The acquisition of Downlink (DL) channel state information at the transmitter (CSIT) is known to be a challenging task in multiuser massive MIMO systems when uplink/downlink channel reciprocity does not hold (e.g., in frequency division duplexing systems). From a coding viewpoint, the DL channel state acquired at the users via DL training can be seen as an information source that must be conveyed to the base station via the UL communication channels. The transmission of a source through a channel can be accomplished either by separate or joint source-channel coding (SSCC or JSCC). In this work, using classical remote distortion-rate (DR) theory, we first provide a theoretical lower bound on the channel estimation meansquare-error (MSE) of both JSCC and SSCC-based feedback schemes, which however requires encoding of large blocks of successive channel states and thus cannot be used in practice since it would incur in an extremely large feedback delay. We then focus on the relevant case of minimal (one slot) feedback delay and propose a practical JSCC-based feedback scheme that fully exploits the channel second-order statistics to optimize the dimension projection in the eigenspace. We analyze the large SNR behavior of the proposed JSCC-based scheme in terms of the quality scaling exponent (QSE). Given the second-order statistics of channel estimation of any feedback scheme, we further derive the closed-form of the lower bound to the ergodic sum-rate for DL data transmission under maximum ratio transmission and zero-forcing precoding. Via extensive numerical results, we show that our proposed JSCC-based scheme outperforms known JSCC, SSCC baseline and deep learning-based schemes and is able to approach the performance of the optimal DR scheme in the range of practical SNR.

In this work, we study and compare two types of CSI feedback schemes in multi-user massive MIMO systems, respectively based on joint and separate source-channel coding (JSCC and SSCC). Using the classical remote distortion-rate (DR) theory, we first provide a theoretical lower bound on the channel estimation mean-square-error (MSE) of any feedback scheme. The DR bound is achieved by using vector quantization applied to long sequences of channel state estimates and requires capacity-achieving channel coding in the uplink, resulting in a large delay in the CSI feedback loop that makes the scheme impractical. Thus we propose a practical JSCC-based feedback scheme that sends the CSI with minimal delay. Unlike previous works that simply apply linear mapping and equal power allocation to generate the feedback signal, our method applies the dimension projection in the eigenspace and optimizes power allocation by fully exploiting the channel second-order statistics. The extensive numerical results show that our proposed JSCC-based scheme not only outperforms the previous JSCC scheme with linear processing but also produces lower channel estimate MSE compared to a standard SSCC-based scheme at practical SNR.

We propose a method for channel training and precoding in FDD massive MIMO based on deep neural networks (DNNs), exploiting Downlink (DL) channel covariance knowledge. The DNN is optimized to maximize the DL multi-user sum-rate, by producing a pre-beamforming matrix based on user channel covariances that maps the original channel vectors to “effective channels”. Measurements of these effective channels are received at the users via common pilot transmission and sent back to the base station (BS) through analog feedback without further processing. The BS estimates the effective channels from received feedback and constructs a linear precoder by concatenating the optimized pre-beamforming matrix with a zero-forcing precoder over the effective channels. We show that the proposed method yields significantly higher sum-rates than the state-of-the-art DNN-based channel training and precoding scheme, especially in scenarios with small pilot and feedback size relative to the channel coherence block length. Unlike many works in the literature, our proposition does not involve deployment of a DNN at the user side, which typically comes at a high computational cost and parameter-transmission overhead on the system, and is therefore considerably more practical.

We study the problem of providing channel state information (CSI) at the transmitter in multi-user “massive” MIMO systems operating in frequency division duplexing (FDD). The wideband MIMO channel is a vector-valued random process correlated in time, space (antennas), and frequency (subcarriers). The base station (BS) broadcasts periodically$\beta _{\mathrm{ tr}}$pilot symbols from its$M$antenna ports to$K$single-antenna users (UEs). Correspondingly, the$K$UEs send feedback messages about their channel state using$\beta _{\mathrm{ fb}}$symbols in the uplink (UL). Using results from remote rate-distortion theory, we show that, as${\sf snr}\to \infty $, the optimal feedback strategy achieves a channel state estimation mean squared error (MSE) that behaves as$\Theta {(}1)$if$\beta _{\mathrm{ tr}} < r$and as$\Theta \left ({{\sf snr}^{-\alpha }}\right)$when$\beta _{\mathrm{ tr}} \ge r$, where$\alpha = \min (\beta _{\mathrm{ fb}}/r, 1)$, where$r$is the rank of the channel covariance matrix. The MSE-optimal rate-distortion strategy implies encoding of long sequences of channel states, which would yield completely stale CSI and therefore poor multiuser precoding performance. Hence, we consider three practical “one-shot” CSI strategies with minimum one-slot delay and analyze their large-SNR channel estimation MSE behavior. These are: (1) digital feedback via entropy-coded scalar quantization (ECSQ), (2) analog feedback (AF), and (3) local channel estimation at the UEs via compressed sensing and digital feedback. These schemes have different requirements in terms of knowledge of the channel statistics at the UE and at the BS. In particular, the latter strategy requires no statistical knowledge and is closely inspired by a CSI feedback scheme currently proposed in 3GPP standardization. It is shown that ECSQ achieves optimal MSE at the price of a slight increase in feedback rate which vanishes for large SNR. AF achieves the optimal MSE decay rate of$\Theta ({\sf snr}^{-1})$whenever$\beta _{\mathrm{ tr}},\beta _{\mathrm{ fb}} \ge r$but is sub-optimal if$\beta \ge r$and$\beta _{\mathrm{ fb}} < r$. The 3GPP-inspired scheme is shown, via numerical simulations, to achieves performance similar to ECSQ and AF when the multipath channel is sufficiently sparse in the angle-delay domain, but suffers from a large performance gap if this requirement is not met.

We consider the problem of estimating the fading coefficients of a frequency-selective, spatially correlated channel via Downlink (DL) training and Uplink (UL) feedback in frequency division duplexing (FDD) massive MIMO systems. Using ratedistortion theory, we derive optimal bounds on the achievable channel state estimation error in terms of the number of training pilots in DL (βtr) and feedback dimension in UL (βfb), with random, spatially isotropic pilots. It is shown that when the number of training pilots exceeds the channel covariance rank (r), the optimal rate-distortion feedback strategy achieves an estimation error decay of ΘpSNR−αq in estimating the channel state, where α = minpβfb{r,1q is the so-called quality scaling exponent (QSE). We then discuss an "analog" feedback strategy, showing that it achieves the optimal QSE for a wide range of training and feedback dimensions with no channel covariance knowledge and simple signal processing at the user side. Our findings are supported by numerical simulations comparing these strategies in terms of channel state mean squared error and achievable ergodic sum-rate in DL with zero-forcing precoding.

Using commodity WiFi data for applications such as indoor localization, object identification and tracking and channel sounding has recently gained considerable attention. We study the problem of channel impulse response (CIR) estimation from commodity WiFi channel state information (CSI). The accuracy of a CIR estimation method in this setup is limited by both the available channel bandwidth as well as various CSI distortions induced by the underlying hardware. We propose a multi-band splicing method that increases channel bandwidth by combining CSI data across multiple frequency bands. In order to compensate for the CSI distortions, we develop a per-band processing algorithm that is able to estimate the distortion parameters and remove them to yield the “clean” CSI. This algorithm incorporates the atomic norm denoising sparse recovery method to exploit channel sparsity. Splicing clean CSI over M frequency bands, we use orthogonal matching pursuit (OMP) as an estimation method to recover the sparse CIR with high (M-fold) resolution. Unlike previous works in the literature, our method does not appeal to any limiting assumption on the CIR (other than the widely accepted sparsity assumption) or any ad hoc processing for distortion removal. We show, empirically, that the proposed method outperforms the state of the art in terms of localization accuracy.

We study the problem of maximum-likelihood (ML) estimation of an approximate common eigenstructure, i.e. an approximate common eigenvectors set (CES), for an ensemble of covariance matrices given a collection of their associated i.i. d vector realizations. This problem has a direct application in multi-user MIMO communications, where the base station (BS) has access to instantaneous user channel vectors through pilot transmission and attempts to perform joint multi-user Downlink (DL) precoding. It is widely accepted that an efficient implementation of this task hinges upon an appropriate design of a set of common “virtual beams” that captures the common eigenstructure among the user channel covariances. In this paper, we propose a novel method for obtaining this common eigenstructure by casting it as an ML estimation problem. We prove that in the special case where the covariances are jointly diagonalizable, the global optimal solution of the proposed ML problem coincides with the common eigenstructure. Then we propose a projected gradient descent (PGD) method to solve the ML optimization problem over the manifold of unitary matrices and prove its convergence to a stationary point. Through exhaustive simulations, we illustrate that in the case of jointly diagonalizable covariances, our proposed method converges to the exact CES. Also, in the general case where the covariances are not jointly diagonalizable, it yields a solution that approximately diagonalizes all covariances. Besides, the empirical results show that our proposed method outperforms the well-known joint approximate diagonalization of eigenmatrices (JADE) method in the literature.

In spatial channel models used in multi-antenna wireless communications, the propagation from a single-antenna transmitter (e.g. a user) to an M-antenna receiver (e.g. a Base Station) occurs through scattering clusters located in the far field of the receiving array. The angular power spread function (APSF) of the corresponding M-dim channel vector describes the angular density of the received signal power at the array. In many applications, such as channel sounding and Uplink Downlink covariance transformation in FDD systems, estimating the APSF is required either implicitly or explicitly. However, the existing literature on the subject has mainly focused on channel covariance estimation from a set of noisy pilot observations. It is also assumed that the APSF consists only of discrete components corresponding to Line-of-Sight (LoS) paths and specular scattering. It turns out that while covariance estimation is a well-posed problem, APSF estimation is a much harder task and is in general ill-posed. The reason is that the propagation environment can also include diffuse scattering elements, resulting in continuous APSF components. Therefore, the APSF is a function belonging to the infinite-dimensional space of nonnegative measures over the angle domain. In this paper, we show that under a geometrically-consistent, group-sparse structure on the APSF, which is prevalent in massive MIMO channels, one is able to estimate the APSF properly. We propose an algorithm based on deep neural networks (DNNs) that learns this structure and yields precise APSF estimates, even when the number of available pilot observations is relatively small. We empirically show that our proposed method outperforms the state-of-the-art method in various performance metrics.

Obtaining channel covariance knowledge is of great importance in various Multiple-Input Multiple-Output MIMO communication applications, including channel estimation and user grouping. Considering recently proposed massive MIMO systems, covariance estimation proves to be challenging due to the large number of antennas (M >> 1) employed in the base station. In this case, the number of pilot transmissions N becomes comparable to the number of antennas and standard estimators, such as the sample covariance, yield a poor estimate of the true covariance and are hence undesirable. In this paper, we propose a Maximum-Likelihood (ML) massive MIMO covariance estimator, based on a parametric representation of the channel angular spread function (ASF). The parametric representation emerges from super-resolving discrete ASF components plus approximating its continuous components using carefully chosen limited-support density function. We maximize the likelihood function using a Concave-Convex procedure, which is initialized via a non-negative least-squares optimization problem. Our simulation results show that the proposed method outperforms the state of the art in various estimation quality metrics.

We propose a novel randomized channel sparsifying hybrid precoding (RCSHP) design to reduce the signaling overhead of channel estimation and the hardware cost and power consumption at the base station (BS), in order to fully harvest benefits of frequency division duplex (FDD) massive multiple-input multiple-output (MIMO) systems. RCSHP allows time-sharing among multiple analog precoders, each serving a compatible user group. The analog precoder is adapted to the channel statistics to properly sparsify the channel for the associated user group, such that the resulting effective channel (product of channel and analog precoder) not only has enough spatial degrees of freedom (DoF) to serve this group of users, but also can be accurately estimated under the limited pilot budget. The digital precoder is adapted to the effective channel based on the duality theory to facilitate the power allocation and exploit the spatial multiplexing gain. We formulate the joint optimization of the time-sharing factors and the associated sets of analog precoders and power allocations as a general utility optimization problem, which considers the impact of effective channel estimation error on the system performance. Then we propose an efficient stochastic successive convex approximation algorithm to provably obtain Karush-Kuhn-Tucker (KKT) points of this problem.

We propose a novel method for massive multiple-input multiple-output (massive MIMO) in frequency division duplexing (FDD) systems. Due to the large frequency separation between uplink (UL) and downlink (DL) in FDD systems, channel reciprocity does not hold. Hence, in order to provide DL channel state information to the base station (BS), closed-loop DL channel probing, and channel state information (CSI) feedback is needed. In massive MIMO, this typically incurs a large training overhead. For example, in a typical configuration with M ≃200 BS antennas and fading coherence block of T ≃ 200 symbols, the resulting rate penalty factor due to the DL training overhead, given by max{0, 1 - M/T }, is close to 0. To reduce this overhead, we build upon the well-known fact that the angular scattering function of the user channels is invariant over frequency intervals whose size is small with respect to the carrier frequency (as in current FDD cellular standards). This allows us to estimate the users' DL channel covariance matrix from UL pilots without additional overhead. Based on this covariance information, we propose a novel sparsifying precoder in order to maximize the rank of the effective sparsified channel matrix subject to the condition that each effective user channel has sparsity not larger than some desired DL pilot dimension Tdl, resulting in the DL training overhead factor max{0, 1 - Tdl/T } and CSI feedback cost of Tdl pilot measurements. The optimization of the sparsifying precoder is formulated as a mixed integer linear program, that can be efficiently solved. Extensive simulation results demonstrate the superiority of the proposed approach with respect to the concurrent state-of-the-art schemes based on compressed sensing or UL/DL dictionary learning.

In this paper, we propose a novel method for efficient implementation of a massive Multiple-Input Multiple- Output (massive MIMO) system with Frequency Division Duplexing (FDD) operation. Our main objective is to reduce the large overhead incurred by Downlink (DL) common training and Uplink (UL) feedback needed to obtain channel state information (CSI) at the base station. Our proposed scheme relies on the fact that the underlying angular distribution of a channel vector, also known as the angular scattering function, is a frequency-invariant entity yielding a ULDL reciprocity and has a limited angular support. We estimate this support from UL CSI and interpolate it to obtain the corresponding angular support of the DL channel. Finally we exploit the estimated support of the DL channel of all the users to design an efficient channel probing and feedback scheme that maximizes the total spectral efficiency of the system. Our method is different from the existing compressed-sensing (CS) based techniques in the literature. Using support information helps reduce the feedback overhead from O(s logM) in CS techniques to O(s) in our proposed method, with s andM being sparsity order of the channel vectors and the number of base station antennas, respectively. Furthermore, in order to control the channel sparsity and therefore the DL common training and UL feedback overhead, we introduce the novel concept of active channel sparsification. In brief, when the fixed pilot dimension is less than the required amount for reliable channel estimation, we introduce a pre-beamforming matrix that artificially reduces the effective channel dimension of each user to be not larger than the DL pilot dimension, while maximizing both the number of served users and the number of probed angles. We provide numerical experiments to assess the performance of our method and compare it with the state-of-the-art CS technique.

In this paper, we study the problem of multiband (frequency-variant) covariance interpolation with a particular emphasis towards massive MIMO applications. In massive MIMO, the communication between each Base Station (BS) with M ≫ 1 antennas and each single-antenna user occurs through a collection of scatterers in the environment, where the channel vector of each user at BS antennas consists in a weighted linear combination of the array responses of the scatterers, where each scatterer has its own angle of arrival (AoA) and complex channel gain. The array response at a given AoA depends on the wavelength of the incoming planar wave and is naturally frequency dependent. While in typical wireless communication applications the signal bandwidth is narrow enough, such that the channel second-order statistics (notably, the channel covariance matrix) can be considered frequency independent, in many other applications such as Frequency Division Duplexing (FDD) the uplink (UL) and the downlink (DL) channels are separated by a large frequency interval, such that the dependence of the channel covariance on frequency cannot be ignored. In this paper, we show that although this dependence is generally negligible for a small number of antennas M, it results in a considerable distortion of the covariance matrix when M → ∞. Moreover, we prove that this frequency-dependent distortion can be fully compensated by a suitable covariance interpolation in frequency. We analyze the covariance interpolation problem mathematically and prove its stability under a very mild reciprocity condition on the angular power spread function (PSF) of the users. We also investigate the validity of our results using numerical simulations.

In this paper, we study the prediction of a circularly symmetric zero-mean stationary Gaussian process from a window of observations consisting of finitely many samples. This is a prevalent problem in a wide range of applications in communication theory and signal processing. Due to stationarity, when the autocorrelation function or equivalently the power spectral density (PSD) of the process is available, the Minimum Mean Squared Error (MMSE) predictor is readily obtained. In particular, it is given by a linear operator that depends on autocorrelation of the process as well as the noise power in the observed samples. The prediction becomes, however, quite challenging when the PSD of the process is unknown. In this paper, we propose a blind predictor that does not require the a priori knowledge of the PSD of the process and compare its performance with that of an MMSE predictor that has a full knowledge of the PSD. To design such a blind predictor, we use the random spectral representation of a stationary Gaussian process. We apply the well-known atomic-norm minimization technique to the observed samples to obtain a discrete quantization of the underlying random spectrum, which we use to predict the process. Our simulation results show that this estimator has a good performance comparable with that of the MMSE estimator.