An Efficient Beam and Channel Acquisition via Sparsity Map and Joint Angle-Delay Power Profile Estimation for Wideband Massive MIMO Systems

10/13/2019
by   Ali O. Kalayci, et al.
0

In this paper, an efficient beam and channel acquisition scheme together with joint angle-delay power profile (JADPP) construction are proposed for single-carrier mm-wave wideband sparse massive multiple-input multiple-output (MIMO) channels when hybrid beamforming architecture is utilized. We consider two different modes of operation, namely slow-time beam acquisition and fast-time instantaneous channel estimation, for training stage of time division duplex based systems. In the first mode, where pre-structured hybrid beams are formed to scan intended angular sectors, the joint angle-delay sparsity map together with power intensities of each user channels are obtained by using a novel constant false alarm rate thresholding algorithm inspired from adaptive radar detection theory. The proposed thresholding algorithm employs a spatio-temporal adaptive matched filter type estimator, taking the strong interference due to simultaneously active multipath components of different user channels into account, in order to estimate JADPP of each user. After applying the proposed thresholding algorithm on the estimated power profile, the angle-delay sparsity map of the massive MIMO channel is constructed, based on which the channel covariance matrices (CCMs) are formed with significantly reduced amount of training snapshots. Then, by using the estimated CCMs, the analog beamformer is reconstructed by means of a virtual sectorization while taking the inter-group and inter-symbol interference into account. Finally, for the second mode of operation, two novel reduced-rank instantaneous channel estimators, operating in a proper beamspace formed by the hybrid structure, are proposed. The proposed beam and channel acquisition techniques attain the channel estimation accuracy of minimum mean square error filter with true knowledge of CCMs.

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I Introduction

Multi-user massive multiple-input multiple-output (MIMO) cellular system is expected to be one of the key technologies for G beyond [andrews14], since it provides large gains in spectral and energy efficiency, high spatial resolution, and allows simple transceiver design [marzetta10], [larsson14], [rusek13]. To capitalize aforementioned gains, instantaneous channel state information (CSI) is requisite for multi-user precoding at downlink or multi-user decoding at uplink in a massive MIMO system for both time division duplex (TDD) and frequency division duplex (FDD) modes [guvensen16_2], [noh14]. However, the CSI acquisition is recognized as a very challenging task for massive MIMO systems due to the high dimensionality of channel matrices, pilot contamination, training overhead, computational complexity and so on [marzetta10]. In FDD mode, CSI is typically obtained through explicit downlink training and uplink limited feedback. However, as the number of antenna elements at base station (BS) increases, the traditional downlink channel estimation strategy for FDD systems becomes infeasible [swindlehurst14]. Contrary to FDD mode, the acquisition of CSI via uplink training in TDD mode is more practical [ashikhmin11].

I-a Related Work

There are various channel estimation algorithms proposed in the literature. Among them, channel covariance matrix (CCM) based methods offer additional statistical knowledge about the channel parameters, and thus achieve much better estimation accuracy when compared to the compressive sensing based approaches, which try to recover CSI from fewer sub-Nyquist sampling points [rao14], [gao15], or the angle-space methods which exploit spatial basis expansion models [Xie16], [Xie17]. To embrace their benefits, however, CCMs need to be acquired first. The acquisition of CCM in full dimension still constitutes a bottleneck for the performance of massive MIMO systems even for TDD mode due to the computational burden of processing large dimensional signals and significant training overhead. There are different approaches to estimate CCMs such as using temporal averaging of received signal snapshots in full dimension [Brennan74], [Li03], employing compressive sensing algorithms [romero16], [park18], and applying approximate maximum likelihood (AML) technique formulated as a semi-definite program in low dimensional subspace [Haghighatshoar18]. In [Xie18], CCMs are constructed by exploiting power angular spectrum and angle parameters of channels. However, the channel estimation accuracy is still far from that of the minimum mean square error (MMSE) estimator with true CCM knowledge. Furthermore, the work in [Xie18] takes only the angular sparsity of the wideband channel into account while overlooking the joint angle-delay sparsity of the mm-wave channels as did by many researchers in the literature so far. In contrast, next generation wireless systems will inevitably be broadband in mm-wave [ghosh14], [swindlehurst14_2] due to the much higher throughput requirements, thus leading the channel to be sparse both in angle and delay domain.

Most of the existing researches for massive MIMO systems adopt flat-fading channel model by considering the use of orthogonal frequency division multiplexing (OFDM) [swindlehurst14]. However, due to the drawbacks of OFDM transmission (e.g., high peak-to-average-power ratio (PAPR)), the use of single-carrier (SC) in massive MIMO systems employing mm-wave bands, exhibiting sparsity both in angle and delay plane, was considered in [Larsson2012], [caire19], [Guvensen16], [kurt19]. In these studies, the mitigation of inter-symbol interference (ISI) via reduced complexity beamspace processing (rather than temporal processing) motivates the use of SC in spatially correlated wideband massive MIMO systems [caire19], [Guvensen16], [kurt19].

Besides CSI estimation problem, one can also exploit CCMs in order to reduce the effective channel dimension so that the computational burden in massive MIMO systems is highly relieved. To facilitate this approach, two-stage beamforming concept under the name of Joint Spatial Division and Multiplexing (JSDM) has been proposed [adhikary13], [nam14]

, where users with approximately same channel covariance eigenspaces are partitioned into multiple groups. Then, a statistical analog beamformer can be constructed only from the long-term parameters, basically CCMs, (instead of using instantaneous CSI) in order to distinguish intra-group signals while suppressing inter-group interference. Furthermore, the training overhead, necessary to learn the effective channel of each user, is reduced considerably

[kurt19]. Here, the JSDM framework motivates the use of hybrid beamforming architectures [Molisch17], [Li18] instead of using fully digital precoding/decoding in mm-wave, where efficient reconfigurable radio frequency (RF) architectures can be implemented at competitive cost, size, and energy.

I-B Contributions

In the light of above discussion, we are seeking for an effective CCM construction technique, yielding high instantaneous CSI accuracy, by exploiting only slowly varying parameters such as angles of arrivals (AoAs), temporal delays, and average power of the multipath components (MPCs). In this paper, efficient algorithms are proposed to estimate the joint angle-delay sparsity map and power profile of SC wideband massive MIMO channel to construct CCMs, based on which an adaptive beam and instantaneous channel acquisition is carried out for JSDM architecture in mm-wave bands. Throughout the paper, we conceive two modes of operation for beam and channel acquisition, namely slow-time beam acquisition and fast-time instantaneous channel estimation, in training stage of TDD based massive MIMO system utilizing hybrid beamforming structure. In the first mode, initially, pre-structured hybrid search beams are utilized to scan the intended angular sector of interest. Then, an adaptive spatio-temporal matched filter (AMF), taking the simultaneously active interfering users into account, is designed to estimate power intensities of active MPCs for each user channel, which is a kind of scatter map on joint angle-delay plane. Following the joint angle-delay power profile (JADPP) estimator, a novel constant false alarm rate (CFAR) algorithm, inspired from adaptive radar detection theory, is applied onto the estimated JADPPs in order to extract joint angle-delay sparsity map, showing the spatio-temporal locations (cells) where the power of each MPC is concentrated. After obtaining the sparsity map and power profiles, the parametric construction of CCMs for each active MPC is realized in full dimension. Here, different from the existing literature [romero16], [park18], heavily based on compressive sensing tools, the proposed technique requires much reduced amount of training data and complexity while allowing simultaneous transmission of all active users in SC mode. This brings significant reduction in the acquisition period for slowly varying channel parameters (i.e., CCMs, JADPPs, sparsity map) by resolving the channel both in angle and delay domain. Finally, in second mode of operation, the instantaneous CSI is acquired in reduced dimensional beamspace formed by the proposed hybrid architecture exploiting the estimated power profiles.

The novelty in this paper lies behind the integration of efficient adaptive detection/estimation algorithms in radar literature with massive MIMO hybrid beamforming in order to extract the scatter map of frequency-selective multi-user channel where the MPCs are resolvable both in angular and temporal domain. The contribution is twofold. First, to the author’s knowledge, there is no such prior work, that obtains the sparsity map both in angle and delay domain via CFAR thresholding for wideband SC transmission. Here, with the help of proposed methodology, which is completely different than the aforementioned studies in the literature, the sparsity map and CCMs are acquired with lowered dimensional observation after hybrid beamforming. Second, based on the estimated CCMs, a novel statistical analog beamformer, suppressing both inter-group interference and ISI, and taking the doubly sparse structure of wideband channel into account, is designed for JSDM framework by inspiring from the work in [Guvensen16] (where a nearly optimal Capon like beamformer for general rank signal model was constructed). While designing the statistical pre-beamformer, the distribution of RF chains among different MPCs is optimized for SC transmission by considering the amount of interference they are subject to. After reducing the dimension via the proposed statistical beamformer, efficient beamspace aware instantaneous CSI estimators are provided. It is shown that the proposed CCM construction and reduced rank channel acquisition techniques necessitate considerably lowered slow and fast time training overhead. Furthermore, the performance benchmark for the estimation of doubly sparse SC wideband massive MIMO channel in mean square error (MSE) sense is achieved via the proposed algorithms which can be regarded as promising beam and channel acquisition techniques in this regard for next generation wireless networks.

Notations:Vectors and matrices are denoted by boldface small and capital letters; the transpose, Hermitian and inverse of the matrix are denoted by , and ; is the entry of ; the entry index of the vector and the matrix starts from 0; is the trace of ; is the determinant of ;

is the identity matrix with appropriate size;

is the statistical expectation; denotes the cardinality of the set ; denotes the Euclidean norm of , and is the Kronecker-Delta function which is equal to if otherwise .

Ii System Model

Fig. 1: Angle-delay power profile of active users in massive MIMO system (left) and beam acquisition scheme (right)

We consider a multi-user massive MIMO system operating at mm-wave bands in TDD mode. The BS is equipped with antennas and serves single-antenna users. At the beginning of every coherence interval, all users transmit training sequences with length . We assume a linear modulation (e.g., PSK or QAM) and a transmission over frequency-selective channel for all user equipments (UEs) with a slow evolution in time relative to the signaling interval (symbol duration). Under such conditions, the baseband equivalent received signal samples, taken at symbol rate () after pulse matched filtering, are expressed as

(1)

for , where is multipath channel vector, namely, the array impulse response of the serving BS stemming from the MPC of user. Here, are the training symbols for the user, is the channel memory of user multipath channels. The symbols at the start of the preamble, prior to the first observation at , are the precursors. Training symbols are selected from a signal constellation and is set to for all . In (1), are the additive complex white Gaussian noise (AWGN) vectors during uplink pilot segment with spatially and temporarily independent and identically distributed (i.i.d.) as , and is the noise power.

Ii-a Statistical Models for MPCs

We assume Rayleigh-correlated MPCs where each user has channel . Then, their corresponding cross-covariance matrices can be expressed in the form of

(2)

by using the uncorrelated local scattering model where all MPCs are assumed to be mutually independent according to the well-known wide sense stationary uncorrelated scattering (WSSUS) model [adhikary14], [swindlehurst16]; the multipath channel vectors are uncorrelated with respect to

, and also mutually uncorrelated with that of the different users. The average received signal-to-noise ratio (SNR) for the

MPC of user can be defined as where 111 It shows the average SNR after maximal ratio combining (MRC) when the beam is steered toward the angular location of MPC of user. Then, can be seen as the average received SNR at each antenna element before beamforming.. Then, the total received SNR of user is where .

In mm-wave bands, an important phenomena is channel sparsity observed both in angular and temporal domain. That is to say, most of the channel power is concentrated in a finite region on angle-delay plane, corresponding to the interaction with physical clusters of scatterers in the real world [adhikary14]. Thus the number of significant MPCs is reduced to a much lower value than that for a microwave system, and these dominant MPCs are seen by the BS under a very constrained angular range (AoA support). Then, CCM of a particular MPC is given by [adhikary13], [Ma19]

(3)

where with is the uniform linear array (ULA) manifold (steering) vector. The steering vector can be expressed as where the antenna spacing is the half of the signal carrier wavelength for . In (3), is the angular spread (AS) of MPC of user with mean look angle and is the angular power density of MPC of user where . The angular power density is non-zero if where is the angular support set of MPC of user. Here, the support set is defined as . Based on (3), it is simple to note that can be expressed in terms of as .

As can be seen from (3), the CCM of a particular MPC is a function of the power intensity which is non-zero only for particular values of among and for a constrained angular range of . These particular values on joint angle-delay plane for which is significantly above the noise level, can be used to construct sparsity map, a matrix composed of ones and zeros only. The non-zero entries of this matrix shows the temporal locations and angular supports of active MPCs for each user channel on joint angle-delay plane. The sparsity map together with power intensities are slowly varying in time as the AoA of each user signal evolves depending on the user mobility, variation rate of the scattering environment characteristics, etc. [adhikary14], [you15], [caire16], [utschick05]. The rate of change of these long term parameters is much smaller than that of the actual small-scale fading process. This fact helps us design channel estimators in hybrid architecture after effectively reducing the signaling dimension via these slowly-varying parameters.

Ii-B Equivalent Multi-Ray Channel Model for MPCs

Our objective is to determine the regions in joint angle-delay power map where is non-zero, thus to estimate based on (3). In order to realize this, the following practical model for the MPC of each user channel, namely in (1), is adopted

(4)

where the propagation from MPC of user to BS is composed of rays, and represents the complex gain of the ray having AoA [Xie17], [Xie18]. Since the WSSUS model is adopted, satisfies the following equation

(5)

Based on the given model in (4), one can validate that the covariance of asymptotically satisfies (3) after Riemann integration when

(6)

Iii Hybrid Beamforming Based Joint Angle-Delay Domain Power Profile (JADPP) Estimation

In order to estimate the CCMs and sparsity map of each users, we need to estimate their JADPPs, i.e., in (3) together with their angular support first. Since hybrid beamforming structure is used, limited number of RF chains () is utilized (). In hybrid structure, before estimating the power profile of each MPCs, initially, the sector of interest, in which the users are to be served, is divided into non-overlapping sub-angular sectors (which are scanned by initial search beams constructed in analog domain). Then, JADPPs and sparsity map are extracted for each user in the sector of interest. This mode of operation is called as slow-time beam acquisition mode (or slow-time training mode) as illustrated in Fig. 1. In this mode, each UE transmits its slow-time training sequence so that the BS estimates the angular locations of each user MPCs in TDD mode. We define pilot (training) sequence vector for MPC of user as follows

(7)

In addition, the angular search sector of interest, , which is defined as the ordered set of look angles (to which the beam is steered towards), is taken as . Then, the angular sub-sectors in Fig. 1 are constructed such that . In slow-time training mode, where ’s are scanned by initial search beams (constructed by the columns of analog beamformer matrix peculiar to each sub-sector in Fig. 1) separately, each active UE repeats its training sequence in (1) at least times so that the power profiles of all users in are acquired by the BS.

Iii-a Discrete Time Spatio-Temporal Domain Signal Model for JADPP Estimation

If the selected look angle is in the support set , and the AS is narrow enough (which is the case for mm-wave channels [guvensen16_2]), we can approximate by assuming in (4) as

(8)

Here, can be regarded as the effective complex channel gain (reflection coefficient) of MPC for user at look angle . Asymptotically, one can calculate the corresponding average channel power as

(9)

when in (4), and it can be noted that if uniform power distribution is assumed, then . During the slow-time training phase, if the BS intends to estimate the effective channel gain of the MPC for the user at look angle , namely , it is useful to construct the following discrete time equivalent signal model in spatio-temporal domain by using (1), (7), and (8):

(10)

where , total interfering component to MPC of the user, is given as

(11)

Note that in (11), is composed of self interference signal stemming from the MPCs of user other than the MPC, inter-user interference signal and AWGN. Based on (10), is to be estimated at preassumed spatio-temporal locations on joint angle-delay map for all active users. That is to say, average power of the MPCs, which are likely to exist, at each angular and temporal delay locations (for and ) is to be estimated for all active users.

Iii-B Initial Beam Acquisition Mode

In hybrid beamforming architecture adopted, the initial analog beamformer matrix, , is constructed to illuminate the intended sub-sector of interest where as shown in Fig. 1222 The initial analog beamformer used to estimate power profile in slow-time training mode needs to satisfy in order for intended sector to be covered properly.. In order to maximize the coverage of the intended sector, the columns of (where ) can be obtained as the most dominant

(number of RF chains) eigenvectors of the following matrix

(12)

Here, can be regarded as the spatial autocorrelation matrix of user channels in sector , and the analog beamforming via in this mode is nothing but the Karhunen-Loeve Transform (KLT) in angular domain [van2004optimum]. Assuming that each user in sector-

having a mean AoA uniformly distributed over the sector of interest,

can also be considered as the initial CCM estimate of each user MPCs in sector-. Similarly, intra-sector digital fine search beams for the look angle shown in Fig. 1, namely ’s are constructed after projection on the range space of . After illuminating the intended sector by , the digital search beams in reduced dimension are steered towards at which in (8) is to be estimated. In order to realize this, a particular angular region (patch) in whose center is the look angle , which we are interested in, is to be selected and illuminated by . Here, can be contemplated as the matrix of the eigenvectors corresponding to the largest eigenvalues of which is defined as the reduced dimensional spatial autocorrelation matrix of the user channels of selected angular patch in whose center is , and is the search dimension (number of digital beams to be constructed). Then, can be constructed for the mean look angle as

(13)

where is the angular width of the selected patch in sector- which is simply called as the look spread. While constructing these initial intra-sector digital beams, ’s can be normalized such that for proper operation. Then, we can define a hybrid beamformer matrix in slow-time training mode as and express the signals of (10) in reduced dimensional digital beamspace as

(14)

Later, is to be updated for fast-time instantaneous channel acquisition (as explained in Section VI-B) by using the estimated CCMs (which are constructed via the JADPP and sparsity map of each user).

Iii-C Proposed JADPP Estimation Techniques

By using the reduced dimensional observations obtained after hybrid beamforming in (14), we propose efficient algorithms to estimate the average channel power in (8) for each look angle and temporal delays . Here, we first conceive that is to be estimated for each preassumed spatio-temporal resolution cell which is defined as the pair on joint angle-delay map by using each slow-time training snapshot in (10). We denote this estimate as . Then, the estimate of at look angle , which is denoted by , is constructed as . Since is proportional with from (9) when is large enough, gives us the estimate of the angular power density at look angle . We develop two different approaches in order to construct for each spatio-temporal resolution cell :

Iii-C1 Spatio-Temporal Adaptive Matched Filter (AMF)

After reducing the dimension of spatio-temporal observation via hybrid beamformer in (14), the maximum likelihood (ML) estimate of non-random parameter at look angle can be obtained as

(15)

whose detailed derivation is given in Appendix I. While obtaining (15), the spatial covariance matrix of interfering MPCs are also assumed to be unknown non-random parameters to be estimated together with . We call this estimator as adaptive matched filter (AMF), inspired from the adaptive detection algorithm in [robey92], since a sample matrix inversion (SMI) type adaptive filtering is utilized in (15) to construct . The spatial autocorrelation matrix of interfering MPCs (given by in (11)) is estimated by means of a simple temporal averaging of the columns of . The most important difference of the proposed AMF from the conventional SMI based detectors in [Brennan74], [robey92] and [kelly86] is that the temporal averaging to form is obtained after projecting the observation signal on the desired signal nullspace (i.e., the null-space of in (7)) in order to eliminate the signal contamination due to desired MPC at for user.

Iii-C2 Spatio-Temporal Matched Filter (MF)

As a special case, one can simplify (15) by taking as , which corresponds to assuming spatially white interfering signal to desired MPC at . In this case, the following estimator, which is a spatio-temporal matched filter (MF) without any adaptive cancellation of interference in (10), is given:

(16)

Then, the outputs of AMF/MF type estimators will be provided to the subsequent thresholding algorithm to construct the joint angle-delay sparsity map of each user in the sector of interest.

Iv User Activity Detection and Sparsity Map Construction via Constant False Alarm Rate (CFAR) Algorithm

Based on the estimated JADPPs, one can construct the sparsity map composed of the spatio-temporal resolution cells in joint angle-delay domain where is determined to be non-zero for and . In order to construct the sparsity map, we apply two-stage adaptive thresholding onto (obtained via AMF/MF type preprocessing in (15) and (16)) by inspiring from the well known cell-averaging CFAR technique in radar literature [richardsbook]. Thus, the regions where the power of MPCs concentrated on joint angle-delay map is determined (for each user). The following adaptive thresholding can be applied to each resolution cell , called as cell-under-test (CUT), on joint angle-delay domain: , and where is a CFAR threshold which is to be adaptively determined.

While constructing , one can use multiple non-coherent snapshots, i.e., independent observations in (10) of the same sector obtained via the slow-time training data. Then, the random fluctuations on can be smoothed out by taking simple averaging over multiple snapshots:

(17)

where is the estimated power profiles obtained from slow-time training snapshot for where is the total number of slow-time training snapshots.

Iv-a Two-Stage CFAR Algorithm

We propose the following Two-Stage CFAR algorithm which can be realized with adaptive thresholding both in temporal and spatial domain for the CUT on joint angle-delay domain:

Iv-A1 Temporal thresholding for selected resolution cell

For each user, we first apply temporal thresholding on at each look angle :

(18)

where

is the desired average false alarm probability of the test. In (

18), the adaptive threshold is obtained by simple averaging over JADPPs of each user for different delay locations other than at the selected look angle similar to the well-known cell-averaging CFAR tests in radar literature [richardsbook]. By assuming that ’s are exponential i.i.d.random variables, one can obtain the threshold level which provides constant false alarm rate despite varying interference power levels333 The false alarm rate is nothing but the probability of declaring an empty resolution cell, where , as an active MPC having non-zero power level.. That is to say, the test in (18) yields average false alarm probability which does not depend on the actual value of interfering signal levels.

Iv-A2 Spatial thresholding for selected resolution cell

Similarly, we can apply adaptive spatial thresholding on estimated JADPPs at each spatio-temporal location on angle-delay domain as

(19)

where , and is the guard interval, which is an angular window with mean angle of . It is taken as where is the length of guard interval, i.e., the number of angular resolution cells around the CUT (which are taken as guard cells, not used in cell averaging). In (19), the adaptive threshold is obtained by averaging over estimated JADPPs of each user for different look angles at selected temporal delay .

Iv-B Joint Angle-Delay Domain Sparsity Map Construction of User Power Profiles

After thresholding, the sparsity map of user, , which is matrix, where is the number of resolution cells in angular domain, is constructed. If there is a detection at a particular CUT, the corresponding entry is set to 1, otherwise 0. That is to say for and :

(20)

Note that, when all elements of is zero, it means that user is not an active user.
An Exemplary Scenario: In Fig. 2, we demonstrate the estimated power levels of each MPCs of active users via the proposed JADPP estimators, namely AMF and MF. We investigate a scenario where a BS with antenna elements in the form of ULA is serving single-antenna users. It is assumed that both users have two active MPCs where the resolution cells with non-zero power levels are and for the first user, and and for the second one respectively. For all MPCs, the AS is taken as and is used in simulations. It is assumed that the received power levels are dB and dB. In Fig. 2, we provide sparsity maps after applying two-stage thresholding on JADPP estimations given in Fig. 2. Here, is set to and is taken as .

Fig. 2: JADPP (left) and sparsity map (right) for user 1 and user 2 (, , dB, dB)

Iv-C Performance Metrics for User Activity Detection

Here, our aim is to find the probability of detecting the active MPCs, which have non-zero power at a given angle-delay resolution cell. We use two performance metrics to compare AMF in (15) and MF in (16), namely the probability of detection () and the probability of false alarm (). By using the constructed sparsity map in (20), the probability of detection for the active MPCs of user can be expressed as

(21)

where is the set of indices of active MPCs having positive , and is the angular index pointing the mean AoA for MPC of user: where is element of . Similarly, the probability of false alarm, showing the probability that inactive MPCs (having zero ) are declared as active for user, can be expressed as

(22)

where is the set of indices such that the look angle is inside the angular support set . It can be given as .

V Sparse CCM Construction

We can construct CCMs purely based on the estimated power levels and sparsity map for each user. It is important to note that the observation signal in (10) is not available in full dimension in our hybrid structure, however each CCM needs to be estimated in full dimension. Also, we need to obtain accurate enough CCM estimates with significantly reduced amount of training snapshots. That is why we need parametric construction of CCM by exploiting its reduced rank property due to sparse nature of the channel. Hence, we can construct CCMs for each user by using sparsity matrix and power estimates as follows

(23)

for and . The proposed construction technique in (23) is completely different from the conventional ones, which are based on SMI type temporal averaging in full dimension necessitating large amount of training snapshots for proper operation [Brennan74].

Asymptotic Convergence of the Proposed CCM Estimation:
It is interesting to see that in (23) converges to true CCM values, in (3) when the sparsity map in (20) is perfectly acquired, i.e., if and otherwise. When (if AS is narrow enough), we can obtain the asymptotic value of by letting in (23), and as follows

(24)

This shows that when the number of angular resolution cells is high enough, and accurate JADPP estimates are available, parametric construction of by (23) in full dimension can be realized efficiently after hybrid beamforming.

Vi Nearly Optimal Covariance-Based Reduced Rank Hybrid Beamformer Design

In this section, based on the estimated CCMs in (23), the analog beamformer, in Fig. 1 can be updated by using the statistical properties of each user channels. In slow-time beam acquisition mode, the long term parameters of each user channels, i.e., JADPPs , the sparsity map , and CCMs are acquired by using slow-time training snapshots as explained in previous chapters. Also, the analog beamformer can be optimized by exploiting these slowly-varying long term parameters. It is important to note that slow-time training data is transmitted much slower compared to the rate of change of instantaneous CSI.

Vi-a User Grouping Stage

In slow-time beam acquisition mode, since no apriori information is assumed related with these slowly varying parameters initially, predetermined search sector beams together with slow-time training data are utilized to extract the spatial signatures of each user (based on AMF in (15) and MF in (16)). This initial step can be regarded as pre-grouping stage. After this initial stage, an efficient reconstruction of analog beamformer in Fig. 1 can be carried out by using the estimated CCMs and sparsity map. This can be realized by means of a virtual sectorization via second-order channel statistics based user-grouping inspired from JSDM framework [adhikary13], [nam14]. In this framework, all active users can be divided into groups based on their spatial information, i.e., the estimated sparsity map and CCMs, by using proper user grouping algorithms known in the literature444 The design of user grouping algorithm is out of scope of this paper. An efficient procedure can be found in [nam14], [Gesbert16]. (as in the case of JSDM). We define as the set of all UEs belonging to group with cardinality , and are UE indices forming , where the users in group are assumed to have statistically i.i.d. channels.
In user grouping stage, one need to construct the common covariance matrix of each MPCs in group , denoted by which can be considered as the common spatial covariance matrix of UEs belonging to group at delay. Instead of using the true covariance matrices of each MPCs, which can not be known accurately, we can construct estimated by using the acquired CCMs in (23) as follows

(25)

where is the estimated covariance matrix for the MPC of the user in group . Similarly, the estimated spatial covariance matrix of received signal in (1) (assuming that the transmitted symbols are i.i.d. with unity power) can be obtained as

(26)

and the estimated covariance matrix of the inter-group interference to group , consisting of the statistical information for all inter-group users interfering with group , can be constructed as

(27)

Vi-B Post-User Grouping Stage

After user grouping stage, an efficient analog beamformer for each user group can be designed via the estimated group covariance matrices given in (25), (26), (27). This stage can be regarded as post-grouping stage. Here, we load in Fig. 1 with the optimized statistical analog beamformer, which is applied in order to distinguish intra-group signal of users in group from other groups by suppressing the inter-group interference while reducing the signaling dimension of in (10). In this stage, a -dimensional space-time vector , where is the number of RF chains assigned to group , can be formed by using

for all groups after the following linear transformation:

(28)

where is an statistical analog beamformer matrix that projects the -dimensional received signal samples in (1) on a suitable -dimensional subspace in spatial domain. Here, since a limited number of RF chains, , are used in hybrid architecture, we have the following constraint: . After constructing the optimal , in Fig. 1 is replaced with .

Vi-B1 MPC Grouping for Efficient Analog Beamformer Design

If there exists a significant overlap among some of the MPCs of group in the angular domain, one can simply form groups of nonresolvable MPCs, and process them jointly. One can group