Applications of Tensor Models in Wireless Communications and Mobile Computing
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Joint DL and UL Channel Estimation for Millimeter Wave MIMO Systems Using Tensor Modeling
Abstract
In this paper, we address the problem of joint downlink (DL) and uplink (UL) channel estimation for millimeter wave (mmWave) multipleinput multipleoutput (MIMO) systems. Assuming a closedloop and multifrequencybased channel training framework in which pilot signals received by multiple antenna mobile stations (MSs) are coded and spread in the frequency domain via multiple adjacent subcarriers, we propose two tensorbased semiblind receivers by capitalizing on the multilinear structure and sparse feature of the received signal at the BS equipped with a hybrid analogdigital beamforming (HB) architecture. As a first processing stage, the joint estimation of the compressed DL and UL channel matrices can be obtained in an iterative way by means of an alternating least squares (ALS) algorithm that capitalizes on a parallel factors model for the received signals. Alternatively, for more restricted scenarios, a closedform solution is also proposed. From the estimated effective channel matrices, the users’ channel parameters such as angles of departure (AoD), angles of arrival (AoA), and path gains are then estimated in a second processing stage by solving independent compressed sensing (CS) problems (one for each MS). In contrast to the classical approach in the literature, in which the DL and UL channel estimation problems are usually considered as two separate problems, our idea is to jointly estimate both the DL and UL channels as a single problem by concentrating most of the processing burden for channel estimation at the BS side. Simulation results demonstrate that the proposed receivers achieve a performance close to the classical approach that is applied on DL and UL communication links separately, with the advantage of avoiding complex computations for channel estimation at the MS side as well as dedicated feedback channels for each MS, which are attractive features for massive MIMO systems.
1. Introduction
In recent years, millimeter wave (mmWave) massive multipleinput multipleoutput (MIMO) technology has been a subject of increasing interest in both academia and industry for future wireless standards due to its great potential to provide substantial gains in data rates and energy efficiency. However, due to the severe path loss over the mmWave frequency bands, large antenna arrays should be deployed at the base station (BS) and mobile stations (MSs) to provide sufficient beamforming gain in mmWave MIMO scenarios [1]. In this context, the implementation of fully digital beamforming architectures becomes prohibitive due its expensive cost, hardware constraints, and power consumption of highresolution analogtodigital converters (ADC) and digitaltoanalog converters (DAC) per antenna port [2, 3]. To overcome these practical limitations, hybrid analogdigital beamforming (HB) architectures that split the signal processing between analog and digital domains using a reduced number of radio frequency (RF) chains (assumed to be smaller than the number of antennas) have been investigated [4, 5]. In the HB architectures, the digital part performs baseband processing using microprocessors, while the analog part can be implemented at the RF domain using different analog approaches such as phaseshifter networks [6], switches [3, 7], or lenses [8].
To fully benefit from the beamforming gains in mmWave MIMO systems, an accurate channel estimation is crucial to realize the hybrid precoding designs in which the analog part is used to improve the signal power, while the digital part is designed to suppress interuser interferences [9–11]. For this purpose, several channel estimation techniques such as [12–16] have been proposed. The authors of [12] proposed an iterative method based on the least squares estimation (LSE) concept and sparse message passing (SMP) algorithm. In this method, the location of nonzero entries of the channel vector is detected through the SMP, while the LSE is used for estimating the channel coefficients at each iteration. Zhu et al. [13] proposed an auxiliary beam pair design for mmWave channel estimation in which the best auxiliary beam pair is fed back to the transmitter via a feedback channel. The method proposed by Ghauch et al. [14] consists of a subspacebased approach that exploits the channel reciprocity in timedivision duplexing (TDD) MIMO systems for hybrid precoding design. It iteratively estimates the dominant singular modes of the channel instead of the entire channel. In contrast, the works [15, 16] explore the angular sparsity of mmWave channels and use compressed sensing (CS) theory to estimate only the channel parameters from which the mmWave channel can be reconstructed. In [15], the angular spreads over the angle of arrival (AoA) and angle of departure (AoD) are considered in the channel modeling, while the lowrank structure of the channel is exploited to reduce the number of samples needed to recover the mmWave channel. Similarly, but disregarding the angular spreads in the spatial domains, the method in [16] uses the 2D unitary ESPRIT algorithm for spatial parameters estimation, while the path gains are estimated by means of the LS criterion.
The researchers [17–21] have proposed CSbased and tensorbased channel estimators for mmWave MIMO systems, respectively. They assume the conventional channel training framework, where the DL and UL channel estimation problems are treated separately (as two decoupled procedures at the MS and BS, respectively). In particular, for frequency division duplexing (FDD) systems, where channel estimation is usually carried at the powerlimited MS side, computational complexity plays a significant role due to the large number of channel coefficients to be estimated. An interesting approach to deal with this problem exploits the poor scattering nature of the mmWave channels via CS techniques [22, 23]. For instance, in [17, 18], the intrinsic sparse feature of the mmWave channel is exploited and CSbased channel estimation algorithms are formulated. However, the adaptive algorithm proposed in [17] can be applied to estimate the DL or UL channel separately. In [19], a layered pilot transmission scheme is proposed to UL channel estimation, while [20, 21] exploit the DL communication of wideband mmWave channels. The main idea in [21] is to divide the overall channel estimation problem into three smaller CS subproblems via tensorbased modeling to estimate the channel parameters (AoDs, AoAs, and delays) with less computational complexity. The system model is formulated as a parallel factors (PARAFAC) decomposition [24], and tensorbased algorithms combined with CS tools are proposed to solve the channel estimation problem. However, Zhou et al. [19] only considered the UL channel estimation, while Zhou et al. and Araújo and de Almeida [20, 21] focused on the DL case. On the other hand, Shen et al. [25] proposed a closedloop based training framework, where the MSs directly feed the received pilots back to the BS without channel estimation. Therein, a simplified approach is adopted, in which the UL channel is modeled as an additive white Gaussian noise (AWGN) term. Moreover, in [25], only the DL channel could be estimated. Different from [25], we are interested in a joint estimation of the UL and DL channels, and a more realistic multipath channel model that fits to mmWave MIMO scenario and HB architecture is considered.
In this paper, we study the problem of joint DL and UL channel estimation in the context of mmWave MIMO systems that employ HB architectures. Initially, we propose a novel closedloop and multifrequencybased channel training framework in which the pilot signals received by multiple MSs are coded and spread in the frequency domain and then fed back to the BS over the same UL resources. Making use of the proposed framework for channel estimation, the received closedloop signal at the BS can be modeled as a threeway array (i.e., a thirdorder tensor) that follows a PARAFAC model. By capitalizing both on the multidimensional and sparse structures of the received signal, we propose two tensorbased semiblind receivers for joint DL and UL channel estimation. The first receiver is an iterative solution based on the alternating least squares (ALS) algorithm [26]. The second is a closedform solution based on the least squares KhatriRao factorization (LSKRF) algorithm [27]. In the proposed receivers, we first obtain joint estimates of the compressed DL and UL channel matrices. Then, we exploit the sparse representation of the DL and UL channels to individually recover the channel parameters (AoDs, AoAs, and path gains) of each user via CSbased techniques. The proposed framework allows concentrating most of the processing burden for channel estimation at the BS side, i.e., avoiding unnecessary computational overhead for channel estimation at the powerlimited MS side. Our simulation results reveal that the proposed receivers achieve a performance close to the classical framework that treats the estimation of DL and UL channels as separate problems.
In summary, the main contributions of this paper can be listed as follows:(i)We propose a novel closedloop and multifrequencybased channel training framework for channel estimation that focuses jointly on the DL and UL communication links. The proposed framework concentrates the processing burden for joint channel estimation at the BS, avoiding processing with high computational cost at the MS side.(ii)We show that, by making use of the proposed framework for channel estimation, the received closedloop signal can be modeled as a thirdorder tensor that follows a PARAFAC model. Then, we formulate two tensorbased semiblind receivers (iterative and closedform ones) for joint DL and UL channel estimation by capitalizing on a tensor structure of the received closedloop signal.(iii)We study the identifiability issues under which the DL and UL channel matrices can be jointly and uniquely estimated using the proposed receivers. Useful lower bounds on the number of subcarriers required to accomplish the joint channel estimation are derived.
The rest of this paper is structured as follows: In Section 2, we provide as a presentation complement some important tensor definitions, tensor algebra operations, and a brief overview on the PARAFAC decomposition. In Section 3, we present the proposed channel training framework, and the system and channel models. Section 4 formulates the two proposed tensorbased semiblind receivers for joint DL and UL channel estimation. In Section 5, we analyze the identifiability conditions of the proposed receivers. Simulation results are provided in Section 6. Conclusions and perspectives for future work are drawn in Section 7.
1.1. Notation and Properties
Scalars, column vectors, matrices, and tensors are denoted by nonbold lowercase letters a, bold lowercase letters , bold uppercase letters , and calligraphic uppercase letters , respectively. The superscripts , , , and denote the transpose, complex conjugate, conjugate transpose, and pseudoinverse operations. represents the Frobenius norm of a matrix or tensor. The th entry of is denoted by . The operator converts into a diagonal matrix, while consists in a diagonal matrix formed by the ith row of . converts to a vector by stacking its columns on top of each other, while converts to a matrix . converts the diagonal elements of into a vector. denotes the outer product operator. The Kronecker and KhatriRao products are denoted by and , respectively. The KhatriRao product between the matrices and corresponds to a columnwise Kronecker product, i.e.,
We shall make use of the following two properties of the Kronecker and KhatriRao products:where is assumed to be a diagonal matrix in (3). In both cases, the matrices have compatible dimensions.
2. Tensor Preliminaries
In order to facilitate the presentation of the proposed receivers, we provide below a brief overview on some important tensor definitions and tensor algebra operations. We also introduce the PARAFAC decomposition and its different representation forms.
2.1. Tensor Definitions and Basic Operations
Throughout this paper, the definitions and operations involving tensors are in accordance with [28, 29]. A tensor is defined here as a multidimensional array. The order of a tensor corresponds to the number of dimensions. It can be seen as a generalization of a matrix to higherorder dimensions. For instance, a scalar is a tensor of order 0, a vector is a tensor of order 1, and a matrix is a tensor of order 2. An nmode fiber of a tensor is a vector obtained by varying the nth index and keeping all the other indexes fixed. Slices are twodimensional sections of a tensor, obtained by fixing all but two indices. The operator denotes the concatenation of two matrices along the nth mode of a tensor. The 1 mode, 2 mode, and 3 mode unfolding matrices of the thirdorder tensor , denoted by , , and , are obtained by collecting all the 1 mode, 2 mode, and 3 mode fibers to be columns of the resulting matrices, respectively. The nmode product between the tensor and a matrix is denoted by , which is equivalent in a matrix fashion to .
2.2. PARAFAC Decomposition
By definition, the PARAlell FACtor (PARAFAC) analysis decomposition of a thirdorder tensor , introduced by [24], is the factorization of in a sum of R thirdorder rankone tensors each one being formed by the outer product of three vectors. Mathematically, the PARAFAC decomposition of is given bywhere R is the rank of the PARAFAC decomposition and is defined as the minimum number of rankone tensors for which holds exactly. The vector denotes the rth column of the factor matrix along the nth mode .
The PARAFAC decomposition can also be represented in terms of the frontal slices of as follows:for _{.}
By using nmode product notation, equations (5) and (6) can be written aswhere denotes a thirdorder identity tensor of size . Its elements are equal to 1 when all indices are equal and 0 elsewhere.
The 1 mode, 2 mode, and 3 mode unfolding matrices of admit the following factorizations with respect to the factor matrices :
3. System and Channel Models
In this section, we introduce the proposed closedloop and multifrequency channel training framework. Then, we formulate our DL and UL signal models. In addition, the considered mmWave massivo MIMO channel model is also presented.
3.1. Downlink Signal Model
Consider a wireless communication system operating in the FDD mode, where a BS equipped with antennas serves simultaneously U MSs equipped with antennas. We assume that the BS employs a hybrid beamforming architecture using RF chains. Due to the different instants of time dedicated to DL and UL communications, the beamforming matrix associated with DL transmission is denoted by , while the beamforming matrix associated with UL reception is denoted by . Note that equal beamforming matrices can also be considered in the transmission and reception phases without loss of generality. The BS transmits a lengthT pilot sequence over the pth spatial direction using the beamforming vector ( and ). The received signal at the uth MS over different directions is given bywhere denotes the DL channel matrix associated with the uth MS, denotes the transmission beamforming matrix, concatenates the pilot sequences to be sent by each transmission beam. The matrix is the additive white Gaussian noise (AWGN) term at the uth MS.
During the training phase, we assume identity matrices for the digital beamforming matrices, while the analog beamforming matrices have constant unit modulus entries with random phases. Thus, the entries of and are chosen uniformly from a unit circle scaled by a constant , i.e.,where and follow a uniform distribution. Since this work deals with the channel estimation problem, the optimum design of the beamforming matrices is not addressed here.
3.2. Uplink Signal Model
The pilot signal (11) received at the uth MS is fed back to the BS after a multifrequency coding operation (i.e., no channel estimation is done at the MS side). More specifically, we assume that () is coded and spread in the frequency domain across K adjacent subcarriers over which the fading channel is assumed to be constant. The coded signal of the uth MS transmitted at the kth subcarrier can be expressed aswhere denotes a known code vector associated with the kth subcarrier and used by uth MS. It is worth noting that the coding vectors used by the different MSs do not need to be orthogonal. As will be discussed later, the linear independence assumption is enough. In practice, this means that these codes can be locally generated at each MS as pseudorandom sequences, i.e., no prior signaling and coordination between MSs is necessary [30].
In the UL communication, the BS employs Q beamforming vectors ( and ) to receive the coded uplink pilot signals over a set of Q different spatial directions. The received closedloop signal at the BS associated with the kth subcarrier is then given bywhere denotes an extended version of the UL channel matrix that concatenates the U UL channel matrices () of all MSs, denotes an extended matrix that concatenates the feedback signals sent by all MSs, is an extended code vector that contains the coding vectors of all MSs with respect to the kth subcarrier, and represents the filtered noise term at the output of the RF chains.
3.3. Conventional Proposed Channel Training Framework
The conventional framework for channel estimation, summarized in Figure 1, assumes channel reciprocity in TDD or treat the DL and UL channel estimation as two separated problems in FDD, i.e., solved independently at the MS and BS, respectively. For the DL channel estimation, the BS first sends pilot signals to all MSs. At the MS side, the DL channel estimation can be performed by means of the stateoftheart least squares (LS), minimum mean square error (MMSE), or CSbased estimators. Then, the estimated DL channel is reported back to the BS via dedicated UL resources [30]. To solve the UL channel estimation problem, a pilot signal is sent to the BS by each MS. Finally, the UL channels of all MSs are estimated by the BS. In practice, the DL channel estimated from UL pilots under the reciprocity assumption may not be accurate due to radio frequency distortions or a different carrier frequency such as in FDD. In addition, the conventional framework for channel estimation may imply a high computational complexity at the MS side, especially for powerlimited devices.
In the proposed framework, summarized in Figure 2, no processing for channel estimation is performed at the MSs side. In contrast to the conventional approach, the received pilot signals at each MS are fed back to the BS after a multifrequency coding operation across a set of adjacent subcarriers. After this closedloop procedure, the joint DL and UL channel estimation can be performed at the BS from the received signal given in (14). We can note that the proposed framework alleviates computational overhead due to channel estimation at the powerlimited devices, by shifting this processing burden to the BS side. Furthermore, it also relaxes channel reciprocity assumptions since the DL and UL channels can be estimated jointly from (14).
After estimating the UL and DL channels, the BS may report the estimated parameters (AoDs, AoAs, and path gains) to the MS. Then, each MS can rebuild an estimation of the DL channel (according to procedure presented in Section 4.5) before decoding the information data. This approach reduces the overhead since a number of DL channel parameters reported to the MS is much smaller with respect to the size of the DL channel matrix in an mmWave MIMO system.
3.4. Channel Model
In (11) and (14), we consider a general formulation in which the DL and UL channels are completely independent. In other words, the channels do not share any reciprocity in the angular or path gain domains. We also assume that the UL channels are constant across the K adjacent subcarriers used in the multifrequency coding operation. Due to the severe path loss, mmWave channels can be modeled by a narrowband clustered channel model with few dominant paths between the uth MS and the BS. The DL channel matrix associated with the uth MS can be written as [19]where denotes the complex path gain of the uth MS related to the lth path in the DL communication. The path gains are modeled as circular symmetric Gaussian random variables with zero mean and unit variance. and are the antenna array response vectors evaluated at the angle of arrival and angle of departure uniformly distributed in the interval . Throughout this paper, we assume uniform linear arrays (ULAs) at the BS and MSs. However, the proposed method can be applied to arbitrary array geometries without loss of generality. For ULA configurations with interantennas spacing equals to , where λ denotes the wavelength of the signal, the array response vectors at the MS and BS can be formulated as
In matrix form, can be rewritten aswhere denotes the vector that contains the path gains in the DL. The array response matrices and at the MS and BS are expressed as
The UL channel matrix from the uth MS to the BS can be represented in a similar way. We define as followswhere and are now functions of the spatial parameters in the UL, while denotes the vector that contains the path gains of the UL channel.
4. Proposed TensorBased Semiblind Receivers for Joint DL and UL Channel Estimation
Our aim is to jointly estimate the DL and UL channel matrices and of each MS () by solving a multiuser channel estimation problem at the BS. To this end, we first rewrite the received closedloop signal (14) using a tensor formalism. Then, by capitalizing on its multidimensional structure, we formulate two tensorbased semiblind receivers to initially obtain the estimates for the compressed channel matrices. As a final step, we exploit the sparse representation of the DL and UL channels to estimate their parameters by decoupling the multiuser channel estimation problem into multiple singleuser ones solved in a parallel way via separate CS problems.
4.1. PARAFAC Modeling
According to (6) and (7), the noiseless term in the received closedloop signal (14) can be interpreted as the kth frontal slice of the following thirdorder PARAFAC decomposition:obtained by concatenating the K signal matrices associated with the different adjacent subcarriers along the third mode of , i.e.,
By analogy with (7), the following correspondence holds:
The three dimensions, or modes, of stands for the number of receive beams, pilot sequence length, and number of subcarriers. The matrix can be seen as a compression matrix associated with the first mode of which reduces the size of the first mode from to Q RF chains (i.e., number of beams). The kth row of the multifrequency coding matrix contains the code values used by the U MSs at the kth subcarrier, i.e.,
Estimates of the channel parameters (AoDs, AoAs, and path gains) that build up the channel matrices and can be obtained by fitting the noisy version of to a PARAFAC decomposition. In the following, we formulate the first stage of the proposed receivers that consists of estimating the factor matrices and in an iterative or closedform way. Once the factor matrices are estimated, the second stage of the proposed receivers is to solve U independent CS problems that yield to channel parameter estimation of each MS, as will be shown later.
4.2. First Stage: Bilinear Alternating Least Squares (BALS Receiver)
According to (8)–(10), we can obtain the following representations for the unfolding matrices , , and of in terms of its factor matrices:where we denote for the simplicity of representation.
Since the multifrequency coding matrix is assumed to be known at the BS while the DL and UL channelstate information are not available, the proposed BALS receiver consists of estimating and in an alternating way from and by optimizing, respectively, the following two nonlinear LS problems:
The solutions of which are given by and , respectively.
Each iteration of the bilinear ALSPARAFAC algorithm contains only two LS updating steps. At each step, one factor matrix is updated, while the other is assumed fixed to its value obtained in the previous step [31]. This procedure is repeated until the convergence of the algorithm, denoted bythe residual error between the received signal tensor and the reconstructed signal tensor at the ith iteration. We declare that the first stage has converged at the ith iteration whenwhere σ is a threshold. In our computational simulations, we set . Convergence to the global minimum is always achieved within a few iterations due to the knowledge of the frequency spread matrix at the BS. The proposed BALS receiver is summarized in Algorithm 1.

4.3. Alternative ClosedForm Solution to the First Stage
In contrast to the BALS receiver, the second proposed receiver named LSKRF is an alternative closedform solution that can be employed in particular cases in which . The idea is to filter the received signal tensor by exploiting the knowledge of the multifrequency coding matrix and then solve a set of rank 1 approximation problems.
Initially, by multiplying both sides of in (26) by the pseudoinverse of from the righthand side, we obtain
According to property in (4), the ith column of (30) can be rewritten aswhich denotes the vectorization operation of the rank 1 matrix . By defining as the singular value decomposition (SVD) of , estimates for and can be obtained by truncating the SVD of to a rank 1 approximation, i.e., [27]where and are the corresponding first left and right singular vectors of and , respectively. denotes the largest singular value of the matrix . Final estimates for and are obtained by repeating this SVD computation times in parallel, one for each column of (30). The pseudocode of the LSKRF receiver is summarized in Algorithm 2.

4.4. Second Stage: Sparse Formulation to DL and UL Channel Parameters Estimation
Once the matrices and are estimated, the second stage of the proposed receivers consists in estimating the channel parameters (AoDs, AoAs, and path gains) to reconstruct the channel matrices and related to each MS. Thanks to the knowledge of the multifrequency coding matrix , the estimated factor matrices are not affected by permutation of columns ambiguity. Therefore, the first stage provides automatic separation of the compressed users’ channels. From and , the multiuser channel estimation problem can be decoupled into U singleuser ones as formulated below.
Let us rewrite the block representation of , defined aswherewhere denotes the estimated UL channel matrix related to the uth MS. By replacing for (19), and then vectorizing (34) according to the property in (3), we have
Using the property in (2), we straightforwardly obtainwhere denotes an identity matrix of size .
The same procedure can directly be applied in the uth block of the estimated factor matrix:where
denotes the estimated DL channel matrix defined above in (17). Similar to (36), we obtain the following vector formulation:where and .
From (36) and (39), two independent CS problems can be formulated to jointly estimate the parameters of the DL and UL channels of the uth MS. We assume that grid quantization errors are neglected, i.e., the AoDs and AoAs are drawn from a uniform angle grid of N points contained in the set , with and . Based on this assumption, we can obtain the following sparse formulations for the vectors and , respectively:where denotes the known dictionary matrix used to solve the sparse signal recovery problem, defined as
The matrices and that make the dictionary are given byand contain all points of the uniform angle grid. The lefthand side matrices and are called measurement matrices of the sparse problems. and are sparse vectors obtained by augmenting the vector gains and with zero elements, respectively.
Estimates for the parameters of the channel matrices and can be obtained by applying CS algorithms in the estimated sparse vectors (40) and (41). Many efficient algorithms such as orthogonal matching pursuit (OMP) [22], structured compressive sampling matching pursuit (SCoSaMP) [25], and fast iterative shrinkagethresholding (FISTA) [23], to name a few, can be used to solve these two sparse signal recovery problems. In a simplified view, the estimates for the path gains and correspond to nonzero entries of the estimated sparse vectors and , while estimation for the spatial parameters (AoDs and AoAs) are obtained by selecting the columns of the dictionary matrix related to the positions of the estimated path gains in the sparse vector. In our numerical results, we adopt the OMP algorithm to estimate the UL and DL channel parameters for simplicity reasons, although any stateoftheart CSbased algorithm is equally applicable to solve problems (40) and (41).
Remark:. Compared to (36), the sparse signal recovery problem formulated from (39) naturally incorporates the DL noise contribution in its structure. For this reason, the proposed closedloop framework for channel estimation can lead to some performance degradation in the DL channel estimation compared to UL channel estimation. Therefore, we can observe a tradeoff between DL channel estimation accuracy performed by the BS and reduction of the processing cost for channel estimation at the MS side. This discussion is reinforced by means of numerical simulations in Section 6.
4.5. Joint DL and UL Channel Estimation
Finally, from the estimated channel parameters (AoDs, AoAs, and path gains), the BS can construct the estimated DL and UL channel matrices and of the uth MS according to relations (17) and (19) as follows:
As previously presented in Section 4.4, the proposed receivers decouple the multiuser channel estimation problem into singleuser ones (U problems dedicated to each communication link) that can be solved independently for each MS. Since digital processing units are available at the BS, the second stage of the proposed receivers can be computed in parallel. Therefore, its processing delay can be kept constant (i.e., it does not increase with the number of MSs), when the BS is equipped with multiple (at least ) digital processing units. The parallelized processing for the second stage of the proposed receivers is illustrated in Figures 3 and 4. In addition, the overall pseudocode of the proposed twostage tensorbased receivers for joint DL and UL channel estimation is summarized in Algorithm 3.

5. Identifiability Issues
In this section, we examine the identifiability issues under which the compressed DL and UL channel matrices and can be jointly and uniquely recovered using the proposed receivers.
5.1. BALS Receiver
Unique LS solutions for the compressed DL and UL channel matrices and obtained from (24) and (25) require that and have full row rank to be right invertible. Hence, the following two conditions must be satisfied:
Combining these conditions yields the following lower bound on the number of subcarriers required for the multifrequency coding at the MS:where denotes the smallest integer number that is greater or equal to x.
5.2. LSKRF Receiver
The LSKRF receiver requires that the following necessary and sufficient uniqueness condition be satisfied:
Note that this condition indicates that the application of the LSKRF receiver requires a more restricted scenario compared to the proposed BALS receiver since the number of frequency resources (subcarriers) increases with the number of antennas at the MSs and active MSs. On the contrary, the LSKRF receiver is a closedform solution in contrast to the iterative BALS receiver.
6. Simulation Results
In this section, we present a set of simulation results to evaluate the performance of the proposed joint DL and UL channel estimator. We compare the proposed channel training framework with the conventional training framework illustrated in Figure 1, where the CSbased OMP algorithm [22] is applied at both MSs and BS to estimate the channel parameters in a decoupled way. The OMP algorithm is also considered as the second stage of our algorithm, according Section 4.4. The MSs and the BS employ uniform linear arrays with halfwavelengthspaced antennas. We set , , , , and equal signaltonoise ratio (SNR) for the DL and UL communications in all experiments. The obtained results are averaged over 1000 independent Monte Carlo runs. At each run, the DL and UL channel matrices with and paths per user and HB matrices are generated in accordance with equations (17), (19), and (12), respectively. The pilot signal is a binary phase shift keying (BPSK) modulated matrix, and the multifrequency coding matrix has random coefficients following a uniform distribution.
The receiver’s performance is evaluated in terms of the normalized mean square error (NMSE) measures between the estimated and true DL and UL channel matrices:
In our experiments, we evaluate the accuracy of channel estimation in terms of the NMSE metric for different values of signaltonoise ratio (SNR), number of transmission (P) and reception (Q) beams, number of training subcarriers (K), and length of the pilot sequences (T).
Figures 5 and 6 show the NMSE as a function of the number of transmission (P) and reception (Q) beams for different values of SNR, and fixed values , , and . According Figure 6, the proposed method outperforms the classical framework to the UL channel estimation, while the DL performance is worse in all the simulated SNR ranges as shown in Figure 5. From this experiment, we can observe the tradeoff between DL channel estimation accuracy and computational complexity. In other words, the proposed framework concentrates most of the processing burden for channel estimation at the BS side, while a better performance of the DL channel estimation comes at the expense of a high computational cost to complex channel estimation processing at the MS side when the conventional framework is utilized. On the contrary, the performance loss at DL is compensated with more accurate estimations at UL. In addition, the NMSE performance is not influenced by the number of RF chains when P and Q are greater than 12. This result reveals that the proposed framework provides a good channel estimation accuracy even when the BS is equipped with a few number of RF chains, which is the case in HB architectures.
In Figures 7 and 8, the NMSE performance is evaluated as a function of the number of subcarriers (K). An increase of K leads to an improved performance only until subcarriers in the DL channel estimation, while for the UL channel estimation this value is approximately equal to subcarriers. This result shows that the proposed closedloop channel training framework can operate with few frequency resources to jointly estimate the MS channels with high accuracy.
Figures 9 and 10 depict the NMSE performance in terms of the length of the pilot sequence (T). In this experiment, we also set , , and . Here, we conclude that short pilot sequences are necessary to estimate the DL and UL channels from the proposed method. Low variability in the NMSE values is observed when . For a massive MIMO scenario, this result implies in a substantial reduction in the pilot overhead to joint DL and UL channel estimation.
7. Conclusion and Perspectives
In this paper, we have addressed the joint DL and UL channel estimation problem for multiuser FDD massive MIMO systems with HB architecture. As contributions of this work, we firstly proposed a novel closedloop and multifrequencybased channel training framework that concentrates most of the processing burden for channel estimation at the BS side. We have shown that making use of the proposed framework, the received closedloop signal follows a thirdorder PARAFAC model, which can be exploited by two tensorbased semiblind receivers followed by compressed sensing recovery of the channel parameters. Additionally, we have also provided an identifiability study. We have compared our proposed approach with the conventional channel training framework, where the DL and UL channel estimation problems are treated as two decoupled problems, i.e., solved by the MSs and BS, separately. Compared to the conventional framework, the proposed receivers have shown a superior performance in the estimation of the UL channel, while the performance of the DL channel estimation exhibits some degradation. It is worth noting that such a degradation is the price to pay for the complexity reduction at the MS by transferring the processing burden associated with the DL channel estimation to the BS. Perspectives include the extension of the proposed modeling to frequency and timeselective channels.
Data Availability
The data used to support the findings of this study are available from the corresponding author upon request.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Acknowledgments
This work was supported by Ericsson Research, Technical Cooperation Contract UFC.47. The authors also thank the partial support of CAPES/PROBRAL (grant no. 88887.144009/201700), CNPq and FUNCAP.
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Copyright © 2019 Paulo R. B. Gomes et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.