# Channel Estimation for 6G V2X Hybrid Systems using Multi-Vehicular Learning

Marouan Mizmizi, Dario Tagliaferri , Damiano Badini, Christian Mazzucco, and
Umberto Spagnolini Marouan Mizmizi, Dario Tagliaferri, and Umberto Spagnolini
are with Dipartimento di Elettronica, Informazione e Bioingegneria,
Politecnico di Milano.Damiano Badini and Christian Mazzucco are with Huawei
Technologies ItaliaUmberto Spagnolini is also Huawei Industry Chair.

###### Abstract

Channel estimation for hybrid Multiple Input Multiple Output (MIMO) systems at
Millimeter-Waves (mmW)/sub-THz is a fundamental, despite challenging,
prerequisite for an efficient design of hybrid MIMO precoding/combining. Most
works propose sequential search algorithms, e.g., Compressive Sensing (CS),
that are most suited to static channels and consequently cannot apply to
highly dynamic scenarios such as Vehicle-to-Everything (V2X). To address the
latter ones, we leverage recurrent vehicle passages to design a novel Multi
Vehicular (MV) hybrid MIMO channel estimation suited for Vehicle-to-
Infrastructure (V2I) and Vehicle-to-Network (V2N) systems. Our approach
derives the analog precoder/combiner through a MV beam alignment procedure.
For the digital precoder/combiner, we adapt the Low-Rank (LR) channel
estimation method to learn the position-dependent eigenmodes of the received
digital signal (after beamforming), which is used to estimate the compressed
channel in the communication phase. Extensive numerical simulations, obtained
with ray-tracing channel data and realistic vehicle trajectories, demonstrate
the benefits of our solution in terms of both achievable Spectral Efficiency
(SE) and Mean Square Error (MSE) compared to the Unconstrained Maximum
Likelihood (U-ML) estimate of the compressed digital channel, making it
suitable for both 5G and future 6G systems. Most notably, in some scenarios,
we obtain the performance of the optimal Fully Digital (FD) systems.

###### Index Terms:

Low-Rank Channel Estimation, Hybrid MIMO systems, Millimeter-Wave, sub-THz,
V2X, 5G New Radio, 6G

##  I Introduction

Recent advances in millimeter-wave (mmW) hardware [1] and the potential
availability of spectrum has encouraged the wireless industry to consider mmW,
for the Fifth Generation of cellular systems (5G) [2] and, in particular, for
Vehicle-to-Everything (V2X) applications [3, 4]. Following the same trend,
sub-THz are envisioned for 6G systems [5, 6, 7]. Due to the increased carrier
frequency, e.g., $24.25-52.6$ GHz for 5G New Radio (NR) Frequency Range 2
(FR2) and $>100$ GHz for sub-THz, mmW/sub-THz signals experience an orders-of-
magnitude increase in free-space path loss compared to the current majority of
wireless systems, resulting in highly sparse channels [8, 9]. Multiple Input
Multiple Output (MIMO) systems are a redeeming solution that can provide a
beamforming gain to overcome the path loss and establish links with a
reasonable Signal-to-Noise Ratio (SNR). Additionally, MIMO systems enable
precoding and combining of multiple data streams which could significantly
improve the achievable data rate [10, 11].

While the fundamental theory of MIMO precoding/combining is the same
regardless of the carrier frequency, the hardware in the mmW/sub-THz band is
subject to a set of non-trivial practical limitations. The processing in
traditional MIMO systems is performed digitally at baseband, which requires a
dedicated Radio Frequency (RF) chain for each antenna element. Unfortunately,
due to the high number of elements required in mmW (even more at sub-THz),
this implies a high cost and power consumption, which makes it unpractical
[12].

A promising solution to these problems lies in the concept of hybrid arrays,
which use a combination of analog beamforming in the RF domain and digital
beamforming in the baseband, with a reduced number of RF chains. Hybrid
Beamforming (HBF) was first introduced and analyzed in [13]. It is driven by
the fact that the number of RF chains is only lower-limited by the number of
transmitted data streams, while the beamforming gain and diversity order is
given by the number of antenna elements if proper precoding/combining is
applied. Analog precoding/combining is often implemented using phase shifters
[14], [15], switches [16], or lenses [17]. A HBF based on phase shifting
network imposes the constraint of constant amplitude on the elements of the RF
precoder. Moreover, there are two main HBF architectures, as shown in Fig. 1.
On one hand, a Fully-Connected (FC-HBF) architecture, where each RF chain
connects to all antenna elements of the array, while on the other hand, a Sub-
Connected (SC-HBF) architecture, where the RF chains connect to disjoint
subarrays, offering a cost-effective solution to HBF. Consequently, deriving
the hybrid precoder/combiner is a complex, non-convex problem and therefore it
is mathematically intractable [18].

### Related Works

Most works on hybrid precoding/combining design [18, 19, 20] require the
knowledge of the full MIMO channel at both Transmitter (Tx) and Receiver (Rx).
However, estimating the MIMO channel in mmW/sub-THz systems is a hard task due
to the low Signal-to-Noise Ratio (SNR) before any beamforming. The presence of
analog precoders/combiners implies that the digitally-observed channel is
limited to a portion of the full MIMO one, introducing an equivalent analog
compression which cannot be handled with conventional channel estimation
approaches [21]. From the mathematical point of view, the channel
decompression can be achieved by applying the hybrid echoing method proposed
in [22], which consists of consecutively transmitting and receiving training
sequences, while using all possible analog precoders/combiners (obtained, for
example, as subset of a Fourier basis) and decompressing the channel after the
concatenation of the received signals for each subset. However, this approach
turns out to be infeasible for practical systems due to (i) mobility of the
terminals and (ii) the low SNR resulting from mismatched Tx-Rx beams.

The authors in [23, 24] propose a grid-based method for FC-HBF architecture,
first estimating the Angles of Arrival/Departure (AoAs/AoDs) of the channel
through a closed-loop beam training, after which the path gain of each pair
AoA/AoD is derived. In [25, 26], a similar approach is proposed for SC-HBF
architecture under practical hardware impairments. In both architectures, the
performance tends to be limited by the codebook resolution, while the
complexity increases with the number of users. A different approach is based
on Compressed Sensing (CS) techniques in [27], imposing a structured sparsity
in the channel estimation problem. In [21], the CS-based open-loop approach is
used to explicitly estimate the full channel, with a dictionary of quantized
AoAs/AoDs. The results show the capability of CS to capture the full MIMO
channel features allowing for the joint optimization of both analog and
digital precoders/combiners. However, the algorithm requires an a-priori
knowledge of the number of channel paths, and its performance is affected by
the true sparsity level of the channel. Moreover, the joint optimization of
both analog and digital precoders/combiners increases the complexity and the
cost of the implementation in practical high-mobility systems, as the channel
is rapidly time-varying. Finally, as any grid-based technique, CS has a
significant drawback in the high sensitivity to array calibrations [28], which
is critical in hybrid systems [29].

Conversely, Low-Rank (LR) methods approach the MIMO channel estimation by
exploiting the invariance of Spatial-Temporal (ST) channel features (i.e,
AoA/AoD and delays) across different MIMO channel realizations, extracting a
modal filtering on the received signal. LR are algebraic-based methods that
leverage on the sparsity of the MIMO channel, as opposite to CS. Originally
proposed in [30, 31, 32, 33] for low-frequency systems, where the channel is
not sufficiently sparse to boost the LR application to practical systems, the
LR has recently been studied for mmW/sub-THz systems, for Fully Digital (FD)
systems only [28, 34]. In particular, the work in [28] demonstrates that LR
methods attain similar performances to CS with lower sensitivity to hardware
impairments. In [28], the LR channel estimation is enabled by consecutive
transmissions of training blocks, that limits the application to static or
low-mobility scenarios.

### Contribution

In mobility, AoAs and AoDs describe an algebraic span of MIMO channel that has
a LR, with a set of subspaces, and for mobile-to-fixed links both approaches
are location dependent. Differently from the position-dependency of the of
AoA/AoD in MIMO channel [34], here we first adapt this concept to Multi-
Vehicular (MV) LR, and then we specialize the estimate to hybrid massive MIMO
systems in mmW/sub-THz bands, considering both FC-HBF and SC-HBF
architectures. Leveraging the algebraic properties of the MIMO channel and the
constraints of hybrid hardware, we propose a two-stage training process for
the algebraic estimation of single-user (e.g., or assuming that multi-users
are allocated on minimally interfering angular, or frequency, or time radio
resources), spatial hybrid MIMO channel in a mobile scenario. The proposed
two-stage method applies to low mobility and high mobility scenarios, e.g.,
Vehicle-to-Infrastructure (V2I) or Vehicle-to-Network (V2N), which is the
focus of this paper since it shows more challenging and interesting
characteristics. In the first stage of pre-training (Section III), we
determine the optimal analog precoder/combiner at the Mobile Station (MS) and
the Base Station (BS) (or Road Side Unit) through a MV codebook-based beam
alignment procedure. In the second stage (Section IV), the LR training system
learns the algebraic channel subspace structure (eigenmodes) from the received
training signal, observed at the digital side, used to obtain the LR-estimated
channel. In particular, we propose two methods: an optimal approach,
exploiting the joint MS and BS spatial subspace, and another sub-optimal
approach, considering the separate MS and BS subspaces. Finally, during the
communication phase (online phase), the MS and the BS use the pre-computed
position-dependent analog precoder/combiner derived in the pre-training to
transmit training sequences for digital LR estimation. Based on the so-called
compressed channel (i.e., after analog BF [22]), the BS derives the digital
precoder and combiner. The novel aspect here is based on the position-
dependency of either the analog precoders/combiners and the LR of the
compressed MIMO channel obtained from multiple repeated vehicle passages on
the same geographical area, where the spatial features of the channel, i.e.,
AoAs/AoDs, are invariant. More specifically, the analog precoder/combiner and
the LR eigenmodes are associated to the specific location in space of the MS.
A notable advantage of the proposed approach is that, at the end of the
training procedure, the BS stores a dataset of optimal analog
precoders/combiners and digital channel eigenmodes, which does not require to
be updated unless macroscopic changes in the environment occur. In this
setting, during the communication phase the beam alignment can be avoided.
This allows to reduce meaningfully the overall training overhead favoring the
applications of the proposed channel estimation method to practical V2I/V2N
systems.

The proposed method is validated numerically, considering a realistic urban
scenario and repeating-passage vehicle trajectories. The information on the
building’s geometry and road network topology is extracted from OpenStreetMap
[35], while, for vehicular mobility, we employ Simulator of Urban MObility
(SUMO) software [36]. Finally, we use the ray-tracing software in [37] to
generate the channel coefficients. Extensive simulations to assess the
behavior of the proposed solution varying the channel (multipath vs. single-
path), the SNR per antenna, the number of vehicles used during the training
steps, and the number of RF chains (HBF setting). We observe that the proposed
LR channel estimation method allows outperforming the Unconstrained Maximum
Likelihood (U-ML) in terms of Mean Square Error (MSE) on channel estimation
and Spectral Efficiency (SE) for both FC-HBF and SC-HBF architectures. In
particular, for a target SE, both architectures and LR methods achieve an SNR
gain up to 15 dB in single-path scenarios, and up to 10 dB in multipath
scenarios. In general, the performance of the proposed channel estimation are
proportional to the sparsity degree of the MIMO digital channel (after analog
beamforming), which is high at mmW and it is even more prominent at sub-THz
[9], making it suitable for 6G systems.

### Organization

The paper is organized as follows: Section II introduces the system and the
channel model that are used throughout the paper. Section III describes the
proposed MV analog beam alignment, while Section IV details the LR approach
for hybrid MIMO systems. Section V reports the numerical results validating
our work. Finally, Section VI draws the conclusions.

### Notation

Bold upper- and lower-case letters describe matrices and column vectors.
$\left[\mathbf{A}\right]_{i,j}$ denotes the $(i,j)$ entry of matrix
$\mathbf{A}$, while $\mathbf{A}^{(i)}$ is the $i$-th column. Matrix
transposition and conjugate transposition is indicated as
$(\cdot)^{\mathrm{T}}$ and $(\cdot)^{\mathrm{H}}$, respectively.
$\left\lVert\cdot\right\rVert$ denotes the Frobenius norm.
$\mathrm{tr}\left(\mathbf{A}\right)$ and
$\mathrm{rank}\left(\mathbf{A}\right)$ extracts the trace and the rank of
matrix $\mathbf{A}$, respectively, while $\mathrm{eig}_{r}(\mathbf{A})$ is the
collection of $r$ eigenvectors of $\mathbf{A}$. $\otimes$, $\diamond$ and
$\odot$ denote, respectively, the Kronecker, the Kathri-Rao and the element-
wise product between two matrices. $\mathrm{vec}(\cdot)$ denotes the
vectorization by columns and $\mathrm{vec}^{-1}(\cdot)$ its inverse operation.
$\mathrm{span}(\mathbf{A})$ denotes the subspace spanned by the columns of
$\mathbf{A}$. $\mathbf{A}^{\dagger}$ is the Moore-Penrose pseudo-inverse of
$\mathbf{A}$. $\mathrm{diag}(\cdot)$ denotes either a diagonal matrix or the
extraction of the diagonal of a matrix. The following properties of the
vectorization are used in the text:
$\mathrm{vec}(\mathbf{A}\mathbf{B}\mathbf{C})=(\mathbf{C}^{\mathrm{T}}\otimes\mathbf{A})\mathrm{vec}(\mathbf{B})$,
$\mathrm{vec}(\mathbf{A}\mathbf{B})=(\mathbf{B}^{\mathrm{T}}\otimes\mathbf{I})\mathrm{vec}(\mathbf{A})$.
With $\mathbf{a}\sim\mathcal{CN}(\boldsymbol{\mu},\mathbf{C})$ we denote a
multi-variate complex Gaussian random variable $\mathbf{a}$ with mean
$\boldsymbol{\mu}$ and covariance $\mathbf{C}$. $\mathbb{E}[\cdot]$ is the
expectation operator, while $\mathbb{R}$ and $\mathbb{C}$ stand for the set of
real and complex numbers, respectively. $\delta_{n}$ is the Kronecker delta.

##  II System and Channel Model

(a)

(b)

(c)

(d)

Figure 1: Block scheme of the FC-HBF (1a,1b) and SC-HBF (1c,1d) hybrid MIMO
system

We consider the single-user hybrid mmWave MIMO system depicted in Fig. 1. The
Tx is equipped with $N_{T}$ antenna elements and $N_{T}^{RF}$ RF chains, that
is communicating $N_{S}$ data streams. The Rx has $N_{R}$ antenna elements and
$N^{RF}_{R}$ RF chains. The hybrid hardware configuration consists in
$N_{T}^{RF}<N_{T}$ and $N_{R}^{RF}<N_{R}$, while the number of parallel data
streams $N_{S}$ is upper-bounded as
$N_{S}\leq\mathrm{min}(N_{R}^{RF},N_{T}^{RF})$. For the sub-connected
configuration, the Tx and Rx antennas are grouped into sub-arrays of
$N_{T}^{B}$ and $N_{R}^{B}$ antennas, respectively, each one connected to a
single RF chain, i.e., $N_{T}^{B}=N_{T}/N_{T}^{RF}$ and
$N_{R}^{B}=N_{R}/N_{R}^{RF}$. The $N_{S}$ complex symbols to be transmitted
are $\mathbf{s}\in\mathbb{C}^{N_{S}\times
1}\sim\mathcal{CN}\left(\mathbf{0},\mathbf{I}_{N_{S}}/N_{S}\right)$, and are
precoded using the cascade of $\mathbf{F}_{BB}\in\mathbb{C}^{N^{RF}_{T}\times
N_{S}}$, obtaining the digital signal vector
$\widetilde{\mathbf{s}}=\mathbf{F}_{BB}\,\mathbf{s}\in\mathbb{C}^{N_{T}^{RF}\times
1}$, and of $\mathbf{F}_{RF}\in\mathbb{C}^{N_{T}\times N^{RF}_{T}}$ in the
analog domain. The discrete-time transmitted signal is therefore given by:

$\mathbf{x}=\mathbf{F}_{RF}\;\widetilde{\mathbf{s}},$ (1)

where $\mathbf{x}\in\mathbb{C}^{N_{T}\times 1}$. For channel estimation, an
orthogonal training sequence $\mathbf{v}\in\mathbb{C}^{N_{T}^{RF}\times 1}$,
detailed in Section IV, is transmitted without the digital precoder
$\mathbf{F}_{BB}$, i.e., $\widetilde{\mathbf{s}}=\mathbf{v}$.

Since $\mathbf{F}_{RF}$ is implemented using analog phased shifters, its
elements are constrained to have the same norm, i.e.,
$[\mathbf{F}^{(i)}_{RF}\mathbf{F}^{(i),\mathrm{H}}_{RF}]_{k,k}=1/N_{T}$, while
the Tx total power constraint is enforced by designing $\mathbf{F}_{BB}$ such
that $\left\lVert\mathbf{F}_{RF}\mathbf{F}_{BB}\right\rVert^{2}=N_{S}$. In the
SC-HBF configuration, the $N_{T}^{B}N_{T}^{RF}\times N_{T}^{RF}$ analog
precoding matrix $\mathbf{F}_{RF}$ is block-diagonal:

$\mathbf{F}_{RF}=\begin{bmatrix}\mathbf{f}^{(1)}_{RF}&\mathbf{0}&\cdots&\mathbf{0}\\\
\mathbf{0}&\mathbf{f}^{(2)}_{RF}&\cdots&\mathbf{0}\\\
\vdots&\vdots&\vdots&\vdots\\\
\mathbf{0}&\cdots&\mathbf{0}&\mathbf{f}^{(N_{T}^{RF})}_{RF}\end{bmatrix},$ (2)

where $\mathbf{f}^{(n)}_{RF}\in\mathbb{C}^{N_{T}^{B}\times 1}$,
$n=1,\dots,N_{T}^{RF}$ is the beamforming vector for the $n$-th Tx sub-array.

The transmitted signal is assumed to propagate in a spatially-sparse channel
$\mathbf{H}\in\mathbb{C}^{N_{R}\times N_{T}}$ affected, for simplicity, by
block-fading [18]. After the time-frequency synchronization, the received
signal is:

$\mathbf{y}=\mathbf{H}\mathbf{x}+\mathbf{n}$ (3)

where noise $\mathbf{n}\sim\mathcal{CN}\left(\mathbf{0},\mathbf{Q}_{n}\right)$
is generally not white due to the presence of directional interference.
Similarly to the Tx, the Rx applies the cascade of analog and digital
combiners, here indicated with $\mathbf{W}_{RF}\in\mathbb{C}^{N_{R}\times
N^{RF}_{R}}$ and $\mathbf{W}_{BB}\in\mathbb{C}^{N_{R}^{RF}\times N_{S}}$,
respectively. The (compressed) digital signal
$\widetilde{\mathbf{y}}\in\mathbb{C}^{N_{R}^{RF}\times 1}$ after the analog
combiner $\mathbf{W}_{RF}$ is:

$\begin{split}\widetilde{\mathbf{y}}=\underbrace{\mathbf{W}^{\mathrm{H}}_{RF}\mathbf{H}\mathbf{F}_{RF}}_{\widetilde{\mathbf{H}}}\,\widetilde{\mathbf{s}}+\widetilde{\mathbf{n}}\end{split}$
(4)

where:

  * •

$\mathbf{W}_{RF}$ compressing the analog signal is subject to the same
constraint of $\mathbf{F}_{RF}$, i.e.,
$[\mathbf{W}^{(j)}_{RF}\mathbf{W}^{(j),\mathrm{H}}_{RF}]_{l,l}=1/N_{R}$;

  * •

$\widetilde{\mathbf{H}}\in\mathbb{C}^{N_{R}^{RF}\times N_{T}^{RF}}$ is the
equivalent and compressed MIMO channel observed at the digital side;

  * •

the noise after the analog beamforming is
$\widetilde{\mathbf{n}}=\mathbf{W}^{\mathrm{H}}_{RF}\,\mathbf{n}\sim\mathcal{CN}(\mathbf{0},\widetilde{\mathbf{Q}}_{n})$,
with
$\widetilde{\mathbf{Q}}_{n}=\mathbf{W}^{\mathrm{H}}_{RF}\mathbf{Q}_{n}\mathbf{W}\\`{\i}_{RF}$.

Similarly to $\mathbf{F}_{RF}$, the analog combiner $\mathbf{W}_{RF}$ for SC-
HBF architectures is block-diagonal.

Finally, the received data flows $\mathbf{z}\in\mathbb{C}^{N_{S}\times 1}$
after the digital combiner $\mathbf{W}_{BB}$ are:

$\begin{split}\mathbf{z}=\mathbf{W}^{\mathrm{H}}_{BB}\widetilde{\mathbf{y}}=\mathbf{W}^{\mathrm{H}}_{BB}\widetilde{\mathbf{H}}\,\widetilde{\mathbf{s}}+\mathbf{W}^{\mathrm{H}}_{BB}\widetilde{\mathbf{n}}.\end{split}$
(5)

Derivation of
$\mathbf{F}_{BB},\mathbf{F}_{RF},\mathbf{W}_{BB},\mathbf{W}_{RF}$ has been
investigated in depth in [18]. Here, the analog precoders/combiners
$\mathbf{F}_{RF}$ and $\mathbf{W}_{RF}$ are derived from a MV codebook-based
beam alignment procedure. After, the digital precoders $\mathbf{F}_{BB}$ and
combiners $\mathbf{W}_{BB}$ are computed employing the LR training in the
second stage, as detailed in Section IV. The aforementioned system model
refers, for instance, to one sub-carrier of an OFDM system and temporal
processing over the sub-carriers is not detailed herein.

###  II-A Channel Model

As customary in mmW/sub-THz links, we consider the spatially-sparse clustered
MIMO channel model [38, 39]. The channel matrix $\mathbf{H}$ can be written as
the sum of $P$ paths as

$\mathbf{H}=\sum_{p=1}^{P}\alpha_{p}\,\mathbf{a}_{R}(\boldsymbol{\vartheta}_{p})\mathbf{a}^{\mathrm{T}}_{T}(\boldsymbol{\psi}_{p}),$
(6)

where: (i) $\alpha_{p}$ is the complex gain of the $p$-th path; (ii)
$\mathbf{a}_{T}(\boldsymbol{\psi}_{p})\in\mathbb{C}^{N_{T}\times 1}$ and
$\mathbf{a}_{R}(\boldsymbol{\vartheta}_{p})\in\mathbb{C}^{N_{R}\times 1}$
represent, respectively, the Tx and Rx and array response vectors to $p$-th
path, function of the AoDs
$\boldsymbol{\psi}_{p}=[\psi_{\mathrm{az},p},\psi_{\mathrm{el},p}]^{\mathrm{T}}$
and the AoAs
$\boldsymbol{\vartheta}_{p}=[\vartheta_{\mathrm{az},p},\vartheta_{\mathrm{el},p}]^{\mathrm{T}}$.

Without loss of generality, we assume the faded channel to be normalized such
that $\mathbb{E}[\left\lVert\mathbf{H}\right\rVert^{2}]=N_{T}N_{R}$. The
channel matrix (6) can be rewritten in compact form as:

$\mathbf{H}=\mathbf{A}_{R}\left(\boldsymbol{\vartheta}\right)\mathbf{D}\,\mathbf{A}_{T}^{\mathrm{T}}\left(\boldsymbol{\psi}\right),$
(7)

where
$\mathbf{A}_{T}\left(\boldsymbol{\psi}\right)=\left[\mathbf{a}_{T}(\boldsymbol{\psi}_{1}),\dots,\mathbf{a}_{T}(\boldsymbol{\psi}_{P})\right]\in\mathbb{C}^{N_{T}\times
P}$ and
$\mathbf{A}_{R}\left(\boldsymbol{\vartheta}\right)=\left[\mathbf{a}_{R}(\boldsymbol{\vartheta}_{1}),\dots,\mathbf{a}_{R}(\boldsymbol{\vartheta}_{P})\right]\in\mathbb{C}^{N_{R}\times
P}$ are two matrices identifying the Tx and Rx beam spaces, and diagonal
matrix $\mathbf{D}\in\mathbb{C}^{P\times
P}=\mathrm{diag}\left(\alpha_{1},\dots,\alpha_{P}\right)$ collects all the
channel amplitudes, obeying the Wide-Sense Stationary Uncorrelated Scattering
(WSSUS) model [40]:

$\mathbb{E}\left[\mathbf{D}_{n}\mathbf{D}^{\mathrm{H}}_{n+m}\right]=\mathbf{P}\delta_{n-m},$
(8)

with $\mathbf{P}=\mathrm{diag}\left(P_{1},\dots,P_{P}\right)$ containing the
paths’ powers, normalized such that $\sum_{p}P_{p}=1$, and $n$, $m$ denoting
two different channel realizations in either time (different fading blocks) or
space (different locations).

Matrices $\mathbf{A}_{T}\left(\boldsymbol{\psi}\right)$ and
$\mathbf{A}_{R}\left(\boldsymbol{\vartheta}\right)$ allow to define the
diversity orders of channel $\mathbf{H}$ for Tx ($r_{T}$) and Rx ($r_{R}$)

$\displaystyle r_{T}$
$\displaystyle=\mathrm{rank}(\mathbf{A}_{T}\left(\boldsymbol{\psi}\right))\leq\mathrm{min}\left(N_{T},P\right),$
(9) $\displaystyle r_{R}$
$\displaystyle=\mathrm{rank}(\mathbf{A}_{R}\left(\boldsymbol{\vartheta}\right))\leq\mathrm{min}\left(N_{R},P\right),$
(10)

i.e., the number of resolvable spatial paths according to the number of Tx and
Rx antennas.

The analog precoder/combiner pair $\mathbf{F}_{RF}$ and $\mathbf{W}_{RF}$
modifies the beam spaces and the diversity orders of the digitally-equivalent
channel $\widetilde{\mathbf{H}}$, whose structure is:

$\widetilde{\mathbf{H}}=\mathbf{W}^{\mathrm{H}}_{RF}\mathbf{A}_{R}(\boldsymbol{\vartheta})\,\mathbf{D}\,\mathbf{A}_{T}^{\mathrm{T}}(\boldsymbol{\psi})\mathbf{F}_{RF}.$
(11)

The diversity orders, namely number of resolvable paths given the Tx and RX
HBF configurations, now become:

$\displaystyle\widetilde{r}_{T}=\mathrm{rank}(\mathbf{A}_{T}^{\mathrm{T}}(\boldsymbol{\psi})\mathbf{F}_{RF})\leq\mathrm{min}\left(\mathrm{rank}(\mathbf{F}_{RF}),r_{T}\right),$
(12)
$\displaystyle\widetilde{r}_{R}=\mathrm{rank}(\mathbf{W}^{\mathrm{H}}_{RF}\mathbf{A}_{R}(\boldsymbol{\vartheta}))\leq\mathrm{min}\left(\mathrm{rank}(\mathbf{W}_{RF}),r_{R}\right),$
(13)

where, in general,

$\displaystyle\mathrm{rank}(\mathbf{F}_{RF})\leq
N_{T}^{RF},\,\,\,\mathrm{rank}(\mathbf{W}_{RF})\leq N_{R}^{RF}.$ (14)

As will be shown in the following, as the full MIMO channel cannot be directly
estimated, [18, 23], we exploit the algebraic structure of the digitally-
observed channel $\widetilde{\mathbf{H}}$ to improve the channel estimation.
As opposite to existing works [23, 22, 24, 21], we propose a learning-based
approach, tailored for both static and dynamic scenarios (e.g., V2I/V2N
scenarios). A MV codebook-based beam alignment procedure selects the analog
precoder/combiner pair $\mathbf{F}_{RF}$, $\mathbf{W}_{RF}$ (Section III);
then, a second MV-LR method learns the algebraic spatial eigenmodes of the
digital compressed channel $\widetilde{\mathbf{H}}$, which are used to derive
digital precoders/combiners $\mathbf{F}_{BB}$ and $\mathbf{W}_{BB}$ from the
LR-estimated equivalent compressed channel (Section IV).

##  III Multi-vehicular Codebook-based Analog Beam Alignment

The hardware constraint and the low SNR in the mmW/sub-THz bands makes the
derivation of analog precoder/combiner in hybrid systems is a complex non-
convex problem [41]. A conventional solution is to use a fixed codebook and a
beam alignment strategy to appropriately scan the full channel (both AoA and
AoD) and to select the best beam pairs that satisfy some criterion, such as to
maximize the received power, the SNR, or the achievable rate. The trade-off
between complexity and resolution must be taken into account when designing
the codebook [42]. We elaborate further below from this beam-alignment
approach.

(a)

(b)

(c)

(d)

Figure 2: Graphical representation of the MV codebook-based beam alignment:
the BS assigns different analog beams to different MSs (2a,2b,2c), collecting
a set of position-related power measurements used to progressively filling
matrix $\mathbf{P}_{R}$. At the end of the procedure, the MV beam alignment
selects the best analog beam pairs (i.e., analog precoder/combiner pair) (2d)

In a quasi-static propagation environment, different vehicles crossing the
same location in space with slightly different co-directed trajectories (as
commonly happens in urban scenarios) experience the same angles (AoD/AoA) in
communicating with the BS and different fading amplitudes made varying by the
Doppler [43]. Therefore, by leveraging this property, we explore a set of MIMO
channel snapshots of recurrent vehicle passages to design a novel multi-
vehicular codebook-based analog beam alignment procedure for dynamic
scenarios, characterized by the mobility of one of the terminals, e.g., in
V2I/V2N communications. We assume the Rx, e.g., a BS, with a fixed position
and a set of collaborative vehicles, both equipped with hybrid antenna arrays.
In particular, all the MSs have the same array equipment, and their positions
$\mathbf{p}_{\ell}$ and headings $\theta_{\ell}$ are known, with a reasonable
accuracy, for each training block ($\ell$-th MS). The way the position and
orientation are obtained is out of the scope but one can use (i) a Radio
Access Technology (RAT)-based localization and tracking algorithms [44], or
(ii) some signaling from the vehicle’s onboard sensors (e.g., Global
Navigation Satellite System (GNSS)).

The Tx analog codebook is designed from a 2D Fourier basis that for a
$N_{1}\times N_{2}$ Uniform Rectangular Array (URA) with half-wavelength
spaced antennas configuration becomes:

$\mathbf{B}_{\mathrm{2D}}\left(N_{1},N_{2}\right)=\mathbf{B}\left(N_{1}\right)\otimes\mathbf{B}\left(N_{2}\right),$
(15)

where $\mathbf{B}\left(N\right)\in\mathbb{C}^{N\times N}$ is the DFT matrix
with entries

$\left[\mathbf{B}\left(N\right)\right]_{m,n}=\frac{1}{\sqrt{N}}e^{-j\frac{2\pi
mn}{N}},$ (16)

and dimensions are

$\begin{split}&N_{1}=N_{T}^{\mathrm{az}},\,\,\,\,N_{2}=N_{T}^{\mathrm{el}}\quad\text{for
FC-HBF},\\\
&N_{1}=N_{T}^{B,\mathrm{az}},\,\,N_{2}=N_{T}^{B,\mathrm{el}}\quad\text{for SC-
HBF}\end{split}$ (17)

in which $N_{T}^{\mathrm{az}}$, $N_{T}^{\mathrm{el}}$ denote the number of Tx
antennas along the azimuth (horizontal) and elevation (vertical) direction of
the URA ($N_{T}=N_{T}^{\mathrm{az}}\times N_{T}^{\mathrm{el}}$), and
$N_{T}^{B,\mathrm{az}}$, $N_{T}^{B,\mathrm{el}}$ the same for each sub-array
($N_{T}^{B}=N_{T}^{B,\mathrm{az}}\times N_{T}^{B,\mathrm{el}}$). The Rx
codebook is analogously obtained.

The learning stage of the MV codebook-based beam alignment procedure is
depicted in Fig. 2 and it consists on the following steps, that involve the
usage of a low-frequency signalling link (e.g., 5G NR FR1):

  1. 1.

the BS commands each collaborative MS entering the BS’s coverage area to use a
certain analog beam $\mathbf{f}_{RF}$ (e.g., codebook index from
$\mathbf{B}_{\mathrm{2D}}\left(N_{1},N_{2}\right)$) and the relative training
sequence. Additional information, such as position $\mathbf{p}_{\ell}$ and
heading $\mathbf{\theta}_{\ell}$, could be requested to the MS by the BS;

  2. 2.

the collaborative $\ell$-th MS, while moving in the BS’s coverage area,
transmits training sequences according to the BS using the fixed analog beam
(see Figs. 2a, 2b, 2c). The position and heading, if requested, are related to
the instant in which the training sequence is transmitted;

  3. 3.

the meantime the MS moves, the BS continuously scans all the analog beams
$\mathbf{w}_{RF}$ of the Rx codebook. In the event of a match between BS and
MSℓ analog beams, the BS stores: (i) the received power $P_{R}$, (ii) the MSℓ
analog beam $\mathbf{f}_{RF}$, (iii) the BS analog beam $\mathbf{w}_{RF}$, and
(iv) position $\mathbf{p}_{\ell}$ and heading $\mathbf{\theta}_{\ell}$;

  4. 4.

the BS, after the training period for multiple MSs, each with different
precoder $\mathbf{f}_{RF}$, groups received powers in clusters based on
positions and headings of the moving MSs and generates the related received
power matrix $\mathbf{P}_{R}\left(\mathbf{\bar{p}},\bar{\theta}\right)$, where
$\mathbf{\bar{p}}$ and $\bar{\theta}$ are the reference position and heading
respectively. Fig. 3 shows an example of
$\mathbf{P}_{R}\left(\mathbf{\bar{p}},\bar{\theta}\right)$ for FC-HBF and SC-
HBF architectures.

The optimal analog precoders/combiners $\mathbf{F}_{RF}$ and $\mathbf{W}_{RF}$
for each position are those maximizing the received power in matrix
$\mathbf{P}_{R}\left(\mathbf{\bar{p}},\bar{\theta}\right)$, learned from
multiple passages illustrated in Figs. 2a, 2b, 2c. The problem consists in
selecting the maximums of
$\mathbf{P}_{R}\left(\mathbf{\bar{p}},\bar{\theta}\right)$ corresponding to
the true channel paths. In hybrid systems, however, as the full channel matrix
$\mathbf{H}$ is unknown and cannot be directly estimated, it is not possible
to approach the maximization analytically. Furthermore, heuristic approaches
are disadvantageous, since
$\mathbf{P}_{R}\left(\mathbf{\bar{p}},\bar{\theta}\right)$ has several local
maximums, due to LoS/NLoS spatial components of the channel (when they match
with the selected beams) and their related grating lobes as can be observed in
Fig.3. Here, we select the set of beam pairs for $\mathbf{F}_{RF}$ and
$\mathbf{W}_{RF}$ by searching for the first $N_{T}^{RF}$ and $N_{R}^{RF}$
maximums over the rows and the columns of
$\mathbf{P}_{R}\left(\mathbf{\bar{p}},\bar{\theta}\right)$ independently. This
ensures that the analog beams at MS and BS are not repeated, i.e., we avoid
rank-deficient precoders/combiners $\mathbf{F}_{RF}$/$\mathbf{W}_{RF}$
matrices.

Finally, the BS defines a list $\mathbf{L}_{F}$ of optimal analog precoders
with the associated reference positions and headings, such that:

$\left[\mathbf{L}_{F}\right]_{k}=\left\\{\mathbf{F}_{RF,k},\left(\mathbf{\bar{p}}_{k},\bar{\theta}_{k}\right)\right\\}$
(18)

and similarly for the optimal combiners, with list $\mathbf{L}_{W}$.

(a) FC-HBF

(b) SC-HBF

Figure 3: Example of $\mathbf{P}_{R}$, with $N_{T}=64$, $N_{R}=128$ and
$N^{RF}_{T}=4$, $N^{RF}_{R}=8$ hybrid array configuration, for FC-HBF (3a) and
SC-HBF (3b) architecture

##  IV Low-Rank Estimation of Digital Compressed Channels

In the second stage of channel estimation, the BS has to learn the eigenmodes
of the equivalent compressed channel $\widetilde{\mathbf{H}}$. Again, we
exploit recurrent vehicle passages. The BS sends the optimal analog precoders
list $\mathbf{L}_{F}$ defined in the first stage to all collaborative MSs
entering its coverage area. The $\ell$-th collaborative MS, while moving,
transmits M consecutive training sequences
$\mathbf{v}_{\ell}\in\mathbb{C}^{N_{T}^{RF}\times 1}$, such that
$\widetilde{\mathbf{S}}_{\ell}=\left[\mathbf{v}_{1,\ell},\mathbf{v}_{2,\ell},\dots,\mathbf{v}_{M,\ell}\right]\in\mathbb{C}^{N_{T}^{RF}\times
M}$. We assume the training sequences are chosen to be uncorrelated in both
space and time and also mutually uncorrelated among different MSs, i.e.,
$\mathbb{E}[\widetilde{\mathbf{S}}_{\ell}\widetilde{\mathbf{S}}^{\mathrm{H}}_{s}]=\sigma^{2}_{s}\mathbf{I}_{N_{T}^{RF}}\delta_{s-\ell}$
and
$\mathbb{E}[\widetilde{\mathbf{S}}^{\mathrm{H}}_{\ell}\widetilde{\mathbf{S}}_{s}]=\mathbf{I}_{M}\delta_{s-\ell}$,
where $\sigma^{2}_{s}$ denotes the Tx power. The optimal analog precoder
$\mathbf{F}_{RF}$ used for transmitting the training sequences is selected
from the received list $\mathbf{L}_{F}$ based on the MS current MS position
$\mathbf{p}_{\ell}$ and heading $\theta_{\ell}$.

The BS selects similarly the optimal analog combiner $\mathbf{W}_{RF}$ from
the list $\mathbf{L}_{W}$ defined in the first stage, obtaining:

$\widetilde{\mathbf{Y}}_{\ell}=\widetilde{\mathbf{H}}_{\ell}\,\widetilde{\mathbf{S}}_{\ell}+\widetilde{\mathbf{N}}_{\ell}$
(19)

where the noise is such that
$\mathbb{E}[\widetilde{\mathbf{N}}_{\ell}\widetilde{\mathbf{N}}^{\mathrm{H}}_{s}]=\widetilde{\mathbf{Q}}_{n}\delta_{s-\ell}$
and
$\mathbb{E}[\widetilde{\mathbf{N}}^{\mathrm{H}}_{\ell}\widetilde{\mathbf{N}}_{s}]=\mathbf{I}_{M}\delta_{s-\ell}$.
At the end of the procedure, the set of received training sequences
$\\{\widetilde{\mathbf{Y}}_{\ell}\\}_{\ell=1}^{\ell=L}$ for each position and
heading is used to retrieve the compressed channel eigenmodes and the LR-
estimated channel through algebraic manipulations, detailed in the following.
In particular, two solutions are provided: (i) optimal LR estimation, i.e.,
Joint Space (JS), and (ii) sub-optimal LR estimation, i.e., Disjoint Space
(DS). In Section V, the performance of the two approaches are compared and
discussed.

###  IV-A Joint Space Low Rank (JS-LR) Estimation

The LR-estimated compressed channel
$\widehat{\mathbf{h}}_{\ell}\in\mathbb{C}^{N_{R}^{RF}N_{T}^{RF}\times 1}$ can
be retrieved from the single received training signal
$\widetilde{\mathbf{Y}}_{\ell}$ as the combination of a training sequence-
dependent matrix $\mathbf{G}_{\ell}$ and another one referred as position-
dependent matrix $\boldsymbol{\Pi}(\mathbf{\bar{p}},\bar{\theta})$, as [28]:

$\widehat{\mathbf{h}}_{\ell}=\boldsymbol{\Pi}(\mathbf{\bar{p}},\bar{\theta})\,\mathbf{G}_{\ell}\,\mathrm{vec}(\widetilde{\mathbf{Y}}_{\ell})=\boldsymbol{\Pi}(\mathbf{\bar{p}},\bar{\theta})\,\overline{\mathbf{y}}_{\ell},$
(20)

where $\overline{\mathbf{y}}_{\ell}\in\mathbb{C}^{N_{R}^{RF}N_{T}^{RF}\times
1}$ is the pre-processed sequence by $\mathbf{G}_{\ell}$. A notable example is
the LS or U-ML channel estimation, where $\mathbf{G}_{\ell}$ is a suitable
rearrangement of known pilot symbols. For instance, for the LS channel
estimation from (19),
$\mathbf{G}_{\ell}=(\widetilde{\mathbf{S}}^{\mathrm{T}}_{\ell}\otimes\mathbf{I}_{N_{R}^{RF}})^{\dagger}$.

The position-dependent linear processing
$\boldsymbol{\Pi}(\mathbf{\bar{p}},\bar{\theta})$ is estimated from an
ensemble of $L$ training sequences
$\\{\overline{\mathbf{y}}_{\ell}\\}_{\ell=1}^{\ell=L}$, originated from
multiple vehicles passing in the same location such that each one has the same
propagation structure with all the others. In the context of (20), matrix
$\boldsymbol{\Pi}(\mathbf{\bar{p}},\bar{\theta})$ operates a modal filtering
on $\overline{\mathbf{y}}_{\ell}$, projecting it onto the propagation subspace
[31].

The first step to obtain the position-dependent
$\boldsymbol{\Pi}(\mathbf{\bar{p}},\bar{\theta})$ is to identify the algebraic
structure of the compressed channel
$\widetilde{\mathbf{h}}=\mathrm{vec}(\widetilde{\mathbf{H}})$, which can be
shown to be:

$\begin{split}\widetilde{\mathbf{h}}&=\left(\mathbf{F}^{\mathrm{T}}_{RF}\otimes\mathbf{W}^{\mathrm{H}}_{RF}\right)\boldsymbol{\mathcal{A}}\left(\boldsymbol{\psi},\boldsymbol{\vartheta}\right)\boldsymbol{\alpha}=\mathbf{T}\left(\boldsymbol{\psi},\boldsymbol{\vartheta}\right)\,\boldsymbol{\alpha}\end{split}$
(21)

where (i)
$\mathbf{T}\left(\boldsymbol{\psi},\boldsymbol{\vartheta}\right)\in\mathbb{C}^{N_{T}^{RF}N_{R}^{RF}\times
P}$ embeds the spatial features of the compressed channel, invariant across
multiple MSs passing the same position; (ii) matrix
$\boldsymbol{\mathcal{A}}\left(\boldsymbol{\psi},\boldsymbol{\vartheta}\right)\in\mathbb{C}^{N_{T}N_{R}\times
P}=\mathbf{A}_{T}\left(\boldsymbol{\psi}\right)\diamond\mathbf{A}_{R}\left(\boldsymbol{\vartheta}\right)$,
and (iii) $\boldsymbol{\alpha}\in\mathbb{C}^{P\times
1}=\left[\alpha_{1},\dots,\alpha_{P}\right]^{\mathrm{T}}$ collects the channel
amplitudes, different from MS to MS but with the same power profile (8).

Let us define the compressed channel correlation
$\widetilde{\mathbf{R}}=\mathbb{E}[\widetilde{\mathbf{h}}\,\widetilde{\mathbf{h}}^{\mathrm{H}}]$,
which can be computed by exploiting the invariance of AoAs/AoDs across
multiple vehicles, as:

$\begin{split}\widetilde{\mathbf{R}}&=\mathbf{T}\left(\boldsymbol{\psi},\boldsymbol{\vartheta}\right)\mathbf{P}\,\mathbf{T}\left(\boldsymbol{\psi},\boldsymbol{\vartheta}\right)^{\mathrm{H}}=\\\
&=\sum_{p=1}^{P}P_{p}\left[\mathbf{F}^{\mathrm{T}}_{RF}\mathbf{R}_{T,p}\mathbf{F}^{*}_{RF}\otimes\mathbf{W}^{\mathrm{H}}_{RF}\mathbf{R}_{R,p}\mathbf{W}_{RF}\right]\end{split}$
(22)

where $\mathbf{R}_{T,p}\in\mathbb{C}^{N_{T}\times
N_{T}}=\mathbf{a}_{T}\left(\boldsymbol{\psi}_{p}\right)\mathbf{a}_{T}\left(\boldsymbol{\psi}_{p}\right)^{\mathrm{H}}$
and $\mathbf{R}_{R,p}\in\mathbb{C}^{N_{R}\times
N_{R}}=\mathbf{a}_{R}\left(\boldsymbol{\vartheta}_{p}\right)\mathbf{a}_{R}\left(\boldsymbol{\vartheta}_{p}\right)^{\mathrm{H}}$.

We can re-parameterize the channel $\widetilde{\mathbf{h}}$ in (21) using the
$\widetilde{r}$ leading eigenvectors of $\widetilde{\mathbf{R}}$, i.e.,
$\widetilde{\mathbf{U}}=\mathrm{eig}_{\widetilde{r}}(\widetilde{\mathbf{R}})$,
such that:

$\begin{split}&\mathrm{span}(\widetilde{\mathbf{U}})=\mathrm{span}(\mathbf{T}\left(\boldsymbol{\psi},\boldsymbol{\vartheta}\right)),\end{split}$
(23)

the orthonormal basis
$\widetilde{\mathbf{U}}\in\mathbb{C}^{N_{T}^{RF}N_{R}^{RF}\times\widetilde{r}}$
span the joint Tx and Rx subspace of the compressed channel, of dimension
$\widetilde{r}=\mathrm{rank}(\widetilde{\mathbf{R}})=\mathrm{rank}(\mathbf{T}\left(\boldsymbol{\psi},\boldsymbol{\vartheta}\right))$.
The latter represents the number of compressed channel paths (diversity order)
that can be resolved by the digital system:

$\widetilde{r}\leq\mathrm{min}\left(\mathrm{rank}(\mathbf{F}_{RF})\mathrm{rank}(\mathbf{W}_{RF}),r\right),$
(24)

where
$r=\mathrm{rank}(\boldsymbol{\mathcal{A}}\left(\boldsymbol{\psi},\boldsymbol{\vartheta}\right))\leq\mathrm{min}(N_{T}N_{R},P)$
is the number of resolvable paths of the full channel
$\mathbf{h}=\boldsymbol{\mathcal{A}}\left(\boldsymbol{\psi},\boldsymbol{\vartheta}\right)\boldsymbol{\alpha}$,
obtained by rearranging (6) similarly to (21).

From the LR contraint (24), the position-dependent matrix
$\boldsymbol{\Pi}(\mathbf{\bar{p}},\bar{\theta})$ is estimated as [28]:

$\widehat{\boldsymbol{\Pi}}(\mathbf{\bar{p}},\bar{\theta})=\mathbf{C}^{\frac{\mathrm{H}}{2}}\,\widehat{\boldsymbol{\Pi}}_{\mathrm{JS}}\,\mathbf{C}^{-\frac{\mathrm{H}}{2}},$
(25)

where:

  * •

$\mathbf{C}$ is the covariance matrix of $\overline{\mathbf{y}}_{\ell}$,
corresponding to the Cramer-Rao Bound (CRB). For this problem, it is
$\mathbf{C}\approx(\mathbf{I}_{N_{T}^{RF}}/\sigma_{s}^{2})\otimes\widetilde{\mathbf{Q}}_{n}$
(asymptotic approximation). Matrix $\mathbf{C}$ is used to perform the
whitening (and de-whitening) of $\overline{\mathbf{y}}_{\ell}$ to optimally
handle any presence of noise correlation (e.g., interference);

  * •

$\widehat{\boldsymbol{\Pi}}_{\mathrm{JS}}=\widehat{\mathbf{U}}\widehat{\mathbf{U}}^{\mathrm{H}}$
is the JS-LR projection matrix onto the propagation subspace spanned by
$\widehat{\mathbf{U}}=\mathrm{eig}_{\widetilde{r}}(\widehat{\mathbf{R}})$,
where

$\widehat{\mathbf{R}}=\frac{1}{L}\sum_{\ell=1}^{L}\overline{\overline{\mathbf{y}}}_{\ell}\overline{\overline{\mathbf{y}}}_{\ell}^{\mathrm{H}}$
(26)

is the sample correlation of whitened sequences
$\overline{\overline{\mathbf{y}}}_{\ell}=\mathbf{C}^{-\frac{\mathrm{H}}{2}}\overline{\mathbf{y}}_{\ell}$,
collected from $L$ different MSs passing the same position.

The performance of the proposed LR channel estimation, hereafter referred to
as Joint-Space LR (JS-LR), provided by the application of
$\boldsymbol{\Pi}(\mathbf{\bar{p}},\bar{\theta})$ in (25) on signal
$\overline{\mathbf{y}}_{\ell}$, depends on the sparsity degree of the
compressed channel $\widetilde{\mathbf{h}}$. The latter is proportional to the
ratio between the effective number of spatially-separable analog beams for MS
and BS, respectively $N_{T}^{\mathrm{beams}}$ and $N_{R}^{\mathrm{beams}}$,
and the number of resolvable paths $\widetilde{r}$ of the compressed channel.

For FC-HBF systems, the number of separable beams are
$N_{T}^{\mathrm{beams}}\leq\mathrm{rank}(\mathbf{F}_{RF})\leq N_{T}^{RF}$,
$N_{R}^{\mathrm{beams}}\leq\mathrm{rank}(\mathbf{W}_{RF})\leq N_{R}^{RF}$ as
the Tx/Rx terminals can, in general, use arbitrary angular separated analog
beams. For the analog beams chosen here as selected from orthogonal codebooks
and not repeated (Section III), we have
$N_{T}^{\mathrm{beams}}=\mathrm{rank}(\mathbf{F}_{RF})=N_{T}^{RF}$ and
$N_{R}^{\mathrm{beams}}=\mathrm{rank}(\mathbf{W}_{RF})=N_{R}^{RF}$, and the
sparsity degree of the compressed channel is maximum.

In SC-HBF architectures, the block-diagonal structure of $\mathbf{F}_{RF}$ and
$\mathbf{W}_{RF}$ leads, in general, to
$N^{\mathrm{beams}}_{T}\leq\mathrm{rank}(\mathbf{F}_{RF})=N_{T}^{RF}$ and
$N^{\mathrm{beams}}_{R}\leq\mathrm{rank}(\mathbf{F}_{RF})=N_{R}^{RF}$, but
again the proposed analog beam alignment ensures that
$N^{\mathrm{beams}}_{T}=\mathrm{rank}(\mathbf{F}_{RF})=N_{T}^{RF}$ and
$N^{\mathrm{beams}}_{R}=\mathrm{rank}(\mathbf{F}_{RF})=N_{R}^{RF}$, as every
RF chain employs a different orthogonal beam, maximizing the channel sparsity.

In this regard, provided that:

$\widetilde{r}<N_{T}^{RF}N_{R}^{RF},$ (27)

the LR provides superior performance compared to conventional approaches
(e.g., LS/U-ML).

###  IV-B Disjoint Space Low Rank Estimation

To reduce the complexity of the JS-LR implementation, mainly due to the large-
matrix eigendecomposition of $\widehat{\mathbf{R}}$ in (26), we propose a sub-
optimal approach, referred herein as Disjoint-Space LR (DS-LR). This assumes
the separability of Tx and Rx spatial subspaces of the compressed channel as
suggested in [31]. In analogy to (22), we leverage the algebraic structure of
$\widetilde{\mathbf{H}}$ in (11) and define the Tx and Rx compressed channel
correlations
$\widetilde{\mathbf{R}}_{T}=\mathbb{E}[\widetilde{\mathbf{H}}^{\mathrm{H}}\widetilde{\mathbf{H}}]$
and
$\widetilde{\mathbf{R}}_{R}=\mathbb{E}[\widetilde{\mathbf{H}}\widetilde{\mathbf{H}}^{\mathrm{H}}]$,
respectively equal to:

$\displaystyle\begin{split}\widetilde{\mathbf{R}}_{T}&=\mathbf{F}_{RF}^{\mathrm{H}}\mathbf{A}^{*}_{T}\left(\boldsymbol{\psi}\right)\,\widetilde{\boldsymbol{\Gamma}}_{T}\,\mathbf{A}^{\mathrm{T}}_{T}\left(\boldsymbol{\psi}\right)\mathbf{F}_{RF},\end{split}$
(28)
$\displaystyle\begin{split}\widetilde{\mathbf{R}}_{R}&=\mathbf{W}_{RF}^{\mathrm{H}}\mathbf{A}_{R}\left(\boldsymbol{\vartheta}\right)\,\widetilde{\boldsymbol{\Gamma}}_{R}\,\mathbf{A}^{\mathrm{H}}_{R}\left(\boldsymbol{\vartheta}\right)\mathbf{W}_{RF},\end{split}$
(29)

where

$\displaystyle\widetilde{\boldsymbol{\Gamma}}_{T}$
$\displaystyle=\mathbf{P}\odot\mathbf{A}^{\mathrm{H}}_{R}\left(\boldsymbol{\vartheta}\right)\mathbf{W}_{RF}\mathbf{W}_{RF}^{\mathrm{H}}\mathbf{A}_{R}\left(\boldsymbol{\vartheta}\right),$
(30) $\displaystyle\widetilde{\boldsymbol{\Gamma}}_{R}$
$\displaystyle=\mathbf{P}\odot\mathbf{A}^{\mathrm{T}}_{T}\left(\boldsymbol{\psi}\right)\mathbf{F}_{RF}\mathbf{F}_{RF}^{\mathrm{H}}\mathbf{A}^{*}_{T}\left(\boldsymbol{\psi}\right),$
(31)

are diagonal matrices of $P\times P$ size. Eq. (30)-(31) highlight how the
analog precoder/combiner pair affects the eigenvalues of the Tx and Rx channel
correlation matrices. The last term in (30), for instance, represents the
overall matching between the steering vectors of the AoAs with the combiner
$\mathbf{W}_{RF}$: for a fixed precoder $\mathbf{F}_{RF}$, the Tx side
experiences different channel gains (sum of eigenvalues) for different
combiners. The same applies for the Rx side, with an optimum precoder/combiner
pair for the best pointing between Tx and Rx.

From the $\widetilde{r}_{T}$ and $\widetilde{r}_{R}$ leading eigenvectors of
$\widetilde{\mathbf{R}}_{T}$ and $\widetilde{\mathbf{R}}_{R}$, i.e.,
$\widetilde{\mathbf{U}}_{T}=\mathrm{eig}_{\widetilde{r}_{T}}(\widetilde{\mathbf{R}}_{T})$
and
$\widetilde{\mathbf{U}}_{R}=\mathrm{eig}_{\widetilde{r}_{R}}(\widetilde{\mathbf{R}}_{R})$,
we have that:

$\displaystyle\mathrm{span}(\widetilde{\mathbf{U}}_{T})$
$\displaystyle=\mathrm{span}\left(\mathbf{A}_{T}^{\mathrm{T}}(\boldsymbol{\psi})\mathbf{F}_{RF}\right),$
(32) $\displaystyle\mathrm{span}(\widetilde{\mathbf{U}}_{R})$
$\displaystyle=\mathrm{span}\left(\mathbf{W}_{RF}^{\mathrm{H}}\mathbf{A}_{R}(\boldsymbol{\vartheta})\right),$
(33)

i.e.,
$\widetilde{\mathbf{U}}_{T}\in\mathbb{C}^{N_{T}^{RF}\times\widetilde{r}_{T}}$
and
$\widetilde{\mathbf{U}}_{R}\in\mathbb{C}^{N_{R}^{RF}\times\widetilde{r}_{R}}$
span the Tx and Rx spatial subspaces of the compressed channel
$\widetilde{\mathbf{H}}$, of dimensions $\widetilde{r}_{T}$ and
$\widetilde{r}_{R}$ (see (12)-(13) in Section II).

The position-dependent matrix for the DS-LR method is:

$\widehat{\boldsymbol{\Pi}}(\mathbf{\bar{p}},\bar{\theta})=\mathbf{C}^{\frac{\mathrm{H}}{2}}\,\widehat{\boldsymbol{\Pi}}_{\mathrm{DS}}\,\mathbf{C}^{-\frac{\mathrm{H}}{2}},$
(34)

where
$\widehat{\boldsymbol{\Pi}}_{\mathrm{DS}}=\widehat{\mathbf{U}}^{*}_{T}\widehat{\mathbf{U}}^{\mathrm{T}}_{T}\otimes\widehat{\mathbf{U}}_{R}\widehat{\mathbf{U}}^{\mathrm{H}}_{R}$
is the DS-LR projector onto to the propagation subspace, represented by basis
$\widehat{\mathbf{U}}^{*}_{T}\otimes\widehat{\mathbf{U}}_{R}$. Notice that the
Kronecker separability of Tx and Rx subspaces is an approximation, as the
Kronecker structure of the digital channel correlation (22) holds for single
paths only. Similarly to JS-LR,
$\widehat{\mathbf{U}}_{T}=\mathrm{eig}_{\widetilde{r}_{T}}(\widehat{\mathbf{R}}_{T})$
and
$\widehat{\mathbf{U}}_{R}=\mathrm{eig}_{\widetilde{r}_{R}}(\widehat{\mathbf{R}}_{R})$
are set from the $\widetilde{r}_{T}$ and $\widetilde{r}_{R}$ leading
eigenvectors of the following sample correlations:

$\displaystyle\widehat{\mathbf{R}}_{T}$
$\displaystyle=\frac{1}{L}\sum_{\ell=1}^{L}\overline{\overline{\mathbf{Y}}}^{\mathrm{H}}_{\ell}\,\overline{\overline{\mathbf{Y}}}_{\ell},$
(35) $\displaystyle\widehat{\mathbf{R}}_{R}$
$\displaystyle=\frac{1}{L}\sum_{\ell=1}^{L}\overline{\overline{\mathbf{Y}}}_{\ell}\,\overline{\overline{\mathbf{Y}}}^{\mathrm{H}}_{\ell},$
(36)

where we indicate with
$\overline{\overline{\mathbf{Y}}}_{\ell}\in\mathbb{C}^{N_{R}^{RF}\times
N_{T}^{RF}}=\mathrm{vec}^{-1}(\overline{\overline{\mathbf{y}}}_{\ell})$ the
whitened sequences in matrix form. It can be demonstrated that, asymptotically
($L\rightarrow\infty$):

$\displaystyle\mathrm{span}(\widehat{\mathbf{U}}_{T})\rightarrow\mathrm{span}(\mathbf{A}_{T}^{\mathrm{T}}(\boldsymbol{\psi})\mathbf{F}_{RF}),$
(37)
$\displaystyle\mathrm{span}(\widehat{\mathbf{U}}_{R})\rightarrow\mathrm{span}(\widetilde{\mathbf{Q}}^{-\frac{\mathrm{H}}{2}}_{n}\mathbf{W}_{RF}^{\mathrm{H}}\mathbf{A}_{R}(\boldsymbol{\vartheta})).$
(38)

The DS-LR channel estimation method provides a performance gain with respect
to conventional approaches when the spatial structure of
$\widetilde{\mathbf{H}}$ is sparse, which is equivalent to state that at least
one of the following conditions hold:

$\displaystyle\widetilde{r}_{T}$ $\displaystyle<N_{T}^{RF},$ (39)
$\displaystyle\widetilde{r}_{R}$ $\displaystyle<N_{R}^{RF}.$ (40)

Compared to JS-LR, the DS-LR method requires a lower number of training
sequences, $L$, to estimate the compressed channel eigenmodes, at the price of
a reduced performance (the sparsity degree of DS-LR is always less than the
JS-LR one).

Remark. Without interference
($\mathbf{Q}_{n}=\sigma_{n}^{2}\mathbf{I}_{N_{R}}$) and with an orthogonal
codebook for $\mathbf{W}_{RF}$ as here, it follows that
$\widetilde{\mathbf{Q}}_{n}\approx N\sigma_{n}^{2}\mathbf{I}_{N^{RF}_{R}}$,
where $\sigma_{n}^{2}$ is the noise power and $N$ accounts for analog
beamforming ($N=N_{R}$ for FC-HBF, $N=N^{B}_{R}$ for SC-HBF). In this setting,
for $L\rightarrow\infty$ we have:

$\displaystyle\mathrm{span}(\widehat{\mathbf{U}})\rightarrow\mathrm{span}(\widetilde{\mathbf{U}}),$
(41)
$\displaystyle\mathrm{span}(\widehat{\mathbf{U}}_{T})\rightarrow\mathrm{span}(\widetilde{\mathbf{U}}_{T}),$
(42)
$\displaystyle\mathrm{span}(\widehat{\mathbf{U}}_{R})\rightarrow\mathrm{span}(\widetilde{\mathbf{U}}_{R}).$
(43)

Therefore, the whitening/de-whitening reduces the position-dependent matrix
$\boldsymbol{\Pi}(\mathbf{\bar{p}},\bar{\theta})$ to the projection matrix
associated to the sample estimates of (22) (JS-LR) and (28)-(29) (DS-LR).

###  IV-C Lossy vs. Lossless Channel Compression for FC-HBF Architectures

By exploiting (12), (13) and (14) we can observe that if both the following
conditions hold

$\displaystyle N_{T}^{RF}\geq r_{T},$ (44) $\displaystyle N_{R}^{RF}\geq
r_{R},$ (45)

this implies that there exist an analog precoder/combiner
$\mathbf{F}_{RF}/\mathbf{W}_{RF}$ with $\mathrm{rank}(\mathbf{F}_{RF})\geq
r_{T}$ and $\mathrm{rank}(\mathbf{W}_{RF})\geq r_{R}$, such that FC-HBF
performance, in terms of Spectral Efficiency (SE), attains the Full-Digital
(FD) one. The first condition, (44), asserts that the overall number of RF
chains at Tx must be larger than the number of Tx-resolvable paths of the full
channel $\mathbf{H}$. This is derived from (12) by noticing that, if (44) does
not hold, $\widetilde{r}_{T}<r_{T}$ would mean that the HBF system cannot
explore the full channel diversity for insufficient number of available beams
at Tx, regardless the choice of $\mathbf{F}_{RF}/\mathbf{W}_{RF}$. This is
equivalent to a lossy compression of the channel. Condition (45) can be
analogously derived from (13). When both (44) and (45) apply,
$\widetilde{r}_{T}\leq r_{T}$ and $\widetilde{r}_{R}\leq r_{R}$, i.e., the
number of resolvable paths at Tx and Rx before and after the analog
compression can be equal when a suitable combination of
$\mathbf{F}_{RF}/\mathbf{W}_{RF}$ is employed (lossless compression of the
channel).

Remark 1: In practical FC-HBF systems, where the analog precoder/combiner are
defined by a codebook, the performances can deteriorate if the resolution is
poor, i.e., low angular sampling interval.

Remark 2: The previous consideration does not apply to SC-HBF systems, unless
a proper Tx power augmentation is considered. Indeed, for a fixed Tx power,
the reduced beamforming gain of SC-HBF compared to FC-HBF does not allow to
reach the performance of FD systems.

###  IV-D Digital Precoders/Combiners Design

The digital precoders/combiners are retrieved from the LR estimated compressed
channel matrix $\widehat{\mathbf{H}}=\mathrm{vec}^{-1}(\widehat{\mathbf{h}})$,
which must be known at the Tx side through a feedback from Rx. The optimal
digital precoder $\mathbf{F}_{BB}$ is [45]:

$\mathbf{F}_{BB}=\mathrm{eig}_{N_{S}}\left(\widehat{\mathbf{H}}^{\mathrm{H}}\widehat{\mathbf{H}}\right),$
(46)

while the digital combiner $\mathbf{W}_{BB}^{\mathrm{H}}$ is derived using the
optimal unconstrained Minimum Mean Squared Error (MMSE) as [46]

$\begin{split}\mathbf{W}_{BB}^{\mathrm{H}}=\bigg{(}\widehat{\mathbf{H}}^{\mathrm{H}}\mathbf{F}^{\mathrm{H}}_{BB}\widetilde{\mathbf{Q}}_{n}^{-1}\mathbf{F}_{BB}&\widehat{\mathbf{H}}+\frac{\mathbf{I}_{N_{S}}}{N_{S}}\bigg{)}^{-1}\widehat{\mathbf{H}}^{\mathrm{H}}\mathbf{F}^{\mathrm{H}}_{BB}\widetilde{\mathbf{Q}}_{n}^{-1}.\end{split}$
(47)

##  V Numerical Results

Figure 4: Urban scenario used in simulations: the solid yellow line represents
the reference vehicle trajectory, the blue triangle the mmW BS (located at 6 m
height from ground) while red and green circles the S1 and S2 locations used
for testing the proposed LR channel estimation. The inset sketches the
trajectories dataset used to simulate multiple vehicle passages in the ray-
tracing software  Figure 5: Number of channel paths vs. distance from the BS:
the blue curve represent the number of paths provided by ray-tracer, the red
one the meaningful ones

(a)

(b)

Figure 6: Hybrid arrays configuration for MS (6a) and BS (6b)

To demonstrate the effectiveness of the proposed channel estimation method, we
present the results obtained through numerical simulations using ray-tracing
channel data and a set of realistic vehicle trajectories. The latter are aimed
at simulating multiple vehicle passages, in a typical urban scenario (Fig. 4).
Two locations are selected for testing: the first (red circle in Fig. 4)
relatively far from the BS, $\approx 60$ m, and the second (green circle) at
$\approx 8$ m (close to the BS). To ease the reader in analyzing the results,
we will refer to these locations as S1 (far from the BS) and S2 (close to the
BS). Fig. 5 shows the number of channel paths as function of the MS-to-BS
distance. The blue curve represents the number of paths provided by the ray-
tracer, while the red curve is the number of meaningful paths, i.e., those
with cumulative power within the $99.9$ percentile. Red and green ellipses in
Fig. 5 identify the S1 and S2 locations used in simulation.

We consider an interference-free MS-to-BS (UL) communication in the 5G NR FR2
band (28 GHz carrier frequency) [47]. The hybrid MIMO setting is such that the
BS is equipped with $N_{R}=128$ ($16\times 8$) antennas and $N_{R}^{RF}=8$ RF
chains, while the MS (i.e., each vehicle) with $N_{T}=64$ ($8\times 8$)
antennas and $N_{T}^{RF}=4$ RF chains. We make use of Altair WinProp ray-
tracing software [37] to generate the required channel data (power, AoDs, AoAs
and scattering amplitude for each ray), whereas the MIMO channel (6) is
obtained by post-processing in far-field (i.e., AoDs/AoAs equal for each Tx/Rx
antennas). The required trajectories (i.e., position, velocity and direction
over time) are instead generated by means of SUMO [36]. In both MV beam
alignment (Section III) and LR training (Section IV) procedures, we consider
the assignment of MSs’ positions according to the spatial granularity of the
experiment. In other words, we exploit multiple vehicle passages in a spatial
region of radius $\rho$, where AoAs/AoDs are invariant. We set $\rho=2$ m for
S1 and $\rho=0.5$ m for S2, determined with ray-tracing. The size of the MV
region plays an important role in the proposed method: if excessive, a
performance penalty is experienced by the system as the channel subspaces
decorrelate (Section V-B).

Unless otherwise specified, the parameters given in Table I are used to
generate the results, while the MS and BS array configurations are in Fig. 6.
Most of the results we present in this section are related to the multipath
scenario S1, while the single-path S2 is used for comparison.

TABLE I: Simulation Parameters Parameter | Symbol | Value(s)
---|---|---
Carrier frequency | $f_{c}$ |  $28$ GHz
Number of channel paths | $P$ | $1-7$
Number of data streams | $N_{S}$ | 1
Number of MS antennas | $N_{T}$ |  $64$ ($8\times 8$)
Number of MS RF chains | $N_{T}^{RF}$ | $4$
Number of MS antennas x sub-array | $N_{T}^{B}$ | $16$
Number of BS antennas | $N_{R}$ |  $128$ ($16\times 8$)
Number of BS RF chains | $N_{R}^{RF}$ | $8$
Number of BS antennas x sub-array | $N_{R}^{B}$ | $16$
BS height from ground | - | 6 m
MV region radius | $\rho$ | 2 m (S1), 0.5 m (S2)

The performance of both JS-LR and DS-LR channel estimation methods are
compared to the U-ML one in terms of Spectral Efficiency (SE) and MSE on
compressed channel estimation. The SE is defined as [48]:

$\begin{split}R=\log_{2}\bigg{|}\mathbf{I}_{N_{S}}+\frac{1}{N_{S}}\mathbf{Q}^{-1}_{\mathrm{eff}}\mathbf{H}_{\mathrm{eff}}\mathbf{H}_{\mathrm{eff}}^{\mathrm{H}}\bigg{|}\end{split}$
(48)

where
$\mathbf{Q}_{\mathrm{eff}}=\mathbf{W}^{\mathrm{H}}_{BB}\widetilde{\mathbf{Q}}_{n}\mathbf{W}_{BB}$
is the covariance of the noise at the decision variable, and
$\mathbf{H}_{\mathrm{eff}}=\mathbf{W}_{BB}^{\mathrm{H}}\widetilde{\mathbf{H}}\mathbf{F}_{BB}$
is the effective end-to-end channel between MS and BS. The MSE is computed as:

$\mathrm{MSE}=\mathbb{E}\left[\left\lVert\widehat{\mathbf{h}}-\widetilde{\mathbf{h}}\right\rVert^{2}\right],$
(49)

and for U-ML method it is compared to the theoretical Cramer-Rao Lower Bound
(CRLB), while for LR it is asymptotically lower-bounded by [28]:

$\begin{split}\mathrm{MSE_{b}}&=\mathrm{tr}\left(\boldsymbol{\Pi}\\{\widehat{r}\\}\,\mathbf{C}\,\boldsymbol{\Pi}^{\mathrm{H}}\\{\widehat{r}\\}\right)+\\\
&+\mathrm{tr}\left(\Delta\boldsymbol{\Pi}\\{\widehat{r}\\}\,\widetilde{\mathbf{R}}\,\Delta\boldsymbol{\Pi}^{\mathrm{H}}\\{\widehat{r}\\}\right)\end{split}$
(50)

where: (i) $\boldsymbol{\Pi}\\{\widehat{r}\\}$ is the asymptotic
($L\rightarrow\infty$) (true) position-dependent matrix
$\boldsymbol{\Pi}(\mathbf{\bar{p}},\bar{\theta})$ evaluated for the estimated
rank, either $\widehat{r}$ (for JS-LR) or $\widehat{r}_{T}$ and
$\widehat{r}_{R}$ (for DS-LR); (ii)
$\Delta\boldsymbol{\Pi}\\{\widehat{r}\\}=\boldsymbol{\Pi}\\{\widetilde{r}\\}-\boldsymbol{\Pi}\\{\widehat{r}\\}$
is the difference between the asymptotic position-dependent matrix computed
for the true channel rank ($\widetilde{r}$ for JS-LR or $\widetilde{r}_{T}$
and $\widetilde{r}_{R}$ for DS-LR) and for the estimated one. Therefore, the
first term accounts for the residual noise contribution, while the second for
misparameterization (errors in the estimated diversity orders of the channel).
Here, we estimate the channel rank from the correlations’ eigenvalues, in
descending order, according to the $99.9$ percentile of their cumulative sum.

###  V-A S1 (multipath scenario far from the BS)

(a)

(b)

Figure 7: Spectral efficiency of FC-HBF vs. SC-HBF (7a) and FC-HBF vs. FD
(7b), with U-ML, JS-LR, and DS-LR channel estimation methods and perfect CSI;
$L=1000$ training blocks (vehicle passages) Figure 8: Spectral efficiency of
FC-HBF and SC-HBF systems varying the number of vehicle passages $L$

(a) FC-HBF

(b) SC-HBF

Figure 9: MSE on compressed channel estimation varying the number of vehicle
passages $L$ for FC-HBF (9a) and SC-HBF (9b) systems, with U-ML, JS-LR, and
DS-LR methods and corresponding theoretical bounds; SNR (per antenna) = $-10$
dB Figure 10: Spectral efficiency of FC-HBF and SC-HBF systems varying the
number of RF chains ($N_{T}^{RF}\times N_{R}^{RF}$), with U-ML, JS-LR, and DS-
LR channel estimation methods and perfect CSI; $L=1000$ vehicle passages, SNR
= $-10$ dB (FD as upper bound)

(a) FC-HBF

(b) SC-HBF

Figure 11: MSE on compressed channel estimation varying the number of RF
chains ($N_{T}^{RF}\times N_{R}^{RF}$) for FC-HBF (11a) and SC-HBF (11b)
systems, with U-ML, JS-LR, and DS-LR channel estimation methods and
corresponding theoretical bounds; $L=1000$ vehicle passages, SNR = $-10$ dB

Fig. 7 shows the achievable SE in (48) varying the SNR per antenna, i.e.,
before analog beamforming, for FC-HBF vs. SC-HBF architectures (Fig. 7a) and
FC-HBF vs. FD (Fig. 7b). The SE is evaluated with four different degrees of
channel knowledge: (i) perfect Channel State Information (CSI) (black lines);
(ii) optimal JS-LR channel estimation (blue lines); (iii) sub-optimal DS-LR
channel estimation (green lines); (iv) U-ML channel estimation (red lines).
The FD performance are computed as benchmark, with precoders and combiners
obtained with (46) and (47), respectively, by using the U-ML-, JS-LR- and DS-
LR-estimated full channel $\widehat{\mathbf{H}}$. In all the LR
implementations, the number of training vehicle passages is $L=1000$. As
expected, the SC-HBF architectures provides worst performance compared to FC-
HBF, as a consequence of the reduced analog beamforming gain. In both FC-HBF
and SC-HBF configurations, however, we notice the remarkable performance gain
compared to U-ML provided by DS-LR and especially JS-LR. For FC-HBF, at a
reference SNR of $-10$ dB, the SE gap amounts to 0.8 bits/s/Hz for DS-LR and
to 2.6 bits/s/Hz for JS-LR (Fig. 7a). It can be appreciated that, for FC-HBF
system, the use of JS-LR channel estimation method allows to practically
approach the perfect CSI case ($R\approx 8$ bits/s/Hz) with 5 dB less of SNR
compared to U-ML. For SC-HBF, instead, the DS-LR provides a SE gain of 0.7
bits/s/Hz and JS-LR 1.9 bits/s/Hz, while the SNR gain is even higher, up to
$\approx 10$ dB for $R\approx 3$ bits/s/Hz, while the perfect CSI case is
attained for $SNR\geq 10$ dB. As can be observed from Fig. 7b, the FC-HBF
(dashed lines) performance practically matches the FD one (solid lines), apart
from a negligible SE penalty due to the fixed spatial sampling provided by the
use of analog codebooks. According to Subsection IV-C, this result is
expected, as $N_{T}^{RF}=4>r_{T}=3$ and $N_{R}^{RF}=8>r_{R}=3$.

Figs. 8 and 9 depict, respectively, the behavior of the SE and of the MSE of
both FC-HBF and SC-HBF systems with respect to the number of vehicle passages
$L$, for SNR = $-10$ dB. The DS-LR method requires a lower number of passages,
approximately $L=50$ for HBF, to converge to its asymptotic MSE bound, whereas
for JS-LR method the convergence is guaranteed for $L=500$ blocks (HBF). It is
important to emphasize that, at the cost of approximately 1.5 bits/s/Hz in SE
(Fig. 8), we have a remarkable gain in complexity, which is approximately
ruled by the computation of the eigenvectors of correlation matrices in (26)
and (35)-(36), since the computational cost of
$\mathrm{eig}(\hat{\mathbf{R}})$ is
$\mathcal{O}((N^{RF}_{T}N^{RF}_{R})^{3})\geq\mathcal{O}((N^{RF}_{T})^{3})+\mathcal{O}((N^{RF}_{R})^{3})$,
as required for $\mathrm{eig}(\hat{\mathbf{R}}_{T})$ and
$\mathrm{eig}(\hat{\mathbf{R}}_{R})$. In general, the results show a
significant performance gain with LR compared to U-ML on whole SNR range.

The last results on S1 are related to the SE and MSE performance of HBF
varying the number of RF chains $N_{R}^{RF}\times N_{T}^{RF}$, summarized in
Figs. 10 and 11, fixing $L=1000$ vehicle passages and SNR per antenna of $-5$
dB. For FC-HBF systems, the SE gap between LR and U-ML goes proportionally to
the number of RF chains. The MSE in Fig. 11a explains the SE performance: by
increasing $N_{R}^{RF}$ and $N_{T}^{RF}$, the MSE of LR decreases with the
increasing sparsity of the compressed channel; conversely, the MSE of U-ML
does not change. For SC-HBF architectures, instead, Fig. 10 shows an
interesting trade-off between having a high analog resolution (few RF chains)
or having a high digital resolution (i.e., approach the FD system, for
$16\times 32$ RF chains). A high analog resolution implies a comparably low LR
gain with respect to U-ML, as the compressed channel sparsity decreases; a
high digital resolution leads to a significant sparsity of
$\widetilde{\mathbf{H}}$ and thus to a huge LR gain. For any HBF configuration
in between, the performance decreases. The MSE in 11b exhibits a similar trend
with respect to Fig. 11a, again explaining the SE gain of LR compared to U-ML.

###  V-B S2 (single-path scenario close to the BS)

(a)

(b)

Figure 12: Spectral efficiency of FC-HBF vs. SC-HBF (12a) and FC-HBF vs. FD
(12b), with U-ML, JS-LR, and DS-LR channel estimation methods and perfect CSI;
$L=1000$ training blocks (vehicle passages) Figure 13: Spectral efficiency of
FC-HBF vs. SC-HBF varying the MV region radius $\rho$, with U-ML, JS-LR, and
DS-LR channel estimation methods and perfect CSI; $L=1000$ training blocks
(vehicle passages), SNR = $-5$ dB Figure 14: MSE on JS-LR channel estimation
of FC-HBF vs. SC-HBF, varying the SNR, function of the MV region radius
$\rho$; $L=1000$ training blocks (vehicle passages), SNR = $-5$ dB

The last set of results are related to the single-path scenario S2 (Fig. 4).
Similarly to S1, we report in Fig. 12 the SE varying the SNR before
beamforming for FC-HBF vs. SC-HBF architectures (Fig. 12a) and FC-HBF vs. FD
(Fig. 12b). Again, we consider U-ML, JS-LR and DS-LR channel estimation
methods, and the perfect CSI case as upper bound, with $L=1000$ vehicle
passages. Compared to the multipath scenario S1, JS-LR and DS-LR channel
estimation methods approach the perfect CSI case, for both FC-HBF and SC-HBF.
This can be explained by considering that for a single-path channel, the
sparsity degrees in (27) and (39)-(40) are maximum, and the residual error on
the LR-estimated channel does not remarkably impact on the SE.

In S2, the proposed system performance is more sensitive to the MV region size
compared to S1. Fig. 13 shows the SE of FC-HBF and SC-HBF for all the channel
estimation methods varying $\rho$ (MV region radius) from 0.5 m to 4 m (the
latter basically considering the whole area of the crossing in Fig. 4), for a
fixed SNR = $-5$ dB. We notice that the JS-LR and DS-LR performance
drastically decrease with $\rho$, especially for FC-HBF, and can be even worse
than the U-ML one. This is a direct consequence of the AoDs/AoAs variation
within the selected MV region, which exceeds the system resolution (spatial
selectivity of BS array) and leads to subspace decorrelation. In other words,
the ensemble of received sequences $\\{\overline{\mathbf{y}}\\}_{\ell=1}^{L}$
(Subsection IV) do not have the same propagation subspace. As the spatial
resolution of FC-HBF systems is higher of SC-HBF one, the effect for the
former is stronger. This is further confirmed by the MSE of JS-LR varying the
SNR before beamforming (Fig. 14), where we notice a progressive deviation from
the asymptotic MSE bound (black, dashed line), proportional to $\rho$ and to
the SNR. For low SNRs, the imperfect modal filtering provided by an excessive
cluster size $\rho$ is negligible for low SNRs, where the noise is dominant,
while is relevant for higher SNR values. This last result shows a tradeoff
between the performance of the HBF system and the available resolution in MS
position accuracy, which allows to set up the MV regions. FC-HBF systems
provide superior performance, attaining FD one, but require a very accurate
positioning, while SC-HBF allows to relax this constraint.

##  VI Conclusion

In this paper, we propose a training-based multi-stage channel estimation
method for hybrid mmWave/sub-THz MIMO systems, based on terminals in mobility
(e.g., in V2I or V2N scenarios), in the 5G and future 6G context. The first
training stage relies on a novel multi-vehicular codebook-based beam alignment
procedure to obtain the optimal analog precoder and combiner. In the second
training stage, we adapt the Low-Rank (LR) channel estimation to hybrid
systems, and we propose two LR methods, namely Joint-Space Low-Rank (JS-LR)
and Disjoint-Space Low-Rank (DS-LR), for deriving the hybrid channel
eigenmodes. Finally, in the last stage, i.e., communications phase, we derive
the digital precoders/combiners based on both the optimal analog
precoder/combiner pair from the first stage and the hybrid channel eigenmodes
from the second stage.

The proposed LR methods are analyzed numerically, but realistically,
considering a V2I/V2N urban scenario based on OpenStreetMap for
roads/buildings topology and SUMO for the vehicular mobility. The channel is
generated by ray-tracing and the performances are compared in terms of
Spectral Efficiency (SE) and Mean Squared Error (MSE) on channel estimation.
The metrics on Full Digital (FD) system are reported as benchmark, as well as
the performance of the Unconstrained Maximum Likelihood (U-ML). The two
proposed solutions, i.e., optimal (JS-LR) and sub-optimal (DS-LR), are
examined for both Fully Connected (FC-HBF) and Sub-Connected (SC-HBF)
architectures varying SNR, training vehicles’ number, RF chains configuration,
and channel configurations, i.e., multipath (S1) and single-path (S2).

The achieved results proved the great advantage of our solution. In
particular, we observed that in the single-path scenario (S2), both JS-LR and
DS-LR solutions attain the SE of the perfect CSI results. Moreover, the FC-HBF
architecture exhibits similar performance to the benchmark (FD). In the
multipath scenario (S1), both solutions show better performance compared to
the U-ML estimator and attain the perfect CSI for an SNR $>-5$ dB for FC-HBF
and SNR $>10$ dB for SC-HBF. In general, we can conclude that, under the same
conditions, the FC-HBF architectures perform better than SC-HBF in terms of SE
and present the same MSE. However, the SC-HBF architectures are less sensitive
to positioning errors, which impacts on the size of the multi-vehicular region
used for training.

Another aspect of interest is that, as the number of RF chains increases, the
performance gap (LR-U-ML) of the FC-HBF architectures increases, while for SC-
HBF architecture, we observe that pursuing a trade-off between digital and
analog resolution is detrimental, and it is more appropriate to consider a
system with high digital resolution (higher number of RF chains), or high
analog resolution (low number of RF chains), with the former being preferable.
Concerning the comparison between JS-LR and DS-LR, we found that, under the
same conditions, the former shows a better SE and MSE. Moreover, the SE gap is
greater especially in the multipath and/or low SNR scenario, and it reduces in
the single-path and/or high SNR scenario. Consequently, in these cases, the
DS-LR method is recommended as it is significantly less complex, requiring
less training vehicles for convergence to the theoretical bound. In real
cases, the presence of neighbouring vehicles (even parked) is expected to make
the estimates to be time-varying, but nevertheless the BS can always command
the MSs to repeat some MV learning steps for refinement of the position-based
estimate.

## Acknowledgment

The research has been carried out in the framework of the Huawei-Politecnico
di Milano Joint Research Lab. The Authors want to acknowledge Huawei Milan
Research Center.

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