# Channel Estimation and Beamforming for Beyond Diagonal Reconfigurable
Intelligent Surfaces

Hongyu Li, , Shanpu Shen, ,
Yumeng Zhang, , and Bruno Clerckx,
Manuscript received; Part of this work has been accepted by IEEE International
Workshop on Computational Advances in Multi-Sensor Adaptive Processing
(CAMSAP), 2023 [1].Hongyu Li, Yumeng Zhang, and Bruno Clerckx are with the
Department of Electrical and Electronic Engineering, Imperial College London,
London SW7 2AZ, U.K. (e-mail:{c.li21, yumeng.zhang19,
b.clerckx}@imperial.ac.uk).Shanpu Shen is with the Department of Electrical
Engineering and Electronics, University of Liverpool, Liverpool L69 3GJ, U.K.
(e-mail: Shanpu.Shen@liverpool.ac.uk).

###### Abstract

Beyond diagonal reconfigurable intelligent surface (BD-RIS) is a new advance
and generalization of the RIS technique. BD-RIS breaks through the isolation
between RIS elements by creatively introducing inter-element connections,
thereby enabling smarter wave manipulation and enlarging coverage. However,
exploring proper channel estimation schemes suitable for BD-RIS aided
communication systems still remains an open problem. In this paper, we study
channel estimation and beamforming design for BD-RIS aided multi-antenna
systems. We first describe the channel estimation strategy based on the least
square (LS) method, derive the mean square error (MSE) of the LS estimation,
and formulate the joint pilot sequence and BD-RIS design problem with unique
constraints induced by BD-RIS architectures. Specifically, we propose an
efficient pilot sequence and BD-RIS design which theoretically guarantees to
achieve the minimum MSE. With the estimated channel, we then consider two BD-
RIS scenarios and propose beamforming design algorithms. Finally, we provide
simulation results to verify the effectiveness of the proposed channel
estimation scheme and beamforming design algorithms. We also show that more
inter-element connections in BD-RIS improves the performance while increasing
the training overhead for channel estimation.

###### Index Terms:

Beyond diagonal reconfigurable intelligent surfaces, beamforming design,
channel estimation.

##  I Introduction

Beyond diagonal reconfigurable intelligent surface (BD-RIS) is an emerging
technique, which generalizes and goes beyond conventional RIS with diagonal
phase shift matrix [2, 3] and generates scattering matrices not limited to
being diagonal, by introducing connections among RIS elements [4]. Thanks to
the flexible inter-element connections, BD-RIS has benefits in providing
smarter wave manipulation and enlarging coverage compared to conventional RIS
[4].

Existing works of BD-RIS have been carried out for modeling [5], beamforming
design [6, 7], and mode/architecture design [8, 9, 10]. The modeling of BD-RIS
and the concept of group- and fully-connected architectures, named according
to the circuit topology of inter-element connections, are first proposed in
[5], followed by the discrete-value design [6] and optimal beamforming design
[7]. Inspired by the group/fully-connected architectures [5] and the concept
of intelligent omni surface (IOS) with enlarged coverage [11, 12], BD-RIS with
hybrid transmitting and reflecting mode [8] and multi-sector mode [9] are
proposed to achieve full-space coverage with enhanced channel gain and system
performance. To find a better performance-complexity trade-off of BD-RIS, BD-
RIS with tree- and forest-connected architectures are proposed in [10], which
are proved to achieve the performance upper bound with the minimum circuit
complexity. The enhanced performance achieved by BD-RIS with different modes
and architectures highly depends on accurate channel state information (CSI),
while none of the above-mentioned works [5, 8, 9, 10, 6, 7] study the channel
estimation of BD-RIS. Therefore, it remains an open problem to effectively
obtain the CSI for BD-RIS aided wireless systems.

In conventional RIS literature on channel estimation, there are in general two
strategies to obtain the instantaneous CSI [13]. The first strategy is to
separately estimate the channels between the base station (BS)/users and the
RIS by partially “activating” RIS elements using RF chains. In this strategy,
the separate channels can be estimated by leveraging channel reciprocity and
time division duplex (TDD) and applying conventional estimation approaches,
such as the estimation of signal parameters via rotational invariance
technique (ESPRIT) and multiple signal classification (MUSIC) [14], and
compressed sensing (CS) [15, 16]. This strategy generally has low training
overhead regardless of the number of RIS elements, and can be directly used in
BD-RIS aided communication systems, since the structures of separate channels
do not depend on RIS architectures. Once the channel estimates are obtained,
the existing BD-RIS design algorithms [7, 8, 9] based on perfect and separate
channels can still be utilized. However, the drawback is that the equipment of
RF chains will result in additional cost and power consumption, which violates
the original motivation of employing RIS in wireless communication systems.
The second strategy is to estimate the cascaded user-RIS-BS channels relying
on purely passive RISs [17]. In this strategy, the BS estimates the cascaded
channel through proper design of the pilot sequence and RIS patterns, such as
the ON/OFF-based design [18] and the orthogonality-based design [19, 20]. To
reduce the training overhead, other estimation schemes, such as a three-phase
framework [21] and an anchor-assisted channel estimation [22], are further
proposed making use of the common BS-RIS channels for different users. With
the same motivation, a novel cascaded channel estimation scheme is proposed by
exploiting the sparsity and correlations of millimeter wave channels [23].
However, adapting this strategy to BD-RIS aided scenarios raises the following
challenges: First, the structure of the cascaded channels is tightly coupled
to the BD-RIS architectures and thus different from conventional RIS cases,
such that the existing channel estimation schemes cannot be directly used for
BD-RIS aided scenarios. Second, the dimension of the cascaded channel is
larger than that for conventional RIS cases due to the inter-element
connections of BD-RIS, such that significant training overhead for channel
estimation is required. Third, the BD-RIS training pattern should also be re-
designed due to the different constraints on the scattering matrix induced by
different BD-RIS architectures. Fourth, existing BD-RIS design algorithms [7,
8] based on separate channels are not applicable, such that algorithms
suitable for cascaded channels should be developed. Therefore, it remains a
challenging problem to develop efficient channel estimation methods for BD-RIS
aided scenarios with affordable overhead and to study corresponding
beamforming design algorithms with cascaded channel estimates.

To address the above challenges, in this work, we study the channel estimation
and beamforming design for BD-RIS aided wireless communication systems.
Specifically, we investigate the cascaded channel estimation for BD-RIS aided
wireless systems. With only the cascaded channel estimate, we re-consider the
BD-RIS design and propose corresponding algorithms for different scenarios.
The contributions of this work are summarized as follows.

First, we propose a novel channel estimation scheme for a reflective BD-RIS
aided multiple input multiple output (MIMO) system, where the joint design of
pilot sequence and BD-RIS matrix during the training process is investigated.
The proposed pilot sequence and BD-RIS design differs from that for
conventional RIS cases due to the new structure of the cascaded channel and
new constraints originating from the unique BD-RIS architectures. The proposed
design is applicable for arbitrary channel fading. Specifically, we propose a
closed-form pilot sequence and BD-RIS design which theoretically guarantees to
achieve the minimum mean square error (MSE) of the least square (LS)
estimator. A comprehensive analysis of the training overhead of the proposed
channel estimation scheme is also provided.

Second, we generalize the proposed channel estimation scheme to different BD-
RIS aided scenarios. On one hand, we show that the proposed channel estimation
scheme can be used for reflective BD-RIS aided multi-user systems. On the
other hand, we illustrate that the proposed scheme is also readily generalized
to BD-RIS with hybrid transmitting and reflecting mode and multi-sector mode
by assuming each sector of the hybrid/multi-sector BD-RIS works in turns.

Third, we investigate the beamforming design for a reflective BD-RIS aided
point-to-point MIMO system based on the cascaded channel estimates. The
proposed design differs from [7] since only the CSI for cascaded channels is
required.

Fourth, we study the beamforming design for a hybrid and multi-sector BD-RIS
aided multi-user multiple input single output (MU-MISO) system with cascaded
channel estimates. The proposed design is a generalization of the algorithm in
[8], with modifications adapting to the cascaded CSI.

Fifth, we present simulation results to verify the effectiveness and accuracy
of the proposed channel estimation scheme and evaluate the performance of the
proposed beamforming design algorithms. Simulation results verify that the
proposed channel estimation scheme achieves the theoretical minimum MSE.
Simulation results also show that with the estimated channel obtained by the
proposed channel estimation scheme, the proposed beamforming design algorithms
can achieve satisfactory performance close to the perfect CSI cases. More
importantly, there exists a practical trade-off between the channel estimation
and the performance for data transmission, indicating that the improved
performance of BD-RIS architectures is achieved at the cost of increasing the
training overhead.

Organization: Section II describes the channel model and transmission protocol
for a BD-RIS aided communication system. Section III introduces the proposed
channel estimation scheme for BD-RIS. Section IV illustrates the beamforming
design of BD-RIS based on estimated channels. Section V evaluates the
performance of the proposed channel estimation and beamforming design. Section
VI concludes this work.

Notations: Boldface lower- and upper-case letters indicate vectors and
matrices, respectively. $\mathbb{C}$, $\mathbb{R}$, and $\mathbb{N}$ denote
the sets of complex numbers, real numbers, and natural numbers, respectively.
$\mathbb{E}\\{\cdot\\}$ represents the statistical expectation.
$\Re\\{\cdot\\}$ denotes the real part of complex numbers. $(\cdot)^{T}$,
$(\cdot)^{*}$, $(\cdot)^{H}$, and $(\cdot)^{-1}$ denote the transpose,
conjugate, conjugate-transpose, and inversion operations, respectively.
$\otimes$ and $\odot$ denote the Kronecker product and Hadamard product,
respectively. $\mathsf{blkdiag}(\cdot)$ represents a block-diagonal matrix.
$|\cdot|$, $\|\cdot\|_{2}$, and $\|\cdot\|_{F}$ denote the absolute-value
norm, the Euclidean norm, and the Frobenius norm, respectively.
$\mathsf{vec}(\cdot)$, $\mathsf{rank}(\cdot)$, and $\mathsf{tr}(\cdot)$,
respectively, are the vectorization, rank, and trace of a matrix.
$\overline{\mathsf{vec}}(\cdot)$ is the reverse operation of the
vectorization, which reshapes the vectorized matrix into the original matrix.
$\mathsf{circshift}(\mathbf{a},N)$ rearranges vector $\mathbf{a}$ by moving
the final $N$ entries to the first $N$ positions. $\mathsf{mod}(M,N)$ returns
the remainder after $M$ is divided by $N$. $\mathbf{I}_{M}$ denotes an
$M\times M$ identity matrix and $\mathbf{1}_{M}$ denotes an $M\times 1$ all-
one vector. $\mathbf{F}_{M}$ and $\mathbf{D}_{M}$, respectively, denote
$M\times M$ discrete Fourier transform (DFT) matrix and Hadamard matrix.
$[\mathbf{A}]_{i:i^{\prime},j:j^{\prime}}$ extracts the $i$-th to
$i^{\prime}$-th rows and the $j$-th to $j^{\prime}$-th columns of
$\mathbf{A}$. $[\mathbf{a}]_{i:i^{\prime}}$ extracts the $i$-th to
$i^{\prime}$-th entries of $\mathbf{a}$.

##  II Channel Model and Transmission Protocol

In this section, we illustrate the uplink and downlink channel models of
general BD-RIS assisted MIMO systems, and introduce the transmission protocol
of the whole system.

###  II-A Channel Model

Figure 1: A paradigm of a BD-RIS assisted MIMO system.

We consider a narrowband system which consists of an $N$-antenna BS, an
$M$-antenna BD-RIS with reflective mode, and a $K$-antenna user, as
illustrated in Fig. 1. The $M$ antennas of the BD-RIS are connected to an
$M$-port group-connected reconfigurable impedance network [5], where the $M$
ports are uniformly divided into $G_{1}$ groups with each containing
$\bar{M}=\frac{M}{G_{1}}$ ports connected to each other. Mathematically, the
BD-RIS with group-connected architecture has a block-diagonal scattering
matrix111The group-connected architecture is a general case including both
single- and fully-connected architectures as special cases. Specifically,
$G_{1}=M$ boils down to single-connected architecture with diagonal scattering
matrices; $G_{1}=1$ represents fully-connected architecture with full
scattering matrices. $\bar{\mathbf{\Phi}}\in\mathbb{C}^{M\times M}$ with each
block $\bar{\mathbf{\Phi}}_{g}\in\mathbb{C}^{\bar{M}\times\bar{M}}$, $\forall
g\in\mathcal{G}_{1}=\\{1,\ldots,G_{1}\\}$, being unitary [5], i.e.,

$\bar{\mathbf{\Phi}}=\mathsf{blkdiag}(\bar{\mathbf{\Phi}}_{1},\ldots,\bar{\mathbf{\Phi}}_{G_{1}}),\bar{\mathbf{\Phi}}_{g}^{H}\bar{\mathbf{\Phi}}_{g}=\mathbf{I}_{\bar{M}},\forall
g\in\mathcal{G}_{1}.$ (1)

In this work, we assume the direct user-BS channel is blocked and focus purely
on the estimation of the cascaded user-RIS-BS channel222When the direct BS-
user channel exists, the direct channel can be effectively obtained by turning
off the BD-RIS and using conventional channel estimation strategies.. In the
uplink scenario, we denote $\mathbf{G}\in\mathbb{C}^{N\times M}$ and
$\mathbf{H}\in\mathbb{C}^{M\times K}$ as the channel from the BD-RIS to BS,
and from the user to BD-RIS, respectively. The uplink user-RIS-BS channel
$\mathbf{H}_{\mathrm{u}}\in\mathbb{C}^{N\times K}$ is

$\displaystyle\mathbf{H}_{\mathrm{u}}$
$\displaystyle=\mathbf{G}\bar{\mathbf{\Phi}}\mathbf{H}=\sum_{g\in\mathcal{G}_{1}}\mathbf{G}_{g}\bar{\mathbf{\Phi}}_{g}\mathbf{H}_{g}$
(2)
$\displaystyle=\overline{\mathsf{vec}}\Big{(}\sum_{g\in\mathcal{G}_{1}}\underbrace{\mathbf{H}_{g}^{T}\otimes\mathbf{G}_{g}}_{=\bar{\mathbf{Q}}_{g}\in\mathbb{C}^{NK\times\bar{M}^{2}}}\mathsf{vec}(\bar{\mathbf{\Phi}}_{g})\Big{)},$

where
$\mathbf{G}_{g}=[\mathbf{G}]_{:,(g-1)\bar{M}+1:g\bar{M}}\in\mathbb{C}^{N\times\bar{M}}$
and
$\mathbf{H}_{g}=[\mathbf{H}]_{(g-1)\bar{M}+1:g\bar{M},:}\in\mathbb{C}^{\bar{M}\times
K}$, $\forall g\in\mathcal{G}_{1}$. Alternatively, in the downlink scenario,
we denote $\mathbf{G}^{\prime}=[\mathbf{G}_{1}^{\prime
T},\ldots,\mathbf{G}_{G_{1}}^{\prime T}]^{T}\in\mathbb{C}^{M\times N}$ and
$\mathbf{H}^{\prime}=[\mathbf{H}_{1}^{\prime},\ldots,\mathbf{H}_{G_{1}}^{\prime}]\in\mathbb{C}^{K\times
M}$, respectively, as the channels from BS to BD-RIS, and from BD-RIS to the
user. The downlink BS-RIS-user channel
$\mathbf{H}_{\mathrm{d}}\in\mathbb{C}^{K\times N}$ is

$\displaystyle\mathbf{H}_{\mathrm{d}}$
$\displaystyle\overset{\text{(a)}}{=}\mathbf{H}^{\prime}\bar{\mathbf{\Theta}}\mathbf{G}^{\prime}=\sum_{g\in\mathcal{G}_{1}}\mathbf{H}_{g}^{\prime}\bar{\mathbf{\Theta}}_{g}\mathbf{G}_{g}^{\prime}$
(3)
$\displaystyle\overset{\text{(b)}}{=}\overline{\mathsf{vec}}^{T}\Big{(}\sum_{g\in\mathcal{G}_{1}}(\mathbf{H}_{g}^{\prime}\otimes\mathbf{G}_{g}^{\prime
T})\mathsf{vec}(\bar{\mathbf{\Theta}}_{g}^{T})\Big{)},$

where (a) holds by defining the BD-RIS matrix for downlink data transmission
as
$\bar{\mathbf{\Theta}}=\mathsf{blkdiag}(\bar{\mathbf{\Theta}}_{1},\ldots,\bar{\mathbf{\Theta}}_{G_{1}})$;
(b) holds by the property of vectorization.

When the communication occurs with TDD and the reciprocity between the uplink
and downlink user-RIS and RIS-BS channels exists, i.e.,
$\mathbf{H}^{\prime}=\mathbf{H}^{T}$ and $\mathbf{G}^{\prime}=\mathbf{G}^{T}$,
we have

$\mathbf{H}_{\mathrm{d}}=\overline{\mathsf{vec}}^{T}\Big{(}\sum_{g\in\mathcal{G}_{1}}\bar{\mathbf{Q}}_{g}\mathsf{vec}(\bar{\mathbf{\Theta}}_{g}^{T})\Big{)},$
(4)

which indicates that the channel estimate
$\bar{\mathbf{Q}}=[\bar{\mathbf{Q}}_{1},\ldots,\bar{\mathbf{Q}}_{G_{1}}]\in\mathbb{C}^{NK\times\bar{M}^{2}G_{1}}$
obtained by the uplink training can be used for downlink data
transmission333We should clarify here that
$\bar{\mathbf{\Theta}}\neq\bar{\mathbf{\Phi}}^{T}$ since the BD-RIS matrices
are designed focusing on different metrics, i.e., minimizing the channel
estimation error and maximizing the data transmission rate.. This motivates us
to estimate $\bar{\mathbf{Q}}$ instead of separate channels $\mathbf{H}$ and
$\mathbf{G}$, and design the BD-RIS beamforming with the knowledge of
$\bar{\mathbf{Q}}$ for data transmission.

###  II-B Tile-Based Channel Construction

Figure 2: Illustration of BD-RIS antenna ports, groups, and tiles. Left: BD-
RIS with $M=36$, $G_{1}=18$ and $G_{2}=9$; right: BD-RIS with $M=36$,
$G_{1}=12$ and $G_{2}=4$. For example, in the left figure, BD-RIS antenna
ports with indices $\\{1,2\\}$ and $\\{3,4\\}$ form two different groups, each
of which consists of antenna ports connected to each other by reconfigurable
impedance components. These two groups further form one tile, within which the
groups share the same BD-RIS pattern.

Estimating $\bar{\mathbf{Q}}$ requires extremely high complexity and training
overhead due to the large number of channel coefficients growing linearly with
$G_{1}$ and quadratically with $\bar{M}$. To effectively reduce the channel
estimation complexity and training overhead, we generalize the methods used in
conventional RIS channel estimation works [18, 24, 25] by introducing the
concept of tile, which consists of adjacent groups, each having $\bar{M}$
ports connected to each other. The groups in the same tile share the same BD-
RIS pattern during the training. To ease understanding, we provide two
examples in Fig. 2 to illustrate antenna ports, groups, and tiles. Here, we
define $G_{2}$ as the number of tiles and $\bar{G}=\frac{G_{1}}{G_{2}}$ as the
tile size. We further define the shared pattern
$\mathbf{\Phi}_{i}\in\mathbb{C}^{\bar{M}\times\bar{M}}$ for tile $i$ with
$\mathbf{\Phi}_{i}^{H}\mathbf{\Phi}_{i}=\mathbf{I}_{\bar{M}}$, $\forall
i\in\mathcal{G}_{2}=\\{1,\ldots,G_{2}\\}$ and rewrite the BD-RIS matrix
$\bar{\mathbf{\Phi}}$ as

$\bar{\mathbf{\Phi}}=\mathsf{blkdiag}(\mathbf{I}_{\bar{G}}\otimes\mathbf{\Phi}_{1},\ldots,\mathbf{I}_{\bar{G}}\otimes\mathbf{\Phi}_{G_{2}}).$
(5)

The user-RIS-BS channel (2) can thus be re-written as

$\mathbf{H}_{\mathrm{u}}=\overline{\mathsf{vec}}\Big{(}\sum_{i\in\mathcal{G}_{2}}\underbrace{\sum\nolimits_{j=1}^{\bar{G}}\bar{\mathbf{Q}}_{(i-1)\bar{G}+j}}_{=\mathbf{Q}_{i}\in\mathbb{C}^{NK\times\bar{M}^{2}}}\mathsf{vec}(\mathbf{\Phi}_{i})\Big{)}.$
(6)

As such, we estimate the combined cascaded channel of each tile, referred to
as the tile-based channel, which leads to the reduction of training overhead
since the channel parameters to be estimated reduces from $\bar{\mathbf{Q}}$
to
$\mathbf{Q}=[\mathbf{Q}_{1},\ldots,\mathbf{Q}_{G_{2}}]\in\mathbb{C}^{NK\times\bar{M}^{2}G_{2}}$.
However, this is achieved at the expense of obtaining reduced CSI, which will
limit the beamforming flexibility of BD-RIS and further degrades the
performance for data transmission. The impact of estimating the tile-based
channel instead of the original cascaded channel on the beamforming
flexibility of BD-RIS is mathematically reflected in the downlink channel
$\mathbf{H}_{\mathrm{d}}$, which is re-written as

$\mathbf{H}_{\mathrm{d}}=\overline{\mathsf{vec}}^{T}\Big{(}\sum_{i\in\mathcal{G}_{2}}\mathbf{Q}_{i}\mathsf{vec}(\mathbf{\Theta}_{i}^{T})\Big{)},$
(7)

with a reformulation of the BD-RIS matrix as

$\bar{\mathbf{\Theta}}=\mathsf{blkdiag}(\mathbf{I}_{\bar{G}}\otimes\mathbf{\Theta}_{1},\ldots,\mathbf{I}_{\bar{G}}\otimes\mathbf{\Theta}_{G_{2}}).$
(8)

This indicates that with the knowledge of $\mathbf{Q}$, we should design
$\mathbf{\Theta}_{i}$, $i\in\mathcal{G}_{2}$, which provides less beam
manipulation flexibility compared with designing $\bar{\mathbf{\Theta}}_{g}$,
$\forall g\in\mathcal{G}_{1}$ as in perfect CSI cases, resulting in
performance loss for downlink data transmission.

Remark 1: The trade-off between the training overhead for channel estimation
and the performance for data transmission comes from twofold perspectives. On
the one hand, due to the tile-based channel construction, the larger $G_{2}$
reduces the dimension of channels to be estimated, i.e., from
$\bar{\mathbf{Q}}$ to $\mathbf{Q}$, which further reduces the overhead.
Meanwhile, the larger $G_{2}$ reduces the dimension of BD-RIS matrices to be
designed, i.e., from $\bar{\mathbf{\Theta}}_{g}$, $\forall
g\in\mathcal{G}_{1}$ to $\mathbf{\Theta}_{i}$, $\forall i\in\mathcal{G}_{2}$.
On the other hand, due to the BD-RIS architectures, the larger $\bar{M}$
increase the dimension of channels to be estimated and further the overhead.
Meanwhile, the larger $\bar{M}$ provides more flexibility to BD-RIS design,
which further enhances the performance. This overhead-performance trade-off
will be qualitatively and quantitatively studied in the following sections.

###  II-C Transmission Protocol

With the tile-based channel construction, we instead estimate the reduced
dimensional cascaded channel $\mathbf{Q}$ and design the beamforming with the
estimate of $\mathbf{Q}$ for data transmission. This results in the following
protocol [18, 17, 20] with each transmission frame divided into three phases
as illustrated in Fig. 3, based on the assumption that user-RIS and RIS-BS
channels remain approximately constant within one transmission frame.

Phase 1: The BS estimates $\mathbf{Q}$ by uplink training, where the pilots
are consecutively transmitted from the user and reflected by the BD-RIS with
varied $\mathbf{\Phi}_{i}$, $\forall i\in\mathcal{G}_{2}$ within $T_{1}$
symbol durations. In this phase, the difference compared to conventional RIS
comes from the design of the varied BD-RIS pattern due to the different
constraint of the scattering matrix, whose details will be given in Section
III.

Phase 2: The BS optimizes the transceiver beamformers, and
$\mathbf{\Theta}_{i}$, $\forall i\in\mathcal{G}_{2}$ based on the estimated
channel $\mathbf{Q}$, and feeds back the optimized $\mathbf{\Theta}_{i}$,
$\forall i\in\mathcal{G}_{2}$ to BD-RIS requiring $T_{2}$ symbol durations.
This phase is also different from conventional RIS cases due to the more
general unitary constraint of the BD-RIS, whose details will be given in
Section IV.

Phase 3: The BS performs the downlink data transmission for $T_{3}$ symbol
durations based on the optimized beamformers and $\mathbf{\Theta}_{i}$,
$\forall i\in\mathcal{G}_{2}$ in Phase 2.

Figure 3: One transmission frame consisting of three phases with
$T=T_{1}+T_{2}+T_{3}$ symbol durations.

##  III Channel Estimation for BD-RIS

In this section, we describe the proposed channel estimation strategy based on
the LS method, formulate the BD-RIS design problem to minimize the MSE of the
LS estimator, and provide the closed-form solution.

###  III-A LS Based Channel Estimation

The channel estimation strategy is described as follows. Assuming the user
sends pilot symbol vector $\bar{\mathbf{x}}_{t}\in\mathbb{C}^{K\times 1}$ with
$|[\bar{\mathbf{x}}_{t}]_{k}|=1$, $\forall k\in\mathcal{K}=\\{1,\ldots,K\\}$,
at time slot $t$, $\forall t\in\mathcal{T}=\\{1,\ldots,T_{1}\\}$, the signal
received at the BS is

$\displaystyle\mathbf{y}_{t}$
$\displaystyle=\sqrt{P_{\mathrm{u}}}\mathbf{G}\bar{\bar{\mathbf{\Phi}}}_{t}\mathbf{H}\bar{\mathbf{x}}_{t}+\mathbf{n}_{t}$
(9)
$\displaystyle=\sqrt{P_{\mathrm{u}}}(\bar{\mathbf{x}}_{t}^{T}\otimes\mathbf{I}_{N})\sum_{i\in\mathcal{G}_{2}}\mathbf{Q}_{i}\mathsf{vec}(\mathbf{\Phi}_{t,i})+\mathbf{n}_{t}$
$\displaystyle=\sqrt{P_{\mathrm{u}}}(\bar{\mathbf{x}}_{t}^{T}\otimes\mathbf{I}_{N})\mathbf{Q}\bar{\bm{\phi}}_{t}+\mathbf{n}_{t}$
$\displaystyle=\sqrt{P_{\mathrm{u}}}\underbrace{(\bar{\bm{\phi}}_{t}^{T}\otimes\bar{\mathbf{x}}_{t}^{T}\otimes\mathbf{I}_{N})}_{=\widehat{\mathbf{\Phi}}_{t}\in\mathbb{C}^{N\times
NK\bar{M}^{2}G_{2}}}\mathsf{vec}(\mathbf{Q})+\mathbf{n}_{t},\forall
t\in\mathcal{T},$

where $P_{\mathrm{u}}$ denotes the transmit power at each user antenna,
$\bar{\bar{\mathbf{\Phi}}}_{t}=\mathsf{blkdiag}(\mathbf{I}_{\bar{G}}\otimes\mathbf{\Phi}_{t,1},\ldots,\mathbf{I}_{\bar{G}}\otimes\mathbf{\Phi}_{t,G_{2}})$
with $\mathbf{\Phi}_{t,i}\in\mathbb{C}^{\bar{M}\times\bar{M}}$ denoting the
$i$-th tile of the BD-RIS matrix at time slot $t$, and
$\mathbf{n}_{t}\in\mathbb{C}^{N\times 1}$ with
$\mathbf{n}_{t}\sim\mathcal{CN}(\mathbf{0},\sigma^{2}\mathbf{I}_{N})$,
$\forall t\in\mathcal{T}$ denotes the noise. The vector $\bar{\bm{\phi}}_{t}$
is defined as
$\bar{\bm{\phi}}_{t}=[\mathsf{vec}^{T}(\mathbf{\Phi}_{t,1}),\ldots,\mathsf{vec}^{T}(\mathbf{\Phi}_{t,G_{2}})]^{T}\in\mathbb{C}^{G_{2}\bar{M}^{2}}$,
$\forall t\in\mathcal{T}$. To uniquely estimate the cascaded channel
$\mathbf{Q}$ in (9), $T_{1}\geq K\bar{M}^{2}G_{2}$ pilot symbols should be
transmitted from the user. Combining the data from such pilots together, we
have

$\displaystyle\mathbf{y}$
$\displaystyle=[\mathbf{y}_{1}^{T},\ldots,\mathbf{y}_{T_{\mathrm{1}}}^{T}]^{T}$
(10)
$\displaystyle=\sqrt{P_{\mathrm{u}}}\underbrace{[\widehat{\mathbf{\Phi}}_{1}^{T},\ldots,\widehat{\mathbf{\Phi}}^{T}_{T_{\mathrm{1}}}]^{T}}_{=\widehat{\mathbf{\Phi}}\in\mathbb{C}^{NT_{1}\times
NK\bar{M}^{2}G_{2}}}\mathbf{q}+\underbrace{[\mathbf{n}_{1}^{T},\ldots,\mathbf{n}_{T_{\mathrm{1}}}^{T}]^{T}}_{=\mathbf{n}\in\mathbb{C}^{NT_{1}\times
1}},$

where $\mathbf{q}=\mathsf{vec}(\mathbf{Q})\in\mathbb{C}^{NK\bar{M}^{2}G_{2}}$.
The simplest way to estimate $\mathbf{q}$ is to use the LS method, yielding
the LS estimator of $\mathbf{q}$ as the following form

$\displaystyle\widehat{\mathbf{q}}$
$\displaystyle=(\sqrt{P_{\mathrm{u}}})^{-1}\widehat{\mathbf{\Phi}}^{\dagger}\mathbf{y}=(\sqrt{P_{\mathrm{u}}})^{-1}(\widehat{\mathbf{\Phi}}^{H}\widehat{\mathbf{\Phi}})^{-1}\widehat{\mathbf{\Phi}}^{H}\mathbf{y}$
(11)
$\displaystyle=\mathbf{q}+\underbrace{(\sqrt{P}_{\mathrm{u}})^{-1}(\widehat{\mathbf{\Phi}}^{H}\widehat{\mathbf{\Phi}})^{-1}\widehat{\mathbf{\Phi}}^{H}\mathbf{n}}_{=\mathbf{q}_{\mathrm{e}}\in\mathbb{C}^{NK\bar{M}^{2}G_{2}\times
1}}.$

Then the MSE of the LS estimator is

$\mathrm{e}_{\widehat{\mathbf{q}}}=\mathbb{E}\\{\|\widehat{\mathbf{q}}-\mathbf{q}\|_{2}^{2}\\}=\mathbb{E}\\{\|\mathbf{q}_{\mathrm{e}}\|_{2}^{2}\\}=\frac{\sigma^{2}}{P_{\mathrm{u}}}\mathsf{tr}((\widehat{\mathbf{\Phi}}^{H}\widehat{\mathbf{\Phi}})^{-1}),$
(12)

which implies that the MSE of channel estimation depends on the values of
$\bar{\mathbf{x}}_{t}$ and $\bar{\bm{\phi}}_{t}$, $\forall t\in\mathcal{T}$,
such that a proper design of them is required. As such, we formulate the
following MSE minimization problem

$\displaystyle\min_{\\{\bar{\mathbf{x}}_{t},\bar{\bm{\phi}}_{t}\\}_{\forall
t}}~{}$
$\displaystyle\mathsf{tr}((\widehat{\mathbf{\Phi}}^{H}\widehat{\mathbf{\Phi}})^{-1})$
(13a) s.t.
$\displaystyle\mathbf{\Phi}_{t,i}^{H}\mathbf{\Phi}_{t,i}=\mathbf{I}_{\bar{M}},\forall
t\in\mathcal{T},\forall i\in\mathcal{G}_{2},$ (13b)
$\displaystyle|[\bar{\mathbf{x}}_{t}]_{k}|=1,\forall t\in\mathcal{T},\forall
k\in\mathcal{K},$ (13c)
$\displaystyle\mathsf{rank}(\widehat{\mathbf{\Phi}})=NK\bar{M}^{2}G_{2},$
(13d)

where we set $T_{1}=K\bar{M}^{2}G_{2}$ to minimize the overhead without
introducing estimation ambiguities, which yields $\widehat{\mathbf{\Phi}}$ a
full-rank square matrix. Problem (13) is difficult to solve due to the inverse
operation in the objective and the non-convex constraints of group-connected
BD-RIS. In the following subsection, we will first simplify the objective
function and then propose an efficient approach for optimal
$\bar{\mathbf{x}}_{t}$ and $\bar{\bm{\phi}}_{t}$, $\forall t\in\mathcal{T}$.

###  III-B Solution to Problem (13)

We start by simplifying the inverse operation in objective (13a). To this end,
we derive the minimum of the objective (13a) based on the following lemma.

Lemma 1: The objective function in (13a) achieves its minimum, i.e.,
$\min\mathsf{tr}((\widehat{\mathbf{\Phi}}^{H}\widehat{\mathbf{\Phi}})^{-1})=N\bar{M}$,
with the condition
$\widehat{\mathbf{\Phi}}^{H}\widehat{\mathbf{\Phi}}=\widehat{\mathbf{\Phi}}\widehat{\mathbf{\Phi}}^{H}=K\bar{M}G_{2}\mathbf{I}_{NK\bar{M}^{2}G_{2}}$.

Proof: Please refer to Appendix A. $\square$

With Lemma 1, we deduce that achieving the minimum MSE in problem (13) is
equivalent to finding a unitary matrix $\widehat{\mathbf{\Phi}}$, i.e.,
$\widehat{\mathbf{\Phi}}^{H}\widehat{\mathbf{\Phi}}=\widehat{\mathbf{\Phi}}\widehat{\mathbf{\Phi}}^{H}=K\bar{M}G_{2}\mathbf{I}_{NK\bar{M}^{2}G_{2}}$.
Problem (13) is thus transformed into a feasibility-check problem

find
$\displaystyle\widehat{\mathbf{\Phi}}\in\mathbb{C}^{NK\bar{M}^{2}G_{2}\times
NK\bar{M}^{2}G_{2}}$ (14a) s.t.
$\displaystyle\widehat{\mathbf{\Phi}}^{H}\widehat{\mathbf{\Phi}}=K\bar{M}G_{2}\mathbf{I}_{NK\bar{M}^{2}G_{2}},\text{(\ref{eq:unitary_constraint}),
(\ref{eq:pilot_constraint})}.$ (14b)

Given the large dimension and complicated structure of matrix
$\widehat{\mathbf{\Phi}}$, which is constructed by a coupling of
$\bar{\bm{\phi}}_{t}$ and $\bar{\mathbf{x}}_{t}$, $\forall t\in\mathcal{T}$,
it is difficult to directly find such a matrix to simultaneously satisfy the
three constraints in problem (14). To decouple the design of
$\bar{\bm{\phi}}_{t}$ and $\bar{\mathbf{x}}_{t}$, $\forall t\in\mathcal{T}$,
we apply the most common training approach [26], which breaks the training
time into $T_{\mathrm{s}}=\frac{T_{1}}{K}=\bar{M}^{2}G_{2}$ blocks of length
$K$, as illustrated in Fig. 4. Specifically, the same BD-RIS pattern,
$\bm{\phi}_{s}$, repeats $K$ times within each block
$s\in\mathcal{S}=\\{1,\ldots,T_{\mathrm{s}}\\}$, while it varies from block to
block. Meanwhile, the pilots
$\mathbf{X}=[\mathbf{x}_{1},\ldots,\mathbf{x}_{K}]^{T}$ are chosen as an
orthogonal sequence, i.e., $\mathbf{X}^{H}\mathbf{X}=K\mathbf{I}_{K}$, and
repeat over the $T_{\mathrm{s}}$ blocks. In other words, we establish the BD-
RIS pattern set $\bm{\phi}_{s}$, $\forall s\in\mathcal{S}$, and the pilot
sequence set $\mathbf{x}_{k}$, $\forall k\in\mathcal{K}$, for periodicity.
Specifically, we have the mappings between $\bar{\bm{\phi}}_{t}$ and
$\bm{\phi}_{s}$, and between $\bar{\mathbf{x}}_{t}$ and $\mathbf{x}_{k}$,
respectively, as

$\bar{\bm{\phi}}_{(s-1)K+k}=\bm{\phi}_{s},\bar{\mathbf{x}}_{(s-1)K+k}=\mathbf{x}_{k},\forall
k\in\mathcal{K},\forall s\in\mathcal{S}.$ (15)

This allows us to rewrite $\widehat{\mathbf{\Phi}}^{H}\widehat{\mathbf{\Phi}}$
in the following form

$\displaystyle\widehat{\mathbf{\Phi}}^{H}\widehat{\mathbf{\Phi}}$
$\displaystyle=\sum_{t=1}^{K\bar{M}^{2}G_{2}}(\bar{\bm{\phi}}_{t}^{*}\bar{\bm{\phi}}_{t}^{T})\otimes(\bar{\mathbf{x}}_{t}^{*}\bar{\mathbf{x}}_{t}^{T})\otimes\mathbf{I}_{N}$
(16)
$\displaystyle\overset{\text{(a)}}{=}\Big{(}\sum_{s=1}^{T_{\mathrm{s}}}\bm{\phi}_{s}^{*}\bm{\phi}_{s}^{T}\Big{)}\otimes\Big{(}\sum_{k=1}^{K}\mathbf{x}_{k}^{*}\mathbf{x}_{k}^{T}\Big{)}\otimes\mathbf{I}_{N}$
$\displaystyle\overset{\text{(b)}}{=}K(\mathbf{\Phi}^{H}\mathbf{\Phi})\otimes\mathbf{I}_{NK},$

where (a) holds due to the mappings in (15); (b) holds with
$\mathbf{X}^{H}\mathbf{X}=K\mathbf{I}_{K}$, and by defining
$\mathbf{\Phi}=[\bm{\phi}_{1},\ldots,\bm{\phi}_{T_{\mathrm{s}}}]^{T}\in\mathbb{C}^{\bar{M}^{2}G_{2}\times\bar{M}^{2}G_{2}}$.
Based on (16), we transform designing $\bar{\bm{\phi}}_{t}$ and
$\bar{\mathbf{x}}_{t}$ in (14) into designing matrices $\mathbf{\Phi}$ and
$\mathbf{X}$. Specifically, the feasible solution of $\mathbf{X}$ can be
easily obtained, such as using DFT or Hadamard matrices. The design of
$\mathbf{\Phi}$, meanwhile, can be formulated as the following feasibility-
check problem

find
$\displaystyle\mathbf{\Phi}\in\mathbb{C}^{\bar{M}^{2}G_{2}\times\bar{M}^{2}G_{2}}$
(17a) s.t.
$\displaystyle\mathbf{\Phi}^{H}\mathbf{\Phi}=\bar{M}G_{2}\mathbf{I}_{\bar{M}^{2}G_{2}},\text{(\ref{eq:unitary_constraint})}.$
(17b)

Problem (17) is still difficult to solve due to the constraint from BD-RIS
architectures and the large dimension of matrix $\mathbf{\Phi}$. To further
simplify the design, we propose to decouple problem (17) into two reduced-
dimensional sub-problems with simpler constraints. This leads to the following
lemma.

Figure 4: Training breaking the training time into $T_{\mathrm{s}}$ blocks of
length $K$.

Lemma 2: The feasible matrix $\mathbf{\Phi}$ from problem (17) can be
constructed as $\mathbf{\Phi}=\mathbf{A}\otimes\breve{\mathbf{\Phi}}$, where
$\mathbf{A}\in\mathbb{C}^{G_{2}\times G_{2}}$ is obtained by solving the
following problem:

find $\displaystyle\mathbf{A}\in\mathbb{C}^{G_{2}\times G_{2}}$ (18a) s.t.
$\displaystyle\mathbf{A}^{H}\mathbf{A}=G_{2}\mathbf{I}_{G_{2}},$ (18b)
$\displaystyle|[\mathbf{A}]_{i,i^{\prime}}|=1,\forall
i,i^{\prime}\in\mathcal{G}_{2}.$ (18c)

$\breve{\mathbf{\Phi}}\in\mathbb{C}^{\bar{M}^{2}\times\bar{M}^{2}}$ is
obtained by solving the following problem:

find
$\displaystyle\breve{\mathbf{\Phi}}\in\mathbb{C}^{\bar{M}^{2}\times\bar{M}^{2}}$
(19a) s.t.
$\displaystyle\breve{\mathbf{\Phi}}^{H}\breve{\mathbf{\Phi}}=\breve{\mathbf{\Phi}}\breve{\mathbf{\Phi}}^{H}=\bar{M}\mathbf{I}_{\bar{M}^{2}},$
(19b)
$\displaystyle\overline{\mathsf{vec}}^{H}([\breve{\mathbf{\Phi}}^{T}]_{:,m})\overline{\mathsf{vec}}([\breve{\mathbf{\Phi}}^{T}]_{:,m})=\mathbf{I}_{\bar{M}},\forall
m\in\bar{\bar{\mathcal{M}}},$ (19c)

where $\bar{\bar{\mathcal{M}}}=\\{1,\ldots,\bar{M}^{2}\\}$.

Proof: Please refer to Appendix B. $\square$

With Lemma 2, we decouple problem (17) into two reduced-dimensional problems
(18) and (19). The feasible solution to problem (18) can be easily obtained by
using a $G_{2}\times G_{2}$ DFT matrix $\mathbf{F}_{G_{2}}$ or a Hadamard
matrix $\mathbf{D}_{G_{2}}$, i.e.,
$\mathbf{A}^{\mathrm{DFT}}=\mathbf{F}_{G_{2}}$ or
$\mathbf{A}^{\mathrm{Hadamard}}=\mathbf{D}_{G_{2}}$. However, the solution to
problem (19) is not that straightforward since we need to find a matrix
$\breve{\mathbf{\Phi}}$ such that each row constructs a unitary matrix, i.e.,
constraint (19c), and that different rows are orthogonal to each other, i.e.,
constraint (19b). In other words, the matrix $\breve{\mathbf{\Phi}}$ should
include two-dimensional (intra- and inter-row) orthogonality, which motivates
us to construct $\breve{\mathbf{\Phi}}$ with two orthogonal bases. This brings
the following theorem.

Theorem 1: The matrix $\breve{\mathbf{\Phi}}$ satisfying (19b) and (19c) can
be constructed such that each row, i.e.,
$[\breve{\mathbf{\Phi}}]_{(m-1)\bar{M}+n,:}=\breve{\bm{\phi}}_{m,n}$, $\forall
m,n\in\bar{\mathcal{M}}=\\{1,\ldots,\bar{M}\\}$, has the following structure:

$\displaystyle\breve{\bm{\phi}}_{m,n}=\mathsf{circshift}(\mathsf{vec}^{T}(\mathbf{Z}_{1}),(n-1)\bar{M})\odot([\mathbf{Z}_{2}]_{m,:}\otimes\mathbf{1}_{\bar{M}}^{T}),$
(20)

where $\mathbf{Z}_{1}\in\mathbb{C}^{\bar{M}\times\bar{M}}$ and
$\mathbf{Z}_{2}\in\mathbb{C}^{\bar{M}\times\bar{M}}$ are referred to as base
matrices satisfying the following constraints:

  1. 1)

$\mathbf{Z}_{1}$ is a scaled unitary matrix, i.e.,
$\mathbf{Z}_{1}^{H}\mathbf{Z}_{1}=\mathbf{Z}_{1}\mathbf{Z}_{1}^{H}=\alpha_{1}\mathbf{I}_{\bar{M}}$
with $\alpha\in\mathbb{R}$, $\alpha_{1}\neq 0$ a scalar;

  2. 2)

$\mathbf{Z}_{2}$ is a scaled unitary matrix whose entries have identical
modulus, i.e.,
$\mathbf{Z}_{2}^{H}\mathbf{Z}_{2}=\mathbf{Z}_{2}\mathbf{Z}_{2}^{H}=\alpha_{2}\mathbf{I}_{\bar{M}}$
with $\alpha_{2}\in\mathbb{R}$, $\alpha_{2}\neq 0$ a scalar,
$|[\mathbf{Z}_{2}]_{m,n}|=\sqrt{\frac{\alpha_{2}}{\bar{M}}}$, $\forall
m,n\in\bar{\mathcal{M}}$;

  3. 3)

The product of two scalars is a constant, i.e.,
$\alpha_{1}\alpha_{2}=\bar{M}$.

Proof: Please refer to Appendix C. $\square$

Based on Theorem 1, we transform problem (19) into finding two base matrices
$\mathbf{Z}_{1}$ and $\mathbf{Z}_{2}$ satisfying the above three constraints
in Theorem 1, which can be easily satisfied using DFT/Hadamard matrices. More
specifically, we can simply set
$\mathbf{Z}_{1}^{\mathrm{DFT}}=\mathbf{F}_{\bar{M}}$ or
$\mathbf{Z}_{1}^{\mathrm{Hadamard}}=\mathbf{D}_{\bar{M}}$ and
$\mathbf{Z}_{2}^{\mathrm{DFT}}=\frac{1}{\sqrt{\bar{M}}}\mathbf{F}_{\bar{M}}$
or
$\mathbf{Z}_{2}^{\mathrm{Hadamard}}=\frac{1}{\sqrt{\bar{M}}}\mathbf{D}_{\bar{M}}$
to construct $\breve{\mathbf{\Phi}}$, and further construct
$\mathbf{\Phi}=\mathbf{A}\otimes\breve{\mathbf{\Phi}}$ with
$\mathbf{A}^{\mathrm{DFT}}$ or $\mathbf{A}^{\mathrm{Hadamard}}$.

With solutions to construct pilot sequence $\mathbf{X}$ and matrix
$\mathbf{\Phi}$, the procedure of the proposed channel estimation scheme is
straightforward, which is summarized as the following steps:

  1. S1:

Generate pilot sequence $\mathbf{X}$ such that
$\mathbf{X}^{H}\mathbf{X}=K\mathbf{I}_{K}$.

  2. S2:

Generate $\mathbf{A}$ by $\mathbf{A}^{\mathrm{DFT}}=\mathbf{F}_{G_{2}}$ or
$\mathbf{A}^{\mathrm{Hadamard}}=\mathbf{D}_{G_{2}}$.

  3. S3:

Set $\mathbf{Z}_{1}$ by $\mathbf{Z}_{1}^{\mathrm{DFT}}=\mathbf{F}_{\bar{M}}$
or $\mathbf{Z}_{1}^{\mathrm{Hadamard}}=\mathbf{D}_{\bar{M}}$.

  4. S4:

Set $\mathbf{Z}_{2}$ by
$\mathbf{Z}_{2}^{\mathrm{DFT}}=\frac{1}{\sqrt{\bar{M}}}\mathbf{F}_{\bar{M}}$
or
$\mathbf{Z}_{2}^{\mathrm{Hadamard}}=\frac{1}{\sqrt{\bar{M}}}\mathbf{D}_{\bar{M}}$.

  5. S5:

Construct $\breve{\mathbf{\Phi}}$ by (20) from Theorem 1.

  6. S6:

Construct $\mathbf{\Phi}$ by
$\mathbf{\Phi}=\mathbf{A}\otimes\breve{\mathbf{\Phi}}$ from Lemma 2.

  7. S7:

Map $\mathbf{X}$ and $\mathbf{\Phi}$ to $\bar{\mathbf{x}}_{t}$ and
$\bar{\bm{\phi}}_{t}$, $\forall t\in\mathcal{T}$ by (15).

  8. S8:

Construct $\widehat{\mathbf{\Phi}}$ by (9) with $\bar{\bm{\phi}}_{t}$ and
$\bar{\mathbf{x}}_{t}$, $\forall t\in\mathcal{T}$.

  9. S9:

Obtain $\widehat{\mathbf{q}}$ by (11).

Remark 2: Note that either using DFT matrices or Hadamard matrices to generate
$\mathbf{\Phi}$, we achieve the minimum MSE of the LS estimation, i.e.,
$\mathrm{e}_{\widehat{\mathbf{q}}}^{\mathrm{min}}=N\sigma^{2}P_{\mathrm{u}}^{-1}\bar{M}$,
and avoid the time-consuming inversion by
$\widehat{\mathbf{\Phi}}^{\dagger}=(K\bar{M}G_{2})^{-1}\widehat{\mathbf{\Phi}}^{H}$.
Nevertheless, the two ways have pros and cons from different aspects. On one
hand, the matrix $\mathbf{\Phi}$ constructed by Hadamard bases has the
additional advantage of requiring only two states of each entry, which
indicates that practical group-connected BD-RIS with discrete values can be
effectively tuned to perform accurate channel estimation. However, the values
of $G_{2}$ and $\bar{M}$ should be chosen from the set
$\\{1,2,4n,n\in\mathbb{N}\\}$ to guarantee the existance of Hadamard matrix.
On the other hand, $\mathbf{\Phi}$ constructed by DFT bases requires more
states of each entry of group-connected BD-RIS, which complicates the
practical implementation of BD-RIS, while there is no such limitation of the
values of $G_{2}$ and $\bar{M}$.

TABLE I: Training Overhead, Estimation MSE, and Circuit Complexity of BD-RIS with Group-Connected Architectures Metrics | Value | Property
---|---|---
Training Overhead | $T_{1}=KM\bar{M}\bar{G}^{-1}$ |  $\uparrow$ with $\bar{M}$; $\downarrow$ with $\bar{G}$
Estimation MSE | $\mathrm{e}^{\mathrm{min}}_{\widehat{\mathbf{q}}}=\frac{\sigma^{2}}{P_{\mathrm{u}}}N\bar{M}$ |  $\uparrow$ with $\bar{M}$
Circuit Complexity |  $C=(\bar{M}+1)\frac{M}{2}$ [5] |  $\uparrow$ with $\bar{M}$

###  III-C Overhead, MSE, and Circuit Complexity

The training overhead $T_{1}$ to estimate $\mathbf{q}$, the estimation error
$\mathrm{e}^{\mathrm{min}}_{\widehat{\mathbf{q}}}$, and the circuit complexity
$C$ of the group-connected BD-RIS indicated by the number of impedances to
realize the circuit topology, are summarized in Table I. We can observe that
with fixed number of RIS antennas $M$ and other parameters, the training
overhead, estimation MSE, and circuit complexity all increase with the group
size $\bar{M}$ of the group-connected architecture. Interestingly, the
training overhead can be effectively reduced by introducing tiles (increasing
tile size $\bar{G}$), although the larger $\bar{M}$ decreases the number of
possible $\bar{G}$. More importantly, the larger $\bar{M}$ provides more beam
manipulation flexibility to BD-RIS to enhance the transmission performance,
while the larger $\bar{G}$ leads to reduced CSI and further limits the
transmission performance. This phenomenon which will be numerically studied in
Section V. Therefore, it is important to properly choose the values of
$\bar{M}$ and $\bar{G}$ to achieve satisfactory channel estimation performance
and data transmission performance with affordable training overhead and
circuit complexity.

###  III-D Multi-User Scenarios

The proposed channel estimation scheme can be easily generalized to multi-user
cases by assuming all users transmit orthogonal pilot sequences such that
channels from all users can be simultaneously estimated by the BS. This can be
simply done by setting $K$ as the total number of antennas at all users.
Alternatively, cascaded channels from different users can be successively
estimated by assuming users transmitting pilots in a TDD manner. In this case,
each user can transmit the same pilots and adopt the proposed channel
estimation scheme to estimate the cascaded channel.

###  III-E BD-RIS with Hybrid/Multi-Sector Modes

While the proposed channel estimation is based on BD-RIS with reflective mode,
that is the user is located at the same side as BS, it is also readily
generalized to BD-RIS with hybrid/multi-sector modes [8, 9]. Assume that an
$L$-sector BD-RIS with $M$ antennas in each sector,
$L\in\mathcal{L}=\\{1,\ldots,L\\},L\geq 2$, is adopted in a downlink MU-MISO
system with an $N$-antenna BS and $K$ single-antenna users. Each sector covers
$K_{l}$ users indexed by
$\mathcal{K}_{l}=\\{\sum_{i=1}^{l-1}K_{i}+1,\cdots,\sum_{i=1}^{l}K_{i}\\}$,
$\sum_{l\in\mathcal{L}}K_{l}=K$. In addition, the $L$-sector BD-RIS with
group-connected architecture is, mathematically, characterized by matrices
$\tilde{\mathbf{\Phi}}_{l}=\mathsf{blkdiag}(\tilde{\mathbf{\Phi}}_{l,1},\ldots,\tilde{\mathbf{\Phi}}_{l,G_{1}})\in\mathbb{C}^{M\times
M},\forall l\in\mathcal{L}$, satisfying the following constraint [9]

$\sum_{l\in\mathcal{L}}\tilde{\mathbf{\Phi}}_{l,g}^{H}\tilde{\mathbf{\Phi}}_{l,g}=\mathbf{I}_{\bar{M}},\forall
g\in\mathcal{G}_{1}.$ (21)

Let
$\mathbf{G}=[\mathbf{G}_{1},\ldots,\mathbf{G}_{G_{1}}]\in\mathbb{C}^{N\times
M}$ and
$\mathbf{h}_{k}=[\mathbf{h}_{k,1}^{T},\ldots,\mathbf{h}_{k,G_{1}}^{T}]^{T}\in\mathbb{C}^{M\times
1}$ denote the channel from BD-RIS to BS, and from user $k$, $\forall
k\in\mathcal{K}$, to BD-RIS, respectively. Then the user-RIS-BS uplink channel
can be expressed as

$\mathbf{h}_{\mathrm{u},k}=\sum_{g\in\mathcal{G}_{1}}\underbrace{\mathbf{h}_{k,g}^{T}\otimes\mathbf{G}_{g}}_{=\bar{\mathbf{Q}}_{k,g}\in\mathbb{C}^{N\times\bar{M}^{2}}}\mathsf{vec}(\tilde{\mathbf{\Phi}}_{l,g}),\forall
k\in\mathcal{K}_{l},\forall l\in\mathcal{L}.$ (22)

Applying the tile-based channel construction444For BD-RIS with hybrid/multi-
sector mode, the group is formed by the antenna ports from the same sector
which are connected to each other. The tile is formed by groups from the same
sector. proposed in Section II-B, we instead estimate the channels

$\mathbf{Q}_{k,i}=\sum_{j=1}^{\bar{G}}\bar{\mathbf{Q}}_{k,(i-1)\bar{G}+j},\forall
i\in\mathcal{G}_{2},\forall k\in\mathcal{K}_{l},\forall l\in\mathcal{L}.$ (23)

This can be done according to the following steps.

  1. S1:

In every $T_{1,l}=K_{l}\bar{M}^{2}G_{2}$ time durations, the $K_{l}$ cascaded
channels $\mathbf{Q}_{k,i},\forall i\in\mathcal{G}_{2},\forall
k\in\mathcal{K}_{l}$, can be simultaneously estimated assuming

    1. a)

that $\tilde{\mathbf{\Phi}}_{l^{\prime}}=\mathbf{0}$, $\forall l^{\prime}\neq
l$, within the $T_{1,l}$ durations,

    2. b)

that $\tilde{\mathbf{\Phi}}_{l}$ and the pilots transmitted by $K_{l}$ users
are designed using the proposed scheme.

This yields the channel estimates $\widehat{\mathbf{Q}}_{k,i}$, $\forall
i\in\mathcal{G}_{2}$, $\forall k\in\mathcal{K}_{l}$.

  2. S2:

Repeat S1 for all $L$ sectors.

To facilitate understanding, we provide a simple example for how to estimate
the cascaded channels for BD-RIS with hybrid mode ($L=2$) as illustrated in
Fig. 5. Based on the above strategy, we can obtain the cascaded channels for
BD-RIS with hybrid/multi-sector modes with a total training overhead
$T_{1}=\sum_{l\in\mathcal{L}}T_{1,l}=\sum_{l\in\mathcal{L}}K_{l}\bar{M}^{2}G_{2}=K\bar{M}^{2}G_{2}=KM\bar{M}\bar{G}^{-1}$.

Figure 5: Illustration of channel estimation for BD-RIS with hybrid mode.

##  IV Beamforming Design for BD-RIS

In this section, we formulate two beamforming design problems for BD-RIS with
reflective and hybrid/multi-sector modes and different scenarios, and propose
efficient beamforming design methods relying on the cascaded channel
estimates. The considered problems are different from that in previous works
[8, 7] since, in this work, only cascaded channel estimate is available, while
the proposed algorithms [8, 7] rely on separated BS-RIS and RIS-user channels.

###  IV-A Reflective BD-RIS Aided Point-to-Point MIMO

We first consider a point-to-point MIMO system aided by a BD-RIS with
reflective mode. Assume the $N$-antenna BS transmits the symbol vector
$\mathbf{s}\in\mathbb{C}^{N_{\mathrm{s}}\times 1}$ to the $K$-antenna user,
$\mathbb{E}\\{\mathbf{s}\mathbf{s}^{H}\\}=\mathbf{I}_{N_{\mathrm{s}}}$, where
$N_{\mathrm{s}}$ denotes the number of data streams,
$N_{\mathrm{s}}\leq\min\\{N,K\\}$. Based on (7) and (8), the processed signal
at the user is given by

$\displaystyle\widehat{\mathbf{s}}$
$\displaystyle=\mathbf{W}^{H}\overline{\mathsf{vec}}^{T}\Big{(}\sum_{i\in\mathcal{G}_{2}}\mathbf{Q}_{i}\mathsf{vec}(\mathbf{\Theta}_{i}^{T})\Big{)}\mathbf{P}\mathbf{s}+\mathbf{W}^{H}\mathbf{n}^{\prime},$
(24)

where $\mathbf{W}\in\mathbb{C}^{K\times N_{\mathrm{s}}}$ and
$\mathbf{P}\in\mathbb{C}^{N\times N_{\mathrm{s}}}$ denote the combiner at the
user and the precoder at the BS, respectively, and
$\mathbf{n}^{\prime}\in\mathbb{C}^{K\times 1}$ with
$\mathbf{n}^{\prime}\sim\mathcal{CN}(\mathbf{0},\sigma^{\prime
2}\mathbf{I}_{K})$ denotes the noise.

####  IV-A.1 Problem Formulation

Based on the channel estimate
$\widehat{\mathbf{q}}=\mathsf{vec}(\widehat{\mathbf{Q}})$,
$\widehat{\mathbf{Q}}=[\widehat{\mathbf{Q}}_{1},\cdots,\widehat{\mathbf{Q}}_{G_{2}}]$
with $\widehat{\mathbf{Q}}_{i}$ being the estimate of $\mathbf{Q}_{i}$, the
joint transceiver and BD-RIS design problem to maximize the spectral
efficiency is given by555Here, and in the following subsection, we involve the
term $1-\frac{T_{1}+T_{2}}{T}$ to account for both channel estimation
efficiency and data transmission accuracy, while we ignore the effect of
channel estimation error for simplicity and directly solve the optimization
problem based on channel estimates. However, the effect of channel estimation
error will be taken into account in the performance evaluations.

$\displaystyle\max_{\bm{\theta},\mathbf{W},\mathbf{P}}~{}~{}$
$\displaystyle\Big{(}1-\frac{T_{1}+T_{2}}{T}\Big{)}\log_{2}\Big{(}\Big{|}\mathbf{I}_{N_{\mathrm{s}}}+(\sigma^{\prime
2}\mathbf{W}^{H}\mathbf{W})^{-1}\mathbf{W}^{H}$
$\displaystyle~{}~{}\times\overline{\mathsf{vec}}^{T}(\widehat{\mathbf{Q}}\bm{\theta})\mathbf{P}\mathbf{P}^{H}\overline{\mathsf{vec}}^{*}(\widehat{\mathbf{Q}}\bm{\theta})\mathbf{W}\Big{|}\Big{)}$
(25a) $\displaystyle\mathrm{s.t.}~{}~{}~{}$
$\displaystyle\mathbf{\Theta}_{i}^{H}\mathbf{\Theta}_{i}=\mathbf{I}_{\bar{M}},\forall
i\in\mathcal{G}_{2},$ (25b) $\displaystyle\|\mathbf{P}\|_{F}^{2}\leq
P_{\mathrm{d}},$ (25c)

where $P_{\mathrm{d}}$ denotes the transmit power at the BS, and
$\bm{\theta}=[\mathsf{vec}^{T}(\mathbf{\Theta}_{1}^{T}),\dots,\mathsf{vec}^{T}(\mathbf{\Theta}_{G_{2}}^{T})]^{T}$.
Problem (25) is not easy to solve due to the non-convex constraint (25b) and
the coupling of the BD-RIS constraint, the precoder, and the combiner. To
simplify the joint design, the objective function is decoupled into two
separate optimizations. Specifically, we first focus on the BD-RIS design to
maximize the channel strength. Then, having the effective channel, we
determine the precoder and the combiner to further maximize the spectral
efficiency.

####  IV-A.2 Solution to Problem (25)

For simplicity, we propose to design the BD-RIS to maximize the channel
strength without considering the inter-stream interference, which will be
further arranged by the precoder and combiner once $\bm{\theta}$ is
determined. This allows us to formulate the BD-RIS design problem

$\displaystyle\bm{\theta}^{\star}$
$\displaystyle=\mathop{\arg~{}\max}_{\mathbf{\Theta}_{i}^{H}\mathbf{\Theta}_{i}=\mathbf{I}_{\bar{M}},\forall
i}~{}\mathsf{tr}(\overline{\mathsf{vec}}^{T}(\widehat{\mathbf{Q}}\bm{\theta})\overline{\mathsf{vec}}^{*}(\widehat{\mathbf{Q}}\bm{\theta}))$
(26a)
$\displaystyle\overset{\text{(a)}}{=}\mathop{\arg~{}\max}_{\mathbf{\Theta}_{i}^{H}\mathbf{\Theta}_{i}=\mathbf{I}_{\bar{M}},\forall
i}~{}\sum_{i^{\prime}\in\mathcal{G}_{2}}\sum_{i\in\mathcal{G}_{2}}\bm{\theta}_{i}^{T}\mathbf{V}_{i,i^{\prime}}\bm{\theta}_{i^{\prime}}^{*},$
(26b)

where (a) follows with $\bm{\theta}_{i}=\mathsf{vec}(\mathbf{\Theta}_{i}^{T})$
and
$\mathbf{V}_{i,i^{\prime}}=[\widehat{\mathbf{Q}}^{T}\widehat{\mathbf{Q}}^{*}]_{\mathcal{I}_{i},\mathcal{I}_{i^{\prime}}}$,
$\mathcal{I}_{i}=(i-1)\bar{M}^{2}+1:i\bar{M}^{2}$, $\forall
i,i^{\prime}\in\mathcal{L}$. Note that $\bm{\theta}_{i}$ and
$\bm{\theta}_{i^{\prime}}$, $\forall i\neq i^{\prime}$,
$i,i^{\prime}\in\mathcal{G}_{2}$, have separate constraints, while they are
coupled in the objective function (26b), which makes it difficult to determine
$\bm{\theta}_{i}$, $\forall i\in\mathcal{G}_{2}$. To simplify the design, we
split the objective with respect to each $\bm{\theta}_{i}$ out from (26b) with
fixed others. This results in the sub-problem for $\bm{\theta}_{i}$ as

$\displaystyle\max_{\bm{\theta}_{i}}~{}$
$\displaystyle\bm{\theta}_{i}^{T}\mathbf{V}_{i,i}\bm{\theta}_{i}^{*}+2\Re\\{\bm{\theta}_{i}^{T}\bm{\chi}_{i}\\}$
(27a) s.t.
$\displaystyle\overline{\mathsf{vec}}^{*}(\bm{\theta}_{i})\overline{\mathsf{vec}}^{T}(\bm{\theta}_{i})=\mathbf{I}_{\bar{M}},$
(27b)

where $\bm{\chi}_{i}=\sum\nolimits_{i^{\prime}\neq
i}\mathbf{V}_{i,i^{\prime}}\bm{\theta}_{i^{\prime}}^{*}\in\mathbb{C}^{\bar{M}^{2}\times
1}$, $\forall i\in\mathcal{G}_{2}$. Constraint (27b) constructs a
$\bar{M}^{2}$-dimensional complex Stiefel manifold such that problem (27) can
be formulated as an unconstrained optimization on the Stiefel manifold. Then,
the problem can be effectively solved by the conjugate-gradient methods on the
manifold space [27].

After the BD-RIS has been designed, we obtain the effective channel as
$\mathbf{H}_{\mathrm{d}}^{\star}=\overline{\mathsf{vec}}^{T}(\widehat{\mathbf{Q}}\bm{\theta}^{\star})$.
To obtain the precoder and combiner, we adopt the singular value decomposition
(SVD) of the effective channel $\mathbf{H}_{\mathrm{d}}^{\star}$ as
$\mathbf{H}_{\mathrm{d}}^{\star}=\mathbf{U}_{1}\mathbf{\Sigma}\mathbf{U}_{2}^{H}$,
where $\mathbf{U}_{1}\in\mathbb{C}^{K\times K}$ and
$\mathbf{U}_{2}\in\mathbb{C}^{N\times N}$ are unitary matrices, and
$\mathbf{\Sigma}\in\mathbb{C}^{K\times N}$ is a rectangular diagonal matrix
containing eigenvalues of $\mathbf{H}_{\mathrm{d}}^{\star}$ on the diagonal.
Accordingly, we have the precoder and combiner as

$\mathbf{P}^{\star}=\sqrt{P_{\mathrm{d}}}\frac{[\mathbf{U}_{1}]_{:,1:N_{\mathrm{s}}}}{\|[\mathbf{U}_{1}]_{:,1:N_{\mathrm{s}}}\|_{F}},~{}~{}\mathbf{W}^{\star}=[\mathbf{U}_{2}]_{:,1:N_{\mathrm{s}}}.$
(28)

For clarity, the above beamforming design algorithm is summarized in the
following steps:

  1. S1:

Initialize $\bm{\theta}$.

  2. S2:

With fixed other terms, update $\bm{\theta}_{i}$ by solving problem (27).

  3. S3:

Repeat S2 until the convergence of $\bm{\theta}$ is guaranteed.

  4. S4:

Obtain $\mathbf{P}^{\star}$ and $\mathbf{W}^{\star}$ by (28).

###  IV-B Hybrid/Multi-Sector BD-RIS Aided MU-MISO

In this subsection, we consider a MU-MISO system aided by a BD-RIS with
hybrid/multi-sector mode, as illustrated in Section III-C. Assume the
$N$-antenna BS transmits the symbols
$\tilde{\mathbf{s}}=[s_{1},\ldots,s_{K}]\in\mathbb{C}^{K\times 1}$ to $K$
single-antenna users,
$\mathbb{E}\\{\tilde{\mathbf{s}}\tilde{\mathbf{s}}^{H}\\}=\mathbf{I}_{K}$.
Based on (22) and (23), the received signal at each user covered by sector
$l$, $\forall l\in\mathcal{L}$, is given by

$\displaystyle y_{k}=$
$\displaystyle\sum_{i\in\mathcal{G}_{2}}\mathsf{vec}^{T}(\mathbf{\Theta}_{l,i}^{T})\mathbf{Q}_{k,i}^{T}\mathbf{p}_{k}s_{k}$
(29) $\displaystyle+\sum_{k^{\prime}\neq
k}\sum_{i\in\mathcal{G}_{2}}\mathsf{vec}^{T}(\mathbf{\Theta}_{l,i}^{T})\mathbf{Q}_{k,i}^{T}\mathbf{p}_{k^{\prime}}s_{k^{\prime}}+n_{k},\forall
k\in\mathcal{K}_{l},$

where $\mathbf{\Theta}_{l,i}\in\mathbb{C}^{\bar{M}\times\bar{M}}$, $\forall
l\in\mathcal{L}$, satisfies
$\sum_{l\in\mathcal{L}}\mathbf{\Theta}_{l,i}^{H}\mathbf{\Theta}_{l,i}=\mathbf{I}_{\bar{M}}$,
$\forall i\in\mathcal{G}_{2}$, $\mathbf{p}_{k}\in\mathbb{C}^{N\times 1}$
denotes the precoder for user $k$, and $n_{k}\in\mathbb{C}$ with
$n_{k}\sim\mathcal{CN}(0,\sigma^{\prime 2})$ denotes the noise.

####  IV-B.1 Problem Formulation

Based on the channel estimates $\widehat{\mathbf{Q}}_{k,i}$, $\forall
k\in\mathcal{K}$, $\forall i\in\mathcal{G}_{2}$, the joint transmitter and BD-
RIS design problem to maximize the sum-rate is given by

$\displaystyle\max_{\tilde{\bm{\theta}}_{l},\forall l,\mathbf{p}_{k},\forall
k}~{}~{}$
$\displaystyle\Big{(}1-\frac{T_{1}+T_{2}}{T}\Big{)}\sum_{l\in\mathcal{L}}\sum_{k\in\mathcal{K}_{l}}\log_{2}(1+\gamma_{k})$
(30a) s.t.
$\displaystyle\sum_{l\in\mathcal{L}}\mathbf{\Theta}_{l,i}^{H}\mathbf{\Theta}_{l,i}=\mathbf{I}_{\bar{M}},\forall
i\in\mathcal{G}_{2}.$ (30b)
$\displaystyle\sum_{k\in\mathcal{K}}\|\mathbf{p}_{k}\|_{2}^{2}\leq
P_{\mathrm{d}},$ (30c)

where
$\gamma_{k}=\frac{|\tilde{\bm{\theta}}_{l}^{T}\tilde{\mathbf{Q}}_{k}^{T}\mathbf{p}_{k}|^{2}}{\sum_{k^{\prime}\neq
k}|\tilde{\bm{\theta}}_{l}^{T}\tilde{\mathbf{Q}}_{k}^{T}\mathbf{p}_{k^{\prime}}|^{2}+\sigma^{\prime
2}}$ denotes the signal-to-interference-plus-noise ratio (SINR) with the fresh
notations
$\tilde{\bm{\theta}}_{l}=[\mathsf{vec}^{T}(\mathbf{\Theta}_{l,1}^{T}),\ldots,\mathsf{vec}^{T}(\mathbf{\Theta}_{l,G_{2}}^{T})]^{T}$
and
$\tilde{\mathbf{Q}}_{k}=[\widehat{\mathbf{Q}}_{k,1},\ldots,\widehat{\mathbf{Q}}_{k,G_{2}}]$,
$\forall k\in\mathcal{K}_{l}$, $\forall l\in\mathcal{L}$. Although problem
(30) is a challenging optimization with complicated ratio terms in the
$\log(\cdot)$ function and non-convex constraints of BD-RIS, we show that the
algorithm proposed in [8] can be easily adapted to solving (30) with slight
modifications.

####  IV-B.2 Solution to Problem (30)

By introducing auxiliary variables $\iota_{k}$ and $\tau_{k}$, $\forall
k\in\mathcal{K}_{l}$, $\forall l\in\mathcal{L}$ to each ratio term in (30a)
based on the fractional programming [28], we can transform (30a) into a four-
block optimization with a more decent objective function as
$f(\\{\tilde{\bm{\theta}}_{l}\\}_{\forall l},\\{\mathbf{p}_{k}\\}_{\forall
k},\\{\iota_{k}\\}_{\forall k},\\{\tau_{k}\\}_{\forall
k})=\sum_{l\in\mathcal{L}}\sum_{k\in\mathcal{K}_{l}}(\log_{2}(1+\iota_{k})-\iota_{k}+2\sqrt{1+\iota_{k}}\Re\\{\tau_{k}^{*}\tilde{\bm{\theta}}_{l}^{T}\tilde{\mathbf{Q}}_{k}^{T}\mathbf{p}_{k}\\}-|\tau_{k}|^{2}(\sum_{k^{\prime}\in\mathcal{K}}|\tilde{\bm{\theta}}_{l}^{T}\tilde{\mathbf{Q}}_{k}^{T}\mathbf{p}_{k^{\prime}}|^{2}+\sigma^{\prime
2}))$. The four blocks are iteratively updated in sequential until
convergence. Under the iterative design framework, the solutions to the blocks
$\\{\iota_{k}\\}_{\forall k}$ and $\\{\tau_{k}\\}_{\forall k}$ can be easily
obtained by checking the first-order conditions of
$f(\\{\tilde{\bm{\theta}}_{l}\\}_{\forall l},\\{\mathbf{p}_{k}\\}_{\forall
k},\\{\iota_{k}\\}_{\forall k},\\{\tau_{k}\\}_{\forall k})$. Meanwhile, the
solution to block $\\{\mathbf{p}_{k}\\}_{\forall k}$ can be obtained based on
the Karush-Kuhn-Tucker (KKT) conditions. The detailed derivations can be done
following [28, 8].

When blocks $\\{\iota_{k}\\}_{\forall k}$, $\\{\tau_{k}\\}_{\forall k}$, and
$\\{\mathbf{p}_{k}\\}_{\forall k}$ are determined, the sub-problem for
designing $\\{\tilde{\bm{\theta}}_{l}\\}_{\forall l}$ can be formulated as

$\displaystyle\\{\tilde{\bm{\theta}}_{l}^{\star}\\}_{\forall l}$
$\displaystyle=\mathop{\arg~{}\max}_{\sum\limits_{l\in\mathcal{L}}\mathbf{\Theta}_{l,i}^{H}\mathbf{\Theta}_{l,i}=\mathbf{I}_{\bar{M}},\forall
i}\sum_{l\in\mathcal{L}}\Big{(}2\Re\\{\tilde{\bm{\theta}}_{l}^{T}\mathbf{v}_{l}\\}-\tilde{\bm{\theta}}_{l}^{T}\mathbf{V}_{l}\tilde{\bm{\theta}}_{l}^{*}\Big{)}$
(31)
$\displaystyle=\mathop{\arg~{}\max}_{\sum\limits_{l\in\mathcal{L}}\mathbf{\Theta}_{l,i}^{H}\mathbf{\Theta}_{l,i}=\mathbf{I}_{\bar{M}},\forall
i}\sum_{l\in\mathcal{L}}\Big{(}2\sum_{i\in\mathcal{G}_{2}}\Re\\{\bm{\theta}_{l,i}^{T}\mathbf{v}_{l,i}\\}$
$\displaystyle~{}~{}~{}~{}~{}~{}-\sum_{i\in\mathcal{G}_{2}}\sum_{i^{\prime}\in\mathcal{G}_{2}}\bm{\theta}_{l,i}^{T}\mathbf{V}_{l,i,i^{\prime}}\bm{\theta}_{l,i^{\prime}}^{*}\Big{)},$

where we define
$\mathbf{v}_{l}=\sum_{k\in\mathcal{K}_{l}}\sqrt{1+\iota_{k}}\tau_{k}^{*}\tilde{\mathbf{Q}}_{k}^{T}\mathbf{p}_{k}$,
$\mathbf{V}_{l}=\sum_{k\in\mathcal{K}_{l}}|\tau_{k}|^{2}\tilde{\mathbf{Q}}_{k}^{T}\sum_{k^{\prime}\in\mathcal{K}}\mathbf{p}_{k^{\prime}}\mathbf{p}_{k^{\prime}}^{H}\tilde{\mathbf{Q}}_{k}^{*}$,
$\forall l\in\mathcal{L}$,
$\bm{\theta}_{l,i}=\mathsf{vec}(\mathbf{\Theta}_{l,i}^{T})$,
$\mathbf{v}_{l,i}=[\mathbf{v}_{l}]_{\mathcal{I}_{i}}$, and
$\mathbf{V}_{l,i,i^{\prime}}=[\mathbf{V}_{l}]_{\mathcal{I}_{i},\mathcal{I}_{i^{\prime}}}$,
$\forall i,i^{\prime}\in\mathcal{G}_{2}$. Analogous to the derivation in
Section IV-A, we propose to split the objective with respect to each
$\\{\bm{\theta}_{l,i}\\}_{\forall l}$ from the objective in (31) with fixed
others. Defining
$\ddot{\bm{\chi}}_{i}=[\bm{\chi}_{1,i}^{T},\ldots,\bm{\chi}_{L,i}^{T}]^{T}$
with $\bm{\chi}_{l,i}=\mathbf{v}_{l,i}-\sum\nolimits_{i^{\prime}\neq
i}\mathbf{V}_{l,i,i^{\prime}}\bm{\theta}_{l,i^{\prime}}^{*}$, and
$\ddot{\mathbf{V}}_{i}=\mathsf{blkdiag}(\mathbf{V}_{1,i,i},\ldots,\mathbf{V}_{L,i,i})$,
we can construct the sub-problem for
$\ddot{\bm{\theta}}_{i}=\mathsf{vec}([\mathbf{\Theta}_{1,i}^{T},\ldots,\mathbf{\Theta}_{L,i}^{T}])$
as

$\displaystyle\max_{\ddot{\bm{\theta}}_{i}}~{}$ $\displaystyle
2\Re\\{\ddot{\bm{\theta}}_{i}^{T}\ddot{\bm{\chi}}_{i}\\}-\ddot{\bm{\theta}}_{i}^{T}\ddot{\mathbf{V}}_{i}\ddot{\bm{\theta}}_{i}^{*}$
(32a) s.t.
$\displaystyle\overline{\mathsf{vec}}^{*}(\ddot{\bm{\theta}}_{i})\overline{\mathsf{vec}}^{T}(\ddot{\bm{\theta}}_{i})=\mathbf{I}_{\bar{M}},$
(32b)

where constraint (32b) constructs a $\bar{M}^{2}L$-dimensional complex Stiefel
manifold. Therefore, problem (32) can be solved by the manifold based methods
[27].

For clarity, the above beamforming design algorithm is summarized in the
following steps:

  1. S1:

Initialize $\tilde{\bm{\theta}}_{l}$, $\forall l\in\mathcal{L}$, and
$\mathbf{p}_{k}$, $\forall k\in\mathcal{K}$.

  2. S2:

With fixed other terms, update $\\{\iota_{k}\\}_{\forall k}$ by solving
problem $\\{\iota_{k}^{\star}\\}_{\forall
k}=\arg\max~{}f(\\{\tilde{\bm{\theta}}_{l}\\}_{\forall
l},\\{\mathbf{p}_{k}\\}_{\forall k},\\{\iota_{k}\\}_{\forall
k},\\{\tau_{k}\\}_{\forall k})$.

  3. S3:

With fixed other terms, update $\\{\tau_{k}\\}_{\forall k}$ by solving problem
$\\{\tau_{k}^{\star}\\}_{\forall
k}=\arg\max~{}f(\\{\tilde{\bm{\theta}}_{l}\\}_{\forall
l},\\{\mathbf{p}_{k}\\}_{\forall k},\\{\iota_{k}\\}_{\forall
k},\\{\tau_{k}\\}_{\forall k})$.

  4. S4:

With fixed other terms, update $\\{\mathbf{p}_{k}\\}_{\forall k}$ by solving
the constrained convex optimization problem
$\\{\mathbf{p}_{k}^{\star}\\}_{\forall
k}=\mathop{\arg\max}\limits_{\sum_{k\in\mathcal{K}}\|\mathbf{p}_{k}\|_{2}^{2}\leq
P_{\mathrm{d}}}f(\\{\tilde{\bm{\theta}}_{l}\\}_{\forall
l},\\{\mathbf{p}_{k}\\}_{\forall k},\\{\iota_{k}\\}_{\forall
k},\\{\tau_{k}\\}_{\forall k})$.

  5. S5:

With fixed other terms, iteratively update $\ddot{\bm{\theta}}_{i}$, $\forall
i\in\mathcal{G}_{2}$, by solving problem (32) until the convergence of
$\\{\ddot{\bm{\theta}}_{i}\\}_{\forall i}$.

  6. S6:

Repeat S2-S5 until the convergence of the objective function
$f(\\{\tilde{\bm{\theta}}_{l}\\}_{\forall l},\\{\mathbf{p}_{k}\\}_{\forall
k},\\{\iota_{k}\\}_{\forall k},\\{\tau_{k}\\}_{\forall k})$ is guaranteed.

##  V Performance Evaluation

In this section, we perform simulation results to verify the effectiveness of
the proposed channel estimation scheme and evaluate the performance of the
proposed beamforming design.

###  V-A Simulation Setup

The simulation setup is illustrated as follows. Both BS-RIS and RIS-user
channels are assumed to have Rician fading with Rician factor $\kappa=0$ dB
accounting for the small-scale fading and distance-dependent path loss model
[9] accounting for the large-scale fading. Specifically, we assume each BD-RIS
antenna has an idealized radiation pattern [9], such that the path loss model
is expressed as666For the case of BD-RIS with reflective mode, we calculate
the path loss by setting the number of sectors as $L=2$, which indicates that
BD-RIS with reflective mode covers half of the space.
$\xi=\varrho(1-\cos\frac{\pi}{L})^{2}$, where
$\varrho=4^{3}\pi^{4}d_{1}^{\epsilon_{1}}d_{2}^{\epsilon_{2}}\lambda^{-4}G_{\mathrm{t}}^{-1}G_{\mathrm{r}}^{-1}$
is calculated by the BS-RIS distance $d_{1}=30$ m and RIS-user distance
$d_{2}=10$ m, path loss exponents $\epsilon_{1}=2.5$ and $\epsilon_{2}=2.5$,
wavelength $\lambda=\frac{c}{f}$ with $f=2.4$ GHz, antenna gains at the BS
side $G_{\mathrm{t}}=1$ and the user side $G_{\mathrm{r}}=1$. More details of
the analysis for the BD-RIS aided path loss model can be found in [9]. The
noise powers for uplink channel estimation and downlink transmission are set
as $\sigma^{2}=\sigma^{\prime 2}=-100$ dBm. In the following simulations, we
set the group size $\bar{M}$ of the group-connected architecture for BD-RIS as
$\bar{M}\in\\{1,2,4\\}$ with an affordable circuit complexity. More
comprehensive studies of different BD-RIS architectures and their circuit
complexities can be found in [10].

###  V-B Performance for Channel Estimation

We evaluate the channel estimation performance by plotting the normalized MSE
versus signal-to-noise ratio (SNR) with different group size of group-
connected BD-RIS and different numbers of BS/user antennas777Here we only plot
the case of BD-RIS with reflective mode since, as explained in Section III-E,
the same scheme can be used in BD-RIS with hybrid/multi-sector modes with the
same closed-form MSE performance. The effect of channel estimation error will
be studied in the performance evaluation of beamforming design.. Specifically,
the normalized MSE is calculated by
$\bar{\mathrm{e}}_{\widehat{\mathbf{q}}}=\mathbb{E}\\{\frac{\|\widehat{\mathbf{q}}-\mathbf{q}\|_{2}^{2}}{\|\mathbf{q}\|_{2}^{2}}\\}$.
For fair comparison, the SNR is defined as the ratio between the total power
received at each antenna from each transmit antenna through each RIS element
for the whole uplink training and the noise power, i.e.,
$\bar{\gamma}=\frac{P_{\mathrm{u}}T_{1}}{\xi\sigma^{2}}$. From Fig. 6, we have
the following observations. First, the matrix $\widehat{\mathbf{\Phi}}$ with
DFT base or Hadamard base to generate $\mathbf{Z}_{1}$, $\mathbf{Z}_{2}$, and
$\mathbf{X}$ achieves exactly the same performance, which verifies Theorem 1.
Second, the matrix $\widehat{\mathbf{\Phi}}$ with DFT/Hadamard bases achieves
exactly the same performance as the theoretical minimum, which verifies Lemma
2. Third, the normalized MSE performance increases with the group size
$\bar{M}$ of group-connected BD-RIS, which verifies Lemma 1. The intuition
behind is that the same amount of power is split to estimate more channel
coefficients, which leads to increasing channel estimation error. Fourth, the
normalized MSE decreases with SNR, which indicates that larger received SNR
leads to smaller channel estimation error. Fifth, the matrix
$\widehat{\mathbf{\Phi}}$ designed by the proposed scheme achieves better MSE
performance than that constructed by random unitary matrices, which
demonstrates the effectiveness of the proposed design for channel estimation.

Figure 6: Normalized MSE versus SNR ($M=32$, $\bar{G}=4$, $N=K=2$). Figure 7:
Spectral efficiency versus the number of BD-RIS elements
$M$($N=K=N_{\mathrm{s}}=2$, $P_{\mathrm{u}}=250$ mW,
$P_{\mathrm{d}}=KP_{\mathrm{u}}$).

###  V-C Performance for Beamforming Design

####  V-C.1 Reflective BD-RIS Aided Point-to-Point MIMO

We start by evaluating the performance of the proposed algorithm from Section
IV-A in Fig. 7. For comparison, we simulate the performance achieved by
perfect CSI $\mathbf{Q}$ as an upper bound. We also ignore the term
$1-\frac{T_{1}+T_{2}}{T}$ in the objective function (25a) for the ease of
analyzing the effect of beamforming design. From Fig. 7, we have the following
observations. First, the performance achieved based on both the perfect CSI
and the estimated CSI increases with increasing group size $\bar{M}$ of the
group-connected architecture, which aligns with the results in [5, 8, 7].
Second, with the fixed group size $\bar{M}$, increasing the tile size
$\bar{G}$ during the channel estimation induces the performance degradation
due to the reduction of CSI, which further limits the BD-RIS design. Third,
the performance gap between the perfect CSI case and the estimated CSI case
increases with increasing number of BD-RIS elements. This comes from the fact
that more BD-RIS elements provide more flexibility in beamforming design,
while the beamforming design performance is more sensitive to CSI conditions.

Figure 8: Sum-rate versus the total number of BD-RIS elements $ML$ ($N=K=4$,
$P_{\mathrm{u}}=250$ mW, $P_{\mathrm{d}}=KP_{\mathrm{u}}$).

####  V-C.2 Hybrid/Multi-Sector BD-RIS Aided MU-MISO

We next evaluate the performance of the proposed algorithm from Section IV-B
in Fig. 8, where the term $1-\frac{T_{1}+T_{2}}{T}$ in the objective function
(30a) is also ignored for the ease of evaluating the beamforming performance.
In addition, in the following simulations, we fix the total number of elements
$ML$ for BD-RIS with hybrid mode ($L=2$) and multi-sector mode ($L=4$). For
BD-RIS with hybrid and multi-sector modes, every $K/L$ users are located
within the coverage of each sector. From Fig. 8, we have the following
observations. First, for both perfect and estimated CSI cases, BD-RIS with
multi-sector mode outperforms BD-RIS with hybrid mode thanks to the use of
antennas with narrower beamwidth in the multi-sector mode, which aligns with
the results in [9]. Second, the performance achieved by BD-RIS with
hybrid/multi-sector modes increases with $\bar{M}$ due to the increasing beam
manipulation flexibility, which aligns with the results in [4].

###  V-D Overhead and Transmission Performance Trade-Off

To get more insights on the trade-off between the training overhead for
channel estimation and the performance for data transmission, we take into
account the term $1-\frac{T_{1}+T_{2}}{T}$ in the following simulations. For
simplicity, we ignore the symbol duration for processing and feedback, i.e.,
$T_{2}=0$, since the channel estimation overhead $T_{1}$ and the symbol
duration for data transmission $T_{3}$ are generally much larger than $T_{2}$.
In addition, we set the symbol duration for one transmission frame $T$ as
$T\in\\{600,1000,2000\\}$. In practice, the value of $T$ depends on multiple
factors, such as the carrier frequency of transmit signals, phase differences
in the multipath propagation, and mobility [29].

(a) Single-Connected ($\bar{M}=1$) (b) Group-Connected ($\bar{M}=2$)

(c) Group-Connected ($\bar{M}=4$)

Figure 9: Spectral efficiency versus tile size $\bar{G}$ ($M=64$,
$N=K=N_{\mathrm{s}}=2$, $P_{\mathrm{u}}=250$ mW,
$P_{\mathrm{d}}=KP_{\mathrm{u}}$).

####  V-D.1 Reflective BD-RIS Aided Point-to-Point MIMO

We first plot the spectral efficiency defined in (25a) versus the tile size
$\bar{G}$ for BD-RIS with different architectures in Fig. 9, from which we
have the following observations. First, for a relatively small symbol
duration, e.g., $T\in\\{600,1000\\}$, the spectral efficiency achieved by the
proposed design in general first increases and then decreases with the
increasing tile size $\bar{G}$. This is because a larger tile size leads to a
smaller training overhead $T_{1}$ but a reduced beam manipulation flexibility
of BD-RIS. For example, when $T=600$, the best spectral efficiency is achieved
at $\bar{G}=2$ for group-connected BD-RIS with $\bar{M}=2$, and at $\bar{G}=8$
with $\bar{M}=4$. In this case, the performance improvement by more flexible
BD-RIS design is overwhelmed by the large training overhead. Second, for a
relatively large symbol duration, e.g. $T=2000$, the spectral efficiency
achieved by the proposed design decreases with the increasing tile size. This
is because the large $T$ weakens the impact of training overhead, while the
beamforming performance keeps decreasing with increasing tile size. Third, for
a relatively small tile size, e.g. $\bar{G}\in\\{1,2\\}$, the spectral
efficiency achieved by BD-RIS with larger group size performs worse due to the
significant traning overhead in channel estimation. Foueth, for a relatively
large tile size, e.g. $\bar{G}\in\\{8,16\\}$, the spectral efficiency achieved
by BD-RIS with larger group size performs better thanks to the additional beam
manipulation flexibility supported by the group-connected architecture.
Therefore, the tile size $\bar{G}$ and $\bar{M}$ should be carefully chosen to
balance the training overhead for channel estimation, the beamforming
performance, and the circuit complexity for BD-RIS implementation.

(a) Single-Connected ($\bar{M}=1$) (b) Group-Connected ($\bar{M}=2$)

Figure 10: Sum-rate versus tile size $\bar{G}$ ($ML=128$, $N=K=4$,
$P_{\mathrm{u}}=250$ mW, $P_{\mathrm{d}}=KP_{\mathrm{u}}$).

####  V-D.2 Hybrid/Multi-Sector BD-RIS Aided MU-MISO

In Fig. 10, we plot the sum-rate defined in (30a) as a function of $\bar{G}$
for BD-RIS with hybrid/multi-sector modes. From Fig. 10 we observe that, in
addition to the findings obtained from Fig. 9, BD-RIS with multi-sector mode
is more robust to the symbol durations $T$ and the tile size. This is because
the significant performance enhancement provided by high-gain antennas
compensates for the effects of the training overhead and the beamforming
flexibility.

##  VI Conclusion

In this paper, we propose a novel channel estimation scheme for BD-RIS aided
multi-antenna systems. Specifically, the BD-RIS has different
(single/group/fully-connected) architectures and (reflective/hybrid/multi-
sector) modes, which, mathematically, generate unique constraints and
complicate the channel estimation. To tackle this difficulty, we propose an
efficient pilot sequence and BD-RIS design to estimate the cascaded BD-RIS
channel, which achieves the minimum MSE of the LS estimator. The analysis and
extensions of the proposed channel estimation scheme are also illustrated.

With the cascaded channel estimates, we consider two BD-RIS scenarios, that is
reflective BD-RIS aided point-to-point MIMO, and hybrid/multi-sector BD-RIS
aided MU-MISO, and propose efficient algorithms to design the transceiver and
BD-RIS with different architectures and modes.

Simulation results demonstrate the advantages of the proposed design.
Specifically, the effectiveness of the proposed channel estimation scheme and
beamforming algorithms are verified by comparing with the design for perfect
CSI cases. In addition, the trade-off between the training overhead for
channel estimation and the performance for data transmission is analyzed with
different parameter settings.

##  Appendix A Proof of Lemma 1

The objective (13a) has the following lower bound

$\displaystyle\mathsf{tr}((\widehat{\mathbf{\Phi}}^{H}\widehat{\mathbf{\Phi}})^{-1})$
(33)
$\displaystyle=\sum_{i=1}^{NK\bar{M}^{2}G_{2}}[(\widehat{\mathbf{\Phi}}^{H}\widehat{\mathbf{\Phi}})^{-1}]_{i,i}\overset{\text{(a)}}{\geq}\sum_{i=1}^{NK\bar{M}^{2}G_{2}}[\widehat{\mathbf{\Phi}}^{H}\widehat{\mathbf{\Phi}}]_{i,i}^{-1}$
$\displaystyle\overset{\text{(b)}}{\geq}\frac{N^{2}K^{2}G_{2}^{2}\bar{M}^{4}}{\sum_{i=1}^{NK\bar{M}^{2}G_{2}}[\widehat{\mathbf{\Phi}}^{H}\widehat{\mathbf{\Phi}}]_{i,i}}=\frac{N^{2}K^{2}G_{2}^{2}\bar{M}^{4}}{\mathsf{tr}(\widehat{\mathbf{\Phi}}^{H}\widehat{\mathbf{\Phi}})}$
$\displaystyle=\frac{N^{2}K^{2}G_{2}^{2}\bar{M}^{4}}{\mathsf{tr}(\sum_{t=1}^{K\bar{M}^{2}G_{2}}(\bar{\bm{\phi}}_{t}^{*}\otimes\bar{\mathbf{x}}_{t}^{*}\otimes\mathbf{I}_{N})(\bar{\bm{\phi}}_{t}^{T}\otimes\bar{\mathbf{x}}_{t}^{T}\otimes\mathbf{I}_{N}))}$
$\displaystyle\overset{\text{(c)}}{=}\frac{N^{2}K^{2}G_{2}^{2}\bar{M}^{4}}{\mathsf{tr}(\sum_{t=1}^{K\bar{M}^{2}G_{2}}(\bar{\bm{\phi}}_{t}^{*}\bar{\bm{\phi}}_{t}^{T})\otimes(\bar{\mathbf{x}}_{t}^{*}\bar{\mathbf{x}}_{t}^{T})\otimes\mathbf{I}_{N})}$
$\displaystyle\overset{\text{(d)}}{=}\frac{NKG_{2}^{2}\bar{M}^{4}}{\sum_{t=1}^{K\bar{M}^{2}G_{2}}\mathsf{tr}(\bar{\bm{\phi}}_{t}^{*}\bar{\bm{\phi}}_{t}^{T})}\overset{\text{(e)}}{=}N\bar{M},$

where the equality of (a) can be attained when
$\widehat{\mathbf{\Phi}}^{H}\widehat{\mathbf{\Phi}}$ is a diagonal matrix
[30]; (b) holds due to the relationship between the harmonic mean and the
arithmetic mean with equality achieved when
$\widehat{\mathbf{\Phi}}^{H}\widehat{\mathbf{\Phi}}$ has identical diagonal
entries; (c) holds by the property of the Kronecker product; (d) holds due to
the property of the trace operation and the assumption that
$|[\mathbf{x}_{t}]_{k}|=1$, $\forall k$, yielding
$\mathsf{tr}(\mathbf{x}_{t}^{*}\mathbf{x}_{t}^{T})=K$, $\forall t$; (e) holds
due to the constraint (13b) which yields
$\mathsf{tr}(\bar{\bm{\phi}}_{t}^{*}\bar{\bm{\phi}}_{t}^{T})=G_{2}\bar{M}$,
$\forall t$. Based on the above derivations, we have that
$\min\mathsf{tr}((\widehat{\mathbf{\Phi}}^{H}\widehat{\mathbf{\Phi}})^{-1})=N\bar{M}$
with the condition
$\widehat{\mathbf{\Phi}}^{H}\widehat{\mathbf{\Phi}}=\widehat{\mathbf{\Phi}}\widehat{\mathbf{\Phi}}^{H}=K\bar{M}G_{2}\mathbf{I}_{NK\bar{M}^{2}G_{2}}$,
which completes the proof.

##  Appendix B Proof of Lemma 2

With $\mathbf{A}$ satisfying (18b), (18c) and $\breve{\mathbf{\Phi}}$
satisfying (19b), (19c), we have
$\mathbf{\Phi}=\mathbf{A}\otimes\breve{\mathbf{\Phi}}$ such that

$\displaystyle\mathbf{\Phi}^{H}\mathbf{\Phi}$
$\displaystyle=(\mathbf{A}^{H}\otimes\breve{\mathbf{\Phi}}^{H})(\mathbf{A}\otimes\breve{\mathbf{\Phi}})=(\mathbf{A}^{H}\mathbf{A})\otimes(\breve{\mathbf{\Phi}}^{H}\breve{\mathbf{\Phi}})$
(34)
$\displaystyle=(G_{2}\mathbf{I}_{G_{2}})\otimes(\bar{M}\mathbf{I}_{\bar{M}^{2}})=G_{2}\bar{M}\mathbf{I}_{G_{2}\bar{M}^{2}},$

which aligns with (14b).

In addition, we define the $i$-th sub-block rows of the $s$-th row of
$\mathbf{\Phi}$ by $\bm{\phi}_{s,i}$, $\forall s\in\mathcal{S}$, $\forall
i\in\mathcal{G}_{2}$, i.e.,

$\bm{\phi}_{s,i}=[\mathbf{\Phi}]_{s,(i-1)\bar{M}^{2}:i\bar{M}^{2}}=[\bm{\phi}_{s}^{T}]_{(i-1)\bar{M}^{2}:i\bar{M}^{2}}.$
(35)

Meanwhile, from (15) we have

$\bm{\phi}_{s,i}=[\bar{\bm{\phi}}_{t}^{T}]_{(i-1)\bar{M}^{2}:i\bar{M}^{2}}=\mathsf{vec}^{T}(\mathbf{\Phi}_{t,i}),$
(36)

where $t=(s-1)K+k$, $\forall k\in\mathcal{K}$. Furthermore, from Lemma 2 we
have
$\bm{\phi}_{s,i}=[\mathbf{A}]_{i^{\prime},i}[\breve{\mathbf{\Phi}}]_{m,:}$,
where $s=(i^{\prime}-1)\bar{M}^{2}+m$, $\forall m\in\bar{\bar{\mathcal{M}}}$.
Hence, (13b) is satisfied, i.e.,

$\displaystyle\mathbf{\Phi}_{t,i}^{H}\mathbf{\Phi}_{t,i}$
$\displaystyle=|[\mathbf{A}]_{i^{\prime},i}|^{2}\overline{\mathsf{vec}}^{H}([\breve{\mathbf{\Phi}}^{T}]_{:,m})\overline{\mathsf{vec}}([\breve{\mathbf{\Phi}}^{T}]_{:,m})=\mathbf{I}_{\bar{M}}.$
(37)

The proof is completed.

##  Appendix C Proof of Theorem 1

We start by proving $\breve{\bm{\phi}}_{m,n}$, $\forall
m,n\in\bar{\mathcal{M}}$ satisfies (19c). To this end, we define
$\breve{\bm{\phi}}_{m,n,i}=[\breve{\bm{\phi}}_{m,n}]_{(i-1)\bar{M}+1:i\bar{M}}$,
$\forall i\in\bar{\mathcal{M}}$. We have

$\breve{\bm{\phi}}_{m,n,i}=[\mathbf{Z}_{2}]_{m,i}[\mathbf{Z}_{1}^{T}]_{\mathsf{mod}(i-n,\bar{M})+1,:}.$
(38)

Then we calculate

$\displaystyle\breve{\bm{\phi}}_{m,n,i}^{*}\breve{\bm{\phi}}_{m,n,i^{\prime}}^{T}=$
$\displaystyle\underbrace{[\mathbf{Z}_{1}^{H}]_{\mathsf{mod}(i-n,\bar{M})+1,:}[\mathbf{Z}_{1}]_{:,\mathsf{mod}(i^{\prime}-n,\bar{M})+1}}_{=z_{1,i.i^{\prime},n}}$
(39)
$\displaystyle\times\underbrace{[\mathbf{Z}_{2}]_{m,i}^{*}[\mathbf{Z}_{2}]_{m,i^{\prime}}}_{=z_{2,i,i^{\prime},m}},$

yielding the following two conditions:

  1. 1)

when $i=i^{\prime}$, we have $z_{1,i,i,n}=\alpha_{1}$ and
$z_{2,i,i,m}=\frac{\alpha_{2}}{\bar{M}}$ such that
$\breve{\bm{\phi}}_{m,n,i}^{*}\breve{\bm{\phi}}_{m,n,i^{\prime}}^{T}=1$;

  2. 2)

when $i\neq i^{\prime}$, we have $z_{1,i,i^{\prime},n}=0$ such that
$\breve{\bm{\phi}}_{m,n,i}^{*}\times\breve{\bm{\phi}}_{m,n,i^{\prime}}^{T}=0$.

Therefore, (19c) is guaranteed.

We next prove $\breve{\mathbf{\Phi}}$ constructed by $\breve{\bm{\phi}}_{m,n}$
satisfies (19b). To this end, we calculate

$\displaystyle\breve{\bm{\phi}}_{m,n}^{*}\breve{\bm{\phi}}_{m^{\prime},n^{\prime}}^{T}=$
$\displaystyle\sum_{i\in\bar{\mathcal{M}}}\breve{\bm{\phi}}_{m,n,i}^{*}\breve{\bm{\phi}}_{m^{\prime},n^{\prime},i}^{T}$
(40) $\displaystyle=$
$\displaystyle\sum_{i\in\bar{\mathcal{M}}}\underbrace{[\mathbf{Z}_{1}^{H}]_{\mathsf{mod}(i-n,\bar{M})+1,:}[\mathbf{Z}_{1}]_{:,\mathsf{mod}(i-n^{\prime},\bar{M})+1}}_{=z_{1^{\prime},n,n^{\prime},i}}$
$\displaystyle~{}~{}~{}~{}~{}\times\underbrace{[\mathbf{Z}_{2}]_{m,i}^{*}[\mathbf{Z}_{2}]_{m^{\prime},i}}_{=z_{2^{\prime},m,m^{\prime},i}},$

yielding the following three conditions:

  1. 1)

when $m=m^{\prime}$ and $n=n^{\prime}$, we have
$z_{1^{\prime},n,n,i}=\alpha_{1}$ and
$z_{2^{\prime},m,m,i}=\frac{\alpha_{2}}{\bar{M}}$ such that
$\breve{\bm{\phi}}_{m,n}^{*}\breve{\bm{\phi}}_{m,n}^{T}=\bar{M}$;

  2. 2)

when $n\neq n^{\prime}$, we have $z_{1^{\prime},n,n^{\prime},i}=0$ such that
$\breve{\bm{\phi}}_{m,n}^{*}\times\breve{\bm{\phi}}_{m^{\prime},n^{\prime}}^{T}=0$;

  3. 3)

when $m\neq m^{\prime}$ and $n=n^{\prime}$, we have
$z_{1^{\prime},n,n,i}=\alpha_{1}$ such that
$\breve{\bm{\phi}}_{m,n}^{*}\breve{\bm{\phi}}_{m^{\prime},n^{\prime}}^{T}=\alpha_{1}\sum_{i\in\bar{\mathcal{M}}}z_{2^{\prime},m,m^{\prime},i}=\alpha_{1}[\mathbf{Z}_{2}^{*}]_{m,:}[\mathbf{Z}_{2}^{T}]_{:,m^{\prime}}=0$.

Therefore, (19b) is guaranteed. The proof is completed.

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