# Model-driven Learning for Generic MIMO Downlink Beamforming With Uplink
Channel Information

Juping Zhang, Minglei You, Gan Zheng, Ioannis Krikidis, and Liqiang Zhao J.
Zhang, M. You and G. Zheng are with the Wolfson School of Mechanical,
Electrical and Manufacturing Engineering, Loughborough University,
Loughborough, LE11 3TU, UK (E-mail: {j.zhang3, m.you, g.zheng}@lboro.ac.uk).I.
Krikidis is with the Department of Electrical and Computer Engineering,
University of Cyprus, 1678 Nicosia, Cyprus (E-mail: krikidis@ucy.ac.cy).L.
Zhao is with the State Key Laboratory of Integrated Service Networks at Xidian
University, Xi’an 710071, China (E-mail: lqzhao@mail.xidian.edu.cn).

###### Abstract

Accurate downlink channel information is crucial to the beamforming design,
but it is difficult to obtain in practice. This paper investigates a deep
learning-based optimization approach of the downlink beamforming to maximize
the system sum rate, when only the uplink channel information is available.
Our main contribution is to propose a model-driven learning technique that
exploits the structure of the optimal downlink beamforming to design an
effective hybrid learning strategy with the aim to maximize the sum rate
performance. This is achieved by jointly considering the learning performance
of the downlink channel, the power and the sum rate in the training stage. The
proposed approach applies to generic cases in which the uplink channel
information is available, but its relation to the downlink channel is unknown
and does not require an explicit downlink channel estimation. We further
extend the developed technique to massive multiple-input multiple-output
scenarios and achieve a distributed learning strategy for multicell systems
without an inter-cell signalling overhead. Simulation results verify that our
proposed method provides the performance close to the state of the art
numerical algorithms with perfect downlink channel information and
significantly outperforms existing data-driven methods in terms of the sum
rate.

###### Index Terms:

Model-driven deep learning, beamforming, massive MIMO, multicell cooperation,
CSI mapping.

##  I Introduction

Beamforming is an important multi-antenna technique to deal with interference
and improve the capacity of multiuser wireless communications systems. Most
beamforming design problems are nonconvex, so early efforts to optimize
beamforming mainly refer to numerical algorithms [1][2], which cause high
complexity for practical implementation. Recently deep learning has been
recognized as a new “learn to optimize” approach to design beamforming [3, 4,
5, 6, 7]. The deep learning approach significantly reduces the optimization
complexity and the resulting beamforming solution can be potentially
implemented in real time.

The optimization and performance of beamforming critically depend on the
availability of perfect channel state information (CSI) at the transmitter.
Existing deep learning based beamforming solutions are mainly based on the
availability of perfect CSI. In the downlink, perfect CSI is difficult to
obtain at the base station (BS) for several reasons. In the time division
duplex (TDD) systems, normally channel reciprocity is assumed, so the downlink
CSI can be estimated from the pilots sent by the mobile users in the uplink.
However, in practice, the channel reciprocity does not hold because the analog
radio front-ends at the BS and the mobile users exhibit non-reciprocity due to
non-identical behavior of the individual transmit and receive chains [8]. In
the frequency division duplex (FDD) systems, for the BS to obtain the CSI, the
BS needs to first transmit pilots to the users for the downlink channel
estimation, and then users feed back the estimated downlink channel to the BS.
As a result, this channel acquisition process incurs a large overhead and
reduces the effective system spectral efficiency.

There has been significant efforts in obtaining the CSI using deep learning
techniques. CsiNet was develop in [9] to learn CSI sensing and recovery in
FDD-based massive multiple-input and multiple-output (MIMO) systems using the
channel structure. A learned denoising-based approximate message passing
neural network was proposed in [10] for beamspace channel estimation in
Millimeter-wave (mmWave) massive MIMO systems. Based on the fact that the
uplink and downlink channels share the same propagation environment, deep
neural network for channel calibration between the two directions was designed
in [11] for a generic massive MIMO system. A sparse complex-valued neural
network was introduced in [12] to approximate the uplink-to-downlink mapping
function in FDD massive MIMO systems. Convolutional neural networks and
generative adversarial networks were used in [13] to infer the downlink CSI by
observing the uplink CSI. The feasibility of channel mapping in space and
frequency was demonstrated in [14], where the channels at one set of antennas
using one frequency band can be learned from the channels at another set of
antennas that use a different frequency band. A comprehensive joint channel
estimation and feedback framework based on deep learning was proposed in [15],
which realizes the estimation, compression, and reconstruction of downlink
channels in FDD massive MIMO systems. There is an emerging direction of
studies that aims to learn the beamforming solutions rather than improve the
CSI accuracy, which is close to the main idea of this work. For instance, the
work in [16] proposed a deep learning based CSI feedback framework for
beamforming design in FDD massive MIMO systems to maximize the beamforming
performance gain. A deep neural network using unsupervised training was
proposed in [17] to map the received uplink pilots to the beamforming matrix
at the BS for the intelligent reflecting surface assuming uplink-downlink
channel reciprocity. A channel sensing and hybrid precoding design was
proposed in [18], by using the received pilots without the intermediate
channel estimation step for TDD massive MIMO systems.

While existing deep learning approaches have achieved success in individual
tasks of channel estimation and beamforming design, they normally require
massive amounts of data and computational resources, and their simple
combination will not guarantee satisfactory end results. This is due to their
inherent limitation of being data-driven and model-agnostic. The optimization
of each separate task only focuses on its own objective assuming the other is
ideal. In reality, the optimization in each task will introduce some error or
imperfection, so the overall performance could deteriorate when they are
simply putting together.

A promising direction to remedy this problem is model-driven deep learning
that combines the data-drive approach with the underlying domain knowledge,
mathematical models and problem structures, etc., to achieve a better
inference with less data. Recent advancements in model-driven deep learning
approaches in physical layer communications were discussed in [19]. In our
previous works [7], [20], we have proposed the model-driven neural network
design for beamforming optimization by exploiting the problem structure. Deep
neural networks that adopt the algorithmic structure and constraints of
adaptive signal processing techniques were proposed in [21] that can
efficiently learn to perform fast high-quality ultrasound beamforming by using
very few training data. Model-driving learning is a new concept that can be
broadly applied in engineering design, and a comprehensive review of leading
approaches for combining model-based algorithms with deep learning can be
found in [22], with detailed signal processing and communications oriented
examples.

In this paper, we aim to design a model-drive deep learning approach to
jointly tackle the challenge of channel estimation and optimize the downlink
beamforming to maximize the sum rate of generic multiuser MIMO systems.
Different from the literature, we assume only information about the uplink
channel is available, without explicit knowledge of the downlink CSI, and the
relation between the downlink and uplink channels is unknown. The uplink
channel information could take the form of either perfect CSI or received
pilots. This method will alleviate the burden of channel estimation at the
user side, reduce the feedback overhead, and it is flexible enough to be used
in both TDD and FDD systems, and in massive MIMO and multicell settings.
Especially for FDD systems, the proposed approach allows to learn the downlink
beamforming directly without the need of sending downlink pilots, uplink
feedback or explicit channel estimation. The novelty of our work is two-fold:
first, we propose to optimize the beamforming in order to maximize the end
performance and therefore bypass the explicit intermediate channel estimation
step; second, we introduce a model-driven deep learning-based approach.
Comparing to existing data-driven beamforming learning, our proposed approach
specifies the most appropriate features to be learned with improved
performance of inference and end performance. Our main contributions are
summarized as follows:

  * •

We exploit the algorithmic structure of beamforming solutions as the useful
model information, and propose a hybrid method for joint learning of the
downlink channel and optimization of beamforming to guarantee the sum rate
performance. To be specific, we design a neural network consisting of two
subnets for learning the downlink channel and the auxiliary power vector,
respectively, from which the downlink beamforming solution can be constructed.
The overall loss function is hybrid and chosen to be a weighted sum of the
loss functions of channel and power learning using supervised training, and
the sum rate using unsupervised training.

  * •

We investigate techniques to further reduce the problem dimension and achieve
near-optimal low-complexity learning, by using the zero-forcing (ZF)
beamforming in the loss function, which is specially appealing for massive
MIMO systems.

  * •

We extend the proposed method to multicell massive MIMO systems in which a BS
in each cell is able to learn the beamforming solution in a distributed manner
without signalling exchange with other cells. To the best of our knowledge,
this is the first distributed learning solution for the optimization of
multicell beamforming.

  * •

Extensive simulations are carried out to evaluate the performance the proposed
algorithms, which show that the proposed algorithm can achieve a sum rate
close to the weighted minimum mean squared error (WMMSE) algorithm [2], and
significantly outperforms existing learning methods.

The remainder of this paper is organized as follows. Section II introduces the
system model and the problem formulation. The uplink to downlink channel
mapping is discussed in Section III. The model-driven hybrid learning approach
for a general downlink is proposed in section IV. Section V presents
techniques to further reduce the training complexity for single-cell massive
MIMO systems. Section VI extends the result to allow distributed learning of
the beamforming solution in a multicell massive MIMO scenario. Simulation
results and conclusions are given in Section VII and Section VIII,
respectively.

Notions: The boldface lower case letters and capital letters are used to
represent column vectors and matrices, respectively. The notation
$\mathbf{A}^{H}$ and $\|\mathbf{A}\|$ denote the transpose conjugate and the
Frobenius-norm of a complex matrix $\mathbf{A}$, respectively. $\mathbb{C}$
denotes the complex field. The operator $\mathcal{CN}(\bf{m},\mathbf{\Theta})$
represents a complex Gaussian vector with mean $\bf{m}$ and covariance matrix
$\mathbf{\Theta}$. $\mathbf{I}_{N}$ denotes an $N\times N$ identity matrix.
$E[\cdot]$ denotes the expectation of a random variable.

##  II System Model and Problem Formulation

We start with a single-cell multi-input single-output (MISO) downlink system
where a BS with $N_{t}$ antennas serves $K$ single-antenna users. The received
signal at the user $k$ can be written as

$\displaystyle y_{k}=\mathbf{h}_{D,k}^{H}\mathbf{w}_{k}s_{k}+n_{k},$ (1)

where $\mathbf{h}_{D,k}\in\mathbb{C}^{N_{t}\times 1}$ denotes the downlink
channel vector from the BS to the user $k$, $\mathbf{w}_{k}$ and $s_{k}$
denote the transmit beamforming vector and the information signal for the user
$k$ with normalized power, respectively. $n_{k}$ is the additive Gaussian
white noise with zero mean and variance of $N_{0}$. The beamforming matrix is
${\bf W}=[{\bf w}_{1},\cdots,{\bf w}_{K}]$ and we collect the downlink CSI
into ${\bf H}_{D}=[{\bf h}_{D,1},\cdots,{\bf h}_{D,K}]$. The received SINR at
user $k$ is expressed as

$\displaystyle\gamma_{k}=\frac{|\mathbf{h}_{D,k}^{H}\mathbf{w}_{k}|^{2}}{\sum_{j=1,j\neq
k}^{K}|\mathbf{h}_{D,k}^{H}\mathbf{w}_{j}|^{2}+N_{0}}.$ (2)

The sum rate is then written as
$R^{\text{sum}}=\sum_{k=1}^{K}\log_{2}(1+\gamma_{k})$. Based on the above
model, the sum rate maximization problem under the total transmit power
constraint $P$ can be formulated as

$\displaystyle\max_{{\bf
W}}R^{\text{sum}},~{}~{}~{}~{}\mathrm{s.t.}~{}\sum_{k=1}^{K}\|\mathbf{w}_{k}\|^{2}\leq
P.$ (3)

The extended system models to massive MIMO and multicell scenarios will be
discussed in Sections V and VI, respectively, and specific techniques to
reduce the training complexity and enable distributed learning will also be
introduced.

When the downlink CSI ${\bf H}_{D}$ is available, there exist numerical
algorithms that can find the locally optimal beamforming solution of problem
(3) such as the WMMSE algorithm [2]. In our recent work [7], we have proposed
a deep learning method to solve this problem with perfect downlink CSI.
However, without downlink CSI ${\bf H}_{D}$, existing numerical or deep
learning algorithms cannot be applied. Therefore, we focus on the design of
downlink beamforming algorithms, when only the uplink channel information is
available, either in the form of perfect CSI ${\bf H}_{U}$ or the received
pilot signal.

##  III Uplink to Downlink Channel Mapping

In this paper, we rely on the uplink channel information to infer the downlink
channel and optimize the downlink beamforming, so we assume that there exists
a deterministic and unique mapping from the uplink channel to the downlink
channel but its explicit form is unknown and can be learned by a deep neural
network. This assumption is based on the fact that a wireless channel between
a transmitter-receiver pair is determined by the positions of the transmitter-
receiver pair, antennas, carrier frequency and the environment in which the
signals propagate including the objectives and their materials and shapes
within the environment. Because both uplink and downlink channels share the
same propagation environment, given the positions of the transceiver pair,
there is an intrinsic mapping between the uplink and the downlink channels.
Below we will give details for the FDD and TDD cases, respectively.

  * •

In a FDD system, consider the single-antenna uplink channel $h_{u}$ and the
downlink channel $h_{d}$ that operate at frequencies $f_{u}$ and $f_{d}$,
respectively. Assume that there are $N$ distinct propagation paths in the
environment, the uplink channel can be written as

$h_{u}=\sum_{n=1}^{N}a_{n}e^{-j2\pi f_{u}\tau_{n}+j\phi_{n}},$ (4)

where $a_{n}$ is the path attenuation, $\tau_{n}$ is the path delay and
$\phi_{n}$ is a frequency-independent phase shift that captures the reflection
and attenuation effects of the signal along the path $n$.

The path attenuation $a_{n}$ depends on the distance between the transceiver
pair, their antenna gains, the carrier frequency and the environment, the
phase shift $\phi_{n}$ depends on the scattering and the path delay $\tau_{n}$
depends on the propagation distance. Therefore when the environment and other
factors are unchanged, there is a deterministic mapping from the positions to
the channel [14]. Next we look at the mapping from the channel to the
positions. Although the mapping from the channel to the positions may not
always be unique, it is unique with a high probability in many practical
wireless communication scenarios especially as the number of antennas
increases which is widely exploited in the wireless fingerprinting [25] and
positioning [26]. In other words, the mapping between positions and channel
can be assumed bijective, so is the mapping between the uplink and downlink
channels.

  * •

In a TDD system, the channel reciprocity is usually assumed but in reality,
the analog radio front-ends at different wireless nodes such as BS and the
mobile users exhibit non-reciprocity due to non-identical behavior of the
individual transmit and receive chains. This is caused by the mismatches in
the frequency-responses of both the BS and user side radio front-ends between
the transmit and receive modes, and the differences in mutual coupling of BS
antenna units and the associated RF transceivers under transmit and receive
modes [27] [28].

Specifically, consider the channels between a BS with $N_{t}$ antennas and a
single-antenna user in linear TDD systems. The uplink channel ${\bf
h}_{u}\in\mathbb{C}^{N_{t}\times 1}$ and downlink channel ${\bf
h}_{d}\in\mathbb{C}^{N_{t}\times 1}$ can be written as [8][11],

$\underline{}{\bf h}_{u}={\bf R}_{u}{\bf h}t_{u},\mbox{~{}~{}~{}~{}~{}and
~{}~{}~{}~{}~{}}{\bf h}_{d}^{T}=r_{d}{\bf h}^{T}{\bf T}_{d},$ (5)

where ${\bf h}\in\mathbb{C}^{N_{t}\times 1}$ is the physical reciprocal
channel, $t_{u}$ and $r_{d}$ are the frequency-responses at the user side in
the transmitting and receiving modes, respectively. Denote ${\bf
L}\in\mathbb{C}^{N_{t}\times N_{t}}$ as the frequency-response matrix and
${\bf M}\in\mathbb{C}^{N_{t}\times N_{t}}$ as the mutual coupling matrix of
the BS, and then ${\bf R}_{u}={\bf L}_{r}{\bf M}_{r}$ and ${\bf T}_{d}={\bf
M}_{t}{\bf L}_{t}$ where the subscripts $t$ and $r$ denote the transmitting
and receiving modes, respectively. The frequency-response matrix ${\bf L}$ is
diagonal but the mutual coupling matrix ${\bf M}$ is not diagonal. In general,
${\bf M}_{r}\neq{\bf M}_{t},{\bf L}_{r}\neq{\bf L}_{t},t_{u}\neq r_{d}$, so
the uplink and dowlink channels are non-reciprocal, and their relation can be
described as

${\bf h}_{d}^{T}=r_{d}({\bf R}_{u}^{-1}{\bf h}_{u}t_{u}^{-1})^{T}{\bf
T}_{d}=r_{d}t_{u}^{-1}({\bf h}_{u})^{T}({\bf R}_{u}^{-1})^{T}{\bf T}_{d},$ (6)

which is a deterministic and unique mapping.

Once the bijective mapping between the uplink and downlink channel is
established, it can be learned by using the deep neural networks based on the
universal approximation theorem [29]. Note that in the above, we have adopted
explicit parametric modelling of uplink and downlink channels with simplifying
assumptions (e.g., not all hardware impairment sources such as power amplifier
distortion, phase noise and quantization noise, are considered), but in
practice such parametric models may not be accurate. In this paper we do not
assume any specific parametric channel models in our theoretical development
and instead we use the model-driven learning approach to learn the best
mapping of the uplink to downlink channel in order to maximize the end
performance.

##  IV The Proposed General Algorithm Framework

Since we consider a generic system which could be either TDD or FDD, the exact
theoretical characterization on the mapping between the uplink and downlink
channel is unknown as long as it is bijective, so the most viable way to
obtain the downlink CSI without user feedback is to learn it from data first,
based on which, the downlink beamforming will be learned or optimized
subsequently to maximize the sum rate. This is a traditional method that
treats the channel learning and the end performance optimization separately.
The main drawback of the separate learning is that the explicit channel
learning process does not take into account the ultimate objective of
maximizing the sum rate, and it also causes error propagation when optimizing
the beamforming. In this paper, we use a deep learning approach to solve the
problem (3) directly from the uplink channel information. We still include
learning the downlink channel from the uplink channel information but it is
only an intermediate step and the focus is not to achieve the best channel
learning performance. The key idea of our proposed method is to exploit the
optimal structure of the beamforming solution as the useful model information,
which then guides the design of a highly efficient neural network to solve
(3), with the assistance of the learned downlink CSI. More details are given
below.

###  IV-A Structure of beamforming

According to [23], the optimal downlink beamforming vectors that maximizes the
sum rate possesses the structure below

${\bf w}_{k}^{\ast}=\sqrt{p_{k}}\frac{({\bf
I}_{N_{t}}+\sum^{K}_{j=1}{\frac{q_{k}}{N_{0}}{\bf h}_{D,j}{\bf
h}_{D,j}^{H}})^{-1}{\bf h}_{D,k}}{\|({\bf
I}_{N_{t}}+\sum^{K}_{j=1}{\frac{q_{k}}{N_{0}}{\bf h}_{D,j}{\bf
h}_{D,j}^{H}})^{-1}{\bf h}_{D,k}\|},\forall k,$ (7)

where $p_{k}$ and $q_{k}$ are positive parameters and satisfy
$\sum_{k=1}^{K}p_{k}=\sum_{k=1}^{K}q_{k}=P$. The parameter vector ${\bf
p}=[p_{1},\ldots,p_{K}]^{T}$ represents the downlink power allocation. ${\bf
q}=[q_{1},\ldots,q_{K}]^{T}$ is an auxiliary vector variable which is useful
to determine the direction of beamforming. Because it needs to satisfy the
same total power constraint as the downlink power vector ${\bf p}$, ${\bf q}$
can be interpreted as the virtual uplink power vector. The advantage of this
representation is that the power vector $[{\bf p};{\bf q}]$ can be regarded as
the key feature of the beamforming solution. Instead of learning the high-
dimensional beamforming matrix ${\bf W}$ directly, (7) allows us to learn the
low-dimensional feature $[{\bf p};{\bf q}]$, which will greatly improve the
learning efficiency and accuracy, and reduce the training complexity.

###  IV-B Key modules of the proposed neural network

Based on the expression in (7), we propose a neural network to jointly learn
the downlink channel, the power feature vector with the end objective of
maximizing the system sum rate, as illustrated in Fig 1.

Figure 1: The proposed model-driven neural network structure.

The proposed neural network takes uplink channel information as input, and its
output is the beamforming matrix ${\bf W}$. The input to the neural network
can be either the perfect uplink CSI ${\bf H}_{U}$ with a dimension of
$2N_{t}K$ by stacking the real and imaginary parts, or the uplink pilot signal
which will be discussed later in this section. The proposed neural network
structure consists of the following three modules:

  * •

CSI-Net. This sub-net aims to learn the downlink channel ${\bf H}_{D}$ from
the uplink channel and its output is separated into the real part
$\mathcal{R}({\bf H}_{D})$ and the imaginary part $\mathcal{I}({\bf H}_{D})$.
When the input is the uplink CSI, it will perform uplink-downlink calibration
for TDD systems, while in FDD systems it will map the channel in the uplink
band to the channel in the downlink band. It is not necessary to specify the
TDD or FDD system as this module learns the downlink channel automatically for
either case. When the input is the uplink pilot signal, it additionally
estimates or refines the uplink channel but this is embedded in the module
implicitly. Suppose we have a channel training dataset in which there are $T$
uplink-downlink CSI pairs. Given the predicted output result of the $t$-th
sample in the CSI-Net is $\hat{{\bf H}}_{D}^{(t)}$ and the target result in
the training dataset is ${\bf H}_{D}^{(t)}$, the mean squared error
(MSE)-based loss function of CSI-Net is defined as

$L_{H}=\frac{1}{2TN_{t}K}\sum_{t=1}^{T}{||{\bf H}_{D}^{(t)}-\hat{{\bf
H}}_{D}^{(t)}||^{2}}.$ (8)

The structure of CSI-Net depends on the specific systems of interest, and in
this paper, we adopt fully connected layers and details will be given in
Section VII. Note that for the uplink CSI input, when the channels for
different users are uncorrelated and the statistics is similar, we can use the
same CSI-Net to learn the downlink channel in a single-user manner. This
effectively increases the amount of training data by a factor of $K$, while
reduces the complexity of the neural networks.

  * •

Power-Net. This sub-net aims to learn the concatenated power vector $[{\bf
p};{\bf q}]$, which is the key feature of the beamforming solution. In the
literature, there is no method available that can find the optimal
$p^{\ast}_{k}$ and $q^{\ast}_{k}$ in (7) to maximize the sum rate with a
reasonable complexity. The WMMSE algorithm [2] is a well known iterative
method to find the locally optimal solutions. It ensures the continuity of the
mapping from the channel to the solution, which can be learned by a neural
network. Therefore, we generate samples of the power allocation vectors ${\bf
p}$ and ${\bf q}$ for training, by using the WMMSE algorithm. The supervised
learning with the following loss function based on the MSE metric will be used
to train the Power-Net,

$L_{P}=\frac{1}{2TK}\sum_{t=1}^{T}{\left(\|{\bf p}^{(t)}-\hat{\bf
p}^{(t)}\|^{2}+\|{{\bf q}}^{(t)}-\hat{\bf q}^{(t)}\|^{2}\right)},$ (9)

where ${\bf p}^{(t)}$ and ${\bf q}^{(t)}$ are the $t$-th training samples of
the downlink and uplink power vectors in the power training database obtained
from the WMMSE algorithm, respectively, and $\hat{\bf p}^{(t)}$ and $\hat{\bf
q}^{(t)}$ are the predicted results of Power-Net. Similar to CSI-Net, the
neural network structure is system-dependent, and in this paper, we adopt
fully-connected layers or convolutional neural network (CNN) layers which will
be specified in simulation results of Section VII.

  * •

Beamforming Recovery Module. This module has two functions. First, it aims to
find the downlink beamforming matrix from the downlink channel output from the
Power-Net and the uplink and downlink power output from the Power-Net using
the structure specified in (7); there is no parameter to optimize in this
module. Second, this module is important to calculate the sum rate using
unsupervised learning which then forms the overall loss function for effective
hybrid training as described in the next subsection.

###  IV-C Hybrid training

To train the proposed neural network, we construct the overall loss function
as the weighted sum of the losses of the CSI-Net, the Power-Net and the sum
rate, i.e.,

$\mbox{Loss}=\alpha_{H}L_{H}+\alpha_{P}L_{P}+\alpha_{R}L_{R},$ (10)

where $L_{R}=-R^{\text{sum}}$, and $\alpha_{i},i\in\\{H,P,R\\}$ is the weight
for each loss component. $L_{H}$ and $L_{P}$ can be obtained by supervised
learning from the CSI-Net and the Power-Net, respectively, while
$R^{\text{sum}}$ can be calculated by using the beamforming matrix obtained
from the above Beamforming Recovery Module and the learned downlink channel
from the CSI-Net based on (2); overall, the training adopts a hybrid
supervised and unsupervised approach. Note that both channel learning and
power learning are auxiliary in our proposed algorithm, and the focus of the
overall learning is to maximize the sum rate but not to achieve the best
learning performance of the downlink channel matrices and power vectors. The
incorporation of the sum rate into the loss function is important because this
ensures that the training of the neural network is guided by the end
performance; this is in stark contrast to the separate training approach which
only focuses on learning the channel or the power vectors but cannot guarantee
the sum rate performance. In addition, the inclusion of the $L_{H}$ and
$L_{P}$ is also important because the downlink channel is unknown and it is
difficult to learn the overall mapping from the uplink channel directly to the
downlink beamforming which leads to unsatisfactory training performance.

We illustrate the advantage of the proposed algorithm over the supervised
learning ($\alpha_{R}=0$) and unsupervised learning methods
($\alpha_{H}=\alpha_{P}=0$) in Fig. 2, where the uplink CSI follows
distribution and the relation between the $i$-th elements of the uplink and
downlink CSI is $h_{d}[i]=(h_{u}[i])^{2}$. As can be seen, the proposed
algorithm outperforms the supervised learning method, and the performance gap
increases as the number of antennas grows. When $N_{t}=K=10$, the proposed
algorithm achieves about 10% higher sum rate than the supervised learning
method, while the unsupervised learning cannot achieve satisfactory sum rate
performance as explained above. The results show that in sharp contrast to the
case where perfect CSI is available, existing methods of supervised and
unsupervised learning could not achieve satisfactory performance when the CSI
needs to be learned.

Figure 2: Comparison of the sum rate performance of the proposed algorithm,
supervised learning and unsupervised learning methods versus the number of
users/antennas ($K=N_{t}$), $P=20$ dB.

###  IV-D Learning from the uplink pilot signal

In this subsection, we introduce the preprocessing when the uplink channel
information is in the form of the received pilot signal. Assume that $K$ users
transmit $L$ pilot symbols. Suppose the pilot signals sent by all users are
collected in ${\bf X}\in\mathbb{C}^{K\times L}$, then the received pilot
signal in the uplink ${\bf Y}\in\mathbb{C}^{N_{t}\times L}$ can be written as

${\bf Y}={\bf H}_{U}{\bf X}+{\bf N},$ (11)

where ${\bf N}\in\mathbb{C}^{N_{t}\times L}$ is the received noise and the
elements of ${\bf N}$ has zero mean and variance of $\sigma^{2}_{n}$. In order
to recover the uplink channel from the pilot signal, we choose the pilot ${\bf
X}$ to be a sub-matrix of scaled discrete Fourier transform (DFT) matrix of
dimension $\max(K,L)\times\max(K,L)$ (multiplied by the square root of the
transmit power), which has orthogonal columns and rows and its elements have
unit amplitude.

For our proposed algorithm, instead of using the received pilot ${\bf Y}$, we
use the least square version $\tilde{\bf Y}\in\mathbb{C}^{N_{t}\times K}$ as
the input to the neural network:

$\tilde{\bf Y}\triangleq\frac{{\bf Y}{\bf X}^{H}}{LP}=\begin{cases}{\bf
H}_{U}+\frac{{\bf N}{\bf X}^{H}}{LP}&L\geq K,\\\ {\bf H}_{U}\frac{{\bf X}{\bf
X}^{H}}{LP}+\frac{{\bf N}{\bf X}^{H}}{LP}&L<K.\end{cases}$ (12)

Clearly when $L\geq K$, ${\bf X}{\bf X}^{H}=LP{\bf I}$ which means in this
case, pilot sequences between users are orthogonal; while when $L<K$, there
exists pilot contamination among users which will degrade the performance of
estimating the uplink channel. Note that in our proposed approach, we do not
need to estimate the uplink channel explicitly, but we learn the downlink
beamforming from the uplink pilot directly.

For comparison, the traditional separate approach would first estimate the
uplink channel, e.g., by using a linear minimum mean-squared error (MMSE)
estimator, and then use it as the input of the neural network. Suppose the
linear MMSE estimator is in the form of

$\hat{\bf H}_{U}={\bf Y}{\bf R}+{\bf B},$ (13)

where ${\bf R}$ and ${\bf B}$ are the weighting coefficients. Then the
corresponding channel estimation can be obtained by solving the following
optimization problem:

$\min_{{\bf R},{\bf B}}\|{\bf Y}{\bf R}+{\bf B}-{\bf H}_{U}\|^{2}.$ (14)

The optimal ${\bf R}$ and ${\bf B}$ are given by

$\displaystyle{\bf R}^{*}$ $\displaystyle=$ $\displaystyle\left({\bf X}{\bf
Q}{\bf X}^{H}+N_{t}\sigma_{n}^{2}{\bf I}\right)^{-1}{\bf X}^{H}{\bf Q},$ (15)
$\displaystyle{\bf B}^{*}$ $\displaystyle=$ $\displaystyle-\bar{\bf
H}_{U}({\bf X}{\bf R}-{\bf I}),$ (16)

and the liner MMSE estimation of ${\bf H}_{U}$ is given by

$\hat{\bf H}_{U}={\bf Y}{\bf R}+{\bf B}=\bar{\bf H}_{U}+({\bf Y}-\bar{\bf
H}_{U}{\bf X})\left({\bf X}{\bf Q}{\bf X}^{H}+N_{t}\sigma_{n}^{2}{\bf
I}\right)^{-1}{\bf X}^{H}{\bf Q},$ (17)

where $\bar{\bf H}_{U}$ is the mean value of ${\bf H}_{U}$, i.e., ${\bf
H}_{U}=\bar{\bf H}_{U}+\Delta{\bf H}_{U}$, $\Delta{\bf H}_{U}$ has zero mean,
and ${\bf Q}\triangleq E[\Delta{\bf H}_{U}^{H}\Delta{\bf H}_{U}]$ which can be
estimated from the training data of the uplink channel. The derivation can be
found in Appendix A.

##  V Low-complexity Implementation for Massive MIMO Downlink

The proposed algorithm framework in Section IV is general so it can be applied
to massive MIMO downlink, straightforwardly. However, the large number of
antennas may introduce a high computational complexity, especially in the
calculation of the loss function during the training. In this section, we
propose two techniques that can reduce the complexity when calculating the
loss function of the sum rate in the training process for massive MIMO
systems, without compromising the end performance, especially when $N_{t}\gg
K$.

###  V-A Massive MIMO channel model

We assume that the BS is equipped with uniform linear array (ULA) antennas,
without loss of generality. The channel between the BS and a user (the user
index is omitted for simplicity) that consists of $L_{p}$ paths is modelled as

${\bf h}=\sum\limits_{l=1}^{L_{p}}\alpha_{l}e^{-j2\pi f\tau_{l}+j\phi_{l}}{\bf
a}(\theta_{l}),$ (18)

where $\alpha_{l},\tau_{l},\phi_{l}$ and $\theta_{l}$ are the attenuation, the
path delay, the phase shift and the angle of arrival (AoA, for the uplink) or
the angle of departure (AoD, for the downlink) of the $l$-th path,
respectively. Moreover, ${\bf a}(\theta)\in\mathbb{C}^{N_{t}\times 1}$ is the
array response vector defined as

${\bf
a}(\theta)=\left[{1,e^{-j\frac{2\pi}{\lambda}d\sin(\theta)},\cdots,e^{-j\frac{2\pi}{\lambda}d(N_{t}-1)\sin(\theta)}}\right]^{T},$
(19)

where $d$ is the antenna spacing and $\lambda$ is the wavelength.

From the structure of the optimal beamforming (7), we can see that it involves
inversion of a matrix of dimension $N_{t}\times N_{t}$. This causes a high
complexity for massive MIMO systems, because the calculation of the sum rate
in the loss function (10) requires the construction of the optimal beamforming
according to (7), which needs to calculate the inversion of a matrix of size
$N_{t}\times N_{t}$.

###  V-B Dimension reduction

In the first technique, we aim to reduce the dimension of the inverse matrix,
when calculating the optimal beamforming using (7). We find the following
proposition is useful which is adopted from the result in interference
channels [30] [31].

_Proposition 1:_ Suppose $N_{t}\geq K$ and the users’ channels are linearly
independent and that ${\bf h}_{i}^{H}{\bf h}_{j}\neq 0,\forall i\neq j$. Then
if ${\bf w}_{k}$ is a beamforming vector for user $k$ that corresponds to a
rate point on the Pareto boundary, there exists complex numbers
$\\{\xi_{ij}\\}^{K}_{i,j=1}$ such that

${\bf w}_{k}=\sum_{k=1}^{K}\xi_{ik}{\bf h}_{i},\sum_{k=1}^{K}\|{\bf
w}_{k}\|^{2}=P.$ (20)

It can be proved using the same method as that in [30] and therefore the proof
is omitted.

Recall that ${\bf H}_{D}=[{\bf h}_{D,1},\cdots,{\bf h}_{D,K}]$, and define
$\bm{\xi}_{k}=[\xi_{1k},\cdots,\xi_{Kk}]^{T}\in\mathbb{C}^{K\times 1}$.
Suppose ${\bf w}_{k}={\bf H}\bm{\xi}_{k}$, then the SINR expression becomes

$\gamma_{k}=\frac{|\mathbf{h}_{D,k}^{H}{\bf
H}_{D}\bm{\xi}_{k}|^{2}}{\sum_{j\neq k}^{K}|\mathbf{h}_{D,k}^{H}{\bf
H}_{D}\bm{\xi}_{j}|^{2}+N_{0}}.$ (21)

Define the eigenvalue decomposition ${\bf H}_{D}^{H}{\bf H}_{D}={\bf
U}\Lambda{\bf U}^{H}$, where ${\bf U}\in\mathbb{C}^{K\times K}$ is the unitary
eigen-matrix and $\Lambda\in\mathbb{C}^{K\times K}$ is the diagonal eigenvalue
matrix. Now define the new beamforming vector ${\bf v}_{k}=\Lambda^{1/2}{\bf
U}^{H}_{D}\bm{\xi}_{k}$ and the new channel vector ${\bf
g}_{k}=\Lambda^{-1/2}{\bf U}^{H}{\bf H}_{D}^{H}{\bf h}_{D,k}$. Then the sum
rate maximization can be written equivalently as

$\displaystyle\max_{{\bf v}_{1},\cdots,{\bf v}_{K}}$
$\displaystyle\sum_{k=1}^{K}\log_{2}(1+\gamma_{k})$ (22)
$\displaystyle\gamma_{k}$ $\displaystyle=$ $\displaystyle\frac{|{\bf
g}_{k}^{H}{\bf v}_{k}|^{2}}{\sum_{j=1,j\neq k}^{K}|{\bf g}_{k}^{H}{\bf
v}_{j}|^{2}+N_{0}},$ $\displaystyle\sum_{k=1}^{K}\|{\bf v}_{k}\|^{2}\leq P.$

We can see from the above new problem formulation (22) that the size of the
new channel matrix $\\{{\bf g}_{k}\\}$ reduces from $N_{t}\times K$ to
$K\times K$. The structure of the optimal beamforming is revised to

${\bf v}_{k}^{\ast}=\sqrt{p_{k}}\frac{({\bf
I}_{K}+\sum^{K}_{j=1}{\frac{q_{k}}{N_{0}}{\bf g}_{j}{\bf g}_{j}^{H}})^{-1}{\bf
g}_{k}}{\|({\bf I}_{K}+\sum^{K}_{j=1}{\frac{q_{k}}{N_{0}}{\bf g}_{j}{\bf
g}_{j}^{H}})^{-1}{\bf g}_{k}\|},\forall k,$ (23)

and as a result the size of the matrix inversion is reduced from $N_{t}\times
N_{t}$ to $K\times K$.

Note that although the above dimension reduction technique reduces the
dimension for matrix inversion significantly, it involves extra matrix
multiplication and eigenvalue decomposition, when constructing the new channel
vectors . It was shown in [32] that standard linear algebra operations for a
square matrix of dimension $n\times n$, including matrix inversion and
eigenvalue decomposition problems have the same time complexity as the matrix
multiplication algorithm and there exists the Coppersmith-Winograd algorithm
for matrix multiplication with a complexity of $\mathcal{O}(n^{2.376})$ [34].
Therefore, the benefit of the dimension reduction technique on the overall
training time is only obvious when $N_{t}\gg K$ and this is verified by the
simulation results in Section VII.

###  V-C ZF beamforming in the loss function

Another technique to reduce the complexity of matrix inversion in (7), when
calculating the sum rate in the loss function, is to use the ZF beamforming in
the following form

${\bf W}=d{\bf H}_{D}({\bf H}_{D}^{H}{\bf H}_{D})^{-1},$ (24)

where $d$ is chosen to satisfy the power constraint, i.e., $\|{\bf
W}\|^{2}=P$. It has been shown in the seminal work on massive MIMO [33] that,
ZF beamforming is near-optimal when the number of antennas is large. From the
complexity’s viewpoint, the advantage of the ZF beamforming is that it only
involves the matrix inversion of ${\bf H}_{D}^{H}{\bf H}_{D}$ which is a
$K\times K$ matrix and reduces the complexity of matrix inversion in (7).

Similar to the dimension reduction technique, the advantage of the ZF
beamforming is more prominent when $N_{t}\gg K$ and diminishes as $K$
increases, but its performance is always close to the WMMSE solution as long
as $N_{t}\gg 1$. These properties are verified by the simulation results in
Section VII.

##  VI Multicell Massive MIMO Downlink

In this section, we apply the proposed model-based learning to multicell
massive MIMO systems, where an example of the multicell massive MIMO system
with seven cells is illustrated in Fig. 3. Specifically, we first introduce
the distributed multicell massive MIMO downlink beamforming, without the need
of signal or data exchange between cells, and then describe the uplink channel
estimation via pilots, followed by the proposed model-based learning method.

Figure 3: An multicell massive MIMO network illustration with 7 cells.

###  VI-A Distributed Optimization of Beamforming in Multicell Massive MIMO
Downlink

Consider an $N_{c}$-cell massive MIMO system, and in each cell a BS with
$N_{t}$ antennas serves $K_{i}$ single-antenna users, $i=1,\dots,N_{c}$.
Denote the beamforming vector for the $k_{j}$-th user in the $j$-th cell as
$\mathbf{w}_{k_{j}}\in\mathbb{C}^{N_{t}\times 1}$, then the received scalar
signal $y_{k_{i}}$ in the $i$-th cell is expressed as follows:

$\displaystyle y_{k_{i}}$ $\displaystyle=$
$\displaystyle{\mathbf{h}^{H}_{D,k_{i},i}\mathbf{w}_{k_{i}}{x}_{k_{i}}}+\underbrace{\sum_{k_{j}=1,k_{j}\neq
k_{i}}^{K_{i}}\mathbf{h}^{H}_{D,k_{i},i}\mathbf{w}_{k_{j}}{x}_{k_{j}}}_{\text{Intra-
cell Interference}}$ (25) $\displaystyle+\underbrace{\sum_{j=1,j\neq
i}^{N_{c}}\sum_{k_{j}=1}^{K_{j}}\mathbf{h}^{H}_{D,k_{i},j}\mathbf{w}_{k_{j}}{x}_{k_{j}}}_{\text{Inter-
cell Interference}}+n_{k_{i}},$

where $\mathbf{h}_{D,k_{i},j}\in\mathbb{C}^{N_{t}\times 1}$ is the downlink
channel from the BS in the $j$-th cell to the $k_{i}$-th user in the $i$-th
cell, ${x}_{k_{j}}$ is the signal for the $k_{j}$-th user in the $j$-th cell.
Therefore the SINR of the $k_{i}$-th user in the $i$-th cell is written as
follows

$\gamma_{\text{SINR},k_{i}}=\frac{|\mathbf{h}^{H}_{D,k_{i},i}\mathbf{w}_{k_{i}}|^{2}}{\sum_{k_{j}=1,k_{j}\neq
k_{i}}^{K_{i}}|\mathbf{h}^{H}_{D,k_{i},i}\mathbf{w}_{k_{j}}|^{2}+\sum_{j=1,j\neq
i}^{N_{c}}\sum_{k_{j}=1}^{K_{j}}|\mathbf{h}^{H}_{D,k_{i},j}\mathbf{w}_{k_{j}}|^{2}+N_{0}}.$
(26)

Based on (26), to calculate $\gamma_{\text{SINR},k_{i}}$ for the $k_{i}$-th
user in the $i$-th cell, it requires not only the beamforming
$\mathbf{w}_{k_{i}}$ and downlink channels $\mathbf{h}_{D,k_{i},i}$, but also
the knowledge of any possible interfering cell $j$ with $j=1,\dots,N_{c}$ and
$j\neq i$. This includes interfering BSs’ downlink channels to the considered
user $k_{i}$ in the $i$-th cell $\mathbf{h}_{D,k_{i},j}$ and all users’
beamforming vectors in those interfering cells $\mathbf{w}_{k_{j}}$ with
$k_{j}=1,\dots,K_{j}$. This means to optimize the SINR in any single cell, the
solution of the beamforming vector in this cell is coupled with the solutions
in other cells. Traditional methods, such as the coordinated transmission,
treat the multicell as a large single cell and use ZF beamforming method to
cancel out the interference term in (26). These methods require the
cooperation between cells such as the exchange of CSI and centralized joint
optimization of beamforming vectors, which results in high signaling overhead
[35].

In order to decouple the beamforming between different cells, the signal to
leakage plus noise ratio (SLNR) is used in this paper as an alternative, where
the SLNR of the $k_{i}$-th user in the $i$-th cell is defined as follows [36]:

$\gamma_{\text{SLNR},k_{i}}=\frac{|\mathbf{h}^{H}_{D,k_{i},i}\mathbf{w}_{k_{i}}|^{2}}{\sum_{j=1}^{N_{c}}\sum_{k_{j}=1,k_{j}\neq
k_{i}}^{K_{j}}|\mathbf{h}^{H}_{D,k_{j},i}\mathbf{w}_{k_{i}}|^{2}+N_{0}}.$ (27)

Note that the key difference between the SLNR in (27) and the SINR in (26) is
the interference item in the denominator, where the SLNR in (27) considers the
‘leaked’ interference power due to the beamforming $\mathbf{w}_{k_{i}}$ from
the $k_{i}$-th user in the $i$-th cell to all other cells’ users, instead of
the received interference in SINR in (26). Traditionally the data rate is
defined based on SINR in (26), but this requires the cooperation between
different cells to estimate the data rate performance of the user. Since the
SLNR definition in (27) shares similarity with the SINR definition in (26),
here we define an approximate data rate based on SLNR the $k_{i}$-th user in
the $i$-th cell as follows:

$R_{\text{SLNR},k_{i}}=\log_{2}(1+\gamma_{\text{SLNR},k_{i}})\approx\log_{2}(\gamma_{\text{SLNR},k_{i}}),$
(28)

where the subscript “SLNR” is used to differentiate it from the original data
rate definition, and the approximation is based on the high SNR assumption.

With the definition of SLNR in (27), the optimization of beamforming vector
$\mathbf{w}_{k_{i}}$ for the $i$-th cell depends only on the downlink channels
from the BS in the $i$-th channel to users in all cells, which can be
estimated via the uplink estimation. More importantly, since the required
channel information can be obtained by the BS in a single cell, the
exploitation of SLNR instead of SINR decouples the beamforming in individual
cells. Therefore the multicell sum rate maximization problem can be decoupled
to the per-cell sum rate maximization problem to be addressed in each cell as

$\max_{\mathbf{w}_{k_{i}}}\sum_{k_{i}=1}^{K_{i}}R_{\text{SLNR},k_{i}},\quad\text{s.t.
}\sum_{k_{i}=1}^{K_{i}}\|\mathbf{w}_{k_{i}}\|^{2}\leq
P,i\in\\{1,\dots,N_{c}\\},$ (29)

where $P$ is the transmit power limit for each BS. To solve the problem in
(29), the beamforming vector $\mathbf{w}_{k_{i}}$ is rewritten as
$\mathbf{w}_{k_{i}}=\sqrt{p_{k_{i}}}\mathbf{u}_{k_{i}}$, where
$\mathbf{u}_{k_{i}}$ satisfies $||\mathbf{u}_{k_{i}}||=1$ for all
$k_{i}=1,\dots,K_{i}$, and represents the direction of the beamforming vector,
while the power is characterized by $p_{k_{i}}$. Then with (28), the objective
function in (29) can be further rewritten as follows:

$\displaystyle\sum_{k_{i}=1}^{K_{i}}R^{\text{SLNR}}_{k_{i}}\approx\sum_{k_{i}=1}^{K_{i}}\log_{2}\left(\gamma_{\text{SLNR},k_{i}}\right)$
(30)
$\displaystyle=\log_{2}\left(\prod_{k_{i}=1}^{K_{i}}\frac{p_{k_{i}}|\mathbf{h}^{H}_{D,k_{i},i}\mathbf{u}_{k_{i}}|^{2}}{\sum_{j=1}^{N_{c}}\sum_{k_{j}=1,k_{j}\neq
k_{i}}^{K_{j}}p_{k_{i}}|\mathbf{h}^{H}_{D,k_{j},i}\mathbf{u}_{k_{i}}|^{2}+N_{0}}\right),$

while the per-cell sum rate maximization problem in (29) can be further
rewritten as follows

$\displaystyle\min_{p_{k_{i}},\mathbf{u}_{k_{i}}}$
$\displaystyle\prod_{k_{i}=1}^{K_{i}}\left(\frac{N_{0}}{|\mathbf{h}^{H}_{D,k_{i},i}\mathbf{u}_{k_{i}}|^{2}}{p_{k_{i}}}^{-1}+\frac{\sum_{j=1}^{N_{c}}\sum_{k_{j}=1,k_{j}\neq
k_{i}}^{K_{j}}|\mathbf{h}^{H}_{D,k_{j},i}\mathbf{u}_{k_{i}}|^{2}}{|\mathbf{h}^{H}_{D,k_{i},i}\mathbf{u}_{k_{i}}|^{2}}\right)$
(31) s.t. $\displaystyle\sum_{k_{i}=1}^{K_{i}}p_{k_{i}}\leq P.$

The problem (LABEL:eq_percell_sum_rate_fin_multicell) is non-convex, so we
propose to use the alternating optimization to solve it. Specifically, with a
given power allocation $p_{k_{i}}$, the optimal beamforming direction
$\mathbf{u}_{k_{i}}$ is solved as:

$\mathbf{u}_{k_{i}}^{*}=\frac{\left(\mathbf{I}_{N_{t}}+\frac{p_{k_{i}}}{N_{0}}\sum_{j=1}^{N_{c}}\sum_{k_{j}=1}^{K_{j}}\mathbf{h}_{D,k_{j},i}\mathbf{h}^{H}_{D,k_{j},i}\right)^{-1}\mathbf{h}_{D,k_{i},i}}{||\left(\mathbf{I}_{N_{t}}+\frac{p_{k_{i}}}{N_{0}}\sum_{j=1}^{N_{c}}\sum_{k_{j}=1}^{K_{j}}\mathbf{h}_{D,k_{j},i}\mathbf{h}^{H}_{D,k_{j},i}\right)^{-1}\mathbf{h}_{D,k_{i},i}||},\forall
k_{i},$ (32)

When the beamforming directions $\mathbf{u}_{k_{i}}$ are fixed, the objective
function of (LABEL:eq_percell_sum_rate_fin_multicell) is a posynomial with
regard to $p_{k_{i}}$, so (LABEL:eq_percell_sum_rate_fin_multicell) can be
solved via geometric programming [37]. Since in each step, the objective
function is nondecreasing, such an alternating optimization algorithm will
converge. Therefore we can use the alternating optimization algorithm to
generate the downlink power solutions as the labelled data in the training
process.

When $N_{t}>\sum_{j=1}^{N_{c}}K_{j}$, the dimension reduction techniques in
Section V.B can be also applied to the multicell scenario to reduce the
training complexity.

Note that the optimal beamforming based on SLNR involves only the downlink
channels from the BS in a single cell to the users in all cells. This enables
the distributed beamforming at each single cell and no cooperation is required
between cells, which helps to reduce the signaling overhead in the multicell
scenario. More importantly, this enables the proposed learning-based method to
be decoupled in the multicell scenario and applicable when the system scales
up.

###  VI-B Learning From Uplink Pilots

The optimization based on SLNR requires the channel from the BS in each cell
to learn the downlink channels to users from the uplink channel information,
which can be achieved by estimation from uplink pilots.

Consider the $i$-th cell surrounded by $N_{c}-1$ neighbouring cells. Assume
that each user in the multicell system transmits pilot symbols of length $L$,
then for the BS in the $i$-th cell, the received pilot signal in the uplink
$\mathbf{Y}_{i}\in\mathbb{C}^{N_{t}\times L}$ is written as

$\mathbf{Y}_{i}=\mathbf{H}_{U,i}\mathbf{X}+\mathbf{N}_{i},$ (33)

where $\mathbf{H}_{U,i}\in\mathbb{C}^{N_{t}\times\sum_{j=1}^{N_{c}}K_{j}}$
denotes the uplink channels from all users to the BS in the $i$-th cell,
$\mathbf{N}_{i}\in\mathbb{C}^{N_{t}\times L}$ is the received noise whose
elements has zero mean and variance of $\sigma_{i}^{2}$, and
$\mathbf{X}\in\mathbb{C}^{\sum_{j=1}^{N_{c}}K_{j}\times L}$ is the pilots sent
by all users. Similar to the single cell scenario, the pilot $\mathbf{X}$
adopts the sub-matrix of a discrete DFT matrix with dimension
$\max\\{\sum_{j=1}^{N_{c}}K_{j},L\\}\times\max\\{\sum_{j=1}^{N_{c}}K_{j},L\\}$,
which is scaled by the square root of the transmit power. Then the least
square version of the uplink channel
$\mathbf{\tilde{Y}}_{i}\in\mathbb{C}^{N_{t}\times\sum_{j=1}^{N_{c}}K_{j}}$ is
used as the input of the neural network, and can be estimated via the received
pilot signal $\mathbf{Y}_{i}$ as follows

$\mathbf{\tilde{Y}}_{i}=\frac{\mathbf{Y}_{i}\mathbf{X}^{H}}{LP}=\left\\{\begin{matrix}\mathbf{H}_{U,i}+\frac{\mathbf{N}_{i}\mathbf{X}^{H}}{{LP}},L\geq\sum_{j=1}^{N_{c}}K_{j},\\\
\mathbf{H}_{U,i}\frac{\mathbf{X}\mathbf{X}^{H}}{LP}+\frac{\mathbf{N}_{i}\mathbf{X}^{H}}{{LP}},L<\sum_{j=1}^{N_{c}}K_{j}.\end{matrix}\right.$
(34)

Clearly to avoid the pilot contamination between users, it requires the
pilot’s length $L$ to be no less than the total number of users, i.e.,
$L\geq\sum_{j=1}^{N_{c}}K_{j}$.

For the traditional separate approach, the uplink channel will be estimated
using the linear MMSE method.

###  VI-C The Proposed Distributed Learning

In this subsection, we will adapt the proposed model-based learning approach
in Section IV to the multicell scenario. We will use a different SLNR
beamforming structure below based on (32), i.e.,

$\mathbf{w}_{k_{i}}^{*}=\sqrt{p_{k_{i}}}\frac{\left(\mathbf{I}_{N_{t}}+\frac{p_{k_{i}}}{N_{0}}\sum_{j=1}^{N_{c}}\sum_{k_{j}=1}^{K_{j}}\mathbf{h}_{D,k_{j},i}\mathbf{h}^{H}_{D,k_{j},i}\right)^{-1}\mathbf{h}_{D,k_{i},i}}{||\left(\mathbf{I}_{N_{t}}+\frac{p_{k_{i}}}{N_{0}}\sum_{j=1}^{N_{c}}\sum_{k_{j}=1}^{K_{j}}\mathbf{h}_{D,k_{j},i}\mathbf{h}^{H}_{D,k_{j},i}\right)^{-1}\mathbf{h}_{D,k_{i},i}||},\forall
k_{i}.$ (35)

From (35), we can see that in order to construct the multicell beamforming, we
need the downlink channel information and downlink power allocation.
Therefore, we can still use the hybrid loss function (10) which is rewritten
below

$\mbox{Loss}=\alpha_{H}L_{H}+\alpha_{P}L_{P}+\alpha_{R}L_{R},$ (36)

except the loss of Power-Net only involves the downlink power, i.e., for the
$i$-th cell, it becomes

$L_{P}=\frac{1}{2TK_{i}}\sum_{t=1}^{T}\sum_{i=1}^{K_{i}}{\left(\|{p_{k_{i}}}^{(t)}-\hat{p}_{k_{i}}^{(t)}\|^{2}\right)}.$
(37)

Note that the proposed method based on SLNR and learning from uplink pilots
can be generalized to larger systems. The above analysis is based on the
$i$-th cell, which treats all other cells as interfering cells. With a
homogeneous assumption that the conditions are similar in each cell in the
whole multicell massive MIMO system, the analysis of $i$-th cell is applicable
to any cell in the system. This means the trained neural network’s parameters
are applicable to all cells in a distributed manner, which is demonstrated in
the simulation results in Section VII.C.

##  VII Simulation Results

In this section, numerical simulations are carried out to evaluate the
performance of the proposed algorithms. $10^{5}$ training samples and $10^{3}$
testing samples are used with a batchsize of 100 and 200 epochs. We use Keras
with Tensorflow 1.15 as the backend to train the proposed neural network on a
Nvidia GP100 GPU card in a High Performance Computing (HPC) system. The weight
factors used in the loss function are hyper-parameters and chosen as
$\alpha_{H}=\alpha_{P}=1,\alpha_{R}=0.001$, by using a trial and error
approach, unless otherwise specified. The much lower value of $\alpha_{R}$ is
to balance each component in the loss function considering typical values of
the sum rate.

For the proposed model-driven solution, when calculating the sum rate in the
loss function, both the original algorithm that uses the beamforming solution
in (7), and the one that uses the simplified ZF beamforming in (24) are
included. For comparison, we consider the following benchmark algorithms:

  * •

The WMMSE solution: this is the solution obtained by using the iterative
algorithm proposed in [2] assuming the downlink channel is available,
therefore it serves as a performance upper bound.

  * •

The Learned Channel and Beamforming Solution: this solution is obtained by
first learning the downlink channel via supervised learning, then using the
learned downlink channel to infer the beamforming solution using unsupervised
training.

  * •

The Learned Channel and ZF Beamforming: this solution is used as the benchmark
solution in the single-cell scenario. It first learns the downlink channel via
supervised learning as the above solution, and then constructs a ZF
beamforming (24) by using the learned channel.

  * •

The Learned Channel and SLNR Beamforming: this solution is used as the
benchmark solution in the multi-cell scenario. It first learns the downlink
channel via supervised learning as the above two solutions, and then
constructs a SLNR beamforming (32) by using the learned channel. Note that the
SLNR based approximate data rate is used only for the purpose of the neural
network training, while the sum rate performance is calculated based on the
actual SINR defined in (26).

Note that the closed-form non-iterative ZF and SLNR based beamforming
solutions are chosen as the benchmarks to ensure the low complexity comparable
to our proposed algorithm. In the following, we will present the simulation
results and analysis for three scenarios: single-cell small-scale MIMO,
single-cell massive MIMO, and multicell massive MIMO systems, as well as the
generalization results.

###  VII-A Small-scale MIMO Scenario

In this scenario, a TDD downlink system in which one BS with equal numbers of
transmit antennas and users is considered, i.e., $N_{t}=K\ngg 1$. We assume
the uplink channel elements follow an independent Rayleigh distribution with
zero mean and unit variance. Following the result in [11], the relation
between the downlink channel and the uplink channel due to the radio front-end
mismatch is modelled as:

${\bf h}_{D}=c\Phi{\bf h}_{U},$ (38)

where the unitary matrix $\Phi$ and $c\sim\mathcal{CN}(0,1)$ are used to model
the mismatches in the frequency-responses of the BS and the user sides,
respectively. This mapping will be learned by the CSI-Net. Suppose the learned
downlink channel is $\hat{\bf H}_{D}$. The learning performance is
characterized by the normalized MSE (NMSE), which is defined as

$\mbox{NMSE}=E\left[\frac{\|\hat{\bf H}_{D}-{\bf H}_{D}\|^{2}}{\|{\bf
H}_{D}\|^{2}}\right].$ (39)

The structure and hyper-parameters of the proposed neural network are as
follows. For the CSI-Net, we use four fully connected layers, each with
$4KN_{t}$ neurons and the ‘tanh’ activation function. No activation function
is employed at the output layer. The Power-Net employs four fully connected
layers, each with $4KN_{t}$ neurons and the ‘relu’ activation function and
batch normalization. The output layer of the Power-Net uses ‘softmax’ as the
activation function. Beamforming learning for the ‘Learned Channel and
Beamforming’ solution uses the same neural network as CSI-Net, except that a
batch normalization is included at each layer. $10^{6}$ training samples are
used for this scenario. We also provide brief analysis of the complexity for
the online prediction. For fully connected layers, suppose the input dimension
is $M$ and the number of neurons in the hidden layer is $N$, then the numbers
of multiplication and addition operations are equal to $MN$. Therefore the
overall neural network has an approximate complexity
$\mathcal{O}\left(N_{t}^{2}K^{2}\right)$ for the online prediction.

We first show the sum rate results of various algorithms in Fig. 4, when the
transmit power is 20 dB and the number of users/BS antennas vary from 2 to 8.
We assume that the perfect uplink channel CSI ${\bf H}_{U}$ is available. It
can be seen that the proposed solution achieves the sum rate close to that of
the WMMSE algorithm , and it significantly outperforms the benchmark schemes.
The performance of both benchmark methods that first learn the downlink
channel is not satisfactory. The scheme of learned channel and beamforming,
achieves the worst performance while the one using a ZF beamforming achieves a
higher data rate but still much lower than the proposed solutions. To further
investigate the reason, we plot the channel learning results of downlink
channel learning in Fig. 4. It is obvious that the benchmark schemes that
explicitly learns the channel first outperforms the proposed solution in terms
of achieving a lower estimation NMSE. However, its end performance, i.e., the
sum rate is much lower than the proposed solution. This confirms the advantage
of the proposed model-driven learning with a hybrid training that can better
exploit the available uplink channel information to optimize the end
performance; the traditional approach that separately learns the channel and
then designs the beamforming is not adequate since it is purely data-driven.

Figure 4: (a) Comparison of the sum rate versus the number of users/antennas,
$P=20$ dB; (b) Comparison of the channel estimation performance versus the
number of users.

###  VII-B Single-cell Massive MIMO Scenario

In this scenario, we adopt the channel model specified in Section V for a FDD
massive MIMO system and assume $N_{t}=64$. The uplink and downlink operate at
2.5 GHz and 2.4 GHz, respectively. The antenna spacing is half wavelength of
the downlink signal. Since the DoA or the AoD $\theta$ is limited within a
certain region of the mean angle $\bar{\theta}$, it is modelled as
$\theta\in[\bar{\theta}-\Delta\theta/2,\bar{\theta}+\Delta\theta/2],-\frac{\pi}{6}\leq\Delta\theta\leq\frac{\pi}{6}$.
The path delay $\tau$ and the phase shift $\phi$ are uniformly distributed in
the ranges of $[0,10^{-4}]$ and $[-\pi,-\pi]$, respectively. We assume the
downlink channel attenuation $\alpha_{d,l}$ of the $l$-th path follows an
independent Rayleigh distribution with zero mean and unit variance. Given the
ULA channel model in (18), the nonlinear relation between the uplink channel
${\bf h}_{D}$ and dowlink channel ${\bf h}_{D}$ are characterized by:

$\displaystyle{\bf h}_{U}$ $\displaystyle=$
$\displaystyle\sum\limits_{l=1}^{L_{p}}\alpha_{u,l}e^{-j2\pi
f\tau_{l}+j\phi_{l}}{\bf a}_{u}(\theta_{l}),~{}~{}~{}~{}$ $\displaystyle{\bf
h}_{D}$ $\displaystyle=$
$\displaystyle\sum\limits_{l=1}^{L_{p}}c\Phi(\alpha_{u,l})^{2}e^{-j2\pi
f\tau_{l}+j\phi_{l}}{\bf a}_{d}(\theta_{l}),$ (40)

where the unitary matrix $\Phi$ and $c\sim\mathcal{CN}(0,1)$ are used to model
the mismatches in the frequency-responses of the BS and the user sides,
respectively, which will be learned by the CSI-Net.

The structure and the hyper-parameters of the proposed neural network are as
follows. The CSI-Net contains three fully connected layers, each with
$2N_{t}K$ neurons each and the ‘tanh’ activation function. The Power-Net uses
two 1-D CNN with filter sizes of 16 and 8, batch normalization and ‘relu’
activation function and a dropout rate of 0.3, followed by a fully connected
layer with 256 neurons and ‘relu’ activation function, and an output layer
that uses ‘softmax’ as the activation function. We provide brief complexity
analysis for the online prediction. For $L$ 1-D convolutional layers, suppose
there are $c_{l}$ kernels of size $a_{l}$ in the $l$-th convolutional layer
and the input dimension is $b\times d_{l}$ (padding is added such that the
first input and output dimensions remain the same across layers, i.e., $b$),
then the output dimension of is $b\times c_{l}$, and the numbers of
multiplication and addition operations are equal to $a_{l}bc_{l-1}c_{l}$ when
$l>1$ and $a_{l}bd_{l}c_{l}$ when $l=1$. Thus, the total complexity of all
convolutional layers measured by the number of multiplications is
$\mathcal{O}\left(\sum_{l=1}^{L}a_{l}bc_{l-1}c_{l}\right)$. Therefore for the
Power-Net, the complexity for the online prediction is approximately
$\mathcal{O}\left(N_{t}K\right)$, while the overall complexity is still
$\mathcal{O}\left(N_{t}^{2}K^{2}\right)$ considering the CSI-Net.

The sum rate results of various algorithms are shown in Fig. 5, when the
transmit power is 10 dB and the number of users vary from 2 to 10. It can be
seen that the proposed solution achieves the sum rate very close to the WMMSE
solution and the use of a ZF beamforming in the loss function has almost no
performance loss. The proposed solutions outperform the learned channel and
the ZF beamforming solution, by about 10%, although the latter performs much
better than the small scale MIMO scenario. The learned channel and beamforming
solution is still the worse because it uses a data-driven approach, without
taking into account the overall design objective when learning the downlink
channel.

Next, we evaluate the required training time of the proposed algorithms when
using the low-complexity implementation discussed in Section V. The training
time of the proposed original algorithm $T_{0}$ and the percentage of the time
required by the reduced dimension and the ZF beamforming techniques in
relation to $T_{0}$ are shown in Table I at the top of the next page. It is
obvious that when the number of users $K$ is small, the reduced dimension in
matrix inversion of the two low-complexity schemes leads to a much shorter
training time. However, as the number of users $K$ increases, the gain of the
low-complexity schemes diminishes. This is because for the reduced dimension
scheme, it involves an extra eigenvalue decomposition and matrix
multiplications. In addition, as the number of users $K$ increases, the
training time is dominated by the width of the fully connected layers which is
$2N_{t}K$, therefore the time saved by matrix inversion becomes insignificant.

The sum rate performance versus the number of pilots is shown in Fig. 5, when
the number of users is $K=6$. Similar to Fig. 5, it is confirmed again that
the use of ZF beamforming in the loss function achieves almost the same
performance as the original algorithm and both are close to the WMMSE solution
when the number of pilot symbols $L\geq 6$. The learned channel and ZF
beamforming achieves good performance in this scenario, which is different
from the small scale scenario. There is still significant gap between the
proposed solutions and the learned channel and beamforming solution and this
verifies the superior performance of the proposed model-driven approach.

Figure 5: Comparison of the sum rate versus the number of users, $N_{t}=32$, $P=10$ dB, versus (a) the number of users and (b) the number of pilots in the single-cell massive MIMO scenario. Proposed Algorithms No. of Users | $K=2$ | $K=4$ | $K=6$ | $K=8$ | $K=10$
---|---|---|---|---|---
Original formulation, $T_{0}$ (s) | 1570 | 2314 | 3388 | 4716 | 6683
Reduced dimension (%) | 67.24% | 80.98% | 102.86% | 95.68% | 99.98%
ZF Loss (%) | 60.11% | 73.69% | 84.02% | 87.78% | 93.13%
TABLE I: Training time of the proposed algorithms for massive MIMO systems,
$N_{t}=64$.

###  VII-C Multicell Massive MIMO Scenario

In the multicell massive MIMO scenario, we consider a TDD system in which the
uplink and the downlink operate at different frequencies of 2.5 GHz and 2.4
GHz, respectively. In the simulations, a multicell system with $N_{c}=7$ cells
is considered, where each cell has one active user as illustrated in Fig. 3.
The total transmit power of the BS is 10 dBm, the bandwidth is 20 MHz and the
noise power spectrum density is -174 dBm/Hz. The radius of each cell is 200 m,
and the users are randomly located in each cell by following a uniform
distribution. The minimum distance between each user and its serving BS is 10
m.

The channel attenuation includes both the small and large scale fading
effects. The relation between small scale channel attentions is the same as
(VII-B) in the single-cell scenario. The large scale fading
$\alpha_{\text{PL}}^{(dB)}$ (measured in dB, and the uplink/downlink subscript
is omitted) between the BS and the user is given as follows

$\alpha_{\text{PL}}^{(dB)}=p_{0}+\beta\log_{10}(d),$ (41)

where $p_{0}$ is the reference path loss gain measured in dB, which includes
the effect of the central frequency $f_{c}$, $\beta$ is related to the path
loss exponent, and $d$ is the distance between the BS and the user measured in
kilometer. In the simulation, we assume that the uplink and downlink path loss
gains are expressed as, respectively,

$\alpha_{\text{PL,u}}^{(dB)}=127+30\log_{10}(d),\mbox{~{}and~{}}\alpha_{\text{PL,d}}^{(dB)}=128.1+37.6\log_{10}(d).$
(42)

The structure and the hyper-parameters of the proposed neural network are as
follows. Three fully connected layers are used in _CSI-Net_ and each has
$2N_{t}N_{c}^{2}$ neurons each and use the ‘tanh’ activation function. The
weight factors used in the loss function are $\alpha_{H}=1,\alpha_{R}=-1$.
During the training procedure, this _CSI-Net_ is reused for each cell to learn
the downlink channels based on their uplink channels. Since the constructed
_CSI-Net_ and the proposed method are for a single cell which do not rely on
signalling exchange with other cells, they can be deployed in a distributed
way at each BS once the training is completed. The complexity for the online
prediction is approximately $\mathcal{O}\left(N_{t}^{2}N_{c}^{2}\right)$.

In the multicell massive MIMO scenarios, the number of transmit antennas is
one of the key factors that determines the users’ end performance, so we first
evaluate the sum rate performance of all users versus different numbers of
antennas in Fig. 6. As seen from Fig. 6, the proposed solution achieves a
tight sum rate performance compared to the WMMSE solution. The proposed
solution outperforms all benchmark algorithms, while the performance
comparison of each benchmark algorithm follows a similar trend as in the
single-cell scenario. It is also noticed that the proposed solution based on
the ZF loss function shows a close performance to the proposed solution based
on the SLNR in the loss function, and the performance gap is generally
reducing as the number of antennas increases. It is also clear that the data-
driven approaches are worse than the model-driven approaches, while the
learned channel and beamforming solution is still the worst.

Figure 6: Comparison of the sum rate in the multicell scenario versus (a) the
number of BS antennas, $P=10$ dBm and (b) the number of pilots in the
multicell scenario, $N_{t}=64$, $P=10$ dBm.

Next we investigate the sum rate performance versus different numbers of pilot
symbols $L$ in Fig. 6, when $N_{t}=64$ and the uplink channels are estimated
via the pilots. It can be seen that the performance of all algorithms improves
as the number of pilots increases, but there is a significant gap from the
WMMSE solution when $L<7$. When $L\geq 7$, the proposed solutions with SLNR
and ZF loss functions achieve close performance to the WMMSE solution and
significantly outperform other benchmark schemes. This observation is expected
as there are a total of seven users in the considered scenario. When the
number of pilots is less than seven, there will be users with non-orthogonal
pilots, which results in a deteriorated accuracy of the recovered uplink
channel from the pilots.

###  VII-D Generalization

In this subsection, we study the generalization performance of the proposed
joint training method in the multi-cell massive MIMO scenario when $N_{t}=32$.
The model to be tested is trained with a total transmit power $P=10$ dBm for
the originally considered 7-cell scenario and no Doppler effect is considered.
First, we evaluate the performance of the trained model under different levels
of transmit power in Fig. 7. It is seen that the inference of this model is
still capable of achieving a tight performance close to the WMMSE solution as
the transmit power is near 10 dBm, but the performance gap increases as the
testing power deviation is large. This shows the proposed method can
generalize to scenarios where there is a small variation of the testing power.

Figure 7: The sum rate performance of the trained model under the
generalization evaluation with (a) different total transmit power and (b)
numbers of cells.

Secondly, we consider the scenario where the system has different number of
cells and the results are presented in Fig. 7. It is seen that both proposed
solutions achieve the sum rate performance close to the WMMSE solution under
different cell numbers, which confirms that the proposed distributed learning
solution generalizes well as the number of cells increases. Finally, we
consider the user mobility characterized by the Doppler effect in Fig. 8. Note
that in practical systems, the Doppler effect may be estimated and compensated
at the receiver side [38]. Here we assumed the Doppler effect is not
compensated so that it will influence the estimated CSI and the sum rate
performance. We consider the maximum Doppler frequency to be in the range of
10 to 150 Hz, which corresponds to a speed range from 1 to 20 m/s. Fig. 8
shows the inference results of the trained model evaluated by considering
different maximum Doppler frequencies. It is seen that the sum rate
performance of the proposed solutions shows moderate degradation compared to
the WMMSE solution as the Doppler frequency increases, but is still
significantly higher than the benchmark solutions.

Figure 8: The sum rate performance of the trained model under the
generalization evaluation with different maximum Doppler frequencies.

##  VIII Conclusions

In this paper, we have proposed a new downlink beamforming optimization
algorithm to maximize the sum rate using deep learning when only the uplink
channel information is available, but its mapping to the downlink channel is
unknown. We introduced a model-driven learning approach by exploiting the
structure of the optimal beamforming solution to facilitate an effective
neural network design. The proposed approach was extended to massive MIMO and
distributed multicell scenarios. Simulation results demonstrated that our
proposed algorithm can approach the performance of the conventional WMMSE
algorithm, and achieves a much higher sum rate than the benchmark schemes.
These results show the importance of model-driven and holistic learning
approaches to optimize downlink beamforming in practical systems.

## Appendix A

From (14), the MSE of channel estimation can be expressed as

$\displaystyle\mbox{MSE}_{H}$ $\displaystyle=$ $\displaystyle\|{\bf Y}{\bf
R}+{\bf B}-{\bf H}_{U}\|^{2}$ (43) $\displaystyle=$ $\displaystyle\|({\bf
H}_{U}{\bf X}+{\bf N}){\bf R}+{\bf B}-{\bf H}_{U}\|^{2}$ $\displaystyle=$
$\displaystyle\|{\bf H}_{U}{\bf X}{\bf R}+{\bf N}{\bf R}+{\bf B}-{\bf
H}_{U}\|^{2}$ $\displaystyle=$ $\displaystyle\|{\bf H}_{U}({\bf X}{\bf R}-{\bf
I})+{\bf N}{\bf R}+{\bf B}\|^{2}.$

Because ${\bf H}_{U}=\bar{\bf H}_{U}+\Delta{\bf H}_{U}$, where $\Delta{\bf
H}_{U}$ has a zero mean and ${\bf Q}=E[\Delta{\bf H}^{H}\Delta{\bf H}]$. Then
(43) can be further written as

MSE $\displaystyle=$ $\displaystyle\|\Delta{\bf H}({\bf X}{\bf R}-{\bf
I})+{\bf N}{\bf R}+{\bf B}+\bar{\bf H}({\bf X}{\bf R}-{\bf I})\|^{2}.$ (44)

Apparently the optimal ${\bf B}$ should satisfy

${\bf B}^{*}=-\bar{\bf H}_{U}({\bf X}{\bf R}-{\bf I}).$ (45)

Then

MSE $\displaystyle=$ $\displaystyle\mathrm{E}\|\Delta{\bf H}_{U}({\bf X}{\bf
R}-{\bf I})+{\bf N}{\bf R}\|^{2}$ (46) $\displaystyle=$
$\displaystyle\mbox{trace}[\Delta{\bf H}_{U}({\bf X}{\bf R}-{\bf I})({\bf
X}{\bf R}-{\bf I})^{H})\Delta{\bf H}_{U}^{H}]+\mbox{trace}({\bf R}^{H}{\bf
N}^{H}{\bf N}{\bf R})$ $\displaystyle=$ $\displaystyle\mbox{trace}[{\bf
R}^{H}({\bf X}{\bf Q}{\bf X}^{H}+N_{t}\sigma_{n}^{2}{\bf I}){\bf X}-{\bf
X}{\bf R}{\bf Q}-{\bf Q}{\bf R}^{H}{\bf X}^{H}+{\bf Q}].$

By setting the first order derivative of $\eqref{mse3}$ to be zero, the
optimal ${\bf R}$ is given by

${\bf R}^{*}=\left({\bf X}{\bf Q}{\bf X}^{H}+N_{t}\sigma_{n}^{2}{\bf
I}\right)^{-1}{\bf X}^{H}{\bf Q}.$ (47)

The optimal ${\bf B}^{*}$ and the linear MMSE estimation $\hat{\bf H}_{U}$ can
be obtained by substituting (47) into (45) and (13), respectively.

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