# Deep-Learning Aided Channel Training and Precoding in FDD Massive MIMO with
Channel Statistics Knowledge

Yi Song, Tianyu Yang, Mahdi Barzegar Khalilsarai, and Giuseppe Caire
Technische Universität Berlin, Berlin, 10623, Germany.
Emails: $\\{$yi.song, tianyu.yang, m.barzegarkhalilsarai, caire$\\}$@tu-
berlin.de

###### Abstract

We propose a method for channel training and precoding in FDD massive MIMO
based on deep neural networks (DNNs), exploiting Downlink (DL) channel
covariance knowledge. The DNN is optimized to maximize the DL multi-user sum-
rate, by producing a pre-beamforming matrix based on user channel covariances
that maps the original channel vectors to “effective channels”. Measurements
of these effective channels are received at the users via common pilot
transmission and sent back to the base station (BS) through analog feedback
without further processing. The BS estimates the effective channels from
received feedback and constructs a linear precoder by concatenating the
optimized pre-beamforming matrix with a zero-forcing precoder over the
effective channels. We show that the proposed method yields significantly
higher sum-rates than the state-of-the-art DNN-based channel training and
precoding scheme, especially in scenarios with small pilot and feedback size
relative to the channel coherence block length. Unlike many works in the
literature, our proposition does not involve deployment of a DNN at the user
side, which typically comes at a high computational cost and parameter-
transmission overhead on the system, and is therefore considerably more
practical.

###### Index Terms:

FDD massive MIMO, channel statistics knowledge, analog feedback, DNN-based
training and precoding.

##  I Introduction

Deep Neural Networks (DNNs) have been recently successfully applied in various
areas of wireless communications such as resource allocation and scheduling
[1, 2], channel estimation [3, 4], beamforming [5], transceiver design [6],
etc. Given sufficient data, a DNN is trained in a (semi-)supervised or
unsupervised fashion to learn mappings from an input space to some desired
output that optimizes a suitable utility metric that is otherwise very hard to
optimize with conventional tools.

In this paper, we propose a DNN-based solution to the problem of channel
training and multi-user precoding in a frequency division duplex (FDD) massive
MIMO system with channel statistics knowledge at the Base Station (BS). It is
well-known that, to achieve the benefits of massive MIMO, the transmitter
needs to obtain fresh Downlink (DL) channel state information (CSI). Unlike
time division duplexing (TDD) systems, where relying on channel reciprocity,
DL channels are directly estimated from Uplink (UL) pilots, in FDD the BS must
train the CSI by broadcasting pilot sequences in DL and receiving user
feedback. This process requires careful design of DL pilots, user feedback
messages, and the precoder based on feedback. In particular, a small pilot
length (in DL) and feedback size (in UL) relative to the channel dimension,
results in poor DL spectral efficiency. This is caused by the large channel
estimation error and the resulting interference due to precoding over
erroneous channels. Incorporating knowledge of channel statistics at the
transmitter in designing the pilots and the precoder can significantly
mitigate this effect. We propose a scheme in which a DNN is trained with the
given constraints on pilot and feedback size (fixed by the standard) to
produce a pre-beamforming matrix as a function of user channel statistics.
Both the pilot vectors (a set of $T_{\rm dl}$ row-vectors in
${\mathbb{C}}^{M}$) and precoding vectors (a set of $K$ row-vectors in
${\mathbb{C}}^{M}$) are chosen from the row-space of this pre-beamforming
matrix. As will be apparent by the signal model in the next sections, the pre-
beamforming matrix maps original channels to “effective channels”, which will
be estimated (through DL training and UL feedback) and over which zero-forcing
(ZF) will be performed. Intuitively, since an “accurate” estimation of the
original channels with limited pilot and feedback resources is infeasible, the
DNN-based transform is employed to manage interference by precoding over
certain effective channels with possibly smaller number of known coefficients
and therefore can be trained with the given pilot/feedback budget. Other
elements of our proposed network include the following. Upon receiving pilots,
the users send them back to the BS after a power normalization via analog
feedback [7], i.e., feeding back complex-valued measurements by modulating
them as quadrature and in-phase components of the baseband signal. The BS then
computes a minimum mean-squared error (MMSE) estimate of the effective
channels. The precoder is then generated as a product of a ZF precoder on the
effective channels and the pre-beamforming matrix and is used to send data to
users in DL. The DNN is optimized end-to-end to yield a pre-beamforming matrix
based on input channel statistics, that maximizes the multi-user sum-rate.

Recently many works have utilized DNNs for channel training and precoding in
massive MIMO. Some have proposed extrapolation of DL channels from UL channels
using DNNs [8, 9, 10]. Albeit highly successful under certain scenarios, these
methods would fail when the channel coherence bandwidth is small relative to
the separation between UL and DL carrier frequencies and explicit DL training
and feedback is necessary. In other works, pilot design and channel estimation
with DNNs is considered [4, 11]. The objective in these works is to minimize
the channel estimation MSE, which is different from maximizing the multi-user
sum-rate as considered in our work. Another category of works focuses on
compression of feedback, where perfect channel state knowledge at the users is
assumed [12, 13, 14]. This assumption is hard to achieve in massive MIMO,
since the channel dimension is typically larger than the pilot length and
channel estimation is carried out via a compressed sensing scheme which not
only may fail depending on the channel sparsity order, but is also
computationally costly and difficult to implement in real time in the user
devices. Finally, [15] proposed a highly successful DNN-based scheme for pilot
sequence design, feedback quantization and DL precoding. This scheme however
involves deployment of the feedback computation layers at the user side, which
requires transfer of a large (in the order of a million) number of parameters
to the users, incurring a huge overhead in DL.

Our proposed method offers the following advantages with respect to the
existing works in the literature.

  1. 1.

Exploiting Channel Statistics: Our proposed DNN utilizes channel statistics to
design DL pilots and the precoder. This results in significantly higher DL
sum-rate, particularly in scenarios where channel training is difficult due to
small pilot and feedback dimensions. Note that, although availability of
channel statistics knowledge at the BS is not always granted, it is justified
by the fact that in FDD systems, DL channel covariances can be estimated from
UL pilots based on what is known as “angle reciprocity” [16, 17, 18].
Therefore, it is reasonable to devise DNN-based solutions that exploit channel
covariance knowledge.

  2. 2.

No DNN at the User Side: Unlike many works in the literature [12, 13, 14, 15],
our proposed method does not involve training a DNN at the user side or
transmission of optimized DNN parameters to it. Users simply send back pilot
measurements to the BS with a power normalization, from which the BS estimates
effective channels.

  3. 3.

Direct Sum-Rate Maximization: The idea of training and precoding in FDD
massive MIMO by pre-conditioning channels with a transform based on statistics
was proposed by some of the authors of the present work in [19]. There, the
transform was optimized to maximize the spatial multiplexing gain, which is
equivalent to the rate pre-log factor in high SNR. In contrast, in the present
work we optimize the transform to directly maximize ergodic sum-rate in a
data-driven fashion and through the DNN. Our simulation results will show that
this approach significantly improves upon [19].

We will show via simulations that our method achieves better performance in
terms of DL sum-rate compared to the state-of-art result in [15] as well as
[19], especially when the pilot and feedback dimensions are small compared to
the channel dimension, proving the applicability of this scheme in FDD massive
MIMO systems.

##  II System model

###  II-A Common Training

We consider a massive MIMO system in FDD mode, where a BS equipped with a
uniform linear array (ULA) of $M$ antennas servers $K$ users with a single
antenna in a cell. Because channel reciprocity does not hold in FDD, the BS
has to train DL channels by broadcasting pilot sequences of length $\beta$
from each of its $M$ antenna ports. We denote these pilot sequences as rows of
a pilot matrix ${\bf X}^{\rm p}\in{\mathbb{C}}^{\beta\times M}$ (the
superscript “p” stands for “pilot”). The pilot signal received at user $k$ is
expressed as

${\bf y}_{k}^{\rm p}={\bf X}^{\rm p}{\bf h}_{k}+{\bf z}_{k}^{\rm
p},\;k\in[K],$ (1)

where ${\bf h}_{k}\sim\mathcal{CN}(\mathbf{0},{\bf C}_{k})$ is the Rayleigh
fading channel vector of user $k$ with covariance ${\bf
C}_{k}={\mathbb{E}}[{\bf h}_{k}{\bf h}_{k}^{{\sf H}}]$, ${\bf z}_{k}^{\rm
p}\sim{{\cal C}{\cal N}}(\mathbf{0},\mathbf{I}_{M})$ is additive white
Gaussian noise (AWGN) with unit variance per element, and for an integer $a$
we define $[a]\triangleq\\{1,2,\dots,a\\}$. Assuming the BS has a total
transmission power of $P_{\rm dl}$, the pilot matrix should satisfy the power
constraint

$\|[{\bf X}^{\rm p}]_{i,\cdot}\|^{2}\leq P_{\rm dl},\;\forall i\in[\beta].$
(2)

where $[{\bf X}^{\rm p}]_{i,\cdot}$ denotes the $i$-th row of ${\bf X}^{\rm
p}$. Also, since the noise variance is normalized to one, we define the SNR in
DL as $\text{SNR}_{\rm dl}=P_{\rm dl}$. For future reference, we define the
effective channel of user $k$ by

${\bf g}_{k}\triangleq{\bf B}{\bf h}_{k},\;\forall k\in[K],$ (3)

where ${\bf B}\in{\mathbb{C}}^{M\times M}$ is the pre-beamforming matrix,
mapping the original channel to the effective channel.

We propose to design ${\bf X}^{\rm p}$ as the product

${\bf X}^{\rm p}={\bf W}{\bf B},$ (4)

where ${\bf W}\in{\mathbb{C}}^{\beta\times M}$ is a an arbitrary full-rank
matrix. While we do not impose any constraints on ${\bf W}$ other than being
full-rank, ${\bf B}$ will be produced by a trained DNN with user channel
covariances as input. This will be explained in Section III. With this
construction, the DL pilot signal in (1) can be equivalently written as ${\bf
y}_{k}^{\rm p}={\bf W}{\bf g}_{k}+{\bf z}_{k}^{\rm p}$, so that received pilot
symbols can be equivalently seen as noisy linear measurements of the effective
channel.

###  II-B Analog Feedback

After receiving pilot signals, each user sends a feedback “message” to the BS
using the UL channel. A common approach known as digital feedback consists of
estimating the channel at the receiver from the pilots, quantizing it and
sending the quantization index to the BS [20]. Alternatively, users can encode
the pilot signal into quantization codewords without explicit channel
estimation. A different approach, known as analog feedback consists of sending
complex-valued feedback symbols to the BS by modulating the quadrature and in-
phase components of the carrier by real and imaginary parts of the feedback
symbol [7]. Analog feedback is simpler and imposes less feedback delay than
digital feedback which requires quantization and channel coding. The feedback
symbols can be estimates of the channel or the received pilot signal itself.
In our proposition, the user sends the power-normalized pilot symbols directly
and without channel estimation to the BS via analog feedback. The feedback
message of user $k$ in this case is given by

${\bf x}^{\rm fb}_{k}=\sqrt{\rho_{k}}{\bf y}^{\rm
p}_{k},~{}~{}\text{with}~{}\rho_{k}=\beta P_{\rm ul}/\|{\bf y}^{\rm
p}_{k}\|^{2}.$ (5)

which satisfies the average power constraint

$\|{\bf x}^{\rm fb}_{k}\|^{2}\leq\beta P_{\rm ul},\;\forall k\in[K],$ (6)

where $P_{\rm ul}$ is the user average transmit power per symbol in UL,
assumed equal among all users. The elements of ${\bf x}^{\rm fb}_{k}$ are sent
back to the BS via analog feedback. This means that, just like the analog QAM
modulation, real and imaginary parts of each complex-valued symbol in the
feedback message modulate carriers that have a 90 degrees phase difference.
These carriers are then combined and the resulting signal is sent to the BS.
To avoid any confusion, we emphasize that the feedback symbols are not
quantized and the user does not use a digital QAM modulation here. Also, note
that there is no factual problem with transmitting unquantized feedback
symbols: even in the prevalent OFDM signaling with digital QAM, the continuous
time-domain I and Q signal after the IFFT is transmitted effectively
unquantized (quantized with 10-12 bits per sample). Besides, we model the UL
channel as an AWGN channel which is orthogonally accessed by the users. Then,
the BS receives the noisy feedback signal as

$\displaystyle{\bf y}^{\rm fb}_{k}={\bf x}^{\rm fb}_{k}+{\bf z}^{\rm fb}_{k},$
(7)

where ${\bf z}^{\rm fb}_{k}\sim\mathcal{CN}(\mathbf{0},\mathbf{I}_{\beta})$ is
the noise vector.

###### Remark 1

The UL channel can be generally modeled as a multiple-access channel (MAC),
but we consider the special case of the AWGN channel with orthogonal access
for simplicity and defer the treatment of more general models to a future
work. Note that most previous works do not discuss the feedback channel model
at all and assume availability of perfect, error-free feedback at the BS [13,
21, 15]. $\lozenge$

###### Remark 2

By adopting the proposed analog feedback strategy, there is no need for
complex processing at the user side. This is in contrast to schemes that
involve deploying a DNN at the user side (see e.g., [15, 4, 14]) that have two
disadvantages: First, the forward pass of a DNN involves consecutive matrix
multiplications and applying activation functions which consume time. Second,
and more importantly, if the DNN is trained at the BS side, its optimized
parameters should be transferred to the user as soon as it enters the cell.
Given that the number of parameters in a DNN can be in the order of millions,
this imposes a large overhead on DL resources. Our scheme avoids both of these
disadvantages by using analog feedback. $\lozenge$

###  II-C Effective Channel Estimation and Precoding

Given the feedback signal, the BS computes an MMSE estimate of effective
channels as

$\widehat{{\bf g}}_{k}={\mathbb{E}}[{\bf g}_{k}|{\bf y}_{k}^{\rm fb}]={\bf
C}_{gy,k}{\bf C}_{yy,k}^{-1}{\bf y}^{\rm fb}_{k},~{}k\in[K],$ (8)

where

$\displaystyle{\bf C}_{gy,k}$ $\displaystyle=\mathbb{E}[{\bf g}_{k}({\bf
y}^{\rm fb}_{k})^{{\sf H}}]=\sqrt{\rho_{k}}{\bf B}{\bf C}_{k}({\bf X}^{\rm
p})^{{\sf H}},$ (9) $\displaystyle{\bf C}_{yy,k}$
$\displaystyle=\mathbb{E}[{\bf y}^{\rm fb}_{k}({\bf y}^{\rm fb}_{k})^{{\sf
H}}]=\rho_{k}{\bf X}^{\rm p}{\bf B}{\bf C}_{k}{\bf B}^{{\sf H}}({\bf X}^{\rm
p})^{{\sf H}}+(1+\rho_{k})\mathbf{I}_{\beta}.$ (10)

Next, the BS transmits data in DL using a linear precoder as follows. Let
${\bf s}=[s_{1},\ldots,s_{K}]$ denote a row vector consisting of the user data
symbols, each satisfying ${\mathbb{E}}[|s_{k}|^{2}]=1$. The precoded data
vector is then given by ${\bf x}^{\rm d}={\bf s}{\bf
V}\in\,{\mathbb{C}}^{1\times M}$, where ${\bf V}$ is a linear precoding matrix
and the superscript “d” stands for “data”. Similar to the design of the pilot
matrix in (4), we propose a construction of the precoder as the product

${\bf V}=\widetilde{{\bf V}}{\bf B},$ (11)

where ${\bf B}$ is the pre-beamforming matrix to be designed and
$\widetilde{{\bf V}}$ is a zero-forcing precoder on the estimated effective
channels. Denoting the estimated effective channels by $\widehat{{\bf
G}}=\left[\widehat{{\bf g}}_{1},\dots,\widehat{{\bf g}}_{K}\right]$, this
precoder is given by

$\widetilde{{\bf V}}=\sqrt{\alpha}\left(\widehat{{\bf G}}^{{\sf
H}}\widehat{{\bf G}}\right)^{-1}\widehat{{\bf G}}^{{\sf
H}}\,\in\,{\mathbb{C}}^{K\times M},$ (12)

where each row represents the precoding vector of a user and $\alpha>0$ is a
scalar that forces the precoder to satisfy

${\rm Tr}({\bf V}{\bf V}^{{\sf H}})\leq P_{\rm dl}.$ (13)

The point of decomposing the precoder in (11) is for it to first map the
original channel to the effective channel through ${\bf B}$ and then apply
zero-forcing on the effective channels. The received data symbol at user $k$
is given as

$\displaystyle y^{\rm d}_{k}$ $\displaystyle={\bf x}^{\rm d}{\bf h}_{k}+z^{\rm
d}_{k}$ (14) $\displaystyle={\bf v}_{k}{\bf
h}_{k}s_{k}+\sum^{K}_{k^{\prime}\neq k}{\bf v}_{k^{\prime}}{\bf
h}_{k}s_{k^{\prime}}+z_{k}^{\rm d},$ (15)

where ${\bf v}_{k}$ is the $k$-th row of ${\bf V}$ and $z_{k}^{\rm
d}\sim{{\cal C}{\cal N}}(0,1)$ is the AWGN. Treating interference as noise and
assuming signal and interference coefficients knowledge at the receiver, the
achievable ergodic sum-rate in DL is given by [22]

$\displaystyle R_{\rm sum}$
$\displaystyle=\sum_{k=1}^{K}{\mathbb{E}}\left[\log_{2}\left(1+\frac{|{\bf
v}_{k}{\bf h}_{k}|^{2}}{1+\sum_{k^{\prime}\neq k}|{\bf v}_{k^{\prime}}{\bf
h}_{k}|^{2}}\right)\right],$ (16)

where the expectation is taken over channel and noise distributions. Note that
the terms $\\{|{\bf v}_{k^{\prime}}{\bf
h}_{k}|^{2}~{}:~{}k,k^{\prime}\in[K],\,k\neq k^{\prime}\\}$ are interference
coefficients between channels of users $k$ and $k^{\prime}$. The precoder
${\bf V}$ is a function of the pre-beamforming matrix ${\bf B}$ through the
pilot matrix ${\bf X}^{\rm p}$ in (4), the channel estimates in (8) and the
resulting precoder in (11). Our goal is to design ${\bf B}$ based on available
DL channel covariances at the BS, such that the ergodic sum-rate is maximized,
i.e., we want to find a mapping ${\cal P}_{{\bf B}}(\cdot)$ from the set of
$K$ user channel covariances to the pre-beamforming matrix ${\bf B}$ that
maximizes ergodic sum-rate. We further denote the mapping described by Eqs.
(8)-(13) from user feedback signals denoted by ${\bf Y}^{\rm fb}=\left[{\bf
y}_{1}^{\rm fb},\ldots,{\bf y}_{K}^{\rm fb}\right]$ to the precoder by $f_{\rm
pc}\left(\,\cdot\,;{\bf B}\right)$, so that ${\bf V}=f_{\rm pc}\left({\bf
Y}^{\rm fb};{\bf B}\right)$. Now, the sum-rate maximization problem can be
posed as:

$\displaystyle\underset{{\cal P}_{{\bf B}}(\cdot)}{\text{maximize}}\quad$
$\displaystyle R_{\rm sum}$ (17a) subject to $\displaystyle{\bf B}={\cal
P}_{{\bf B}}\left(\left\\{{\bf C}_{k}\right\\}_{k=1}^{K}\right),$ (17b)
$\displaystyle{\bf V}=f_{\rm pc}({\bf Y}^{\rm fb};{\bf B}),$ (17c)
$\displaystyle\|[{\bf W}{\bf B}]_{i,\cdot}\|^{2}\leq P_{\rm dl},\;\forall
i\in[\beta],$ (17d) $\displaystyle\eqref{eq:power_fb},\eqref{eq:power_V}.$
(17e)

##  III Pre-Beamforming Based on Channel Statistics

In massive MIMO, it is typical to have a pilot length that is small relative
to the channel dimension ($\beta<M$). This results, from (1) in an
underdetermined system of noisy linear equations from which the effective
channel ${\bf g}_{k}={\bf B}{\bf h}_{k}$ must be estimated. Because the system
is underdetermined, the channel estimation error can be high even with the
MMSE estimator. Given the effective channel covariance ${\mathbb{E}}[{\bf
g}_{k}{\bf g}_{k}^{{\sf H}}]={\bf B}{\bf C}_{k}{\bf B}^{{\sf H}}$, it is shown
in [19] that if $\beta<\text{rank}\left({\bf B}{\bf C}_{k}{\bf B}^{{\sf
H}}\right)$, then the effective channel estimation MSE scales as $O(1)$ when
$\text{SNR}_{\rm dl}\to\infty$. This means that a small pilot dimension leads
to a constant channel estimation error which is independent of SNR. In
addition, when the channel estimation error is large, naive zero-forcing
results in large interference coefficients between the users in the
denominator of (16) and reduces the ergodic sum-rate. Thus, the pre-
beamforming matrix should be designed such that the rank of the effective
channel covariance ${\bf B}{\bf C}_{k}{\bf B}^{{\sf H}}$ becomes smaller to
reduce the estimation error with a given pilot dimension. On the other hand,
the effective rank should not reduce too much because then the signal
coefficient $|{\bf v}_{k}{\bf h}_{k}|^{2}$ in the numerator of (16) reduce,
resulting in smaller ergodic sum-rate. In the extreme case, if ${\bf
B}=\mathbf{0}$ then the sum-rate will be zero. These two effects imply that
the pre-beamforming matrix ${\bf B}$ should be a transformation that reduces
the inherent dimension (i.e., channel covariance rank) of effective channels
down to a certain value to achieve a favourable trade-off in minimizing
interference and maximizing signal coefficients.

We simplify the design problem by exploiting properties of the channel
covariance. It is known that the covariance of a ULA channel with large $M$ is
(approximately) diagonalized by the DFT matrix, thanks to the similarity of
large Toeplitz matrices to their Circulant equivalents and the famous Szegö’s
theorem [23, 24]. In other words, the channel covariance can be approximately
decomposed as

${\bf C}_{k}\approx{\bf F}\text{diag}(\hbox{\boldmath$\gamma$}_{k}){\bf
F}^{{\sf H}},$ (18)

where $\hbox{\boldmath$\gamma$}_{k}\in{\mathbb{R}}_{+}^{M}$ is the vector of
channel covariance eigenvalues of user $k$ and ${\bf F}\in\mathbb{C}^{M\times
M}$ is the DFT matrix whose $(m,n)$-th entry is given by $[{\bf
F}]_{m,n}=\frac{1}{\sqrt{M}}e^{-j2\pi\frac{mn}{M}},m,n\in[M]$. We simplify
design of ${\bf B}$ by restricting it to belong to the set

$\mathcal{B}\triangleq\\{{\rm diag}(\hbox{\boldmath$\lambda$}){\bf F}^{{\sf
H}}\;:\;\hbox{\boldmath$\lambda$}\in[0,1]^{M}\\}.$ (19)

Then the covariance of the effective channel is given by

${\bf B}{\bf C}_{k}{\bf B}^{{\sf
H}}\approx\text{diag}\left(\hbox{\boldmath$\lambda$}^{2}\odot\hbox{\boldmath$\gamma$}_{k}\right)$
(20)

where $\hbox{\boldmath$\lambda$}^{2}$ denotes element-wise square of $\lambda$
and $\odot$ denotes element-wise product. Essentially with this design choice,
the “effective rank” of the covariance is equivalent to the number of large
coefficients in $\lambda$ and therefore this vector controls the inherent
dimension of effective channels. From a different perspective, $\lambda$ can
be seen as “beam-selection” vector, since the DFT columns are equivalent to
the array steering vectors of a ULA evaluated on a grid of angle-of-departures
(AoDs). If the $m$-th coordinate of $\lambda$ ($\lambda_{m}\in[0,1]$) is small
($\lambda_{m}\to 0$), then the contribution of the $m$-th beam in the
effective channel of all users will be eliminated. In this sense, the present
work is similar to the active channel sparsification (ACS) method in [19]
which proposed beam-selection with the objective of maximizing the
multiplexing gain. However, the DNN-based method proposed here aims to
directly maximize the sum-rate and in this sense extends the idea presented in
[19].

###  III-A DNN-Based Optimization

We employ a DNN to produce $\lambda$ vector based on input channel covariances
$\\{{\bf C}_{k}\\}_{k=1}^{K}$. Based on the $K$ covariances, the given pilot
dimension and SNR, the network is trained to output a $\lambda$ that maximizes
the sum-rate. Note that the channel covariance of a ULA is a Toeplitz
Hermitian matrix that is fully determined by its first column. Denoting the
first columns of the $K$ covariances by ${\bf c}_{1},\ldots,{\bf c}_{K}$, we
define the matrix $\hbox{\boldmath$\Sigma$}=\left[{\bf c}_{1},\ldots,{\bf
c}_{K}\right]\in{\mathbb{C}}^{M\times K}$. Then the optimization problem (17)
can be reformulated as

$\displaystyle\underset{{\hbox{\boldmath$\Theta$}}}{\text{maximize}}\;$
$\displaystyle R_{\rm sum}$ (21a) subject to
$\displaystyle\hbox{\boldmath$\lambda$}={\cal
P}_{\hbox{\boldmath$\lambda$}}\left(\hbox{\boldmath$\Sigma$};\hbox{\boldmath$\Theta$}\right),$
(21b) $\displaystyle{\bf B}={\rm diag}(\hbox{\boldmath$\lambda$}){\bf F}^{{\sf
H}},$ (21c)
$\displaystyle\eqref{eq:f_pc},\eqref{eq:power_WB},\eqref{eq:power_fb},\eqref{eq:power_V},$
(21d)

where ${\cal
P}_{\hbox{\boldmath$\lambda$}}(\cdot;\hbox{\boldmath$\Theta$}):{\mathbb{C}}^{M\times
K}\to{\mathbb{C}}^{M}$ is the mapping from the covariance first columns to the
beam-selection vector associated with the DNN with parameters $\Theta$. The
proposed architecture is illustrated in Fig. 1.

We solve (21) in a data-driven fashion to optimize network parameters by
generating random realizations of $\Sigma$ according to a distribution ${\cal
D}$. This distribution is typically based on geometric properties of the
scattering environment, such as the number of paths, the distribution of AoDs
and their associated powers. In practice, random realizations of this
distribution can be collected at the BS at different times for $K$ randomly
located users in the cell. In our simulation results, we consider random
samples of ${\cal D}$ to be generated from a multipath scattering model that
is independent across users and is parametrized by the number of paths,
uniformly distributed AoDs and powers. Then, each random sample of ${\cal D}$
is given as input to the DNN. The expected value in the objective function
(21a) is replaced by an empirical mean obtained by generating many independent
samples of the DL channel for each user.

###### Remark 3

The main difference between our design and the DNN-based scheme in [15] is
that we learn a mapping between $\Sigma$ and ${\bf B}$, exploiting the channel
second order statistics for different users. In contrast, [15] proposes to
learn a pilot matrix that should “fit” all the user channel statistics from a
large ensemble, and not the specific statistics of the $K$ users that are
scheduled to be served in a single frame. $\lozenge$

###  III-B DNN Implementation Details

Our DNN consists of three fully-connected layers, where the number of hidden
neurons per layer are $[\ell_{1},\ell_{2},\ell_{3}]=[1024,512,M]$. We use ReLU
activation functions in all hidden layers. In order to produce $\lambda$ in
$[0,1]^{M}$, we use $tanh$ activation in the output layer and scale its output
to $[0,1]$ as $0.5(\tanh(\cdot)+1)$. We implement the network in PyTorch [25]
with the Adam optimizer [26] with a batch size of $1024$ and initial learning
rate of $10^{-4}$. For fast convergence, a batch normalization layer is added
before each linear layer [27].

Figure 1: System schematic for DNN aided FDD multi-user training and precoding
with channel statistics knowledge. The proposed system takes DL covariance
matrices $\Sigma$ as input and train with DL channels $\\{{\bf h}_{k}\\}$ to
output the precoder ${\bf V}$.

##  IV Numerical Results

For the simulations, we consider $M=64$ antennas, $K=6$ users, $\beta=8$
pilots. We stress the point that in general, estimating a set of $6$,
$64$-dimensional channels from a DL pilot of length $8$ is extremely difficult
and a successful performance in such a setup should be noticed. We set the DL
SNR to $P_{\rm dl}=20$ and consider a scattering channel model with $L$ paths.
The DL channel covariance of user $k$ is given by

${\bf C}_{k}=\sum_{\ell=1}^{L}\eta_{k,\ell}{\bf a}(\theta_{k,\ell}){\bf
a}^{{\sf H}}(\theta_{k,\ell}),\;\forall k\in[K],$ (22)

where $\eta_{k,\ell}$ and $\theta_{k,\ell}$ are the power and the AoD of the
$\ell$-th channel path of user $k$, and where ${\bf
a}(\theta)\in{\mathbb{C}}^{M}$ is the steering vector of a ULA, whose $m$-th
entry is given by $[{\bf a}(\theta)]_{m}=e^{j\frac{2\pi
d}{\lambda^{\prime}}(m-1)\sin(\theta)}$, $m\in[M]$ where $d$ is the antenna
spacing and $\lambda^{\prime}$ is the carrier wavelength. We assume the
maximum array angular aperture to be given by $\theta_{\rm max}=60^{\circ}$
and assume that the antenna spacing is set to
$d=\frac{\lambda^{\prime}}{2\sin\theta_{\rm max}}$. The user AoDs are
generated independently from a uniform distribution, i.e.,
$\theta_{k,\ell}\sim\mathcal{U}(-\theta_{\rm max},\theta_{\rm max})$. The path
powers are randomly and uniformly generated in the real interval $[0.4,0.8]$
and then scaled to sum to one, i.e., $\sum^{L}_{\ell=1}\eta_{k,\ell}=1,\forall
k\in[K]$. Choosing powers as such is not necessary, and is simply to avoid
path powers close to zero. We recall that the input of the DNN is the matrix
$\Sigma$ containing the first covariance columns of all users as its columns.
When generated according to the distribution of AoDs and powers as above,
$\Sigma$ follows a distribution ${\cal D}(L)$, parameterized by the number of
paths $L$. This specific characterization is just used here to perform the
simulations. In general, one can choose any family of distributions to
generate $\Sigma$ and train the DNN accordingly. In the upcoming simulations,
we provide results for two important scenarios: (a) sparse scattering with
$L=2$ paths per user, and (b) rich scattering with $L=20$ paths. Note that the
number of paths is equivalent to the channel covariance rank. Given that the
pilot length is $\beta=8$, training channels with a large $L$ is more
difficult than those with a small $L$. For each case, we generate the training
and testing data with randomly generated samples of $\Sigma$ according to
${\cal D}(L)$. The training data is per epoch randomly generated with a fixed
series of random seeds. The testing data contains $1000$ randomly samples of
$\Sigma$, and for each random sample of covariance, we generate $10$ random
instantaneous channel samples as well as DL and UL additive noise vectors. The
same testing data is used to produce results for all the baseline methods.

###  IV-A Comparison Baselines

We compare our proposed scheme with the state-of-the-art DNN-based design in
[15] that is under digital feedback and without channel statistic
knowledge.111We have trained the DNN proposed in [15] according to their
public code. Our code can be found in https://github.com/YiSongTUBerlin/DL-
Aided-Channel-Training-and-Precoding-in-FDD-Massive-MIMO-with-Channel-
Statistics-Knowledge.git The number of feedback symbols in analog feedback is
$\beta$. Considering a UL channel capacity of $C_{\rm ul}=\log_{2}(1+P_{\rm
ul})$ bits per channel use, this translates to $B=\beta C_{\rm ul}$ feedback
bits. In order to make a fair comparison between analog and digital feedback,
we set the UL transmit power to $P_{\rm ul}=2^{B/\beta}-1$ so that both
strategies feed back the same amount of data. Additionally, we provide results
of maximum ratio transmission (MRT) precoding and ZF precoding under perfect
DL CSI. The precoder of MRT and ZF are respectively obtained by ${\bf V}_{\rm
MRT}=J_{\rm MRT}{\bf H}^{{\sf H}}$ and ${\bf V}_{\rm ZF}=J_{\rm ZF}({\bf
H}^{{{\sf H}}}{\bf H})^{-1}{\bf H}^{{\sf H}}$, where ${\bf H}=[{\bf
h}_{1},\dots,{\bf h}_{K}]$ and $J_{\rm MRT}$ and $J_{\rm ZF}$ are power
normalization scalars to satisfy the power constraint (13). Furthermore, we
also provide results for the case of training and precoding without the pre-
beamforming matrix. This is equivalent to setting ${\bf B}={\bf F}^{{\sf H}}$
which performs only a rotation on the channel and is the same as setting
$\hbox{\boldmath$\lambda$}=\mathbf{1}$ (a vector of all ones). Comparing to
this case, the performance improvement by optimizing $\lambda$ will become
clear. Finally, both with and without pre-beamforming, the matrix constituent
${\bf W}$ of the pilot matrix in (4) is generated randomly with standard
normal elements and will be fixed in training. We noted earlier that the
choice of this matrix is arbitrary as long as it is full-rank (which is the
case, with probability 1, when each element generated as a standard normal
random variable). We have tried optimizing this matrix, jointly with the rest
of the network, but this did not result in noticeable gain in performance and
therefore was ignored.

###  IV-B Performance Comparison

The sum-rate performance vs feedback capacity (in bits) for sparse scattering
with $L=2$ is illustrated in Fig. 2. We observe that our proposed DNN-based
technique outperforms all rival methods (except ZF with perfect CSI). In
particular, we see a significant performance advantage in comparison to the
DNN-based method in [15], especially for small feedback sizes. This should be
mainly attributed to the fact that our proposed scheme exploits channel
statistics knowledge at the BS. Even when there is practically no feedback
($B\to 0$), our scheme is able to achieve a relatively large sum-rate because
one component of the designed precoder in (11), namely the pre-beamforming
matrix ${\bf B}$ depends only on channel statistics and not the feedback. In
this case, our DNN is essentially performing a kind of statistical beamforming
with (almost) no CSI. Interestingly, statistical beamforming is shown to be
very effective in the case of sparse channels [28], which supports the
observed behavior here. The proposed method also outperforms MRT, which is due
to the use of ZF precoding in our architecture. The advantage with respect to
the case with no pre-beamforming matrix (the red curve) is rather small. This
is due to the fact that the channels are sparse and $L<\beta$, and therefore
with sufficient feedback no beam-selection is necessary. Here, the optimized
beam-selection vector is $\hbox{\boldmath$\lambda$}\approx\mathbf{1}$, which
is very close to no pre-beamforming with ${\bf B}={\bf F}^{{\sf H}}$. Finally,
we see a similar advantage in comparison to the ACS method in [19], resulting
from the direct maximization of sum-rate rather than multiplexing gain.

Figure 2: Sum-rate v.s. feedback capacity $B$ with $L=2$

T

Figure 3: Sum-rate v.s. feedback capacity $B$ with $L=20$

The methods are compared in the rich scattering scenario with $L=20$ in Fig.
3. Here since $\beta<L$, we expect that using an optimized pre-beamforming
matrix yields a large performance gain. This is confirmed by the results of
Fig. 3, where we see that the proposed DNN-based method clearly outperforms
the competitor methods. The performance advantage with respect to the case
with no pre-beamforming is clearly observed and shows that even with channel
statistics knowledge, the system under-performs because of the interference
cause by MMSE channel estimation with $\beta<L$. The propsed method also
performs better than the ACS method in [19], since it directly maximizes sum-
rate instead of multiplexing gain. Finally, both channel statistics knowledge
and optimized pre-beamforming results in the much higher sum-rate values
achieved by our method and the DNN-based method proposed in [15] for all
feedback sizes. We point out that ZF precoding with perfect CSI yields a sum-
rate of $\approx 60$ bits/s/Hz, which is much larger than the rest and is
omitted from Fig. 3 for a better representation of the results.

In Fig. 4, we present a heat map of the optimized $\lambda$ of the proposed
scheme for $50$ random realizations of ${\cal D}(L)$ (stacked as rows), for
three different combinations of parameters, namely $(L=2,B=1)$ (sparse
scattering, small feedback) in Fig. 4(a), $(L=20,B=40)$ (rich scattering,
large feedback) in Fig. 4(b), and $(L=20,B=1)$ (rich scattering, small
feedback) in Fig. 4(c). First, we observe in Fig. 4(a) that under $L=2$ the
learned $\lambda$ are almost all ones because the channels are sparse enough
that no pre-beamforming is needed and agrees with the sum-rate performance
presented in Fig. 2. In Fig. 4(b), since $\beta<L$, the DNN produces beam-
selection vectors that contain many zeros, meaning that many beams are not
selected in the pre-beamformer. If we decrease the feedback size from $B=40$
to $B=1$ bits, we have the results of Fig. 4(c) where even less beams are
selected (more elements in $\lambda$ turn out to be zero) because feedback
size is extremely small and the DNN chooses accordingly to train effective
channels with fewer coefficients.

##  V Conclusion

We proposed a DNN-based channel training and precoding scheme with channel
statistics knowledge at the BS, for FDD massive MIMO systems. The DNN is
trained for an ensemble of channel statistics (provided by the cell geometric
environment), and generates, for any given input of user channel statistics a
pre-beamforming matrix that maps original channels to effective channels such
that the DL sum-rate is maximized. The proposed system works with analog
feedback and requires no DNN implemented at the user side, which makes it far
more practical than most DNN-based approaches in the literature. Our numerical
results showed the significant advantage offered by this architecture for both
sparse and rich scattering scenarios and various feedback sizes.

(a) $L=2,B=1$

(b) $L=20,B=40$

(c) $L=20,B=1$

Figure 4: Optimized $\lambda$ instances of the proposed DNN for 50 covariance
realizations (stacked as rows)

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