1. Introduction
As a key technology for 5G wireless communications, millimeter wave (mmWave) massive multiple-input multiple-output (MIMO) has received significant attention in recent years [
1,
2]. mmWave techniques provide the ability of fast speed data delivery and wide bandwidth [
3], making them valued in information ages. The large number of spatial degrees of freedom created by the massive MIMO contribute to high user throughput and high spectral efficiency for extended mobile broadband (eMBB). Furthermore, massive MIMO can be used to efficiently support ultra-reliable low-latency communication (URLLC) and massive machine-type communication (mMTC) in the various types of Internet of Things (IoT) connectivities [
4,
5,
6]. In mmWave massive MIMO systems, large antenna arrays implemented at the base station (BS) and mobile station (MS) are utilized to supply adequate beamforming gains, which offers a compensation for high signal attenuation at mmWave frequencies [
7,
8,
9]. In such a context, the number of radio frequency (RF) chains is much smaller than that of antennas due to lower hardware cost and power consumption. Considering the RF limitations, the hybrid analog/digital precoding scheme [
10,
11,
12,
13,
14] is widely adopted to provide larger precoding gains. However, the overall channel state information is hard to observe because of massive antennas and hybrid precoding structures. In practice, mmWave channels exhibit the characteristic of broad-band frequency-selective fading (FSF) owing to the large bandwidth and distinct multipath delays [
15]. While the whole bandwidth may be subject to FSF, each sub-band undergoes flat fading. When orthogonal frequency division multiplexing (OFDM) is integrated into massive MIMO architectures, mmWave channels are viewed as the aggregation of a number of parallel flat fading channels, which urges us to achieve large multiplexing and diversity gains [
16,
17,
18] with a large number of spatial degrees of freedom. Nevertheless, the overall system performance still hugely depends on the channel estimation, which is a tricky task for the channel subspace sampling limitation [
19,
20].
In the aspect of channel estimation in massive MIMO systems, several relevant schemes [
21,
22,
23,
24,
25] have been developed in various scenarios. In general, massive MIMO channels are efficiently parameterized with a series of associated parameters including fading coefficients, angles of arrival or departure (AoAs/AoDs) and delays. Owing to the intrinsic sparsity of mmWave channels, some approaches [
10], Refs. [
26,
27,
28,
29] based on compressing sensing (CS) techniques convert the issue of channel estimation into the recovery of a line sparse signal. For example, a conventional CS-based algorithm is proposed for channel estimation with much training overhead reduction in a multiuser mmWave system [
28]. Further in the work [
10], with a novel hierarchical multi-resolution codebook, an adaptive CS-based algorithm is proposed to estimate the channel parameters both in single-path and multipath mmWave environments. According to the CS-based methods, the continuous parameter space is discretized into a finite of grid points [
30,
31,
32], and the estimated angles are assumed to lie on the prespecified grid points. In fact, actual angles may be not in accordance with this assumption, which seriously degrades the channel estimation performance. In addition, some tensor decomposition-based methods stand out in the fields of mmWave channel estimation. The work [
33] achieves the joint channel parameter estimation via parallel factor (PARAFAC) analysis in massive MIMO systems. As the improvement of [
33], Ref. [
34] proposes a more accurate PARAFAC-based estimator in the presence of pilot contamination. For the joint multiuser channel estimation, [
35] proposes a layered pilot transmission scheme and decomposes the problem of multiuser channel estimation into a number of problems of single-user channel estimation. In the aforementioned scenarios [
33,
34,
35], simulations demonstrate that the tensor-based algorithms have better estimation performance compared with these CS-based methods. However, most of the existing works concentrate on the narrowband systems while realistic mmWave channels exhibit the broad-band FSF. The work [
36] proposes a simultaneous orthogonal matching pursuit (SOMP) scheme and regards the broadband mmWave channel estimation as a multiple measurement vector (MMV) problem with a common support (i.e., the channel support at different frequencies is assumed to be the same). Unlike the proposed scheme in [
36], the work [
37] estimates the support of the angle-domain channel at some frequencies independently by the orthogonal matching pursuit (OMP) algorithm and combines them into the common support at all frequencies. Moreover, a CANDECOMP/PARAFAC (CP) decomposition-based method for downlink channel estimation is proposed in a mmWave MIMO-OFDM system, where the received signal at individual user is formulated as a low-rank PARAFAC tensor. So far, to the best of our knowledge, there are few tensor decomposition-based methods for joint multiuser mmWave channel estimation, especially over the FSF channels.
In our paper, we consider the broad-band FSF channel and address the problem of multiuser uplink channel estimation in a massive MIMO system, where the hybrid precoding strategy is adopted. Due to the sparsity of mmWave channels, a PARAFAC decomposition-based scheme is proposed for multiuser mmWave channel estimation. In the successive time slots, all the MSs simultaneously transmit respective pilots to a common BS, and the signals received at the BS can be fitted in a low-rank PARAFAC structure.
Table 1 presents some reviewed works along their main distinctive characteristics to better position our present work. Unlike the system [
38] where individual users are considered for downlink transmission, our proposed algorithm concentrates on the joint estimation of multiuser uplink channels, which leads to wider identifiability and more relaxed requirements for unique PARAFAC decomposition. The main contributions of our work are summarized as follows.
(1) For the sparsity of mmWave FSF channels, the signals transmitted from multiple users can be modeled as a low-rank PARAFAC structure at the BS, which promotes the joint channel estimation in contrast with [
38]. It’s worth noting that the massive MIMO channel is characterized with path gains, AoAs/AoDs and delays. This tensor modeling not only takes full advantage of diversities in space, time and frequency dimensions, but also has the benefits of multidimensional structures.
(2) In this context, a PARAFAC decomposition-based algorithm is proposed for accurate estimates of multiuser mmWave channels parameters including path gains, AoAs/AoDs and delays. The proposed algorithm consists of two stages. In the first stage, the accelerated trilinear alternating least squares (ATALS) algorithm is put forward for efficient PARAFAC decomposition. In the second stage, we make further efforts to extract unknown channel parameters of multiple users from factor matrices. The uniqueness of PARAFAC decomposition is also studied, which gives the guidance on the design of precoding and combining matrices. What’s more, our proposed scheme has wider identifiability in terms of unique PARAFAC decomposition.
(3) To provide a benchmark for system evaluation, the Cramér -Rao Bound (CRB) results for channel parameters of multiple users are derived. Simulations reveal that the mean square errors (MSEs) of channel parameters obtained by our algorithm are nearly close to their corresponding CRB results, which indicates our proposed PARAFAC-based algorithm yields accurate channel estimation. While our proposed scheme is put forward at the assumption of uniform linear array (ULA) antennas, it is also suitable for uniform planar array (UPA) antennas.
(4) The computational complexity of the proposed scheme is analyzed. For comparison, we also analyze the computational complexities of some CS-based channel estimation schemes including the SOMP [
36], OMP [
37] and adaptive CS [
10] algorithms. Numerical results demonstrate that our proposed algorithm outperforms its counterparts in estimation accuracy and computational complexity, particularly with a substantial training overhead reduction. Specially, our proposed scheme still performs well, even though the number of users is large.
The rest of this paper is organized as follows.
Section 2 first presents the fundamentals of PARAFAC model and formulates the PARAFAC model of multiuser mmWave massive MIMO-OFDM architecture. The uniquess issue of constructed PARARFAC model is also discussed in
Section 2. In
Section 3, a two-staged PARAFAC-based algorithm is proposed for multiuser mmWave channel estimation. The CRB derivation are given in
Section 4.
Section 5 presents some numerical results for the system performance analysis. Finally, some conclusions are drawn in
Section 6.
Notation: Scalars, vectors, matrices and tensors are denoted by lowercase, boldface lower case, boldface uppercase and calligraphic uppercase letters, e.g., a, , and , respectively. , , and denote the transpose, conjugate transpose, inverse and pseudo-inverse of matrix , respectively. The operators ⊙, ⊗ and ∘ denote the Khatri-Rao, Kronecker and outer products, respectively. denotes the identity matrix and denotes the Frobenius norm. The operator constructs a diagonal matrix with the vector and the operator changes a matrix into a column vector by stacking the columns. For convenience, all the abbreviations used in this paper are given in Abbreviations.
2. System Model
As shown in
Figure 1, we consider a multiuser mmWave massive MIMO-OFDM system composed of BS and
U MSs. The BS is equipped with
N antennas and
R RF chains, and the
u-th MS is equipped with
antennas and
RF chain. Since hybrid precoding structure is adopted in this scenario, the number of RF chains is less than that of antennas, i.e.,
,
. Specially, the number of antennas for each MS is assumed to be equal, i.e.,
,
. The number of subcarriers for transmission is assumed to be
K.
In the uplink, all the MSs simultaneously transmit respective pilot signals to the common BS via a set of
K subcarriers. As mmWave channels exhibit the inherent feature of sparsity, a geometric channel model with finite scatterings is utilized to account for mmWave channels. In terms of the uplink transmission via the
k-th subcarrier, the channel of
scatterings from the
u-th MS to BS can be expressed as
in the frequency domain
with the
-th path
for
,
, where
is the sampling rate,
denotes the number of propagations and
denotes the time delay of the
-th path. In this setting of
scatterings, a single propagation path
is parameterized by a set of channel parameters
, where
,
and
denote the complex gain, AoA and AoD of the
-th path, respectively. The vectors
and
are the receive and transmit steering vectors, respectively. For the ULA, the steering vectors are given by
where
is signal wavelength and
d is the antenna spacing. Assume that
,
,
with the term
, we derive
2.1. PARAFAC Model
In this subsection, we give a brief overview of PARAFAC model and discuss the unique issue of PARAFAC decomposition. The PARAFAC decomposition amounts to decomposing the
N-way tensor
into a sum of components, each component being a rank-one tensor. For a three-way tensor
of
R components, PARAFAC decomposition can be expressed as
where
,
and
. For brevity, we have the expression
. Alternatively, it can be written as the mode-
n product:
where
is the three-way identity tensor of dimensions
which represents a diagonal hypercubic tensor with its nonzero elements
when and only when
. Furthermore, slicing
along each dimension comes to three distinct slabs
,
and
. Arranging the slices in some manner leads to the three corresponding matrix unfoldings:
,
and
.
Extended to a
N-way tensor
of rank
R, PARAFAC analysis has the form:
with its element
, where
,
,
,
. Furthermore, the columns of matrix
are assumed to be normalized, i.e., with a unit norm, for
, we derive that
where the identity tensor
is replaced by the diagonal tensor
whose nonzero elements are equal to scaling factors
lying on its diagonal.
Then we investigate the uniqueness of PARAFAC-
N model in (9). Since the condition for essential uniqueness of PARAFAC model involves the notion of
k-rank of a matrix, we first recall the definition of
k-rank. The
k-rank of a matrix
is denoted by
, where
on condition that any
r columns of
are linearly independent, where
k-rank stands for Kruskal-rank. It can be shown by using the identifiability theorem for the PARAFAC-
N model if
then
is essentially unique and its loading matrices
,
are unique to column permutation and scaling. Hence, there exists an alternative set of matrices
with the relation
, for
, where
is a permutation matrix and
is a nonsingular diagonal matrix such as
.
2.2. Constructed PARAFAC Model
According to the upstream communication protocol, multiple MSs encode their respective pilots to the BS through the FSF channels via
K subcarriers in
P successive time slots. At the destination, the BS employs the combining vectors to detect the transmitted signals from the MSs. Define
as the pilot for the
u-th MS with
k-th the subcarrier, and
as the precoding vector in the
p-th time slot, the signal vector received at the BS in the
p-th time slot can be expressed as
where
is additive white Gaussian noise. Considering the hybrid precoding strcuture, the BS employs the combining vectors
to detect the received signal
. For the
r-th RF chain at the BS, we derive
where
is the received signal observed from the
r-th RF via the
k-th subcarrier in the
p-th time slot, and
is additive White Gaussian noise. Arrange the
R combining vectors at the BS to form the composite combining matrix
, i.e.,
, which is composed of baseband combiner
and RF combiner
, i.e.,
. Similarly, the combined precoder
at the
u-th MS is composed of baseband precoder
and RF precoder
, i.e.,
. It should be mentioned that the elements of RF precoding/combining matrices
and
meet the constant modulus property. In addition, all
K subcarriers share the same RF precoding/combining in consideration of flat analog precoders which are fixed over the band.
Our goal is to achieve reliable estimation of channel parameters
from
with as few pilots as possible. Since the elements of matrices
and
are randomly chosen from a unit cicrle, we assume that
and
. Substituting (5) into (12) yields
where
,
,
,
. Define
,
corresponds to the frontal slice of third order PARAFAC model
. We derive that
where
. Clearly, the aggregate channel tensor
includes all the channels of
U MSs via
K subcarriers with its three modes standing for the number of antennas at the BS, antennas at the MS and subcarriers for transmission.
Collecting the received signal
from
R RF chains contributes to
, and arranging the vectors
in columns leads to the received signal at the
k-th subcarrier. By concatenating the received data via
K subcarriers, a three-way signal tensor
can be expressed as
where
,
,
and
. Notice that
and
are the signal and noise tensors characterized with three dimensions standing for numbers of RF chains, time slots and subcarriers in space, time and frequency domains, respectively. Moreover, the three forms of unfolding
,
and
are given by
where the slices
,
and
represent the three forms of unfolding of tensor
.
Since the number of resolvable paths
from MS to BS approximately satisfies the Poisson distribution
with
when mmWave works at 28 GHz [
39], each MS can be assumed to have 1 or 2 scatterings in the mmWave communications. Thus, the value of
is usually relatively smaller to the value of each dimension
, which enables the signal tensor
to exhibit a low-rank PARAFAC structure.
2.3. Uniqueness Issue
According to the above derivations, it has been proven that the received signal tensor has a low-rank PARAFAC structure. In this subsection, we study the uniqueness of PARAFAC analysis and provide some sufficient conditions for the unique decomposition of .
Inequality (10) establishes the sufficient condition for the uniqueness of PARAFAC-
N model. In the case of our mmWave massive MIMO architecture, if the sum of
k-ranks satisfies
there exists the unique triple
up to permutation and scaling ambiguities, i.e.,
,
,
, where
and
,
and
are unique estimates of
,
and
. In the generic case, the sufficient condition becomes
This indicates that matrices , and must be of full k-rank for the reason that the k-rank is always less than or equal to the rank, i.e., , for any matrix .
Then we make a profound study of the k-ranks of . Considering the ULA implemented at the BS and MSs, matrices and have the Vandermonde structure, and thus we have , . For the requirement of full k-rank of and , an ingenious design of combining matrix and precoding matrix becomes an indispensable factor. In practice, both and are accomplished by analog phase shifters, which only change the phase of signals. Therefore, the constant modulus constraint is imposed on the elements of and . Since the total transmission power constraint is enforced by normalizing such that , we assume that the entries of combining matrix and precoding matrix are chosen uniformly from a unit circle by a constant, i.e., , , , , , , where and follow the independent and identically distribution (i.i.d) uniform distribution . Thus, the entries of and are viewed as i.i.d Gaussian variables, and we have , . In addition, due to the shift-invariant vector of all U MSs, is full k-rank, i.e., , where can be seen as a frequency-domain steering vector pointing towards the delay .
In practice, the number of propagation paths
is largely limited for mmWave circumstances of poor scatterings. Moreover, the scatterings appear in groups with similar delays, AoAs and AoDs because of massive antennas, which limits effective numbers of active paths. Generally, the number of subcarriers
K is larger than
, which turns the uniqueness condition into
From this condition, we can conclude that the BS has the potiential to achieve the acurrate channel estimation in as few as two successive time slots, with the help of combining vectors. Hence, our proposed scheme can realize the accurate multiuser channel estimation with much reduced training overhead. However, a slightly larger pair of values for is chosen for better performance in our simulations.
3. Proposed PARAFAC Decomposition-Based Channel Estimation Scheme
In the presence of inescapable white noise, we are seeking for some appropriate solutions to the channel estimation issue. To make the most of low-rank PARAFAC strcuture of tesnsor , an efficient and low-complexity multiuser channel estimation via PARAFAC decomposition is put forward. The algorithm is divided into two stages. In the first stage, three factor matrices including channel parameters are uniquely decomposed from by the proposed ATALS algorithm. In the second stage, channel parameters are extracted from these factor matrices through the one-dimensional search method.
3.1. The ATALS Algorithm
As the workhorse technique for low-rank decomposition of three and higher dimensions, the alternating least squares (ALS) algorithm is widely applied in the field of PARAFAC analysis. In order to estimate the channel associated matrices , and in an efficient way, the ATALS algorithm is proposed as a speed-up version of the classical ALS algorithm. In our simulations, all the channel state information is assumed to be unknown from the MSs or the BS, and the numbers of propagation paths at each MS, the matrices and are provided in advance.
In terms of the ALS algorithm, the criteria is that when estimating each matrix in the LS algorithm, the other matrices should maintain the previous estimation until the following cost function converges to the minimum value. At the
i-th iteration, the cost function
is expressed as:
where
,
and
are the estimated values of
,
and
acquired from the
i-th iteration, respectively. At the
i-th iteration, the factor matrices are updated as follows:
where
,
and
are the estimated values of
,
and
at the
-th iteration, respectively. However, in some cases, simulations reveal that the convergence is slow and the estimates of
,
and
show an increasing tendency towards a certain direction.
To speed up the convergence of the ALS algorithm, a line search process of calculating the linear interpolations for
is carried out before the ALS algorithm. The interpolated values for the
i-th iteration are expressed as
where
is the relaxation factor and the differences
,
and
denote the search directions at the
i-th iteration. To speed up the convergence, we choose an appropriate step size
[
40], which is viewed as
in the classic ALS algorithm. Assume that
and
, the cost function comes to
The relaxation factor is accepted on the condition that , and then , and are chosen for the i-th iteration in the following ALS algorithm. Otherwise, , and are chosen for further estimates at the i-th iteration. Nevertheless, if this acceleration experiences three consecutive failures, a smaller step size should be chosen as .
3.2. Channel Parameter Extraction
On the basis of estimates
from the previous stage, we aim to extract the channel parameters from these estimated factor matrices in this subsection. With the permutation and scaling ambiguities, we have
where
is the permutation ambiguity.
,
and
are the diagonal scaling matrices with
. To achieve the estimation of the channel parameters of all
U MSs, we resort to a one-dimensional search. For the
-th path of the
u-th MS, we obtain the estimated channel parameters
:
where
is the
l-th unit coordinate vector with
. Relying on the estimates
and
of
U MSs, we can obtain the scaling matrices
and
. According to the equality
, matrix
can be computed. With the known parameters
and
, the complex gain
fianlly can be obtained. So far, we are capable of estimating the channel parameters
of
U MSs, and then the channel tensor
can be recovered. The detailed steps of our proposed algorithm are given in Algorithm 1:
Algorithm 1 PARAFAC decomposition-based channel estimation algorithm |
Input: Received tensor , precoding matrix , combining matrix , one-dimensional search number J, the number of paths of U MSs and error threshold . |
Output: Estimates , , and . |
First stage (the ATALS algorithm): |
Initialization: Randomly initialize , , , , , . Set , and . |
Step 1.1. Compute from (25) with , and compare it with from (20). |
If , construct , , with ; else, construct , , with . |
Step 1.2. Update using , , by . |
Step 1.3. Update using , , by . |
Step 1.4. Update using , , by . |
Step 1.5. Compute the new error . If , then end. |
Otherwise, set and go to Step 1.1.. |
Second stage (Channel parameter extraction via one-dimension search): |
for, and for |
Step 2.1. Estimate the AoA by (27) with , . |
Step 2.2. Estimate the AoD by (28) with , . |
Step 2.3. Estimate the delay by (29) with , . |
Step 2.4. Compute the diagonal matrices , and . |
Step 2.5. Estimate the path gain according to . |