3GPP short=3GPP, long=3rd generation partnership project ADC short=ADC,
long=analog-to-digital converter AMP short=AMP, long=approximate message
passing ANN short=ANN, long=artificial neural network AoA short=AoA,
long=angle-of-arrival AoD short=AoD, long=angle-of-departure APS short=APS,
long=azimuth power spectrum AR short=AR, long=augmented reality AV short=AV,
long=autonomous vehicle BM short=BM, long=beam management BS short=BS,
long=base station BSM short=BSM, long=basic safety message CDF short=CDF,
long=cumulative distribution function CP short=CP, long=cyclic-prefix CSI-RS
short=CSI-RS, long=Channel State Information Reference Signal DFT short=DFT,
long=discrete Fourier transform DL short=DL, long=downlink EKF short=EKF,
long=extended Kalman filter DSRC short=DSRC, long=dedicated short-range
communication FDD short=FDD, long=frequency division duplex FMCW short=FMCW,
long=frequency modulated continuous wave FoV short=FoV, long=field-of-view
GNSS short=GNSS, long=global navigation satellite system IA short=IA,
long=initial access IMU short=IMU, long=inertial measurement unit lidar
short=lidar, long=light detection and ranging LOS short=LOS, long=line-of-
sight LPF short=LPF, long=low pass filter LTE short=LTE, long=long term
evolution MIMO short=MIMO, long=multiple-input multiple-output ML short=ML,
long=machine learning mmWave short=mmWave, long=millimeter wave MRR short=MRR,
long=medium range radar NLOS short=NLOS, long=non-line-of-sight NB short=NB,
long=narrow beam NR short=NR, long=new radio OFDM short=OFDM, long=orthogonal
frequency-division multiplexing ppm short=ppm, long=parts-per-million PF
short=PF, long=particle filter RMS short=RMS, long=root-mean-square RPE
short=RPE, long=relative precoding efficiency RS short=RS, long=reference
signal RSRP short=RSRP, long=reference signal received power RSU short=RSU,
long=roadside unit SCS short=SCS, long=subcarrier spacing SNR short=SNR,
long=signal-to-noise ratio SRS short=SRS, long=Sounding Reference Signal SSB
short=SSB, long=Synchronization Signal Block THz short=THz, long=terahertz UAV
short=UAV, long=unmanned aerial vehicle UE short=UE, long=user equipment UKF
short=UKF, long=unscented Kalman filter UL short=UL, long=uplink ULA
short=ULA, long=uniform linear array V2I short=V2I, long=vehicle-to-
infrastructure V2V short=V2V, long=vehicle-to-vehicle V2X short=V2X,
long=vehicle-to-everything VR short=VR, long=virtual reality VRU short=VRU,
long=vulnerable road user WB short=WB, long=wide beam BF short=BF,
long=beamforming CS short=CS, long=compressive sensing NN short=NN,
long=neural network RF short=RF, long=radio frequency MISO short=MISO,
long=multiple-input single-output SIMO short=MISO, long=single-input multiple-
output MLP short=MLP, long=multilayer perceptron AMCF short=AMCF,
long=alternative minimization method with a closed-form expression AWGN
short=AWGN, long=additive white Gaussian noise t-SNE short=t-SNE,
long=t-distributed stochastic neighbor embedding TDD short=TDD, long=time
division duplex mMTC short=mMTC, long=massive Machine Type Communications

# Learning Site-Specific Probing Beams for Fast mmWave Beam Alignment

Yuqiang Heng, Jianhua Mo and Jeffrey G. Andrews Yuqiang Heng and Jeffrey G.
Andrews are with the Wireless Networking and Communications Group (WNCG), The
University of Texas at Austin, Austin, TX 78701 USA. Email:
<EMAIL_ADDRESS>jandrews@ece.utexas.edu).Jianhua Mo is with Samsung
Research America, Plano, TX 75023, USA. Email<EMAIL_ADDRESS>conference version of this work is submitted to IEEE GLOBECOM 2021 [1].

###### Abstract

Beam alignment – the process of finding an optimal directional beam pair – is
a challenging procedure crucial to mmWave communication systems. We propose a
novel beam alignment method that learns a site-specific probing codebook and
uses the probing codebook measurements to predict the optimal narrow beam. An
end-to-end NN architecture is designed to jointly learn the probing codebook
and the beam predictor. The learned codebook consists of site-specific probing
beams that can capture particular characteristics of the propagation
environment. The proposed method relies on beam sweeping of the learned
probing codebook, does not require additional context information, and is
compatible with the beam sweeping-based beam alignment framework in 5G. Using
realistic ray-tracing datasets, we demonstrate that the proposed method can
achieve high beam alignment accuracy and SNR while significantly – by roughly
a factor of 3 in our setting – reducing the beam sweeping complexity and
latency.

###### Index Terms:

5G mobile communication, Beam steering, Beam management, Beam codebook,
Machine learning, Millimeter wave communication, Supervised learning.

##  I Introduction

Cellular systems will increasingly tap into the mmWave spectrum to provide
higher data rates and to support a wide range of emerging use cases. For
example, the current release of 5G adopts several mmWave bands between 24.25
GHz and 52.6 GHz, while future releases are expected to further expand the
spectrum to 71 GHz and even the so-called “Terahertz” bands extending up to
300 GHz [2]. While these high carrier frequencies allow much larger
bandwidths, they also impose harsher propagation conditions and utilize large
arrays of very small antenna elements, and thus rely on highly directional BF
to maintain viable received signal strength. Meanwhile, these directional
links are highly sensitive to blockage and reflections, so beam alignment –
finding and maintaining near-optimal analog BF weights, including for NLOS
paths – is essential. MmWave devices typically adopt codebooks of indexed
analog beams to allow good beams to be identified by the receiver and fedback
to the transmitter. These codebooks will contain much more numerous and much
narrower beams as higher carrier frequencies are adopted, making the latency
and beam sweeping overhead of traditional beam searches prohibitive. As a
result, beam alignment will become an increasingly critical bottleneck in the
future.

###  I-A Background and Related Work

The current release of 5G adopts a beam alignment framework based on beam
sweeping, measurements and reporting [3],[4],[5]. In the DL, the BS transmits
RS such as SSB and CSI-RS using different beams to sweep the angular space, as
illustrated in Fig. 1. The UE uses a quasi-omnidirectional beam or sweeps its
beam codebook using different receiving beams, measures the receive signal
power, then reports the RS measurements to the BS. With exhaustive beam
sweeping, the BS and the UE need to search all combinations of beam pairs,
resulting in significant beam sweeping overhead and latency. The SSB are
transmitted periodically and are “always-on”. They are also used in cell
discovery and IA for new UE. In order for an unconnected UE to achieve
synchronization before accessing the network, it needs to measure the SSB
transmitted by the BS, find one associated with a good beam and derive the
necessary information from that SSB. Since beam sweeping is essential for both
beam alignment and cell search, a beam sweeping-based framework is likely
going to stay in future releases of 5G.

Figure 1: Beam sweeping-based beam alignment framework in 5G. The BS transmits
beamformed SSB and CSI-RS. The UE measures and reports the quality of the RS.
The BS selects a beam for future data or control transmission.

Hierarchical beam searches have been proposed to reduce the beam sweeping
complexity [6],[7]. The BS and the UE, equipped with multiple-tier codebooks,
sweep wider beams first and iteratively thin the search space for the best
narrow beam. Since mmWave systems often employ analog or hybrid BF, the
hardware constraints need to be considered when designing the wide beams in
these hierarchical codebooks. Different hierarchical codebook design
techniques have been proposed in recent works such as [8] and [9]. While the
hierarchical search reduces the number of beams swept compared to an
exhaustive one, the search procedure needs to be repeated for each UE,
marginalizing the gain for multiple UEs. They are also more susceptible to
search errors caused by noise in received signal and imperfect wide-beam
patterns. The hierarchical search method proposed in [9] uses wide beams with
multiple mainlobes to reduce the beam sweeping overhead for multiple UEs.
However, the intermediate-layer beams need to be dynamically generated based
on measurements of upper-layer beams, which lacks standardization support from
5G and also significantly increases the size of the effective hierarchical
codebook.

In addition to the beam sweeping-based approaches, beam alignment methods that
utilize context information has been explored. In [10],[11] and [12], the
location information of UEs are used to reduce the beam search space. Beam
alignment methods that utilizes sub-6 GHz measurements are proposed in [13],
[14] and [15]. In [16], omni-directionally received sounding signals are used
to predict the optimal beam. A beam alignment method assisted by radar
measurements is proposed in [17]. However, such context information can be
hard to obtain since mmWave devices need to be equipped with the required
additional sensors. The feedback of such context information also incurs
additional overhead and sometimes requires a more robust sub-6 GHz link
between the BS and the UE.

ML solutions have been explored for the beam alignment problem. The pattern
extraction and function approximation powers of ML models make them
particularly suitable for processing a wide range of context information, such
as location [11], sub-6 GHz channels [14] and omni-directional sounding
signals [16]. A joint BF, power control and interference coordination method
using reinforcement learning (RL) is proposed in [18]. A beam alignment method
that uses CS to leverage channel sparsity is proposed in [19]. In [20], a deep
learning architecture is used to learn CS matrices and predict the best beams.

Compared to the beam sweeping-based approaches, beam alignment solutions that
rely on context information often require an additional cell search procedure
to discover unconnected new UE regardless of whether traditional ML or deep
learning techniques are used. The feedback of additional context information
requires the UE to be connected to the network through mmWave links or sub-6
GHz side links, which can be problematic during IA. Furthermore, solutions
that do not adopt beam sweeping are not compatible with the beam alignment
framework in 5G. Significant modifications to the 5G standard is required to
accommodate these approaches.

The NN architecture proposed in this work consists of a complex layer which
represents the analog beam codebook and an MLP which acts as the beam
selector. The complex-NN layer used in this work was first proposed in [21]
for computer vision and audio-related tasks. A similar complex fully-connected
layer is used in [22], where the authors optimize beam patterns for particular
environments and hardware imperfections. This work focuses on finding an
optimal beam from a large predefined narrow-beam codebook and differs from
[22] which focuses on direct codebook learning.

In [23], a beam alignment method that trains a NN to predict optimal beams
using UL measurements from a sparse probing codebook is proposed. However, the
probing codebooks used in [23] are predetermined undersampled DFT codebooks
with evenly spaced narrow beams, whereas the probing codebook in our proposed
method are site-specific and learned using a complex-valued NN module. The NN
in [23] requires knowledge of the complex received signals, whereas our
proposed method only need the received power. We demonstrate in Section V-F
that our learned probing codebooks are much more effective at capturing
characteristics of the environment and providing useful information to the
beam predictor.

###  I-B Contributions

In this work, we propose a beam alignment method that uses the beam sweeping
measurements of a probing codebook to predict the optimal narrow beam. The
proposed method is based on beam sweeping and does not require any additional
context information, which is compatible with the beam alignment framework in
5G. By jointly training the probing codebook and the beam predictor using a NN
in an end-to-end fashion, the probing codebook is able to learn particular
characteristics of the propagation environment and optimize its beam patterns
accordingly. Some key features of the proposed method are summarized as
follows.

Trainable site-specific probing codebook: A complex-NN module is used to
parameterize the probing codebook during training so that the BF weights can
be extracted and implemented using actual RF chains during deployment. The
probing codebook is able to learn particular characteristics of the
propagation environment and optimize its beams to capture the channel
information effectively. The proposed architecture can be adopted by BS in
various deployment scenarios with arbitrary array geometry.

Compatibility with 5G framework: The proposed method can be directly adopted
without modifications to the 5G standard. It does not require the collection
and feedback of hard-to-obtain context information such as UE location or out-
of-band information, which needs additional standardization support. Instead,
the proposed method uses beam sweeping measurements of a probing codebook,
which is exactly compatible with the beam sweeping-based framework currently
adopted in 5G. The probing beams can be transmitted using SSB, which can also
be used for cell discovery and IA.

High beam alignment accuracy and SNR: We demonstrate using multiple realistic
ray-tracing datasets that the proposed method can achieve high beam alignment
accuracy and SNR, beating the hierarchical beam search baselines. For
instance, the proposed method can achieve a beam alignment accuracy of over
90% and can outperform even the exhaustive search in terms of the average SNR.

Reduced beam sweeping overhead: The proposed method has lower beam sweeping
overhead compared to exhaustive and hierarchical beam searches, especially
when considering beam alignment for multiple UE. For instance, when
considering simultaneous beam alignment for 10 UE, the proposed method is
about 3$\times$ faster compared to exhaustive and hierarchical beam searches.

Applicable to a wide range of propagation scenarios: Multiple accurate ray-
tracing datasets modelling a wide range of propagation environments are used
to evaluate the performance of the proposed method. The proposed beam
alignment approach consistently achieves high accuracy and SNR in indoor and
outdoor environments, for LOS and NLOS UE, and with 28 GHz and 60 GHz carrier
frequencies.

The rest of this article is organized as follows. The system model is
described in Section II. The proposed beam alignment approach, the appropriate
metrics and the baselines of comparison are explained in Section III. The
datasets used are described in Section IV. The simulation results are
presented in Section V. Finally, the conclusion and final remarks are provided
in Section VI.

##  II System Model

A DL MISO system is considered, where each BS has an antenna array of $N_{t}$
elements and each UE has a single antenna. While UE typically have antenna
arrays also, we consider a MISO scenario where beam alignment is only
performed on the BS side for simplicity. The MISO model is also applicable to
the mMTC use case of 5G, where the each sensor would likely use a single
antenna and an isotropic beam pattern. The extension to receive beam alignment
on the UE side is left to future work. A ray-based narrowband block-fading
mmWave channel model with $N_{p}$ paths is considered [24]:

$\displaystyle\mathbf{h}=\sum_{l=1}^{N_{p}}\alpha_{l}\mathbf{a}(\phi_{l}^{D},\theta_{l}^{D}).$
(1)

For each path $l$, its complex gain is $\alpha_{l}$, the azimuth and elevation
angles of departure are $\phi_{l}^{D}$ and $\theta_{l}^{D}$, and the array
steering vector at these angles is denoted by
$\mathbf{a}(\phi_{l}^{D},\theta_{l}^{D})$. For a ULA with $N_{t}$ antenna
elements on the $y$-axis, its beam steering is limited to the azimuth domain
and its steering vector can be written as

$\mathbf{a_{ULA}}(\phi_{l})=\frac{1}{\sqrt{N_{t}}}\begin{bmatrix}1&e^{j\frac{2\pi}{\lambda}d\sin\phi_{l}}&\cdots&e^{j(N_{t}-1)\frac{2\pi}{\lambda}d\sin\phi_{l}}\end{bmatrix}^{T},$
(2)

where $\lambda$ is the carrier wavelength and d is the antenna spacing [25].
While a ULA is considered instead of a planar array for simplicity, the
proposed beam alignment approach is array-geometry agnostic and can be applied
to arrays of arbitrary geometry, as we will explain in Section III-A1.

Due to the cost and complexity of fully digital BF at mmWave frequencies, each
BS is assumed to perform analog-only or hybrid BF. For the purpose of beam
alignment, only the RF domain processing is considered. For a BS-UE pair, the
BS is assumed to employ a single RF chain to which all antenna elements are
connected. Analog BF is assumed to be implemented using phase shifters
connected to each antenna element. The BF vector can be written as

$\mathbf{v}=\frac{1}{\sqrt{N_{t}}}\begin{bmatrix}e^{j\theta_{1}}&e^{j\theta_{2}}&\cdots&e^{j\theta_{N_{t}}}\end{bmatrix}^{T},$
(3)

where $\mathbf{v}$ satisfies the power constraint and each element of
$\mathbf{v}$ satisfies the constant modulus constraint.

In the DL, the BS transmits a symbol $s\in\mathbb{C}$ satisfying average power
constraint $\mathbb{E}[|s|^{2}]=1$ to the UE using a BF vector $\mathbf{v}$.
The received signal at the UE can be written as

$y=\sqrt{P_{T}}\mathbf{h}^{H}\mathbf{v}s+n,$ (4)

where $P_{T}$ is the transmit power, $\mathbf{h}\in\mathbbm{C}^{N_{t}\times
1}$ is the channel vector and $n$ is the complex additive noise with noise
power $\sigma_{n}^{2}$.

The SNR for a UE with channel $\mathbf{h}$ and using a BF vector $\mathbf{v}$
can be written as

$\text{SNR}=\frac{P_{T}|\mathbf{h}^{H}\mathbf{v}|^{2}}{\sigma_{n}^{2}}.$ (5)

The BS has a codebook $\mathbf{V}\in\mathbbm{C}^{N_{t}\times N_{\mathbf{V}}}$
of predefined narrow analog beams that are used for the data or the control
channel, where each column of $\mathbf{V}$ represents the BF weights of a
beam. The size of the narrow-beam codebook $N_{\mathbf{V}}$ is assumed to be
large since $\mathbf{V}$ needs to cover the entire angular space. For a BS and
a UE, the optimal narrow beam index is the one that achieves the maximum SNR:

$i_{\mathbf{v}}^{*}=\mathop{\rm
arg\,max}_{i\in\left\\{1,2,\cdots,N_{\mathbf{V}}\right\\}}\left(\frac{|\mathbf{h}^{H}\mathbf{v}_{i}|^{2}P_{T}}{\sigma_{n}^{2}}\right)=\mathop{\rm
arg\,max}_{i\in\left\\{1,2,\cdots,N_{\mathbf{V}}\right\\}}(|\mathbf{h}^{H}\mathbf{v}_{i}|^{2}),$
(6)

where $\mathbf{v}_{i}$ is the $i$th beam, i.e., the $i$th column of
$\mathbf{V}$.

##  III The Proposed Method, Metrics and Baselines

We propose a beam alignment method that is compatible with the beam sweeping-
based framework in 5G so that it can serve to achieve both beam alignment for
connected UE and IA for unconnected UE using only beam sweeping measurements.
With the proposed method, the BS first sweeps a small probing codebook to
gather information about the channel then selects candidate narrow beams based
on the probing-codebook measurements. In addition to the size-$N_{\mathbf{V}}$
narrow-beam codebook $\mathbf{V}\in\mathbbm{C}^{N_{t}\times N_{\mathbf{V}}}$
that is used for the data or the control channel, the BS also has a probing
codebook $\mathbf{W}\in\mathbbm{C}^{N_{t}\times N_{\mathbf{W}}}$ with
$N_{\mathbf{W}}$ beams. The size of the probing codebook $N_{\mathbf{W}}$ is
much smaller than $N_{\mathbf{V}}$. The BS first sweeps its probing codebook
$\mathbf{W}$. All UE connected to the BS measure and report the received power
of the probing signals. The beam sweeping, measurement and reporting is
assumed to be completed within the coherence time during which the channel
remains the same. The reported beam sweeping results for each UE $u$ can be
written as

$\displaystyle\mathbf{x}$
$\displaystyle=\begin{bmatrix}|y_{1}|^{2}&\cdots&|y_{N_{\mathbf{W}}}|^{2}\end{bmatrix}^{T},$
(7)

where $y_{i}=\sqrt{P_{T}}\mathbf{h}^{H}\mathbf{w}_{i}s+n_{i}$ is the received
signal using the $i$th probing beam (the $i$th column of $\mathbf{W}$). Given
the reported power of received probing signals $\mathbf{x}$ of a UE, the BS
then predicts the narrow beam index
$i_{\mathbf{v}}\in\left\\{1,2,\cdots,N_{\mathbf{V}}\right\\}$ using a function
$f:\mathbf{x}\rightarrow i_{\mathbf{v}}$. Overall, this problem can be
formulated as

$\begin{array}[]{rrclcl}\displaystyle\max_{\mathbf{W},f}&\lx@intercol{\mathop{\mathbb{E}}_{\mathbf{h}\in\mathcal{H}}}[|\mathbf{h}^{H}\mathbf{v}_{i_{\mathbf{v}}}|^{2}]\hfil\lx@intercol\\\
\textrm{s.t.}&i_{\mathbf{v}}&=&f(\mathbf{x})\\\
&|\mathbf{[W]}_{i,j}|&=&\frac{1}{\sqrt{N_{t}}},\forall
i=1,\cdots,N_{t},\forall j=1,\cdots,N_{\mathbf{W}}.\\\ \end{array}$ (8)

The optimization problem in (8) is non-convex and difficult to solve due to
the constant-modulus constraint of the probing BF weights and the unknown
function $f$. The proposed method is analogous to a hierarchical beam search
with 2 tiers. The probing codebook is similar to the wide-beam codebook in a
hierarchical search in that they both provide rough information about the
channel. Unlike the hierarchical search, the probing codebook consists of beam
patterns adapted to the environment, which are not limited to wide beams. In a
hierarchical method, the narrow-beam selection function $f$ picks the best
child beam of the best wide beam, which incurs another round of beam
measurement and report. In the proposed method, the narrow-beam selection
function $f$ predicts good narrow beams by intelligently utilizing
measurements of all probing beams instead of using a simple heuristic, e.g.,
picking the narrow beams pointing to the directions of the probing beam with
the best measurement.

###  III-A The proposed NN architecture

The probing codebook $\mathbf{W}$ needs to be designed so that its beam
sweeping measurements provide useful information regarding which narrow beam
in $\mathbf{V}$ to select. A good probing codebook is site-specific and should
capture particular characteristics of the propagation environment. The beam
selection function $f$ is also optimized for that particular BS and needs to
be designed so that it picks narrow beams from $\mathbf{V}$ which tend to
maximize the average SNR. Since the probing codebook $\mathbf{W}$ and the beam
selection function $f$ are interdependent, they are parameterized with
different NN modules – a complex-NN module and a MLP classifier – and jointly
trained in an end-to-end fashion. The overall architecture is illustrated in
Fig. 2.

####  III-A1 The trainable probing codebook

Beam sweeping using the probing codebook is modeled with a complex-NN module
that computes the complex received BF signals and their power. The input to
this NN module is the channel vector $\mathbf{h}$. The complex layer in the
complex-NN module implements the complex arithmetic of analog BF using real
arithmetic. When parameterizing a $N_{\mathbf{W}}$-beam codebook, the
trainable weights of the complex layer are elements of
$\Theta\in\mathbbm{R}^{N_{t}\times N_{\mathbf{W}}}$, which are the phase shift
values applied to each antenna element. The complex BF weights
$\mathbf{W}\in\mathbbm{C}^{N_{t}\times N_{\mathbf{W}}}$ can then be computed
as

$\mathbf{W}=\frac{1}{\sqrt{N_{t}}}(\cos{\Theta}+j\cdot\sin{\Theta}).$ (9)

The complex matrix multiplication

$\mathbf{z}=\mathbf{W}^{H}\mathbf{h}$ (10)

can be expressed as a real matrix multiplication

$\begin{bmatrix}\mathbf{z}^{real}\\\
\mathbf{z}^{imag}\end{bmatrix}=\begin{bmatrix}\mathbf{W}^{real}&-\mathbf{W}^{imag}\\\
\mathbf{W}^{imag}&\mathbf{W}^{real}\end{bmatrix}^{T}\begin{bmatrix}\mathbf{h}^{real}\\\
\mathbf{h}^{imag}\end{bmatrix},$ (11)

where $\mathbf{h}\in\mathbbm{C}^{N_{t}\times 1}$ is the channel vector,
$\mathbf{z}\in\mathbbm{C}^{N_{\mathbf{W}}\times 1}$ is the BF output, and we
express $\mathbf{z},\mathbf{W}$ and $\mathbf{h}$ in terms of their real and
imaginary parts. The BF signal power can then be computed as

$|\mathbf{z}|^{2}=\left[(\mathbf{z}_{1}^{real})^{2}+(\mathbf{z}_{1}^{imag})^{2},\cdots,(\mathbf{z}_{N_{\mathbf{W}}}^{real})^{2}+(\mathbf{z}_{N_{\mathbf{W}}}^{imag})^{2}\right]^{T}.$
(12)

While $|\mathbf{z}|^{2}$ is not complex differentiable with respect to
$\mathbf{z}$, backpropagation can be enabled by treating the real and
imaginary parts of $\mathbf{z}$ independently and compute
$\frac{\partial|\mathbf{z}|^{2}}{\partial\mathbf{z}^{real}}$ and
$\frac{\partial|\mathbf{z}|^{2}}{\partial\mathbf{z}^{imag}}$. The phase-shift
values $\Theta$ can then be updated using the chain rule and backpropagation.
Since only the dimension of $\Theta$ needs to be specified when initializing
the NN, the complex-NN module only needs to know the number of antenna
elements and not the exact array geometry. The array-geometry information is
embedded in the input channel vectors so that the complex-NN module can
automatically learn the optimal phase-shift values to apply at each antenna
element. This allows the architecture to be flexibly adopted by BS with
different antenna arrays.

The complex-NN module computes the received signal power in (7). Since updates
are made to the phase-shift values $\Theta$ during training, this architecture
enforces the constant-modulus constraint of phase-shifter-only analog BF. Note
that $\Theta$ can be extracted from the NN module at any time and be
implemented as an analog codebook. After training, the complex-NN module can
be discarded so that (7) can be computed using an actual RF chain and with an
analog BF codebook derived from $\Theta$.

####  III-A2 The MLP beam selection function

The beam selection function $f$ is modeled using an MLP classifier, which is a
feedforward fully connected NN with non-linear activation functions. The input
to the MLP is the power of the complex received signals of all beams in
$\mathbf{W}$, which is calculated using the complex-NN module during training
or through beam sweeping during deployment. The MLP consists of several hidden
layers before the output layer to increase its approximation power. The output
of an MLP with 1 hidden layer can be written as

$g(\mathbf{x})=b_{1}+A_{1}\sigma(b_{0}+A_{0}\mathbf{x}),$ (13)

where the input feature vector is $\mathbf{x}$, the output vector is
$g(\mathbf{x})$, the biases and weights of the hidden layer are $b_{0}$ and
$A_{0}$, the biases and weights of the output layer are $b_{1}$ and $A_{1}$,
and the non-linear activation function of the hidden layer is $\sigma$. The
biases and weights models a trainable affine transformation, while the non-
linear activation function allows the MLP to approximate a wide range of non-
linear functions. The biases and weights of the NN can be updated through
backpropagation to optimize some given objective function. One obvious way to
optimize $f$ is to design it to predict the optimal narrow beam
$\mathbf{v}^{*}$ which achieves the highest SNR with the current channel.
Hence, the final softmax layer of the MLP outputs the predicted posterior
probability distribution of each narrow beam in $\mathbf{V}$ being the optimal
beam. The MLP is a powerful function approximator and can produce good
estimates of posterior class probabilities. The BS can select the narrow beam
with the highest predicted posterior probability. To increase the beam
alignment robustness, the BS can also use the output of the MLP to reduce the
search space and sweep the top-$k$ narrow beams with the highest predicted
posterior probabilities.

The complex-NN architecture can be used to optimize a wide range of objective
functions since it essentially implements analog BF while allowing gradient
descent updates that respect the phase-shifter-only constraints. For instance,
it is used to directly minimize the the mean squared error (MSE) between the
gain of the strongest beam in the codebook and the equal gain combining (EGC)
gain in [22]. It is also shown to be robust against hardware impairment [22].
In order to select an optimal beam from a given narrow-beam codebook, the
probing codebook should provide useful information to the beam selection
function based on the entire environment. Hence the MLP beam selection
function is stacked after the complex-NN module and the entire NN is trained
in an end-to-end fashion instead of directly optimizing the BF gain of the
probing beams. The cross-entropy between the predicted optimal-beam
distribution and the true optimal-beam distribution is used as the loss
function. The partial derivative of the loss function with respect to the MLP
biases and weights as well as $\Theta$ in the complex-NN module can be
computed so that the MLP and the complex-NN module can be updated during
training. The probing codebook is optimized implicitly to assist the
downstream beam selection function. Interestingly, it still learns to capture
particular characteristics of the propagation environment, as will be
discussed in Section V-C.

###  III-B Practicality of the proposed method in 5G

The proposed beam alignment method requires an offline training phase and a
deployment phase. During the training phase, the BS optimizes the probing
codebook and the beam selection function by learning from training data and
updating the NN. The training data consists of the channel vectors for a BS
and its potential UE. Operators can obtain the channel vectors through ray-
tracing simulations of the site prior to deployment. Alternatively, or for
further refinement of the probing codebook, the BS can begin with a default
codebook, and then gradually develop a site-specific probing codebook through
interaction with its UE. In a typical TDD scenario, the BS can directly
estimate the UL channel by receiving the SRS transmitted by UE and assume the
DL channel is same as the estimated UL channel. If the DL and UL channel
reciprocity does not exist, the UE can estimate the DL channel by receiving
the SSB and CSI-RS transmitted by the BS and then feed back the estimated
channel.

While the probing codebook is parameterized using a NN during training, the
complex-NN module can be discarded in the deployment phase. As illustrated in
Fig. 3, the BF weights of the probing codebook can be extracted from the
complex-NN module and implemented as an analog beam codebook at the BS after
training. During the deployment phase, the BS periodically sweeps the learned
probing codebook by transmitting a sequence of SSB using different probing
beams. Each UE measures all the SSB and reports the received signal power to
the BS. The received signal power vector $\mathbf{x}$ is fed into the MLP beam
predictor at the BS, which then selects the optimal narrow beam or a few
candidate beams to try according to the predicted posterior probability
distribution. If the BS chooses to search the top-$k$ predicted narrow beams
for additional robustness, it can do so by sweeping those beams using the
aperiodic CSI-RS, which can be independently configured for each UE. The
proposed beam alignment method is adapted to characteristics of the
propagation environment such as the distribution of UE and the location of the
scatterers. If the environment changes, the probing codebook as well as the
MLP beam predictor need to retrained. The NN modules can be trained from
scratch, or be initialized with the existing probing codebook and MLP weights
and be refined using data from the new environment. The retraining can be
triggered if the beam alignment performance is below a threshold. Since such
macroscopic characteristics of the environment are expected to evolve slowly,
the retraining should occur infrequently.

The proposed beam alignment method essentially consists of periodic beam
sweeping by the BS and beam measurement and reporting by the UE during the
deployment phase. This beam sweeping, measurement and reporting process
precisely fits into the beam sweeping-based framework currently adopted in 5G,
as discussed in Section I-A. Instead of sweeping a general codebook using SSB
to cover the entire angular space, the proposed method sweeps a site-specific
probing codebook that learns to strategically place beams in directions that
can effectively capture characteristics of the environment and provide useful
information to the downstream beam selector. Since the learned probing
codebook replaces traditional full-coverage codebooks, its beams can be
transmitted using “always-on” SSB and thus can also be used by unconnected UE
for cell discovery and IA. Instead of selecting the narrow beam with the
highest reported power and ignoring the measurement of the rest of the
codebook, the proposed method predicts good candidate beams using a NN that
intelligently utilizes the measurement of all probing beams. This is also
supported in 5G since the BS can request additional beam reports from UE to
obtain measurements of all probing beams. Overall, the proposed method can be
directly adopted in 5G and does not require any modification to the existing
5G standard.

Figure 2: The architecture of the proposed NN, including the probing codebook
$\mathbf{W}$ and the beam selection function $f$. Figure 3: Proposed beam
alignment architecture during the deployment phase.

###  III-C Baselines and Metrics

The proposed beam alignment method selects the optimal narrow beam based on
measurements of a probing codebook. It is analogous to a hierarchical beam
search where the optimal narrow child beam is determined based on measurements
of wider parent beams. In a traditional hierarchical beam search, the BS needs
to sweep all child beams of the best parent beam at each layer of the
codebook. The parent beam needs to have a wider beam width and should contain
the coverage areas of its child narrower beams. Four baselines of comparison
are considered, including 2 hierarchical beam searches, an exhaustive beam
search and a genie. In all baselines, the BS has the same narrow-beam codebook
$\mathbf{V}\in\mathbbm{C}^{N_{t}\times N_{\mathbf{V}}}$ with $N_{\mathbf{V}}$
beams from which it needs to select a beam for the data or control channel.
When comparing the performance of different beam alignment methods, the beam
alignment accuracy and the SNR are two key metrics. The beam alignment
accuracy is the probability or relative frequency that the BS selects the
optimal narrow beam from the codebook $\mathbf{V}$. The SNR is calculated as
in (5).

2-Tier Hierarchical Beam Search The 2-tier hierarchical beam search uses a
wide-beam codebook with $N_{\mathbf{W}}$ beams and a narrow-beam codebook with
$N_{\mathbf{V}}\gg N_{\mathbf{W}}$ beams, both covering the same angular
space. Each narrow beam is the child beam of one of the wide beams. The
coverage area of each wide beam contains the coverage areas of all of its
children beams. The BS first sweeps the $N_{\mathbf{W}}$ wide beams then
sweeps the children beams of the best wide beam. The final selected beam is
the best child beam of the best wide beam. The wide-beam codebook is analogous
to the proposed probing codebook in that they both provide rough information
about the channel for beam selection.

Binary Hierarchical Beam Search The binary hierarchical beam search is a
generalized version of the 2-tier beam search. It performs a binary tree
search on the narrow beam codebook $\mathbf{V}$. Starting with a search space
equal to the entire angular space, the BS repeatedly splits the search space
into two partitions and sweeps two wide beams each covering one of the
partitions until reaching one of the narrow beams in the final codebook
$\mathbf{V}$. With $N_{\mathbf{V}}$ narrow beams, each beam search consists of
$\log_{2}N_{\mathbf{V}}$ layers. With more hierarchical search layers, the
binary beam search is more susceptible to search errors compared to the 2-tier
search. If a sub-optimal wide beam is chosen in any of the upper layers due to
noise or imperfect wide-beam patterns, the error will propagate forward and
affect the selected narrow beam in $\mathbf{V}$.

Exhaustive Beam Search The BS exhaustively sweeps the narrow beam codebook
$\mathbf{V}$ with $N_{\mathbf{V}}$ beams and selects the beam with the highest
received power. Compared to a hierarchical beam search, the exhaustive search
directly measures the narrow-beam codebook instead of some wide-beam
intermediate codebooks. The best beam in the narrow-beam codebook has larger
directionality gain compared to the wide beams used in the hierarchical
methods. As a result, the exhaustive search is less susceptible to noise in
the beam measurements.

Genie (Upper Bound) The BS has knowledge of the true BF gain of each narrow
beam in the codebook $\mathbf{V}$ and always selects the best beam. While the
hierarchical searches and the exhaustive search are all susceptible to search
errors caused by noise in the received BF signal, the genie method is not. If
the receive noise power is zero, the genie is equivalent to the exhaustive
beam search; else it is strictly better than exhaustive search. Given a
narrow-beam codebook $\mathbf{V}$, the genie method achieves a perfect beam
alignment accuracy of 100% and provides a performance upper bound.

The hierarchical beam search approaches require wide beams that contain the
coverage area of their children narrower beams. Multiple works have studied
hierarchical codebook designs. Synthesizing wide beams often requires multiple
RF chains or antenna activation, such as in [26], [27] and [8]. Since a single
RF chain and analog BF only is assumed in this work, the AMCF algorithm
proposed in [9] is used to generate the wide beams in the hierarchical
codebooks.

##  IV Dataset

Realistic and accurate data is essential to learning good NN models. Ray
tracing is able to achieve high accuracy and maintain spatial consistency when
modeling mmWave channels. A state-of-the-art commercial-grade ray-tracing
software called Wireless InSite [28] is used to generate the channel data. The
ray-tracing software simulates rays emitting from the transmitter at all
directions in the angular space and computes their interaction with the
environment along their paths before reaching the receiver, including
scattering, reflection and blockage. An environment needs to be constructed in
the ray-tracing software, specifying the terrain, the scatterers and their
dielectric properties.

Four different ray-tracing scenarios are considered to capture a wide range of
propagation environments for mmWave: a dense urban outdoor area, an urban
street, an indoor conference room with hallways and an urban street with
severe blockage and reflections. The ray-tracing scenarios include both LOS
and NLOS UE and cover two different mmWave carrier frequencies: 28 GHz and 60
GHz. The ray-tracing simulation parameters are summarized in Table I.

Rosslyn Experiment The Rosslyn dataset captures an outdoor dense urban
environment located in downtown Rosslyn, Virginia, USA. It was created with
our own experiments and was published in [11]. A 3-D render of the environment
is shown in Fig. 4. The Rosslyn environment has multiple buildings surrounding
an intersection. A BS is placed at the center of the intersection, elevated by
10 meters above the ground. A total of 73,884 UE positions are placed
uniformly 0.35 meter apart and 2 meters above the terrain surface. The entire
simulated area is around 90 meters $\times$ 90 meters. The Rosslyn environment
consists of mostly LOS UE and uses a carrier frequency of 28 GHz.

DeepMIMO O1_28 Experiment The DeepMIMO O1_28 dataset captures an outdoor
street environment and is available in the public DeepMIMO dataset [29]. A
portion of the original dataset corresponding to BS #3 and UE in row #800 to
row #1200 is selected. The environment consists of a street with buildings on
both sides. The BS is placed on one side of the street with an elevation of 6
meters. A total of 72,581 UE positions are placed uniformly on the street 20
centimeters apart. The environment consists of mostly LOS UE and uses a
carrier frequency of 28 GHz.

DeepMIMO I3 Experiment The DeepMIMO I3 dataset models an indoor conference
room and its hallways and is available in the public DeepMIMO dataset [29]. A
3-D view of the environment is shown in Fig. 5. A BS is placed 2 meters high
on the wall inside the conference room. A total of 118,959 UE positions are
placed inside two grids: one LOS grid inside the conference room and one NLOS
grid in the hallway. The carrier frequency is 60 GHz.

DeepMIMO O1_28B Experiment The DeepMIMO O1_28B outdoor street environment is
similar to the O1_28 scenario [29] but with severe blockage and reflections. A
2-D illustration of the environment is shown in Fig. 6. A 24-meter-wide metal
screen is placed in front of the BS and two reflectors are placed on both
sides. A total of 497,931 UE positions are placed uniformly on the street 20
centimeters apart. This environment includes both LOS and NLOS UE and uses a
carrier frequency of 28 GHz.

Figure 4: 3-D view of the Rosslyn environment. The green block represents the
BS. Points on the yellow grid represent the UEs. Figure 5: 3-D view of the
DeepMIMO I3 environment. Adapted: [30] Figure 6: 2-D illustration of the
DeepMIMO O1_28B environment. Dimensions are not to scale.

##  V Evaluation

Accurate ray-tracing channel data is used in our experiments as described in
Section IV. In all experiments, 60% of the data is used for training, 20% of
the data is used for validation and the remaining 20% is used for testing. The
training dataset is used to optimize the NN weights. The hyperparameters of
the NN are tuned by performing a grid search over a set of predefined values
and comparing their performance on the validation set. The test set is used to
evaluate the final performance of each beam alignment method. The MLP module
in the proposed NN architecture has 2 hidden layers with rectified linear unit
(ReLU) activation. The NN is trained for 200 epochs using the Adam optimizer
[31]. The training and validation loss after each training epoch is examined
to ensure that the NN has converged. A complex AWGN is assumed. The noise
power in the received BF signal in (4) is -81 dBm unless otherwise specified.
The simulation parameters are summarized in Table I. To make training more
stable and efficient, the channel vectors are normalized by the maximum
magnitude of the elements in the dataset:
$\max_{\mathbf{h}\in\mathcal{H}}|\mathbf{h}_{i,j}|$. Similar normalization
techniques are adopted in [16] and [22]. The noise is also scaled
appropriately according to the normalization factor. Since the normalization
factor is a predetermined constant that only depends on the underlying
environment, it should not affect the practicality of the proposed method. The
final narrow beam codebook $\mathbf{V}$ is a 128-beam DFT codebook.

TABLE I: Simulation Parameters BS Antenna |  $64\times 1$ ULA
---|---
UE Antenna | Single
Narrow beam codebook size $N_{\mathbf{V}}$ | 128
Carrier Frequency |  Rosslyn, DeepMIMO O1_28, O1_28B: 28 GHz DeepMIMO I3: 60 GHz
Bandwidth ($B$) | 100 MHz
Transmit Power ($P_{T}$) |  Rosslyn, DeepMIMO O1_28, I3: 10 dBm DeepMIMO O1_28B: 20 dBm
Noise power spectral density (PSD) | -161 dBm / Hz
Number of Rays | 25

###  V-A Can the proposed beam alignment method achieve good accuracy?

The accuracy of the proposed method and the baselines with increasing probing
codebook size is shown in Fig. 7. The genie always has a perfect accuracy of 1
and is not shown in the figure. The accuracy of the proposed method increases
as the number of probing beams increases, which is expected since a larger
probing codebook allows the BS to obtain more information about the channel
for a UE. Among the traditional beam sweeping-based baselines, the exhaustive
search performs the best and the binary search performs the worst. This is
expected since a method with more layers in the hierarchical search structure
is more vulnerable to noise in the received signal. Instead of directly
choosing the predicted optimal beam, the accuracy of the proposed method can
be improved significantly by searching a few additional candidate beams. In
all 4 environments, the proposed method can achieve a beam alignment accuracy
of at least 85% with just 14 probing beams and searching an additional $k$ = 3
narrow beams, outperforming both the binary and the 2-tier hierarchical beam
search baselines. In the 2 LOS environments (Rosslyn and DeepMIMO O1_28), the
proposed method can beat the binary beam search with 10 probing beams and $k$
= 3. With 16 probing beams, it can even outperform the exhaustive search by
sweeping an additional $k$ = 3 narrow beams. In the 2 environments with NLOS
UE (DeepMIMO I3 and O1_28B), traditional beam sweeping-based baselines perform
considerably worse compared to in the LOS environments. The exhaustive search
can only achieve an accuracy of around 80% compared to around 90% in the LOS
environments. The hierarchical beam searches suffer from even worse accuracy
degradation, with the 2-tier hierarchical beam search achieving accuracies of
around 65% compared to around 80% in the LOS environments. Clearly, beam
alignment for the NLOS UE is more challenging for traditional beam sweeping-
based baselines. On the other hand, the proposed method shines in these
environments with NLOS UE. With just 8 probing beams and $k$ = 3, it
outperforms any beam sweeping-based baseline, including the exhaustive search.
With 12 probing beams and $k$ = 3, the proposed method can achieve an accuracy
of over 84% in the I3 environment and over 88% in the O1_28B environment.
Overall, the proposed method can outperform the traditional beam sweeping-
based baselines with a moderate probing codebook size, particularly in
environments with NLOS UE which are usually challenging for beam alignment.

(a) Rosslyn

(b) DeepMIMO O1_28

(c) DeepMIMO I3

(d) DeepMIMO O1_28B

Figure 7: Beam alignment accuracy vs. probing codebook size.

###  V-B Can the proposed beam alignment method achieve good SNR?

The beam alignment accuracy considers the probability of finding the optimal
narrow beam. With a large, oversampled codebook, adjacent narrow beams may
have similar BF gains. As a result, operators may be more interested in the
SNR achieved after beam alignment. The average SNR of the proposed method and
the baselines with increasing probing codebook size is shown in Fig. 8. In the
LOS environments (Rosslyn and DeepMIMO O1_28), the proposed method outperforms
the binary beam search with just 10 probing beams and $k$ = 2 but is worse
than both the exhaustive and the 2-tier hierarchical methods. It is able to
match the 2-tier hierarchical baseline in terms of average SNR with 20 probing
beams and $k$ = 3. The exhaustive search achieves close-to-optimal average SNR
while its accuracy is just around 90%. Similar to the accuracy performance,
the proposed method shines in the challenging NLOS environments in terms of
average SNR. With just 8 probing beams and without additional narrow-beam
sweeping, it outperforms both hierarchical search methods. With 12 probing
beams and $k$ = 3, the proposed method can achieve similar if not better
average SNR compared to the exhaustive search. Overall, with 12 probing beams
and $k$ = 3, the gap in the average SNR of the proposed method from the genie
upper bound is 3.73 dB in the Rosslyn environment, 2.86 dB in the DeepMIMO
O1_28 environment, 1.36 dB in the I3 environment and 2.15 dB in the O1_28B
environment.

(a) Rosslyn

(b) DeepMIMO O1_28

(c) DeepMIMO I3

(d) DeepMIMO O1_28B

Figure 8: Average SNR vs. probing codebook size.

###  V-C Can the proposed NN learn meaningful probing codebooks?

The learned probing codebook should provide meaningful and helpful information
to the downstream MLP beam predictor. In the proposed end-to-end training
procedure, the complex-NN probing codebook and the MLP beam predictor are
jointly trained. Intuitively, the complex-NN module should learn beam patterns
that can effectively capture the characteristics of the underlying
environment. The DeepMIMO O1_28 and O1_28B environments provide a good case
study. Both environments feature similar topologies where a roadside BS serves
UE located on the street. While all UE are LOS in the O1_28 environment, a
significant portion of the UE are NLOS due to blockage by a metal screen
placed in front of the BS in the O1_28B environment. In order to provide
coverage to the NLOS UE, the BS in the O1_28B environment needs to steer beams
towards the two reflectors on both sides of the street. The majority of the
LOS UE are also distributed on both sides of the BS.

The learned probing codebook patterns in both environments is shown in Fig. 9.
Firstly, the learned radiation patterns are similar with increasing probing
codebook sizes $N_{\mathbf{W}}$ in either environment. Regardless of the
codebook size, the probing codebook consistently learns to focus energy on
specific areas. This indicates that the complex-NN module can consistently
learn probing codebook patterns in a given environment. Unlike conventional
DFT beams which have a single main lobe, the learned beams often have multiple
main lobes. Such beam patterns can likely capture more information about the
propagation environment given the small number of probing beams allowed.
Furthermore, the learned probing codebooks are adapted to the particular
characteristics of different environments. The learned beam patterns in the
O1_28 environment are drastically different from those in O1_28B. In the
O1_28B environment, the codebooks are optimized to focus energy in the angular
regions close to $\pm 90^{\circ}$, corresponding to the positions of the LOS
UE and the reflectors. In comparison, the codebooks learned in the O1_28
environment spread the energy much more evenly in the broadside direction,
which is consistent with the even distribution of LOS UE in front of the BS in
this environment. While the complex-NN module is not explicitly optimized to
leverage spatial patterns of the environment, it can nevertheless consistently
learn probing beams that captures particular characteristics of the
environment.

(a) O1_28, $N_{\mathbf{W}}=6$

(b) O1_28, $N_{\mathbf{W}}=8$

(c) O1_28, $N_{\mathbf{W}}=10$

(d) O1_28, $N_{\mathbf{W}}=12$

(e) O1_28B, $N_{\mathbf{W}}=6$

(f) O1_28B, $N_{\mathbf{W}}=8$

(g) O1_28B, $N_{\mathbf{W}}=10$

(h) O1_28B, $N_{\mathbf{W}}=12$

Figure 9: Learned beam patterns with different probing codebook sizes in the
DeepMIMO O1_28 and O1_28B environments.

###  V-D Does the proposed method achieve lower beam sweeping complexity?

With the proposed beam alignment method, all UE can measure the probing beams
simultaneously when the BS sweeps the probing codebook. If the BS choose to
sweep the top-$k$ predicted beams, those beams may be different for each UE.
Hence the beam sweeping complexity is $N_{\mathbf{W}}+K\cdot
k_{\mathbbm{1}_{\\{k>1\\}}}$ for $K$ UE. Each UE needs to feedback the
received signal power of the $N_{\mathbf{W}}$ probing beams. If the BS choose
to sweep additional beams, each UE only needs to feedback the index of the
best beam. With the 2-tier hierarchical beam search, the 1st-tier wide beams
can be transmitted using SSB and be measured by all UE simultaneously, while
different 2nd-tier children beams need to be swept for each UE. On average,
the beam sweeping complexity is
$N_{\mathbf{W}}+K\frac{N_{\mathbf{V}}}{N_{\mathbf{W}}}$ for $K$ UE. Each UE
needs to feedback the index of the best beam in each tier. With the binary
hierarchical beam search, the two first layer beams can be measured
simultaneously by all UE while the subsequent beam sweeping needs to be done
for each different UE. Hence the beam sweeping complexity is
$2+2K\log_{2}\frac{N_{\mathbf{V}}}{2}$ for $K$ UE. Each UE needs to feedback
the index of the best beam in each level of the binary search. With the
exhaustive beam search, the $N_{\mathbf{V}}$ beams can be measured by all UEs
simultaneously. The beam complexity is $N_{\mathbf{V}}$ regardless of the
number of UE. Each UE needs to feedback the index of the best beam. A summary
of the beam sweeping and feedback complexity of the proposed method and the
baselines is shown in Table II.

TABLE II: Beam Sweeping Complexity for $K$ UEs Beam alignment method | Beam sweeping complexity | Feedback complexity
---|---|---
Proposed method | $N_{\mathbf{W}}+K\cdot k_{\mathbbm{1}_{\\{k>1\\}}}$ |  $KN_{\mathbf{W}}$ received signal power \+ $K\cdot\mathbbm{1}_{\\{k>1\\}}$ beam indices
2-tier hierarchical search | $N_{\mathbf{W}}+K\frac{N_{\mathbf{V}}}{N_{\mathbf{W}}}$ |  $2K$ beam indices
Binary hierarchical search | $2+2K\log_{2}\frac{N_{\mathbf{V}}}{2}$ |  $K\log_{2}N_{\mathbf{V}}$ beam indices
Exhaustive search | $N_{\mathbf{V}}$ |  $K$ beam indices

A comparison of the beam sweeping complexity with 1,5,10 and 15 UE is shown in
Fig.10. When considering a single UE, the proposed method has lower beam
sweeping complexity compared to the exhaustive search and the 2-tier
hierarchical beam search. With fewer than 11 probing beams, the proposed
method also incurs lower beam sweeping complexity than the binary beam search
does even when sweeping 2 or 3 additional beams. When considering simultaneous
beam alignment for multiple UE such as 5, 10 and 15 UE, the beam sweeping
complexity of the proposed method is lower than that of any baseline. In the 2
NLOS environments (DeepMIMO I3 and O1_28B), when considering simultaneous beam
alignment for 10 UE, the proposed method with 12 probing beams and $k=3$ can
achieve an average SNR similar to that of the exhaustive beam search, at least
3.65 dB better than that of the 2-tier hierarchical beam search and at least
7.71 dB better than that of the binary beam search, while incurring less than
35.4% of the beam sweeping complexity of any baseline. With 12 probing beams
but without additional beam sweeping ($k=1$), the proposed method can still
beat the hierarchical search baselines while reducing the beam sweeping
complexity by 10$\times$.

(a) 1 UE

(b) 5 UEs

(c) 10 UEs

(d) 15 UEs

Figure 10: Beam sweeping complexity vs. probing codebook size.

###  V-E Is the proposed beam alignment method robust to noise?

The proposed beam alignment method, like any beam sweeping-based approach,
relies on measurements of the received power of the BF signals. As a result,
noise in the received BF signal may have significant impacts on the beam
alignment performance. We compare the beam alignment accuracy at various noise
levels to that when there is no noise. The accuracy degradation is defined as
the absolute difference between the accuracy with no noise and the accuracy at
a certain SNR level. The accuracy degradation at various SNR levels is shown
in Fig. 11. In the LOS environments (Rosslyn and DeepMIMO O1_28), the accuracy
degradation of all compared methods is minimal when the SNR is over 35 dB.
When the SNR is between 5 dB and 25 dB, the exhaustive search experiences the
least amount of accuracy drop. The proposed method with $k$ = 3 experiences
similar levels of degradation compared to the hierarchical search baselines.
In the NLOS environments (DeepMIMO I3 and O1_28B), the accuracy degradation is
much more noticeable even at high SNR levels of over 35 dB. The proposed
method also performs more favorably in terms of accuracy drop. With $k$ = 3,
it experiences the least amount of accuracy degradation when the SNR is over
15 dB in the DeepMIMO I3 environment. In the O1_28B environment, the proposed
method experiences less degradation than any other baseline at all SNR levels.

(a) Rosslyn

(b) DeepMIMO O1_28

(c) DeepMIMO I3

(d) DeepMIMO O1_28B

Figure 11: Beam alignment accuracy gap vs. SNR. The vertical axis represents
the accuracy degradation from when there is no noise in the received signal.
The horizontal axis represents the average SNR of the optimal narrow beams of
all UEs.

###  V-F How does the learned probing codebook help the beam predictor?

The trainable probing codebook is an important part of the proposed bean
alignment method and should be optimized to help the subsequent optimal-beam
classification task. To verify this, the performance of the MLP classifier is
evaluated while the trainable probing codebook is replaced with a
predetermined probing codebook. Two predetermined probing codebooks are
considered: a DFT codebook with $N_{\mathbf{W}}$ evenly-spaced narrow beams
which is similar to the sparse codebook used in [23], and a wide-beam codebook
whose $N_{\mathbf{W}}$ evenly-spaced wide beams are generated using the AMCF
algorithm. The MLP is trained from scratch using the received signal power of
each predetermined probing codebook. A beam alignment accuracy comparison of
the learned probing codebook and the predetermined ones in all 4 environments
is shown in Fig. 12. In all 4 environments, the learned probing codebook
achieves significantly better beam alignment accuracy. By placing beams
strategically according to the propagation environment instead of evenly in
the angular space regardless of the environment, the learned probing codebook
is much more effective at capturing channel characteristics, which greatly
benefits the downstream classification task.

(a) Rosslyn

(b) DeepMIMO O1_28

(c) DeepMIMO I3

(d) DeepMIMO O1_28B

Figure 12: Accuracy vs. probing codebook size.

The probing codebook can also be viewed through the lens of data clustering
and representation learning, which provides a further explanation of how the
learned codebook help select the optimal beam. By performing beam sweeping
using the probing codebook, the high-dimensional channel vector
$\mathbf{h}\in\mathbbm{C}^{N_{t}\times 1}$ is transformed into a feature
vector of received signal power values
$\mathbf{x}\in\mathbbm{R}^{N_{\mathbf{W}}\times 1}$ lying in a lower-
dimensional subspace determined by the probing codebook. If the transformed
feature vectors with the same optimal narrow beam are assigned to the same
cluster, a good probing codebook should intuitively make clusters
corresponding to different narrow beams well separated so that the MLP can
more easily predict the optimal beam. After beam sweeping using the probing
codebook, channel realizations with the same optimal narrow beam should be
close to each other in the transformed subspace, while those with different
optimal narrow beams should be farther apart. This is similar to the
representation learning problem in ML, which often seeks to learn low-
dimensional representations of high-dimensional data that exhibits natural
clustering according to the data labels [32]. One measure of the clustering
quality is the silhouette coefficient [33]. For a dataset $\mathcal{D}$, its
silhouette coefficient $S(\mathcal{D})\in[-1,1]$ is

$S(\mathcal{D})=\mathop{\mathbb{E}}_{i\in\mathcal{D}}\Big{[}\frac{b(i)-a(i)}{\max\\{a(i),b(i)\\}}\Big{]},$
(14)

where $a(i)$ is the mean intra-cluster distance of a data sample $i$ and
$b(i)$ is its distance to the nearest cluster of which it is not a part of. A
higher silhouette coefficient indicates better clustering and better
separability of the data, which will likely make classifying the optimal beam
easier. The silhouette coefficients of the received signal power vector
$\mathbf{x}$ in all 4 environments are shown in Table III. Compared to the
predefined AMCF and DFT probing codebooks, the learned codebook consistently
achieves better silhouette coefficients regardless of the environment and the
number of probing beams, thus explaining its superior beam alignment
performance.

To further visualize the clustering effect of the probing codebooks, 2-D
embeddings of the probing codebook measurements $\mathbf{x}$ are learned using
the t-SNE algorithm. The t-SNE algorithm [34] is commonly used to learn low-
dimensional embeddings of high-dimensional data while preserving its
distribution in the high-dimensional space, so that similar data samples are
more likely to be closer together and dissimilar ones are more likely to be
farther apart in the embedding space. Two environments – Rosslyn and DeepMIMO
O1_28 – are selected as case studies, and their t-SNE visualizations are shown
in Fig. 13. With the AMCF and the DFT probing codebooks, the clusters have
elongated and twisted shapes. With the learned codebook, the data samples
within each cluster are more tightly packed. The learned probing codebook
allows the channel realizations to form better-shaped clusters which will
likely make the data easier to classify.

TABLE III: Silhouette Coefficients of Beam Sweeping Received Signal Power Environment |  Probing codebook | Probing codebook size
---|---|---
6 | 8 | 10 | 12 | 14 | 16 | 18 | 20
Rosslyn | learned | -0.179 | -0.144 | -0.118 | -0.099 | -0.078 | -0.071 | -0.051 | -0.035
AMCF | -0.333 | -0.266 | -0.262 | -0.242 | -0.215 | -0.196 | -0.172 | -0.160
DFT | -0.376 | -0.404 | -0.320 | -0.281 | -0.231 | -0.231 | -0.189 | -0.199
DeepMIMO O1_28 | learned | -0.256 | -0.231 | -0.196 | -0.173 | -0.175 | -0.151 | -0.161 | -0.119
AMCF | -0.367 | -0.337 | -0.316 | -0.290 | -0.281 | -0.283 | -0.250 | -0.238
DFT | -0.426 | -0.380 | -0.353 | -0.340 | -0.329 | -0.302 | -0.272 | -0.292
DeepMIMO I3 | learned | -0.302 | -0.255 | -0.223 | -0.218 | -0.206 | -0.191 | -0.190 | -0.177
AMCF | -0.505 | -0.477 | -0.440 | -0.406 | -0.391 | -0.390 | -0.366 | -0.349
DFT | -0.503 | -0.480 | -0.416 | -0.393 | -0.385 | -0.399 | -0.361 | -0.364
DeepMIMO O1_28B | learned | -0.597 | -0.584 | -0.588 | -0.573 | -0.573 | -0.572 | -0.571 | -0.571
AMCF | -0.623 | -0.618 | -0.616 | -0.612 | -0.608 | -0.611 | -0.604 | -0.601
DFT | -0.746 | -0.742 | -0.696 | -0.683 | -0.668 | -0.662 | -0.641 | -0.634

(a) Rosslyn, learned

(b) Rosslyn, AMCF

(c) Rosslyn, DFT

(d) DeepMIMO O1_28, learned

(e) DeepMIMO O1_28, AMCF

(f) DeepMIMO O1_28, DFT

Figure 13: t-SNE visualization of the received signal power of different
probing codebooks with $N_{\mathbf{W}}=16$ beams.

##  VI Conclusion

We propose a mmWave beam alignment method that uses ML to predict the optimal
narrow beam using measurements of a learned probing codebook. We design a NN
architecture that optimizes the site-specific probing codebook so that it can
capture particular characteristics of the propagation environment. After an
offline training phase, operators can implement the learned probing codebook
using an RF chain and use its beam sweeping results to select an optimal
narrow beam or a few candidate beams to try, which is compatible with the beam
alignment framework in 5G. The proposed method can outperform hierarchical
beam search baselines and even the exhaustive beam search, particularly in
challenging environments with NLOS UE, while significantly reducing the beam
sweeping overhead. We also provide an explanation of why the learned probing
codebook is beneficial to the beam alignment task through the lens of data
clustering and representation learning. The proposed method uses channel
information during its offline training phase. Future works may consider beam
prediction without offline training or explicit channel knowledge. The
complex-NN architecture may also be extended to consider hybrid BF. The
extension to receive beam alignment on the UE side is another promising
direction.

##  VII Acknowledgements

The authors thank V. Va, A. Ali and B.L. Ng from Samsung Research America for
their valuable feedback and discussion.

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