# ML Codebook Design for Initial Access and CSI Type-II Feedback in Sub-6GHz
5G NR

Ryan M. Dreifuerst  Robert W. Heath Jr Ryan M. Dreifuerst and Robert W. Heath
Jr. are with North Carolina State University, Raleigh, NC 27695 {rmdreifu,
<EMAIL_ADDRESS>This material is based upon work supported by the
National Science Foundation under grant nos. NSF-ECCS-2153698, NSF-
CCF-2225555, NSF-CNS-2147955 and is supported in part by funds from federal
agency and industry partners as specified in the Resilient & Intelligent NextG
Systems (RINGS) program.

###### Abstract

Beam codebooks are a recent feature to enable high dimension multiple-input
multiple-output (MIMO) in 5G new radio (NR). Codebooks comprised of
customizable beamforming weights can be used to transmit reference signals and
aid the channel state information (CSI) acquisition process. Codebooks are
also used for quantizing feedback following CSI acquisition. In this paper, we
characterize the role of each codebook used during the beam management process
and design a neural network to find codebooks that improve overall system
performance. Evaluating a codebook is not purely about maximizing signal
power. Instead, it is about considering the system-level dependency between
the codebooks, feedback, overhead, and spectral efficiency. The proposed
neural network is built on translating codebook and feedback knowledge into a
consistent beamspace basis similar to a virtual channel model. Then, the
beamspace is fed into the neural network to generate initial access codebooks.
This beamspace codebook algorithm is designed to directly integrate with
current 5G beam management standards (Releases 15-17), without changing the
feedback format, beam training timing, or requiring additional side
information. Our simulations show that the neural network codebooks improve
over traditional codebooks in received signal power, even in dispersive
sub-6GHz environments. We further use our simulation framework to evaluate
type-II CSI feedback formats with regard to effective multi-user spectral
efficiency. Our results suggest that optimizing codebook performance can
provide valuable spectral efficiency improvements, but optimizing the feedback
configuration can be equally important for multi-user performance in sub-6GHz
bands.

##  I Introduction

Fifth-generation (5G) cellular systems have adopted a beam-based approach for
MIMO communications. In such systems, broadcast control signals are beamformed
to provide array gain [1]. This helps increase coverage and meet link budgets
at high carrier frequencies. While the motivation for beam-based systems was
to provide array gain in millimeter-wave (mmWave) frequency bands, the same
framework is also used at low frequencies. Unfortunately, optimization of the
beamforming framework, and subsequent feedback strategies, has not been
extensively investigated for sub-6GHz multi-user MIMO (MU-MIMO), despite the
importance of low- and mid-band MIMO in current and anticipated future
standards [2].

Codebook-based beam management is a framework for beamforming and feedback
that enables transmitting beamformed reference signals and multiple forms of
CSI acquisition [1]. In early 3GPP releases up to Release $12$, channel state
information was obtained by estimating the channel from reference signals for
every transmitter-receiver beam pair using unprecoded signals. Unprecoded
reference signals were used up through 4G LTE Release $14$ because multi-
antenna beamforming was not needed to achieve higher signal-to-noise ratio
(SNR) for synchronization and channel estimation [3]. Beamformed control
signals have become a necessity in mmWave bands as a result of the increasing
bandwidths (which reduce the SNR for a constant transmit power) and shrinking
antennas sizes–resulting in pre-beamformed SNR on the order of $-15$dB or less
[4]. By beamforming the control and reference signals, receivers are more
likely to detect initial access signals and obtain more accurate
synchronization and channel measurements.

The process of obtaining CSI at the transmitter involves multiple steps and
codebooks to balance the overhead, latency, and accuracy of the CSI [5, 6]. To
begin the process, the base station (BS) transmits a synchronization signal
block (SSB) with a beamformer selected from a codebook (SSB codebook) that
enables synchronization and random access. The SSB codebook is typically small
and all of the codewords in the codebook are used during each SSB period. The
user equipment (UE) will provide a small amount of feedback to conclude the
initial access that includes a beam selection from the SSB codebook. The BS
will use this information to select a subset of beamformers from a large
codebook, different from the SSB codebook, for transmitting channel state
information reference signals (CSI-RS). We assign this codebook the name “CSI-
RS codebook”. The relationship between the SSB and CSI-RS codebook can be
understood from the perspective of a hierarchical beam search [7]. The CSI-RS
enable hybrid beam training by providing multiple precise beams for analog
beam training–often referred to as beam refinement–and pilots for digital
channel estimation. The UE will provide an in-depth packet of feedback
following CSI-RS transmission that includes the CSI-RS codebook beamformer
selection as well as information based on the channel estimation. While the
first iterations of beam management used wide and narrow discrete Fourier
transform (DFT) beamformers [4], there is no requirement that the beamformers
be “wide” and “narrow” for the SSB and CSI-RS codebooks. In fact, the
codebooks can be arbitrary since the UEs do not need knowledge of the
beamformers during initial access, beam refinement, or even data transmission.

While the SSB and CSI-RS codebooks can be arbitrary, the third codebook used
for quantizing the feedback (denoted as the FB codebook) is precisely defined
depending on the format (type-I, type-II). The FB codebook is defined for
specific BS and UE array geometries so that the CSI can be represented and fed
back as indices from the codebook. This codebook must be known at the BS and
UE. It is not strictly necessary that even the feedback codebook is used when
reciprocity exists, as reciprocity between the uplink and downlink channel can
be exploited in time-division-duplexing systems. Feedback from downlink
training, though, is still beneficial in these systems due to asynchronous
link budgets [8] or when the number of transmitting and receiving antennas on
a UE is not equal [9].

Our work focuses on a realistic and system-level evaluation of the codebooks
used during the beam management framework in sub-6GHz bands. In particular, we
unify the two stages of codebook design with the feedback framework and MU-
MIMO data transmission to achieve good network performance. Our contributions
in this manuscript are summarized as follows:

  * •

First, we precisely describe the beam management and CSI type-II feedback
approach for 5G networks. There are many misconceptions on how the codebooks,
SSB, and CSI-RS processes are used for obtaining different forms of feedback.
We explain how the various codebooks are used, what is specified in the
standards versus left up to implementations, and provide an efficient
implementation for feedback quantization in CSI type-II.

  * •

Second, we present a novel neural network architecture, Beamspace-Codex (BSC),
and feedback processing technique for SSB codebook generation. We constrain
the system to follow 5G beam management timing which presents a significant
constraint for dynamic SSB and CSI-RS codebook generation. To mitigate this
constraint, the SSB codebook is dynamically generated–which is manageable at
the SSB periodicity–and the CSI-RS codebook is determined as a precomputed DFT
decomposition of the SSB codebook. With this integration, we unify the
arbitrary codebooks learned via deep learning with the feedback quantization
technique to facilitate a hierarchical beam search that improves the beam
management process used in 5G NR.

  * •

Third, we present the results of an extensive evaluation of our proposed
method with benchmark DFT codebooks and CSI-based eigen-beamforming solutions
using various levels of feedback and overhead. We show that our proposed
solution achieves significant performance improvements. We also show that MU-
MIMO performance is dramatically limited by the quantization resolution
techniques in low-band 5G. In contrast, the frequency selectivity and resource
allocation have little impact in the current framework and scattering
environment.

There is significant work on beam training and beam alignment. Here we review
a set of relevant prior work relating to machine learning and beam training.
In [10], a gradient-optimization approach, not using deep learning, is shown
to be successful in aiding the beam search process with respect to maximizing
the SNR in $60$GHz personal networks. Another approach using classical data-
driven techniques is presented in [11] for vehicular networks based on
historical beam training results. While the codebooks and algorithms are
relatively straightforward, the idea of “learning” an efficient codebook from
site-specific data is a cornerstone of many modern beam alignment techniques
[12, 13, 14]. These papers [10, 11, 12, 13, 14] all focus on mmWave channels
and often exploit the sparsity of such environments to aid the beam training
process. Furthermore, all of these papers focus exclusively on beam alignment
with standard orthogonal codebooks like DFT and phase-shifted codebooks. While
DFT codebooks are sensible for certain array geometries and limited scattering
environments at mmWave bands, these codebooks do not necessarily work well in
rich scattering. Furthermore, none of these papers consider a MIMO format with
multiple UE antennas and streams of data. In this paper, we consider a full
MU-MIMO OFDM system in a rich scattering environment and use dynamic
codebooks.

In another line of related work, algorithms have been proposed that design
beamformers or codebooks [15, 7, 16, 17, 18, 13, 19, 20]. Hierarchical
codebooks and the associated design process were proposed in [15, 7, 16] which
separated the beam training into multiple stages of wide-beam and narrow-beam
codebooks. A similar strategy was adopted by 5G using the SSB and CSI-RS
codebooks and plays an important role in systems designed for 5G deployment.
Deep learning was applied to the task in [17], which uses deep reinforcement
learning to design broadcast beam patterns. While designed for small-scale
MIMO, the results suggest that arbitrary learned beam patterns can be an
effective way to initiate communication in broadband networks. A supervised
and an unsupervised learning approach were presented in [18], which showed
promising results when used to design beamforming vectors from channel
measurements. These results were based on a simplified channel model, though,
and perfect CSI was assumed at the BS. Recently, a deep, federated,
reinforcement learning strategy was proposed [13] that jointly trained a beam
management model over a network using user location information. The focus was
on deciding which sectors would be active, thereby narrowing the “codebook,”
although the paper ignores all beamforming besides sector-level association
and large-scale pathloss. Another paper considered the task of learning
probing beams [19], which is one of the only papers to consider both the SSB
and CSI-RS codebooks. The results show significant gains can be achieved over
a basic hierarchical search, although perfect CSI is assumed to be available
for training and the channels are assumed to be narrowband. The results from
[19] provide motivation for researching realistic deep learning codebook
methods. In our previous work we designed a neural network for SSB codebook
generation in narrowband channels as well but trained using an angular
representation of the channel [20]. The results, however, did not extend well
to wideband systems and did not consider the impact that the SSB codebook has
on system-level performance. In this investigation, we now model wideband
channels and fully incorporate the entire beam management and data
transmission into our evaluation.

An important part of the 5G beam management framework is the connection
between beam training and feedback. An enhanced CSI feedback strategy was
proposed for full-dimension MIMO [3] in 4G Release $13$ which is based on a
linear combination of DFT codewords. This strategy is very similar to the
type-II feedback format introduced in 5G Release $15$. One key difference is
that the initial idea assumed the users must know the beamforming vector used
during pilot transmission, which was not adopted into the standards. There has
also been growing interest in using machine learning to design or modify the
feedback [21, 22, 23]. All that work assumes that both the network and the
users can share models for encoding and decoding the feedback, which is not
supported or easily introduced into network operation. In this paper, we do
not use deep learning to change the feedback methods. Instead, we characterize
the relationship and limitations the feedback and beam management strategy
have on MU-MIMO performance.

Figure 1: Beamforming codebooks are used to transmit reference signals for
synchronization from the SSB codebook. Based on the SSB report, additional
reference signals are beamformed using the CSI-RS codebook for channel
estimation by the user equipment (UEs). Using the CSI-RS report, the base
station can design a precoder (i.e. RZF precoder) that attempts to maximize
the spectral efficiency. The proposed BSC algorithm updates the SSB codebook,
which in turn affects the CSI-RS codebook and feedback.

$\mathbf{Notation}$: $\mathbf{A}$ is a matrix, $\mathbf{a}$ and
$\\{a[i]\\}_{i=1}^{N}$ are column vectors and $a,A$ denote scalars.
$\mathbf{A}^{T}$, $\overline{\mathbf{A}}$, $\mathbf{A}^{*}$, and
$\mathbf{A}^{\dagger}$ represent the transpose, conjugate, conjugate
transpose, and psuedo-inverse of $\mathbf{A}$. $\mathbf{A}[k,\ell]$ denotes
the entry of $\mathbf{A}$ in the $k^{\text{th}}$ row and the
$\ell^{\text{th}}$ column. The same meaning is also associated with
$\mathbf{A}_{k,\ell}$. Similarly, $\mathbf{A}[:,k]$ refers to the
$k^{\text{th}}$ column of $\mathbf{A}$. Unspecified norm equations are
$\left\lVert\mathbf{a}\right\rVert_{2}=\mathbf{a}^{*}\mathbf{a}$ for vectors
and the Frobenius norm
$\left\lVert\mathbf{A}\right\rVert_{F}=\sqrt{\text{Tr}(\mathbf{A}\mathbf{A}^{*})}$
for matrices. We make use of the disjoint operator $\mathbf{A}\ \backslash\
\bm{\alpha}$ to describe the matrix (or vector) $\mathbf{A}$ excluding the
subset (or element) $\bm{\alpha}$. We define $\mathsf{j}=\sqrt{-1}$. The
operator $\mathbb{E}[\cdot]$ is used for the expectation of a random variable.
Due to the notational complexity of MU-MIMO with OFDM, we will always use $u$
to refer to a specific UE, $t$ as a specific time, $k$ as a specific frequency
resource, and $n_{\text{t}}/n_{\text{r}}$ to refer to a specific transmit or
receive antenna.

The remainder of our paper is organized as follows. First, we introduce the
system model beginning from an arbitrary MU-MIMO OFDM channel and continuing
through the SSB and CSI-RS beam management processes. We finish this section
with a description of the channel estimation, feedback, and data transmission
stages. Next, we introduce the beamspace observation and neural network
architecture that form the basis of our work. In the final sections, we
present the simulation setup and evaluate the various codebooks and feedback
formats to address the lack of prior work when it comes to codebooks and CSI
formats in sub-6GHz 5G NR.

##  II System model

We begin the investigation with an overview of the channel definition and
assumptions before introducing the beam management steps including SSB
transmission, CSI-RS transmission, feedback, and downlink data transmission,
as shown in Figure 1. Afterward, we outline the problem formulation that
influences the neural architecture. Throughout this paper, we will limit the
system configuration to a single cell with multiple UEs each equipped with
multiple antennas and all arrays are fully digital and single-polarization.
While the inclusion of multiple cells is more realistic, the impact of multi-
cell interference on beam management and feedback can be mitigated with
limited coordination between nearby cells. The use of single-polarized arrays
is also a simplification, although during beam training the second
polarization operates as a simple extension with little impact on the overall
system operation. We will investigate multiple BS deployments with dual-
polarized arrays in future work.

###  II-A MU-MIMO channel model

We model the system so that it is representative of real-world conditions to
evaluate performance in a realistic beam training scenario for sub-6GHz
massive MIMO. We do not explicitly specify TDD or FDD because this work does
not rely on channel reciprocity. In addition, the channel will always be
handled in the frequency domain as the multi-user downlink OFDM channel,
however, we do not assume any specific channel model at this point. The OFDM
MU-MIMO channel is defined for $U$ users, over $T$ time slots, and $K$
subcarriers between each of the $N_{\text{R}}$ receive antennas and
$N_{\text{T}}$ transmit antennas as $\mathbf{H}\in\mathbb{C}^{U\times T\times
K\times N_{\text{R}}\times N_{\text{T}}}$. Note that we can generate this
channel model using a clustered channel model as typically done by QuaDRiGa
[24], 3GPP, and other raytracing or statistical models with the array response
vectors
$\mathbf{a}_{N}(\theta)=\frac{1}{\sqrt{N}}[1,e^{j\pi\cos{\theta}},e^{j2\pi\cos{\theta}},...,\
e^{j(N-1)\pi\cos{\theta}}]^{T}$. Defining the time and frequency responses for
every antenna pair is critical to accurately model the beam training process
with specific resource elements allocated for pilots (reference signals) and
data. With this setup in mind, a generic received signal model for signal
$\mathbf{s}$ transmitted with precoder/beamformer $\mathbf{F}$ and received
with a combiner $\mathbf{W}$ over a band of subcarriers $K$ is

$\displaystyle\mathbf{y}_{t,k}^{(u\color[rgb]{0,0,0})}$
$\displaystyle=\frac{1}{\sqrt{KN_{\text{T}}}}\mathbf{W}_{t,k}^{(u\color[rgb]{0,0,0})}\mathbf{H}_{u\color[rgb]{0,0,0},t,k}\sum_{\text{u}=0}^{U-1}\mathbf{F}_{\text{u},t,k}\mathbf{s}_{\text{u},t,k}+\mathbf{W}_{t,k}^{(u\color[rgb]{0,0,0})}\mathbf{N}_{t,k}.$
(1)

The noise $\mathbf{N}$ is modeled as IID, complex Gaussian random values to
account for thermal noise and the noise figure of the receiver. Note that
because users do not share information, the combiners and received signals
have superscript $(u\color[rgb]{0,0,0})$, while the beamformers and channels
have subscript $u\color[rgb]{0,0,0}$ to imply that an aggregation is possible
i.e. the BS has knowledge of the precoders for all users. During the next
subsections, we will specify how
$\mathbf{W}^{(u\color[rgb]{0,0,0})},\mathbf{F}_{u\color[rgb]{0,0,0}},$ and
$\mathbf{s}_{u\color[rgb]{0,0,0}}$ are used during the SSB, CSI-RS, and data
transmission stages. We will assume throughout the paper that all beamformers
are normalized according to a per-symbol power constraint, i.e.
$\left\lVert\mathbf{F}\right\rVert^{2}=\ N_{\text{T}}$. In the SSB and CSI-RS
stages, $\mathbf{F}$ is selected from one or more codebooks that play critical
roles in beam management.

###  II-B Codebooks

Communication protocols employ codebooks in many different ways for beam
training, feedback representations, information quantization, and more.
Generally, codebooks are understood as an ordered set of elements called
codewords. Throughout this work on beam management, codewords will correspond
to a vector of complex values and we will treat codebooks as matrices to
simplify notation in later steps. There are three codebooks that we will
reference and directly impact beam management: a codebook of beamforming
vectors for SSB transmission (SSB codebook), a codebook for beamforming CSI-RS
transmissions (CSI-RS codebook), and a codebook for feedback quantization (FB
codebook). Each of these codebooks is designed to serve different purposes,
although there are many inter-relationships that make designing and evaluating
codebooks challenging.

The sizes of the codebooks are based on the supported usage for each one. The
SSB codebook $\mathbf{B}_{\text{SSB}}\in\mathbb{C}^{N_{\text{P}}\times
L_{\text{{max}}}}$ has $L_{\text{{max}}}$ codewords which is between $1$ and
$64$ depending on the carrier frequency [1]. The CSI-RS codebook
$\mathbf{B}_{\text{CSI-RS}}\in\mathbb{C}^{N_{\text{P}}\times N}$ is at least
the same size, where $N$ is typically large but only a small subset of
beamformers is selected to be used in one period. The FB codebook
$\mathbf{B}_{\text{DFT}}\in\mathbb{C}^{N_{\text{P}}\times
N_{\text{X}}O_{\text{H}}N_{\text{Y}}O_{\text{V}}}$ is an oversampled DFT
codebook with horizontal and vertical oversampling factors $O_{\text{H}}$ and
$O_{\text{V}}$. The number of logical ports, $N_{\text{P}}$, determines the
digital precoding dimension and is equal to $N_{\text{X}}N_{\text{Y}}$ in a
fully digital array. The first two codebooks are transparent to the UE,
meaning that they are configurable and any arbitrary beamforming codebooks can
be employed. The FB codebook must stay constant and is fully determined by the
BS logical array size and oversampling factors, which are sent to the UE
during initial access so that both the BS and the UE can use the same codebook
for feedback representation. Previously, 4G used a feedback codebook based
upon predefined matrices called CSI type-I precoding matrix indicator (PMI).
5G augmented the feedback to include type-II PMI which corresponds to a set of
directional array responses with amplitude and phase combining factors.

###  II-C SSB transmission and reception

Initial access is used in mobile networks to obtain limited channel
information, achieve synchronization, and set up random access procedures. In
5G NR, a cell may initiate the initial access period at regular intervals of
$\\{5,10,20,40,80,160\\}$ms [6], to control the periodicity a UE must be
active (not in power-saving mode) and provide feedback to the network. During
this period, the cell will transmit SSBs that contain primary and secondary
synchronization signals (PSS, SSS) as well as demodulation reference signals
(DMRS) [6]. These SSBs are beamformed using a specific beamforming vector
selected from $\mathbf{B}_{\text{SSB}}$ and associated with a beam index. Each
SSB is transmitted sequentially in time, using $20$ contiguous resource
blocks. This is a much smaller bandwidth than the downlink data transmission
and is a key limitation of relying on SSB feedback to obtain CSI in frequency-
selective channels. Furthermore, the resource allocation of SSBs is
essentially fixed to assist new users with joining a network. Depending on the
cell carrier frequency and subcarrier spacing, a cell may transmit up to
$L_{\text{{max}}}=\\{1,4,8,64\\}$ beams in a period, and all cells must
transmit at least one beam. During transmission, the UE will measure the
Reference Signal Received Power (RSRP) and report the measurement using the
random access channel slot corresponding to the index of the beam with the
highest RSRP. This way the base station knows how strong the signal was and to
which of the SSB codebook’s codewords it corresponds.

The SSB is transmitted over a band of $K_{\text{SSB}}$ frequency resources at
times $T_{{i},\text{SSB}}$ for the $i^{\text{th}}$ SSB using beamformer
$\mathbf{f}_{i}\in\mathbf{B}_{\text{SSB}}$. While transmitting the reference
signal, the rest of the band is still available for data transmission, so the
system power is shared over all $K$ resource blocks. In total, all
$L_{\text{{max}}}$ SSBs are transmitted from the codebook
$\mathbf{B}_{\text{SSB}}$. During SSB reception, the generic received signal
model from (1) is modified to account for the single data stream

$\displaystyle
y_{\text{SSB}_{i},t,k}^{(u\color[rgb]{0,0,0})}=\frac{1}{\sqrt{KN_{\text{T}}}}\mathbf{w}_{t,k}^{(u\color[rgb]{0,0,0})*}\mathbf{H}_{u\color[rgb]{0,0,0},t,k}\mathbf{f}_{i}s_{t,k}+\mathbf{w}_{t,k}^{(u\color[rgb]{0,0,0})*}\mathbf{N}_{t,k}.$
(2)

In mmWave bands, it is common that $\mathbf{w}$ is also determined through
beam training. In sub-6GHz bands, the receiver instead uses antenna selection
or simply the first antenna for power saving [25]. With the received signal
model (2) in mind, we can now define an important quantity provided as
feedback during initial access, the RSRP.

###  II-D Reference signal receive power

RSRP is one of the primary metrics that the receiver will measure during
initial access for determining the channel quality. The base station will use
the RSRP feedback to determine the strength of the signal, the initial code
rate to be used, and what CSI-RS beamformers should be employed at the next
iteration. The RSRP measures the received power during a given reference
signal (SSB or CSI-RS). In the SSB case, the demodulation reference signal
(DMRS), known at the transmitter and receiver, is a pilot signal used for RSRP
measurement and decoding the master information block provided in the SSB. We
will assume antenna selection for the
$n_{\text{r}}\in\\{0,..N_{\text{R}}-1\\}$ antenna during SSB reception, but
because it is fully digital the UE can determine the antenna selection after
receiving the full SSB and selecting the strongest receive branch. The RSRP is
then

$\displaystyle\mathbf{y}_{\text{SSB}_{i},t,k}^{(u\color[rgb]{0,0,0})}$
$\displaystyle=\
\frac{1}{\sqrt{KN_{\text{T}}}}\mathbf{H}_{u\color[rgb]{0,0,0},t,k}\mathbf{f}_{i}s_{t,k}+\mathbf{N}_{t,k}$
(3) $\displaystyle\text{RSRP}_{i}^{(u\color[rgb]{0,0,0})}$ $\displaystyle=\
\max_{n_{\text{r}}}\sum_{k\in
K_{\text{DMRS}}}\sum_{t\in\mathbf{T}_{{i},\text{DMRS}}}\left\lVert\mathbf{y}_{\text{SSB}_{i},t,k,n_{\text{r}}}^{(u\color[rgb]{0,0,0})}\right\rVert^{2}.$
(4)

Note that (4) is for a specific UE $(u\color[rgb]{0,0,0})$ and SSB $(i)$, so
the total SSB information aggregated at the base station is comprised of the
strongest RSRP and corresponding SSB index (SSBRI) for each of the UEs,
denoted as $(\mathbf{p},\mathbf{m})$, and defined as

$\displaystyle\mathbf{p}$ $\displaystyle=\
\left\\{\max_{i}{\text{RSRP}_{i}^{(u\color[rgb]{0,0,0})}}\right\\}^{U-1}_{u\color[rgb]{0,0,0}=0}$
(5) $\displaystyle\mathbf{m}$ $\displaystyle=\
\left\\{\operatorname*{argmax}_{i}{\text{RSRP}_{i}^{(u\color[rgb]{0,0,0})}}\right\\}^{U-1}_{u\color[rgb]{0,0,0}=0}.$
(6)

It is worth noting that the actual SSBRI is not transmitted by the users; the
random access slot that the UE responds during will correspond to a specific
SSBRI. This is an efficient way to reduce the amount of feedback, allow the
base station to perform receive combining for the UE feedback, and still share
the necessary information.

Figure 2: The data processing flow from data streams (layers) to antenna
outputs in 5G. First, the streams (and DMRS) are multiplexed according to the
precoding matrix, which essentially assigns the layers to the ports. Then, the
result, along with CSI-RS, is beamformed to the logical ports. The logical
processes are then mapped onto OFDM resource grids and potentially beamformed
again onto the corresponding physical antennas.

###  II-E CSI-RS transmission and reception

Beam refinement is the second step of the hierarchical beam management process
that is used to obtain better beam alignment and enable wideband channel
estimation. Beam refinement is done through the transmission of CSI-RS which
is an encompassing term used for the reference pilots, the beamformed
reference signal, and the logical ports that the pilots are sent on. In
contrast to SSB signals, CSI-RS are extremely flexible with many combinations
of time and frequency resources allocatable. There are some limitations on how
often CSI-RS can be transmitted, but the default periodicity a UE expects when
joining the network is $80$ms and the symbols must span a larger bandwidth
than the SSB–at least $24$ resource blocks and up to the entire bandwidth
should contain CSI-RS symbols for a measurement report. Furthermore, CSI-RS
are also transmitted both periodically and aperiodically to assist beam
tracking with highly mobile users. All told, the beam refinement stage
requires more resources than initial access, which is why a hierarchical beam
search is used to limit the number of CSI-RS processes needed to achieve a
strong connection.

Once a BS receives the SSB feedback for a new user, it can attempt to gather
more precise CSI through channel state information reference signals. The CSI-
RS (and associated CSI-RS beamformers $\mathbf{F}_{\text{CSI-RS}}$) are used
for refined channel estimation, although the estimation is performed on the
beamformed channel $\mathbf{H}_{u\color[rgb]{0,0,0}}\mathbf{F}_{\text{CSI-
RS}}$, rather than the physical channel. This enables larger hybrid arrays and
improves the SNR for more accurate estimation. Readers are encouraged to
review [6, 5, 9] for more details of CSI-RS. The BS can use the SSB feedback
to determine and transmit a set of CSI-RS signals with beamformers that are
intended to further increase the RSRP compared to the SSB.

The design of CSI-RS codebooks has been studied to some degree, but DFT beams
are common [3, 13, 26] (and references therein). Often, it is assumed that the
set of beamformers is selected from the same codebook as the codebook used for
feedback, which is specified as an oversampled DFT codebook. There is no
specification or need for the SSB or CSI-RS codebooks to be DFT codebooks,
although using DFT beamformers during CSI-RS has advantages related to fast
and efficient processing. For example, the CSI-RS beamformers can be
efficiently selected as the largest DFT components of the SSB beamformers. We
will assume the CSI-RS codebook, $\mathbf{B}_{\text{CSI-RS}}$, is comprised of
$N_{\text{X}}N_{\text{Y}}O_{\text{H}}O_{\text{V}}$ oversampled DFT vectors
throughout this paper, although additional investigations into joint SSB–CSI-
RS codebook algorithms are planned in future work. The selection of the CSI-RS
beams from SSB decomposition is further refined in Section III.

The CSI-RS process involves transmitting a set of pilots $\mathbf{s}$ on the
$N_{\text{P}}$ ports along with $P_{\text{CSI}}$ beamformers
$\mathbf{F}_{\text{CSI-
RS}}=\\{\mathbf{f}_{i}\\}_{i=0}^{P_{\text{CSI}}}\subseteq\mathbf{B}_{\text{CSI-
RS}}$ chosen by the serving base station to improve the UE channel estimation
SNR for the effective channel. The beamformers are used during non-overlapping
time-frequency resources
($\mathbf{T}_{i,\text{CSI}},\mathbf{K}_{i,\text{CSI}}$), with the aggregate
set of beamformers corresponding to what we define as $\mathbf{F}_{\text{CSI-
RS}}$. Each CSI-RS allocation will have at least $P_{\text{CSI}}$ OFDM
resource elements allocated for channel estimation in one resource block and
only one CSI-RS beam is active for a given resource to allow for a similar,
interference-free beam selection as the SSB process. This is done by setting
all other CSI-RS resources and ports to be zero-power CSI-RS except the active
one for the specified resources. In a dual-polarized system, the two
polarizations are transmitted together to estimate a co-polarization factor
that would be fed back during CSI reporting. The whole CSI-RS process is
allocated over NRB resource blocks, which is a configurable parameter to use
wider bandwidths and noise averaging during channel estimation. The beam
reception and selection process is identical to the RSRP and SSB selection in
the previous subsection. For more information on the CSI-RS beam and port
allocation please see [3] and references therein. At this stage, the entire UE
antenna array is active and the UE can perform maximum ratio combining
$\mathbf{w}_{i,t,k}^{(u\color[rgb]{0,0,0})}=\left(\widehat{\mathbf{H}_{u\color[rgb]{0,0,0},t,k}\mathbf{f}_{i}}\right)^{*}$
based on the estimated beamformed channel and appropriate normalization. The
estimated SNR at the receiver is

$\text{SNR}^{(u\color[rgb]{0,0,0})}=\max_{c\in
P_{\text{CSI}}}\frac{1}{K}\sum_{k\in\mathbf{K}_{c,\text{CSI}}}\sum_{t\in\mathbf{T}_{c,\text{CSI}}}\frac{1}{N_{T}}\frac{\left\lVert\widehat{\mathbf{H}_{u\color[rgb]{0,0,0},t,k}\mathbf{f}_{c}}\right\rVert^{2}}{\mathbb{E}[\left\lVert\mathbf{N}_{t,k}\right\rVert^{2}]}.$
(7)

Note that the noise power
$\mathbb{E}[\left\lVert\mathbf{N}_{t,k}\right\rVert^{2}]$ can be estimated
using zero-power CSI-RS resources. The SNR can be reported during feedback,
although one of the most important aspects of the SNR is how the channel
estimation (and subsequent feedback) is impacted. For the next part of this
section, we will focus on a single user ($u\color[rgb]{0,0,0}$) with SNR
corresponding to the channel estimation SNR for the beamformed channel
$(\mathbf{H}_{u\color[rgb]{0,0,0},t,k}\mathbf{F}_{\text{CSI-RS}})$.

###  II-F Channel estimation

Channel estimation accuracy depends on the number of pilot symbols and the
received SNR of the pilots. The algorithm for UE channel estimation is not
defined in the specification and is not critical for our system-level
analysis. For simplicity, we will assume channel estimation via a simple least
squares algorithm so that only knowledge of the pilots is required. Now, we
motivate the inclusion of channel estimation in beam management by considering
the relationship between the channel estimation error, the codebook-dependent-
SNR, and the number of pilot resources $N_{\text{pilots}}$. The resulting mean
squared error (MSE) of the channel estimate is [27, section 3.7]

$\text{MSE}=\frac{1}{\frac{N_{\text{pilots}}}{N_{T}}\text{SNR}}.$ (8)

Therefore, we can see that a primary feature of the beam management process is
to enable more accurate channel estimation by beamforming the reference pilot
signals. It might seem that the overhead of using multiple beams is wasted
given that the number of pilots could alternatively be increased according to
(8). The issue with such logic is that: 1) it assumes that the signal is
always detectable over the noise floor, which is not necessarily the case for
non-beamformed signals and 2) the beamforming can provide a gain on the order
of $20$dB or more, yet that would require $100$ times more pilot symbols, far
larger than the number of beams used.

Calculating the estimated effective channel, $\widehat{\mathbf{H}\mathbf{F}}$,
which may not be frequency selective, is the first step in determining the
precoder matrix information (PMI), which is one of the quantized CSI elements
provided during feedback. The PMI can correspond to different aspects of the
estimated channel because it is left to operator implementation as part of the
PMI selection procedure. For example, the user could provide PMI according to
its desired beamformers (calculated via singular value decomposition of
$\widehat{\mathbf{H}\mathbf{F}}$). The base station could then use this
information to beamform to a given user and null steer [28] the beams for
other users. This puts the majority of the computational burden, however, on
the UE. Instead, we follow models similar to [29] where the user provides PMI
corresponding to the effective channel estimate, rather than the SVD of the
effective channel estimate. This way the BS handles most of the computation,
channel reconstruction, and precoder determination.

###  II-G CSI feedback and reconstruction

We will focus on CSI type-II feedback (Release $16$), as our feedback
baseline, which is intended for MU-MIMO with up to $2$ layers per user [25],
and is also compliant with Release $17$. The UE quantizes the channel estimate
according to the FB codebook for a specified $L_{\text{{CSI}}}$ beams per rank
up to the rank indicator, which is the maximum supported MIMO rank, $R$. A
straightforward method for selecting the PMI would be to exhaustively try
every possible combination of $L_{\text{{CSI}}}$ beams from the codebook and
select the best one (according to some metric). Assuming the feedback codebook
is an oversampled DFT codebook, we can make use of efficient algorithms for
quantizing the beamformed channel estimate into the PMI. Although not
presented here, codebooks can also be used for SU-MIMO feedback; we use this
form of feedback when comparing against the SU-MIMO results in Section IV.

First, we define the oversampled DFT codebook $\mathbf{B}_{\text{DFT}}$
beginning with the (non-oversampled) DFT codebook for a uniform linear array
$\mathbf{U}_{N}$ defined in the same way as e.g. [30]. Then we can define a
diagonal oversampling matrix $\mathbf{D}_{O,N}$ for an oversampling $O$ and
$N$ elements as

$\displaystyle\mathbf{D}_{O,N}=\text{diag}\left(1,\
\exp{\left(-\frac{\mathsf{j}2\pi}{ON}\right)},...,\exp{\left(-\frac{\mathsf{j}2\pi(N-1)}{ON}\right)}\right).$
(9)

Then, the oversampled DFT codebook for a one-dimensional array can be
concisely written as

$\displaystyle\mathbf{B}_{O,N}=[\mathbf{U}_{N},\mathbf{D}_{O,N}\mathbf{U}_{N},\mathbf{D}_{O,N}^{2}\mathbf{U}_{N}...,\mathbf{D}_{O,N}^{O-1}\mathbf{U}_{N}]\in\mathbb{C}^{N\times
ON}.$ (10)

Finally, the UPA oversampling codebook can be defined as the kronecker product
of the $N_{\text{X}}$ and $N_{\text{Y}}$ dimensions

$\displaystyle\mathbf{B}_{\text{DFT}}=\mathbf{B}_{O_{\text{H}},N_{\text{X}}}\otimes\mathbf{B}_{O_{\text{V}},N_{\text{Y}}}\in\mathbb{C}^{N_{\text{X}}N_{\text{Y}}\times
O_{\text{H}}N_{\text{X}}O_{\text{V}}N_{\text{Y}}}.$ (11)

With the FB codebook defined (11), the PMI selection procedure can begin. The
UE first calculates the inner product for all beams from the oversampled
codebook $\mathbf{B}_{\text{DFT}}$ and selects the first beam (and by
association the oversampling basis) as

$\displaystyle\mathbf{C}$
$\displaystyle=\widehat{\mathbf{H}\mathbf{F}}\mathbf{B}_{\text{DFT}}\
\in\mathbb{C}^{N_{\text{R}}\times
N_{\text{X}}O_{\text{H}}N_{\text{Y}}O_{\text{V}}}$ (12) $\displaystyle Q_{0}$
$\displaystyle=\operatorname*{argmax}|{\mathbf{C}}|$ (13)

where the argmax is determined from the absolute value of all elements of
$\mathbf{C}$ and the resulting $Q_{0}$ is an index corresponding to
oversampling powers $Q_{0,X}$ and $Q_{0,Y}$. Because (non-oversampled) DFT
beams are orthogonal, the PMI can be calculated iteratively by selecting the
first beam from the entire oversampled codebook $\mathbf{B}_{\text{DFT}}$,
reducing the codebook to just the orthogonal subset, and selecting the beams
from the subset. The orthogonal subset can be understood as the kronecker
product of the oversampling-shifted DFT codebooks

$\displaystyle\mathbf{B}_{\text{ortho},Q_{0}}=\mathbf{D}_{O_{\text{H}},N_{\text{X}}}^{Q_{0,X}}\mathbf{U}_{N_{\text{X}}}\otimes\mathbf{D}_{O_{\text{H}},N_{\text{X}}}^{Q_{0,Y}}\mathbf{U}_{N_{\text{Y}}}$
(14)

where $Q_{0,X}$ and $Q_{0,Y}$ are the oversampling powers in the X and Y
directions of the codebook index $Q_{0}$. With the orthogonal subset, the next
step is to select the $L_{\text{{CSI}}}$ beam indices
$\mathbf{Q}\in\mathbb{Z}_{+}^{N_{\text{R}}\times L_{\text{{CSI}}}}$ and
combining coefficients $\mathbf{A}\in\mathbb{C}^{N_{\text{R}}\times
L_{\text{{CSI}}}}$

$\displaystyle\mathbf{c}_{\text{ortho}}$
$\displaystyle=\widehat{\mathbf{H}\mathbf{F}}_{n_{\text{r}}}\mathbf{B}_{\text{ortho},Q_{0}}\
\in\mathbb{C}^{N_{\text{X}}N_{\text{Y}}}$ (15)
$\displaystyle\mathbf{Q}_{n_{\text{r}}}$
$\displaystyle=(L_{\text{{CSI}}})\text{-}\operatorname*{argmax}|{\mathbf{\mathbf{c}_{\text{ortho}}}}|\quad\forall
n_{\text{r}}$ (16) $\displaystyle\mathbf{A}_{n_{\text{r}}}$
$\displaystyle=\mathbf{c}_{\text{ortho}}\left[\mathbf{Q}_{{n_{\text{r}}}}\right]\quad\forall
n_{\text{r}}$ (17) $\displaystyle\mathbf{A}_{n_{\text{r}}}$
$\displaystyle=\frac{\mathbf{A}_{n_{\text{r}}}}{\mathbf{A}_{n_{\text{r}}}[\operatorname*{argmax}{|\mathbf{A}_{n_{\text{r}}}|}]}\quad\forall
n_{\text{r}}.$ (18)

First, the orthogonal beams are “evaluated” in (15), then we make use of the
$k\text{-}\operatorname*{argmax}$ operation defined as the $k$ indices for the
$k$ largest values in (16) to select the $L_{\text{{CSI}}}$ beam indices.
Finally, the combining coefficients are determined as the complex values of
the beam products (17) and normalized according to the largest coefficient
(18). The oversampling basis $Q_{0,X},Q_{0,Y}$ is reported back in the PMI so
that the overall beam selections can be reconstructed at the BS.

The CSI report contains the CSI-RS indicator (CRI) which is the strong CSI-RS
beamforming codeword, the rank of the PMI, the RSRP or SNR, and the PMI
feedback beam indices $\mathbf{Q}$, as well as $4$ bit amplitude and $8$-PSK
phase values. The amplitude and phase values come from $\mathbf{A}$ which are
the corresponding complex values for each $[n_{r},\ell]$ feedback beam. There
is an additional overhead reduction step in the specification where instead of
indexing each element of $\mathbf{Q}$ from the entire codebook, the set of
indices is translated to one of the ${N_{\text{X}}N_{\text{Y}}\choose
L_{\text{{CSI}}}}$ combinations and one index corresponding to the
oversampling factors of the strongest index $(Q_{0,X},Q_{0,Y})$ is used. In
this paper, we will ignore the amplitude and phase quantization (assuming
perfect amplitude and phase knowledge) to focus on the effects of the
codebooks, SNR, and feedback. The CSI report can also contain both wideband
and narrowband components for a degree of frequency selective precoding. The
number of narrowband components is small, at most $\text{BWP}=8$ in FR1, to
mitigate the growing feedback and complexity. All of the previous and
subsequent steps are defined for $\text{BWP}=1$, but extending to frequency
selective feedback and precoding is straightforward. We investigate how the
amount of feedback ($L_{\text{{CSI}}},\text{BWP}$) impacts performance in
Section IV.

###  II-H MU-MIMO data transmission

After the CSI-RS period, the base station must determine the precoder that
serves a group of users, often with the goal of maximizing the sum rate or
proportional fair rate. Until this stage, all of the downlink transmissions
have been multi-cast in the sense that the same pilots and data are intended
for all users. Therefore, there is no interference assuming a well-designed
OFDM cyclic prefix and subcarrier spacing is used with accurate
synchronization. In contrast, the downlink data transmission carries different
data for each user, resulting in interference between data streams. The
remainder of this section outlines the channel reconstruction process,
precoder selection, and achievable spectral efficiency in the downlink
channel.

First, because the feedback is type-II, the BS can reconstruct the PMI
(corresponding to the beamformed channel estimate here) with the corresponding
complex scaling $\mathbf{A}$ as

$\widehat{\widehat{\mathbf{HF}}}[n_{r}]=\sum_{\ell=0}^{L_{\text{{CSI}}}-1}\mathbf{A}[n_{r},\ell]\mathbf{B}_{\text{DFT}}[:,\mathbf{Q}_{n_{r},\ell}]\quad\forall\
n_{\text{r}}.$ (19)

Then, assuming the beamformers were well-chosen such that
$\mathbf{F}_{\text{CSI-RS}}$ is right-invertible, channel reconstruction can
be performed successfully. We reintroduce the UE $(u\color[rgb]{0,0,0})$
notation here to allow for aggregating all user channels in subsequent steps
by

$\displaystyle\widehat{\widehat{\mathbf{H}}}_{u\color[rgb]{0,0,0}}$
$\displaystyle=\widehat{\widehat{\mathbf{H}_{u\color[rgb]{0,0,0}}\mathbf{F}}}\mathbf{F}_{\text{CSI-
RS}}^{\dagger}.$ (20)

In the case of DFT beams, this process simply becomes an IDFT operation [3],
which is significantly more efficient than the matrix inverse of an
$N_{\text{P}}\times N_{\text{P}}$ matrix. The efficiency at this step is
critical to quickly using the feedback to serve users and is one reason we
will tailor our algorithm to integrate with a DFT codebook for CSI-RS.
Aggregating the results for each user, the base station obtains the downlink
channel estimate $\widehat{\widehat{\mathbf{H}}}\in\mathbb{C}^{U\times
N_{\text{R}}\times N_{\text{P}}}$

$\displaystyle\widehat{\widehat{\mathbf{H}}}$
$\displaystyle=\left[\widehat{\widehat{\mathbf{H}}}_{0},\widehat{\widehat{\mathbf{H}}}_{1},...,\widehat{\widehat{\mathbf{H}}}_{U-1}\right].$
(21)

The base station can design the precoders for the set of estimated channels
using any form of precoding. To reasonably evaluate the system we will assume
a regularized zero forcing precoder is used with regularization determined to
minimize the signal-to-leakage noise ratio [27, section 9.9]. The RZF precoder
is built up for each user as

$\displaystyle\mathbf{F}_{u\color[rgb]{0,0,0}}=\sqrt{\frac{N_{\text{T}}}{U}}\frac{(\sum_{i=0}^{U-1}\widehat{\widehat{\mathbf{H}}}_{i}^{*}\widehat{\widehat{\mathbf{H}}}_{i}+UN_{\text{T}}\mathbb{E}[\mathbf{N}_{t,k,u}^{2}])^{-1}\widehat{\widehat{\mathbf{H}}}_{u\color[rgb]{0,0,0}}^{*}}{\left\lVert(\sum_{i=0}^{U-1}\widehat{\widehat{\mathbf{H}}}_{i}^{*}\widehat{\widehat{\mathbf{H}}}_{i}+UN_{\text{T}}\mathbb{E}[\mathbf{N}_{t,k,u}^{2}])^{-1}\widehat{\widehat{\mathbf{H}}}_{u\color[rgb]{0,0,0}}^{*}\right\rVert}.$
(22)

The noise factor, $\mathbb{E}[\mathbf{N}_{t,k,u}^{2}]$ can be estimated based
on the RSRP/SNR reported and the bandwidth of the downlink signal. The UE will
also perform a combining strategy based on the previously determined rank. In
this work, we will assume an LMMSE receiver is used to maximize the signal-to-
interference noise ratio (SINR) for any precoder, using the embedded DMRS to
determine the combined channel and precoder. The resulting SINR at resource
element ($t,k$) for a given user $u\color[rgb]{0,0,0}$ and layer $r$ with
equal power allocation per user is obtained as

$\displaystyle\mathbf{I}_{t,k,u}$
$\displaystyle=\bigg{(}\sum_{i=0}^{U-1}\mathbf{H}_{u,t,k}\mathbf{F}_{i}\mathbf{F}_{i}^{*}\mathbf{H}_{u,t,k}^{*}+UN_{\text{T}}\mathbb{E}[\mathbf{N}_{t,k,u}^{2}]\bigg{)}^{-1}$
(23) $\displaystyle\text{SINR}_{t,k,u\color[rgb]{0,0,0},r}$
$\displaystyle=\frac{1}{K}\frac{\mathbf{F}^{*}_{u\color[rgb]{0,0,0}}[:,r]\mathbf{H}_{u,t,k}^{*}\mathbf{I}_{t,k,u}\mathbf{H}_{u,t,k}\mathbf{F}_{u\color[rgb]{0,0,0}}[:,r]}{1-\mathbf{F}^{*}_{u\color[rgb]{0,0,0}}[:,r]\mathbf{H}_{u,t,k}^{*}\mathbf{I}_{t,k,u}\mathbf{H}_{u,t,k}\mathbf{F}_{u\color[rgb]{0,0,0}}[:,r]}.$
(24)

While the SINR expression (24) appears complicated, it is a simplified ratio
between the signal power for the data stream $(u\color[rgb]{0,0,0},r)$ versus
the interference power of all other streams. The SINR at this step is not
calculated in a physical system, but it is necessary for evaluating the
achievable spectral efficiency.

Finally, the most critical metric in the wireless network is the sum SE or sum
rate when applied to a specific bandwidth. A fairness constraint can also be
applied, however, we have not specified any form of scheduling so we are most
interested in maximizing the sum spectral efficiency or rate. Assuming
Gaussian signaling and treating interference as Gaussian noise, the achievable
spectral efficiency, $\text{SE}_{u\color[rgb]{0,0,0},t}$, is

$\displaystyle\text{SE}_{u\color[rgb]{0,0,0},t}=\sum_{k=0}^{K-1}\sum_{r=0}^{R-1}{\log_{2}(1+\text{SINR}_{u\color[rgb]{0,0,0},t,k,r}}).$
(25)

In the final post-processing, we consider the effective sum SE, which accounts
for the overhead due to beam training by removing the corresponding
time/frequency resources due to training and feedback of the beam management
system $(T_{\text{BM}},K_{\text{BM}})$ from the spectral efficiency
calculation

$\displaystyle\text{Eff-SSE}=\sum^{U-1}_{u\color[rgb]{0,0,0}=0}\sum_{t\notin
T_{\text{BM}}}\sum_{k\notin
K_{\text{BM}}}\sum_{r=0}^{R-1}\log_{2}(1+\text{SINR}_{u\color[rgb]{0,0,0},t,k,r}).$
(26)

With this goal and the beam management framework in mind, we can now define
the beamspace processing and machine learning codebook design algorithm.

Figure 3: A timing diagram of the process for codebook learning and
evaluation. The AI/ML engine provides new codebooks at each SSB interval for
the base station. At each CSI-RS period, the BS transmits pilots using the
CSI-RS codebook, which is determined based on the SSB codebook and feedback.
The CSI-RS feedback is used to determine the precoder used for data
transmission. Figure 4: A depiction of the BSC pre-processing, neural
architecture, and post-processing. The feedback is converted to an angular
representation (beamspace) and the strongest directions are identified based
on the feedback reported. The neural network jointly processes the SSB
feedback and beamspace representation to produce a new beamspace codebook that
is then converted back to complex beamforming coefficients. This unifies the
various codebooks with a consistent input dimensionality even with varying
numbers of users or even antenna sizes.

##  III Beamspace-Codex

In this section, we present the Beamspace-Codex which is a neural network
architecture and processing setup that is used to generate the SSB codebooks,
with considerations for joint SSB-CSI-RS codebook design left to future work.
We start by introducing the beamspace observation that provides a consistent
basis for a learning algorithm using dynamic codebooks. We then present the
supervised learning formulation that enables convergent and dynamic codebooks.
Finally, we define the neural network architecture that enables learning the
underlying relationship between the beamspace observation and the SSB
codebook. The full methodology–processing, architecture, training, and
evaluation–is critical to integrating wireless domain knowledge and physical
structure into the learning algorithm.

One of the primary steps in designing an appropriate neural network is
formulating the input-output relationship to be conducive and consistent for
learning. There are two problems with neural networks within the context of 5G
beam management. First, the SSBRI feedback corresponds to a beamformer from a
codebook that was used previously, which does not provide a consistent
representation or meaning with dynamic codebooks. Second, the number of users
is not constant within a cell, so the size of the feedback and data dimension
changes over time. Although there are specific ways to overcome dynamic data
sizes, it often requires sacrificing hardware optimizations and impedes
inference and training times.

We propose transforming the codebook into the beamspace domain. The beamspace
is an angular representation of a beamformer corresponding to the array factor
evaluated over a range of directions [31]. This ensures that regardless of the
specific beamforming codeword, or even physical antenna dimension, the
corresponding input from a set of previous beamformers still represents a two-
dimensional grid of projections. The beamspace conversion for $N_{x0}$ azimuth
directions and $N_{y0}$ elevation direction is calculated as

$\displaystyle\bm{\theta}_{N_{m}}$
$\displaystyle=\frac{1}{\pi}[0,1,...,N_{m}-1]^{T}$ (27)
$\displaystyle\mathbf{U}_{N_{s},N_{m}}$
$\displaystyle\triangleq[\mathbf{a}_{N_{s}}(\bm{\theta}_{0}),...\mathbf{a}_{N_{s}}(\bm{\theta}_{N_{m}-1})]\in\mathbb{C}^{N_{s}\times
N_{m}}$ (28)
$\displaystyle\mathbf{F}_{\text{SSB}}^{(N_{\text{X}},N_{\text{Y}})}[i]$
$\displaystyle=[\mathbf{F}_{\text{SSB}}[i,0:N_{\text{X}}]^{T},\mathbf{F}_{\text{SSB}}[i,N_{\text{X}}:2N_{\text{X}}]^{T},...,\mathbf{F}_{\text{SSB}}[i,(N_{\text{X}}-1)N_{\text{Y}}:N_{\text{X}}N_{\text{Y}}]^{T}]$
(29) $\displaystyle\mathbf{O}_{\text{BSC}}[i]$
$\displaystyle=\mathbf{U}_{N_{x0},N_{\text{X}}}\mathbf{F}_{\text{SSB}}^{(N_{\text{X}},N_{\text{Y}})}[i]\mathbf{U}_{N_{\text{Y}},N_{y0}}\quad\forall
i\in\\{1,2,...L_{\text{{max}}}\\}.$ (30)

The conversion starts by generating an angular matrix from (28) for the
azimuth directions $\mathbf{U}_{N_{x0},N_{\text{X}}}$ and elevation directions
$\mathbf{U}_{N_{\text{Y}},N_{y0}}$ as a series of array responses. Then, the
codebook must be reshaped from a vector of size $N_{\text{T}}$ to the planar
dimensions $N_{X}\times N_{Y}$ in (29) before the beamspace conversion in (30)
that produces the beamspace observation, $\mathbf{O}_{\text{BSC}}$. In
addition to the angular representation, the input is also concatenated with
the feedback corresponding to the number of users reporting each beam and the
sum RSRP after min-max normalization. Note that the beamspace conversion is a
reversible operation, so we also train the network to predict the beamspace of
the desired output, rather than direct beamforming coefficients. The predicted
beamformers are obtained in post-processing, and the computational complexity
can be controlled by changing the observation sizes $N_{x0}$ and $N_{y0}$.
With the pre/post-processing defined, we can now define the supervised
learning environment and desired outputs used for training.

Supervised learning provides many advantages over unsupervised or
reinforcement learning when there is a desired output compared to just a
black-box metric. The task of determining beamforming vectors that maximize
the power to a set of users has an optimal solution if CSI is available via
singular value decomposition (SVD) beamforming. To that end, we assume a
dataset of channels has been built up or simulated to allow for offline
training, although our site-specific results in Section IV suggest that
transferring a learned model between sites is still more effective than DFT
codebooks. Then, the solution to maximize the power received for a user is the
right singular vectors (RSV) of the SVD of the channel [4]. In the MU-MIMO
OFDM case, this means taking the RSVs corresponding to the $L_{\text{{max}}}$
strongest singular values of all users. Additionally, the timing of the beams
is important and each beam has a specified set of resources on which to be
transmitted. We must sacrifice the time relationship of the data, however, to
build a true supervised setting. In particular, we do not use information from
the past SSB feedback beyond the previous step. This limitation removes the
capability of using historical data as inputs but significantly improves
convergence compared to a reinforcement learning setting. The process is
restricted to a two-step relationship where we include the previous codebook
and feedback to help design the next codebook. We now have a supervised
framework and pre/post-processing step that converts the previous codebook and
RSV codebook to the beamspace domain. The final component of the Beamspace-
Codex is the neural architecture that facilitates efficient learning.

Determining the best neural network architecture is often challenging even
with hyperparameter tuning and an intuitive understanding of the problem.
Using the beamspace conversion, the problem can roughly be seen as a
translation task from the initial beamspace to the RSV beamspace. It is not
necessarily obvious how the limited feedback information could be used to
recover the RSV codebook. There is some underlying information available,
however, by the selection of one beam over $L_{\text{{max}}}-1$ other options.
Additionally, the environment and user densities follow a pattern that is also
possible to exploit. At the same time, the inputs and outputs are similar to
an image of the angular directions of transmitted power. Therefore, we
considered traditional architectures (fully-connected networks), image-based
architectures (convolutional and diffusion [32] networks), and translation-
based architectures (transformers [33] and autoencoders). Although the problem
has many similarities with both vision and translation problems, state-of-the-
art architectures were not necessarily best under reasonable computational
limits i.e. $24$GB of VRAM and $5$ms inference time. Throughout our testing,
the fully connected architecture showed the strongest results with an average
of $1$dB higher average RSRP than other architectures, while also being the
most computationally efficient one.

We propose a multi-layer fully connected network with significant
regularization through dropout comprised of $6$ densely connected layers with
parameters and sizes determined through Bayesian hyperparameter tuning and
defined in Table I. The model is trained using an Adam optimizer [34] with
early stopping and cosine learning rate decay. Training is performed with a
$550,000$ sample training dataset following the setup in Section V with
validation and test sets each from an unseen portion of the data corresponding
to $160,000$ and $80,000$ samples respectively. Gradients are determined by
the cosine distance
($d_{\text{cos}}(\mathbf{a},\mathbf{b})=1-\mathbf{a}^{*}\mathbf{b}/(\left\lVert\mathbf{a}\right\rVert\left\lVert\mathbf{b}\right\rVert)$)
between the RSV beamspace and the neural network-predicted beamspace. The
choice of the loss function is critical for performance as the same network
architectures trained on mean squared error results in significantly worse
($>5$dB) performance due to focusing on the pixel-wise magnitude, rather than
the relative magnitude or direction of the RSV beamspace.

TABLE I: Beamspace-Codex network parameters Layer | Primary Parameter | Activation Func. | Output Dimension
---|---|---|---
Flatten |  |  | ($2L_{\text{{max}}}(N_{x0}+2)(N_{y0}+2)$)
Fully Conn. | 144 Neurons | ReLU | (144)
Dropout | 0.2 Rate |  | (144)
Fully Conn. | 1808 Neurons | ReLU | (1808)
Dropout | 0.4 Rate |  | (1808)
Fully Conn. | 272 Neurons | ReLU | (272)
Dropout | 0.4 Rate |  | (272)
Fully Conn. | 240 Neurons | ReLU | (240)
Fully Conn. | 80 Neurons | ReLU | (80)
Fully Conn. | ($2N_{x0}N_{y0}L_{\text{{max}}}$) Neurons |  | ($N_{x0},N_{y0},2L_{\text{{max}}}$)
Reshape |  |  | ($N_{x0}$, $N_{y0}$, $2L_{\text{{max}}}$)

##  IV Data generation setup

We require an accurate channel simulator coupled with a flexible framework
that retains the timing of the 5G beam training process to fairly evaluate the
performance of ML-designed codebooks. We integrate the channel generation from
QuaDRiGa [24] with a post-processing suite that enables an initial access
scenario, beamspace generation, codebook determination, and evaluation of the
network performance after accounting for beam management overhead. QuaDRiGa
generates channels through a stochastic process with additional features for
spatial consistency, correlations, and spherical wave modeling. In our
simulations, we simulate $200$ users scattered over a base station sector
site. The BS is equipped with an $N_{\text{X}}=4$, $N_{\text{Y}}=8$ planar
array using a 3GPP 3D antenna model with half-wavelength spacing. Each user
has $N_{\text{R}}=4$ antennas all tuned for a 3.5GHz carrier frequency. $10\%$
of users are given a vehicular mobility pattern such that they travel along a
roadway through the center of the cell with speeds normally distributed with
mean $25$m/s and variance $5$m/s. The remaining users are uniformly scattered
within $450$m and travel in any direction with speeds uniformly drawn from an
interval of $[0,3]$m/s. Channel distributions follow a 3GPP urban macrocell
environment and channels are sampled over a bandwidth of $100$MHz every $1$ms
for $2$s. A large number of users ($\approx 200$) are simulated to build up a
database that is spatially and environmentally consistent. In the final
section of the results, we address how the model generalizes to a new setting
with different environmental parameters and different roadway locations.

Once the channels are generated, post-processing is necessary to produce a
realistic setting and build up the algorithm’s datasets. The process starts by
randomly selecting a number of active users (uniformly between $[4,12]$) and
gathering the channels and time slots from the channel set. Then randomly
selecting $L_{\text{{max}}}$ DFT beams for the ‘prior’ codebook and
calculating the RSV codebook. The codebooks are converted to beamspace
representations and stored in a dataset along with the SSB feedback from a
subset of the active users where each active user is assigned an $80\%$
probability of its feedback being included. This represents an initial access
channel where not all users are previously known and introduces an additional
regularization term into our algorithm to prevent over-focusing on known
users. It is assumed that UEs use antenna selection for the RSRP reception
during the SSB process, digital combining for CSI-RS, and LMMSE combining
during data transmission as defined in the system model.

##  V Simulation results

It is critical to extensively evaluate a neural network model to fairly
represent and compare it against traditional methods. In our previous work
[35], we have considered a learning threshold as a minimal condition for
determining if meaningful relationships have been learned by the network. In
realistic wireless channels, such settings are much harder to characterize, so
we instead benchmark the performance against industry-standard DFT techniques
and RSV codebooks assuming CSI were known perfectly. This helps to identify
the upper-performance limits (with respect to maximum RSRP) as well as the
minimal performance needed to justify moving beyond traditional codebooks. We
then use our robust simulation framework to better understand how the various
codebooks and type-II CSI feedback parameters affect network performance in
realistic FR1 channels. In the first result, the RSRP performance of the SSB
process with various codebooks is plotted. Then the subsequent SNR achieved
during CSI-RS is evaluated, where the CSI-RS beams are determined by
decomposing the SSB beam into a subset of DFT beams with an equal proportion
to the number of users selecting the beam during SSB.

###  V-A RSRP and SNR

The RSRP is a basic metric that can be used to directly demonstrate the
ability of the neural network to output a valuable SSB codebook. Following the
SSB process, we also highlight the performance that the dynamic codebook
provides when selecting CSI-RS beams as a decomposition of the SSB codebook.
While these two metrics–RSRP and SNR–do not ultimately characterize the
network performance as a result of the codebook algorithm, they do provide
meaningful insight into the performance of various codebook methods. Figure 5a
shows the empirical cumulative distribution function (CDF) of the RSRP
reported using various codebook/beamforming methods compared to our proposed
BSC algorithm. Without any beamforming, more than $50\%$ of users would be
below the minimum RSRP for signal detection, which is often around $-120$dBm.
The SSB codebooks are decomposed into CSI-RS codebooks in Figure 5b, where we
see the RSV codebook is not consistently the best SSB codebook when directly
decomposed into CSI-RS codebooks, although the decomposed BSC is not
universally better than decomposed DFT codebooks either. We can understand
this as a result of the rich scattering environment that causes the RSV to not
effectively decompose into a small number of incoherent DFT components. While
RSV beamformers maximize the received power for a user, this maximization is
based on the signal coherently combining and is not well-represented by a
narrow DFT beam. From these results, we can see that the BSC algorithm is
advantageous compared to traditional methods, but we also see that restricting
the CSI-RS codebook to oversampled DFT beams is a significant limitation.

(a) SSB RSRP  (b) CSI-RS SNR

Figure 5: Two plots of the codebook performance measuring (a) RSRP received
during SSB and (b) SNR during CSI-RS reception. During the SSB stage, each of
the codebooks is used directly and then in (b) the SSB codebooks are
decomposed to form the active CSI-RS codebook subset. Non-beamformed
transmission presents a significant loss in performance shown by the “gain of
BF” arrow, while the learned BSC codebook shows gains of almost $3$dB over
traditional DFT beams. While RSV beamforming is RSRP-maximizing during the
SSB, the DFT decomposition of an RSV codebook is not necessarily SNR-
maximizing due to multipath propagation.

###  V-B CSI-RS reporting

In this subsection, we provide an in-depth evaluation of the CSI-RS process
without BSC codebooks. We evaluate performance based on the effective downlink
SE after overhead due to beam training, uplink feedback, and an additional
signaling overhead factor of $10\%$. Uplink feedback is assumed to be the only
resource allocated for uplink transmission, and we assume a worst-case
scenario where users transmit with MCS0 from Table 5.1.3.1-1 [25] with users
allocated at the resource block level to account for additional overhead. User
selection during the data phase is performed based only on the knowledge at
the BS by first estimating the sum spectral efficiency achieved with the
imperfect CSI for all users with $\\{1,2\\}$ data layers. The best estimated
SE combination is selected and used. Such a format is naturally suboptimal,
even with perfect CSI, due to a lack of rank and power adaption. User
selection is not the focus of this investigation, so we only use an exhaustive
search over the reconstructed channel knowledge available at the BS to provide
a basic but reasonable scheduling algorithm. The SE is also compared with
rank-1 non-PMI beamforming where RZF is not employed, the channel is not
estimated, and the only feedback is obtained from the CRI, with user selection
and precoder determination simply being the CSI-RS beamforming codewords with
the strongest reported user RSRP. This is often an envisioned beam management
strategy for sparse mmWave channels with short, low-latency feedback, but is
not assumed to be effective in sub-6GHz channels due to the rich scattering
environment.

The size of the CSI-RS codebook is related to the channel estimation SNR
because larger CSI-RS codebooks, made of orthogonal DFT beamformers, will
result in equivalent or higher SNR compared to smaller codebooks with the same
channels and feedback. We characterize the performance of larger or smaller
CSI-RS codebooks in Figure 6a. Although the estimation SNR, and by extension
performance before quantization, will always improve with a larger codebook,
there is also a modest amount of overhead with larger $P_{\text{CSI}}$ so that
there is little spectral efficiency gain beyond $P_{\text{CSI}}\geq 8$.

In the next comparison, the feedback quantization is evaluated by modifying
the parameter $L_{\text{{CSI}}}$ in type-II formats. We can see from Fig. 6b
that higher resolution feedback is beneficial, with the effective data rate
exceeding $2$ times higher when the resolution is improved from
$L_{\text{{CSI}}}=1$ to $L_{\text{{CSI}}}=32$. From there on, the other
parameters CSI-RS resource block allocation (NRB) and the number of frequency
selective resources (BWP), are less influential compared to the quantization
resolution. Most of the modest performance gains achieved with multiple
frequency selective precoders and larger downlink CSI-RS resource usage are
offset by the increasing overhead resulting in negligible or even reductive
performance impacts. Joining the information from Figures 6a-7a, a good
balance of feedback and performance can be achieved with a setting of
$L_{\text{{CSI}}}=32$, $\text{BWP}=1$, $\text{NRB}=24$, $P_{\text{CSI}}=8$. It
is important to note that the current 5G Release 16, even with enhanced type-
II feedback, only supports $L_{\text{{CSI}}}<=6$, which means most situations
will only yield about $1.5$ times higher effective spectral efficiency
relative to SU-MIMO, and essentially equivalent performance to the non-PMI
beamforming. Perhaps even more surprising is that using a single CSI-RS beam
($P_{\text{CSI}}=1$) has less degradation than using $L_{\text{{CSI}}}=8$
feedback beams per rank, relative to the best performance. This suggests that
feedback limitations are more likely to restrict MU-MIMO performance in
sub-6GHz systems compared to codebook optimization or accurate channel
estimation.

(a) Codebook size $P_{\text{CSI}}$ (b) Quantization paths $L_{\text{{CSI}}}$

Figure 6: Evaluation of the effects of active CSI-RS codebook size (a) and
quantization using $L_{\text{{CSI}}}$ feedback components (b). We can use the
size of the codebook as a means of looking at how codebook optimization
improves performance because the overhead with larger $P_{\text{CSI}}$ is very
small compared to the overhead of type-II feedback. SU-MIMO results using
type-I PMI are shown by dashed lines while MU-MIMO results are shown by solid
lines and Non-PMI refers to rank $1$ beamforming using the CRI. SU-MIMO does
not include multipath feedback so only a single SU-MIMO result is shown in
(b). Other parameters are set to $L_{\text{{CSI}}}=32$ for (a),
$P_{\text{CSI}}=16$ for (b), $\text{BWP}=1$, $\text{NRB}=24$. We can see that
the loss from using $L_{\text{{CSI}}}=4$ (3GPP Release 16 maximum support)
compared to $L_{\text{{CSI}}}=32$ is more detrimental than restricting the
system to a single CSI-RS beamformer.

(a) BWP  (b) NRB

Figure 7: A comparison of the effective spectral efficiency with different
numbers of frequency selective precoders BWP and varying the number of
resource blocks (NRB) per BWP. It is assumed that $L_{\text{{CSI}}}=32$,
$P_{\text{CSI}}=16$, $\text{NRB}=24\times\text{BWP}$ in (a) and $\text{BWP}=1$
in (b). While increasing the number of resource blocks, therefore increasing
the number of pilots, also increases the channel estimation performance, the
rising feedback overhead causes the performance to decrease in (b). In
comparison, the number of BWP increases the overhead linearly, yet the
performance is essentially equivalent in (a). Therefore we see that frequency
selective feedback and beamforming is not necessarily advantageous when
overhead is accounted for.

###  V-C Site generalization

Up to now, we have used a test set of data drawn from the same statistical
environment as the training set. While a network operator will likely gather
data for a specific site, here we examine the impact of testing on a new
distribution of data. We simulate a new environment with completely different
user distributions and mobility patterns. Now $50\%$ of users are vehicular
with roadways defined in new locations and a new stochastic realization of the
channel environment. In Figure 8 the agnostic (orange) data is obtained
without updating the model, whereas the fine-tuned data (blue) shows the
results after we allow for $60$s of retraining on data from the new
environment. Our time limit is arbitrary and could ultimately be orders of
magnitude shorter than the timescale a network operator might use to update
the model, but corresponds to approximately $1\%$ of the training time used in
our setup for a balance of fine-tuning and generalization. It can be seen in
Figure 8 the performance in the new environment (agnostic) is significantly
worse relative to the RSV. The performance loss is noticeably improved during
fine-tuning, especially for the lowest $10\%$ of users.

Figure 8: Histogram representing the performance delta between BSC and RSV
codebooks in the agnostic setting and after fine-tuning. It can be seen that
the worst-case differences are resolved and in some cases (when more users are
present than SSB beams) the RSV codebook can be outperformed based on the
higher proportion of the blue columns to the right of the orange columns.

##  VI Conclusion

In this paper, we presented a novel framework for learning dynamic codebooks
for sub-6GHz 5G NR. We set up a system with multiple UEs each equipped with a
uniform linear array to receive from a base station with a fully digital
planar array. We simulate realistic sub-6GHz channels using QuaDRiGa and build
a post-processing framework for evaluating the system performance of SSB and
CSI-RS codebooks and feedback. Using this framework we address two questions
related to codebook performance in 5G: 1) Can machine learning design more
effective SSB codebooks? 2) How should feedback be configured for MU-MIMO in
sub-6GHz environments?

We first proposed a codebook transformation based on beamspace projections
that allowed for efficiently and consistently representing SSB codebooks in a
format conducive to learning. This beamspace representation can circumvent
challenges with dynamic user numbers, changing codebooks, and even deploying
across different array sizes. Using the beamspace as a translation tool, we
designed a neural architecture, Beamspace-Codex, to output new codebooks from
SSB feedback. The proposed method directly integrates with current 5G
standards and improves SSB RSRP by $2$-$3$dB on average. We also show that
only $1\%$ of the training compute budget is needed for retraining in new
environments to significantly improve the worst-case performance. Secondly, we
found that the PMI quantization resolution, determined by the number of
feedback DFT beams, is a limiting factor in the performance of MU-MIMO in
current standards.

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