# Novel Active Sensing and Inference for mmWave Beam Alignment Using Single RF
Chain Systems ††thanks: This research was supported by National Science
Foundation (NSF) under Grant CCF-2124929 and Grant CCF-2225617.††thanks: The
authors are with the Department of Electrical and Computer Engineering,
University of California, San Diego, La Jolla, CA 92092, USA<EMAIL_ADDRESS>brao@ucsd.edu).

Rohan R. Pote  ECE Department, UC San Diego Bhaskar D. Rao  ECE Department, UC
San Diego

###### Abstract

We propose a novel sensing approach for the beam alignment problem in
millimeter wave systems using a single Radio Frequency (RF) chain.
Conventionally, beam alignment using a single phased array involves comparing
beamformer output power across different spatial regions. This incurs large
training overhead due to the need to perform the beam scan operation. The
proposed Synthesis of Virtual Array Manifold (SVAM) sensing methodology is
inspired from synthetic aperture radar systems and realizes a virtual array
geometry over temporal measurements. We demonstrate the benefits of SVAM using
Cramér-Rao bound (CRB) analysis over schemes that repeat beam pattern to boost
signal-to-noise (SNR) ratio. We also showcase versatile applicability of the
proposed SVAM sensing by incorporating it within existing beam alignment
procedures that assume perfect knowledge of the small-scale fading
coefficient. We further consider the practical scenario wherein we estimate
the fading coefficient and propose a novel beam alignment procedure based on
efficient computation of an approximate posterior density on dominant path
angle. We provide numerical experiments to study the impact of parameters
involved in the procedure. The performance of the proposed sensing and beam
alignment algorithm is empirically observed to approach the fading
coefficient-perfectly known performance, even at low SNR.

###### Index Terms:

Single radio frequency chain, virtual array manifold synthesis, coherence
interval, active sensing, direction of arrival, hierarchical codebook,
Bayesian estimation

##  I Introduction

Millimeter wave (mmWave) technology is essential for expanding the existing
capabilities of cellular networks[1, 2]. The availability of the large
spectrum in the 30-300 GHz spectrum range, and the ability to place many more
antennas in the same form factor on a device are very promising avenues for
the next generation wireless systems. This has propelled the interest, both in
the industry and academia. The envisioned benefits include much higher
throughput and low latency. The impact of this technology can be gauged from
the numerous use cases enabled by the mmWave technology, which includes
industrial-IoT, virtual/augmented reality, biomedical applications, and non-
terrestrial networks[3, 4].

MmWave technology also faces many challenges. The mmWave channel incurs large
propagation losses thereby restricting coverage per base station (BS), and
requiring additional infrastructure compared to legacy cellular networks. A
second challenge is the _sparse_ nature of mmWave channel consequently
requiring accurate beam alignment. This challenge is only further exacerbated
by the narrow beamwidths and consequent large codebook size due to the large
antenna array dimensions. Several options are being considered for enhancing
coverage at low-cost such as integrated access backhaul and intelligent
reflective surfaces. On the other hand, reducing the beam alignment phase
duration is a critical and active area of research. Hardware cost also impacts
the ability of the transceivers to sense the mmWave channel. The large number
of antenna elements are typically supported by only a few Radio Frequency (RF)
chains[5], and thus necessitates for a low-dimensional projection of the
received signal at the antennas. Beam alignment using such a low-dimensional
signal is a challenging problem, and it is the main focus of this work.

There is a need to build a better sensing approach coupled with efficient
inference mechanisms that exploit the array geometry and channel
characteristics under the hardware constraints. An interesting direction
adopted in [6] formulates the beam alignment problem under the posterior
matching framework, and actively learns the single path Direction of Arrival
(DoA). The work is shown to improve over the detection-based algorithm in [7].
However, the authors in [6] assume that the small-scale fading coefficient is
perfectly known. Subsequent effort build upon this work, and estimate both the
DoA as well as the fading coefficient [8]. The Kalman filter-based posterior
matching algorithm in [8] still requires good prior density on the small-scale
fading coefficient. Similar assumptions on availability of good prior density
was made in the variational hierarchical posterior matching algorithm proposed
in [9].

In this work, we reflect on the beam alignment problem from the perspective of
active sensing for improved estimation performance. We do so without relying
on additional information such as good prior knowledge about the small-scale
fading coefficient. The contributions of this work are as follows:

  * •

A novel sensing methodology, inspired from Synthetic Aperture Radar (SAR), is
proposed for the single RF chain mmWave systems. Under the proposed sensing
approach, a virtual Uniform Linear Array (ULA) manifold is synthesized over
temporal measurements. Extension to construct a virtual arbitrary array
geometry such as Sparse Linear Array (SLA) is also discussed. The proposed
sensing is useful even in the presence of multipaths and its operation under
such scenarios is briefly discussed.

  * •

Benefits of the proposed sensing are described, when the channel small-scale
fading coefficient is known, in terms of i. its impact on the Cramér-Rao lower
Bound (CRB) on the variance of estimation of the unknown dominant path angle,
when compared to a benchmark scheme, and ii. its ability to be incorporated
within existing active beam alignment procedures. The improvement in terms of
lower training overhead is demonstrated using numerical experiments.

  * •

A novel beam alignment procedure is proposed which adapts the beamformer based
on the current estimate of the posterior on the unknown angle. The proposed
algorithm estimates a posterior on the small-scale fading along with the
angular posterior. Both, flexible and hierarchical codebook-based beam
alignment procedures are presented.

  * •

Finally, the proposed sensing and beam alignment procedures are empirically
studied, and compared with the performance using perfect knowledge of the
channel state information. Impact of the parameters involved in the estimation
procedure is also studied, which also reveal the ability of the adaptive beam
alignment procedure to self-correct in case of premature misalignment during
the early phase of the training period.

###  I-A Relevant Prior Work

A virtual array synthesis from spatial measurements under the reduced number
of RF chains constraint for mmWave systems was proposed in [10]. A similar
sensing scheme as in [10], was proposed in [11] for mmWave multipath angle
estimation and in [12] for the DoA estimation problem. In [11], the authors
proposed using random precoders and combiners that are submatrices of banded
Toeplitz matrices. The work focuses on mmWave systems with multiple RF chains,
and the ideas are extended to the single RF chain case. In [12], the authors
investigated the applicability of root-MUSIC and ESPRIT algorithms, as a
consequence of preserving the Vandermonde structure and the shift-invariance
under the virtual array synthesis procedure. These sensing methodologies can
be traced back to Silverstein[13] and Tkacenko [14]. The sensing scheme
proposed in this work synthesizes a virtual array over _temporal measurements_
, and considers the practical single phased array system. To the best of our
knowledge, the presented adaptive sensing methodology is the first of its
kind.

Many non-adaptive and adaptive beamforming approaches have been proposed for
the mmWave beam alignment problem in the past. Random beamforming was proposed
in [15, 16, 17], wherein the inference was carried out using compressed
sensing algorithms. This approach does not exploit beamforming gain needed to
combat large path loss in the mmWave channels. An exhaustive beam steering
approach proposed in IEEE 802.11ad and 5G standards improves the beamforming
gain, but can be slow in selecting the appropriate beam. The acquisition time
is improved in literature by replacing the _non-adaptive_ linear search with
an _adaptive_ binary (or in general $n$-ary, $n\geq 2$) search within a
hierarchical codebook [18, 7, 19, 20]. A comparison in terms of asymptotic
misalignment probability between the exhaustive search and the hierarchical
search was studied in [21]. For adaptive schemes using hierarchical codebook,
the inference at each hierarchical level is typically carried out by comparing
power at the output of different beams within a hierarchical node. The
inference mechanism was improved in [6] by computing posterior on the dominant
path angle and selecting next beam based on the posterior within the
hierarchical codebook of [7]. The work assumes that the small-scale fading
coefficient is perfectly known. However, such assumption is difficult to
satisfy in practice. The problem of estimating small-scale fading coefficient
(along with the unknown path angle) was considered in later works [8, 9]. A
grid-approach was also proposed in [8], but the relevant issue on how to
choose appropriate grid on the small-scale fading coefficient was not
addressed. An adaptive beam search algorithm scheme based on posterior
computation to compare beams was proposed in [22]. Many learning-based
approaches have been proposed in literature as well[23, 24, 25]. These include
approaches that frame the beam alignment problem as multi-armed bandit
problem, or train an end-to-end neural network to design a model-free or
codebook-free architecture. In this work, we emphasize the model and propose
novel sensing (for improved acquisition) and inference procedures to estimate
the unknown parameters. The paper builds on our previous work in [26].

###  I-B Organization of the Paper and Notations

We describe the problem tackled in this work, and introduce the new sensing
approach in Section II. In Section III, we investigate the impact of the
proposed sensing for the case when the fading coefficient is perfectly known,
and only the unknown DoA is to be recovered. The more practical case, when the
fading coefficient needs to be estimated along with the DoA is discussed in
Section IV. We provide empirical results in Section V, and present our
conclusions in Section VI.

Notations: We represent scalars, vectors, and matrices by lowercase, boldface-
lowercase, and boldface-uppercase letters, respectively. Sets are represented
using blackboard bold letters. $(.)^{T},(.)^{H},(.)^{c}$ denotes transpose,
Hermitian, and complex conjugate operation respectively. $\otimes$ denotes
matrix Kronecker product, and $\odot$ denotes Hadamard product of two
conformable matrices. $*$ denotes convolution operation.
$[M]=\\{0,1,\ldots,M-1\\},M\in\mathbb{Z}^{+}$.

##  II Problem Statement & Proposed Novel Sensing

We consider a receiver (base station or user equipment) equipped with a
Uniform Linear antenna Array (ULA) of size $N$ and a single RF chain. We
assume a flat fading channel, with a single dominant path between the
transmitter and receiver. We further assume that the channel remains coherent
within the training duration due to low receiver mobility.

###  II-A Problem Statement

Figure 1: A (virtual) ULA segment of size $N_{v}=4$ is created using $4$
snapshots. The beamformer, $\mathbf{f}_{t(l)}$, is adapted once in every
segment duration.

The received signal at the antennas at instant $l$,
$\mathbf{x}_{l}\in\mathbb{C}^{N}$, is given by

$\mathbf{x}_{l}=\sqrt{P_{s}}\alpha\bm{\phi}_{N}(u)+\bar{\mathbf{n}}_{l},\quad
l\in[L],$ (1)

where $L$ denotes the total training duration. $P_{s}\geq 0$ denotes the
combined contribution of transmitted power and the large-scale fading (path
loss and shadowing), $\alpha\in\mathbb{C}$ is the _unknown_ small-scale fading
coefficient. Since the transmitted symbol is known, it can be easily absorbed
within the signal term of $\mathbf{x}_{l}$ in (1). Thus, we assume the
transmitted symbol value to be $1$ without loss of generality.
$\bm{\phi}_{N}(u)$ is the array manifold or response vector for an incoming
narrowband signal along the angle $u$; $u=\sin\theta,u\in[-1,1),$ where
$\theta\in[-\frac{\pi}{2},\frac{\pi}{2})$ denotes physical DoA. Noise,
$\bar{\mathbf{n}}_{l}\in\mathbb{C}^{N}$, is distributed as
$\mathcal{CN}(\mathbf{0},\sigma_{n}^{2}\mathbf{I})$ and i.i.d. over time.
Since there is a single RF chain, the received signal is processed using an
analog combiner, $\mathbf{w}_{l}\in\mathbb{C}^{N}$. The output,
$y_{l}\in\mathbb{C}$, available for inference is given by

$\displaystyle y_{l}=\mathbf{w}_{l}^{H}\mathbf{x}_{l}$
$\displaystyle=\sqrt{P_{s}}\alpha\mathbf{w}_{l}^{H}\bm{\phi}_{N}(u)+\mathbf{w}_{l}^{H}\bar{\mathbf{n}}_{l}$
(2)
$\displaystyle=\sqrt{P_{s}}\alpha\mathbf{w}_{l}^{H}\bm{\phi}_{N}(u)+{n}_{l}.$

The goal is to design $\mathbf{w}_{l}$ and infer $u$; $\mathbf{w}_{l}$ can be
adapted over time to improve the inference. For a ULA with $\lambda/2$ inter-
element spacing111$\lambda/2$ inter-element spacing prevents ambiguity in
angular estimation., where $\lambda$ denotes the wavelength of the received
signal, we have

$\bm{\phi}_{N}(u)=[1\>\exp{(j\pi u)}\>\cdots\>\exp{(j\pi(N-1)u)}]^{T}.$ (3)

In this paper, we consider the ULA geometry for easier exposition of ideas.
However, these ideas can be extended to planar geometries as well, such as
uniform rectangular arrays[27]. Next, we describe a high-level structure that
we impose when designing the beamformer, $\mathbf{w}_{l}$.

###  II-B Synthesis of Virtual Array Manifold (SVAM) Sensing

The sensing methodology is inspired from SAR systems used in remote sensing
and automotive radar (see [28, 29] and references within them). In typical SAR
systems, the sensor motion allows synthesis of larger aperture than physical
antenna, which helps to improve resolution. In this work, we mimic the sensor
motion by designing $\mathbf{w}_{l}$ appropriately. We exploit the coherence
interval to synthesize virtual apertures _over time_. An important consequence
is that, such measurements preserve phase information from the physical
antenna, which captures rich information about the DoA of the incoming signal.
The proposed sensing can be applied more broadly to multi-path angles.
Moreover, leveraging the complex exponential structure present at the combiner
output $y_{l},l\in[L]$, it is possible to apply an _unlimited_ number of
digital filters on these measurements.

####  II-B1 Constructing a Virtual ULA with $\lambda/2$ Inter-element Spacing

Let $N_{v}$ denote the number of antenna elements in the _virtual_ ULA we wish
to create. We assume that the total training duration, $L$, is divisible by
$N_{v}$ for simplicity. Let $t(l)=\mathrm{floor}(l/N_{v})$ denote222
$\mathrm{floor}(\cdot)$ denotes the floor function. the (ULA) segment index
(see Fig. 1). We design a beamformer of size $M=N-N_{v}+1$,
$\mathbf{f}_{t(l)}\in\mathbb{C}^{M}$, such that
$\lVert\mathbf{f}_{t(l)}\rVert_{2}=1$, initially spanning the region of
interest (RoI). A RoI incorporates any prior information available about the
DoA. In certain scenarios the receiver may be interested in identifying paths
in a narrow region, for example, due to restricted mobility patterns for the
transmitter. As the system gathers information about the unknown DoA, $u$, the
design of the beamformer, $\mathbf{f}_{t(l)}$, is adapted. The analog combiner
at instant $l$ is given by

$\mathbf{w}_{l}=\left[\begin{array}[]{ccc}\mathbf{0}^{T}_{\mathrm{mod}(l,N_{v})}&\mathbf{f}_{t(l)}^{T}&\mathbf{0}_{N_{v}-\mathrm{mod}(l,N_{v})-1}^{T}\end{array}\right]^{T}.$
(4)

Thus, within a segment duration, the beamformer slides along the antenna
aperture. In contrast to the work in [10], here only a single RF chain is
available and thus a virtual ULA segment is synthesized over time. Let

$\beta_{t(l)}(u)=\mathbf{f}_{t(l)}^{H}\bm{\phi}_{M}(u),$ (5)

denote the complex gain of the beamformer along the angle $u$. The signal
$y_{l}$ post-combining can be expressed as

$y_{l}=\mathbf{w}_{l}^{H}\mathbf{x}_{l}=\sqrt{P_{s}}\alpha\beta_{t(l)}(u)\cdot\exp{\left(j\pi
u\mathrm{mod}(l,N_{v})\right)}+n_{l}.$ (6)

Note that the gain $\beta_{t(l)}(u)$ does not change within a segment, but
‘$\exp{\left(j\pi u\mathrm{mod}(l,N_{v})\right)}$’ varies within the segment.

######  Remark 1.

The beamforming gain, measured in terms of $|\beta_{t(l)}(u)|^{2}$, in the
passband depends primarily on the beamwidth of the beamformer,
$\mathbf{f}_{t(l)}$. For an ideal beamformer design, the gain in the
beamformer passband corresponding to a beamwidth of $\frac{2}{R},R\geq 1,$ in
$u$-space is given by $|\beta_{t(l)}(u)|^{2}=R$. As the beamformer size, $M$,
increases the beamformer response approaches the ideal response.

We drop the notation for dependence of $t$ on $l$ for simplicity. We stack the
measurements within a segment duration to form
$\mathbf{y}_{t}=\left[y_{tN_{v}}\>\>y_{tN_{v}+1}\>\>\cdots\>\>y_{(t+1)N_{v}-1}\right]^{T}\in\mathbb{C}^{N_{v}},t\in[L/N_{v}]$,

$\displaystyle\mathbf{y}_{t}$
$\displaystyle=\sqrt{P_{s}}\alpha\beta_{t}(u)\left[1\>\exp{(j\pi
u)}\cdots\exp{(j\pi(N_{v}-1)u)}\right]^{T}+\mathbf{n}_{t}$ (7)
$\displaystyle=\sqrt{P_{s}}\alpha\beta_{t}(u)\bm{\phi}_{N_{v}}(u)+\mathbf{n}_{t}.$

We identify the following design parameters: a) $N_{v}\in\\{1,2,\ldots,N\\}$,
the virtual ULA size, and b) beamformer, $\mathbf{f}_{t}$, design, which
includes the beam direction and beamwidth. $N_{v}=1$ reduces to the
conventional beam design. Thus, the proposed sensing strategy includes the
methodology adopted for sensing in [7, 6] as a special case. The hierarchical
codebook in [7] designed using a least squared error criterion imposes a
constant amplitude and phase in the passband. The inference is improved by
relaxing the constant phase requirement in the passband. Thus, in this work we
design the beamformers as _linear_ -phase Finite Impulse Response (FIR) filter
using the Parks-McClellan algorithm[30].

Let
$\tilde{\mathbf{y}}_{t}=[\mathbf{y}_{0}^{T},\ldots,\mathbf{y}_{t}^{T}]^{T}\in\mathbb{C}^{(t+1)N_{v}}$
denote the measurements until snapshot index, $l=(t+1)N_{v}-1$. Then

$\displaystyle\tilde{\mathbf{y}}_{t}=\left[\begin{array}[]{c}\mathbf{y}_{0}\\\
\mathbf{y}_{1}\\\ \vdots\\\ \mathbf{y}_{t}\end{array}\right]$
$\displaystyle=\sqrt{P_{s}}\alpha\left[\begin{array}[]{c}\beta_{0}(u)\bm{\phi}_{N_{v}}(u)\\\
\beta_{1}(u)\bm{\phi}_{N_{v}}(u)\\\ \vdots\\\
\beta_{t}(u)\bm{\phi}_{N_{v}}(u)\end{array}\right]+\left[\begin{array}[]{c}\mathbf{n}_{0}\\\
\mathbf{n}_{1}\\\ \vdots\\\ \mathbf{n}_{t}\end{array}\right]$ (20)
$\displaystyle=\sqrt{P_{s}}\alpha\left(\tilde{\bm{\beta}}_{t}(u)\otimes\bm{\phi}_{N_{v}}(u)\right)+\tilde{\mathbf{n}}_{t},$
(21)

where
$\tilde{\bm{\beta}}_{t}(u)=\left[\beta_{0}(u),\beta_{1}(u),\ldots,\beta_{t}(u)\right]^{T}$
and
$\mathbf{\tilde{n}}_{t}=[\mathbf{n}_{0}^{T},\mathbf{n}_{1}^{T},\ldots,\mathbf{n}_{t}^{T}]^{T}$.
In Section IV we discuss how to estimate $u$ along with $\alpha$ using
$\tilde{\mathbf{y}}_{t}$, and how to adaptively design the beamformer i.e.,
$\mathbf{f}_{t+1}$ for the next segment.

######  Remark 2.

It is important to highlight the significance of the proposed sensing
methodology. Given just two measurements, it is possible to construct a
virtual ULA under the proposed sensing with $N_{v}=2$. This is equivalent to a
contrived _single_ snapshot measurement from a physical array of size $2$.
Owing to the rich (array) geometrical information preserved in the
measurements, it is thus possible to _estimate_ the dominant path DoA in a
_gridless_ manner using existing techniques[31]. In contrast, the beam scan
operation using $N_{v}=1$ requires as many measurements as the codebook size
to _detect_ the DoA.

####  II-B2 Constructing a Virtual Sparse Linear Array

The construction presented in the previous subsection can be extended to form
virtual ULAs with more than $\lambda/2$ spacing333Any ambiguity in angular
estimation can be resolved if the RoI is an appropriate fraction of the
spatial region.. More generally, a Sparse Linear Array (SLA) can also be
realized as the virtual array geometry, for example, minimum redundancy
arrays[32], nested arrays[33] or co-prime arrays[34]. These can help to
increase the virtual aperture and improve resolution for the same segment
duration. Let $N_{v}$ denote the number of antenna elements in the virtual SLA
we wish to construct over time. Let $\mathbb{P}=\\{P_{i}:0\leq
P_{i}<N,P_{i}\in\mathbb{Z},i\in[N_{v}]\\}$ denote the set of sensor positions
in the SLA ordered in an increasing manner; $P_{0}=0$, without loss of
generality. We design a beamformer, $\mathbf{f}_{t}$, of length
$M=N-P_{N_{v}-1}$. The analog combiner at time $l$, in the case of SLA, is
given by

$\mathbf{w}_{l}=\left[\begin{array}[]{ccc}\mathbf{0}_{P_{\mathrm{mod}(l,N_{v})}}^{T}&\mathbf{f}_{t}^{T}&\mathbf{0}_{N-M-
P_{\mathrm{mod}(l,N_{v})}}^{T}\end{array}\right]^{T}.$ (22)

Using identical notation to describe the complex gain, $\beta_{t}(u)$, as in
(5), the signal $y_{l}$ post-combining can be expressed as

$y_{l}=\mathbf{w}_{l}^{H}\mathbf{x}_{l}=\sqrt{P_{s}}\alpha\beta_{t}(u)\cdot\exp{\left(j\pi
uP_{\mathrm{mod}(l,N_{v})}\right)}+n_{l}.$ (23)

Note that (23) generalizes (6) for the SLA case. Finally, the measurements
within the $t$-th SLA segment can be stacked as

$\displaystyle\mathbf{y}_{t}$
$\displaystyle=\sqrt{P_{s}}\alpha\beta_{t}(u)[1\>\exp{(j\pi
P_{1}u)}\cdots\exp{(j\pi P_{N_{v}-1}u)}]^{T}+\mathbf{n}_{t}$ (24)
$\displaystyle=\sqrt{P_{s}}\alpha\beta_{t}(u)\mathbf{S}_{\mathbb{P}}\bm{\phi}_{N}(u)+\mathbf{n}_{t},$

where $\mathbf{S}_{\mathbb{P}}\in\mathbb{R}^{N_{v}\times N}$ is a binary
sampling matrix given by

$[\mathbf{S}_{\mathbb{P}}]_{m,n}=\left\\{\begin{array}[]{cc}1&\mbox{if
}n=P_{m}\\\ 0&\mbox{otherwise}\end{array}\right.,m\in[N_{v}],n\in[N].$ (25)

In the remainder of the work, we focus on the virtual ULA with $\lambda/2$
spacing-based sensing for ease of exposition, but the ideas presented can be
easily extended to the virtual SLA case.

We refer to the sensing methodology described in this section as Synthesis of
Virtual Array Manifold (SVAM). Furthermore, we describe the SVAM sensing in
conjunction with the virtual ULA size (with $\lambda/2$ spacing) by
SVAM-$N_{v}$. For example, SVAM-$2$ indicates the sensing methodology employed
is as described in subsection II-B1 with $N_{v}=2$.

###  II-C Implications of Using SVAM sensing to multipath channels

The proposed SVAM sensing approach can be applied to scenarios including
multiple paths. In this subsection, we briefly take a detour from (1) and
consider the following more general measurement model corresponding to a
channel with $K$ paths ($K\geq 1$)

$\mathbf{x}_{l}=\sqrt{P_{s}}\sum_{k=1}^{K}\alpha_{k}\bm{\phi}_{N}(u_{k})+\bar{\mathbf{n}}_{l},\quad
l\in[L],$ (26)

where the notations simply extend for $K$ paths compared to (1) and $P_{s}$
denotes the average power value for the $K$ paths. A similar development as
shown in Subsection II-B1 leads to the following received measurement vector
after beamforming in the $t$-th virtual ULA segment duration

$\mathbf{y}_{t}=\sqrt{P_{s}}\sum_{k=1}^{K}\alpha_{k}\beta_{t}(u_{k})\bm{\phi}_{N_{v}}(u_{k})+\mathbf{n}_{t}$
(27)

compared to (7). The combined measurements after $(t+1)N_{v}$ snapshots is
given by

$\displaystyle\tilde{\mathbf{y}}_{t}=\left[\begin{array}[]{c}\mathbf{y}_{0}\\\
\mathbf{y}_{1}\\\ \vdots\\\ \mathbf{y}_{t}\end{array}\right]$
$\displaystyle=\sqrt{P_{s}}\sum_{k=1}^{K}\alpha_{k}\left(\tilde{\bm{\beta}}_{t}(u_{k})\otimes\bm{\phi}_{N_{v}}(u_{k})\right)+\tilde{\mathbf{n}}_{t},$
(32)

compared to (21), where
$\tilde{\bm{\beta}}_{t}(u)=\left[\beta_{0}(u),\beta_{1}(u),\ldots,\beta_{t}(u_{k})\right]^{T}$
and
$\mathbf{\tilde{n}}_{t}=[\mathbf{n}_{0}^{T},\mathbf{n}_{1}^{T},\ldots,\mathbf{n}_{t}^{T}]^{T}$.
The beamforming gain for each path is a function of the beamformer design
$\mathbf{f}_{t}$ and the path angle. Thus, all the paths within the passband
of the beamformer $\mathbf{f}_{t}$ are boosted. This demonstrates the
applicability of the proposed SVAM sensing for more general channel models
beyond the scenario considered in (1). The remainder of this work specializes
to the single dominant path model in (1) for tractability and ease of
exposition. The more general scenarios involving multipaths (such as [35, 36,
37, 38]) is left as future work.

##  III Benefits of SVAM For Beam Alignment

We discuss some of the benefits of the proposed SVAM sensing for estimating
the DoA, $u$. We demonstrate the benefits in two settings: i. agnostic to the
adaptive scheme used, ii. when hierarchical posterior matching (hiePM)[6]
scheme is used. In both settings we assume that $\alpha$ is known. We also
briefly discuss the role of virtual antenna size $N_{v}$ for improving the
estimation performance. The more practical scenario, where $\alpha$ is
unknown, is discussed in the next section.

###  III-A Adaptive Scheme-Agnostic Analysis

We begin by first deriving the CRB after $L$ snapshots, assuming $\alpha$ is
known. Let
$\mathbf{W}=[\mathbf{w}_{0},\ldots,\mathbf{w}_{L-1}]\in\mathbb{C}^{N\times
L},\lVert\mathbf{w}_{l}\rVert_{2}=1$ denote the matrix of beamformers used to
generate measurements $\mathbf{y}=[y_{0},\ldots,y_{L-1}]^{T}\in\mathbb{C}^{L}$
as in (2). Note that $\mathbf{w}_{l}$’s may be designed generally, and not
necessarily under SVAM for the following result to hold.

######  Theorem 1.

The $\mathrm{CRB}(u)$ on the variance for estimating $u$ using the beamformer
matrix
$\mathbf{W}=[\mathbf{w}_{0},\ldots,\mathbf{w}_{L-1}],\lVert\mathbf{w}_{l}\rVert_{2}=1$
as in (2) over $L$ snapshots when $\alpha$ is known is given by

$\mathrm{CRB}(u)=\frac{\sigma_{n}^{2}}{2P_{s}|\alpha|^{2}}\left\\{\left(\frac{\partial}{\partial
u}\bm{\phi}_{N}(u)\right)^{H}\mathbf{W}\mathbf{W}^{H}\frac{\partial}{\partial
u}\bm{\phi}_{N}(u)\right\\}^{-1}.$ (33)

###### Proof.

The proof follows standard steps for deriving CRB and provided in [39]. ∎

In this subsection, we compare two sensing strategies - both deterministically
modify the beamformer design after every $N_{v}$ snapshots. The proposed
sensing strategy involves a shift in space as described in (4), and requires
the beamformer to be fixed for $N_{v}$ snapshots by design. The alternative
(_benchmark_) strategy designs the beamformer $\mathbf{w}_{l}$ of size $N$
without inserting $0$’s - used in SVAM to effect a _linear time invariant_
operation. The beamformer for the two strategies considered here are to be
designed with identical specifications except the length. The SVAM beamformer
$\mathbf{f}_{t}$ has a size of $M(=N-N_{v}+1)$, whereas the alternative
strategy utilizes the total antenna aperture of size $N$. We expect that for a
large antenna size $N$, which is typical in mmWave systems, the slightly
different length of the beamformers will have negligible impact. The rest of
the beamformer specifications, which can change every $N_{v}$ snapshots, may
be chosen arbitrarily for the discussion in this subsection. We defer the
discussion that involves using the specific adaptive scheme - HiePM[6] to the
subsection III-C. We specialize the CRB expression in Theorem 33 for the
benchmark strategy and the proposed sensing strategy. This exercise helps to
understand: _how informative are the measurements available post analog
combining about the unknown DoA using either of the two techniques for
designing the analog combiners?_ For both the cases, we treat as if the same
ordered set of $L$ received signal snapshots
$\mathbf{x}_{l}\in\mathbb{C}^{N},l\in[L]$ were available at the antenna.

####  III-A1 CRB for the Benchmark Strategy

Let
$\mathbf{W}^{\mathrm{B}}=[\mathbf{w}^{\mathrm{B}}_{0},\mathbf{w}^{\mathrm{B}}_{1},\ldots,\mathbf{w}^{\mathrm{B}}_{L-1}]\in\mathbb{C}^{N\times
L}$ be the $L$ beamformers used to generate the measurements as in (2);
superscript $\mathrm{B}$ highlights the benchmark sensing strategy. Note that

$\mathbf{w}^{\mathrm{B}}_{l}=\mathbf{w}^{\mathrm{B}}_{N_{v}\times\mathrm{floor}(l/N_{v})},$
(34)

under the benchmark scheme. Let
$\mathbf{F}^{\mathrm{B}}=[\mathbf{w}^{\mathbf{B}}_{0},\mathbf{w}^{\mathbf{B}}_{N_{v}},\ldots,\mathbf{w}^{\mathbf{B}}_{L-N_{v}}]\in\mathbb{C}^{N\times\frac{L}{N_{v}}}$
denote the matrix of unique beamformers from $\mathbf{W}^{\mathrm{B}}$. We
have the following result.

######  Corollary 1.1.

The $\mathrm{CRB}(u)$ using measurements in (2) from the beamformer matrix
$\mathbf{W}^{\mathrm{B}}$ under the constraint in (34) over $L$ snapshots when
$\alpha$ is known is given by

$\displaystyle\mathrm{CRB}(u)$
$\displaystyle=\frac{1}{N_{v}}\frac{\sigma_{n}^{2}}{2P_{s}|\alpha|^{2}}\left\\{\left(\frac{\partial}{\partial
u}\bm{\phi}_{N}(u)\right)^{H}\mathbf{F}^{\mathrm{B}}(\mathbf{F}^{\mathrm{B}})^{H}\frac{\partial}{\partial
u}\bm{\phi}_{N}(u)\right\\}^{-1}.$ (35)

###### Proof.

Proof follows from simplifying (33) using (34), and is provided in [39]. ∎

####  III-A2 CRB for the Proposed Sensing Strategy

Let
$\mathbf{W}^{\mathrm{P}}=[\mathbf{w}^{\mathrm{P}}_{0},\mathbf{w}^{\mathrm{P}}_{1},\ldots,\mathbf{w}^{\mathrm{P}}_{L-1}]\in\mathbb{C}^{N\times
L}$ be the $L$ beamformers designed under SVAM as described in (4);
superscript $\mathrm{P}$ highlights the proposed SVAM sensing. Let
$\mathbf{F}^{\mathrm{P}}=[\mathbf{f}_{0},\mathbf{f}_{1},\ldots,\mathbf{f}_{L/N_{v}-1}]\in\mathbb{C}^{M\times\frac{L}{N_{v}}}$
denote the matrix of SVAM beamformers. We have the following result.

######  Corollary 1.2.

The $\mathrm{CRB}(u)$ using measurements in (2) from the beamformer matrix
$\mathbf{W}^{\mathrm{P}}$ under the construction in (4) over $L$ snapshots
when $\alpha$ is known is given by

$\displaystyle\mathrm{CRB}(u)$
$\displaystyle=\frac{1}{N_{v}}\frac{\sigma_{n}^{2}}{2P_{s}|\alpha|^{2}}\left\\{\left(\frac{\partial}{\partial
u}\bm{\phi}_{M}(u)\right)^{H}\mathbf{F}^{\mathrm{P}}(\mathbf{F}^{\mathrm{P}})^{H}\frac{\partial}{\partial
u}\bm{\phi}_{M}(u)\right.$ (36)
$\displaystyle\quad+\Biggl{.}G\Biggr{\\}}^{-1},\mbox{ where}$ $\displaystyle
G$
$\displaystyle=\frac{\pi^{2}(N_{v}-1)(2N_{v}-1)}{6}\bm{\phi}_{M}(u)^{H}\mathbf{F}^{\mathrm{P}}(\mathbf{F}^{\mathrm{P}})^{H}\bm{\phi}_{M}(u)$
(37)
$\displaystyle\quad-\pi(N_{v}-1)\mathrm{Im}\left\\{\left(\frac{\partial}{\partial
u}\bm{\phi}_{M}(u)\right)^{H}\mathbf{F}^{\mathrm{P}}(\mathbf{F}^{\mathrm{P}})^{H}\bm{\phi}_{M}(u)\right\\}$

###### Proof.

Proof follows from simplifying (33) using (4), and is provided in [39]. ∎

The CRB expression in (36) differs from the expression in (35) in two ways.
The first term in (36) involves an array of dimension $M=N-N_{v}+1$ instead of
$N$ in (35). For large array sizes, $N$, which are typical in mmWave systems,
we expect this term to be similar to that in (35). Secondly, the denominator
in (36) has an additional second term ‘$+G$’. We show that the conditions
under which $G\geq 0$ is not difficult to satisfy, by deriving a sufficient
condition to ensure the same.

######  Theorem 2.

$G$ in (37) is non-negative if

$\frac{\bm{\phi}^{H}_{M}(u)\mathbf{F}^{\mathrm{P}}(\mathbf{F}^{\mathrm{P}})^{H}\bm{\phi}_{M}(u)}{\lVert\bm{\phi}_{M}(u)\rVert^{2}}\geq\frac{\lambda_{\mathrm{max}}\left((\mathbf{F}^{\mathrm{P}})^{H}\mathbf{P}_{u,\perp}\mathbf{F}^{\mathrm{P}}\right)}{4},$
(38)

where $\lambda_{\mathrm{max}}(\mathbf{X})$ denotes the largest eigenvalue of
the matrix $\mathbf{X}$.
$\mathbf{P}_{u,\perp}=[\bm{\phi}_{M}(u)\>\bm{\phi}^{\perp}_{M}(u)]\left[\begin{array}[]{cc}\lVert\bm{\phi}_{M}(u)\rVert^{2}&0\\\
0&\lVert\bm{\phi}^{\perp}_{M}(u)\rVert^{2}\end{array}\right]^{-1}$
$\times[\bm{\phi}_{M}(u)\>\>\bm{\phi}^{\perp}_{M}(u)]^{H}$ denotes a
projection onto the subspace of orthogonal vectors $\bm{\phi}_{M}(u)$ and
$\bm{\phi}^{\perp}_{M}(u)=\left[\begin{array}[]{cccc}-\frac{(M-1)}{2}&(1-\frac{(M-1)}{2})&\ldots&\frac{(M-1)}{2}\end{array}\right]^{T}\odot\bm{\phi}_{M}(u)$.

###### Proof.

The proof is provided in Appendix section VII-A. ∎

The following remark discusses the implication of Theorem 2.

######  Remark 3.

If the left singular vectors of $\mathbf{F}^{\mathrm{P}}$ includes
$\bm{\phi}_{M}(u)$ and $\bm{\phi}^{\perp}_{M}(u)$ (post-normalization), we can
further simplify (38). If $\bm{\phi}_{M}(u)$ leads $\bm{\phi}^{\perp}_{M}(u)$,
then (38) is trivially satisfied. If the opposite is true, in that,
$\bm{\phi}^{\perp}_{M}(u)$ leads $\bm{\phi}_{M}(u)$, then (38) describes the
required gap in the two corresponding singular values within
$\mathbf{F}^{\mathrm{P}}$. In practice, we expect an adaptive scheme to choose
beamformers close to the direction of the DoA. Also, it was found out that
$G\geq 0$ very often even when $\mathbf{F}^{\mathrm{P}}$ contained i.i.d.
complex Gaussian random entries.

|  |
---|---|---
(a) Snapshots, $L=60$ | (b) SNR = $-10$ dB | (c) SNR = $-5$ dB
Figure 2:  (a) RMSE vs SNR: Proposed SVAM sensing with modified-hiePM improves
over hiePM[6] when $N_{v}\in\\{2,3,4,5\\}$. (b) RMSE vs number of snapshots at
SNR=$-10$ dB: Proposed sensing reduces the duration needed for beam alignment
when $N_{v}=2$. (c) RMSE vs number of snapshots at SNR=$-5$ dB: At high SNR,
the proposed scheme still improves over the benchmark scheme, but takes
slightly more time to converge.

Theorem 2 emphasizes that the sufficient condition for ensuring $G\geq 0$ are
not difficult to satisfy and highlights the benefit from using SVAM sensing
compared to the benchmark scheme. The benchmark analysis can be extended to
include more general scenario, wherein it need not repeat the beamformer in
$N_{v}$ snapshots. Since the contribution from each snapshot appears
independent of other snapshots, as a linear sum in the denominator in the
general CRB expression in (33), any time a beamformer is repeated in a
benchmark scheme, it can be replaced with the proposed scheme. A similar
impact and analysis can be carried out to reveal ensuing benefits.

###  III-B Case Study: Combining SVAM with HiePM Framework

The hiePM algorithm[6] processes each new snapshot and updates the beamformer
based on the current estimate of the posterior density on the unknown DoA. To
incorporate the proposed sensing, one approach is to update the beamformer
after every $N_{v}$ snapshots, where $N_{v}$ denotes the size of the virtual
ULA. Within each $N_{v}$ interval, a SVAM beamformer, $\mathbf{f}_{t}$, is
designed; $\mathbf{f}_{t}$ is simply a codeword from the hierarchical codebook
that satisfies the selection criteria within hiePM framework. The beamfomer
(of physical antenna size, $N$) for $N_{v}$ snapshots within this interval is
constructed as in (4). The remainder steps in Algorithm 1 in [6] are
compatible with the proposed sensing. This modification also becomes crucial
when $\alpha$ is unknown; this is discussed in Section IV.

######  Remark 4.

The impact of the modification can be understood in the following manner. The
hiePM strategy in [6] may repeat the same beamformer multiple times until the
posterior condition triggers a new beamformer. While it repeats the
beamformer, it gains only in terms of the Signal-to-Noise Ratio (SNR), which
is also evident from the CRB analysis for the benchmark scheme in (35).
$N_{v}$ multiplied with $P_{s}$ in (35) indicates a $N_{v}$-fold SNR boost.
Instead of a repeated beamformer, the proposed sensing aims to achieve both, a
SNR boost and at the same time an _(virtual) aperture_ gain (‘$+G$’ term in
the CRB analysis). Since the adaptive beamformer sequence is unknown apriori,
analysis of the proposed modification to hiePM in this subsection is more
involved. Instead, we provide empirical studies in the next subsection.

######  Remark 5.

The proposed sensing can be incorporated in other beam alignment procedures as
well. The inference procedure and adaptive strategy may require suitable
modifications, or be left unaltered as demonstrated using the hiePM case study
in this subsection.

###  III-C Numerical Results

We numerically analyze the benefits of the proposed sensing when used with the
hiePM framework in [6], and when $\alpha$ is assumed to be known. We compare
the hiePM[6] approach with i. the proposed modified algorithm in Section
III-B, and ii. the benchmark scheme where the beamformer is simply repeated,
as described in Section III-A. We also plot the CRB444Note that since the
posterior density is computed on a grid which includes the ground truth angle,
the performance may be biased. Thus, CRB is not a valid lower bound but it is
provided for insight. conditioned on the beamformers selected by the modified
algorithm with $N_{v}=2$. We assume that the DoA, $u$, lies in between $[0,1)$
i.e., RoI is $\frac{1}{2}$ of the entire space. The information about the RoI
is incorporated into the posterior calculation. We set the following
parameters as: Physical antenna size: $N=64$, number of grid points uniformly
spaced in RoI: $G=64$, number of random realizations for averaging: $Q=1000$.
The main metric employed for comparison is the root mean squared error (RMSE)
computed in $u$-space as
$\mbox{RMSE}=\sqrt{\frac{1}{Q}\sum_{q=1}^{Q}\left(\hat{u}_{q}-u_{q}\right)^{2}}$,
$u_{q}$ and $\hat{u}_{q}$ denotes the true and the estimated angle
respectively, in the $q-$th realization. The estimated angle is the on-grid
angle corresponding to the mode of the posterior density estimated by the
algorithm for each curve. In this work we define SNR as the signal power to
noise ratio at each antenna i.e., before combining.

####  III-C1 Performance as a function of SNR

In Fig. 2 (a) we plot the RMSE as a function of SNR for the modified-hiePM
under the proposed SVAM sensing using different virtual ULA sizes. As observed
in this plot, given $L=60$ snapshots, the curves using $N_{v}\in\\{2,3,4,5\\}$
improve over the hiePM in [6]. As the virtual aperture increases, the number
of beamformer updates, given by ‘$L/N_{v}$’, reduces. For
$N_{v}\in\\{10,15\\}$, the reduced number of updates or longer acquisition
time before a beamformer update, is seen to negatively impact the performance.
Studying the tradeoff between the virtual antenna size and the frequency of
update as they impact the beam alignment performance is an important future
direction.

####  III-C2 Performance as a function of number of snapshots

In Fig. 2 (b) and (c), we plot the RMSE as a function of number of snapshots
at SNR=$-10$ dB and $-5$ dB, respectively. As observed from the Fig. 2 (b),
the performance of hiePM improves under the proposed sensing when $N_{v}$ is
set to $2$. Setting it to a higher value such as $N_{v}=10$ degrades the
performance, for reasons similar to those discussed for Fig. 2 (a). In red
dashed curve we plot the performance for hiePM with benchmark sensing scheme
(in Section III-A). Although the benchmark scheme improves the beam alignment
duration over hiePM (solid blue curve), it still is much higher compared to
setting $N_{v}=2$ in the proposed sensing. This indicates that the improvement
under SVAM is not merely due to more reliable beamformer adaptations caused by
basing adaptations on more measurements. It is inherent to the _phase
response_ characteristics of the beamformer under SVAM, as described in (6).
Note that as we increase SNR to $-5$ dB, the performance gains can be limited
as seen from Fig. 2(b). This highlights that SVAM is beneficial in reducing
the training time at low SNR. Also it is again observed that SVAM improves
over the benchmark scheme. Thus, a dynamic method to adapt the virtual ULA
size is an interesting future direction.

##  IV Beam Alignment Algorithm with Unknown $\alpha$

We begin with a wide RoI,
$u\in[u_{l},u_{r}],u_{l}<u_{r},u_{l},u_{r}\in[-1,1)$, and presume that the DoA
lies within the RoI. Therefore, we initially set the SVAM beamformer to span
this region and suppress any interference coming from outside the RoI. This
ensures that we incorporate any prior information available about the DoA. The
approach is also practical, as base stations are typically deployed with
dedicated antennas to serve a specific RoI. We collect measurements over time
and process them to compute an approximate posterior on the unknown angle. We
adapt SVAM beamformer once we accrue enough posterior mass around the mode of
the posterior. We now describe the methodology adopted in this work to
estimate the posterior and to adapt the SVAM beamformer over time.

###  IV-A Algorithm Preliminaries: Initializing Angular Grid and Stochastic
Modeling of both $\alpha$ and $u$

Since $\alpha$ is unknown but assumed to be fixed during the training phase,
it can be estimated along with the DoA given _two_ or more measurements. We
observe this from the conditional CRB on variance for angle estimation when
$\alpha$ is unknown. Let
$\mathbf{W}=[\mathbf{w}_{0},\ldots,\mathbf{w}_{L-1}]\in\mathbb{C}^{N\times L}$
denote the beamformer matrix used to generate measurements
$\mathbf{y}=[y_{0},\ldots,y_{L-1}]^{T}\in\mathbb{C}^{L}$ as in (2). The
beamformers may be designed generally, and not necessarily under SVAM. The
$\mathrm{CRB}(u)$ is given by

$\displaystyle\mathrm{CRB}(u)$
$\displaystyle=\frac{\sigma_{n}^{2}}{2P_{s}|\alpha|^{2}}\left\\{\left(\frac{\partial}{\partial
u}\bm{\phi}_{N}(u)\right)^{H}\mathbf{W}\right.$
$\displaystyle\>\>\left.\times\left(\mathbf{I}-\frac{\mathbf{W}^{H}\bm{\phi}_{N}(u)\bm{\phi}_{N}(u)^{H}\mathbf{W}}{\bm{\phi}_{N}(u)^{H}\mathbf{W}\mathbf{W}^{H}\bm{\phi}_{N}(u)}\right)\mathbf{W}^{H}\frac{\partial}{\partial
u}\bm{\phi}_{N}(u)\right\\}^{-1},$ (39)

and can be derived from the conditional CRB expression (Theorem 4.1) in [40]
by stacking all $L$ snapshots along a column and replacing $\bm{\phi}_{N}(u)$
and $\frac{\partial}{\partial u}\bm{\phi}_{N}(u)$ by
$\mathbf{W}^{H}\bm{\phi}_{N}(u)$ and $\mathbf{W}^{H}\frac{\partial}{\partial
u}\bm{\phi}_{N}(u)$, respectively. For $L=1$, we can see that the CRB is
infinite. Furthermore, if the beamformer is kept fixed over time i.e., if
$\mathbf{w}_{l}=\mathbf{w}_{0},l\in[L]$, or in general if $\mathbf{W}$ is
rank-one, even then the CRB is infinite. This exercise highlights a key
requirement for being able to estimate both DoA and $\alpha$ simultaneously.
The magnitude or phase response of the beamformer $\mathbf{w}_{l}$ should have
variation over time, and it must vary differently for different angles. The
requirement is not satisfied in the case of hiePM algorithm, if initial
measurements are all taken using a fixed codeword, which is possible, early
on, within the hiePM framework555The CRB analysis is applicable when the
unknowns $(\alpha,u)$ are estimated in the maximum likelihood sense. The
issues can be circumvented by imposing a grid or strong prior information on
$\alpha$[8].. In the proposed sensing scheme this condition is naturally
prevented, as the phase response (over time) leads to the virtual array
manifold, which is sufficient to estimate both $\alpha$ and $u$ when $N_{v}>1$
and given at least two measurements.

The two basic pre-requisities for adapting the SVAM beamformer are the next
beam direction to steer towards, and the beamwidth. Since the problem-at-hand
considers the single path scenario, the beam direction can be estimated by
_matching_ -based criteria which results in a maximum likelihood estimate
(MLE), and it also yields a _deterministic_ estimate for $\alpha$. However,
for a good beamwidth selection we need to account for the uncertainty in the
estimation procedure. The bounds for the CRB analysis can be used. In this
work, we achieve this by modeling both $\alpha$ and the unknown DoA, $u$, as
_stochastic_ variables. For the latter, we impose a uniform prior distribution
in the wide RoI. We model $\alpha$ with a complex circular Gaussian
distribution with a parameterized prior.

We introduce a uniform grid, $\mathbf{u}_{\mathrm{grid}}$, of size $G$ within
the RoI, $[u_{l},u_{r}]$. Let $u_{i},i\in[G]$, denote the $i$-th grid point.
The initial grid can be refined as we successively reduce uncertainty of the
estimate for $u$ over snapshots. The grid refinement aspect is not focused in
this paper, and left as future work. For each candidate DoA, $u_{i}$, we
estimate a corresponding value for the complex path gain; let us denote the
same as $\alpha_{i}$. We impose the following parameterized prior on
$\alpha_{i}$

$\alpha_{i}\sim\mathcal{CN}(0,\gamma_{i}).$ (40)

######  Remark 6.

Assigning a different complex gain, $\alpha_{i}$, per candidate angle,
$u_{i}$, is a non-trivial choice. By doing so, we allow each candidate $u_{i}$
to _explain_ the measurements at its best. Going forward, we explicitly
utilize the presence of a single path by identifying suitable value for
hyperparameters, $\gamma_{i}$, separately, instead of jointly. A joint
optimization is widely adopted in the _absence_ of knowledge of the model
order[41].

A consequence of the prior imposed in (40) is that the marginalized pdf of the
measurements $\tilde{\mathbf{y}}_{t}$ in (21) after $(t+1)N_{v}$ snapshots,
conditioned on angle $u=u_{i}$ is given by

$\displaystyle f(\tilde{\mathbf{y}}_{t}\mid u=u_{i};\gamma_{i})$
$\displaystyle\quad=\int f(\tilde{\mathbf{y}}_{t}\mid
u=u_{i},\alpha_{i})f(\alpha_{i};\gamma_{i})\>d\alpha_{i}$
$\displaystyle\quad=\frac{1}{\pi^{(t+1)N_{v}}\mathrm{det}(\bm{\Sigma}_{i,t})}\exp{\left(-\tilde{\mathbf{y}}_{t}^{H}\bm{\Sigma}_{i,t}^{-1}\tilde{\mathbf{y}}_{t}\right)},\mbox{
where}$ (41) $\displaystyle\bm{\Sigma}_{i,t}$
$\displaystyle=P_{s}\gamma_{i}\left(\tilde{\bm{\beta}}_{t}(u_{i})\otimes\bm{\phi}_{N_{v}}(u_{i})\right)\left(\tilde{\bm{\beta}}_{t}(u_{i})\otimes\bm{\phi}_{N_{v}}(u_{i})\right)^{H}+\sigma_{n}^{2}\mathbf{I}$
$\displaystyle=P_{s}\gamma_{i}\left(\tilde{\bm{\beta}}_{t}(u_{i})\tilde{\bm{\beta}}_{t}(u_{i})^{H}\right)\otimes\left(\bm{\phi}_{N_{v}}(u_{i})\bm{\phi}_{N_{v}}(u_{i})^{H}\right)+\sigma_{n}^{2}\mathbf{I},$
(42)

where in the last line, we use the mixed product property of Kronecker
product, namely if matrix products $\mathbf{AC}$ and $\mathbf{BD}$ are
defined, then
$(\mathbf{A}\otimes\mathbf{B})(\mathbf{C}\otimes\mathbf{D})=\mathbf{AC}\otimes\mathbf{BD}$.
Note that $\sigma_{n}^{2}$ is assumed to be known666This assumption is also
made in other works[6, 8].. In the absence of such knowledge, it can be
included in the estimation procedure (see [41]). We focus on the estimation of
$\alpha$ in this work, and study the impact of imperfect knowledge of
$\sigma_{n}^{2}$ in the simulation section.

###  IV-B Estimation of Hyperparameters under the MLE Framework

We find the hyperparameters of the imposed prior, namely $\gamma_{i},i\in[G]$,
in the MLE sense. The derivation is simple, and similar to the work in [42,
43]. Since $\bm{\Sigma}_{i,t}$ in (42) can be expressed as a rank-one
perturbation to the noise covariance matrix, the determinant can be expressed
in closed-form as

$\mathrm{det}(\bm{\Sigma}_{i,t})=(P_{s}\gamma_{i}g_{t}(u_{i})\lVert\bm{\phi}_{N_{v}}(u_{i})\rVert_{2}^{2}+\sigma_{n}^{2})\cdot(\sigma_{n}^{2})^{(t+1)N_{v}-1},$
(43)

where $g_{t}(u_{i})=\sum_{t^{\prime}=0}^{t}|\beta_{t^{\prime}}(u_{i})|^{2}$.
Also, using matrix-inversion lemma, we can simplify
$\tilde{\mathbf{y}}_{t}^{H}\bm{\Sigma}_{i,t}^{-1}\tilde{\mathbf{y}}_{t}$ as

$\displaystyle\tilde{\mathbf{y}}_{t}^{H}\bm{\Sigma}_{i,t}^{-1}\tilde{\mathbf{y}}_{t}$
$\displaystyle=\sigma_{n}^{-2}\lVert\tilde{\mathbf{y}}_{t}\rVert_{2}^{2}-\frac{P_{s}\gamma_{i}}{\sigma_{n}^{2}}$
(44)
$\displaystyle\quad\times\frac{|\bm{\phi}^{H}_{N_{v}}(u_{i})\sum_{t^{\prime}=0}^{t}\beta^{c}_{t^{\prime}}(u_{i})\mathbf{y}_{t^{\prime}}|^{2}}{(P_{s}\gamma_{i}g_{t}(u_{i})\lVert\bm{\phi}_{N_{v}}(u_{i})\rVert_{2}^{2}+\sigma_{n}^{2})}.$

Using (43) and (44), we maximize $f(\tilde{\mathbf{y}}_{t}\mid
u=u_{i};\gamma_{i})$ over $\gamma_{i}$ by setting the derivative w.r.t. the
same as zero. We get

$\displaystyle\hat{\gamma}_{i,t}=\max\Biggl{\\{}$ $\displaystyle
0,\frac{1}{P_{s}g_{t}(u_{i})\lVert\bm{\phi}_{N_{v}}(u_{i})\rVert_{2}^{2}}\Bigr{.}$
(45)
$\displaystyle\left.\times\left(\left|\frac{\left(\tilde{\bm{\beta}}_{t}(u_{i})\otimes\bm{\phi}_{N_{v}}(u_{i})\right)^{H}}{\lVert\tilde{\bm{\beta}}_{t}(u_{i})\otimes\bm{\phi}_{N_{v}}(u_{i})\rVert_{2}}\tilde{\mathbf{y}}_{t}\right|^{2}-\sigma_{n}^{2}\right)\right\\}.$

The above can be interpreted as normalized beamforming output power that is
compensated for the noise power. The beamformer in this case is the
conventional beamformer[27] and takes into account the complex gain,
$\tilde{\bm{\beta}}_{t}(u_{i})$, from the SVAM beamformer
$\mathbf{f}_{t^{\prime}},t^{\prime}\in[t+1]$. The posterior density of
$\alpha_{i}$ is also complex circular Gaussian, and can be computed as

$f(\alpha_{i}\mid\tilde{\mathbf{y}}_{t},u_{i};\hat{\gamma}_{i,t})=\frac{1}{\pi\hat{\sigma}^{2}_{\alpha_{i},t}}\exp{\left(-\frac{|\alpha_{i}-\hat{\mu}_{\alpha_{i},t}|^{2}}{\hat{\sigma}^{2}_{\alpha_{i},t}}\right)},$
(46) $\displaystyle\begin{aligned} \mbox{where
}\hat{\mu}_{\alpha_{i},t}&=\sqrt{P_{s}}\hat{\gamma}_{i,t}\frac{\left(\tilde{\bm{\beta}}_{t}(u_{i})\otimes\bm{\phi}_{N_{v}}(u_{i})\right)^{H}\tilde{\mathbf{y}}_{t}}{P_{s}\hat{\gamma}_{i,t}g_{t}(u_{i})\lVert\bm{\phi}_{N_{v}}(u_{i})\rVert_{2}^{2}+\sigma_{n}^{2}}\\\
\hat{\sigma}^{2}_{\alpha_{i},t}&=\hat{\gamma}_{i,t}\frac{\sigma_{n}^{2}}{P_{s}\hat{\gamma}_{i,t}g_{t}(u_{i})\lVert\bm{\phi}_{N_{v}}(u_{i})\rVert_{2}^{2}+\sigma_{n}^{2}}.\end{aligned}$
(47)

###  IV-C Computing Posterior on $u$

We begin with describing the posterior probability,
$p(u=u_{i}\mid\tilde{\mathbf{y}}_{t})$, and identifying the missing pieces for
computing this posterior. Using Bayes’ rule, we can write

$\displaystyle p(u=u_{i}\mid\tilde{\mathbf{y}}_{t})$
$\displaystyle=\frac{p(u=u_{i})f(\tilde{\mathbf{y}}_{t}\mid
u_{i})}{f(\tilde{\mathbf{y}}_{t})}\overset{(a)}{\propto}f(\tilde{\mathbf{y}}_{t}\mid
u_{i}),$ (48)

where $(a)$ follows from two facts: i. we rely on discrete _uniform_ (prior)
distribution on $u$, ii. $f(\tilde{\mathbf{y}}_{t})$ does not depend on $u$.
Consequently, the goal is to be able to compute $f(\tilde{\mathbf{y}}_{t}\mid
u_{i})$ as accurately possible. We explore this direction next.

Using marginalization and Bayes’ rule we get

$\displaystyle f(\tilde{\mathbf{y}}_{t}\mid u_{i})$ $\displaystyle=\int
f(\tilde{\mathbf{y}}_{t},\alpha\mid u_{i})d\alpha$ (49) $\displaystyle=\int
f(\alpha\mid u_{i})f(\tilde{\mathbf{y}}_{t}\mid\alpha,u_{i})d\alpha.$

The integrand consists of two factors, of which, the second factor,
$f(\tilde{\mathbf{y}}_{t}\mid\alpha,u_{i})$, is well-defined, provided the
distribution of noise. However, the first factor, $f(\alpha\mid u_{i})$, is
unavailable.

In the absence of this knowledge, we approximate the unknown distribution with
the best proxy available at-hand that uses all the available measurements
i.e., we replace $f(\alpha\mid u_{i})$ with
$f(\alpha_{i}\mid\tilde{\mathbf{y}}_{t},u_{i};\hat{\gamma}_{i,t})$. The
required quantity,
$f(\alpha_{i}\mid\tilde{\mathbf{y}}_{t},u_{i};\hat{\gamma}_{i,t})$, was
computed in the previous subsection, and we update this proxy as we collect
more measurements. In other words, we compute the following _approximate_
likelihood

$\displaystyle\hat{f}_{t}(\tilde{\mathbf{y}}_{t}\mid u_{i})$
$\displaystyle=\int
f(\alpha_{i}\mid\tilde{\mathbf{y}}_{t},u_{i};\hat{\gamma}_{i,t})f(\tilde{\mathbf{y}}_{t}\mid\alpha_{i},u_{i})d\alpha_{i}$
(50)
$\displaystyle=\frac{1}{\pi^{(t+1)N_{v}}\mathrm{det}(\tilde{\bm{\Sigma}}_{i,t})}\exp{\left(-\tilde{\mathbf{e}}_{i,t}^{H}\tilde{\bm{\Sigma}}^{-1}_{i,t}\tilde{\mathbf{e}}_{i,t}\right)},$

where
$\tilde{\mathbf{e}}_{i,t}=\tilde{\mathbf{y}}_{t}-\tilde{\bm{\mu}}_{i,t}$,
$\tilde{\bm{\mu}}_{i,t}=\sqrt{P_{s}}\hat{\mu}_{\alpha_{i},t}\left(\tilde{\bm{\beta}}_{t}(u_{i})\otimes\bm{\phi}_{N_{v}}(u_{i})\right)$
and
$\tilde{\bm{\Sigma}}_{i,t}=P_{s}\hat{\sigma}^{2}_{\alpha_{i},t}\left(\tilde{\bm{\beta}}_{t}(u_{i})\tilde{\bm{\beta}}_{t}(u_{i})^{H}\right)\otimes\left(\bm{\phi}_{N_{v}}(u_{i})\bm{\phi}_{N_{v}}(u_{i})^{H}\right)$
$+\sigma_{n}^{2}\mathbf{I}$.
$(\hat{\mu}_{\alpha_{i},t},\hat{\sigma}^{2}_{\alpha_{i},t})$ were estimated in
the previous subsection.

Next, we demonstrate the procedure to compute the required quantities in (50)
efficiently. We provide the final result below

$\displaystyle\mathrm{det}(\tilde{\bm{\Sigma}}_{i,t})$
$\displaystyle=(P_{s}\sigma^{2}_{\alpha_{i},t}g_{t}(u_{i})\lVert\bm{\phi}_{N_{v}}(u_{i})\rVert_{2}^{2}+\sigma_{n}^{2})$
(51) $\displaystyle\quad\times(\sigma_{n}^{2})^{(t+1)N_{v}-1}$
$\displaystyle\tilde{\mathbf{e}}_{i,t}^{H}\tilde{\bm{\Sigma}}_{i,t}^{-1}\tilde{\mathbf{e}}_{i,t}$
$\displaystyle=\sigma_{n}^{-2}\lVert\tilde{\mathbf{e}}_{i,t}\rVert_{2}^{2}-\frac{P_{s}\sigma^{2}_{\alpha_{i},t}}{\sigma_{n}^{2}}$
(52)
$\displaystyle\quad\times\frac{|\bm{\phi}^{H}_{N_{v}}(u_{i})\sum_{t^{\prime}=0}^{t}\beta^{c}_{t^{\prime}}(u_{i})\tilde{\mathbf{e}}_{i,t^{\prime}}|^{2}}{(P_{s}\sigma^{2}_{\alpha_{i},t}g_{t}(u_{i})\lVert\bm{\phi}_{N_{v}}(u_{i})\rVert_{2}^{2}+\sigma_{n}^{2})}.$

Once the required quantities in (50) are computed, we compute the likelihood
estimate, $\hat{f}_{t}(\tilde{\mathbf{y}}_{t}\mid u_{i})$, for all grid points
$u_{i},i\in[G]$, in the RoI. We get the estimate for the posterior as

$\hat{p}_{t}(u=u_{i}\mid\tilde{\mathbf{y}}_{t})=\frac{\hat{f}_{t}(\tilde{\mathbf{y}}_{t}\mid
u_{i})}{\sum_{i^{\prime}}\hat{f}_{t}(\tilde{\mathbf{y}}_{t}\mid
u_{i^{\prime}})}.$ (53)

Let
$\hat{\mathbf{p}}_{t}=[\hat{p}_{t}(u_{0}\mid\tilde{\mathbf{y}}_{t}),\hat{p}_{t}(u_{1}\mid\tilde{\mathbf{y}}_{t}),\ldots,\hat{p}_{t}(u_{G-1}\mid\tilde{\mathbf{y}}_{t})]^{T}$.
We adapt the SVAM beamformer, $\mathbf{f}_{t+1}$, for the next snapshot based
on the computed posterior in (53). We discuss this step next.

###  IV-D Proposed Adaptive SVAM Beamforming Algorithm

The aim is to ensure that the DoA lies within the passband of the SVAM
beamformer, $\mathbf{f}_{t}$, so that effective received SNR for inference is
high. A high effective SNR helps further to ensure a successful beam
alignment. The presented approach still may not ensure that the DoA always
stays in the passband. We circumvent this issue by evaluating the posterior
$\hat{p}_{t}(u_{i}\mid\tilde{\mathbf{y}}_{t})$ over the entire RoI. This
allows the posterior mass to move freely and helps to recover when the SVAM
beamformer adapted prematurely in the incorrect region. In other words, even
if we adapt the SVAM beamformer, the RoI is kept fixed.

Result: Beam direction: $u^{*}$

Input:
$L,N_{v},P_{s},\sigma_{n}^{2},\mathrm{beam\\_dir},\mathrm{BW}_{\mathbf{f}}:=\mathrm{BW}_{\mathrm{initial}},p_{\mathrm{thresh}}$

1 Initialize:
$\mathrm{beam\\_spec\\_initial}:=\\{\mathrm{beam\\_dir,\mathrm{BW}_{\mathbf{f}}}\\},\mathbf{f}_{0}:=\mathrm{beamformer\\_design}(\mathrm{beam\\_spec\\_initial})$

2 for  _$l:=0\ >\mathrm{to\>}L-1$_ do

3 $\mathbf{y}_{l}=\mathbf{w}_{l}^{H}\mathbf{x}_{l}$ (New measurement,
$\mathbf{w}_{l}$ as in (4))

4 if  _$\mathrm{mod}(l+1,N_{v})=0$_ then

5 $t:=(l+1)/N_{v}-1$

6 Form $\mathbf{y}_{t}$, then $\tilde{\mathbf{y}}_{t}$ as in (7) and (21)
respectively

7 ($\alpha$ Prior): Compute $\hat{\gamma}_{i,t}$ as in (45), $i\in[G]$

8 ($\alpha$ Posterior): Compute
$\hat{\mu}_{\alpha_{i},t},\hat{\sigma}^{2}_{\alpha_{i},t}$ as in (47)

9 (Likelihood $\mid u_{i}$): Compute likelihood in (50)

10 ($u_{i}$ Posterior): $\hat{\mathbf{p}}_{t}:=$Compute pmf in (53)

11 $\mathrm{BW}_{\mathrm{check}}:=0.5\times\mathrm{BW}_{\mathbf{f}}$

12
$[\mathrm{peak\\_prob},\mathrm{beam\\_spec}]:=\mathrm{cumul\\_peak}(\hat{\mathbf{p}}_{t},\mathrm{BW}_{\mathrm{check}})$
(Algorithm 2)

13 while  _$\mathrm{peak\\_prob} <p_{\mathrm{thresh}}$_ do

14 $\mathrm{BW}_{\mathrm{check}}:=2\times\mathrm{BW}_{\mathrm{check}}$

15
$[\mathrm{peak\\_prob},\mathrm{beam\\_spec}]:=\mathrm{cumul\\_peak}(\hat{\mathbf{p}}_{t},\mathrm{BW}_{\mathrm{check}})$
(Algorithm 2)

16 end while

17 $\mathrm{BW}_{\mathbf{f}}:=\mathrm{BW}_{\mathrm{check}}$

18 $\mathbf{f}_{t+1}$:=$\mathrm{beamformer\\_design}(\mathrm{beam\\_spec})$

19 end if

20

21 end for

22$i^{*}:=\arg\max_{i}\>\hat{p}_{L/N_{v}-1}(u_{i}\mid\tilde{\mathbf{y}}_{L/N_{v}-1}),u^{*}:=u_{\mathrm{i^{*}}}$

Algorithm 1 Proposed Adaptive SVAM Beamforming

####  IV-D1 Proposed Algorithm

We initially set the beam direction of the SVAM beamformer, $\mathbf{f}_{0}$,
to the centre of the RoI, and the beamwidth to
$\mathrm{BW}_{\mathbf{f}}=\mathrm{BW}_{\mathrm{initial}}<2$ in $u$-space;
$\mathrm{BW}_{\mathrm{initial}}$ chosen to cover the RoI. We adapt the SVAM
beamformer only if there is a sufficient posterior mass _concentrated_ around
the posterior mode. More specifically, we adapt the beamformer if the peak of
the posterior mass in any contiguous span of a specific beamwidth, that
includes the posterior mode, exceeds a fixed threshold $p_{\mathrm{thresh}}$.
We begin with setting the beamwidth for this search to
$\mathrm{BW}_{\mathrm{check}}=0.5\times\mathrm{BW}_{\mathbf{f}}$ i.e., half of
the current beamwidth for the SVAM sensing. If the threshold condition is not
satisfied, $\mathrm{BW}_{\mathrm{check}}$ is doubled. This continues until the
posterior threshold condition is met. Note that in the default scenario, the
next beam resorts to the initial beam specifications. When the posterior
threshold condition is met, the corresponding contiguous span of angular grid
points is selected, and the next SVAM beamformer, $\mathbf{f}_{t+1}$, is
designed to cover the selected region. The threshold parameter
$p_{\mathrm{thresh}}$ may be set to a fixed value or chosen dynamically. A
lower value for $p_{\mathrm{thresh}}$ allows the SVAM beamformer to adapt
often, whereas a higher value makes the adaptations more cautious. We discuss
the role of this crucial parameter in detail in Section V. Algorithm 1
summarizes the overall approach.

######  Remark 7.

The proposed adaptive beam search mechanism has the ability to both refine, as
well as correct erroneous adaptations. In the case when the beam is adapted in
the wrong portion of the spatial region, the posterior mass along with the
mode is expected to either shift or widen as more measurements are collected.
The beam gets rectified within the proposed scheme as the search is carried
out around the updated posterior mode, and it has the ability to widen the
beamwidth beyond the $\mathrm{BW}_{\mathbf{f}}$ presently in use for the
sensing.

The presented approach to adapt the beamformer has an equivalent
representation within the _compact_ hierarchical codebook, and is discussed
next.

Result:
$\mathrm{peak\\_prob},\mathrm{beam\\_spec}:=\\{\mathrm{BD\\_sel},\mathrm{BW}^{\prime}\\}$

Input: $\mathbf{p},\mathrm{BW}^{\prime}$

1 Initialize: $\mathbf{1}_{\mathrm{BW}^{\prime}}$ (vector of $1$’s of size
$\equiv\mathrm{BW}^{\prime}$)

2 $\mathrm{mode}:=\arg\max_{j}\>\mathbf{p}(j)$

3 $\mathrm{cumul\\_prob}:=\mathbf{p}*\mathbf{1}_{\mathrm{BW}^{\prime}}$
(‘$*$’: convolution operation)

4
$[\mathrm{peak\\_prob},k]:=\max(\mathrm{cumul\\_prob}(\mathrm{mode}:\mathrm{mode}+\mathrm{size}(\mathbf{1}_{\mathrm{BW}^{\prime}})-1)$

$\mathrm{BD\\_sel}:=\mathbf{u}_{\mathrm{grid}}(\mathrm{mode}+k)-\mathrm{BW}^{\prime}/2$

Algorithm 2 Cumulative Posterior Peak

Result:
$l^{h}_{\mathrm{final}}\in[\log_{2}G],k^{h}_{\mathrm{final}}\in[2^{l^{h}_{\mathrm{final}}}-1]$

Input: $l^{h}_{\mathrm{init}}$ (current level in hierarchy), $\mathbf{p}$
(posterior pmf), $G$, $p_{\mathrm{thresh}}$

1 Initialize: $l^{h}:=l^{h}_{\mathrm{init}}+1,G_{l^{h}}:=2^{l^{h}}$

2 $\mathrm{mode}:=\arg\max_{j}\>\mathbf{p}(j)$

3 (node at level $l^{h}$):
$k^{h}:=\mathrm{floor}\left(\mathrm{mode}\times\frac{G_{l^{h}}}{G}\right)$

4 for  _$\bar{l}^{h}:=l^{h}\ >\mathrm{to}\>0$ (descending)_ do

5
$\mathrm{node\\_k\\_ind}:=\frac{G}{G_{l^{h}}}k^{h}:\frac{G}{G_{l^{h}}}(k^{h}+1)-1$

6 if  _$\mathrm{sum}(\mathbf{p}(\mathrm{node\\_k\\_ind}))\geq
p_{\mathrm{thresh}}$_ then

7 break

8 else

9 $k^{h}:=\mathrm{floor}(k^{h}/2)$

10 $G_{l^{h}}:=G_{l^{h}}/2$

11 end if

12

13 end for

$l^{h}_{\mathrm{final}}:=l^{h},k^{h}_{\mathrm{final}}:=k^{h}$

Algorithm 3 Beam Search In Hierarchical Codebook

####  IV-D2 Using Hierarchical Codebook

The proposed scheme in Algorithm 1 adopts a flexible beam design which
requires beam direction and beamwidth to design the SVAM beamformer. In some
cases, such flexible beam designs may not be possible due to complexity, and a
smaller codebook may be desired. We take an example of the (binary)
hierarchical codebook[7] and demonstrate the procedure to select a codeword
based on the posterior computed in (53). The procedure closely mimics the
approach taken in Algorithm 1, while constraining the beam to belong to the
hierarchical codebook.

The basic idea is to begin at one level below in hierarchy compared to the
current level used for SVAM sensing, and traverse _up_ the hierarchy to
satisfy the posterior condition. At one-level below current, we begin with
that node which contains the posterior mode. The principle in doing so, is to
try and ensure that the DoA is included in the next beam. We summarize this
beam search in Algorithm 3 which can replace line $11-18$ in Algorithm 1. We
also study the impact of using a hierarchical codebook on the performance of
the proposed adaptive scheme in simulation Section V-E.

######  Remark 8.

A variation on the beam search Algorithm 3 is to initialize the search at a
deeper hierarchical level, which can be favourable to converge early at high
SNRs, but may lead to premature beam focusing at low SNRs. Similar idea can be
incorporated with Algorithm 1 as well. This is not explored further in this
paper, but left as future work.

####  IV-D3 On HiePM-based Beam Alignment

The posterior computed in (50) involves approximating the posterior on
$\alpha$, $f(\alpha\mid u_{i})$, using a Gaussian density,
$f(\alpha_{i}\mid\tilde{\mathbf{y}}_{t},u_{i};\hat{\gamma}_{i,t})$. This
allows to bypass a grid approach on $\alpha$, which was explored in [8] under
Algorithm 1. Although the grid approach can be more accurate, it is
computationally expensive. Also initializing an appropriate grid on $\alpha$
is a non-trivial problem.

Although the adopted prior on $\alpha$ in this work and within Alg. 2 in [8],
both are Gaussian distributed, there is an important distinction. The
presented approach utilizes a _parameterized_ Gaussian prior unlike the fixed
Gaussian prior in [8]. This has a couple of implications: a) We allow the
framework to _learn_ the relevant parameters directly from the measurements
without relying on any additional information such as prior mean or variance,
b) The choice of prior here allows the framework to fit the best Gaussian
posterior on $\alpha$ using the measurements, in contrast to the fixed prior
case where only the posterior mean is learned from measurements and the
variance in the passband depends only on the prior and the noise variance
level. It is known that the inference is less sensitive to the values of
higher-level hyperparameters than the values for the $\alpha$-prior
distribution (see [44] and references therein). This leads to more _robust_
learning.

Most importantly, the proposed approach processes measurements _jointly_ for
computing the posterior on $u$ (see (48)). In contrast, in [8] the
$u$-posterior is updated _sequentially_. Since $\alpha$ is considered static
in time, both in this work and within Algorithm 2 in [8], and because the
posteriors are approximate, it was observed that processing measurements
_jointly_ instead of _sequentially_ performs better. In other words, the
$u$-posterior at previous snapshot is not a sufficient statistic when computed
_approximately_. The impact is studied in Section V-A.

##  V Numerical Results

In this section we perform numerical experiments to study the performance of
the proposed sensing and beam alignment Algorithm 1 under different scenarios,
and the impact of the parameters involved. The parameters that are central to
the performance include i. Size of the virtual ULA, $N_{v}$, ii. noise
variance parameter $\sigma_{n}^{2}$, iii. posterior threshold,
$p_{\mathrm{thresh}}$. In Section III-C, we studied the role of $N_{v}$; we
focus on the parameters $\sigma_{n}^{2}$ and $p_{\mathrm{thresh}}$ in this
section. We assume that the DoA lies in between $[0,1)$ i.e., RoI is
$\frac{1}{2}$ of the entire space. The information about the RoI is
incorporated into the posterior calculation. Note that the RoI can be
arbitrary, and one may choose a wider or narrower RoI depending on the use
case specifications. The virtual ULA constructed here has the same inter-
element spacing as the physical antenna. Depending on the RoI, a wider virtual
inter-element spacing can be realized777For e.g., here a $\lambda$-spacing
instead of $\lambda/2$ may be realized without causing ambiguity in angular
estimation, since the DoA lies in $\frac{1}{2}$ of the spatial region., but in
the following simulation we do not exploit it further. We set the following
parameters, unless otherwise specified as: Physical antenna size: $N=64$,
number of grid points uniformly spaced in RoI: $G=64$, virtual ULA size:
$N_{v}=4$, total number of snapshots: $L=120$, number of random realizations
for averaging: $Q=100$. The SVAM beamformer is designed using the Parks-
McClellan algorithm[30] with the passband edge set to realize the desired
beamwidth, and the transition width set as a fraction of the desired
beamwidth.

###  V-A Performance of Algorithm 1 as a function of SNR

---
Figure 3: RMSE as a function of SNR. Performance of Algorithm 1 is observed to
be close to the performance of algorithms using true $\alpha$.

In Fig. 3, we study the RMSE in $u$-space as a function of SNR in dB. The red
curve (with pentagram markers) plots the performance of hiePM with the SVAM
sensing scheme (Section III-B). It is seen to improve over the hiePM algorithm
in blue curve with hexagram markers, at SNR=$\\{-15,-12.5\\}$ dB. These two
curves assume that $\alpha$ is known. At high SNR, the schemes incur zero
error in this experiment as the DoA is on-grid. The green curves plots the
performance of the proposed approach when $\alpha$ is unknown. The different
curves correspond to different posterior threshold, $p_{\mathrm{thresh}}$,
levels for adapting the beamformer. At lower thresholds, the algorithm has the
ability to adapt often. This may lead to slightly unstable performance,
especially at low SNR, which is observed in the green-dashed curve with
$\circ$ markers where $p_{\mathrm{thresh}}=0.5$ is used. Setting a high
$p_{\mathrm{thresh}}$ requires a higher posterior mass to be accumulated for
the beamformer to adapt. This adds more stability to the algorithm, but may
incur slow beamformer adaptations, especially at high SNR. This is seen in
green-dashed curve with $\mathrm{x}$ markers where $p_{\mathrm{thresh}}=0.8$
is set. Note that all the green curves are already close to the blue and red
curves, without assuming any knowledge of $\alpha$. The curve for the
Algorithm 2 in [8] is plotted in purple with $\Delta$ markers. The algorithm
is provided with the true value of $\alpha$ as mean along with variance set to
$1$ for the Gaussian $\alpha$-prior. It is competitive at low SNR as it hinges
on the already good prior information, whereas at high SNR it disregards the
provided prior. Another important reason behind the large gap between this
curve, and the blue curve with hexagram markers is the fact that the static
$\alpha$-case considered in [8] is not exploited888Posterior on $\alpha$ at
current snapshot is used to compute likelihood of current measurement, but not
utilized to update likelihoods of previous measurements.. The proposed beam
alignment procedure exploits the same, and thus demonstrates a drastic
improvement as the resulting posterior on angle is more accurate. We plot the
(conditional) CRB on variance of angular estimation for two schemes i. using a
non-adaptive SVAM beamformer with beam direction as $0.5$ and beamwidth as $1$
ii. using adaptive SVAM beamformers chosen during runtime of Algorithm 1 with
$p_{\mathrm{thresh}}=0.6$. As seen in Fig. 3, the curves using proposed
approach attains (& surpasses) the CRBs as SNR increases. Note that since the
DoA is on grid which is also employed by the algorithms, they are biased.
Thus, CRB is not a valid lower bound. Still it provides useful insight about
the performance of the proposed inference procedure.

###  V-B Beamforming gain over time using Algorithm 1

(a) $p_{\mathrm{thresh}}=0.6$ | (b) $p_{\mathrm{thresh}}=0.7$
---|---
Figure 4: Beamforming gain over time (SNR=$-10$ dB).

In Fig. 4, we study the beamforming gain experienced by the DoA using
Algorithm 1 at SNR of $-10$ dB. We plot the mean, maximum and the minimum
beamforming gain over time (in terms of the virtual ULA segment index $t$)
over $100$ random realizations. In Fig. 4 (a) and (b) we plot the results when
$p_{\mathrm{thresh}}$ is set to $0.6$ and $0.7$, respectively. As expected, at
a lower posterior threshold the algorithm is more flexible, and thus the
maximum beamforming gain is achieved earlier than compared to the case when a
higher posterior threshold is set. In contrast, the minimum beamforming gain
may be less stable when the beamformer is adapted frequently at such low SNR.
This is seen in plot (a) where the minimum gain fluctuates even after $25$
virtual ULA segments i.e., $25\times 4=100$ snapshots. The solid curve
representing the mean beamforming gain indicates how the gain improves over
time as the beamwidth adaptively narrows. We also plot the beamforming gain
achieved by an ideal beamformer. A hierarchical level
$l^{h}\in\\{1,2,\ldots,5\\}$ indicates a beam focused in $2/2^{l^{h}+1}$ of
the spatial region and consequently achieves a beamforming gain of
$2^{l^{h}+1}/2$, in absolute scale. As observed in both the plots, beginning
with a beamformer focusing on $\frac{1}{2}$ of the space (equivalently
$10\log_{10}(2)\approx 3$ dB gain), the beamformer is able to adapt to
$1/32$-fraction, garnering a beamforming gain of around
$10\log_{10}(32)\approx 15$ dB. Thus, the proposed algorithm adds around $12$
dB gain at $-10$ dB SNR.

###  V-C Performance as a function of number of snapshots

In Fig. 5 we plot the RMSE over time (in terms of the number of virtual ULA
measurements $t$). We plot the hiePM algorithms with $\alpha$ perfectly known
for comparison. In Fig. 5 (a) and (b), we plot curves corresponding to
SNR$=-10$ dB and $0$ dB, respectively. We study two curves corresponding to
the proposed scheme i. $p_{\mathrm{thresh}}=0.6$ (green curve with
$\mathrm{x}$ markers) ii. $p_{\mathrm{thresh}}=0.8$ (purple curve with
$\diamond$ markers). The former setting allows the beamformer to aggressively
adapt, which can be rewarding at high SNR, but can lead to poor beamforming
gain in the low SNR regime as it adapts to incorrect regions. This is seen in
the two plots, at high SNR the curve corresponding to
$p_{\mathrm{thresh}}=0.6$ converges quickly, whereas at low SNR it leads to
larger error and variance initially, compared to setting
$p_{\mathrm{thresh}}=0.8$. Note that even at low SNR and low threshold setting
(Fig. 5 (a)), the yellow curve manages to adapt back to the correct spatial
region, which is indicated by a drop in RMSE around $t=25$ and is able to
retain the improvement over time. Overall, it is observed that a high
$p_{\mathrm{thresh}}$ leads to stable yet slow convergence, which is useful
when the SNR is low and if the training duration is short. At high SNR or if
the training duration is longer, a low $p_{\mathrm{thresh}}$ pays off, as the
_algorithm is robust to recover from its mistakes_!

One may dynamically select $p_{\mathrm{thresh}}$ such that it is set to higher
value initially during the training phase, but it may be reduced over time as
the beamformer narrows further and enjoys a high beamforming gain. This is
left for future work.

###  V-D Studying Impact of Noise Variance Parameter

|
---|---
(a) SNR $=-10$ dB | (a) SNR $=0$ dB
Figure 5: RMSE over time. Shaded region covers the RMSE $\pm$ root standard
deviation of squared error over time at every time index along x-axis. Figure
6: RMSE vs. $x\times\sigma_{n}^{2},x\in\\{0.1,1/8,1/4,1/2,1,2,4,8,10\\}$.
SNR$=-10$ dB, $L=200$. As the posterior threshold increases, the optimal value
for the noise variance parameter is observed to be lower than the true value.

We plot the RMSE as a function of different settings for the noise variance
parameter in Fig. 6. The optimal value is observed to be around the true noise
variance value. We also plot curves corresponding to different posterior
thresholds. For a low posterior threshold, the optimal value of the noise
variance parameter is observed to be slightly higher ($\times 2$) than or
equal to the true value. This is due to the aggressive nature of the algorithm
to adapt, which can be compensated by setting a slightly higher noise variance
parameter. On the other hand, a high posterior threshold already imparts more
stable adaptations, and consequently does not need setting a higher noise
variance parameter. In fact, at such large thresholds, the algorithm benefits
from lower noise variance parameter setting, which intuitively offsets the
conservative posterior threshold setting. This is observed by the curve
corresponding to $p_{\mathrm{thresh}}=0.9$, where setting noise variance to
$0.5\times\sigma_{n}^{2}$ leads to much better performance than setting it to
the true value.

###  V-E Impact of compact hierarchical codebook (Algorithm 3)

Figure 7: RMSE as a function of SNR. A hierarchical codebook is used, which
results in little to no performance loss, indicating that the overall
performance gain is unaffected by the compact hierarchical codebook.

In Fig. 7, we plot the curve corresponding to using a hierarchical codebook
(see purple curve with $\diamond$ markers). The approach is described in
Algorithm 3. We set $p_{\mathrm{thresh}}=0.6$ and compare the performance with
using the flexible codebook (see green curve with $\circ$ markers). As
observed in Fig. 7, the two implementations have very similar performance.
This suggests that the superior performance is due to other ingredients,
namely i. improved sensing (and filter design), ii. good posterior estimate,
and iii. good adaptive strategy. A large codebook size helps, but is not the
key to the performance improvement demonstrated in this paper.

##  VI Conclusion

We proposed a novel Synthesis of Virtual Array Manifold (SVAM) sensing
approach for the mmWave single RF chain systems and discussed the ensuing
benefits. More specifically, the proposed sensing is demonstrated to lead to
faster and more robust beam alignment. We believe this contribution will have
significant impact on the traditional paradigm for sensing in mmWave systems.
We also proposed a novel inference scheme that estimates a posterior density
on the small-scale fading coefficient and the unknown dominant path angle.
Based on the proposed inference procedure, an adaptive beamforming scheme is
provided that aims to collect high SNR measurements. Finally, the performance
of the proposed active sensing scheme is evaluated under different scenarios,
and a significant improvement over various benchmarks is demonstrated. The
empirical study also reveals the impact of the different design parameters on
the beam alignment performance.

##  VII Appendix

###  VII-A Proof of Theorem 2

###### Proof.

We first note that the following holds

$\frac{\partial}{\partial
u}\bm{\phi}_{M}(u)=j\pi\left\\{\frac{(M-1)}{2}\bm{\phi}_{m}(u)+\bm{\phi}^{\perp}_{M}(u)\right\\},\vspace{-0.5em}$
(54)

where
$\bm{\phi}^{\perp}_{M}(u)=\left[\begin{array}[]{cccc}-\frac{(M-1)}{2}&(1-\frac{(M-1)}{2})&\ldots&\frac{(M-1)}{2}\end{array}\right]^{T}$
$\odot\bm{\phi}_{M}(u)$, ‘$\odot$’ denotes Hadamard product. Using the result
in (54) we simplify the second term in $G$ as

$\displaystyle-\pi(N_{v}-1)\mathrm{Im}\left\\{\left(\frac{\partial}{\partial
u}\bm{\phi}_{M}(u)\right)^{H}\mathbf{F}^{\mathrm{P}}(\mathbf{F}^{\mathrm{P}})^{H}\bm{\phi}_{M}(u)\right\\}$
$\displaystyle=\pi^{2}(N_{v}-1)\mathrm{Re}\biggl{\\{}\left(\frac{(M-1)}{2}\bm{\phi}^{H}_{M}(u)+\left(\bm{\phi}^{\perp}_{M}(u)\right)^{H}\right)\biggr{.}$
$\displaystyle\quad\times\biggl{.}\mathbf{F}^{\mathrm{P}}(\mathbf{F}^{\mathrm{P}})^{H}\bm{\phi}_{M}(u)\biggr{\\}}$
$\displaystyle=\pi^{2}(N_{v}-1)\biggl{\\{}\frac{(M-1)}{2}\bm{\phi}^{H}_{M}(u)\mathbf{F}^{\mathrm{P}}(\mathbf{F}^{\mathrm{P}})^{H}\bm{\phi}_{M}(u)\biggr{.}$
$\displaystyle\biggl{.}\quad+\mathrm{Re}\left\\{\left(\left(\bm{\phi}^{\perp}_{M}(u)\right)^{H}\right)\mathbf{F}^{\mathrm{P}}(\mathbf{F}^{\mathrm{P}})^{H}\bm{\phi}_{M}(u)\right\\}\biggr{\\}}$
$\displaystyle\geq\pi^{2}(N_{v}-1)\biggl{\\{}\frac{(M-1)}{2}\bm{\phi}^{H}_{M}(u)\mathbf{F}^{\mathrm{P}}(\mathbf{F}^{\mathrm{P}})^{H}\bm{\phi}_{M}(u)\biggr{.}$
$\displaystyle\biggl{.}\quad-\left|\bm{\phi}^{H}_{M}(u)\mathbf{F}^{\mathrm{P}}(\mathbf{F}^{\mathrm{P}})^{H}\bm{\phi}^{\perp}_{M}(u)\right|\biggr{\\}}$
$\displaystyle=\pi^{2}(N_{v}-1)\bm{\phi}^{H}_{M}(u)\mathbf{F}^{\mathrm{P}}(\mathbf{F}^{\mathrm{P}})^{H}\bm{\phi}_{M}(u)\biggl{\\{}\frac{(M-1)}{2}\biggr{.}$
$\displaystyle\biggl{.}\quad-\frac{\left|\bm{\phi}^{H}_{M}(u)\mathbf{F}^{\mathrm{P}}(\mathbf{F}^{\mathrm{P}})^{H}\bm{\phi}^{\perp}_{M}(u)\right|}{\bm{\phi}^{H}_{M}(u)\mathbf{F}^{\mathrm{P}}(\mathbf{F}^{\mathrm{P}})^{H}\bm{\phi}_{M}(u)}\biggr{\\}}.$
(55)

Thus to ensure

$\displaystyle G$ $\displaystyle\geq 0$ $\displaystyle\implies$ $\displaystyle
c\biggl{\\{}\frac{(2N_{v}-1)}{6}+\frac{(M-1)}{2}\biggr{.}$ $\displaystyle\geq
0$ (56)
$\displaystyle\biggl{.}-\frac{\left|\bm{\phi}^{H}_{M}(u)\mathbf{F}^{\mathrm{P}}(\mathbf{F}^{\mathrm{P}})^{H}\bm{\phi}^{\perp}_{M}(u)\right|}{\bm{\phi}^{H}_{M}(u)\mathbf{F}^{\mathrm{P}}(\mathbf{F}^{\mathrm{P}})^{H}\bm{\phi}_{M}(u)}\biggr{\\}}$

where
$c=\pi^{2}(N_{v}-1)\bm{\phi}^{H}_{M}(u)\mathbf{F}^{\mathrm{P}}(\mathbf{F}^{\mathrm{P}})^{H}\bm{\phi}_{M}(u)$.
This implies

$\displaystyle\frac{\left|\bm{\phi}^{H}_{M}(u)\mathbf{F}^{\mathrm{P}}(\mathbf{F}^{\mathrm{P}})^{H}\bm{\phi}^{\perp}_{M}(u)\right|}{\bm{\phi}^{H}_{M}(u)\mathbf{F}^{\mathrm{P}}(\mathbf{F}^{\mathrm{P}})^{H}\bm{\phi}_{M}(u)}$
$\displaystyle\leq\biggl{\\{}\frac{(3M+2N_{v}-4)}{6}\biggr{\\}}$
$\displaystyle\frac{\left|\bm{\phi}^{H}_{M}(u)\mathbf{F}^{\mathrm{P}}(\mathbf{F}^{\mathrm{P}})^{H}\bm{\phi}^{\perp}_{M}(u)\frac{\lVert\bm{\phi}_{M}(u)\rVert}{\lVert\bm{\phi}^{\perp}_{M}(u)\rVert}\right|}{\bm{\phi}^{H}_{M}(u)\mathbf{F}^{\mathrm{P}}(\mathbf{F}^{\mathrm{P}})^{H}\bm{\phi}_{M}(u)}$
$\displaystyle\leq C(N,N_{v}),$ (57)

where
$C(N,N_{v})=\frac{\lVert\bm{\phi}_{M}(u)\rVert}{\lVert\bm{\phi}^{\perp}_{M}(u)\rVert}\biggl{\\{}\frac{(3M+2N_{v}-4)}{6}\biggr{\\}},\lVert\bm{\phi}_{M}(u)\rVert=\sqrt{M},\lVert\bm{\phi}^{\perp}_{M}(u)\rVert=\frac{\sqrt{M(M^{2}-1)}}{2\sqrt{3}}$.
Note that $C(N,N_{v})\geq\sqrt{3}$ when $N_{v}>1$. Let
$\bm{\phi}^{\perp}_{M,\mathrm{n}}(u)=\bm{\phi}^{\perp}_{M}(u)\frac{\lVert\bm{\phi}_{M}(u)\rVert}{\lVert\bm{\phi}^{\perp}_{M}(u)\rVert}$.
The LHS can be further simplified to get

$\displaystyle\frac{\left|\bm{\phi}^{H}_{M}(u)\mathbf{F}^{\mathrm{P}}(\mathbf{F}^{\mathrm{P}})^{H}\bm{\phi}^{\perp}_{M}(u)\frac{\lVert\bm{\phi}_{M}(u)\rVert}{\lVert\bm{\phi}^{\perp}_{M}(u)\rVert}\right|}{\bm{\phi}^{H}_{M}(u)\mathbf{F}^{\mathrm{P}}(\mathbf{F}^{\mathrm{P}})^{H}\bm{\phi}_{M}(u)}$
$\displaystyle=\frac{\sqrt{\bm{\phi}^{H}_{M}(u)\mathbf{F}^{\mathrm{P}}(\mathbf{F}^{\mathrm{P}})^{H}\bm{\phi}^{\perp}_{M,\mathrm{n}}(u)(\bm{\phi}^{\perp}_{M,\mathrm{n}}(u))^{H}\mathbf{F}^{\mathrm{P}}(\mathbf{F}^{\mathrm{P}})^{H}\bm{\phi}_{M}(u)}}{\bm{\phi}^{H}_{M}(u)\mathbf{F}^{\mathrm{P}}(\mathbf{F}^{\mathrm{P}})^{H}\bm{\phi}_{M}(u)}$
$\displaystyle\overset{(a)}{=}\sqrt{\frac{\lVert\bm{\phi}_{M}(u)\rVert^{2}\bm{\phi}^{H}_{M}(u)\mathbf{F}^{\mathrm{P}}(\mathbf{F}^{\mathrm{P}})^{H}\mathbf{P}_{u,\perp}\mathbf{F}^{\mathrm{P}}(\mathbf{F}^{\mathrm{P}})^{H}\bm{\phi}_{M}(u)}{(\bm{\phi}^{H}_{M}(u)\mathbf{F}^{\mathrm{P}}(\mathbf{F}^{\mathrm{P}})^{H}\bm{\phi}_{M}(u))^{2}}-1}$
$\displaystyle\overset{(b)}{\leq}\sqrt{\frac{\lVert\bm{\phi}_{M}(u)\rVert^{2}\lambda_{\mathrm{max}}\left((\mathbf{F}^{\mathrm{P}})^{H}\mathbf{P}_{u,\perp}\mathbf{F}^{\mathrm{P}}\right)}{\bm{\phi}^{H}_{M}(u)\mathbf{F}^{\mathrm{P}}(\mathbf{F}^{\mathrm{P}})^{H}\bm{\phi}_{M}(u)}-1}.$
(58)

where in $(a)$ we express
$\frac{\bm{\phi}^{\perp}_{M,\mathrm{n}}(u)(\bm{\phi}^{\perp}_{M,\mathrm{n}}(u))^{H}}{\lVert\bm{\phi}^{\perp}_{M,\mathrm{n}}(u)\rVert^{2}}=\mathbf{P}_{u,\perp}-\frac{\bm{\phi}_{M}(u)\bm{\phi}_{M}(u)^{H}}{\lVert\bm{\phi}_{M}(u)\rVert^{2}}$.
$\mathbf{P}_{u,\perp}=[\bm{\phi}_{M}(u)\>\>\bm{\phi}^{\perp}_{M}(u)]$
$\times\left[\begin{array}[]{cc}\lVert\bm{\phi}_{M}(u)\rVert^{2}&0\\\
0&\lVert\bm{\phi}^{\perp}_{M}(u)\rVert^{2}\end{array}\right]^{-1}[\bm{\phi}_{M}(u)\>\>\bm{\phi}^{\perp}_{M}(u)]$.
In $(b)$ we upper bound the quadratic term using the largest eigenvalue,
$\lambda_{\mathrm{max}}\left((\mathbf{F}^{\mathrm{P}})^{H}\mathbf{P}_{u,\perp}\mathbf{F}^{\mathrm{P}}\right)$,
of the Hermitian-symmetric matrix
$(\mathbf{F}^{\mathrm{P}})^{H}\mathbf{P}_{u,\perp}\mathbf{F}^{\mathrm{P}}$.
Thus a more stricter condition that satisfies the inequality in (57) is given
by

$\displaystyle\sqrt{\frac{\lVert\bm{\phi}_{M}(u)\rVert^{2}\lambda_{\mathrm{max}}\left((\mathbf{F}^{\mathrm{P}})^{H}\mathbf{P}_{u,\perp}\mathbf{F}^{\mathrm{P}}\right)}{\bm{\phi}^{H}_{M}(u)\mathbf{F}^{\mathrm{P}}(\mathbf{F}^{\mathrm{P}})^{H}\bm{\phi}_{M}(u)}-1}$
$\displaystyle\leq C(N,N_{v})$ (59)

which implies

$\displaystyle\frac{\bm{\phi}^{H}_{M}(u)\mathbf{F}^{\mathrm{P}}(\mathbf{F}^{\mathrm{P}})^{H}\bm{\phi}_{M}(u)}{\lVert\bm{\phi}_{M}(u)\rVert^{2}}\geq\frac{\lambda_{\mathrm{max}}\left((\mathbf{F}^{\mathrm{P}})^{H}\mathbf{P}_{u,\perp}\mathbf{F}^{\mathrm{P}}\right)}{C^{2}(N,N_{v})+1}.$
(60)

Since $C^{2}(N,N_{v})\geq 3$, an even stricter but simplified condition than
(60) is

$\frac{\bm{\phi}^{H}_{M}(u)\mathbf{F}^{\mathrm{P}}(\mathbf{F}^{\mathrm{P}})^{H}\bm{\phi}_{M}(u)}{\lVert\bm{\phi}_{M}(u)\rVert^{2}}\geq\frac{\lambda_{\mathrm{max}}\left((\mathbf{F}^{\mathrm{P}})^{H}\mathbf{P}_{u,\perp}\mathbf{F}^{\mathrm{P}}\right)}{4}.$
(61)

This concludes the proof. ∎

## References

  * [1] Z. Pi and F. Khan,  “An introduction to millimeter-wave mobile broadband systems,”  IEEE Communications Magazine, vol. 49, no. 6, pp. 101–107, June 2011.
  * [2] T. S. Rappaport, S. Sun, R. Mayzus, H. Zhao, Y. Azar, K. Wang, G. N. Wong, J. K. Schulz, M. Samimi, and F. Gutierrez,  “Millimeter wave mobile communications for 5G cellular: It will work!,”  IEEE Access, vol. 1, pp. 335–349, 2013.
  * [3] E. Bastug, M. Bennis, M. Medard, and M. Debbah,  “Toward interconnected virtual reality: Opportunities, challenges, and enablers,”  IEEE Communications Magazine, vol. 55, no. 6, pp. 110–117, 2017\.
  * [4] A. Ghosh, A. Maeder, M. Baker, and D. Chandramouli,  “5g evolution: A view on 5g cellular technology beyond 3gpp release 15,”  IEEE Access, vol. 7, pp. 127639–127651, 2019.
  * [5] C.H. Doan, S. Emami, D.A. Sobel, A.M. Niknejad, and R.W. Brodersen,  “Design considerations for 60 ghz cmos radios,”  IEEE Communications Magazine, vol. 42, no. 12, pp. 132–140, 2004\.
  * [6] S. Chiu, N. Ronquillo, and T. Javidi,  “Active learning and csi acquisition for mmwave initial alignment,”  IEEE Journal on Selected Areas in Communications, vol. 37, no. 11, pp. 2474–2489, 2019.
  * [7] A. Alkhateeb, O. El Ayach, G. Leus, and R. W. Heath,  “Channel estimation and hybrid precoding for millimeter wave cellular systems,”  IEEE Journal of Selected Topics in Signal Processing, vol. 8, no. 5, pp. 831–846, Oct 2014.
  * [8] N. Ronquillo, S. Chiu, and T. Javidi,  “Sequential learning of csi for mmwave initial alignment,”  in 2019 53rd Asilomar Conference on Signals, Systems, and Computers, 2019, pp. 1278–1283.
  * [9] N. Akdim, C. N. Manchón, M. Benjillali, and P. Duhamel,  “Variational hierarchical posterior matching for mmwave wireless channels online learning,”  in 2020 IEEE 21st International Workshop on Signal Processing Advances in Wireless Communications (SPAWC), 2020, pp. 1–5.
  * [10] R. R. Pote and B. D. Rao,  “Reduced dimension beamspace design incorporating nested array for mmwave channel estimation,”  in Asilomar Conference on Signals, Systems, and Computers, 2019, pp. 1212–1216.
  * [11] Y. Lin and T. Yang,  “Random sbt precoding for angle estimation of mmwave massive mimo systems using sparse arrays spacing,”  IEEE Access, vol. 8, pp. 163380–163393, 2020.
  * [12] P. Chen and P. P. Vaidyanathan,  “Convolutional beamspace for linear arrays,”  IEEE Trans. on Signal Proc., vol. 68, pp. 5395–5410, 2020.
  * [13] S.D. Silverstein, W.E. Engeler, and J.A. Tardif,  “Parallel architectures for multirate superresolution spectrum analyzers,”  IEEE Transactions on Circuits and Systems, vol. 38, no. 4, pp. 449–453, 1991.
  * [14] A. Tkacenko and P.P. Vaidyanathan,  “The role of filter banks in sinusoidal frequency estimation,”  Journal of the Franklin Institute, vol. 338, no. 5, pp. 517–547, 2001.
  * [15] D. Ramasamy, S. Venkateswaran, and U. Madhow,  “Compressive adaptation of large steerable arrays,”  in 2012 Information Theory and Applications Workshop, 2012, pp. 234–239.
  * [16] D. E. Berraki, S. M. D. Armour, and A. R. Nix,  “Application of compressive sensing in sparse spatial channel recovery for beamforming in mmwave outdoor systems,”  in 2014 IEEE Wireless Communications and Networking Conference (WCNC), 2014, pp. 887–892.
  * [17] A. Alkhateeb, G. Leus, and R. W. Heath,  “Compressed sensing based multi-user millimeter wave systems: How many measurements are needed?,”  in 2015 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP), 2015, pp. 2909–2913.
  * [18] S. Hur, T. Kim, D. J. Love, J. V. Krogmeier, T. A. Thomas, and A. Ghosh,  “Millimeter wave beamforming for wireless backhaul and access in small cell networks,”  IEEE Transactions on Communications, vol. 61, no. 10, pp. 4391–4403, 2013.
  * [19] Z. Xiao, T. He, P. Xia, and X. Xia,  “Hierarchical codebook design for beamforming training in millimeter-wave communication,”  IEEE Trans. on Wireless Communications, vol. 15, no. 5, pp. 3380–3392, 2016.
  * [20] J. Zhang, Y. Huang, Q. Shi, J. Wang, and L. Yang,  “Codebook design for beam alignment in millimeter wave communication systems,”  IEEE Transactions on Communications, vol. 65, no. 11, pp. 4980–4995, 2017.
  * [21] C. Liu, M. Li, S. V. Hanly, I. B. Collings, and P. Whiting,  “Millimeter wave beam alignment: Large deviations analysis and design insights,”  IEEE Journal on Selected Areas in Communications, vol. 35, no. 7, pp. 1619–1631, 2017.
  * [22] C. Liu, L. Zhao, M. Li, and L. Yang,  “Adaptive beam search for initial beam alignment in millimetre-wave communications,”  IEEE Trans. on Vehicular Technology, vol. 71, no. 6, pp. 6801–6806, 2022.
  * [23] F. Sohrabi, T. Jiang, W. Cui, and W. Yu,  “Active sensing for communications by learning,”  IEEE Journal on Selected Areas in Communications, vol. 40, no. 6, pp. 1780–1794, 2022.
  * [24] Y. Wei, Z. Zhong, and V. Y. F. Tan,  “Fast beam alignment via pure exploration in multi-armed bandits,”  IEEE Transactions on Wireless Communications, vol. 22, no. 5, pp. 3264–3279, 2023.
  * [25] M. Hussain and N. Michelusi,  “Second-best beam-alignment via bayesian multi-armed bandits,”  in 2019 IEEE Global Communications Conference (GLOBECOM), 2019, pp. 1–6.
  * [26] R. R. Pote and B. D. Rao,  “Novel sensing methodology for initial alignment using mmwave phased arrays,”  in Asilomar Conference on Signals, Systems, and Computers, 2023\.
  * [27] H. L. Van Trees,  “Optimum array processing: Part IV of Detection, Estimation, and Modulation Theory,”  John Wiley & Sons, Ltd, 2002.
  * [28] Alberto Moreira, Pau Prats-Iraola, Marwan Younis, Gerhard Krieger, Irena Hajnsek, and Konstantinos P. Papathanassiou,  “A tutorial on synthetic aperture radar,”  IEEE Geoscience and Remote Sensing Magazine, vol. 1, no. 1, pp. 6–43, 2013.
  * [29] C. Waldschmidt, J. Hasch, and W. Menzel,  “Automotive radar — from first efforts to future systems,”  IEEE Journal of Microwaves, vol. 1, no. 1, pp. 135–148, 2021.
  * [30] A. V. Oppenheim and R. W. Schafer,  Discrete-Time Signal Processing,  Prentice Hall Press, USA, 3rd edition, 2009.
  * [31] R. R. Pote and B. D. Rao,  “Maximum likelihood-based gridless doa estimation using structured covariance matrix recovery and sbl with grid refinement,”  IEEE Trans. on Signal Proc., vol. 71, pp. 802–815, 2023.
  * [32] A. Moffet,  “Minimum-redundancy linear arrays,”  IEEE Transactions on Antennas and Propagation, vol. 16, no. 2, pp. 172–175, 1968.
  * [33] P. Pal and P. P. Vaidyanathan,  “Nested arrays: A novel approach to array processing with enhanced degrees of freedom,”  IEEE Transactions on Signal Processing, vol. 58, no. 8, pp. 4167–4181, 2010.
  * [34] P. P. Vaidyanathan and P. Pal,  “Sparse sensing with co-prime samplers and arrays,”  IEEE Transactions on Signal Processing, vol. 59, no. 2, pp. 573–586, 2011.
  * [35] Vutha Va, Haris Vikalo, and Robert W. Heath,  “Beam tracking for mobile millimeter wave communication systems,”  in 2016 IEEE Global Conference on Signal and Information Processing (GlobalSIP), 2016, pp. 743–747.
  * [36] X. Song, S. Haghighatshoar, and G. Caire,  “Efficient beam alignment for millimeter wave single-carrier systems with hybrid mimo transceivers,”  IEEE Transactions on Wireless Communications, vol. 18, no. 3, pp. 1518–1533, 2019.
  * [37] Y. Yang, S. Dang, M. Wen, S. Mumtaz, and M. Guizani,  “Bayesian beamforming for mobile millimeter wave channel tracking in the presence of doa uncertainty,”  IEEE Transactions on Communications, vol. 68, no. 12, pp. 7547–7562, 2020.
  * [38] E. Khordad, I. B. Collings, S. V. Hanly, and G. Caire,  “Compressive sensing-based beam alignment schemes for time-varying millimeter-wave channels,”  IEEE Transactions on Wireless Communications, vol. 22, no. 3, pp. 1604–1617, 2023.
  * [39] R. R. Pote,  “Efficient techniques for millimeter wave sensing and beam alignment, sparse recovery, and doa estimation,”  in Ph.D. dissertation, University of California, San Diego, 2023\.
  * [40] P. Stoica and Arye Nehorai,  “Music, maximum likelihood, and cramer-rao bound,”  IEEE Transactions on Acoustics, Speech, and Signal Processing, vol. 37, no. 5, pp. 720–741, 1989.
  * [41] D. Wipf and B. D. Rao,  “Sparse Bayesian learning for basis selection,”  IEEE Trans. on Signal Proc., vol. 52, no. 8, pp. 2153–2164, Aug 2004.
  * [42] R. R. Pote and B. D. Rao,  “Light-weight sequential sbl algorithm: An alternative to omp,”  in IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP), 2023, pp. 1–5.
  * [43] M. E. Tipping and A. C. Faul,  “Fast marginal likelihood maximisation for sparse Bayesian models,”  in Proceedings of the Ninth International Workshop on AI and Statistics. 2003, vol. R4, pp. 276–283, PMLR.
  * [44] R. Giri and B. Rao,  “Type i and type ii bayesian methods for sparse signal recovery using scale mixtures,”  IEEE Transactions on Signal Processing, vol. 64, no. 13, pp. 3418–3428, 2016.