[acronym]long-short

# Channel Shaping Using Beyond Diagonal Reconfigurable Intelligent Surface:
Analysis, Optimization, and Enhanced Flexibility

Yang Zhao, Hongyu Li,
Massimo Franceschetti, and Bruno Clerckx  Yang Zhao, Hongyu Li, and Bruno
Clerckx are with the Department of Electrical and Electronic Engineering,
Imperial College London, London SW7 2AZ, U.K. (e-mail: {yang.zhao18, c.li21,
b.clerckx}@imperial.ac.uk). Massimo Franceschetti is with the Department of
Electrical and Computer Engineering, University of California, San Diego, La
Jolla CA 92093, USA (e-mail: massimo@ece.ucsd.edu). This work has been
partially supported by UKRI grant EP/Y004086/1, EP/X040569/1, EP/Y037197/1,
EP/X04047X/1, EP/Y037243/1.

###### Abstract

This paper investigates the capability of a passive Reconfigurable Intelligent
Surface (RIS) to redistribute the singular values of point-to-point Multiple-
Input Multiple-Output (MIMO) channels for achieving power and rate gains. We
depart from the conventional Diagonal (D)-RIS with diagonal phase shift matrix
and adopt a Beyond Diagonal (BD) architecture that offers greater wave
manipulation flexibility through element-wise connections. Specifically, we
first provide shaping insights by characterizing the channel singular value
regions attainable by D-RIS and BD-RIS via a novel geodesic optimization.
Analytical singular value bounds are then derived to explore their shaping
limits in typical deployment scenarios. As a side product, we tackle BD-RIS-
aided MIMO rate maximization problem by a local-optimal Alternating
Optimization (AO) and a shaping-inspired low-complexity approach. Results show
that compared to D-RIS, BD-RIS significantly improves the dynamic range of all
channel singular values, the trade-off in manipulating them, and thus the
channel power and achievable rate. Those observations become more pronounced
when the number of RIS elements and MIMO dimensions increase. Of particular
interest, BD-RIS is shown to activate multi-stream transmission at lower
transmit power than D-RIS, hence achieving the asymptotic Degrees of Freedom
(DoF) at low Signal-to-Noise Ratio (SNR) thanks to its higher flexibility of
shaping the distribution of channel singular values.

###### Index Terms:

Reconfigurable intelligent surface, channel singular value redistribution,
rate maximization, manifold optimization.

##  I Introduction

###  I-A Background

Today we are witnessing a paradigm shift from connectivity to intelligence,
where the wireless environment is no longer a chaotic medium but a conscious
agent that can serve on demand. This is empowered by recent advances in
Reconfigurable Intelligent Surface (RIS), a programmable passive planar
surface that recycles and redistributes ambient electromagnetic waves for
improved wireless performance. A typical RIS consists of numerous low-power
sub-wavelength non-resonant scattering elements, whose response can be
engineered in real-time to manipulate the amplitude, phase, frequency, and
polarization of the scattered waves [1]. It enables low-noise full-duplex
operation, featuring better flexibility than reflectarrays, lighter footprint
than relays, and greater scalability than Multiple-Input Multiple-Output
(MIMO) systems. One popular RIS research direction is _joint passive and
active beamforming_ design with transceivers to enhance a specific performance
measure, which has attracted significant interests in wireless communication
[2, 3, 4], sensing [5, 6, 7], and power transfer literature [8, 9, 10]. While
passive beamforming at RIS suffers attenuation from double fading, it offers
better asymptotic behaviors than active beamforming at transceivers (e.g.,
second-order array gain and fourth-order harvested power [10]). Another RIS
application is _information modulation_ by periodically switching its
reflection pattern within the channel coherence time. This creates a free-ride
message stream with dual benefits: integrating with legacy transmitter for
enhanced channel capacity [11, 12, 13], or serving as individual source for
low-power uplink communication [14, 15, 16]. Different from above, _channel
shaping_ exploits RIS as a stand-alone device to modify the inherent
properties of the wireless environment, for example, compensate for the
Doppler effect [17], flatten frequency-selective channels [18], improve MIMO
channel rank [19], and artificially diversify channel over time for orthogonal
[20] and non-orthogonal [21] multiple access schemes. This helps to decouple
joint beamforming problems into a channel shaping stage and a legacy
transceiver design stage, providing a versatile solution for various wireless
applications.

###  I-B Related Works

At a specific resource block, channel shaping metrics can be classified into
two categories:

  * •

_Singular value:_ The impact of RIS has been studied in terms of minimum
singular value [22], effective rank [22, 23], condition number [24, 25], and
degree of freedom [26, 27, 28]. Those are closely related to performance
measures (e.g., achievable rate and harvested power [29]) but sensitive to
minor perturbations of the channel matrix.

  * •

_Power:_ The impact of RIS has been studied in terms of channel power gain [2,
30, 31, 32, 33] in point-to-point channels and leakage interference [34] in
interference channels. Those second-order metrics are less informative in MIMO
but easier to analyze and optimize.

Although above works offered inspiring glimpses into the channel shaping
potential of passive RIS, none attempted to disclose the entire attainable
channel singular value region. Most relevant literature [22, 23, 24, 25, 26,
27, 28, 2, 34] have also been limited to a Diagonal (D)-RIS model where each
element is connected to a dedicated impedance and disconnected from others. As
such, wave impinging on one element is entirely scattered by the same element.
This simple architecture is modeled by a diagonal scattering matrix with unit-
magnitude diagonal entries, which only applies a phase shift to the incoming
signal. The idea was soon generalized to Beyond Diagonal (BD)-RIS with group-
connected architecture [30], where adjacent elements within the same group are
connected via passive reconfigurable components111Those components can be
either symmetric (e.g., capacitors and inductors) or asymmetric (e.g., ring
hybrids and branch-line hybrids) [35], resulting in symmetric and asymmetric
scattering matrices, respectively.. This allows wave impinging on one element
to propagate within the circuit and depart partially from any element in the
same group. It can thus manipulate both amplitude and phase of the scattered
wave while remaining passive. Such a powerful model can be realized at reduced
hardware cost using tree- and forest-connected architectures by graph theory
[32]. BD-RIS can also function in multi-sector mode [36] for full-space
coverage and multi-user support. Practical challenges such as channel
estimation [37], mutual coupling [38], and wideband modelling [39] have also
been studied in recent literature. Its beamforming superiority over D-RIS has
been proved in Single-Input Single-Output (SISO) and Multiple-Input Single-
Output (MISO) systems [30, 31, 32, 33, 40, 41, 36, 42], however, the interplay
between BD-RIS and MIMO is still in the infancy stage. The authors of [43]
investigated the rate-optimal joint beamforming design for a fully-connected
BD-RIS-aided MIMO system where the direct link is blocked. A transmitter-side
BD-RIS was introduced to massive MIMO systems that exploits statistical
Channel State Information (CSI) for improved spectral efficiency [44], which
again assumed blocked direct channel and fully-connected BD-RIS. Received
power maximization with continuous-valued and discrete-valued BD-RIS have been
tackled respectively in closed form [31] and by machine learning approach
[45], but the corresponding single-stream transceiver is rate-suboptimal.

###  I-C Contributions

This paper is motivated by a fundamental question: _What is the singular value
(and power gain) shaping capability of a passive RIS in point-to-point MIMO
channels?_ We aim for a comprehensive answer via analysis and optimization.
The contributions are summarized below.

First, we pioneer BD-RIS study in general MIMO channels and interpret its
shaping potential as channel rearrangement and space alignment. Channel
rearrangement refers to rearranging and recombining the entries of backward
(i.e., RIS-receiver) and forward (i.e., transmitter-RIS) channel matrices by
their strength. Space alignment generalizes the phase matching of direct and
indirect channels in SISO and MISO to the singular vector space in MIMO. The
former is uniquely attributed to the off-diagonal entries of the scattering
matrix of BD-RIS.

Second, we propose a novel BD-RIS design method that allows reshaping of the
available channels through singular values manipulation. Our Riemannian
Conjugate Gradient (RCG) method compares favorably with respect to existing
ones in that the updates are along the geodesics (i.e., the shortest path
between two points in a Riemannian manifold) of the feasible domain to
accelerate convergence. It also works for general design problems of group-
connected BD-RIS.

Third, we provide a numerical answer to the shaping question by characterizing
the Pareto frontiers of channel singular values. The enclosed region
generalizes most relevant metrics and provides an intuitive shaping benchmark.
Results show that increasing BD-RIS group size enlarges this region, improving
the dynamic range of all singular values and the trade-off in manipulating
them.

Fourth, we provide an analytical answer to the shaping question in typical
deployment scenarios. For rank-deficient forward/backward channel, we derive
singular value bounds applying to D\- and BD-RIS with asymptotically large
number of elements. For blocked direct channel, we derive singular value
bounds applying to fully-connected BD-RIS with arbitrary number of elements.
Those bounds are validated by comparing with the numerical results above.
Results show that for a fixed number of elements, BD-RIS can approach the
asymptotic bounds better than D-RIS.

Fifth, we tackle BD-RIS-aided MIMO rate maximization problem by a local-
optimal Alternating Optimization (AO) and a shaping-inspired low-complexity
approach. The former iteratively updates active beamforming by eigenmode
transmission and passive beamforming by geodesic RCG, until convergence. The
latter exploits the BD-RIS to shape the channel for maximum power gain then
performs eigenmode transmission. Interestingly, the rate deficit from the
shaping-inspired approach diminishes as the BD-RIS evolves towards fully-
connected. We conclude that: 1) channel shaping decouples joint beamforming
for reduced design complexity; 2) the power and rate gains of BD-RIS over
D-RIS increase with the number of scattering elements and MIMO dimensions; 3)
at low transmit power, BD-RIS can activate more streams than D-RIS and achieve
the asymptotic Degrees of Freedom (DoF), thanks to its higher flexibility of
shaping the distribution of channel singular values.

_Notation:_ Italic, bold lower-case, and bold upper-case letters indicate
scalars, vectors and matrices, respectively. $\jmath$ denotes the imaginary
unit. $\mathbb{C}$ represents the set of complex numbers. $\mathbb{H}^{n\times
n}$ and $\mathbb{U}^{n\times n}$ denotes the set of $n\times n$ Hermitian and
unitary matrices, respectively. $\mathbf{0}$ and $\mathbf{I}$ are the all-zero
and identity matrices with appropriate size, respectively. $\Re\\{\cdot\\}$
takes the real part of a complex number. $\mathrm{arg}(\cdot)$ gives the
argument of a complex number. $\mathrm{tr}(\cdot)$ and $\det(\cdot)$ evaluates
the trace and determinant of a square matrix, respectively.
$\mathrm{diag}(\cdot)$ constructs a square matrix with arguments on the main
(block) diagonal and zeros elsewhere. $\mathrm{sv}(\cdot)$ returns the
singular value vector. $\sigma_{n}(\cdot)$ and $\lambda_{n}(\cdot)$ is the
$n$-th largest singular value and eigenvalue, respectively. $(\cdot)^{*}$,
$(\cdot)^{\mathsf{T}}$, $(\cdot)^{\mathsf{H}}$, $(\cdot)^{\dagger}$
$(\cdot)^{(r)}$, $(\cdot)^{\star}$ denote the conjugate, transpose, conjugate
transpose (Hermitian), Moore-Penrose inverse, $r$-th iterated point, and
stationary point, respectively. $(\cdot)_{[x:y]}$ is a shortcut for
$(\cdot)_{x},(\cdot)_{x+1},\ldots,(\cdot)_{y}$. $\lvert\cdot\rvert$,
$\lVert\cdot\rVert$, and $\lVert\cdot\rVert_{\mathrm{F}}$ denote the absolute
value, Euclidean norm, and Frobenius norm, respectively. $\odot$ represents
the element-wise (Hadamard) product.
$\mathcal{CN}(\mathbf{0},\mathbf{\Sigma})$ is the multivariate Circularly
Symmetric Complex Gaussian (CSCG) distribution with mean $\mathbf{0}$ and
covariance $\mathbf{\Sigma}$. $\sim$ means “distributed as”.

##  II bd-ris Model

Consider a BD-RIS aided point-to-point MIMO system with $N_{\mathrm{T}}$ and
$N_{\mathrm{R}}$ transmit and receive antennas, respectively, and
$N_{\mathrm{S}}$ scattering elements at the BD-RIS. This configuration is
denoted as $N_{\mathrm{T}}\times N_{\mathrm{S}}\times N_{\mathrm{R}}$ in the
following context. The BD-RIS can be modeled as an $N_{\mathrm{S}}$-port
network [46] that further divides into $G$ individual groups, each containing
$L\triangleq N_{\mathrm{S}}/G$ elements interconnected by real-time
reconfigurable components [30]. To simplify the analysis, we assume a lossless
asymmetric network without mutual coupling between scattering elements, as
have been previously considered in [47, 36, 43]. The overall scattering matrix
of the BD-RIS is block-diagonal with unitary diagonal blocks222Following
footnote 1, we do not limit the scattering matrix to be symmetric. One can
enforce symmetry over the result by
$\mathbf{\Theta}\leftarrow(\mathbf{\Theta}+\mathbf{\Theta}^{\mathsf{T}})/2$ if
necessary.

$\mathbf{\Theta}=\mathrm{diag}(\mathbf{\Theta}_{1},\ldots,\mathbf{\Theta}_{G}),$
(1)

where $\mathbf{\Theta}_{g}\in\mathbb{U}^{L\times L}$ is the $g$-th unitary
block (i.e., $\mathbf{\Theta}_{g}^{\mathsf{H}}\mathbf{\Theta}_{g}=\mathbf{I}$)
that describes the response of group
$g\in\mathcal{G}\triangleq\\{1,\ldots,G\\}$. Note that D-RIS can be regarded
as its extreme case with group size $L=1$. Some potential physical
architectures of BD-RIS are illustrated in [30, Fig. 3], [36, Fig. 5], and
[32, Fig. 2], where the circuit topology has been modeled in the scattering
matrix. Let $\mathbf{H}_{\mathrm{D}}\in\mathbb{C}^{N_{\mathrm{R}}\times
N_{\mathrm{T}}}$, $\mathbf{H}_{\mathrm{B}}\in\mathbb{C}^{N_{\mathrm{R}}\times
N_{\mathrm{S}}}$, $\mathbf{H}_{\mathrm{F}}\in\mathbb{C}^{N_{\mathrm{S}}\times
N_{\mathrm{T}}}$ denote the direct (i.e., transmitter-receiver), backward
(i.e., RIS-receiver), and forward (i.e., transmitter-RIS) channels,
respectively. The equivalent channel is a function of the scattering matrix

$\mathbf{H}=\mathbf{H}_{\mathrm{D}}+\mathbf{H}_{\mathrm{B}}\mathbf{\Theta}\mathbf{H}_{\mathrm{F}}=\mathbf{H}_{\mathrm{D}}+\sum_{g}\underbrace{\mathbf{H}_{\mathrm{B},g}\mathbf{\Theta}_{g}\mathbf{H}_{\mathrm{F},g}}_{\triangleq\mathbf{H}_{g}},$
(2)

where $\mathbf{H}_{\mathrm{B},g}\in\mathbb{C}^{N_{\mathrm{R}}\times L}$ and
$\mathbf{H}_{\mathrm{F},g}\in\mathbb{C}^{L\times N_{\mathrm{T}}}$ are the
backward and forward channels associated with BD-RIS group $g$, corresponding
to the $(g{-}1)L{+}1$ to $gL$ columns of $\mathbf{H}_{\mathrm{B}}$ and rows of
$\mathbf{H}_{\mathrm{F}}$, respectively. Let
$\mathbf{H}_{g}\triangleq\mathbf{H}_{\mathrm{B},g}\mathbf{\Theta}_{g}\mathbf{H}_{\mathrm{F},g}$
be the indirect channel via BD-RIS group $g$. Since unitary matrices
constitute an algebraic group with respect to multiplication, the scattering
matrix of group $g$ can be decomposed as

$\mathbf{\Theta}_{g}=\mathbf{L}_{g}\mathbf{R}_{g}^{\mathsf{H}},$ (3)

where $\mathbf{L}_{g},\mathbf{R}_{g}\in\mathbb{U}^{L\times L}$ are two unitary
factor matrices. Let
$\mathbf{H}_{\mathrm{B},g}=\mathbf{U}_{\mathrm{B},g}\mathbf{\Sigma}_{\mathrm{B},g}\mathbf{V}_{\mathrm{B},g}^{\mathsf{H}}$
and
$\mathbf{H}_{\mathrm{F},g}=\mathbf{U}_{\mathrm{F},g}\mathbf{\Sigma}_{\mathrm{F},g}\mathbf{V}_{\mathrm{F},g}^{\mathsf{H}}$
be the compact Singular Value Decomposition (SVD) of the backward and forward
channels, respectively. The equivalent channel can thus be rewritten as

$\mathbf{H}=\overbrace{\mathbf{H}_{\mathrm{D}}+\sum_{g}\mathbf{U}_{\mathrm{B},g}\mathbf{\Sigma}_{\mathrm{B},g}\underbrace{\mathbf{V}_{\mathrm{B},g}^{\mathsf{H}}\mathbf{L}_{g}\mathbf{R}_{g}^{\mathsf{H}}\mathbf{U}_{\mathrm{F},g}}_{\text{backward-
forward}}\mathbf{\Sigma}_{\mathrm{F},g}\mathbf{V}_{\mathrm{F},g}^{\mathsf{H}}}^{\text{direct-
indirect}}.$ (4)

By analyzing (4), we conclude that the off-diagonal entries of the BD-RIS
scattering matrix provide two key potentials for MIMO channel shaping:

  * •

_Channel rearrangement:_ It refers to rearranging and recombining the backward
and forward channel branches (i.e., entries of $\mathbf{H}_{\mathrm{B}}$ and
$\mathbf{H}_{\mathrm{F}}$) by their strength. In SISO 333Following previous
notation, here we denote the direct channel as $h_{\mathrm{D}}$, the backward
and forward channels as
$\mathbf{h}_{\mathrm{B}}\in\mathbb{C}^{N_{\mathrm{S}}\times 1}$ and
$\mathbf{h}_{\mathrm{F}}^{\mathsf{H}}\in\mathbb{C}^{1\times N_{\mathrm{S}}}$,
whose $n$-th entries are $h_{\mathrm{B},n}$ and $h_{\mathrm{F},n}$,
respectively. The channels associated with BD-RIS group $g$ are also denoted
as $\mathbf{h}_{\mathrm{B},g}\in\mathbb{C}^{L\times 1}$ and
$\mathbf{h}_{\mathrm{F},g}^{\mathsf{H}}\in\mathbb{C}^{1\times L}$, whose
$l$-th entries are $h_{\mathrm{B},g,l}$ and $h_{\mathrm{F},g,l}$,
respectively., D-RIS with perfect phase matching can provide a maximum
indirect channel gain $g_{\mathrm{D}}=\sum_{n=1}^{N_{\mathrm{S}}}\lvert
h_{\mathrm{B},n}\rvert\lvert h_{\mathrm{F},n}\rvert$, while BD-RIS can exploit
in-group connections and generalize it to
$g_{\mathrm{BD}}=\sum_{g=1}^{G}\sum_{l=1}^{L}\lvert
h_{\mathrm{B},g,\pi_{\mathrm{B},g}(l)}\rvert\lvert
h_{\mathrm{F},g,\pi_{\mathrm{F},g}(l)}\rvert$, where $\pi_{\mathrm{B},g}$ and
$\pi_{\mathrm{F},g}$ are arbitrary permutations of
$\mathcal{L}\triangleq\\{1,\ldots,L\\}$. Note the first summation is over
groups and the second summation is over permuted channels. By rearrangement
inequality [48, Inequality 10.2.1], for BD-RIS we have

$g_{\mathrm{BD}}\leq\sum_{g=1}^{G}\sum_{l=1}^{L}\lvert
h_{\mathrm{B},g,\pi^{\downarrow}_{\mathrm{B},g}(l)}\rvert\lvert
h_{\mathrm{F},g,\pi^{\downarrow}_{\mathrm{F},g}(l)}\rvert,$ (5)

where $\pi^{\downarrow}_{\mathrm{B},g}$ and $\pi^{\downarrow}_{\mathrm{F},g}$
sort $\\{\lvert h_{\mathrm{B},g,l}\rvert\\}_{l=1}^{L}$ and $\\{\lvert
h_{\mathrm{F},g,l}\rvert\\}_{l=1}^{L}$ in descending orders, respectively.
That is to say, when the direct channel is blocked, the maximum SISO channel
gain is attained if the BD-RIS pairs the $l$-th strongest backward and forward
branches within each group. Since the number of channels associated with each
group is proportional to $N_{\mathrm{T}}N_{\mathrm{R}}$, we conclude that the
advantage of BD-RIS in channel rearrangement scales with MIMO dimensions.

  * •

_Space alignment:_ It refers to aligning the singular vector spaces of the
direct, forward, and backward channels. The BD-RIS needs to strike a balance
between the space alignment of backward-forward (intra-group, multiplicative)
channels and direct-indirect (inter-group, additive) channels. In SISO,
singular vectors reduce to scalars and space alignment reduce to phase
matching, such that the scattering matrix of group $g$ maximizing the channel
gain is

$\mathbf{\Theta}_{g}^{\star}=\exp\bigl{(}\jmath\mathrm{arg}(h_{\mathrm{D}})\bigr{)}\mathbf{V}_{\mathrm{B},g}\mathbf{U}_{\mathrm{F},g}^{\mathsf{H}},$
(6)

where
$\mathbf{V}_{\mathrm{B},g}=\bigl{[}\mathbf{h}_{\mathrm{B},g}/\lVert\mathbf{h}_{\mathrm{B},g}\rVert,\mathbf{N}_{\mathrm{B},g}\bigr{]}\in\mathbb{U}^{L\times
L}$,
$\mathbf{U}_{\mathrm{F},g}=\bigl{[}\mathbf{h}_{\mathrm{F},g}/\lVert\mathbf{h}_{\mathrm{F},g}\rVert,\mathbf{N}_{\mathrm{F},g}\bigr{]}\in\mathbb{U}^{L\times
L}$, and
$\mathbf{N}_{\mathrm{B},g},\mathbf{N}_{\mathrm{F},g}\in\mathbb{C}^{L\times(L-1)}$
are the orthonormal bases of the null spaces of $\mathbf{h}_{\mathrm{B},g}$
and $\mathbf{h}_{\mathrm{F},g}$, respectively. We notice that D-RIS (i.e.,
$L=1$) suffices for perfect phase matching in SISO. When it comes to MIMO,
each D-RIS element can only apply a scalar phase shift to the corresponding
indirect channel matrix of size $N_{\mathrm{R}}\times N_{\mathrm{T}}$ defined
in (2). We thus conclude that the disadvantage of D-RIS in space alignment
deteriorates with MIMO dimensions.

##  III bd-ris Design Frameworks

This section aims for an efficient yet universal optimization framework for
group-connected BD-RIS design. Most relevant optimization problems can be
formulated as Θf(Θ) Θ_g^H Θ_g=I, ∀g, where the objective function
$f(\mathbf{\Theta})$ can be any function of the BD-RIS scattering matrix, for
example, channel singular value (covered in Section IV), channel power gain
(covered in Section V), and achievable rate (covered in Section V). The
feasible domain of each group is a $L$-dimensional Stiefel manifold
$\mathbf{\Theta}_{g}\in\mathbb{U}^{L\times L}$ that is non-convex and non-
Euclidean. Therefore, most BD-RIS optimization problems are solved by _relax-
then-project_ methods [40] or _general manifold RCG_ [47, 36, 41]. The former
solves unconstrained problem (III) by quasi-Newton methods then projects the
solution back to domain (III), which often ends up far from optimal. The
latter generalizes the conjugate gradient methods to Riemannian manifolds and
iteratively updates the solution by addition and retraction. In the following
context, we briefly review the general RCG method and discuss its drawbacks
inherited from the non-geodesic nature, then propose a novel group-wise
geodesic RCG method that operates directly on the Stiefel manifold for faster
convergence.

###  III-A General (Non-Geodesic) rcg

A geodesic is a curve representing the shortest path between two points in a
Riemannian manifold, whose tangent vectors remain parallel when transporting
along the curve. The general RCG method proposed in [49, 50] is applicable to
optimization problems over arbitrary manifolds. The idea is to perform
additive updates along the conjugate direction guided by the Riemannian
gradient, project the solution back onto the manifold, and repeat until
convergence. For optimization problem (III), the steps for BD-RIS group $g$ at
iteration $r$ are summarized below:

  1. 1.

_Compute the Euclidean gradient:_ The gradient of $f$ with respect to
$\mathbf{\Theta}_{g}^{*}$ in the Euclidean space is

$\nabla_{\mathrm{E},g}^{(r)}=\frac{\partial
f(\mathbf{\Theta}_{g}^{(r)})}{\partial\mathbf{\Theta}_{g}^{*}};$ (7)

  2. 2.

_Translate to the Riemannian gradient[49]:_ At point $\mathbf{\Theta}^{(r)}$,
the Riemannian gradient lies in the tangent space of the Stiefel manifold
$\mathcal{T}_{\mathbf{\Theta}_{g}^{(r)}}\mathbb{U}^{L\times
L}\triangleq\\{\mathbf{M}\in\mathbb{C}^{L\times
L}\mid\mathbf{M}^{\mathsf{H}}\mathbf{\Theta}_{g}^{(r)}+{\mathbf{\Theta}_{g}^{(r)\mathsf{H}}}\mathbf{M}=\mathbf{0}\\}$.
It gives the steepest ascent direction of the objective on the manifold can be
obtained by projecting the Euclidean gradient onto the tangent space:

$\nabla_{\mathrm{R},g}^{(r)}=\nabla_{\mathrm{E},g}^{(r)}-\mathbf{\Theta}_{g}^{(r)}{\nabla_{\mathrm{E},g}^{(r)\mathsf{H}}}\mathbf{\Theta}_{g}^{(r)};$
(8)

  3. 3.

_Determine the conjugate direction[51]:_ The conjugate direction is obtained
over the Riemannian gradient and previous direction as

$\mathbf{D}_{g}^{(r)}=\nabla_{\mathrm{R},g}^{(r)}+\gamma_{g}^{(r)}\mathbf{D}_{g}^{(r-1)},$
(9)

where $\gamma_{g}^{(r)}$ is the parameter that deviates the conjugate
direction from the tangent space for accelerated convergence. A popular choice
is the Polak-Ribière formula

$\gamma_{g}^{(r)}=\frac{\mathrm{tr}\bigl{(}(\nabla_{\mathrm{R},g}^{(r)}-\nabla_{\mathrm{R},g}^{(r-1)}){\nabla_{\mathrm{R},g}^{(r)\mathsf{H}}}\bigr{)}}{\mathrm{tr}\bigl{(}\nabla_{\mathrm{R},g}^{(r-1)}{\nabla_{\mathrm{R},g}^{(r-1)\mathsf{H}}}\bigr{)}};$
(10)

  4. 4.

_Perform additive update[50]:_ The point is updated by moving along a straight
path in the conjugate direction

$\bar{\mathbf{\Theta}}_{g}^{(r+1)}=\mathbf{\Theta}_{g}^{(r)}+\mu\mathbf{D}_{g}^{(r)},$
(11)

where $\mu$ is the step size refinable by the Armijo rule [52];

  5. 5.

_Retract for feasibility[49, 47]:_ The resulting point needs to be projected
to the closest point (in terms of Euclidean distance) on the Stiefel manifold
by

$\mathbf{\Theta}_{g}^{(r+1)}=\bar{\mathbf{\Theta}}_{g}^{(r+1)}\bigl{(}{\bar{\mathbf{\Theta}}_{g}^{(r+1)\mathsf{H}}}\bar{\mathbf{\Theta}}_{g}^{(r+1)}\bigr{)}^{-1/2}.$
(12)

One can also combine the addition (11) and retraction (12) in one step

$\mathbf{\Theta}_{g}^{(r+1)}=\bigl{(}\mathbf{\Theta}_{g}^{(r)}+\mu\mathbf{D}_{g}^{(r)}\bigr{)}\bigl{(}\mathbf{I}+\mu^{2}{\mathbf{D}_{g}^{(r)\mathsf{H}}}\mathbf{D}_{g}^{(r)}\bigr{)}^{-1/2},$
(13)

and determine the step size therein.

The method is called non-geodesic since the addition (11) and retraction (12)
constitute a zigzag path departing from and returning to the manifold. It
converges to stationary points of the original problem but usually requires a
large number of iterations due to inefficient operations in the Euclidean
space.

###  III-B Geodesic rcg

Before introducing geodesic RCG, we revisit some basic concepts in
differential geometry. A Lie group is simultaneously a continuous group and a
differentiable manifold. Lie algebra refers to the tangent space of the Lie
group at the identity element. The exponential map acts as a bridge between
the Lie algebra and Lie group, which allows one to recapture the local group
structure using linear algebra techniques. The set of unitary matrices
$\mathbb{U}^{L\times L}$ forms a Lie group $U(L)$ under multiplication, and
the corresponding Lie algebra
$\mathfrak{u}(L)\triangleq\mathcal{T}_{\mathbf{I}}\mathbb{U}^{L\times
L}=\\{\mathbf{M}\in\mathbb{C}^{L\times
L}\mid\mathbf{M}^{\mathsf{H}}+\mathbf{M}=\mathbf{0}\\}$ consists of skew-
Hermitian matrices. A geodesic emanating from the identity with velocity
$\mathbf{D}\in\mathfrak{u}(L)$ can be described by [53]

$\mathbf{G}_{\mathbf{I}}(\mu)=\exp(\mu\mathbf{D}),$ (14)

where $\exp(\mathbf{A})=\sum_{k=0}^{\infty}(\mathbf{A}^{k}/k!)$ is the matrix
exponential and $\mu$ is the step size (i.e., magnitude of the tangent
vector). Note that the right translation is an isometry in $U(L)$. During the
optimization of group $g$, the geodesic evaluated at the identity (14) should
be translated to $\mathbf{\Theta}_{g}^{(r)}$ for successive updates [54]

$\mathbf{G}_{g}^{(r)}(\mu)=\mathbf{G}_{\mathbf{I}}(\mu)\mathbf{\Theta}_{g}^{(r)}=\exp(\mu\mathbf{D}_{g}^{(r)})\mathbf{\Theta}_{g}^{(r)},$
(15)

while the Riemannian gradient evaluated at $\mathbf{\Theta}_{g}^{(r)}$ (8)
should be translated back to the identity for exploiting the Lie algebra [54]

$\tilde{\nabla}_{\mathrm{R},g}^{(r)}=\nabla_{\mathrm{R},g}^{(r)}\mathbf{\Theta}_{g}^{(r)\mathsf{H}}=\nabla_{\mathrm{E},g}^{(r)}\mathbf{\Theta}_{g}^{(r)\mathsf{H}}-\mathbf{\Theta}_{g}^{(r)}{\nabla_{\mathrm{E},g}^{(r)\mathsf{H}}}.$
(16)

After gradient translation, the deviation parameter and conjugate direction
can be determined similarly to (10) and (9)

$\tilde{\gamma}_{g}^{(r)}=\frac{\mathrm{tr}\bigl{(}(\tilde{\nabla}_{\mathrm{R},g}^{(r)}-\tilde{\nabla}_{\mathrm{R},g}^{(r-1)}){\tilde{\nabla}_{\mathrm{R},g}^{(r)\mathsf{H}}}\bigr{)}}{\mathrm{tr}\bigl{(}\tilde{\nabla}_{\mathrm{R},g}^{(r-1)}{\tilde{\nabla}_{\mathrm{R},g}^{(r-1)\mathsf{H}}}\bigr{)}}.$
(17)
${\mathbf{D}}_{g}^{(r)}=\tilde{\nabla}_{\mathrm{R},g}^{(r)}+\tilde{\gamma}_{g}^{(r)}{\mathbf{D}}_{g}^{(r-1)},$
(18)

The solution can thus be updated along the geodesic in a multiplicative
rotational manner

$\mathbf{\Theta}_{g}^{(r+1)}=\mathbf{G}_{g}^{(r)}(\mu)=\exp(\mu\mathbf{D}_{g}^{(r)})\mathbf{\Theta}_{g}^{(r)},$
(19)

where an appropriate $\mu$ may be obtained by the Armijo rule. To double the
step size, one can simply square the rotation matrix instead of recomputing
the matrix exponential, that is,
$\exp^{2}(\mu\mathbf{D}_{g}^{(r)})=\exp(2\mu\mathbf{D}_{g}^{(r)})$. We
highlight that the proposed geodesic RCG method has been tailored to the
Stiefel manifold.

1: Group-wise geodesic RCG for BD-RIS design

1:$f(\mathbf{\Theta})$, $G$

2:$\mathbf{\Theta}^{\star}$

3:Initialize $r\leftarrow 0$, $\mathbf{\Theta}^{(0)}$

4:Repeat

5: For $g\leftarrow 1$ to $G$

6: $\nabla_{\mathrm{E},g}^{(r)}\leftarrow$ (7)

7: $\tilde{\nabla}_{\mathrm{R},g}^{(r)}\leftarrow$ (16)

8: $\tilde{\gamma}_{g}^{(r)}\leftarrow$ (17)

9: $\mathbf{D}_{g}^{(r)}\leftarrow$ (18)

10: If
$\Re\bigl{\\{}\mathrm{tr}({\mathbf{D}_{g}^{(r)\mathsf{H}}}\tilde{\nabla}_{\mathrm{R},g}^{(r)})\bigr{\\}}<0$
$\triangleright$ not an ascent direction

11: $\mathbf{D}_{g}^{(r)}\leftarrow\tilde{\nabla}_{\mathrm{R},g}^{(r)}$

12: End If

13: $\mu\leftarrow 1$

14: $\mathbf{G}_{g}^{(r)}(\mu)\leftarrow$ (15)

15: While
$f\bigl{(}\mathbf{G}_{g}^{(r)}(2\mu)\bigr{)}-f(\mathbf{\Theta}_{g}^{(r)})\geq\mu\cdot\mathrm{tr}(\mathbf{D}_{g}^{(r)}{\mathbf{D}_{g}^{(r)\mathsf{H}}})/2$

16: $\mu\leftarrow 2\mu$

17: End While

18: While
$f\bigl{(}\mathbf{G}_{g}^{(r)}(\mu)\bigr{)}-f(\mathbf{\Theta}_{g}^{(r)})<\mu/2\cdot\mathrm{tr}(\mathbf{D}_{g}^{(r)}{\mathbf{D}_{g}^{(r)\mathsf{H}}})/2$

19: $\mu\leftarrow\mu/2$

20: End While

21: $\mathbf{\Theta}_{g}^{(r+1)}\leftarrow$ (19)

22: End For

23: $r\leftarrow r+1$

24:Until $\lvert
f(\mathbf{\Theta}^{(r)})-f(\mathbf{\Theta}^{(r-1)})\rvert/f(\mathbf{\Theta}^{(r-1)})\leq\epsilon$

Algorithm 1 summarizes the proposed BD-RIS design framework based on group-
wise geodesic RCG. Compared to the general non-geodesic approach, it leverages
the Lie group properties to replace the add-then-retract update (13) with a
multiplicative rotational update (19) along geodesics of the Stiefel manifold.
This leads to faster convergence and simplifies the step size tuning thanks to
appropriate parameter space. Convergence to a local optimum is still
guaranteed if not initialized at a stationary point. Note that the group-wise
updates can be performed in parallel to facilitate large-scale BD-RIS
optimization problems. Since the set of block-unitary matrices are
algebraically closed under multiplication, one can also avoid group-wise
updates by directly operating on $\mathbf{\Theta}$ and pinching (i.e., keeping
the main block diagonal and nulling others) the Euclidean gradient (7), which
can further accelerate the algorithm when the number of group is large.

##  IV Channel Singular Values Redistribution

In this section, we first provide a toy example to illustrate the channel
shaping advantage of BD-RIS architecture. Next, we numerically characterize
the attainable channel singular value region on top of Algorithm 1. Finally,
we derive some analytical singular value bounds in specific channel
conditions.

###  IV-A Toy Example

We showcase the channel shaping capabilities of different RIS models by a toy
example. Consider a $2\times 2\times 2$ setup where the direct link is
blocked. The D-RIS is modeled by
$\mathbf{\Theta}_{\mathrm{D}}=\mathrm{diag}(e^{\jmath\theta_{1}},e^{\jmath\theta_{2}})$
while the fully-connected BD-RIS can be modeled by 4 independent angular
parameters

$\mathbf{\Theta}_{\mathrm{U}}=e^{\jmath\phi}\begin{bmatrix}e^{\jmath\alpha}\cos\psi&e^{\jmath\beta}\sin\psi\\\
-e^{-\jmath\beta}\sin\psi&e^{-\jmath\alpha}\cos\psi\end{bmatrix}.$ (20)

It is worth noting that $\phi$ has no impact on the singular value because
$\mathrm{sv}(e^{\jmath\phi}\mathbf{A})=\mathrm{sv}(\mathbf{A})$. To simplify
the analysis, we also enforce symmetry
$\mathbf{\Theta}_{\mathrm{U}}=\mathbf{\Theta}_{\mathrm{U}}^{\mathsf{T}}$ by
$\beta=\pi/2$ such that both architectures have the same number of variables
in the scattering matrix.

Figure 1: $2\times 2\times 2$ channel singular value shaping by D-RIS and
symmetric fully-connected BD-RIS when the direct channel is blocked.
$\sigma_{1}(\mathbf{H})$ and $\sigma_{2}(\mathbf{H})$ refer to the most and
least dominant singular values, respectively. Their maximum and minimum have
been marked explicitly on the plot.

Fig. 1 compares the achievable singular values of one typical channel
realization, where the backward and forward channels are given by

$\begin{split}\mathbf{H}_{\mathrm{B}}&=\begin{bmatrix}-0.2059+0.5914\jmath&-0.0909+0.5861\jmath\\\
0.4131+0.2651\jmath&-0.1960+0.4650\jmath\end{bmatrix},\\\
\mathbf{H}_{\mathrm{F}}&=\begin{bmatrix}-0.6362+0.1332\jmath&-0.1572+1.5538\jmath\\\
0.0196+0.4011\jmath&-0.3170-0.2303\jmath\end{bmatrix}.\end{split}$

The results are obtained by an exhaustive grid search over
$(\theta_{1},\theta_{2})$ for D-RIS and $(\alpha,\psi)$ for symmetric BD-RIS.
In this example, we observe that both singular values can be manipulated up
to444The percentage for manipulating $\sigma_{n}(\mathbf{H})$ is calculated by
$\eta_{n}^{+}=\frac{\max\sigma_{n}(\mathbf{H})-\mathrm{avg}\sigma_{n}(\mathbf{H})}{\mathrm{avg}\sigma_{n}(\mathbf{H})}\times
100\%$ and
$\eta_{n}^{-}=\frac{\min\sigma_{n}(\mathbf{H})-\mathrm{avg}\sigma_{n}(\mathbf{H})}{\mathrm{avg}\sigma_{n}(\mathbf{H})}\times
100\%$. $\pm 9\%$ by D-RIS (using 2 reconfigurable impedance) and $\pm 42\%$
by symmetric fully-connected BD-RIS (using 3 reconfigurable impedance). Such a
large gain comes from simply connecting both elements, and it can be even
larger using an asymmetric component. This example shows that BD-RIS can
provide a wider dynamic range of channel singular values and motivates further
studies on channel shaping.

###  IV-B Pareto Frontier Characterization

We now characterize the Pareto frontier of singular values of a general
$N_{\mathrm{T}}\times N_{\mathrm{S}}\times N_{\mathrm{R}}$ channel (2) by
maximizing their weighted sum Θ∑_n ρ_n σ_n(H) Θ_g^H Θ_g=I, ∀g, where
$n\in\mathcal{N}\triangleq\\{1,\ldots,N\\}$,
$N\triangleq\min(N_{\mathrm{T}},N_{\mathrm{R}})$ is the maximum channel rank,
and $\rho_{n}$ is the weight of the $n$-th singular value that can be
positive, zero, or negative. Varying $\\{\rho_{n}\\}_{n\in\mathcal{N}}$
characterizes the Pareto frontier that encloses the entire singular value
region. Thus, we claim problem (IV-B) generalizes most singular value shaping
problems. It can be solved optimally by Algorithm 1 with the Euclidean
gradient given by Lemma 1.

######  Lemma 1.

The Euclidean gradient of (IV-B) with respect to BD-RIS group $g$ is

$\frac{\partial\sum_{n}\rho_{n}\sigma_{n}(\mathbf{H})}{\partial\mathbf{\Theta}_{g}^{*}}=\mathbf{H}_{\mathrm{B},g}^{\mathsf{H}}\mathbf{U}\mathrm{diag}(\rho_{1},\ldots,\rho_{N})\mathbf{V}^{\mathsf{H}}\mathbf{H}_{\mathrm{F},g}^{\mathsf{H}},$
(21)

where $\mathbf{U}$ and $\mathbf{V}$ are the left and right compact singular
matrices of $\mathbf{H}$, respectively.

###### Proof.

Please refer to Appendix -A. ∎

We now analyze the computational complexity of solving Pareto singular value
problem (IV-B) by Algorithm 1. To update each BD-RIS group, compact SVD of
$\mathbf{H}$ requires $\mathcal{O}(NN_{\mathrm{T}}N_{\mathrm{R}})$, Euclidean
gradient (21) requires
$\mathcal{O}\bigl{(}LN(N_{\mathrm{T}}+N_{\mathrm{R}}+L+1)\bigr{)}$, Riemannian
gradient translation (16) requires $\mathcal{O}(L^{3})$, deviation parameter
(17) and conjugate direction (18) together require $\mathcal{O}(L^{2})$, and
matrix exponential (19) requires $\mathcal{O}(L^{3})$ operations [55]. The
overall complexity is thus
$\mathcal{O}\bigl{(}I_{\text{RCG}}G\bigl{(}NN_{\mathrm{T}}N_{\mathrm{R}}+LN(N_{\mathrm{T}}+N_{\mathrm{R}}+L+1)+I_{\text{BLS}}L^{3}\bigr{)}\bigr{)}$,
where $I_{\text{RCG}}$ and $I_{\text{BLS}}$ are the number of iterations for
geodesic RCG and backtracking line search (i.e., line 15 – 20 of Algorithm 1),
respectively.

###  IV-C Some Analytical Bounds

We also provide some analytical bounds related to channel singular values,
which help us to explore the limits of channel shaping using passive RIS.

######  Definition 1 (Degrees of freedom).

DoF (a.k.a. multiplexing gain) refers to the maximum number of independent
streams that can be transmitted in parallel over a MIMO channel, which is
defined asymptotically as

$d=\lim_{\rho\to\infty}\frac{\log\det(\mathbf{I}+\rho\mathbf{H}\mathbf{H}^{\mathsf{H}})}{\log\rho},$
(22)

where $\rho$ is the Signal-to-Noise Ratio (SNR).

######  Proposition 1 (Degrees of freedom).

In point-to-point MIMO, BD-RIS cannot achieve a larger number of DoF than
D-RIS.

###### Proof.

Please refer to Appendix -B. ∎

Proposition 1 suggests that we cannot hope for a DoF gain (i.e., more parallel
channels) by simply connecting the RIS elements. We thus focus on the question
of reshaping the available channels through manipulation of the singular
values.

######  Proposition 2 (Rank-deficient channel).

If the minimum rank of backward and forward channels is $k$ ($k\leq N$), then
for D-RIS or BD-RIS of arbitrary number of elements, the $n$-th singular value
of the equivalent channel is bounded above and below respectively by

$\displaystyle\sigma_{n}(\mathbf{H})$
$\displaystyle\leq\sigma_{n-k}(\mathbf{T}),$ $\displaystyle\text{if }n>k,$
(23a) $\displaystyle\sigma_{n}(\mathbf{H})$
$\displaystyle\geq\sigma_{n}(\mathbf{T}),$ $\displaystyle\text{if }n<N-k+1,$
(23b)

where $\mathbf{T}$ is an auxiliary matrix satisfying

$\mathbf{T}\mathbf{T}^{\mathsf{H}}=\begin{cases}\mathbf{H}_{\mathrm{D}}(\mathbf{I}-\mathbf{V}_{\mathrm{F}}\mathbf{V}_{\mathrm{F}}^{\mathsf{H}})\mathbf{H}_{\mathrm{D}}^{\mathsf{H}},&\text{if
}\mathrm{rank}(\mathbf{H}_{\mathrm{F}})=k,\\\
\mathbf{H}_{\mathrm{D}}^{\mathsf{H}}(\mathbf{I}-\mathbf{U}_{\mathrm{B}}\mathbf{U}_{\mathrm{B}}^{\mathsf{H}})\mathbf{H}_{\mathrm{D}},&\text{if
}\mathrm{rank}(\mathbf{H}_{\mathrm{B}})=k,\end{cases}$ (24)

and $\mathbf{V}_{\mathrm{F}}$ and $\mathbf{U}_{\mathrm{B}}$ are the right and
left compact singular matrices of $\mathbf{H}_{\mathrm{F}}$ and
$\mathbf{H}_{\mathrm{B}}$, respectively.

###### Proof.

Please refer to Appendix -C. ∎

Inequality (23a) states that if the backward and backward channels are at
least rank $k$, then the $n$-th singular value of $\mathbf{H}$ can be enlarged
to the $(n-k)$-th singular value of $\mathbf{T}$, or suppressed to the $n$-th
singular value of $\mathbf{T}$. Moreover, the first $k$ channel singular
values are unbounded above555The energy conservation law
$\sum_{n}\sigma_{n}^{2}(\mathbf{H})\leq 1$ still has to be respected. This
constraint is omitted in the following context for brevity. while the last $k$
channel singular values can be suppressed to zero. An example is given in
Corollary 2.1 for Line-of-Sight (LoS) channels (i.e., $k=1$)666A similar
eigenvalue result for D-RIS has been derived in [56]..

######  Corollary 2.1 (LoS channel).

If at least one of backward and forward channels is LoS, then a D-RIS or BD-
RIS can at most enlarge the $n$-th ($n\geq 2$) channel singular value to the
$(n-1)$-th singular value of $\mathbf{T}$, or suppress the $n$-th channel
singular value to the $n$-th singular value of $\mathbf{T}$. That is,

$\sigma_{1}(\mathbf{H})\geq\sigma_{1}(\mathbf{T})\geq{\sigma_{2}(\mathbf{H})}\geq\ldots\geq\sigma_{N-1}(\mathbf{T})\geq{\sigma_{N}(\mathbf{H})}\geq\sigma_{N}(\mathbf{T}).$
(25)

###### Proof.

This is a direct result of (23) with $k=1$. ∎

We highlight that Proposition 2 is an asymptotic result and the inequalities
apply to D\- and BD-RIS with arbitrary number of elements, regardless of the
presence of direct channel. The bounds are asymptotically tight when the
number of scattering elements approaches infinity. On the other hand, we will
show in Section VI that with a finite number of elements, BD-RIS can approach
those bounds better than D-RIS. Proposition 2 evaluates the ultimate channel
shaping limits of passive RIS and provides a reference for choosing the number
of scattering elements in a practical deployment scenario.

######  Proposition 3 (Blocked direct channel).

If the direct link is blocked and the BD-RIS is fully-connected, then the
channel singular values can be manipulated up to

$\mathrm{sv}(\mathbf{H})=\mathrm{sv}(\mathbf{BF}),$ (26)

where $\mathbf{B}$ and $\mathbf{F}$ are arbitrary matrices with
$\mathrm{sv}(\mathbf{B})=\mathrm{sv}(\mathbf{H}_{\mathrm{B}})$ and
$\mathrm{sv}(\mathbf{F})=\mathrm{sv}(\mathbf{H}_{\mathrm{F}})$.

###### Proof.

Please refer to Appendix -D. ∎

Proposition 3 states that if the direct channel is blocked and the BD-RIS is
fully-connected, the only singular value bounds on the equivalent channel are
the singular value bounds on the product of unitary-transformed backward and
forward channels. For example, consider a $2\times 2\times 2$ setup with

$\mathbf{H}_{\mathrm{B}}=\begin{bmatrix}3&0\\\
0&2\end{bmatrix},\quad\mathbf{H}_{\mathrm{F}}=\begin{bmatrix}1&0\\\
0&4\end{bmatrix}.$

It is obvious that any D-RIS can only attain
$\mathrm{sv}(\mathbf{H})=[8,3]^{\mathsf{T}}$ while a BD-RIS
$\mathbf{\Theta}=\begin{bmatrix}0&1\\\ 1&0\end{bmatrix}$ results in
$\mathrm{sv}(\mathbf{H})=[12,2]^{\mathsf{T}}$. That is, fully-connected BD-RIS
applies right unitary transformation on $\mathbf{H}_{\mathrm{B}}$ and left
unitary transformation on $\mathbf{H}_{\mathrm{F}}$, widening the singular
values to $8\leq\sigma_{1}(\mathbf{H})\leq 12$ and
$2\leq\sigma_{2}(\mathbf{H})\leq 3$.

Proposition 3 is important since it transforms the channel shaping question to
a linear algebra question: _How the singular values of matrix product are
bounded by the singular values of its individual factors?_ Let
$\bar{N}=\max(N_{\mathrm{T}},N_{\mathrm{S}},N_{\mathrm{R}})$ and
$\sigma_{n}(\mathbf{H})=\sigma_{n}(\mathbf{H}_{\mathrm{F}})=\sigma_{n}(\mathbf{H}_{\mathrm{B}})=0$
for $N<n\leq\bar{N}$. We have the following corollaries.

######  Corollary 3.1 (Generic singular value bounds).

$\prod_{k\in{K}}\sigma_{k}(\mathbf{H})\leq\prod_{i\in{I}}\sigma_{i}(\mathbf{H}_{\mathrm{B}})\prod_{j\in{J}}\sigma_{j}(\mathbf{H}_{\mathrm{F}}),$
(27)

for all admissible triples $(I,J,K)\in T_{r}^{\bar{N}}$ with $r<\bar{N}$,
where

$\begin{gathered}T_{r}^{\bar{N}}\triangleq\Bigl{\\{}(I,J,K)\in
U_{r}^{\bar{N}}\mid\forall p<r,(F,G,H)\in T_{p}^{r},\\\ \sum_{f\in
F}i_{f}+\sum_{g\in G}j_{g}\leq\sum_{h\in
H}k_{h}+p(p+1)/2\Bigr{\\}},\end{gathered}$
$U_{r}^{\bar{N}}\triangleq\Bigl{\\{}(I,J,K)\mid\sum_{i\in I}i+\sum_{j\in
J}j=\sum_{k\in K}k+r(r+1)/2\Bigr{\\}}.$

###### Proof.

Please refer to [57, Theorem 8]. ∎

Corollary (3.1), also recognized as a variation of Horn’s inequality [58], is
the most comprehensive channel singular value bound attainable by passive RIS
when the direct channel is blocked. We observe that the number of admissible
triples (and thus bounds) grows exponentially with $\bar{N}$. For example, the
number of inequalities described by (27) grows from 12 to 2062 when $\bar{N}$
increases from 3 to 7. This renders the analytic approach computationally
expensive for large-scale MIMO systems. Next, we showcase some useful
inequalities enclosed by (27). Interested readers are referred to [59, Chapter
16, 24] for more information.

######  Corollary 3.2 (Upper bound on the largest singular value).

$\sigma_{1}(\mathbf{H})\leq\sigma_{1}(\mathbf{H}_{\mathrm{B}})\sigma_{1}(\mathbf{H}_{\mathrm{F}}).$
(28)

###### Proof.

This is a direct result of (27) with $r=1$. ∎

######  Corollary 3.3 (Lower bound on the smallest singular value).

$\sigma_{\bar{N}}(\mathbf{H})\geq\sigma_{\bar{N}}(\mathbf{H}_{\mathrm{B}})\sigma_{\bar{N}}(\mathbf{H}_{\mathrm{F}}).$
(29)

###### Proof.

This can be deducted from (27) with $r_{1}=\bar{N}{-}1$ and $r_{2}=\bar{N}$. ∎

######  Corollary 3.4 (Upper bound on the product of first $k$ singular
values).

$\prod_{n=1}^{k}\sigma_{n}(\mathbf{H})\leq\prod_{n=1}^{k}\sigma_{n}(\mathbf{H}_{\mathrm{B}})\prod_{n=1}^{k}\sigma_{n}(\mathbf{H}_{\mathrm{F}}).$
(30)

###### Proof.

This is a direct result of (27) with $r=k$. ∎

######  Corollary 3.5 (Lower bound on the product of last $k$ singular
values).

777Corollaries 3.3 and 3.5 are less informative when $\bar{N}\neq N$ (i.e.,
$N_{\mathrm{T}}=N_{\mathrm{S}}=N_{\mathrm{R}}$ being false) as the lower
bounds would coincide at zero.

$\prod_{n=\bar{N}}^{\bar{N}-k+1}\sigma_{n}(\mathbf{H})\geq\prod_{n=\bar{N}}^{\bar{N}-k+1}\sigma_{n}(\mathbf{H}_{\mathrm{B}})\prod_{n=\bar{N}}^{\bar{N}-k+1}\sigma_{n}(\mathbf{H}_{\mathrm{F}}).$
(31)

###### Proof.

This can be deducted from (27) with $r_{1}=\bar{N}-k$ and $r_{2}=\bar{N}$. ∎

######  Corollary 3.6 (Upper bound on the channel power gain).

The channel power gain is upper bounded by the sum of element-wise product of
squared singular values of backward and forward channels

$\lVert\mathbf{H}\rVert_{\mathrm{F}}^{2}=\sum_{n=1}^{N}\sigma_{n}^{2}(\mathbf{H})\leq\sum_{n=1}^{N}\sigma_{n}^{2}(\mathbf{H}_{\mathrm{B}})\sigma_{n}^{2}(\mathbf{H}_{\mathrm{F}}).$
(32)

###### Proof.

Please refer to [59, Inequality 24.4.7]. ∎

To achieve equalities in Corollaries 3.2 – 3.6, the RIS needs to completely
align the singular vector spaces of $\mathbf{H}_{\mathrm{B}}$ and
$\mathbf{H}_{\mathrm{F}}$. The resulting scattering matrix is generally
required to be unitary

$\mathbf{\Theta}^{\star}=\mathbf{V}_{\mathrm{B}}\mathbf{U}_{\mathrm{F}}^{\mathsf{H}},$
(33)

which can be concluded from (46) and (47) in Appendix -D. We notice that D-RIS
can attain those equalities if and only if $\mathbf{H}_{\mathrm{B}}$ and
$\mathbf{H}_{\mathrm{F}}$ are both rank-1. In such case, the equivalent
channel reduces to
$\mathbf{H}=\sigma_{\mathrm{B}}\sigma_{\mathrm{F}}\mathbf{u}_{\mathrm{B}}\mathbf{v}_{\mathrm{B}}^{\mathsf{H}}\mathbf{\Theta}\mathbf{u}_{\mathrm{F}}\mathbf{v}_{\mathrm{F}}^{\mathsf{H}}$
and the RIS only needs to align $\mathbf{v}_{\mathrm{B}}^{\mathsf{H}}$ and
$\mathbf{u}_{\mathrm{F}}$ by

$\mathbf{\Theta}^{\star}=\mathbf{v}_{\mathrm{B}}\mathbf{u}_{\mathrm{F}}^{\mathsf{H}}\odot\mathbf{I},$
(34)

which becomes a special case of (6). However, in general D-RIS is not
sufficient to achieve the upper bound of channel power gain. This again
highlights the benefit of BD-RIS in reshaping available channels. As a side
note, when both $\mathbf{H}_{\mathrm{B}}$ and $\mathbf{H}_{\mathrm{F}}$ follow
Rayleigh fading, the maximum expected channel power gain can be numerically
evaluated as

$\mathbb{E}\bigl{\\{}\lVert\mathbf{H}\rVert_{\mathrm{F}}^{2}\bigr{\\}}=\sum_{n=1}^{N}\int_{0}^{\infty}f_{\lambda_{n}^{\min(N_{\mathrm{R}},N_{\mathrm{S}})}}(x)dx\int_{0}^{\infty}f_{\lambda_{n}^{\min(N_{\mathrm{S}},N_{\mathrm{T}})}}(x)dx,$
(35)

where $\lambda_{n}^{K}$ is the $n$-th eigenvalue of the complex $K\times K$
Wishart matrix with probability density function $f_{\lambda_{n}^{K}}(x)$
given by [60, Equation 51]. We note that (35) generalizes the SISO power gain
attainable by BD-RIS [30, Equation 58] to MIMO and models the contribution of
transmit and receive antennas under double Rayleigh fading. In Section VI, we
will numerically evaluate the channel power gain and compare it with the
theoretical upper bound.

Proposition 3 and Corollaries 3.1 – 3.6 provide a comprehensive answer to the
shaping capability question in terms of singular value bounds and channel
power gain limits. Extending those results to the case of direct channel is
non-trivial since the singular vector spaces of direct-indirect channels and
backward-forward channels cannot be aligned at the same time. In such case, we
can invoke optimization approaches from a singular value perspective (Section
IV-B) or a power gain perspective (Section V-A).

##  V Power Gain and Achievable Rate Maximization

In this section, we solve channel power gain maximization problem in closed-
form and then extend the discussion to achievable rate maximization problem.
The implication of the former on the latter is investigated.

###  V-A Channel Power Gain

The MIMO channel power gain maximization problem is formulated with respect to
the BD-RIS scattering matrix Θ∥H_D + H_B Θ H_F ∥_F^2 Θ_g^H Θ_g=I, ∀g, which
generalizes the case of SISO [30], MISO [33, 40], single-stream MIMO [31, 45],
and direct link-blocked MIMO with fully-connected BD-RIS (33). Solving (V-A)
calls for a balance between the additive direct-indirect combining and the
multiplicative backward-forward combining.

######  Remark 1.

Interestingly, in terms of maximizing the inner product
$\langle\mathbf{H}_{\mathrm{D}},\mathbf{H}_{\mathrm{B}}\mathbf{\Theta}\mathbf{H}_{\mathrm{F}}\rangle$,
(V-A) is reminiscent of the weighted orthogonal Procrustes problem [61] Θ∥H_D
\- H_B Θ H_F ∥_F^2 Θ^H Θ=I, which relaxes the block-unitary constraint (V-A)
to unitary (1) but still has no trivial solution. One lossy transformation
exploits the Moore-Penrose inverse and moves $\mathbf{\Theta}$ to one side of
the product [62], formulating two standard orthogonal Procrustes problems
Θ∥H_B^†H_D \- Θ H_F ∥_F^2  or  ∥H_D H_F^†\- H_B Θ ∥_F^2 Θ^H Θ=I, which have
global optimal solutions

$\mathbf{\Theta}=\mathbf{U}\mathbf{V}^{\mathsf{H}},$ (36)

where $\mathbf{U}$ and $\mathbf{V}$ are respectively the left and right
compact singular matrices of
$\mathbf{H}_{\mathrm{B}}^{\dagger}\mathbf{H}_{\mathrm{D}}\mathbf{H}_{\mathrm{F}}^{\mathsf{H}}$
or
$\mathbf{H}_{\mathrm{B}}^{\mathsf{H}}\mathbf{H}_{\mathrm{D}}\mathbf{H}_{\mathrm{F}}^{\dagger}$
[63].

It is worth noting that (33) and (36) are valid fully-connected BD-RIS
solutions to (V-A) when the direct link is blocked and present, respectively.
However, the latter is neither optimal nor a generalization of the former due
to the lossy transformation. Inspired by [64], we propose an optimal solution
to problem (V-A) with arbitrary group size. The idea is to successively
approximate the quadratic objective (V-A) by local Taylor expansions and solve
each step in closed form by group-wise SVD.

######  Proposition 4.

Starting from any feasible $\mathbf{\Theta}^{(0)}$, the sequence

$\mathbf{\Theta}_{g}^{(r+1)}=\mathbf{U}_{g}^{(r)}\mathbf{V}_{g}^{(r)},\quad\forall
g.$ (37)

converges to a stationary point of (V-A), where $\mathbf{U}_{g}^{(r)}$ and
$\mathbf{V}_{g}^{(r)}$ are the left and right compact singular matrices of

$\mathbf{M}_{g}^{(r)}=\mathbf{H}_{\mathrm{B},g}^{\mathsf{H}}\Bigl{(}\mathbf{H}_{\mathrm{D}}+\mathbf{H}_{\mathrm{B}}\mathrm{diag}\bigl{(}\mathbf{\Theta}_{[1:g-1]}^{(r+1)},\mathbf{\Theta}_{[g:G]}^{(r)}\bigr{)}\mathbf{H}_{\mathrm{F}}\Bigr{)}\mathbf{H}_{\mathrm{F},g}^{\mathsf{H}}$
(38)

###### Proof.

Please refer to Appendix -F. ∎

We now analyze the computational complexity of solving channel gain
maximization problem (V-A) by Proposition 4. To update each BD-RIS group,
matrix multiplication (38) requires
$\mathcal{O}\bigl{(}N_{\mathrm{T}}N_{\mathrm{R}}+(G+1)(NL^{2}+N_{\mathrm{T}}N_{\mathrm{R}}L)\bigr{)}$
operations and its compact SVD requires $\mathcal{O}(L^{3})$ operations. The
overall complexity is thus
$\mathcal{O}\bigl{(}I_{\text{SAA}}G\bigl{(}N_{\mathrm{T}}N_{\mathrm{R}}+(G+1)(NL^{2}+N_{\mathrm{T}}N_{\mathrm{R}}L)+L^{3}\bigr{)}\bigr{)}$,
where $I_{\text{SAA}}$ is the number iterations for successive affine
approximation.

###  V-B Achievable Rate Maximization

We aim to maximize the achievable rate of the BD-RIS-aided MIMO system by
jointly optimizing the active and passive beamforming W,ΘR = logdet(I \+
WHHHHWη) ∥W ∥_F^2≤P Θ_g^H Θ_g=I, ∀g, where $\mathbf{W}$ is the transmit
precoder, $R$ is the achievable rate, $\eta$ is the average noise power, and
$P$ is the transmit power constraint. Problem (V-B) is non-convex due to the
block-unitary constraint (V-B) and the coupling between variables. We propose
a local-optimal approach via AO and a low-complexity approach based on channel
shaping.

####  V-B1 Alternating Optimization

This approach updates $\mathbf{\Theta}$ and $\mathbf{W}$ iteratively until
convergence. For a given $\mathbf{W}$, the passive beamforming subproblem is
Θlogdet(I \+ H Q HHη) Θ_g^H Θ_g=I, ∀g, where
$\mathbf{Q}\triangleq\mathbf{W}\mathbf{W}^{\mathsf{H}}$ is the transmit
covariance matrix. Problem (V-B1) can be solved optimally by Algorithm 1 with
the Euclidean gradient given by Lemma 2.

######  Lemma 2.

The Euclidean gradient of (V-B1) with respect to BD-RIS block $g$ is

$\frac{\partial
R}{\partial\mathbf{\Theta}_{g}^{*}}=\frac{1}{\eta}\mathbf{H}_{\mathrm{B},g}^{\mathsf{H}}\biggl{(}\mathbf{I}+\frac{\mathbf{H}\mathbf{Q}\mathbf{H}^{\mathsf{H}}}{\eta}\biggr{)}^{-1}\mathbf{H}\mathbf{Q}\mathbf{H}_{\mathrm{F},g}^{\mathsf{H}}.$
(39)

###### Proof.

Please refer to Appendix -E. ∎

For a given $\mathbf{\Theta}$, the global optimal transmit precoder is given
by eigenmode transmission [65]

$\mathbf{W}^{\star}=\mathbf{V}{\mathrm{diag}(\mathbf{s}^{\star})}^{1/2},$ (40)

where $\mathbf{V}$ is the right singular matrix of the equivalent channel and
$\mathbf{s}^{\star}$ is the optimal water-filling power allocation obtainable
by the iterative method [66].

The AO algorithm is guaranteed to converge to local-optimal points of problem
(V-B) since each subproblem is solved optimally and the objective is bounded
above. Similar to the analysis in Section IV-B, the computational complexity
of solving subproblem (V-B1) by geodesic RCG is
$\mathcal{O}\bigl{(}I_{\text{RCG}}G(NL^{2}+LN_{\mathrm{T}}N_{\mathrm{R}}+N_{\mathrm{T}}^{2}N_{\mathrm{R}}+N_{\mathrm{T}}N_{\mathrm{R}}^{2}+N_{\mathrm{R}}^{3}+I_{\text{BLS}}L^{3})\bigr{)}$.
On the other hand, the complexity of solving active beamforming subproblem by
(40) is $\mathcal{O}\bigl{(}NN_{\mathrm{T}}N_{\mathrm{R}}\bigr{)}$. The
overall complexity is thus
$\mathcal{O}\bigl{(}I_{\text{AO}}\bigl{(}I_{\text{RCG}}G(NL^{2}+LN_{\mathrm{T}}N_{\mathrm{R}}+N_{\mathrm{T}}^{2}N_{\mathrm{R}}+N_{\mathrm{T}}N_{\mathrm{R}}^{2}+N_{\mathrm{R}}^{3}+I_{\text{BLS}}L^{3})+NN_{\mathrm{T}}N_{\mathrm{R}}\bigr{)}\bigr{)}$,
where $I_{\text{AO}}$ is the number of iterations for AO.

####  V-B2 Low-Complexity Solution

We finally propose a suboptimal two-stage solution to problem (V-B) that
decouples the joint RIS-transceiver design. The idea is to first consider
channel shaping and replace the rate maximization subproblem (V-B1) by channel
power gain maximization problem (V-A), then proceed to conventional eigenmode
transmission (40). Both steps are solved in closed form and the computational
complexity is
$\mathcal{O}\bigl{(}I_{\text{SAA}}G\bigl{(}N_{\mathrm{T}}N_{\mathrm{R}}+(G+1)(NL^{2}+N_{\mathrm{T}}N_{\mathrm{R}}L)+L^{3}\bigr{)}+NN_{\mathrm{T}}N_{\mathrm{R}}\bigr{)}$.
While suboptimal, this shaping-inspired solution avoids outer iterations and
efficiently handles inner iterations.

##  VI Simulation Results

In this section, we provide numerical results to evaluate the proposed BD-RIS
designs.888The simulation code is publicly available at
https://github.com/snowztail/channel-shaping. Consider a distance-dependent
path loss model $\Lambda(d)=\Lambda_{0}d^{-\gamma}$ where $\Lambda_{0}$ is the
reference path loss at distance 1m, $d$ is the propagation distance, and
$\gamma$ is the path loss exponent. The small-scale fading model is
$\mathbf{H}=\sqrt{\kappa/(1+\kappa)}\mathbf{H}_{\text{LoS}}+\sqrt{1/(1+\kappa)}\mathbf{H}_{\text{NLoS}}$,
where $\kappa$ is the Rician K-factor, $\mathbf{H}_{\text{LoS}}$ is the
deterministic LoS component, and
$\mathbf{H}_{\text{NLoS}}\sim\mathcal{CN}(\mathbf{0},\mathbf{I})$ is the
Rayleigh component. We set $\Lambda_{0}=\qty{-30}{dB}$,
$d_{\mathrm{D}}=\qty{14.7}{m}$, $d_{\mathrm{F}}=\qty{10}{m}$,
$d_{\mathrm{B}}=\qty{6.3}{m}$, $\gamma_{\mathrm{D}}=3$,
$\gamma_{\mathrm{F}}=2.4$ and $\gamma_{\mathrm{B}}=2$ for reference, which
corresponds to a typical indoor environment with
$\Lambda_{\mathrm{D}}=\qty{-65}{dB}$, $\Lambda_{\mathrm{F}}=\qty{-54}{dB}$,
$\Lambda_{\mathrm{B}}=\qty{-46}{dB}$. The indirect path via RIS is thus 35
weaker than the direct path. Rayleigh fading (i.e., $\kappa=0$) is assumed for
all channels unless otherwise specified.

###  VI-A Algorithm Evaluation

TABLE I: Average Performance of Geodesic and Non-Geodesic RCG Algorithms on Problem (IV-B) RCG path | $N_{\mathrm{S}}=16$ | $N_{\mathrm{S}}=256$
---|---|---
Objective | Iterations | Time [s] | Objective | Iterations | Time [s]
Geodesic | $4.359\text{\times}{10}^{-3}$ | 11.59 | $1.839\text{\times}{10}^{-2}$ | $1.163\text{\times}{10}^{-2}$ | 25.58 | 3.461
Non-geodesic | $4.329\text{\times}{10}^{-3}$ | 30.92 | $5.743\text{\times}{10}^{-2}$ | $1.116\text{\times}{10}^{-2}$ | 61.40 | 13.50

We first compare in Table I the geodesic and non-geodesic RCG algorithm on
problem (IV-B) in an $N_{\mathrm{T}}=N_{\mathrm{R}}=4$ system with BD-RIS
group size $L=4$. The statistics are averaged over $100$ independent runs. It
is observed that the geodesic RCG method achieves a slightly higher objective
value with significantly (down to 1/3) lower number of iterations and shorter
(down to 1/4) computational time than the non-geodesic method. The results
demonstrate the efficiency of the proposed geodesic RCG algorithm especially
for large-scale BD-RIS design problems.

###  VI-B Channel Singular Values Redistribution

####  VI-B1 Pareto Frontier

$0$$0.5$$1$$1.5$$2$$2.5$$3$$3.5$$\cdot 10^{-4}$$0$$1$$2$$3$$\cdot
10^{-4}$$\sigma_{1}(\mathbf{H})$$\sigma_{2}(\mathbf{H})$Direct$L=1$$L=4$$L=16$$L=32$

(a) $2\times 32\times 2$ (no direct)

$0.9$$1$$1.1$$1.2$$1.3$$1.4$$\cdot 10^{-3}$$2$$4$$6$$\cdot
10^{-4}$$\sigma_{1}(\mathbf{H})$$\sigma_{2}(\mathbf{H})$Direct$L=1$$L=4$$L=16$$L=32$

(b) $2\times 32\times 2$

$0.8$$1$$1.2$$1.4$$1.6$$1.8$$\cdot 10^{-3}$$0$$2$$4$$6$$8$$\cdot
10^{-4}$$\sigma_{1}(\mathbf{H})$$\sigma_{2}(\mathbf{H})$Direct$L=1$$L=4$$L=16$$L=64$

(c) $2\times 64\times 2$

$0$$0.5$$1$$1.5$$2$$\cdot 10^{-3}$$0$$0.5$$1$$1.5$$\cdot
10^{-3}$$\sigma_{1}(\mathbf{H})$$\sigma_{2}(\mathbf{H})$Direct$L=1$$L=4$$L=16$$L=128$

(d) $2\times 128\times 2$

Figure 2: Pareto frontiers of singular values of an
$N_{\mathrm{T}}=N_{\mathrm{R}}=2$ channel reshaped by BD-RIS.

Fig. 2 shows the Pareto singular values of an
$N_{\mathrm{T}}=N_{\mathrm{R}}=2$ MIMO reshaped by a RIS. When the direct link
is blocked, the achievable regions in Fig.
LABEL:sub@fg:singular_pareto_sx32_nd are shaped like pizza slices. This is
because $\sigma_{1}(\mathbf{H})\geq\sigma_{2}(\mathbf{H})\geq 0$ and there
exists a trade-off between the alignment of two spaces. We observe that the
smallest singular value can be enhanced up to $2\text{\times}{10}^{-4}$ by
D-RIS and $3\text{\times}{10}^{-4}$ by fully-connected BD-RIS, corresponding
to a 50 gain. When the direct link is present, the shape of the singular value
region depends heavily on the relative strength of the indirect link. In Fig.
LABEL:sub@fg:singular_pareto_sx32, a 32-element RIS is insufficient to
compensate the 35dB path loss imbalance and results in a limited singular
value region that is symmetric around the direct point. As the group size $L$
increases, the shape of the region evolves from elliptical to square. This
transformation not only improves the dynamic range of $\sigma_{1}(\mathbf{H})$
and $\sigma_{2}(\mathbf{H})$ by 22 and 38, but also provides a better trade-
off in manipulating both singular values. It suggests the design flexibility
from larger group size allows better alignment of multiple singular vector
spaces simultaneously. The singular value region also enlarges as the number
of scattering elements $N_{\mathrm{S}}$ increases. In particular, Fig.
LABEL:sub@fg:singular_pareto_sx128 shows that the equivalent channel can be
completely nulled (corresponding to the origin) by a 128-element BD-RIS thanks
to its superior channel shaping capability, but not by a diagonal one. Those
results demonstrate the superior channel shaping capability of BD-RIS and
emphasizes the importance of reconfigurable inter-connections between
elements.

####  VI-B2 Analytical Bounds and Numerical Results

$\sigma_{1}(\mathbf{H})$$\sigma_{2}(\mathbf{H})$$\sigma_{3}(\mathbf{H})$$\sigma_{4}(\mathbf{H})$$0$$0.2$$0.4$$0.6$$0.8$$1$$1.2$$1.4$$1.6$$1.8$$2$$\cdot
10^{-3}$AmplitudeD-min$\sigma_{1}(\mathbf{T})$D-max$\sigma_{2}(\mathbf{T})$BD-
min$\sigma_{3}(\mathbf{T})$BD-max$\sigma_{4}(\mathbf{T})$

(a) $4\times 32\times 4$ (rank-1)

$\sigma_{1}(\mathbf{H})$$\sigma_{2}(\mathbf{H})$$\sigma_{3}(\mathbf{H})$$\sigma_{4}(\mathbf{H})$$0$$0.5$$1$$1.5$$2$$2.5$$3$$\cdot
10^{-3}$AmplitudeD-min$\sigma_{1}(\mathbf{T})$D-max$\sigma_{2}(\mathbf{T})$BD-
min$\sigma_{3}(\mathbf{T})$BD-max$\sigma_{4}(\mathbf{T})$

(b) $4\times 64\times 4$ (rank-1)

$\sigma_{1}(\mathbf{H})$$\sigma_{2}(\mathbf{H})$$\sigma_{3}(\mathbf{H})$$\sigma_{4}(\mathbf{H})$$0$$0.5$$1$$1.5$$2$$2.5$$3$$3.5$$4$$\cdot
10^{-3}$AmplitudeD-min$\sigma_{1}(\mathbf{T})$D-max$\sigma_{2}(\mathbf{T})$BD-
min$\sigma_{3}(\mathbf{T})$BD-max$\sigma_{4}(\mathbf{T})$

(c) $4\times 128\times 4$ (rank-2)

$\sigma_{1}(\mathbf{H})$$\sigma_{2}(\mathbf{H})$$\sigma_{3}(\mathbf{H})$$\sigma_{4}(\mathbf{H})$$0$$1$$2$$3$$4$$5$$\cdot
10^{-3}$AmplitudeD-min$\sigma_{1}(\mathbf{T})$D-max$\sigma_{2}(\mathbf{T})$BD-
min$\sigma_{3}(\mathbf{T})$BD-max$\sigma_{4}(\mathbf{T})$

(d) $4\times 256\times 4$ (rank-4)

Figure 3:  Achievable channel singular values: analytical bounds (green lines)
and numerical optimization results (blue and red bars). The intersections of
the blue and red bars denote the singular values of the direct channel. The
blue (resp. red) bars are obtained by solving problem (IV-B) with
$\rho_{n}=-1$ (resp. $+1$) and $\rho_{n^{\prime}}=0$, $\forall n^{\prime}\neq
n$. ‘D’ means D-RIS and ‘BD’ refers to fully-connected BD-RIS. ‘rank-$k$’
refers to the rank of the forward channel.

Fig. 3 illustrates the analytical singular value bounds in Proposition 2 and
the numerical results obtained by solving problem (IV-B) with $\rho_{n}=\pm 1$
and $\rho_{n^{\prime}}=0$, $\forall n^{\prime}\neq n$. Here we assume a
rank-$k$ forward channel without loss of generality. When the RIS is in the
vicinity of the transmitter, Figs. LABEL:sub@fg:singular_bound_rank1_sx32 and
LABEL:sub@fg:singular_bound_rank1_sx64 show that the achievable channel
singular values indeed satisfy Corollary 2.1, namely
$\sigma_{1}(\mathbf{H})\geq\sigma_{1}(\mathbf{T})$,
$\sigma_{2}(\mathbf{T})\leq\sigma_{2}(\mathbf{H})\leq\sigma_{1}(\mathbf{T})$,
etc. It is obvious that BD-RIS can approach those bounds better than D-RIS
especially for a small $N_{\mathrm{S}}$. Another example is given in Fig.
LABEL:sub@fg:singular_bound_rank2_sx128 with rank-2 forward channel. The first
two channel singular values are unbounded above and bounded below by the first
two singular values of $\mathbf{T}$, while the last two singular values can be
suppressed to zero and bounded above by the first two singular values of
$\mathbf{T}$. Those observations align with Proposition 2. Finally, Fig.
LABEL:sub@fg:singular_bound_rank4_sx256 confirms there are no extra singular
value bounds when both backward and forward channels are full-rank. This can
be predicted from (24) where the compact singular matrix
$\mathbf{V}_{\mathrm{F}}$ becomes unitary and $\mathbf{T}=\mathbf{0}$. The
numerical results are consistent with the analytical bounds, and we conclude
that the channel shaping advantage of BD-RIS over D-RIS scales with backward
and forward channel ranks.

141664256$0$$2$$4$$6$$\cdot 10^{-6}$RIS Group SizeChannel Power
[W]EquivalentCascaded

(a) $1\times 256\times 1$ (no direct)

141664256$0$$0.5$$1$$1.5$$2$$2.5$$\cdot 10^{-5}$RIS Group SizeChannel Power
[W]EquivalentCascaded

(b) $4\times 256\times 4$ (no direct)

Figure 4:  Average maximum channel power gain versus BD-RIS group size and
MIMO dimensions. The direct channel is blocked. ‘Cascaded’ refers to the
available power of the cascaded channel, i.e., the sum of (sorted) element-
wise power product of backward and forward channels.

Fig. 4 compares the analytical channel power bound in Corollary 3.6 and the
numerical results obtained by solving problem (V-A) when the direct link is
blocked. Here, a fully-connected BD-RIS can attain the upper bound either in
closed form (33) or via optimization approach (37). For the SISO case in Fig.
LABEL:sub@fg:power_bond_txrx1_nd, the maximum channel power gain is
approximately $4\text{\times}{10}^{-6}$ by D-RIS and
$6.5\text{\times}{10}^{-6}$ by fully-connected BD-RIS, corresponding to a 62.5
gain. It comes purely from channel rearrangement in (5) and aligns with the
asymptotic BD-RIS scaling law derived for SISO in [30, Equation 30].
Interestingly, this relative power gain surges to 270 in
$N_{\mathrm{T}}=N_{\mathrm{R}}=4$ MIMO as shown in Fig.
LABEL:sub@fg:power_bond_txrx4_nd, which can also be predicted from the
expectation analysis (35). We thus conclude that the power gain of BD-RIS
scales with group size and MIMO dimensions.

###  VI-C Power Gain and Achievable Rate Maximization

$2^{0}$$2^{1}$$2^{2}$$2^{3}$$2^{4}$$2^{5}$$2^{6}$$2^{7}$$2^{8}$$10^{-6}$$10^{-5}$$10^{-4}$RIS
Group SizeChannel Power
[W]$N_{\mathrm{S}}=16$$N_{\mathrm{S}}=32$$N_{\mathrm{S}}=64$$N_{\mathrm{S}}=128$$N_{\mathrm{S}}=256$

(a) $16\times N_{\mathrm{S}}\times 16$ (no direct)

$2^{0}$$2^{1}$$2^{2}$$2^{3}$$2^{4}$$2^{5}$$2^{6}$$2^{7}$$2^{8}$$10^{-4}$$10^{-3.8}$$10^{-3.6}$RIS
Group SizeChannel Power [W]No RISOP-leftOP-
right$N_{\mathrm{S}}=16$$N_{\mathrm{S}}=32$$N_{\mathrm{S}}=64$$N_{\mathrm{S}}=128$$N_{\mathrm{S}}=256$

(b) $16\times N_{\mathrm{S}}\times 16$

Figure 5:  Average maximum channel power gain versus RIS configuration. ‘OP-
left’ and ‘OP-right’ refer to the suboptimal solutions to problem (V-A) by
lossy transformation (1) where $\mathbf{\Theta}$ is to the left and right of
the product, respectively.

We first focus on channel power gain maximization problem (V-A). Fig. 5 shows
the maximum channel power under different RIS configurations. An interesting
observation is that the relative power gain of BD-RIS over D-RIS is even
larger with direct link. For example, a 64-element fully BD-RIS can almost
provide the same channel power gain as a 256-element D-RIS in Fig.
LABEL:sub@fg:power_sx_txrx16, but not in Fig. LABEL:sub@fg:power_sx_txrx16_nd.
This is because the RIS needs to balance the multiplicative forward-backward
combining and the additive direct-indirect combining, such that the space
alignment advantage of BD-RIS becomes more pronounced. We also notice that the
suboptimal solutions (36) for fully-connected BD-RIS by lossy transformation
(1) are very close to optimal especially for a large $N_{\mathrm{S}}$.

$-20$$-15$$-10$$-5$$0$$5$$10$$15$$20$$0$$10$$20$$30$$40$$50$Transmit Power
[dB]Achievable Rate [bit/s/Hz]No RISAlternate: $L=1$Decouple: $L=1$Alternate:
$L=4$Decouple: $L=4$Alternate: $L=128$Decouple: $L=128$

(a) $4\times 128\times 4$

$-20$$-15$$-10$$-5$$0$$5$$10$$15$$20$$0$$20$$40$$60$$80$$100$$120$$140$$160$$180$Transmit
Power [dB]Achievable Rate [bit/s/Hz]D: $N_{\mathrm{T}}=N_{\mathrm{R}}=1$BD:
$N_{\mathrm{T}}=N_{\mathrm{R}}=1$D: $N_{\mathrm{T}}=N_{\mathrm{R}}=4$BD:
$N_{\mathrm{T}}=N_{\mathrm{R}}=4$D: $N_{\mathrm{T}}=N_{\mathrm{R}}=16$BD:
$N_{\mathrm{T}}=N_{\mathrm{R}}=16$

(b) $N_{\mathrm{T}}\times 128\times N_{\mathrm{R}}$

$-20$$-15$$-10$$-5$$0$$5$$10$$15$$20$$0$$10$$20$$30$$40$$50$$60$Transmit Power
[dB]Achievable Rate [bit/s/Hz]No RISD: $N_{\mathrm{S}}=16$BD:
$N_{\mathrm{S}}=16$D: $N_{\mathrm{S}}=64$BD: $N_{\mathrm{S}}=64$D:
$N_{\mathrm{S}}=256$BD: $N_{\mathrm{S}}=256$

(c) $4\times N_{\mathrm{S}}\times 4$

$-20$$-15$$-10$$-5$$0$$5$$10$$15$$20$$0$$10$$20$$30$$40$$50$Transmit Power
[dB]Achievable Rate [bit/s/Hz]D:
$\kappa_{\mathrm{F}}=\kappa_{\mathrm{B}}=0$BD:
$\kappa_{\mathrm{F}}=\kappa_{\mathrm{B}}=0$D:
$\kappa_{\mathrm{F}}=\kappa_{\mathrm{B}}=10$BD:
$\kappa_{\mathrm{F}}=\kappa_{\mathrm{B}}=10$D:
$\kappa_{\mathrm{F}}=\kappa_{\mathrm{B}}\to\infty$BD:
$\kappa_{\mathrm{F}}=\kappa_{\mathrm{B}}\to\infty$

(d) $4\times 128\times 4$

Figure 6:  Average achievable rate versus MIMO and RIS configurations. The
noise power is $\eta=\qty{-75}{dB}$, corresponding to a direct SNR of $-10$ to
30dB. ‘Alternate’ refers to the alternating optimization and ‘Decouple’ refers
to the low-complexity design. ‘D’ means D-RIS and ‘BD’ refers to fully-
connected BD-RIS.

Fig. 6 presents the achievable rate under different MIMO and RIS
configurations. At a transmit power $P=\qty{10}{dB}$, Fig.
LABEL:sub@fg:rate_beamforming shows that introducing a 128-element D-RIS to
$N_{\mathrm{T}}=N_{\mathrm{R}}=4$ MIMO can improve the achievable rate from
22.2bps/Hz to 29.2bps/Hz ($+\qty{31.5}{}$). A BD-RIS of group size 4 and 128
can further elevate those to 32.1bps/Hz ($+\qty{44.6}{}$) and 34bps/Hz
($+\qty{53.2}{}$), respectively. An interesting observation is that the rate
gap between the optimal AO approach (V-B1) – (40) and the shaping-inspired
solution (37), (40) narrows at larger group size and completely vanishes for a
fully-connected BD-RIS. This implies that joint RIS-transceiver designs can be
decoupled by first shaping the wireless channel and then optimizing the active
beamformer, which significantly simplifies the process at marginal performance
cost. Figs. LABEL:sub@fg:rate_txrx and LABEL:sub@fg:rate_sx also show that
both _absolute and relative_ rate gains of BD-RIS versus D-RIS increases with
the number of transmit and receive antennas and scattering elements,
especially at high SNR. For $N_{\mathrm{S}}=128$ and $P=\qty{20}{dB}$, the
achievable rate ratio of BD-RIS over D-RIS is $1.04$, $1.11$, and $1.13$ for
$N_{\mathrm{T}}=N_{\mathrm{R}}=1$, $4$, and $16$, respectively. For
$N_{\mathrm{T}}=N_{\mathrm{R}}=4$ and $P=\qty{20}{dB}$, this ratio amounts to
$1.03$, $1.08$, and $1.13$ for $N_{\mathrm{S}}=16$, $64$, and $256$,
respectively. Those observations align with the power gain results in Fig. 5
and highlight the rate benefits of BD-RIS over D-RIS in large-scale MIMO
systems. In the low power regime ($-20$ to -10dB), we also notice that the
slope of the achievable rate of BD-RIS is steeper than that of D-RIS. That is,
BD-RIS can help to activate more streams and achieve the asymptotic DoF at a
low transmit SNR. This is particularly visible in Fig. LABEL:sub@fg:rate_sx
where the topmost curve is almost a linear function of the transmit power. It
is expected from the shaping results in Fig. 2 that BD-RIS can significantly
enlarge all channel singular values for higher receive SNR. Finally, Fig.
LABEL:sub@fg:rate_kfactor shows that the gap between D\- and BD-RIS narrows as
the Rician K-factor increases and becomes indistinguishable in LoS
environment. The observation is expected from previous studies [30, 47, 31]
and aligns with Corollary 2.1, which suggests that the BD-RIS should be
deployed in rich-scattering environments to exploit its channel shaping
potential.

##  VII Conclusion

This paper analyzes the channel shaping capability of RIS in terms of singular
values redistribution. We consider a general BD architecture that allows
elements within the same group to interact, enabling more sophisticated
manipulation than D-RIS. This translates to a wider dynamic range of and
better tradeoff between singular values and significant power and rate gains,
especially in large-scale MIMO systems. We characterize the Pareto frontiers
of channel singular values via optimization approach and provide analytical
bounds for practical deployment scenarios. Specifically, the former is done by
proposing an efficient RCG algorithm for BD-RIS optimization problems, which
converges much faster than existing methods. We also present two beamforming
designs for rate maximization problem, one for optimal performance and the
other exploits channel shaping for lower complexity. Extensive simulations
show that the advantage of BD-RIS stems from its superior space alignment and
channel rearrangement capability, which scales with the number of elements,
group size, and MIMO dimensions.

###  -A Proof of Lemma 1

Let $\mathbf{H}=\sum_{n}\mathbf{u}_{n}\sigma_{n}\mathbf{v}_{n}^{\mathsf{H}}$
be the compact SVD of the equivalent channel. Since the singular vectors are
orthonormal, the $n$-th singular value can be expressed as

$\sigma_{n}=\mathbf{u}_{n}^{\mathsf{H}}\mathbf{H}\mathbf{v}_{n}=\mathbf{u}_{n}^{\mathsf{T}}\mathbf{H}^{*}\mathbf{v}_{n}^{*},$
(41)

whose differential with respect to $\mathbf{\Theta}_{g}^{*}$ is

$\displaystyle\partial\sigma_{n}$
$\displaystyle=\partial\mathbf{u}_{n}^{\mathsf{T}}\underbrace{\mathbf{H}^{*}\mathbf{v}_{n}^{*}}_{\sum_{m}\mathbf{u}_{m}^{*}\sigma_{m}\mathbf{v}_{m}^{\mathsf{T}}\mathbf{v}_{n}}+\mathbf{u}_{n}^{\mathsf{T}}\cdot\partial\mathbf{H}^{*}\cdot\mathbf{v}_{n}^{*}+\underbrace{\mathbf{u}_{n}^{\mathsf{T}}\mathbf{H}^{*}}_{\mathbf{u}_{n}^{\mathsf{T}}\sum_{m}\mathbf{u}_{m}^{*}\sigma_{m}\mathbf{v}_{m}^{\mathsf{T}}}\partial\mathbf{v}_{n}^{*}$
$\displaystyle=\underbrace{\partial\mathbf{u}_{n}^{\mathsf{T}}\mathbf{u}_{n}^{*}}_{\partial
1=0}\cdot\sigma_{n}+\mathbf{u}_{n}^{\mathsf{T}}\cdot\partial\mathbf{H}^{*}\cdot\mathbf{v}_{n}^{*}+\sigma_{n}\cdot\underbrace{\mathbf{v}_{n}^{\mathsf{T}}\partial\mathbf{v}_{n}^{*}}_{\partial
1=0}$
$\displaystyle=\mathbf{u}_{n}^{\mathsf{T}}\mathbf{H}_{\mathrm{B},g}^{*}\cdot\partial\mathbf{\Theta}_{g}^{*}\cdot\mathbf{H}_{\mathrm{F},g}^{*}\mathbf{v}_{n}^{*}$
$\displaystyle=\mathrm{tr}(\mathbf{H}_{\mathrm{F},g}^{*}\mathbf{v}_{n}^{*}\mathbf{u}_{n}^{\mathsf{T}}\mathbf{H}_{\mathrm{B},g}^{*}\cdot\partial\mathbf{\Theta}_{g}^{*}).$

According to [67], the corresponding complex derivative is

$\frac{\partial\sigma_{n}}{\partial\mathbf{\Theta}_{g}^{*}}=\mathbf{H}_{\mathrm{B},g}^{\mathsf{H}}\mathbf{u}_{n}\mathbf{v}_{n}^{\mathsf{H}}\mathbf{H}_{\mathrm{F},g}^{\mathsf{H}}.$
(42)

A linear combination of (42) yields (21).

###  -B Proof of Proposition 1

The scattering matrix of BD-RIS can be decomposed as

$\mathbf{\Theta}=\mathbf{L}\mathbf{\Theta}_{\mathrm{D}}\mathbf{R}^{\mathsf{H}},$
(43)

where $\mathbf{\Theta}_{\mathrm{D}}\in\mathbb{U}^{N_{\mathrm{S}}\times
N_{\mathrm{S}}}$ corresponds to D-RIS and
$\mathbf{L},\mathbf{R}\in\mathbb{U}^{N_{\mathrm{S}}\times N_{\mathrm{S}}}$ are
block-diagonal matrices of $L\times L$ unitary blocks. Manipulating
$\mathbf{L}$ and $\mathbf{R}$ rotates the linear spans of
$\bar{\mathbf{H}}_{\mathrm{B}}\triangleq\mathbf{H}_{\mathrm{B}}\mathbf{L}$ and
$\bar{\mathbf{H}}_{\mathrm{F}}\triangleq\mathbf{R}^{\mathsf{H}}\mathbf{H}_{\mathrm{F}}$
and maintains their rank. On the other hand, there exists a
$\mathbf{\Theta}_{\mathrm{D}}$ such that

$\begin{split}\mathrm{rank}(\mathbf{H}_{\mathrm{B}}\mathbf{\Theta}_{\mathrm{D}}\mathbf{H}_{\mathrm{F}})&=\min\bigl{(}\mathrm{rank}(\mathbf{H}_{\mathrm{B}}),\mathrm{rank}(\mathbf{\Theta}_{\mathrm{D}}),\mathrm{rank}(\mathbf{H}_{\mathrm{F}})\bigr{)}\\\
&=\min\bigl{(}\mathrm{rank}(\bar{\mathbf{H}}_{\mathrm{B}}),N_{\mathrm{S}},\mathrm{rank}(\bar{\mathbf{H}}_{\mathrm{F}})\bigr{)}\\\
&=\max_{\mathbf{\Theta}}\
\mathrm{rank}(\mathbf{H}_{\mathrm{B}}\mathbf{\Theta}\mathbf{H}_{\mathrm{F}})\end{split}$

The same result holds if the direct link is present.

###  -C Proof of Proposition 2

We consider rank-$k$ forward channel and the proof follows similarly for
rank-$k$ backward channel. Let
$\mathbf{H}_{\mathrm{F}}=\mathbf{U}_{\mathrm{F}}\mathbf{\Sigma}_{\mathrm{F}}\mathbf{V}_{\mathrm{F}}^{\mathsf{H}}$
be the compact SVD of the forward channel. The channel Gram matrix
$\mathbf{G}\triangleq\mathbf{H}\mathbf{H}^{\mathsf{H}}$ can be written as

$\begin{split}\mathbf{G}&=\mathbf{H}_{\mathrm{D}}\mathbf{H}_{\mathrm{D}}^{\mathsf{H}}+\mathbf{H}_{\mathrm{B}}\mathbf{\Theta}\mathbf{U}_{\mathrm{F}}\mathbf{\Sigma}_{\mathrm{F}}\mathbf{\Sigma}_{\mathrm{F}}^{\mathsf{H}}\mathbf{U}_{\mathrm{F}}^{\mathsf{H}}\mathbf{\Theta}^{\mathsf{H}}\mathbf{H}_{\mathrm{B}}^{\mathsf{H}}\\\
&\quad+\mathbf{H}_{\mathrm{B}}\mathbf{\Theta}\mathbf{U}_{\mathrm{F}}\mathbf{\Sigma}_{\mathrm{F}}\mathbf{V}_{\mathrm{F}}^{\mathsf{H}}\mathbf{H}_{\mathrm{D}}^{\mathsf{H}}+\mathbf{H}_{\mathrm{D}}\mathbf{V}_{\mathrm{F}}\mathbf{\Sigma}_{\mathrm{F}}\mathbf{U}_{\mathrm{F}}^{\mathsf{H}}\mathbf{\Theta}^{\mathsf{H}}\mathbf{H}_{\mathrm{B}}^{\mathsf{H}}\\\
&=\mathbf{H}_{\mathrm{D}}(\mathbf{I}-\mathbf{V}_{\mathrm{F}}\mathbf{V}_{\mathrm{F}}^{\mathsf{H}})\mathbf{H}_{\mathrm{D}}^{\mathsf{H}}\\\
&\quad+(\mathbf{H}_{\mathrm{B}}\mathbf{\Theta}\mathbf{U}_{\mathrm{F}}\mathbf{\Sigma}_{\mathrm{F}}+\mathbf{H}_{\mathrm{D}}\mathbf{V}_{\mathrm{F}})(\mathbf{\Sigma}_{\mathrm{F}}\mathbf{U}_{\mathrm{F}}^{\mathsf{H}}\mathbf{\Theta}^{\mathsf{H}}\mathbf{H}_{\mathrm{B}}^{\mathsf{H}}+\mathbf{V}_{\mathrm{F}}^{\mathsf{H}}\mathbf{H}_{\mathrm{D}}^{\mathsf{H}})\\\
&=\mathbf{Y}+\mathbf{Z}\mathbf{Z}^{\mathsf{H}},\end{split}$

where we define
$\mathbf{Y}\triangleq\mathbf{H}_{\mathrm{D}}(\mathbf{I}-\mathbf{V}_{\mathrm{F}}\mathbf{V}_{\mathrm{F}}^{\mathsf{H}})\mathbf{H}_{\mathrm{D}}^{\mathsf{H}}\in\mathbb{H}^{N_{\mathrm{R}}\times
N_{\mathrm{R}}}$ and
$\mathbf{Z}\triangleq\mathbf{H}_{\mathrm{B}}\mathbf{\Theta}\mathbf{U}_{\mathrm{F}}\mathbf{\Sigma}_{\mathrm{F}}+\mathbf{H}_{\mathrm{D}}\mathbf{V}_{\mathrm{F}}\in\mathbb{C}^{N_{\mathrm{R}}\times
k}$. That is to say, $\mathbf{G}$ can be expressed as a Hermitian matrix plus
$k$ rank-1 perturbations. According to the Cauchy interlacing formula [63],
the $n$-th eigenvalue of $\mathbf{G}$ is bounded by

$\displaystyle\lambda_{n}(\mathbf{G})$
$\displaystyle\leq\lambda_{n-k}(\mathbf{Y}),$ $\displaystyle\text{if }n>k,$
(44) $\displaystyle\lambda_{n}(\mathbf{G})$
$\displaystyle\geq\lambda_{n}(\mathbf{Y}),$ $\displaystyle\text{if }n<N-k+1.$
(45)

Since $\mathbf{Y}=\mathbf{T}\mathbf{T}^{\mathsf{H}}$ is positive semi-
definite, taking the square roots of (44) and (45) gives (23a) and (23b).

###  -D Proof of Proposition 3

Let
$\mathbf{H}_{\mathrm{B}}=\mathbf{U}_{\mathrm{B}}\mathbf{\Sigma}_{\mathrm{B}}\mathbf{V}_{\mathrm{B}}^{\mathsf{H}}$
and
$\mathbf{H}_{\mathrm{F}}=\mathbf{U}_{\mathrm{F}}\mathbf{\Sigma}_{\mathrm{F}}\mathbf{V}_{\mathrm{F}}^{\mathsf{H}}$
be the SVD of the backward and forward channels, respectively. The scattering
matrix of fully-connected BD-RIS can be decomposed as

$\mathbf{\Theta}=\mathbf{V}_{\mathrm{B}}\mathbf{X}\mathbf{U}_{\mathrm{F}}^{\mathsf{H}},$
(46)

where $\mathbf{X}\in\mathbb{U}^{N_{\mathrm{S}}\times N_{\mathrm{S}}}$ is a
unitary matrix to be designed. The equivalent channel is thus a function of
$\mathbf{X}$

$\mathbf{H}=\mathbf{H}_{\mathrm{B}}\mathbf{\Theta}\mathbf{H}_{\mathrm{F}}=\mathbf{U}_{\mathrm{B}}\mathbf{\Sigma}_{\mathrm{B}}\mathbf{X}\mathbf{\Sigma}_{\mathrm{F}}\mathbf{V}_{\mathrm{F}}^{\mathsf{H}}.$
(47)

Since
$\mathrm{sv}(\mathbf{U}\mathbf{A}\mathbf{V}^{\mathsf{H}})=\mathrm{sv}(\mathbf{A})$
for unitary $\mathbf{U}$ and $\mathbf{V}$, we have

$\displaystyle\mathrm{sv}(\mathbf{H})$
$\displaystyle=\mathrm{sv}(\mathbf{U}_{\mathrm{B}}\mathbf{\Sigma}_{\mathrm{B}}\mathbf{X}\mathbf{\Sigma}_{\mathrm{F}}\mathbf{V}_{\mathrm{F}}^{\mathsf{H}})$
$\displaystyle=\mathrm{sv}(\mathbf{\Sigma}_{\mathrm{B}}\mathbf{X}\mathbf{\Sigma}_{\mathrm{F}})$
$\displaystyle=\mathrm{sv}(\bar{\mathbf{U}}_{\mathrm{B}}\mathbf{\Sigma}_{\mathrm{B}}\mathbf{\bar{V}}_{\mathrm{B}}^{\mathsf{H}}\bar{\mathbf{U}}_{\mathrm{F}}\mathbf{\Sigma}_{\mathrm{F}}\mathbf{\bar{V}}_{\mathrm{F}}^{\mathsf{H}})$
$\displaystyle=\mathrm{sv}(\mathbf{BF}),$

where $\bar{\mathbf{U}}_{\mathrm{B}/\mathrm{F}}$ and
$\bar{\mathbf{V}}_{\mathrm{B}/\mathrm{F}}$ are arbitrary unitary matrices.

###  -E Proof of Lemma 2

The differential of $R$ with respect to $\mathbf{\Theta}_{g}^{*}$ is [67]

$\displaystyle\partial R$
$\displaystyle=\frac{1}{\eta}\mathrm{tr}\biggl{\\{}\partial\mathbf{H}^{*}\cdot\mathbf{Q}^{\mathsf{T}}\mathbf{H}^{\mathsf{T}}\Bigl{(}\mathbf{I}+\frac{\mathbf{H}^{*}\mathbf{Q}^{\mathsf{T}}\mathbf{H}^{\mathsf{T}}}{\eta}\Bigr{)}^{-1}\biggr{\\}}$
$\displaystyle=\frac{1}{\eta}\mathrm{tr}\biggl{\\{}\mathbf{H}_{\mathrm{B},g}^{*}\cdot\partial\mathbf{\Theta}_{g}^{*}\cdot\mathbf{H}_{\mathrm{F},g}^{*}\mathbf{Q}^{\mathsf{T}}\mathbf{H}^{\mathsf{T}}\Bigl{(}\mathbf{I}+\frac{\mathbf{H}^{*}\mathbf{Q}^{\mathsf{T}}\mathbf{H}^{\mathsf{T}}}{\eta}\Bigr{)}^{-1}\biggr{\\}}$
$\displaystyle=\frac{1}{\eta}\mathrm{tr}\biggl{\\{}\mathbf{H}_{\mathrm{F},g}^{*}\mathbf{Q}^{\mathsf{T}}\mathbf{H}^{\mathsf{T}}\Bigl{(}\mathbf{I}+\frac{\mathbf{H}^{*}\mathbf{Q}^{\mathsf{T}}\mathbf{H}^{\mathsf{T}}}{\eta}\Bigr{)}^{-1}\mathbf{H}_{\mathrm{B},g}^{*}\cdot\partial\mathbf{\Theta}_{g}^{*}\biggr{\\}},$

and the corresponding complex derivative is (39).

###  -F Proof of Proposition 4

The differential of (V-A) with respect to $\mathbf{\Theta}_{g}^{*}$ is

$\displaystyle\partial\lVert\mathbf{H}\rVert_{\mathrm{F}}^{2}$
$\displaystyle=\mathrm{tr}\bigl{(}\mathbf{H}_{\mathrm{B},g}^{*}\cdot\partial\mathbf{\Theta}_{g}^{*}\cdot\mathbf{H}_{\mathrm{F},g}^{*}(\mathbf{H}_{\mathrm{D}}^{\mathsf{T}}+\mathbf{H}_{\mathrm{F}}^{\mathsf{T}}\mathbf{\Theta}^{\mathsf{T}}\mathbf{H}_{\mathrm{B}}^{\mathsf{T}})\bigr{)}$
$\displaystyle=\mathrm{tr}\bigl{(}\mathbf{H}_{\mathrm{F},g}^{*}(\mathbf{H}_{\mathrm{D}}^{\mathsf{T}}+\mathbf{H}_{\mathrm{F}}^{\mathsf{T}}\mathbf{\Theta}^{\mathsf{T}}\mathbf{H}_{\mathrm{B}}^{\mathsf{T}})\mathbf{H}_{\mathrm{B},g}^{*}\cdot\partial\mathbf{\Theta}_{g}^{*}\bigr{)}$

and the corresponding complex derivative is

$\frac{\partial\lVert\mathbf{H}\rVert_{\mathrm{F}}^{2}}{\partial\mathbf{\Theta}_{g}^{*}}=\mathbf{H}_{\mathrm{B},g}^{\mathsf{H}}(\mathbf{H}_{\mathrm{D}}+\mathbf{H}_{\mathrm{B}}\mathbf{\Theta}\mathbf{H}_{\mathrm{F}})\mathbf{H}_{\mathrm{F},g}^{\mathsf{H}}=\mathbf{M}_{g}.$
(48)

First, we approximate the quadratic objective (V-A) by its local Taylor
expansion Θ∑_g 2 ℜ{ tr(Θ_g^H M_g) } Θ_g^H Θ_g=I, ∀g. Let
$\mathbf{M}_{g}=\mathbf{U}_{g}\mathbf{\Sigma}_{g}\mathbf{V}_{g}^{\mathsf{H}}$
be the compact SVD of $\mathbf{M}_{g}$. We have

$\Re\bigl{\\{}\mathrm{tr}(\mathbf{\Theta}_{g}^{\mathsf{H}}\mathbf{M}_{g})\bigr{\\}}=\Re\bigl{\\{}\mathrm{tr}(\mathbf{\Sigma}_{g}\mathbf{V}_{g}^{\mathsf{H}}\mathbf{\Theta}_{g}^{\mathsf{H}}\mathbf{U}_{g})\bigr{\\}}\leq\mathrm{tr}(\mathbf{\Sigma}_{g}).$
(49)

The upper bound is tight when
$\mathbf{V}_{g}^{\mathsf{H}}\mathbf{\Theta}_{g}^{\mathsf{H}}\mathbf{U}_{g}=\mathbf{I}$,
which implies the optimal solution of (-F) is
$\tilde{\mathbf{\Theta}}_{g}=\mathbf{U}_{g}\mathbf{V}_{g}^{\mathsf{H}}$,
$\forall g$.

Next, we prove that solving (-F) successively does not decrease (V-A). Since
$\tilde{\mathbf{\Theta}}$ optimal for problem (-F), we have
$\sum_{g}2\Re\bigl{\\{}\mathrm{tr}(\tilde{\mathbf{\Theta}}_{g}^{\mathsf{H}}\mathbf{M}_{g})\bigr{\\}}\geq\sum_{g}2\Re\bigl{\\{}\mathrm{tr}(\mathbf{\Theta}_{g}^{\mathsf{H}}\mathbf{M}_{g})\bigr{\\}}$
which is explicitly expressed by (51). On the other hand, expanding
$\lVert\sum_{g}\mathbf{H}_{\mathrm{B},g}\tilde{\mathbf{\Theta}}_{g}\mathbf{H}_{\mathrm{F},g}-\sum_{g}\mathbf{H}_{\mathrm{B},g}\mathbf{\Theta}_{g}\mathbf{H}_{\mathrm{F},g}\rVert_{\mathrm{F}}^{2}\geq
0$ gives (52). Adding (51) and (52), we have

$2\Re\Bigl{\\{}\mathrm{tr}(\tilde{\mathbf{\Theta}}^{\mathsf{H}}\mathbf{H}_{\mathrm{B}}^{\mathsf{H}}\mathbf{H}_{\mathrm{D}}\mathbf{H}_{\mathrm{F}}^{\mathsf{H}})\Bigr{\\}}+\mathrm{tr}(\mathbf{H}_{\mathrm{F}}^{\mathsf{H}}\tilde{\mathbf{\Theta}}^{\mathsf{H}}\mathbf{H}_{\mathrm{B}}^{\mathsf{H}}\mathbf{H}_{\mathrm{B}}\tilde{\mathbf{\Theta}}\mathbf{H}_{\mathrm{F}})\\\
\geq
2\Re\Bigl{\\{}\mathrm{tr}({\mathbf{\Theta}}^{\mathsf{H}}\mathbf{H}_{\mathrm{B}}^{\mathsf{H}}\mathbf{H}_{\mathrm{D}}\mathbf{H}_{\mathrm{F}}^{\mathsf{H}})\Bigr{\\}}+\mathrm{tr}(\mathbf{H}_{\mathrm{F}}^{\mathsf{H}}{\mathbf{\Theta}}^{\mathsf{H}}\mathbf{H}_{\mathrm{B}}^{\mathsf{H}}\mathbf{H}_{\mathrm{B}}{\mathbf{\Theta}}\mathbf{H}_{\mathrm{F}}),$
(50)

which suggests that updating $\tilde{\mathbf{\Theta}}$ does not decrease
(V-A).

$2\Re\Bigl{\\{}\sum\limits_{g}\mathrm{tr}(\tilde{\mathbf{\Theta}}_{g}^{\mathsf{H}}\mathbf{H}_{\mathrm{B},g}^{\mathsf{H}}\mathbf{H}_{\mathrm{D}}\mathbf{H}_{\mathrm{F},g}^{\mathsf{H}})+\sum\limits_{g_{1},g_{2}}\mathrm{tr}(\tilde{\mathbf{\Theta}}_{g_{1}}^{\mathsf{H}}\mathbf{H}_{\mathrm{B},g_{1}}^{\mathsf{H}}\mathbf{H}_{\mathrm{B},g_{2}}\mathbf{\Theta}_{g_{2}}\mathbf{H}_{\mathrm{F},g_{2}}\mathbf{H}_{\mathrm{F},g_{1}}^{\mathsf{H}})\Bigr{\\}}\geq
2\Re\Bigl{\\{}\sum\limits_{g}\mathrm{tr}({\mathbf{\Theta}}_{g}^{\mathsf{H}}\mathbf{H}_{\mathrm{B},g}^{\mathsf{H}}\mathbf{H}_{\mathrm{D}}\mathbf{H}_{\mathrm{F},g}^{\mathsf{H}})+\sum\limits_{g_{1},g_{2}}\mathrm{tr}({\mathbf{\Theta}}_{g_{1}}^{\mathsf{H}}\mathbf{H}_{\mathrm{B},g_{1}}^{\mathsf{H}}\mathbf{H}_{\mathrm{B},g_{2}}\mathbf{\Theta}_{g_{2}}\mathbf{H}_{\mathrm{F},g_{2}}\mathbf{H}_{\mathrm{F},g_{1}}^{\mathsf{H}})\Bigr{\\}}$

(51)

$\sum\limits_{g_{1},g_{2}}\mathrm{tr}(\mathbf{H}_{\mathrm{F},g_{1}}^{\mathsf{H}}\tilde{\mathbf{\Theta}}_{g_{1}}^{\mathsf{H}}\mathbf{H}_{\mathrm{B},g_{1}}^{\mathsf{H}}\mathbf{H}_{\mathrm{B},g_{2}}\tilde{\mathbf{\Theta}}_{g_{2}}\mathbf{H}_{\mathrm{F},g_{2}})-2\Re\Bigl{\\{}\sum\limits_{g_{1},g_{2}}\mathrm{tr}(\mathbf{H}_{\mathrm{F},g_{1}}^{\mathsf{H}}\tilde{\mathbf{\Theta}}_{g_{1}}^{\mathsf{H}}\mathbf{H}_{\mathrm{B},g_{1}}^{\mathsf{H}}\mathbf{H}_{\mathrm{B},g_{2}}\mathbf{\Theta}_{g_{2}}\mathbf{H}_{\mathrm{F},g_{2}})\Bigr{\\}}+\sum\limits_{g_{1},g_{2}}\mathrm{tr}(\mathbf{H}_{\mathrm{F},g_{1}}^{\mathsf{H}}{\mathbf{\Theta}}_{g_{1}}^{\mathsf{H}}\mathbf{H}_{\mathrm{B},g_{1}}^{\mathsf{H}}\mathbf{H}_{\mathrm{B},g_{2}}{\mathbf{\Theta}}_{g_{2}}\mathbf{H}_{\mathrm{F},g_{2}})\geq
0$

(52)

Finally, we prove that the converging point of (-F), denoted by
$\tilde{\mathbf{\Theta}}^{?}$, is a stationary point of (V-A). The Karush-
Kuhn-Tucker (KKT) conditions of (V-A) and (-F) are equivalent in terms of
primal/dual feasibility and complementary slackness, while the stationary
conditions are respectively, $\forall g$,

$\displaystyle\mathbf{H}_{\mathrm{B},g}^{\mathsf{H}}(\mathbf{H}_{\mathrm{D}}+\mathbf{H}_{\mathrm{B}}\mathbf{\Theta}^{\star}\mathbf{H}_{\mathrm{F}})\mathbf{H}_{\mathrm{F},g}^{\mathsf{H}}-\mathbf{\Theta}_{g}^{\star}\mathbf{\Lambda}_{g}^{\mathsf{H}}=0,$
(53)
$\displaystyle\mathbf{M}_{g}-\mathbf{\Theta}_{g}^{\star}\mathbf{\Lambda}_{g}^{\mathsf{H}}=0.$
(54)

On convergence, (54) becomes
$\mathbf{H}_{\mathrm{B},g}^{\mathsf{H}}(\mathbf{H}_{\mathrm{D}}+\mathbf{H}_{\mathrm{B}}\mathbf{\Theta}^{?}\mathbf{H}_{\mathrm{F}})\mathbf{H}_{\mathrm{F},g}^{\mathsf{H}}-\mathbf{\Theta}_{g}^{?}\mathbf{\Lambda}_{g}^{\mathsf{H}}=0$
and reduces to (53). The proof is thus completed.

## References

  * [1] E. Basar, M. D. Renzo, J. D. Rosny, M. Debbah, M.-S. Alouini, and R. Zhang, “Wireless communications through reconfigurable intelligent surfaces,” _IEEE Access_ , vol. 7, pp. 116 753–116 773, 2019.
  * [2] Q. Wu and R. Zhang, “Intelligent reflecting surface enhanced wireless network via joint active and passive beamforming,” _IEEE Transactions on Wireless Communications_ , vol. 18, pp. 5394–5409, Nov 2019.
  * [3] H. Guo, Y.-C. Liang, J. Chen, and E. G. Larsson, “Weighted sum-rate maximization for reconfigurable intelligent surface aided wireless networks,” _IEEE Transactions on Wireless Communications_ , vol. 19, pp. 3064–3076, May 2020.
  * [4] Y. Liu, Y. Zhang, X. Zhao, S. Geng, P. Qin, and Z. Zhou, “Dynamic-controlled RIS assisted multi-user MISO downlink system: Joint beamforming design,” _IEEE Transactions on Green Communications and Networking_ , vol. 6, pp. 1069–1081, Jun 2022.
  * [5] Y. He, Y. Cai, H. Mao, and G. Yu, “RIS-assisted communication radar coexistence: Joint beamforming design and analysis,” _IEEE Journal on Selected Areas in Communications_ , vol. 40, pp. 2131–2145, Jul 2022.
  * [6] H. Luo, R. Liu, M. Li, Y. Liu, and Q. Liu, “Joint beamforming design for RIS-assisted integrated sensing and communication systems,” _IEEE Transactions on Vehicular Technology_ , vol. 71, pp. 13 393–13 397, Dec 2022.
  * [7] M. Hua, Q. Wu, C. He, S. Ma, and W. Chen, “Joint active and passive beamforming design for IRS-aided radar-communication,” _IEEE Transactions on Wireless Communications_ , vol. 22, pp. 2278–2294, Apr 2023.
  * [8] Q. Wu and R. Zhang, “Joint active and passive beamforming optimization for intelligent reflecting surface assisted SWIPT under QoS constraints,” _IEEE Journal on Selected Areas in Communications_ , vol. 38, no. 8, pp. 1735–1748, Aug 2020.
  * [9] Z. Feng, B. Clerckx, and Y. Zhao, “Waveform and beamforming design for intelligent reflecting surface aided wireless power transfer: Single-user and multi-user solutions,” _IEEE Transactions on Wireless Communications_ , 2022.
  * [10] Y. Zhao, B. Clerckx, and Z. Feng, “IRS-aided SWIPT: Joint waveform, active and passive beamforming design under nonlinear harvester model,” _IEEE Transactions on Communications_ , vol. 70, pp. 1345–1359, 2022.
  * [11] R. Karasik, O. Simeone, M. D. Renzo, and S. S. Shitz, “Beyond max-SNR: Joint encoding for reconfigurable intelligent surfaces,” in _2020 IEEE International Symposium on Information Theory (ISIT)_ , Jun 2020, pp. 2965–2970.
  * [12] E. Basar, “Reconfigurable intelligent surface-based index modulation: A new beyond MIMO paradigm for 6G,” _IEEE Transactions on Communications_ , vol. 68, pp. 3187–3196, May 2020.
  * [13] J. Ye, S. Guo, S. Dang, B. Shihada, and M.-S. Alouini, “On the capacity of reconfigurable intelligent surface assisted MIMO symbiotic communications,” _IEEE Transactions on Wireless Communications_ , vol. 21, pp. 1943–1959, Mar 2022.
  * [14] Y.-C. Liang, Q. Zhang, E. G. Larsson, and G. Y. Li, “Symbiotic radio: Cognitive backscattering communications for future wireless networks,” _IEEE Transactions on Cognitive Communications and Networking_ , vol. 6, pp. 1242–1255, Dec 2020.
  * [15] Y. Zhao and B. Clerckx, “RIScatter: Unifying backscatter communication and reconfigurable intelligent surface,” _IEEE Journal on Selected Areas in Communications_ , pp. 1–1, Dec 2024.
  * [16] H. Yang, H. Ding, K. Cao, M. Elkashlan, H. Li, and K. Xin, “A RIS-segmented symbiotic ambient backscatter communication system,” _IEEE Transactions on Vehicular Technology_ , vol. 73, pp. 812–825, Jan 2024.
  * [17] E. Basar, “Reconfigurable intelligent surfaces for doppler effect and multipath fading mitigation,” _Frontiers in Communications and Networks_ , vol. 2, May 2021.
  * [18] E. Arslan, I. Yildirim, F. Kilinc, and E. Basar, “Over-the-air equalization with reconfigurable intelligent surfaces,” _IET Communications_ , vol. 16, pp. 1486–1497, Aug 2022.
  * [19] O. Ozdogan, E. Bjornson, and E. G. Larsson, “Using intelligent reflecting surfaces for rank improvement in MIMO communications,” in _ICASSP 2020 - 2020 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP)_ , May 2020, pp. 9160–9164.
  * [20] Y. Yang, B. Zheng, S. Zhang, and R. Zhang, “Intelligent reflecting surface meets OFDM: Protocol design and rate maximization,” _IEEE Transactions on Communications_ , vol. 68, pp. 4522–4535, Jul 2020.
  * [21] G. Chen and Q. Wu, “Fundamental limits of intelligent reflecting surface aided multiuser broadcast channel,” _IEEE Transactions on Communications_ , vol. 71, pp. 5904–5919, Oct 2023.
  * [22] M. A. ElMossallamy, H. Zhang, R. Sultan, K. G. Seddik, L. Song, G. Y. Li, and Z. Han, “On spatial multiplexing using reconfigurable intelligent surfaces,” _IEEE Wireless Communications Letters_ , vol. 10, pp. 226–230, Feb 2021.
  * [23] S. Meng, W. Tang, W. Chen, J. Lan, Q. Y. Zhou, Y. Han, X. Li, and S. Jin, “Rank optimization for MIMO channel with RIS: Simulation and measurement,” _IEEE Wireless Communications Letters_ , vol. 13, pp. 437–441, Feb 2024.
  * [24] Y. Zheng, T. Lin, and Y. Zhu, “Passive beamforming for IRS-assisted MU-MIMO systems with one-bit ADCs: An SER minimization design approach,” _IEEE Communications Letters_ , vol. 26, pp. 1101–1105, May 2022.
  * [25] W. Huang, B. Lei, S. He, C. Kai, and C. Li, “Condition number improvement of IRS-aided near-field MIMO channels,” in _2023 IEEE International Conference on Communications Workshops (ICC Workshops)_ , May 2023, pp. 1210–1215.
  * [26] A. H. Bafghi, V. Jamali, M. Nasiri-Kenari, and R. Schober, “Degrees of freedom of the K-user interference channel assisted by active and passive IRSs,” _IEEE Transactions on Communications_ , vol. 70, pp. 3063–3080, May 2022.
  * [27] S. Zheng, B. Lv, T. Zhang, Y. Xu, G. Chen, R. Wang, and P. C. Ching, “On DoF of active RIS-assisted MIMO interference channel with arbitrary antenna configurations: When will RIS help?” _IEEE Transactions on Vehicular Technology_ , Dec 2023.
  * [28] S. H. Chae and K. Lee, “Cooperative communication for the rank-deficient MIMO interference channel with a reconfigurable intelligent surface,” _IEEE Transactions on Wireless Communications_ , vol. 22, pp. 2099–2112, Mar 2023.
  * [29] S. Shen and B. Clerckx, “Beamforming optimization for MIMO wireless power transfer with nonlinear energy harvesting: RF combining versus DC combining,” _IEEE Transactions on Wireless Communications_ , vol. 20, pp. 199–213, Jan 2021.
  * [30] S. Shen, B. Clerckx, and R. Murch, “Modeling and architecture design of reconfigurable intelligent surfaces using scattering parameter network analysis,” _IEEE Transactions on Wireless Communications_ , vol. 21, pp. 1229–1243, Feb 2022.
  * [31] M. Nerini, S. Shen, and B. Clerckx, “Closed-form global optimization of beyond diagonal reconfigurable intelligent surfaces,” _IEEE Transactions on Wireless Communications_ , vol. 23, pp. 1037–1051, Feb 2024.
  * [32] M. Nerini, S. Shen, H. Li, and B. Clerckx, “Beyond diagonal reconfigurable intelligent surfaces utilizing graph theory: Modeling, architecture design, and optimization,” _IEEE Transactions on Wireless Communications_ , pp. 1–1, May 2024.
  * [33] I. Santamaria, M. Soleymani, E. Jorswieck, and J. Gutiérrez, “SNR maximization in beyond diagonal RIS-assisted single and multiple antenna links,” _IEEE Signal Processing Letters_ , vol. 30, pp. 923–926, 2023.
  * [34] ——, “Interference leakage minimization in RIS-assisted MIMO interference channels,” in _ICASSP 2023 - 2023 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP)_ , vol. 39, Jun 2023, pp. 1–5.
  * [35] H.-R. Ahn, _Asymmetric Passive Components in Microwave Integrated Circuits_. Hoboken, NJ, USA: Wiley, 2006.
  * [36] H. Li, S. Shen, and B. Clerckx, “Beyond diagonal reconfigurable intelligent surfaces: A multi-sector mode enabling highly directional full-space wireless coverage,” _IEEE Journal on Selected Areas in Communications_ , vol. 41, pp. 2446–2460, Aug 2023.
  * [37] H. Li, S. Shen, Y. Zhang, and B. Clerckx, “Channel estimation and beamforming for beyond diagonal reconfigurable intelligent surfaces,” _arXiv:2403.18087_ , 2024.
  * [38] H. Li, S. Shen, M. Nerini, M. D. Renzo, and B. Clerckx, “Beyond diagonal reconfigurable intelligent surfaces with mutual coupling: Modeling and optimization,” _IEEE Communications Letters_ , pp. 1–1, Oct 2024.
  * [39] H. Li, M. Nerini, S. Shen, and B. Clerckx, “Wideband modeling and beamforming for beyond diagonal reconfigurable intelligent surfaces,” _arXiv:2403.12893_ , 2024.
  * [40] T. Fang and Y. Mao, “A low-complexity beamforming design for beyond-diagonal RIS aided multi-user networks,” _IEEE Communications Letters_ , pp. 1–1, Jul 2023.
  * [41] Y. Zhou, Y. Liu, H. Li, Q. Wu, S. Shen, and B. Clerckx, “Optimizing power consumption, energy efficiency and sum-rate using beyond diagonal RIS — a unified approach,” _IEEE Transactions on Wireless Communications_ , pp. 1–1, 2023.
  * [42] M. Soleymani, I. Santamaria, E. Jorswieck, and B. Clerckx, “Optimization of rate-splitting multiple access in beyond diagonal RIS-assisted URLLC systems,” _IEEE Transactions on Wireless Communications_ , pp. 1–1, Jul 2024.
  * [43] G. Bartoli, A. Abrardo, N. Decarli, D. Dardari, and M. D. Renzo, “Spatial multiplexing in near field MIMO channels with reconfigurable intelligent surfaces,” _IET Signal Processing_ , vol. 17, Mar 2023.
  * [44] A. Mishra, Y. Mao, C. D’Andrea, S. Buzzi, and B. Clerckx, “Transmitter side beyond-diagonal reconfigurable intelligent surface for massive MIMO networks,” _IEEE Wireless Communications Letters_ , vol. 13, pp. 352–356, Feb 2024.
  * [45] M. Nerini, S. Shen, and B. Clerckx, “Discrete-value group and fully connected architectures for beyond diagonal reconfigurable intelligent surfaces,” _IEEE Transactions on Vehicular Technology_ , vol. 72, pp. 16 354–16 368, Dec 2023.
  * [46] M. T. Ivrlac and J. A. Nossek, “Toward a circuit theory of communication,” _IEEE Transactions on Circuits and Systems I: Regular Papers_ , vol. 57, pp. 1663–1683, Jul 2010.
  * [47] H. Li, S. Shen, and B. Clerckx, “Beyond diagonal reconfigurable intelligent surfaces: From transmitting and reflecting modes to single-, group-, and fully-connected architectures,” _IEEE Transactions on Wireless Communications_ , vol. 22, pp. 2311–2324, Apr 2023.
  * [48] G. H. Hardy, J. E. Littlewood, and G. Pólya, _Inequalities_. Cambridge, UK: Cambridge University Press, 1952.
  * [49] P.-A. Absil, R. Mahony, and R. Sepulchre, _Optimization Algorithms on Matrix Manifolds_. Princeton, NJ, USA: Princeton University Press, 2009.
  * [50] C. Pan, G. Zhou, K. Zhi, S. Hong, T. Wu, Y. Pan, H. Ren, M. D. Renzo, A. L. Swindlehurst, R. Zhang, and A. Y. Zhang, “An overview of signal processing techniques for RIS/IRS-aided wireless systems,” _IEEE Journal of Selected Topics in Signal Processing_ , vol. 16, pp. 883–917, Aug 2022.
  * [51] J. Nocedal and S. J. Wright, _Numerical Optimization_. Springer, Sep 2006.
  * [52] L. Armijo, “Minimization of functions having lipschitz continuous first partial derivatives,” _Pacific Journal of Mathematics_ , vol. 16, pp. 1–3, Jan 1966.
  * [53] A. Edelman, T. A. Arias, and S. T. Smith, “The geometry of algorithms with orthogonality constraints,” _SIAM Journal on Matrix Analysis and Applications_ , vol. 20, pp. 303–353, Jan 1998.
  * [54] T. E. Abrudan, J. Eriksson, and V. Koivunen, “Steepest descent algorithms for optimization under unitary matrix constraint,” _IEEE Transactions on Signal Processing_ , vol. 56, pp. 1134–1147, Mar 2008.
  * [55] C. Moler and C. V. Loan, “Nineteen dubious ways to compute the exponential of a matrix, twenty-five years later,” _SIAM Review_ , vol. 45, pp. 3–49, Jan 2003.
  * [56] D. Semmler, M. Joham, and W. Utschick, “High SNR analysis of RIS-aided MIMO broadcast channels,” in _2023 IEEE 24th International Workshop on Signal Processing Advances in Wireless Communications (SPAWC)_ , Sep 2023, pp. 221–225.
  * [57] W. Fulton, “Eigenvalues, invariant factors, highest weights, and schubert calculus,” _Bulletin of the American Mathematical Society_ , vol. 37, pp. 209–249, Apr 2000.
  * [58] R. Bhatia, “Linear algebra to quantum cohomology: The story of alfred horn’s inequalities,” _The American Mathematical Monthly_ , vol. 108, pp. 289–318, Apr 2001.
  * [59] L. Hogben, Ed., _Handbook of Linear Algebra_. Boca Raton, FL, USA: CRC press, 2013.
  * [60] A. Zanella, M. Chiani, and M. Win, “On the marginal distribution of the eigenvalues of wishart matrices,” _IEEE Transactions on Communications_ , vol. 57, pp. 1050–1060, Apr 2009.
  * [61] J. C. Gower and G. B. Dijksterhuis, _Procrustes Problems_. Oxford, UK: Oxford University Press, 2004.
  * [62] T. Bell, “Global positioning system-based attitude determination and the orthogonal procrustes problem,” _Journal of Guidance, Control, and Dynamics_ , vol. 26, pp. 820–822, Sep 2003.
  * [63] G. H. Golub and C. F. V. Loan, _Matrix Computations_. Baltimore, MD, USA: Johns Hopkins University Press, 2013.
  * [64] F. Nie, R. Zhang, and X. Li, “A generalized power iteration method for solving quadratic problem on the Stiefel manifold,” _Science China Information Sciences_ , vol. 60, p. 112101, Nov 2017.
  * [65] B. Clerckx and C. Oestges, _MIMO Wireless Networks: Channels, Techniques and Standards for Multi-Antenna, Multi-User and Multi-Cell Systems_. Waltham, MA, USA: Academic Press, 2013.
  * [66] D. Tse and P. Viswanath, _Fundamentals of Wireless Communication_. Cambridge, UK: Cambridge University Press, May 2005.
  * [67] A. Hjorungnes and D. Gesbert, “Complex-valued matrix differentiation: Techniques and key results,” _IEEE Transactions on Signal Processing_ , vol. 55, pp. 2740–2746, Jun 2007.