Massive MIMO is a variant of multiuser MIMO, where the number of antennas $M$ at the base-station is large, and generally much larger than the number of spatially multiplexed data streams to/from the users. It has been observed that in many realistic propagation scenarios as well as in spatially correlated channel models used in standardizations, although the user channel vectors have a very high-dim $M$, they lie on low-dim subspaces due to their limited angular spread. This low-dim subspace structure remains stable across many coherence blocks and can be exploited in several ways to improve the system performance. A main challenge, however, is to estimate this signal subspace from samples of users' channel vectors as fast and efficiently as possible. In a recent work, we addressed this problem and proposed a very effective novel algorithm referred to as Approximate Maximum-Likelihood (AML), which was formulated as a semi-definite program (SDP). In this paper, we address two problems left open in our previous work: computational complexity and tracking. The algorithm proposed in this paper is reminiscent of Multiple Measurement Vectors (MMV) problem in Compressed Sensing and is proved to be equivalent to the AML Algorithm for sufficiently dense angular grids. It has also a very low computational complexity and is able to track sharp transitions in the channel statistics very quickly. Although mainly motivated by massive MIMO applications, our proposed algorithm is of independent interest in other related subspace estimation applications. We assess the estimation/tracking performance of our proposed algorithm empirically via numerical simulations, especially in practically relevant situations where a direct implementation of the SDP would be infeasible in real-time. We also compare the performance of our algorithm with other related subspace estimation algorithms in the literature.
Iowa State University（爱荷华州立大学）；Université de La Rochelle（拉罗谢尔大学）；University of Warwick（华威大学）
PCA is one of the most widely used dimension reduction techniques. A related easier problem is "subspace learning" or "subspace estimation". Given relatively clean data, both are easily solved via singular value decomposition (SVD). The problem of subspace learning or PCA in the presence of outliers is called robust subspace learning or robust PCA (RPCA). For long data sequences, if one tries to use a single lower dimensional subspace to represent the data, the required subspace dimension may end up being quite large. For such data, a better model is to assume that it lies in a low-dimensional…
Many real world data sets exhibit an embedding of low-dimensional structure in a high-dimensional manifold. Examples include images, videos and internet traffic data. It is of great significance to reduce the storage requirements and computational complexity when the data dimension is high. Therefore we consider the problem of reconstructing a data stream from a small subset of its entries, where the data is assumed to lie in a low-dimensional linear subspace, possibly corrupted by noise. We further consider tracking the change of the underlying subspace, which can be applied to applications s…
In this work, we study the robust subspace tracking (RST) problem and obtain one of the first two provable guarantees for it. The goal of RST is to track sequentially arriving data vectors that lie in a slowly changing low-dimensional subspace, while being robust to corruption by additive sparse outliers. It can also be interpreted as a dynamic (time-varying) extension of robust PCA (RPCA), with the minor difference that RST also requires a short tracking delay. We develop a recursive projected compressive sensing algorithm that we call Nearly Optimal RST via ReProCS (ReProCS-NORST) because it…
This work studies the robust subspace tracking (ST) problem. Robust ST can be simply understood as a (slow) time-varying subspace extension of robust PCA. It assumes that the true data lies in a low-dimensional subspace that is either fixed or changes slowly with time. The goal is to track the changing subspaces over time in the presence of additive sparse outliers and to do this quickly (with a short delay). We introduce a "fast" mini-batch robust ST solution that is provably correct under mild assumptions. Here "fast" means two things: (i) the subspace changes can be detected and the subspac…
Online Robust Subspace Tracking from Partial Information
Nanjing University of Information Science and Technology（南京信息工程大学）；University of Wisconsin-Madison（威斯康星大学麦迪逊分校）；The Chinese University of Hong Kong（香港中文大学）
Dynamic robust PCA refers to the dynamic (time-varying) extension of robust PCA (RPCA). It assumes that the true (uncorrupted) data lies in a low-dimensional subspace that can change with time, albeit slowly. The goal is to track this changing subspace over time in the presence of sparse outliers. We develop and study a novel algorithm, that we call simple-ReProCS, based on the recently introduced Recursive Projected Compressive Sensing (ReProCS) framework. Our work provides the first guarantee for dynamic RPCA that holds under weakened versions of standard RPCA assumptions, slow subspace chan…
University of Michigan（密歇根大学）；Carnegie Mellon University（卡内基梅隆大学）；Harvard University（哈佛大学）
For many modern applications in science and engineering, data are collected in a streaming fashion carrying time-varying information, and practitioners need to process them with a limited amount of memory and computational resources in a timely manner for decision making. This often is coupled with the missing data problem, such that only a small fraction of data attributes are observed. These complications impose significant, and unconventional, constraints on the problem of streaming Principal Component Analysis (PCA) and subspace tracking, which is an essential building block for many infer…
We study the problem of subspace tracking in the presence of missing data (ST-miss). In recent work, we studied a related problem called robust ST. In this work, we show that a simple modification of our robust ST solution also provably solves ST-miss and robust ST-miss. To our knowledge, our result is the first `complete' guarantee for ST-miss. This means that we can prove that under assumptions on only the algorithm inputs, the output subspace estimates are close to the true data subspaces at all times. Our guarantees hold under mild and easily interpretable assumptions, and allow the underl…
Many applications in data analysis rely on the decomposition of a data matrix into a low-rank and a sparse component. Existing methods that tackle this task use the nuclear norm and L1-cost functions as convex relaxations of the rank constraint and the sparsity measure, respectively, or employ thresholding techniques. We propose a method that allows for reconstructing and tracking a subspace of upper-bounded dimension from incomplete and corrupted observations. It does not require any a priori information about the number of outliers. The core of our algorithm is an intrinsic Conjugate Gradien…
The problem of efficient modulation classification (MC) in multiple-input multiple-output (MIMO) systems is considered. Per-layer likelihood-based MC is proposed by employing subspace decomposition to partially decouple the transmitted streams. When detecting the modulation type of the stream of interest, a dense constellation is assumed on all remaining streams. The proposed classifier outperforms existing MC schemes at a lower complexity cost, and can be efficiently implemented in the context of joint MC and subspace data detection.
We consider channel/subspace tracking systems for temporally correlated millimeter wave (e.g., E-band) multiple-input multiple-output (MIMO) channels. Our focus is given to the tracking algorithm in the non-line-of-sight (NLoS) environment, where the transmitter and the receiver are equipped with hybrid analog/digital precoder and combiner, respectively. In the absence of straightforward time-correlated channel model in the millimeter wave MIMO literature, we present a temporal MIMO channel evolution model for NLoS millimeter wave scenarios. Considering that conventional MIMO channel tracking …
In this work, we address the problem of channel estimation and precoding / combining for the so-called hybrid millimeter wave (mmWave) MIMO architecture. Our proposed channel estimation scheme exploits channel reciprocity in TDD MIMO systems, by using echoing, thereby allowing us to implement Krylov subspace methods in a fully distributed way. The latter results in estimating the right (resp. left) singular subspace of the channel at the transmitter (resp. receiver). Moreover, we also tackle the problem of subspace decomposition whereby the estimated right (resp. left) singular subspaces are a…
This paper proposes the use of subspace tracking algorithms for performing MIMO channel estimation at millimeter wave (mmWave) frequencies. Using a subspace approach, we develop a protocol enabling the estimation of the right (resp. left) singular vectors at the transmitter (resp. receiver) side; then, we adapt the projection approximation subspace tracking with deflation (PASTd) and the orthogonal Oja (OOJA) algorithms to our framework and obtain two channel estimation algorithms. The hybrid analog/digital nature of the beamformer is also explicitly taken into account at the algorithm design …
Channel feedback is essential in frequency division duplexing (FDD) massive multiple-input multiple-output (MIMO) systems. Unfortunately, previous work on multiuser MIMO has shown that the codebook size for channel feedback should scale exponentially with the number of base station (BS) antennas, which is greatly increased in massive MIMO systems. To reduce the codebook size and feedback overhead, we propose an angle-of-departure (AoD)-adaptive subspace codebook for channel feedback in FDD massive MIMO systems. Our key insight is to leverage the observation that path AoDs vary more slowly than…
We propose a subspace constrained precoding scheme that exploits the spatial channel correlation structure in massive MIMO cellular systems to fully unleash the tremendous gain provided by massive antenna array with reduced channel state information (CSI) signaling overhead. The MIMO precoder at each base station (BS) is partitioned into an inner precoder and a Transmit (Tx) subspace control matrix. The inner precoder is adaptive to the local CSI at each BS for spatial multiplexing gain. The Tx subspace control is adaptive to the channel statistics for inter-cell interference mitigation and Qu…
Channel estimation and precoding in hybrid analog-digital millimeter-wave (mmWave) MIMO systems is a fundamental problem that has yet to be addressed, before any of the promised gains can be harnessed. For that matter, we propose a method (based on the well-known Arnoldi iteration) exploiting channel reciprocity in TDD systems and the sparsity of the channel's eigenmodes, to estimate the right (resp. left) singular subspaces of the channel, at the BS (resp. MS). We first describe the algorithm in the context of conventional MIMO systems, and derive bounds on the estimation error in the presenc…
Millimeter-wave (mm-Wave) cellular systems are a promising option for a very high data rate communication because of the large bandwidth available at mm-Wave frequencies. Due to the large path-loss exponent in the mm-Wave range of the spectrum, directional beamforming with a large antenna gain is necessary at the transmitter, the receiver or both for capturing sufficient signal power. This in turn implies that fast and robust channel estimation plays a central role in systems performance since without a reliable estimate of the channel state the received signal-to-noise ratio (SNR) would be mu…
The problem of MIMO channel estimation at millimeter wave frequencies, both in a single-user and in a multi-user setting, is tackled in this paper. Using a subspace approach, we develop a protocol enabling the estimation of the right (resp. left) singular vectors at the transmitter (resp. receiver) side; then, we adapt the projection approximation subspace tracking with deflation and the orthogonal Oja algorithms to our framework and obtain two channel estimation algorithms. We also present an alternative algorithm based on the least squares approach. The hybrid analog/digital nature of the be…
Hybrid beamforming (HB) has been widely studied for reducing the number of costly radio frequency (RF) chains in massive multiple-input multiple-output (MIMO) systems. However, previous works on HB are limited to a single user equipment (UE) or a single group of UEs, employing the frequency-flat first-level analog beamforming (AB) that cannot be applied to multiple groups of UEs served in different frequency resources in an orthogonal frequency-division multiplexing (OFDM) system. In this paper, a novel HB algorithm with unified AB based on the spatial covariance matrix (SCM) knowledge of all …
This paper concerns the maximum-likelihood channel estimation for MIMO systems with orthogonal space-time block codes when the finite alphabet constraint of the signal constellation is relaxed. We study the channel coefficients estimation subspace generated by this method. We provide an algebraic characterisation of this subspace which turns the optimization problem into a purely algebraic one and more importantly, leads to several interesting analytical proofs. We prove that with probability one, the dimension of the estimation subspace for the channel coefficients is deterministic and it dec…
In downlink multiuser multiple-input multiple-output (MU-MIMO) systems, users are practically heterogeneous in nature. However, most of the existing user scheduling algorithms are designed with an implicit assumption that the users are homogeneous. In this paper, we revisit the problem by exploring the characteristics of heterogeneous users from a subspace point of view. With an objective of minimizing interference non-orthogonality among users, three new angular-based user scheduling criteria that can be applied in various user scheduling algorithms are proposed. While the first criterion is …
The accurate estimation of Channel State Information (CSI) is of crucial importance for the successful operation of Multiple-Input Multiple-Output (MIMO) communication systems, especially in a Multi-User (MU) time-varying environment and when employing the emerging technology of Reconfigurable Intelligent Surfaces (RISs). Their predominantly passive nature renders the estimation of the channels involved in the user-RIS-base station link a quite challenging problem. Moreover, the time-varying nature of most of the realistic wireless channels drives up the cost of real-time channel tracking sign…
The number of radio frequency (RF) chains can be reduced through beam selection in lens-based millimeter-wave (mmWave) massive MIMO systems, where the equivalent channel between RF chains and multiple users is required at the BS to achieve the multi-user multiplexing gain. However, to the best of our knowledge, there is no dedicated codebook for the equivalent channel feedback in such systems. In this paper, we propose the dimension-reduced subspace codebook, which achieves a significant reduction of the feedback overhead and codebook size. Specifically, we firstly utilize the limited scatteri…
Terahertz (THz) massive multiple-input multipleoutput (MIMO) has been considered as one of the promising technologies for future 6G wireless communications. It is essential to obtain channel information by beam tracking scheme to track mobile users in THz massive MIMO systems. However, the existing beam tracking schemes designed for narrowband systems with the traditional hybrid precoding structure suffer from a severe performance loss caused by the beam split effect, and thus cannot be directly applied to wideband THz massive MIMO systems. To solve this problem, in this paper we propose a bea…
This paper proposes a Bayesian downlink channel estimation algorithm for time-varying massive MIMO networks. In particular, the quantization effects at the receiver are considered. In order to fully exploit the sparsity and time correlations of channels, we formulate the time-varying massive MIMO channel as the simultaneously sparse signal model. Then, we propose a sparse Bayesian learning (SBL) framework to learn the model parameters of the sparse virtual channel. To reduce complexity, we employ the expectation maximization (EM) algorithm to achieve the approximated solution. Specifically, th…
Blind Null Space Learning (BNSL) has recently been proposed for fast and accurate learning of the null-space associated with the channel matrix between a secondary transmitter and a primary receiver. In this paper we propose a channel tracking enhancement of the algorithm, namely the Blind Null Space Tracking (BNST) algorithm that allows transmission of information to the Secondary Receiver (SR) while simultaneously learning the null-space of the time-varying target channel. Specifically, the enhanced algorithm initially performs a BNSL sweep in order to acquire the null space. Then, it perfor…
Canonical Massive MIMO uses time division duplex (TDD) to exploit channel reciprocity within the coherence time, avoiding feedback of channel state information (CSI), as is required for precoding at the base station. We extend the idea of exploiting reciprocity to the coherence bandwidth, allocating subcarriers of a multicarrier (OFDM)-based system in an interlaced fashion to up- and downlink, respectively, referred to as (OFDM-)subcarrier interlaced FDD (IFDD). Exploiting this "two-dimensional" channel reciprocity within both time and frequency coherence does not require any CSI feedback and,…
Subspace methods are essential to high-resolution environment sensing in the emerging unmanned systems, if further combined with the millimeter-wave (mm-Wave) massive multi-input multi-output (MIMO) technique. The estimation of signal/noise subspace, as one critical step, is yet computationally complex and presents a particular challenge when developing high-resolution yet low-complexity automotive radars. In this work, we develop a fast randomized-MUSIC (R-MUSIC) algorithm, which exploits the random matrix sketching to estimate the signal subspace via approximated computation. Our new approac…
The design of modulation schemes for the physical layer network-coded two-way MIMO relaying scenario is considered, with $n_R$ antennas at the relay R, $n_A$ and $n_B$ antennas respectively at the end nodes A and B. We consider the denoise-and-forward (DNF) protocol which employs two phases: Multiple access (MA) phase and Broadcast (BC) phase. It is known for the network-coded SISO two-way relaying that adaptively changing the networking coding map used at the relay, also known as the denoising map, according to the channel conditions greatly reduces the impact of multiple access interference …
In this paper, we firstly exploit the inter-user interference (IUI) and inter-cell interference (ICI) as useful references to develop a robust transceiver design based on interference alignment for a downlink multi-user multi-cell multiple-input multiple-output (MIMO) interference network under channel estimation error. At transmitters, we propose a two-tier transmit beamforming strategy, we first achieve the inner beamforming direction and allocated power by minimizing the interference leakage as well as maximizing the system energy efficiency, respectively. Then, for the outer beamformer des…
In this paper, we resort to the graph neural network (GNN) and propose the new channel tracking method for the massive multiple-input multiple-output networks under the high mobility scenario. We first utilize a small number of pilots to achieve the initial channel estimation. Then, we represent the obtained channel data in the form of graphs and describe the channel spatial correlation by the weights along the edges of the graph. Furthermore, we introduce the computation steps of the main unit for the GNN and design a GNN-based channel tracking framework, which includes an encoder, a core net…
Millimeter wave signals and large antenna arrays are considered enabling technologies for future 5G networks. Despite their benefits for achieving high data rate communications, their potential advantages for tracking of the location of the user terminals are largely undiscovered. In this paper, we propose a novel support detection-based channel training method for frequency selective millimeter-wave (mm-wave) multiple-input-multiple-output system with lens antenna arrays. We show that accurate position and orientation estimation and tracking is possible using signals from a single transmitter…
In this paper, we present an Integrated Sensing and Communications (ISAC) system enabled by in-band Full Duplex (FD) radios, where a massive Multiple-Input Multiple-Output (MIMO) base station equipped with hybrid Analog and Digital (A/D) beamformers is communicating with multiple DownLink (DL) users, and simultaneously estimates via the same signaling waveforms the Direction of Arrival (DoA) as well as the range of radar targets randomly distributed within its coverage area. Capitalizing on a recent reduced-complexity FD hybrid A/D beamforming architecture, we devise a joint radar target track…
A subspace method for channel estimation has been recently proposed [1] for tackling the pilot contamination effect, which is regarded by some researchers as a bottleneck in massive MIMO systems. It was shown in [1] that if the power ratio between the desired signal and interference is kept above a certain value, the received signal spectrum splits into signal and interference eigenvalues, namely, the "pilot contamination" effect can be completely eliminated. However, [1] assumes an independently distributed (i.d.) channel, which is actually not much the case in practice. Considering this, a m…
We consider interference alignment (IA) over K-user Gaussian MIMO interference channel (MIMO-IC) when the SNR is not asymptotically high. We introduce a generalization of IA which enables receive diversity inside the interference-free subspace. We generalize the existence criterion of an IA solution proposed by Yetis et al. to this case, thereby establishing a multi-user diversity-multiplexing trade-off (DMT) for the interference channel. Furthermore, we derive a closed-form tight lower-bound for the ergodic mutual information achievable using IA over a Gaussian MIMO-IC with Gaussian i.i.d. ch…
This paper proposes a novel maximum Doppler spread estimation algorithm for OFDM systems with the comb-type pilot pattern. By tracking the drifting delay subspace of the multipath channel, the time correlation function is measured at a high accuracy, which accordingly improves the estimation accuracy of the maximum Doppler spread considerably.
A novel stochastic technique is presented to directly model singular vectors and singular values of a multiple input multiple output channel. Thus the component smodeled directly in the eigen domain can be adapted to exhibit realistic physical domain behavior when assembled. The model exploites natural paths of eigenmodes, such that a simple Doppler filter generator process can be used. Furthermore it is possible to directly manipulate the singular vector dynamics in a way that an unrealistic "stress channel" can be modeled in the eigen domain. This is particularly useful for testing the eigen…
We consider the problem of efficient ultra-massive multiple-input multiple-output (UM-MIMO) data detection in terahertz (THz)-band non-orthogonal multiple access (NOMA) systems. We argue that the most common THz NOMA configuration is power-domain superposition coding over quasi-optical doubly-massive MIMO channels. We propose spatial tuning techniques that modify antenna subarray arrangements to enhance channel conditions. Towards recovering the superposed data at the receiver side, we propose a family of data detectors based on low-complexity channel matrix puncturing, in which higher-order d…
This paper focuses on multiuser MIMO channel estimation and data transmission at millimeter wave (mmWave) frequencies. The proposed approach relies on the time-division-duplex (TDD) protocol and is based on two distinct phases. First of all, the Base Station (BS) sends a suitable probing signal so that all the Mobile Stations (MSs), using a subspace tracking algorithm, can estimate the dominant left singular vectors of their BS-to-MS propagation channel. Then, each MS, using the estimated dominant left singular vectors as pre-coding beamformers, sends a suitable pilot sequence so that the BS c…
We study a multi-antenna broadcast channel with two legitimate receivers and an external eavesdropper. We assume that the channel matrix of the eavesdropper is unknown to the legitimate terminals but satisfies a maximum rank constraint. As our main result we characterize the associated secrecy degrees of freedom for the broadcast channel with common and private messages. We show that a direct extension of the single-user wiretap codebook does not achieve the secrecy degrees of freedom. Our proposed optimal scheme involves decomposing the signal space into a common subspace, which can be observ…
A novel maximum Doppler spread estimation algorithm for OFDM systems with comb-type pilot pattern is presented in this paper. By tracking the drifting delay subspace of time-varying multipath channels, a Doppler dependent parameter can be accurately measured and further expanded and transformed into a non-linear high-order polynomial equation, from which the maximum Doppler spread is readily solved by resorting to the Newton's method. Its performance is demonstrated by simulations.
In this paper, the required amount of feedback overhead for multiple-input multiple-output (MIMO) beamforming over time-varying channels is presented in terms of the entropy of the feedback messages. In the case that each transmit antenna has its own power amplifier which has individual power limit, it has been known that only phase steering information is necessary to form the optimal transmit beamforming vector. Since temporal correlation exists for wireless fading channels, one can utilize the previous reported feedback messages as prior information to efficiently encode the current feedbac…
In this paper, a joint design of instantaneous channel estimation, beam tracking, and adaptive beamformer construction for a massive multiple-input multiple-output (MIMO) system is proposed. This design focuses on efficiency in terms of performance and computational complexity under the adverse effects of time variation and mobility of sources, the presence of multiuser and multipath components, or simply multi-clusters, and the near-far effect. The design is also suitable for hybrid beamforming and frequency-selective channels. In the proposed system, channel parameters are estimated in time-…
This work takes the first steps towards solving the "phaseless subspace tracking" (PST) problem. PST involves recovering a time sequence of signals (or images) from phaseless linear projections of each signal under the following structural assumption: the signal sequence is generated from a much lower dimensional subspace (than the signal dimension) and this subspace can change over time, albeit gradually. It can be simply understood as a dynamic (time-varying subspace) extension of the low-rank phase retrieval problem studied in recent work.
In this paper, we establish a general framework on the reduced dimensional channel state information (CSI) estimation and pre-beamformer design for frequency-selective massive multiple-input multiple-output MIMO systems employing single-carrier (SC) modulation in time division duplex (TDD) mode by exploiting the joint angle-delay domain channel sparsity in millimeter (mm) wave frequencies. First, based on a generic subspace projection taking the joint angle-delay power profile and user-grouping into account, the reduced rank minimum mean square error (RR-MMSE) instantaneous CSI estimator is de…
We study the degrees of freedom (DoF) of MIMO two-way X relay channels. Previous work studied the case $N < 2M$, where $N$ and $M$ denote the number of antennas at the relay and each source, respectively, and showed that the maximum DoF of $2N$ is achievable when $N \leq \lfloor\frac{8M}{5}\rfloor$ by applying signal alignment (SA) for network coding and interference cancelation. This work considers the case $N>2M$ where the performance is limited by the number of antennas at each source node and conventional SA is not feasible. We propose a \textit{generalized signal alignment} (GSA) based tr…
