arXiv subjects

Saeid Haghighatshoar

Publications and source records attributed to Saeid Haghighatshoar.

At least 37 records · Page 2Linked to original sources

arXiv2018

Hierarchical Sparse Channel Estimation for Massive MIMO

The problem of wideband massive MIMO channel estimation is considered. Targeting for low complexity algorithms as well as small training overhead, a compressive sensing (CS) approach is pursued. Unfortunately, due to the Kronecker-type sensing (measurement) matrix corresponding to this setup, application of standard CS algorithms and analysis methodology does not apply. By recognizing that the channel possesses a special structure, termed hierarchical sparsity, we propose an efficient algorithm that explicitly takes into account this property. In addition, by extending the standard CS analysis methodology to hierarchical sparse vectors, we provide a rigorous analysis of the algorithm performance in terms of estimation error as well as number of pilot subcarriers required to achieve it. Small training overhead, in turn, means higher number of supported users in a cell and potentially improved pilot decontamination. We believe, that this is the first paper that draws a rigorous connection between the hierarchical framework and Kronecker measurements. Numerical results verify the advantage of employing the proposed approach in this setting instead of standard CS algorithms.

cs.IT↗

arXiv2018

Multi-Band Covariance Interpolation with Applications in Massive MIMO

In this paper, we study the problem of multi-band (frequency-variant) covariance interpolation with a particular emphasis towards massive MIMO applications. In a massive MIMO system, the communication between each BS with $M \gg 1$ 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. This results in a frequency-dependent distortion where the second order statistics, i.e., the covariance matrix, of the channel vectors varies with frequency. In this paper, we show that although this effect is generally negligible for a small number of antennas $M$, it results in a considerable distortion of the covariance matrix and especially its dominant signal subspace in the massive MIMO regime where $M \to \infty$, and can generally incur a serious degradation of the performance especially in frequency division duplexing (FDD) massive MIMO systems where the uplink (UL) and the downlink (DL) communication occur over different frequency bands. We propose a novel UL-DL covariance interpolation technique that is able to recover the covariance matrix in the DL from an estimate of the covariance matrix in the UL under a mild reciprocity condition on the angular power spread function (PSF) of the users. We analyze the performance of our proposed scheme mathematically and prove its robustness under a sufficiently large spatial oversampling of the array. We also propose several simple off-the-shelf algorithms for UL-DL covariance interpolation and evaluate their performance via numerical simulations.

cs.IT↗

arXiv2017

A Robust Time-Domain Beam Alignment Scheme for Multi-User Wideband mmWave Systems

Millimeter wave (mmWave) communication with large array gains is a key ingredient of next generation (5G) wireless networks. Effective communication in mmWaves usually depends on the knowledge of the channel. We refer to the problem of finding a narrow beam pair at the transmitter and at the receiver, yielding high Signal to Noise Ratio (SNR) as Beam Alignment (BA). Prior BA schemes typically considered deterministic channels, where the instantaneous channel coefficients are assumed to stay constant for a long time. In this paper, in contrast, we propose a time-domain BA scheme for wideband mmWave systems, where the channel is characterized by multi-path components, different delays, Angle-of-Arrivals/Angle-of-Departures (AoAs/AoDs), and Doppler shifts. In our proposed scheme, the Base Station (BS) probes the channel in the downlink by some sequences with good autocorrelation property (e.g., Pseudo-Noise (PN) sequences), letting each user estimate its best AoA-AoD that connects the user to the BS with two-sided high beamforming gain. We leverage the sparse nature of mmWaves in the AoA-AoD-time domain, and formulate the BA problem as a Compressed Sensing (CS) of a non-negative sparse vector. We use the recently developed Non-Negative Least Squares (NNLS) technique to efficiently find the strongest path connecting the BS and each user. Simulation results show that the proposed scheme outperforms its counterpart in terms of the training overhead and robustness to fast channel variations.

cs.IT↗

arXiv2017

Estimation of Angles of Arrival Through Superresolution -- A Soft Recovery Approach for General Antenna Geometries

The estimation of direction of arrivals with help of $TV$-minimization is studied. Contrary to prior work in this direction, which has only considered certain antenna placement designs, we consider general antenna geometries. Applying the soft-recovery framework, we are able to derive a theoretic guarantee for a certain direction of arrival to be approximately recovered. We discuss the impact of the recovery guarantee for a few concrete antenna designs. Additionally, numerical simulations supporting the findings of the theoretical part are performed.

eess.SP↗

arXiv2017

FDD Massive MIMO: Efficient Downlink Probing and Uplink Feedback via Active Channel Sparsification

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 UL-DL 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*log M) in CS techniques to O(s) in our proposed method, with $s$ and $M$ 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 compare our method with the state-of-the-art CS technique.

cs.IT↗

arXiv2017

A Scalable and Statistically Robust Beam Alignment Technique for mm-Wave Systems

Millimeter-Wave (mm-Wave) frequency bands provide an opportunity for much wider channel bandwidth compared with the traditional sub-6 GHz band. Communication at mm-Waves is, however, quite challenging due to the severe propagation path loss. To cope with this problem, directional beamforming both at the Base Station (BS) side and at the user side is necessary in order to establish a strong path conveying enough signal power. Finding such beamforming directions is referred to as the Beam Alignment (BA) and is known to be a challenging problem. This paper presents a new scheme for efficient BA, based on the estimated second order channel statistics. As a result, our proposed algorithm is highly robust to variations of the channel time-dynamics compared with other proposed approaches based on the estimation of the channel coefficients, rather than of their second-order statistics. In the proposed scheme, the BS probes the channel in the Downlink (DL) letting each user to estimate its own path direction. All the users within the BS coverage are trained simultaneously, without requiring "beam refinement" with multiple interactive rounds of Downlink/Uplink (DL/UL) transmissions, as done in other schemes. Thus, the training overhead of the proposed BA scheme is independent of the number of users in the system. We pose the channel estimation at the user side as a Compressed Sensing (CS) of a non-negative signal and use the recently developed Non-Negative Least Squares(NNLS) technique to solve it efficiently. The performance of our proposed algorithm is assessed via computer simulation in a relevant mm-Wave scenario. The results illustrate that our approach is superior to the state-of-the-art BA schemes proposed in the literature in terms of training overhead in multi-user scenarios and robustness to variations in the channel dynamics.

cs.IT↗

arXiv2017

Signal Recovery from Unlabeled Samples

In this paper, we study the recovery of a signal from a set of noisy linear projections (measurements), when such projections are unlabeled, that is, the correspondence between the measurements and the set of projection vectors (i.e., the rows of the measurement matrix) is not known a priori. We consider a special case of unlabeled sensing referred to as Unlabeled Ordered Sampling (UOS) where the ordering of the measurements is preserved. We identify a natural duality between this problem and classical Compressed Sensing (CS), where we show that the unknown support (location of nonzero elements) of a sparse signal in CS corresponds to the unknown indices of the measurements in UOS. While in CS it is possible to recover a sparse signal from an under-determined set of linear equations (less equations than the signal dimension), successful recovery in UOS requires taking more samples than the dimension of the signal. Motivated by this duality, we develop a Restricted Isometry Property (RIP) similar to that in CS. We also design a low-complexity Alternating Minimization algorithm that achieves a stable signal recovery under the established RIP. We analyze our proposed algorithm for different signal dimensions and number of measurements theoretically and investigate its performance empirically via simulations. The results are reminiscent of phase-transition similar to that occurring in CS.

cs.IT↗

arXiv2017

Polarization of the Renyi Information Dimension with Applications to Compressed Sensing

In this paper, we show that the Hadamard matrix acts as an extractor over the reals of the Renyi information dimension (RID), in an analogous way to how it acts as an extractor of the discrete entropy over finite fields. More precisely, we prove that the RID of an i.i.d. sequence of mixture random variables polarizes to the extremal values of 0 and 1 (corresponding to discrete and continuous distributions) when transformed by a Hadamard matrix. Further, we prove that the polarization pattern of the RID admits a closed form expression and follows exactly the Binary Erasure Channel (BEC) polarization pattern in the discrete setting. We also extend the results from the single- to the multi-terminal setting, obtaining a Slepian-Wolf counterpart of the RID polarization. We discuss applications of the RID polarization to Compressed Sensing of i.i.d. sources. In particular, we use the RID polarization to construct a family of deterministic $\pm 1$-valued sensing matrices for Compressed Sensing. We run numerical simulations to compare the performance of the resulting matrices with that of random Gaussian and random Hadamard matrices. The results indicate that the proposed matrices afford competitive performances while being explicitly constructed.

cs.IT↗

arXiv2017

Efficient Downlink Channel Probing and Uplink Feedback in FDD Massive MIMO Systems

Massive Multiple-Input Multiple-Output (massive MIMO) is a variant of multi-user MIMO in which the number of antennas at each Base Station (BS) is very large and typically much larger than the number of users simultaneously served. Massive MIMO can be implemented with Time Division Duplexing (TDD) or Frequency Division Duplexing (FDD) operation. FDD massive MIMO systems are particularly desirable due to their implementation in current wireless networks and their efficiency in situations with symmetric traffic and delay-sensitive applications. However, implementing FDD massive MIMO systems is known to be challenging since it imposes a large feedback overhead in the Uplink (UL) to obtain channel state information for the Downlink (DL). In recent years, a considerable amount of research is dedicated to developing methods to reduce the feedback overhead in such systems. In this paper, we use the sparse spatial scattering properties of the environment to achieve this goal. The idea is to estimate the support of the continuous, frequency-invariant scattering function from UL channel observations and use this estimate to obtain the support of the DL channel vector via appropriate interpolation. We use the resulting support estimate to design an efficient DL probing and UL feedback scheme in which the feedback dimension scales proportionally with the sparsity order of DL channel vectors. Since the sparsity order is much less than the number of BS antennas in almost all practically relevant scenarios, our method incurs much less feedback overhead compared with the currently proposed methods in the literature, such as those based on compressed-sensing. We use numerical simulations to assess the performance of our probing-feedback algorithm and compare it with these methods.

cs.IT↗

arXiv2017

Low-Complexity Massive MIMO Subspace Estimation and Tracking from Low-Dimensional Projections

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.

cs.IT↗

arXiv2017

Massive MIMO Pilot Decontamination and Channel Interpolation via Wideband Sparse Channel Estimation

We consider a massive MIMO system based on Time Division Duplexing (TDD) and channel reciprocity, where the base stations (BSs) learn the channel vectors of their users via the pilots transmitted by the users in the uplink (UL). It is well-known that, in the limit of very large number of BS antennas, the system performance is limited by pilot contamination, due to the fact that the same set of orthogonal pilots is reused in multiple cells. In this paper, we propose a low-complexity algorithm that uses the received UL wideband pilot snapshots in an observation window comprising several coherence blocks (CBs) to obtain an estimate of the angle-delay Power Spread Function (PSF) of the received signal. This is generally given by the sum of the angle-delay PSF of the desired user and the angle-delay PSFs of the copilot users (CPUs), i.e., the users re-using the same pilot dimensions in other cells/sectors. We propose supervised and unsupervised clustering algorithms to decompose the estimated PSF and isolate the part corresponding to the desired user only. We use this decomposition to obtain an estimate of the covariance matrix of the user wideband channel vector, which we exploit to decontaminate the desired user channel estimate by applying Minimum Mean Squared Error (MMSE) smoothing filter, i.e., the optimal channel interpolator in the MMSE sense. We also propose an effective low-complexity approximation/implementation of this smoothing filter. We use numerical simulations to assess the performance of our proposed method, and compare it with other recently proposed schemes that use the same idea of separability of users in the angle-delay domain.

cs.IT↗

arXiv2017

Compressive Estimation of a Stochastic Process with Unknown Autocorrelation Function

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.

cs.IT↗

arXiv2016

Channel Vector Subspace Estimation from Low-Dimensional Projections

Massive MIMO is a variant of multiuser MIMO where the number of base-station antennas $M$ is very large (typically 100), and generally much larger than the number of spatially multiplexed data streams (typically 10). Unfortunately, the front-end A/D conversion necessary to drive hundreds of antennas, with a signal bandwidth of the order of 10 to 100 MHz, requires very large sampling bit-rate and power consumption. In order to reduce such implementation requirements, Hybrid Digital-Analog architectures have been proposed. In particular, our work in this paper is motivated by one of such schemes named Joint Spatial Division and Multiplexing (JSDM), where the downlink precoder (resp., uplink linear receiver) is split into the product of a baseband linear projection (digital) and an RF reconfigurable beamforming network (analog), such that only a reduced number $m \ll M$ of A/D converters and RF modulation/demodulation chains is needed. In JSDM, users are grouped according to the similarity of their channel dominant subspaces, and these groups are separated by the analog beamforming stage, where the multiplexing gain in each group is achieved using the digital precoder. Therefore, it is apparent that extracting the channel subspace information of the $M$-dim channel vectors from snapshots of $m$-dim projections, with $m \ll M$, plays a fundamental role in JSDM implementation. In this paper, we develop novel efficient algorithms that require sampling only $m = O(2\sqrt{M})$ specific array elements according to a coprime sampling scheme, and for a given $p \ll M$, return a $p$-dim beamformer that has a performance comparable with the best p-dim beamformer that can be designed from the full knowledge of the exact channel covariance matrix. We assess the performance of our proposed estimators both analytically and empirically via numerical simulations.

cs.IT↗

arXiv2016

Capacity and Degree-of-Freedom of OFDM Channels with Amplitude Constraint

In this paper, we study the capacity and degree-of-freedom (DoF) scaling for the continuous-time amplitude limited AWGN channels in radio frequency (RF) and intensity modulated optical communication (OC) channels. More precisely, we study how the capacity varies in terms of the OFDM block transmission time $T$, bandwidth $W$, amplitude $A$, and the noise spectral density $N_0$. We first find suitable discrete encoding spaces for both cases, and prove that they are convex sets that have a semi-definite programming (SDP) representation. Using tools from convex geometry, we find lower and upper bounds on the volume of these encoding sets, which we exploit to drive pretty sharp lower and upper bounds on the capacity. We also study a practical Tone-Reservation (TR) encoding algorithm and prove that its performance can be characterized by the statistical width of an appropriate convex set. Recently, it has been observed that in high-dimensional estimation problems under constraints such as those arisen in Compressed Sensing (CS) statistical width plays a crucial role. We discuss some of the implications of the resulting statistical width on the performance of the TR. We also provide numerical simulations to validate these observations.

cs.IT↗

arXiv2016

Enhancing the Estimation of mm-Wave Large Array Channels by Exploiting Spatio-Temporal Correlation and Sparse Scattering

In order to cope with the large path-loss exponent of mm-Wave channels, a high beamforming gain is needed. This can be achieved with small hardware complexity and high hardware power efficiency by Hybrid Digital-Analog (HDA) beamforming, where a very large number $M\gg 1$ of antenna array elements requires only a relatively small $m\ll M$ number of A/D converters and modulators/demodulators. As such, the estimation of mm-Wave MIMO channels must deal with two specific problems: 1) high Doppler, due to the large carrier frequency; 2) impossibility of observing directly the M-dimensional channel vector at the antenna array elements, due to the mentioned HDA implementation. In this paper, we consider a novel scheme inspired by recent results on gridless multiple measurement vectors problem in compressed sensing, that is able to exploit the inherent mm-Wave channel sparsity in the angular domain in order to cope with both the above problems simultaneously. Our scheme uses past pilot-symbol observations in a window of length $T$ in order to estimate a low-dimensional subspace that approximately contains the channel vector at the current time. This subspace information can be used directly, in order to separate users in the spatial domain, or indirectly, in order to improve the estimate of the user channel vector from the current pilot-symbol observation.

cs.IT↗

arXiv2015

Unlabeled Sensing with Random Linear Measurements

We study the problem of solving a linear sensing system when the observations are unlabeled. Specifically we seek a solution to a linear system of equations y = Ax when the order of the observations in the vector y is unknown. Focusing on the setting in which A is a random matrix with i.i.d. entries, we show that if the sensing matrix A admits an oversampling ratio of 2 or higher, then with probability 1 it is possible to recover x exactly without the knowledge of the order of the observations in y. Furthermore, if x is of dimension K, then any 2K entries of y are sufficient to recover x. This result implies the existence of deterministic unlabeled sensing matrices with an oversampling factor of 2 that admit perfect reconstruction. The result is universal in that recovery is guaranteed for all possible choices of x. While the proof is constructive, it uses a combinatorial algorithm which is not practical, leaving the question of complexity open. We also analyze a noisy version of the problem and show that local stability is guaranteed by the solution. In particular, for every x, the recovery error tends to zero as the signal-to-noise-ratio tends to infinity. The question of universal stability is unclear. We also obtain a converse of the result in the noiseless case: If the number of observations in y is less than 2K, then with probability 1, universal recovery fails, i.e., with probability 1, there exists distinct choices of x which lead to the same unordered list of observations in y. In terms of applications, the unlabeled sensing problem is related to data association problems encountered in different domains including robotics where it is appears in a method called "simultaneous localization and mapping" (SLAM), multi-target tracking applications, and in sampling signals in the presence of jitter.

cs.IT↗

arXiv2015

An Active-Sensing Approach to Channel Vector Subspace Estimation in mm-Wave Massive MIMO Systems

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 much lower than the minimum necessary for a reliable communication. In this paper, we mainly focus on single-antenna users and a multi-antenna base-station. We propose an adaptive sampling scheme to speed up the user's signal subspace estimation. In our scheme, the beamforming vector for taking every new sample is adaptively selected based on all the previous beamforming vectors and the resulting output observations. We apply the theory of optimal design of experiments in statistics to design an adaptive algorithm for estimating the signal subspace of each user. The resulting subspace estimates for different users can be exploited to efficiently communicate to the users and to manage the interference. We cast our proposed algorithm as low-complexity optimization problems, and illustrate its efficiency via numerical simulations.

cs.IT↗

arXiv2015

A New Identity for the Least-square Solution of Overdetermined Set of Linear Equations

In this paper, we prove a new identity for the least-square solution of an over-determined set of linear equation $Ax=b$, where $A$ is an $m\times n$ full-rank matrix, $b$ is a column-vector of dimension $m$, and $m$ (the number of equations) is larger than or equal to $n$ (the dimension of the unknown vector $x$). Generally, the equations are inconsistent and there is no feasible solution for $x$ unless $b$ belongs to the column-span of $A$. In the least-square approach, a candidate solution is found as the unique $x$ that minimizes the error function $\|Ax-b\|_2$. We propose a more general approach that consist in considering all the consistent subset of the equations, finding their solutions, and taking a weighted average of them to build a candidate solution. In particular, we show that by weighting the solutions with the squared determinant of their coefficient matrix, the resulting candidate solution coincides with the least square solution.

math.OC↗