This paper considers recovering a continuous angular power spectrum (APS) from the channel covariance. Building on the projection-onto-linear-variety (PLV) algorithm, an affine-projection approach introduced by Miretti \emph{et. al.}, we analyze PLV in a well-defined \emph{weighted} Fourier-domain to emphasize its geometric interpretability. This yields an explicit fixed-dimensional trigonometric-polynomial representation and a closed-form solution via a positive-definite matrix, which directly implies uniqueness. We further establish an exact energy identity that yields the APS reconstruction error and leads to a sharp identifiability/resolution characterization: PLV achieves perfect recovery if and only if the ground-truth APS lies in the identified trigonometric-polynomial subspace; otherwise it returns the minimum-energy APS among all covariance-consistent spectra.

Massive multiple-input multiple-output (MIMO) systems employing one-bit digital-to-analog converters offer a hardware-efficient solution for wireless communications. However, the one-bit constraint poses significant challenges for precoding design, as it transforms the problem into a discrete and nonconvex optimization task. In this paper, we investigate a widely adopted ``convex-relaxation-then-quantization" approach for nonlinear symbol-level one-bit precoding. Specifically, we first solve a convex relaxation of the discrete minimum mean square error precoding problem, and then quantize the solution to satisfy the one-bit constraint. Focusing on a real-valued system with an independently and identically distributed (i.i.d.) Gaussian channel, we develop a novel analytical framework based on approximate message passing (AMP) to characterize the high-dimensional asymptotic performance of the considered scheme. The key technical ingredient is an auxiliary AMP iteration that dedicatedly incorporates the nonlinear quantization function into the state evolution analysis. With the proposed framework, we derive a closed-form expression for the symbol error probability (SEP) at the receiver side in the large-system limit, which provides a quantitative characterization of how model and system parameters affect the SEP performance. Our empirical results suggest that the $\ell_\infty^2$ regularizer, when paired with an optimally chosen regularization parameter, achieves optimal SEP performance within a broad class of convex regularization functions. As a first step towards a theoretical justification, we prove the optimality of the $\ell_\infty^2$ regularizer within the mixed $\ell_\infty^2$-$\ell_2^2$ regularization functions.

This paper studies the device activity detection problem in a massive multiple-input multiple-output (MIMO) system for near-field communications (NFC). In this system, active devices transmit their signature sequences to the base station (BS), which detects the active devices based on the received signal. In this paper, we model the near-field channels as correlated Rician fading channels and formulate the device activity detection problem as a maximum likelihood estimation (MLE) problem. Compared to the traditional uncorrelated channel model, the correlation of channels complicates both algorithm design and theoretical analysis of the MLE problem. On the algorithmic side, we present the classical exact coordinate descent (CD) algorithm for solving the MLE problem, which suffers from numerical instability when applied to correlated channels. We propose a computationally efficient inexact CD algorithm by approximating the objective function, which approximately solves the one-dimensional subproblem and improves both computational efficiency and numerical stability. Additionally, we analyze the detection performance of the MLE problem under correlated channels by comparing it with the case of uncorrelated channels. The analysis shows that when the overall number of devicesNis large or the signature sequence lengthLis small, the detection performance of MLE under correlated channels tends to be better than that under uncorrelated channels. Conversely, whenNis small orLis large, MLE performs better under uncorrelated channels than under correlated ones. Finally, we study the MLE model in the joint device activity and data detection context. Simulation results demonstrate the computational performance of the presented algorithms and verify the correctness of the analysis.

Compressive Robust Principal Component Analysis (CRPCA) naturally arises in various applications as a means to recover a low-rank matrix low-rank matrix$\boldsymbol {L}$and a sparse matrix$\boldsymbol {S}$from compressive measurements. In this paper, we approach the problem from a Bayesian inference perspective. We establish a probabilistic model for the problem and develop an improved turbo message passing (ITMP) algorithm based on the sum-product rule and the appropriate approximations. Additionally, we establish a state evolution framework to characterize the asymptotic behavior of the ITMP algorithm in the large-system limit. By analyzing the established state evolution, we further propose sufficient conditions for the global convergence of our algorithm. Our numerical results validate the theoretical results, demonstrating that the proposed asymptotic framework accurately characterize the dynamical behavior of the ITMP algorithm, and the phase transition curve specified by the sufficient condition agrees well with numerical simulations.

In this paper, we consider the recovery of low-rank matrices from noisy observations using spectral denoisers, where the singular values are denoised through an identical scalar smoothing function. We explore the asymptotic mean squared error (AMSE) of these denoisers within a framework where the rank of the matrix to be recovered grows linearly with the matrix size. We demonstrate that, under arbitrary i.i.d. noise and some mild regularity assumptions, the AMSE converges in probability to a deterministic function of the noise power. Our results are applicable to commonly used denoisers, including the best-rank-r denoiser, the singular-value soft-threshold denoiser, and the singular-value hard-threshold denoiser. To the best of our knowledge, this is the first study to establish an analytical expression for the asymptotic MSE under arbitrary i.i.d. noise. The derived analytical expression depends solely on the empirical distribution of the singular values of the low-rank matrix and the specific form of the spectral denoiser employed.

We consider the identifiability issue of maximum-likelihood based activity detection in massive MIMO-based grant-free random access. An intriguing observation by (Chen et al., 2022) indicates that the identifiability undergoes a phase transition for commonly-used random user signatures as$L^{2}$,$N$and$K$tend to infinity with fixed ratios, where$L$,$N$and$K$denote the user signature length, the total number of users, and the number of active users, respectively. In this letter, we provide a precise analytical characterization of the phase transition based on a spectral universality conjecture. Numerical results demonstrate excellent agreement between our theoretical predictions and the empirical phase transitions.

We consider an inverse problem$\boldsymbol {y}= f(\boldsymbol {Ax})$, where$\boldsymbol {x}\in \mathbb {R}^{n}$is the signal of interest,$\boldsymbol {A}$is the sensing matrix,$f$is a nonlinear function and$\boldsymbol {y} \in \mathbb {R}^{m}$is the measurement vector. In many applications, we have some level of freedom to design the sensing matrix$\boldsymbol {A}$, and in such circumstances we could optimize$\boldsymbol {A}$to achieve better reconstruction performance. As a first step towards optimal design, it is important to understand the impact of the sensing matrix on the difficulty of recovering$\boldsymbol {x}$from$\boldsymbol {y}$. In this paper, we study the performance of one of the most successful recovery methods, i.e., the expectation propagation (EP) algorithm. We define a notion of spikiness for the spectrum of$\boldsymbol {A}$and show the importance of this measure for the performance of EP. We show that whether a spikier spectrum can hurt or help the recovery performance depends on$f$. Based on our framework, we are able to show that, in phase-retrieval problems, matrices with spikier spectrums are better for EP, while in 1-bit compressed sensing problems, less spiky spectrums lead to better performance. Our results unify and substantially generalize existing results that compare Gaussian and orthogonal matrices, and provide a platform towards designing optimal sensing systems.

A promising approach to deal with the high hardware cost and energy consumption of massive MIMO transmitters is to use low-resolution digital-to-analog converters (DACs) at each antenna element. This leads to a transmission scheme where the transmitted signals are restricted to a finite set of voltage levels. This paper is concerned with the analysis and optimization of a low-cost quantized precoding strategy, referred to as linear-quantized precoding, for a downlink massive MIMO system under Rayleigh fading. In linear-quantized precoding, the signals are first processed by a linear precoding matrix and subsequently quantized component-wise by the DAC. In this paper, we analyze both the signal-to-interference-plus-noise ratio (SINR) and the symbol error probability (SEP) performances of such linear-quantized precoding schemes in an asymptotic framework where the number of transmit antennas and the number of users grow large with a fixed ratio. Our results provide a rigorous justification for the heuristic arguments based on the Bussgang decomposition that are commonly used in prior works. Based on the asymptotic analysis, we further derive the optimal precoder within a class of linear-quantized precoders that includes several popular precoders as special cases. Our numerical results demonstrate the excellent accuracy of the asymptotic analysis for finite systems and the optimality of the derived precoder.

This paper studies a large random matrix system (LRMS) model involving an arbitrary signal distribution and forward error control (FEC) coding. We establish an area property based on the approximate message passing (AMP) algorithm. Under the assumption that the state evolution for AMP is correct for the coded system, the achievable rate of AMP is analyzed. We prove that AMP achieves the constrained capacity of the LRMS with an arbitrary signal distribution provided that a matching condition is satisfied. We provide related numerical results of binary signaling using irregular low-density parity-check (LDPC) codes. We show that the optimized codes demonstrate significantly better performance over unmatched ones under AMP. For quadrature phase shift keying (QPSK) modulation, bit error rate (BER) performance within 1 dB from the constrained capacity limit is observed.

We consider a nonlinear inverse problem $\mathbf{y}= f(\mathbf{Ax})$, where observations $\mathbf{y} \in \mathbb{R}^m$ are the componentwise nonlinear transformation of $\mathbf{Ax} \in \mathbb{R}^m$, $\mathbf{x} \in \mathbb{R}^n$ is the signal of interest and $\mathbf{A}$ is a known linear mapping. By properly specifying the nonlinear processing function, this model can be particularized to many signal processing problems, including compressed sensing and phase retrieval. Our main goal in this paper is to understand the impact of sensing matrices, or more specifically the spectrum of sensing matrices, on the difficulty of recovering $\mathbf{x}$ from $\mathbf{y}$. Towards this goal, we study the performance of one of the most successful recovery methods, i.e. the expectation propagation algorithm (EP). We define a notion for the spikiness of the spectrum of $\mathbf{A}$ and show the importance of this measure in the performance of the EP. Whether the spikiness of the spectrum can hurt or help the recovery performance of EP depends on $f$. We define certain quantities based on the function $f$ that enables us to describe the impact of the spikiness of the spectrum on EP recovery. Based on our framework, we are able to show that for instance, in phase-retrieval problems, matrices with spikier spectrums are better for EP, while in 1-bit compressed sensing problems, less spiky (flatter) spectrums offer better recoveries. Our results unify and substantially generalize the existing results that compare sub-Gaussian and orthogonal matrices, and provide a platform toward designing optimal sensing systems.

This paper studies a large random matrix system (LRMS) model involving an arbitrary signal distribution and forward error control (FEC) coding. We establish an area property based on the approximate message passing (AMP) algorithm. Under the assumption that the state evolution for AMP is correct for the coded system, the achievable rate of AMP is analyzed. We prove that AMP achieves the constrained capacity of the LRMS with an arbitrary signal distribution provided that a matching condition is satisfied. As a byproduct, we provide an alternative derivation for the constraint capacity of an LRMS using a proved property of AMP. We discuss realization techniques for the matching principle of binary signaling using irregular low-density parity-check (LDPC) codes and provide related numerical results. We show that the optimized codes demonstrate significantly better performance over un-matched ones under AMP. For quadrature phase shift keying (QPSK) modulation, bit error rate (BER) performance within 1 dB from the constrained capacity limit is observed.

Phase retrieval refers to the problem of recovering a signal$ {x}_{\star }\in \mathbb {C}^{n}$from its phaseless measurements$\text {y}_{\text {i}}=| {a}_{i}^{ \mathsf {H}} {x}_{\star }|$, where$\{ {a}_{\text {i}}\}_{\text {i}=1}^{ {m}}$are the measurement vectors. Spectral method is widely used for initialization in many phase retrieval algorithms. The quality of spectral initialization can have a major impact on the overall algorithm. In this paper, we focus on the model where$ {A}=[ {a}_{1},\ldots, {a}_{ {m}}]^{ \mathsf {H}}$has orthonormal columns, and study the spectral initialization under the asymptotic setting$ {m}, {n}\to \infty $with$ {m}/ {n}\to \delta \in (1,\infty)$. We use the expectation propagation framework to characterize the performance of spectral initialization for Haar distributed matrices. Our numerical results confirm that the predictions of the EP method are accurate for not-only Haar distributed matrices, but also for realistic Fourier based models (e.g. the coded diffraction model). The main findings of this paper are the following: 1) There exists a threshold on$\delta $(denoted as$\delta _{ \mathrm {weak}}$) below which the spectral method cannot produce a meaningful estimate. We show that$\delta _{ \mathrm {weak}}=2$for the column-orthonormal model. In contrast, previous results by Mondelli and Montanari show that$\delta _{ \mathrm {weak}}=1$for the i.i.d. Gaussian model. 2) The optimal design for the spectral method coincides with that for the i.i.d. Gaussian model, where the latter was recently introduced by Luo, Alghamdi and Lu.

This paper considers a compressed-coding scheme that combines compressed sensing with forward error control coding. Approximate message passing (AMP) is used to decode the message. Based on the state evolution analysis of AMP, we derive the performance limit of compressed-coding. We show that compressed-coding can approach Gaussian capacity at a very low compression ratio. Further, the results are extended to systems involving non-linear effects such as clipping. We show that the capacity approaching property can still be maintained when generalized AMP is used to decode the message. To approach the capacity, a low-rate underlying code should be designed according to the curve matching principle, which is complicated in practice. Instead, analog spatial-coupling is used to avoid sophisticated low-rate code design.

This paper considers a compressed-coding scheme that combines compressed sensing with forward error control coding. Approximate message passing (AMP) is used to decode the message. Based on the state evolution analysis of AMP, we derive the performance limit of compressed-coding. We show that compressed-coding can approach Gaussian capacity at a very low compression ratio. Further, the results are extended to systems involving non-linear effects such as clipping. We show that the capacity approaching property can still be maintained when generalized AMP is used to decode the message. To approach the capacity, a low-rate underlying code should be designed according to the curve matching principle, which is complicated in practice. Instead, analog spatial-coupling is used to avoid sophisticated low-rate code design. In the end, we study the coupled scheme in a multiuser environment, where analog spatial-coupling can be realized in a distributive way. The overall block length can be shared by many users, which reduces block length per-user.

Phase retrieval refers to algorithmic methods for recovering a signal from its phaseless measurements. There has been recent interest in understanding the performance of local search algorithms that work directly on the non-convex formulation of the problem. Due to the non-convexity of the problem, the success of these local search algorithms depends heavily on their starting points. The most widely used initialization scheme is the spectral method, in which the leading eigenvector of a data-dependent matrix is used as a starting point. Recently, the performance of the spectral initialization was characterized accurately for measurement matrices with independent and identically distributed entries. This paper aims to obtain the same level of knowledge for isotropically random column-orthogonal matrices, which are substantially better models for practical phase retrieval systems. Towards this goal, we consider the asymptotic setting in which the number of measurements m, and the dimension of the signal, n, diverge to infinity with m/n = δ ∈ (1, ∞), and obtain a simple expression for the overlap between the spectral estimator and the true signal vector.

We study information theoretic limits of recovering an unknown n dimensional, complex signal vector x*with unit norm from m magnitude-only measurements of the form yi= |(Ax*)i|2, i = 1, 2 ..., m, where A is the sensing matrix. This is known as the Phase Retrieval problem and models practical imaging systems where measuring the phase of the observations is difficult. Since in a number of applications, the sensing matrix has orthogonal columns, we model the sensing matrix as a subsampled Haar matrix formed by picking n columns of a uniformly random m X m unitary matrix. We study this problem in the high dimensional asymptotic regime, where m, n → ∞, while m/n → δ with δ being a fixed number, and show that if mn(1)) · n, then any estimator is asymptotically orthogonal to the true signal vector x*. This lower bound is sharp since when m > (2 + on(1)) · n, estimators that achieve a non trivial asymptotic correlation with the signal vector are known from previous works.

We consider an ℓ2-regularized non-convex optimization problem for recovering signals from their noisy phaseless observations. We design and study the performance of a message passing algorithm that aims to solve this optimization problem. We consider the asymptotic setting m, n → ∞, m/n → δ and obtain sharp performance bounds, where m is the number of measurements and n is the signal dimension. We show that for complex signals, the algorithm can perform accurate recovery with only m = ((64/π2) - 4)n ≈ 2.5n measurements. Also, we provide a sharp analysis on the sensitivity of the algorithm to noise. We highlight the following facts about our message passing algorithm: 1) adding ℓ2regularization to the non-convex loss function can be beneficial and 2) spectral initialization has a marginal impact on the performance of the algorithm. The sharp analyses, in this paper, not only enable us to compare the performance of our method with other phase recovery schemes but also shed light on designing better iterative algorithms for other non-convex optimization problems.

Linear minimum mean square error (LMMSE) estimation based turbo detection has been extensively studied for coded linear systems since the seminal work of Wang and Poor (WP). The WP algorithm operates iteratively between a linear detector (LD) and a nonlinear detector (NLD): the LD suppresses the interference based on LMMSE filtering, and the NLD decodes the data by treating the output of the LD as an observation from an additive white Gaussian noise (AWGN) channel. In WP, the messages exchanged between LD and NLD are required to beextrinsic. For the NLD, the extrinsic message comes from the constraint imposed on feedforward error correction (FEC) codes. Therefore, WP does not work in an un-coded linear system. Recently, we proposed an orthogonal approximate message passing (OAMP) algorithm, which only requires the input/output error terms of LD and NLD to beorthogonal. We conjectured that for un-coded linear systems that involve certain large random matrices, the dynamics of OAMP can be accurately characterized by state evolution (SE). In this paper, we consider a coded linear system and develop an extrinsic message aided OAMP (EMA-OAMP) algorithm. Similar to the un-coded case, EMA-OAMP relaxes the requirements on output messages to be orthogonal instead of extrinsic. We derive an SE procedure to characterize the performance of OAMP in coded systems. We conjecture that this SE procedure is accurate, which is verified by simulation results. Under this conjecture, we show that EMA-OAMP can outperform WP under certain standard assumptions for iterative decoding. Extensive simulations results are provided to verify the advantages of OAMP in coded MIMO systems.

This paper is concerned with the affine rank minimization (ARM) problem for low-rank matrix recovery purposes. Inspired by the recently proposed Turbo-CS algorithm in the field of compressed sensing, we propose a turbo-type algorithm for ARM, termed Turbo-ARM (TARM). For matrix recovery problems with a large class of random measurement matrices, the performance of TARM can be analyzed via the state evolution framework. Our numerical results show that TARM achieves state-of-the-art reconstruction performance, and our results are further confirmed by state evolution analysis.

We consider an $\ell_2$-regularized non-convex optimization problem for recovering signals from their noisy phaseless observations. We design and study the performance of a message passing algorithm that aims to solve this optimization problem. We consider the asymptotic setting $m,n \rightarrow \infty$, $m/n \rightarrow \delta$ and obtain sharp performance bounds, where $m$ is the number of measurements and $n$ is the signal dimension. We show that for complex signals the algorithm can perform accurate recovery with only $m=\left ( \frac{64}{\pi^2}-4\right)n\approx 2.5n$ measurements. Also, we provide sharp analysis on the sensitivity of the algorithm to noise. We highlight the following facts about our message passing algorithm: (i) Adding $\ell_2$ regularization to the non-convex loss function can be beneficial even in the noiseless setting; (ii) spectral initialization has marginal impact on the performance of the algorithm.

This paper is concerned with pilot transmission schemes in a large antenna system with non-orthogonal multiple-access (NOMA). We investigate two pilot structures-orthogonal pilot (OP) and superimposed pilot (SP). In OP, pilots occupy dedicated time (or frequency) slots, while in SP, pilots are superimposed with data. We study an iterative data-aided channel estimation (IDACE) receiver, where partially decoded data are used to refine channel estimation. We analyze the achievable rates for systems with IDACE receivers for both OP and SP. We show that the optimal portion of pilot power tends to zero for SP with Gaussian signaling. This result is consistent with existing findings obtained via the replica method in statistical physics. The latter involves multiple codes, which is convenient for theoretical analysis but difficult to implement. As a comparison, IDACE is potentially implementable in practice. We demonstrate that, with code optimization, SP can outperform OP in a high mobility environment with a large number of users. We provide numerical examples to verify our analysis.

Approximate message passing (AMP) is a low-cost iterative signal recovery algorithm for compressed sensing. For sensing matrices with independent identically distributed (IID) Gaussian entries, the performance of AMP can be asymptotically characterized by a simple scaler recursion called state evolution (SE). SE analysis shows that AMP can potentially approach the optimal minimum mean squared-error (MMSE) limit. However, SE may become unreliable for other matrix ensembles, especially for ill-conditioned ones. In this paper, we propose an orthogonal AMP (OAMP) algorithm based on de-correlated linear estimation (LE) and divergence-free non-linear estimation (NLE). The Onsager term in standard AMP vanishes as a result of the divergence-free constraint on NLE. We develop an SE procedure for OAMP and show numerically that the SE for OAMP is accurate for a wide range of sensing matrices, including IID Gaussian matrices, partial orthogonal matrices, and general unitarily-invariant matrices. We further derive optimized options for OAMP and show that the corresponding SE fixed point coincides with the optimal performance obtained via the replica method.

In this letter, we propose a turbo compressed sensing algorithm with partial discrete Fourier transform (DFT) sensing matrices. Interestingly, the state evolution of the proposed algorithm is shown to be consistent with that derived using the replica method. Numerical results demonstrate that the proposed algorithm outperforms the well-known approximate message passing (AMP) algorithm when a partial DFT sensing matrix is involved.

This letter is on the performance of the turbo signal recovery (TSR) algorithm for partial discrete Fourier transform (DFT) matrices based compressed sensing. Based on state evolution analysis, we prove that TSR with a partial DFT sensing matrix outperforms the well-known approximate message passing (AMP) algorithm with an independent identically distributed (IID) sensing matrix.

This paper is concerned with the uplink in a multi-cell large antenna system. We study a channel estimation scheme where partially decoded data is used to estimate the channel. We show that there are two types of interference components in this scheme that do not vanish even when the number of antennas grows to infinity: cross-contamination and self-contamination. Cross contamination is in principle similar to pilot contamination in a conventional pilot-based channel estimation scheme, while self-contamination is unique for the data-aided scheme. The data-aided scheme can effectively suppress the contamination effect by increasing the data frame length without causing rate loss. This is confirmed by both analysis and simulation results.

In this paper, we develop a novel iterative equalization algorithm for energy-spreading-transform (EST) based multiple-input multiple-output (MIMO) systems. We show that the proposed scheme significantly outperforms the existing non-linear MIMO equalizers in various system setups. We further investigate the precoder design based on the signal-to-interference-plus-noise-ratio (SINR) variance evolution technique, so as to exploit the available channel state information at the transmitter (CSIT). We derive the optimal precoding directions, and show that the precoder optimization then boils down to a simple power allocation problem that is solvable using convex programming. Numerical results demonstrate that the optimized precoder can achieve a significant power gain, as compared with the non-optimized scheme.

This paper is concerned with precoder design for multiple-input multiple-output (MIMO) systems with iterative equalization. We first consider the case of no channel state information at the transmitter (CSIT). Based on evolution analysis, we derive the optimized precoder that minimizes the bit error rate (BER) of the system. We show that, with the optimized precoder, the linear precoding and iterative equalization scheme can achieve a genie-aided performance upper bound at high signal-to-noise ratio (SNR). We further consider the precoder design with perfect CSIT. We show that the precoder design problem reduces to a convex power-allocation problem that can be efficiently solved using standard convex programming tools. Numerical results are provided to demonstrate the performance advantages of the proposed scheme over its counterparts.

In this paper, we study the equalization problem for linearly precoded multiple-input multiple-output (MIMO) systems. We develop novel iterative equalization algorithms based on message-passing principles. We establish an evolution technique to analyze the performance of the proposed iterative equalization algorithms. We show by numerical results that the simulated performance of the proposed linear precoding and iterative equalization scheme agrees well with the evolution analysis, and that the proposed scheme can significantly outperform the existing schemes. It is worth noting that the performance advantage of the proposed scheme is achieved without exploiting any knowledge of channel state information at the transmitter (CSIT).