerical experiments on both synthetic and real data are performed, demonstrating the near CRB performance of the proposed VALSE-EP for line spectral estimation from quantized samples.

Grid-less Variational Bayesian Inference of Line Spectral from Quantized Samples

The fundamental multidimensional line spectral estimation problem is addressed utilizing the Bayesian methods. Motivated by the recently proposed variational line spectral estimation (VALSE) algorithm, multidimensional VALSE (MDVALSE) is developed. MDVALSE inherits the advantages of VALSE such as automatically estimating the model order, noise variance and providing uncertain degrees of frequency estimates. Compared to VALSE, the multidimensional frequencies of a single component is treated as a whole, and the probability density function (PDF) is projected as independent univariate von Mises distribution to perform tractable inference. Besides, for the initialization, efficient fast Fourier transform (FFT) is adopted to approximate the marginal posterior PDF of frequencies. Numerical results demonstrate the effectiveness of the MDVALSE, compared to state-of-art methods.

Multidimensional Variational Line Spectra Estimation

In this paper, we consider a general form of noisy compressive sensing (CS) where the sensing matrix is not precisely known. Such cases exist when there are imperfections or unknown calibration parameters during the measurement process. Particularly, the sensing matrix may have some structure, which makes the perturbation follow a fixed pattern. While previous work has focused on extending the approximate message passing (AMP) and LASSO algorithm to deal with the independent and identically distributed (i.i.d.) perturbation, we propose the robust variant vector approximate message passing (VAMP) algorithm with the perturbation being structured, based on the recent VAMP algorithm. The performance of the robust version of VAMP is demonstrated numerically.

Qi Zhang

Papers

Grid-less variational Bayesian line spectral estimation with multiple measurement vectors

Off-grid Variational Bayesian Inference of Line Spectral Estimation from One-bit Samples.

Grid-less Variational Bayesian Inference of Line Spectral from Quantized Samples

Multidimensional Variational Line Spectra Estimation

Vector Approximate Message Passing Algorithm for Structured Perturbed Sensing Matrix