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In this paper, we address the problem of joint downlink (DL) and uplink (UL) channel estimation for millimeter wave (mmWave) multiple-input multiple-output (MIMO) systems. Assuming a closed-loop and multifrequency-based 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 tensor-based semiblind receivers by capitalizing on the multilinear structure and sparse feature of the received signal at the BS equipped with a hybrid analog-digital 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 closed-form 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.

In recent years, millimeter wave (mmWave) massive multiple-input multiple-output (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 [

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 [

The researchers [

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 closed-loop and multifrequency-based 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 closed-loop signal at the BS can be modeled as a three-way array (i.e., a third-order tensor) that follows a PARAFAC model. By capitalizing both on the multidimensional and sparse structures of the received signal, we propose two tensor-based semiblind receivers for joint DL and UL channel estimation. The first receiver is an iterative solution based on the alternating least squares (ALS) algorithm [

In summary, the main contributions of this paper can be listed as follows:

We propose a novel closed-loop and multifrequency-based 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.

We show that, by making use of the proposed framework for channel estimation, the received closed-loop signal can be modeled as a third-order tensor that follows a PARAFAC model. Then, we formulate two tensor-based semiblind receivers (iterative and closed-form ones) for joint DL and UL channel estimation by capitalizing on a tensor structure of the received closed-loop signal.

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

Scalars, column vectors, matrices, and tensors are denoted by nonbold lowercase letters

We shall make use of the following two properties of the Kronecker and Khatri-Rao products:

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.

Throughout this paper, the definitions and operations involving tensors are in accordance with [

By definition, the PARAlell FACtor (PARAFAC) analysis decomposition of a third-order tensor

The PARAFAC decomposition can also be represented in terms of the frontal slices of _{.}

By using

The 1 mode, 2 mode, and 3 mode unfolding matrices of

In this section, we introduce the proposed closed-loop 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.

Consider a wireless communication system operating in the FDD mode, where a BS equipped with

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

The pilot signal (

In the UL communication, the BS employs

The conventional framework for channel estimation, summarized in Figure

Conventional training framework. The DL and UL channel estimation problems are solved independently. The BS first transmits pilot sequences. Then, the DL channel is estimated at the MS side. The estimated DL channel is fed back to the BS via dedicated uplink resources. The UL channel is estimated at the BS side. The text in blue refer to DL communication, while the text in red refer to UL communication.

In the proposed framework, summarized in Figure

Proposed closed-loop and multifrequency-based training framework. The DL and UL channels are jointly estimated. The BS first transmits pilot sequences. The MSs encode the received pilots and then feed them back to the BS. The BS jointly estimates the DL and UL channels. The text in blue refer to DL communication, while the text in red refer to UL communication.

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

In (

In matrix form,

The UL channel matrix

Our aim is to jointly estimate the DL and UL channel matrices

According to (

By analogy with (

The three dimensions, or modes, of

Estimates of the channel parameters (AoDs, AoAs, and path gains) that build up the channel matrices

According to (

Since the multifrequency coding matrix

The solutions of which are given by

Each iteration of the bilinear ALS-PARAFAC 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 [

Set

Initialize randomly the factor matrix

From

From

Repeat steps 2–4 until convergence. The convergence is achieved when

In contrast to the B-ALS receiver, the second proposed receiver named LS-KRF is an alternative closed-form solution that can be employed in particular cases in which

Initially, by multiplying both sides of

According to property in (

Apply the

Compute the SVD

where

Repeat steps 1-2 for all columns of (

Once the matrices

Let us rewrite the block representation of

Using the property in (

The same procedure can directly be applied in the

From (

The matrices

Estimates for the parameters of the channel matrices

Compared to (

Finally, from the estimated channel parameters (AoDs, AoAs, and path gains), the BS can construct the estimated DL and UL channel matrices

As previously presented in Section

Illustration of the parallelized processing for the estimation of U uplink channels. The

Illustration of the parallelized processing for the estimation of the U downlink channels. The

From the received signal tensor

From

The BS constructs the estimated DL and UL channel matrices

In this section, we examine the identifiability issues under which the compressed DL and UL channel matrices

Unique LS solutions for the compressed DL and UL channel matrices

Combining these conditions yields the following lower bound on the number of subcarriers required for the multifrequency coding at the MS:

The LS-KRF receiver requires that the following necessary and sufficient uniqueness condition be satisfied:

Note that this condition indicates that the application of the LS-KRF receiver requires a more restricted scenario compared to the proposed B-ALS receiver since the number of frequency resources (subcarriers) increases with the number of antennas at the MSs and active MSs. On the contrary, the LS-KRF receiver is a closed-form solution in contrast to the iterative B-ALS receiver.

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

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 signal-to-noise ratio (SNR), number of transmission (

Figures

NMSE vs. number of transmission beams P for the DL channel estimation:

NMSE vs. number of reception beams

In Figures

NMSE vs. number of subcarriers K for the DL channel estimation:

NMSE vs. number of subcarriers K for the UL channel estimation:

Figures

NMSE vs. length of the training sequence

NMSE vs. length of the training sequence

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 closed-loop and multifrequency-based 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 closed-loop signal follows a third-order PARAFAC model, which can be exploited by two tensor-based 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 time-selective channels.

The data used to support the findings of this study are available from the corresponding author upon request.

The authors declare that they have no conflicts of interest.

This work was supported by Ericsson Research, Technical Cooperation Contract UFC.47. The authors also thank the partial support of CAPES/PROBRAL (grant no. 88887.144009/2017-00), CNPq and FUNCAP.