# Simultaneous Indoor and Outdoor 3D Localization with STAR-RIS-Assisted
Millimeter Wave Systems

Jiguang He1, Aymen Fakhreddine1, and George C. Alexandropoulos1,2 1Technology
Innovation Institute, 9639 Masdar City, Abu Dhabi, United Arab Emirates
2Department of Informatics and Telecommunications, National and Kapodistrian
University of Athens
Panepistimiopolis Ilissia, 15784 Athens, Greece  e-mails: {jiguang.he,
<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

Simultaneously transmitting (refracting) and reflecting reconfigurable
intelligent surfaces (STAR-RISs) have been recently identified to improve the
spectrum/energy efficiency and extend the communication range. However, their
potential for enhanced concurrent indoor and outdoor localization has not yet
been explored. In this paper, we study the fundamental limits, i.e., the
Cramér Rao lower bounds (CRLBs) via Fisher information analyses, on the three-
dimensional (3D) localization performance with a STAR-RIS at millimeter wave
frequencies. The effect of the power splitting between refraction and
reflection at the STAR-RIS as well as the power allocation between the two
mobile stations (MSs) are investigated. By maximizing the principal angle
between the two subspaces corresponding to the STAR-RIS reflection and
refraction matrices, we are able to find the optimal solutions for these
simultaneous operations. We verify that high-accuracy 3D localization can be
achieved for both indoor and outdoor MSs when the system parameters are well
optimized.

###### Index Terms:

Reconfigurable intelligent surface, STAR-RIS, 3D localization, principal
angle, Cramér Rao bound.

##  I Introduction

Reconfigurable intelligent surfaces (RISs) have recently been introduced for
improved energy efficiency (EE), spectrum efficiency (SE), localization
accuracy, sensing capability, as well as network/physical-layer security [1,
2, 3, 4, 5]. The RIS, either being passive, active, or hybrid, is used to
smartly control the radio propagation environment, by virtue of multi-function
capabilities, e.g., reflection, refraction, diffraction, polarization,
scattering, and even absorption [6, 1]. In the literature, the RIS is commonly
used as an intelligent reflector, which breaks the well-known law of
reflection [7], to mitigate the blockage and shadowing effect, and expand the
communication coverage, especially for millimeter wave (mmWave) and Terahertz
(THz) communications.

Recently, there appears another new type of RIS, termed as simultaneously
transmitting and reflecting RIS (STAR-RIS) [8], which can be regarded as a
promising bridge linking indoor and outdoor connectivity. Unlike the
conventional RIS, the STAR-RIS can simultaneously realize two different
functionalities, e.g., reflection and refraction. Namely, mobile stations
(MSs) on both sides of the STAR-RIS, for instance, one is indoor and the other
is outdoor, can be served by it at the same time. In addition, the two
functionalities can also be realized in different manners, e.g., energy
splitting, time switching, and model switching.

Even though the potential of traditional reflective RISs for radio
localization has been studied in the literature [9, 10], we envision that the
localization capability can be even further extended by the introduction of
one or multiple STAR-RISs. With the aid of such a surfaces, an outdoor base
station (BS) is capable of localizing an indoor user, in addition to an
outdoor user, via uplink sounding reference signal (SRS) transmission, not
only at sub-6GHz but also mmWave frequency bands. In this paper, we study the
Cramér Rao lower bounds (CRLBs) for the intermediate channel parameter
estimation based on the Fisher information analysis, and extend it to three-
dimensional (3D) localization of one indoor and one outdoor MSs by virtue of
the Jacobian matrix. We also examine the effect of the energy splitting
between the two STAR-RIS functionalities and power allocation between the two
MSs during SRS transmission. Besides, we find the optimal design of the STAR-
RIS based on the CRLBs, which offers better performance than other alternative
designs.

Notations: A bold lowercase letter ${\mathbf{a}}$ denotes a vector, and a bold
capital letter ${\mathbf{A}}$ denotes a matrix. $(\cdot)^{\mathsf{T}}$ and
$(\cdot)^{\mathsf{H}}$ denote the matrix or vector transpose and Hermitian
transpose, respectively. $(\cdot)^{-1}$ denotes inverse of a matrix,
$\mathrm{tr}(\cdot)$ denotes the trace operator, $\mathrm{diag}({\mathbf{a}})$
denotes a square diagonal matrix with the entries of ${\mathbf{a}}$ on its
diagonal, ${\mathbf{A}}\otimes{\mathbf{B}}$ and
${\mathbf{A}}\diamond{\mathbf{B}}$ denote the Kronecker and Khatri-Rao
products of ${\mathbf{A}}$ and ${\mathbf{B}}$, respectively,
$\mathbb{E}[\cdot]$ and $\mathrm{var}(\cdot)$ are the expectation and variance
operators, $\mathbf{1}$ is the all-one vector, ${\mathbf{I}}_{M}$ denotes the
$M\times M$ identity matrix, $j=\sqrt{-1}$, and $\|\cdot\|_{2}$ denotes the
Euclidean norm of a vector. $[{\mathbf{a}}]_{i}$, $[{\mathbf{A}}]_{ij}$, and
$[{\mathbf{A}}]_{i:j,i:j}$ denote the $i$th element of ${\mathbf{a}}$, the
$(i,j)$th element of ${\mathbf{A}}$, and the submatrix of ${\mathbf{A}}$
formed by rows $i,i+1,\ldots,j$ and columns $i,i+1,\ldots,j$. Finally,
$|\cdot|$ returns the absolute value of a complex number.

##  II System Model

Figure 1: The considered STAR-RIS-assisted mmWave MIMO systems for
simultaneous indoor and outdoor 3D localization.

We consider a nearly-passive STAR-RIS-aided mmWave multiple-input multiple-
output (MIMO) system, which consists of one multi-antenna BS, one multi-
element STAR-RIS, one indoor MS, and one outdoor MS, which is intended for
simultaneous indoor and outdoor 3D localization, as shown in Fig. 1. The BS
and STAR-RIS are equipped with $M$ antennas and $N$ almost passive scattering
elements, respectively, while each MS is equipped with a single antenna.
Extension to multi-antenna MSs is feasible. Both the BS and STAR-RIS employ
the uniform planar array (UPA) structure parallel to the $x-z$ plane. The
STAR-RIS has two operation functionalities, i.e., reflection and refraction,
which can be realized simultaneously.

###  II-A Channel Model

The system is assumed to operate in the mmWave frequency band and we consider
the Saleh-Valenzuela parametric channel model to construct all four individual
channels. The direct line-of-sight (LoS) channel between the outdoor MS and
the $M$-antenna cellular BS is denoted as
${\mathbf{h}}_{1}\in\mathbb{C}^{M\times 1}$ and is mathematically expressed as
follows:

${\mathbf{h}}_{1}=\frac{e^{-j2\pi
d_{1}/\lambda}}{\sqrt{\rho_{1}}}\boldsymbol{\alpha}_{x}(\theta_{1},\phi_{1})\otimes\boldsymbol{\alpha}_{z}(\phi_{1}),$
(1)

where $d_{1}$ (in meters) and $\rho_{1}$ (for the sake of simplicity, we
assume that $\rho_{1}=d_{1}^{2}$) are the distance and path loss between the
outdoor MS and BS, respectively, $\lambda$ is the wavelength of the carrier
frequency, $\theta_{1}$ and $\phi_{1}$ are the azimuth and elevation angles of
arrival associated with ${\mathbf{h}}_{1}$, respectively.111In fact, we can
also consider the free-space path loss, which is modeled:
$\rho_{1}=d_{1}^{2}f_{c}^{2}/10^{8.755},$ where $f_{c}$ (in KHz) is the
carrier frequency, defined as $f_{c}=\frac{c}{\lambda}$ with $c$ being the
speed of light ($3\times 10^{8}$ m/s). Alternatively, the standard 3GPP urban
micro (UMi) path loss model can be considered, according to which holds:
$\rho_{1}=10^{2.27}d_{1}^{3.67}f_{c}^{2.6},$ where $f_{c}$ needs to be
included in GHz [11].  The steering vectors
$\boldsymbol{\alpha}_{x}(\theta_{1},\phi_{1})$ and
$\boldsymbol{\alpha}_{z}(\phi_{1})$ can be written as [12]:

$\displaystyle\boldsymbol{\alpha}_{x}(\theta_{1},\phi_{1})=$
$\displaystyle\Big{[}e^{-j\frac{2\pi
d_{x}}{\lambda}(\frac{M_{x}-1}{2})\cos(\theta_{1})\sin(\phi_{1})},$
$\displaystyle\cdots,e^{j\frac{2\pi
d_{x}}{\lambda}(\frac{M_{x}-1}{2})\cos(\theta_{1})\sin(\phi_{1})}\Big{]}^{\mathsf{T}},$
(2) $\displaystyle\boldsymbol{\alpha}_{z}(\phi_{1})=$
$\displaystyle\Big{[}e^{-j\frac{2\pi
d_{z}}{\lambda}(\frac{M_{z}-1}{2})\cos(\phi_{1})},$
$\displaystyle\cdots,e^{j\frac{2\pi
d_{z}}{\lambda}(\frac{M_{z}-1}{2})\cos(\phi_{1})}\Big{]}^{\mathsf{T}},$ (3)

where $M=M_{x}M_{z}$ with $M_{x}$ and $M_{z}$ being the number of horizontal
and vertical BS antennas, respectively, and $d_{x}$ and $d_{z}$ denote their
inter-element spacing in the horizontal and vertical axes, which were set as
half-wavelength without loss of generality. Similarly, the other channels,
e.g., ${\mathbf{h}}_{2}\in\mathbb{C}^{N\times 1}$ and
${\mathbf{h}}_{3}\in\mathbb{C}^{N\times 1}$, can be presented in the same
manner, as:

${\mathbf{h}}_{i}=\frac{e^{-j2\pi
d_{i}/\lambda}}{\sqrt{\rho_{i}}}\boldsymbol{\alpha}_{x}(\theta_{i},\phi_{i})\otimes\boldsymbol{\alpha}_{z}(\phi_{i}),$
(4)

for $i=2$ and $3$, where $N=N_{x}N_{z}$ is the number of STAR-RIS elements
with $N_{x}$ and $N_{z}$ denoting the numbers in the horizontal and vertical
axes, respectively. Note that the array response vectors
$\boldsymbol{\alpha}_{x}(\cdot)$ and $\boldsymbol{\alpha}_{z}(\cdot)$ may have
a different dimension compared to that in (1) but possess the same form.
Finally, $\rho_{2}$ and $\rho_{3}$ follow the same assumption as $\rho_{1}$.

The wireless channel between the STAR-RIS and BS, i.e.,
${\mathbf{H}}_{4}\in\mathbb{C}^{M\times N}$, is expressed as follows:

${\mathbf{H}}_{4}=\frac{e^{-j2\pi
d_{4}/\lambda}}{\sqrt{\rho_{4}}}\boldsymbol{\alpha}_{x}(\theta_{4},\phi_{4})\otimes\boldsymbol{\alpha}_{z}(\phi_{4})(\boldsymbol{\alpha}_{x}(\theta_{4},\phi_{4})\otimes\boldsymbol{\alpha}_{z}(\phi_{4}))^{\mathsf{H}},$
(5)

assuming that the two UPAs (i.e., one for the BS and the other for the STAR-
RIS) are deployed in parallel without any biased orientation. Due to the wall
in between the BS and the indoor MS, there does not exist a direct LoS path
between them. The only path is the reflection route via the STAR-RIS. In this
work, we consider only LoS paths in all the individual channels; the extension
for the multipath scenario and for arbitrary orientations between the UPAs is
left for future work.

###  II-B Geometric Relationship

The Cartesian coordinates of the BS and STAR-RIS as well as the outdoor and
indoor MSs are
${\mathbf{p}}_{\text{B}}=(x_{\text{B}},y_{\text{B}},z_{\text{B}})^{\mathsf{T}}$,
${\mathbf{p}}_{\text{R}}=(x_{\text{R}},y_{\text{R}},z_{\text{R}})^{\mathsf{T}}$,
${\mathbf{p}}_{\text{U},1}=(x_{\text{U},1},y_{\text{U},1},z_{\text{U},1})^{\mathsf{T}}$,
and
${\mathbf{p}}_{\text{U},2}=(x_{\text{U},2},y_{\text{U},2},z_{\text{U},2})^{\mathsf{T}}$,
respectively. The relationship between the distances and a pair of Cartesian
coordinates are listed below:

$\displaystyle d_{1}$
$\displaystyle=\|{\mathbf{p}}_{\text{B}}-{\mathbf{p}}_{\text{U},1}\|_{2},$ (6)
$\displaystyle d_{i}$
$\displaystyle=\|{\mathbf{p}}_{\text{R}}-{\mathbf{p}}_{\text{U},i-1}\|_{2},\;\text{for}\;i=2,3,$
(7) $\displaystyle d_{4}$
$\displaystyle=\|{\mathbf{p}}_{\text{B}}-{\mathbf{p}}_{\text{R}}\|_{2}.$ (8)

By introducing the three-element vector
$\boldsymbol{\xi}_{i}\triangleq[\cos(\theta_{i})\cos(\phi_{i}),\sin(\theta_{i})\cos(\phi_{i}),\sin(\phi_{i})]^{\mathsf{T}}$
for $i=1,2,3,4$, the geometric relationship between the angular parameters and
the Cartesian coordinates of the nodes can be expressed as

$\displaystyle{\mathbf{p}}_{\text{R}}$
$\displaystyle={\mathbf{p}}_{\text{B}}+d_{4}\boldsymbol{\xi}_{4},$ (9)
$\displaystyle{\mathbf{p}}_{\text{U},1}$
$\displaystyle={\mathbf{p}}_{\text{B}}+d_{1}\boldsymbol{\xi}_{1}={\mathbf{p}}_{\text{R}}+d_{2}\boldsymbol{\xi}_{2},$
(10) $\displaystyle{\mathbf{p}}_{\text{U},2}$
$\displaystyle={\mathbf{p}}_{\text{R}}+d_{3}\boldsymbol{\xi}_{3}.$ (11)

###  II-C Signal Model

Recall that the STAR-RIS has two operation functionalities, i.e., reflection
and refraction. Two separate series of phase shifters are leveraged for
controlling them, so each one is represented by a phase control matrix, i.e.,
$\boldsymbol{\Omega}_{1}$ for controlling refraction and
$\boldsymbol{\Omega}_{2}$ for controlling reflection.
$\boldsymbol{\Omega}_{1}$ and $\boldsymbol{\Omega}_{2}$ are diagonal matrices
with each diagonal element satisfying the unit-modulus constraints, e.g.,
$|[\boldsymbol{\Omega}_{1}]_{jj}|=|[\boldsymbol{\Omega}_{2}]_{jj}|=1$,
$\forall j=1,2,\ldots,N$. We consider the 3D localization via the uplink
transmission, where the two users send SRSs towards the BS simultaneously. The
received signal during the $k$th time slot, for $k=1,2,\ldots,K$, can be
mathematically expressed as

${\mathbf{y}}_{k}={\mathbf{h}}_{1}x_{1,k}+\epsilon_{2}{\mathbf{H}}_{4}\boldsymbol{\Omega}_{2,k}{\mathbf{h}}_{2}x_{1,k}+\epsilon_{1}{\mathbf{H}}_{4}\boldsymbol{\Omega}_{1,k}{\mathbf{h}}_{3}x_{2,k}+{\mathbf{n}}_{k},$
(12)

where $x_{1,k}$ is the SRS from the outdoor MS, $x_{2,k}$ is the SRS from the
indoor MS, as well as $\epsilon_{1}$ (for refraction) and $\epsilon_{2}$ (for
reflection) are used to control the energy splitting for the two different
operational modes of the STAR-RIS, satisfying the following condition:
$\epsilon_{1}^{2}+\epsilon_{2}^{2}=1$. The received signal at the BS is
further corrupted by the white Gaussian noise ${\mathbf{n}}_{k}$, and each
element of ${\mathbf{n}}_{k}$ follows $\mathcal{CN}(0,\sigma^{2})$. During the
$k$th time slot, the refraction matrix $\boldsymbol{\Omega}_{1,k}$ and the
reflection matrix $\boldsymbol{\Omega}_{2,k}$ are considered. In order to
ensure good estimates, $\boldsymbol{\Omega}_{1,k}$ and
$\boldsymbol{\Omega}_{2,k}$ vary from one time slot to another, i.e.,
$\boldsymbol{\Omega}_{1,1}\neq\boldsymbol{\Omega}_{1,2}\neq\cdots\neq\boldsymbol{\Omega}_{1,K}$,
and
$\boldsymbol{\Omega}_{2,1}\neq\boldsymbol{\Omega}_{2,2}\neq\cdots\neq\boldsymbol{\Omega}_{2,K}$.
This refractive/reflective beam sweeping was verified through our numerical
results on the STAR-RIS design in Section IV-C.

Based on the received signals across $K$ time slots, the BS estimates the
Cartesian coordinates of both the indoor and outdoor users, enabling 3D
localization. Without loss of generality, we assume that the sum power
constraint is applied for each time slot, i.e.,
$\mathbb{E}[|x_{1,k}|^{2}]+\mathbb{E}[|x_{1,k}|^{2}]=P$, $\forall k$. The
vector ${\mathbf{y}}_{k}$ in (12) can be further expressed as

$\displaystyle{\mathbf{y}}_{k}=$
$\displaystyle{\mathbf{h}}_{1}x_{1,k}+\epsilon_{2}{\mathbf{H}}_{4}\mathrm{diag}({\mathbf{h}}_{2})\boldsymbol{\omega}_{2,k}x_{1,k}$
$\displaystyle+\epsilon_{1}{\mathbf{H}}_{4}\mathrm{diag}({\mathbf{h}}_{3})\boldsymbol{\omega}_{1,k}x_{2,k}+{\mathbf{n}}_{k},$
(13)

where $\boldsymbol{\Omega}_{1,k}=\mathrm{diag}(\boldsymbol{\omega}_{1,k})$ and
$\boldsymbol{\Omega}_{2,k}=\mathrm{diag}(\boldsymbol{\omega}_{2,k})$, $\forall
k$. By stacking all ${\mathbf{y}}_{k}$’s column by column, we get the
expression:

$\displaystyle{\mathbf{Y}}=$
$\displaystyle\eta_{1}\sqrt{P}{\mathbf{h}}_{1}\mathbf{1}^{\mathsf{T}}+\eta_{1}\sqrt{P}\epsilon_{2}{\mathbf{H}}_{4}\mathrm{diag}({\mathbf{h}}_{2})\bar{\boldsymbol{\Omega}}_{2}$
$\displaystyle+\eta_{2}\sqrt{P}\epsilon_{1}{\mathbf{H}}_{4}\mathrm{diag}({\mathbf{h}}_{3})\bar{\boldsymbol{\Omega}}_{1}+{\mathbf{N}},$
(14)

where $\mathbf{1}$ denotes the $K$-element all-one vector,
${\mathbf{Y}}=[{\mathbf{y}}_{1},\cdots,{\mathbf{y}}_{K}]$,
${\mathbf{N}}=[{\mathbf{n}}_{1},\cdots,{\mathbf{n}}_{K}]$,
$\bar{\boldsymbol{\Omega}}_{1}=[\boldsymbol{\omega}_{1,1},\cdots,\boldsymbol{\omega}_{1,K}]$,
and
$\bar{\boldsymbol{\Omega}}_{2}=[\boldsymbol{\omega}_{2,1},\cdots,\boldsymbol{\omega}_{2,K}]$.
We have also set $|x_{1,k}|^{2}=\eta_{1}^{2}P$, $|x_{2,k}|^{2}=\eta_{2}^{2}P$,
where $\eta_{1}^{2}+\eta_{2}^{2}=1$. Applying vectorization to ${\mathbf{Y}}$
in (II-C), the following expression is deduced:

$\displaystyle{\mathbf{y}}=$
$\displaystyle\eta_{1}\sqrt{P}(\mathbf{1}\otimes{\mathbf{I}}_{M}){\mathbf{h}}_{1}+\eta_{1}\sqrt{P}\epsilon_{2}(\bar{\boldsymbol{\Omega}}_{2}^{\mathsf{T}}\otimes{\mathbf{I}}_{M})({\mathbf{I}}_{N}\diamond{\mathbf{H}}_{4}){\mathbf{h}}_{2}$
$\displaystyle+\eta_{2}\sqrt{P}\epsilon_{1}(\bar{\boldsymbol{\Omega}}_{1}^{\mathsf{T}}\otimes{\mathbf{I}}_{M})({\mathbf{I}}_{N}\diamond{\mathbf{H}}_{4}){\mathbf{h}}_{3}+{\mathbf{n}},$
(15)

where ${\mathbf{y}}=\mathrm{vec}({\mathbf{Y}})$ and
${\mathbf{n}}=\mathrm{vec}({\mathbf{N}})$. We further introduce the following
new notations to simplify the analyses that follows:

$\displaystyle{\mathbf{A}}_{1}$
$\displaystyle=(\mathbf{1}\otimes{\mathbf{I}}_{M}),$ (16)
$\displaystyle{\mathbf{A}}_{2}$
$\displaystyle=(\bar{\boldsymbol{\Omega}}_{2}^{\mathsf{T}}\otimes{\mathbf{I}}_{M})({\mathbf{I}}_{N}\diamond{\mathbf{H}}_{4}),$
(17) $\displaystyle{\mathbf{A}}_{3}$
$\displaystyle=(\bar{\boldsymbol{\Omega}}_{1}^{\mathsf{T}}\otimes{\mathbf{I}}_{M})({\mathbf{I}}_{N}\diamond{\mathbf{H}}_{4}).$
(18)

To this end, expression in (II-C) can be re-written as:

${\mathbf{y}}=\sqrt{P}\eta_{1}{\mathbf{A}}_{1}{\mathbf{h}}_{1}+\sqrt{P}\eta_{1}\epsilon_{2}{\mathbf{A}}_{2}{\mathbf{h}}_{2}+\sqrt{P}\eta_{2}\epsilon_{1}{\mathbf{A}}_{3}{\mathbf{h}}_{3}+{\mathbf{n}}.$
(19)

Upon the STAR-RIS deployment, we assume that the BS knows the exact/precise
location of the STAR-RIS. Thus, we assume that the BS has exact information on
${\mathbf{H}}_{4}$ in terms of the parameters $\theta_{4}$, $\phi_{4}$, and
$d_{4}$.222The assumption needs to be further relaxed since perfect
information is usually infeasible in practice. Therefore, ${\mathbf{A}}_{1}$,
${\mathbf{A}}_{2}$, and ${\mathbf{A}}_{3}$ in (19) are known measurement
matrices to the BS (the BS also knows the refractive/reflective phase
configurations due to its interaction with the STAR-RIS controller).

##  III Cramér Rao Lower Bound Analyses

In this section, we will provide the CRLBs on the estimation of the
intermediate channel parameters, followed by the 3D Cartesian coordinates’
estimation. We also present the refractive/reflective optimization of the
STAR-RIS for the indoor/outdoor localization objective.

###  III-A Estimation of Channel Parameters

The channel parameters to be estimated are those included in
${\mathbf{h}}_{1}$, ${\mathbf{h}}_{2}$, and ${\mathbf{h}}_{3}$, i.e., the
nine-tuple
$\boldsymbol{\nu}\triangleq[\theta_{1},\phi_{1},d_{1},\theta_{2},\phi_{2},d_{2},\theta_{3},\phi_{3},d_{3}]^{\mathsf{T}}$.
Since the additive noise is complex Gaussian distributed, by introducing
$\boldsymbol{\mu}(\boldsymbol{\nu})\triangleq\eta_{1}{\mathbf{A}}_{1}{\mathbf{h}}_{1}+\eta_{1}\epsilon_{2}{\mathbf{A}}_{2}{\mathbf{h}}_{2}+\eta_{2}\epsilon_{1}{\mathbf{A}}_{3}{\mathbf{h}}_{3}$
for (19), the Fisher information matrix for $\boldsymbol{\nu}$ is obtained as:

$[{\mathbf{J}}(\boldsymbol{\nu})]_{i,j}=\frac{P}{\sigma^{2}}\Re\Big{\\{}\frac{\partial\boldsymbol{\mu}^{\mathsf{H}}}{\partial\nu_{i}}\frac{\partial\boldsymbol{\mu}}{\partial\nu_{j}}\Big{\\}}.$
(20)

The detailed information on the partial derivatives in (20) related to
parameters in ${\mathbf{h}}_{1}$, ${\mathbf{h}}_{2}$, and ${\mathbf{h}}_{3}$
is provided below:

$\displaystyle\frac{\partial\boldsymbol{\mu}}{\partial\theta_{1}}$
$\displaystyle=\eta_{1}{\mathbf{A}}_{1}({\mathbf{D}}_{11}\otimes{\mathbf{I}}_{M_{x}}){\mathbf{h}}_{1},$
(21) $\displaystyle\frac{\partial\boldsymbol{\mu}}{\partial\phi_{1}}$
$\displaystyle=\eta_{1}{\mathbf{A}}_{1}({\mathbf{D}}_{12}\otimes{\mathbf{I}}_{M_{x}}){\mathbf{h}}_{1}+\eta_{1}{\mathbf{A}}_{1}({\mathbf{I}}_{M_{z}}\otimes{\mathbf{D}}_{13}){\mathbf{h}}_{1},$
$\displaystyle=\eta_{1}{\mathbf{A}}_{1}({\mathbf{D}}_{12}\otimes{\mathbf{I}}_{M_{x}}+{\mathbf{I}}_{M_{z}}\otimes{\mathbf{D}}_{13}){\mathbf{h}}_{1},$
(22) $\displaystyle\frac{\partial\boldsymbol{\mu}}{\partial d_{1}}$
$\displaystyle=\eta_{1}\frac{-j2\pi
d_{1}/\lambda-1}{d_{1}}{\mathbf{A}}_{1}{\mathbf{h}}_{1},$ (23)
$\displaystyle\frac{\partial\boldsymbol{\mu}}{\partial\theta_{2}}$
$\displaystyle=\eta_{1}\epsilon_{2}{\mathbf{A}}_{2}({\mathbf{D}}_{21}\otimes{\mathbf{I}}_{N_{x}}){\mathbf{h}}_{2},$
(24) $\displaystyle\frac{\partial\boldsymbol{\mu}}{\partial\phi_{2}}$
$\displaystyle=\eta_{1}\epsilon_{2}{\mathbf{A}}_{2}({\mathbf{D}}_{22}\otimes{\mathbf{I}}_{N_{x}}){\mathbf{h}}_{2}+\eta_{1}\epsilon_{2}{\mathbf{A}}_{2}({\mathbf{I}}_{N_{z}}\otimes{\mathbf{D}}_{23}){\mathbf{h}}_{2},$
$\displaystyle=\eta_{1}\epsilon_{2}{\mathbf{A}}_{2}({\mathbf{D}}_{22}\otimes{\mathbf{I}}_{N_{x}}+{\mathbf{I}}_{N_{z}}\otimes{\mathbf{D}}_{23}){\mathbf{h}}_{2},$
(25) $\displaystyle\frac{\partial\boldsymbol{\mu}}{\partial d_{2}}$
$\displaystyle=\eta_{1}\epsilon_{2}\frac{-j2\pi
d_{2}/\lambda-1}{d_{2}}{\mathbf{A}}_{2}{\mathbf{h}}_{2},$ (26)
$\displaystyle\frac{\partial\boldsymbol{\mu}}{\partial\theta_{3}}$
$\displaystyle=\eta_{2}\epsilon_{1}{\mathbf{A}}_{3}({\mathbf{D}}_{31}\otimes{\mathbf{I}}_{N_{x}}){\mathbf{h}}_{3},$
(27) $\displaystyle\frac{\partial\boldsymbol{\mu}}{\partial\phi_{3}}$
$\displaystyle=\eta_{2}\epsilon_{1}{\mathbf{A}}_{3}({\mathbf{D}}_{32}\otimes{\mathbf{I}}_{N_{x}}){\mathbf{h}}_{3}+\eta_{2}\epsilon_{1}{\mathbf{A}}_{3}({\mathbf{I}}_{N_{z}}\otimes{\mathbf{D}}_{33}){\mathbf{h}}_{3},$
$\displaystyle=\eta_{2}\epsilon_{1}{\mathbf{A}}_{3}({\mathbf{D}}_{32}\otimes{\mathbf{I}}_{N_{x}}+{\mathbf{I}}_{N_{z}}\otimes{\mathbf{D}}_{33}){\mathbf{h}}_{3},$
(28) $\displaystyle\frac{\partial\boldsymbol{\mu}}{\partial d_{3}}$
$\displaystyle=\eta_{2}\epsilon_{1}\frac{-j2\pi
d_{3}/\lambda-1}{d_{3}}{\mathbf{A}}_{3}{\mathbf{h}}_{3},$ (29)

where we have used the following matrix definitions:

$\displaystyle{\mathbf{D}}_{11}=$
$\displaystyle\mathrm{diag}\Big{(}\Big{[}-j\pi\Big{(}-\frac{M_{x}-1}{2}\Big{)}\sin(\theta_{1})\sin(\phi_{1}),\cdots,$
$\displaystyle-j\pi\Big{(}\frac{M_{x}-1}{2}\Big{)}\sin(\theta_{1})\sin(\phi_{1})\Big{]}\Big{)},$
(30) $\displaystyle{\mathbf{D}}_{12}=$
$\displaystyle\mathrm{diag}\Big{(}\Big{[}j\pi\Big{(}-\frac{M_{x}-1}{2}\Big{)}\cos(\theta_{1})\cos(\phi_{1}),\cdots,$
$\displaystyle
j\pi\Big{(}\frac{M_{x}-1}{2}\Big{)}\cos(\theta_{1})\cos(\phi_{1})\Big{]}\Big{)},$
(31) $\displaystyle{\mathbf{D}}_{13}$
$\displaystyle=\mathrm{diag}\Big{(}\Big{[}-j\pi\Big{(}-\frac{M_{x}-1}{2}\Big{)}\sin(\phi_{1}),\cdots,$
$\displaystyle-j\pi\Big{(}\frac{M_{x}-1}{2}\Big{)}\sin(\phi_{1})\Big{]}\Big{)},$
(32) $\displaystyle{\mathbf{D}}_{i1}=$
$\displaystyle\mathrm{diag}\Big{(}\Big{[}-j\pi\Big{(}-\frac{N_{x}-1}{2}\Big{)}\sin(\theta_{i})\sin(\phi_{i}),\cdots,$
$\displaystyle-j\pi\Big{(}\frac{N_{x}-1}{2}\Big{)}\sin(\theta_{i})\sin(\phi_{i})\Big{]}\Big{)},\;\text{for}\;i=2,3,$
(33) $\displaystyle{\mathbf{D}}_{i2}=$
$\displaystyle\mathrm{diag}\Big{(}\Big{[}j\pi\Big{(}-\frac{N_{x}-1}{2}\Big{)}\cos(\theta_{i})\cos(\phi_{i}),\cdots,$
$\displaystyle
j\pi\Big{(}\frac{N_{x}-1}{2}\Big{)}\cos(\theta_{i})\cos(\phi_{i})\Big{]}\Big{)},\;\text{for}\;i=2,3,$
(34) $\displaystyle{\mathbf{D}}_{i3}=$
$\displaystyle\mathrm{diag}\Big{(}\Big{[}-j\pi\Big{(}-\frac{N_{x}-1}{2}\Big{)}\sin(\phi_{i}),\cdots,$
$\displaystyle-j\pi\Big{(}\frac{N_{x}-1}{2}\Big{)}\sin(\phi_{i})\Big{]}\Big{)},\;\text{for}\;i=2,3,$
(35)

under the assumption of half-wavelength inter-element spacing for both the BS
and the STAR-RIS UPAs.

For the above estimators of the channel parameters (unbiased
$\hat{\boldsymbol{\nu}}({\mathbf{y}})$), we can calculate the CRLB on the
error covariance matrix as follows:

$\mathbb{E}\\{(\boldsymbol{\nu}-\hat{\boldsymbol{\nu}}({\mathbf{y}}))(\boldsymbol{\nu}-\hat{\boldsymbol{\nu}}({\mathbf{y}}))^{\mathsf{H}}\\}\succeq{\mathbf{J}}^{-1}(\boldsymbol{\nu}),$
(36)

where the notation ${\mathbf{A}}\succeq{\mathbf{B}}$ for square matrices
${\mathbf{A}}$ and ${\mathbf{B}}$ means
${\mathbf{a}}^{\mathsf{H}}{\mathbf{A}}{\mathbf{a}}\geq{\mathbf{a}}^{\mathsf{H}}{\mathbf{B}}{\mathbf{a}}$
for any valid vector ${\mathbf{a}}$.

###  III-B Estimation of 3D Cartesian Coordinates

We are interested in estimating the 3D Cartesian coordinates of the indoor and
outdoor MSs. Therefore, after estimating the channel parameters, we need to
map them to 3D Cartesian coordinates, e.g.,
$\boldsymbol{\kappa}=[x_{\text{U},1},y_{\text{U},1},z_{\text{U},1},x_{\text{U},2},y_{\text{U},2},z_{\text{U},2}]^{\mathsf{T}}$,
based on the geometrical relationship among the BS, the STAR-RIS, and the two
MSs. For the CRLB evaluation of $\boldsymbol{\kappa}$, we resort to the
Jacobian matrix ${\mathbf{T}}$, which links the connection between the channel
parameters $\boldsymbol{\nu}$ and the 3D Cartesian coordinates of the two MSs
$\boldsymbol{\kappa}$. Each $(i,j)$th element of ${\mathbf{T}}$ is expressed
as:

$[{\mathbf{T}}]_{ij}=\frac{\partial[\boldsymbol{\nu}]_{j}}{\partial[\boldsymbol{\kappa}]_{i}}.$
(37)

We provide the following derivatives for the calculation of the submatrix of
the Jacobian matrix related to the outdoor MS:

$\displaystyle\partial\theta_{i}/\partial x_{\text{U},1}$
$\displaystyle=-\frac{\sin(\theta_{i})}{d_{i}\cos(\phi_{i})},$ (38)
$\displaystyle\partial\theta_{i}/\partial y_{\text{U},1}$
$\displaystyle=\frac{\cos(\theta_{i})}{d_{i}\cos(\phi_{i})},$ (39)
$\displaystyle\partial\theta_{i}/\partial z_{\text{U},1}$ $\displaystyle=0,$
(40) $\displaystyle\partial\phi_{i}/\partial x_{\text{U},1}$
$\displaystyle=-\frac{\cos(\theta_{i})\sin(\phi_{i})}{d_{i}},$ (41)
$\displaystyle\partial\phi_{i}/\partial y_{\text{U},1}$
$\displaystyle=-\frac{\sin(\theta_{i})\sin(\phi_{i})}{d_{i}},$ (42)
$\displaystyle\partial\phi_{i}/\partial z_{\text{U},1}$
$\displaystyle=\frac{\cos(\phi_{i})}{d_{i}},$ (43) $\displaystyle\partial
d_{i}/\partial x_{\text{U},1}$ $\displaystyle=\cos(\theta_{i})\cos(\phi_{i}),$
(44) $\displaystyle\partial d_{i}/\partial y_{\text{U},1}$
$\displaystyle=\sin(\theta_{i})\cos(\phi_{i}),$ (45) $\displaystyle\partial
d_{i}/\partial z_{\text{U},1}$ $\displaystyle=\sin(\phi_{i}),$ (46)

for $i=1$ and $2$. The submatrix of ${\mathbf{T}}$ related to the indoor MS
can be computed in the same manner. In addition, it can be easily seen that
only the channel parameters
$\\{\theta_{1},\phi_{1},d_{1},\theta_{2},\phi_{2},d_{2}\\}$ are related to the
coordinates $(x_{\text{U},1},y_{\text{U},1},z_{\text{U},1})$ of the outdoor
MS, and only the parameters $\\{\theta_{3},\phi_{3},d_{3}\\}$ are related to
the coordinates $(x_{\text{U},2},y_{\text{U},2},z_{\text{U},2})$ of the indoor
MS, as concluded from (10) and (11). Therefore, the Jacobian matrix
${\mathbf{T}}$ has the following form:

${\mathbf{T}}=\begin{bmatrix}{\mathbf{T}}_{1}&\mathbf{0}\\\
\mathbf{0}&{\mathbf{T}}_{2}\end{bmatrix},$ (47)

where the submatrix ${\mathbf{T}}_{1}\in\mathbb{R}^{3\times 6}$ consists of
the partial derivatives from (38) to (46), and the submatrix
${\mathbf{T}}_{2}\in\mathbb{R}^{3\times 3}$ consists of the partial
derivatives related to the indoor MS; this part is omitted here due to the
space limitation.

The Fisher information of $\boldsymbol{\kappa}$ can be then expressed as [10]

${\mathbf{J}}(\boldsymbol{\kappa})={\mathbf{T}}{\mathbf{J}}(\boldsymbol{\nu}){\mathbf{T}}^{\mathsf{T}}.$
(48)

Similar to (36), we have the inequality for the CLRB:

$\mathbb{E}\\{(\boldsymbol{\kappa}-\hat{\boldsymbol{\kappa}}({\mathbf{y}}))(\boldsymbol{\kappa}-\hat{\boldsymbol{\kappa}}({\mathbf{y}}))^{\mathsf{T}}\\}\succeq{\mathbf{J}}^{-1}(\boldsymbol{\kappa}).$
(49)

The lower bounds on the root mean square error (RMSE) of the position
estimation of the outdoor and indoor MSs are:

$\displaystyle\text{RMSE}_{\text{U},1}=\sqrt{\mathrm{var}(\hat{{\mathbf{x}}}_{\text{U},1})}$
$\displaystyle\geq\sqrt{\mathrm{tr}\\{[{\mathbf{J}}^{-1}(\boldsymbol{\kappa})]_{1:3,1:3}\\}},$
(50)
$\displaystyle\text{RMSE}_{\text{U},2}=\sqrt{\mathrm{var}(\hat{{\mathbf{x}}}_{\text{U},2})}$
$\displaystyle\geq\sqrt{\mathrm{tr}\\{[{\mathbf{J}}^{-1}(\boldsymbol{\kappa})]_{4:6,4:6}\\}}.$
(51)

###  III-C Localization-Optimal Design for the STAR-RIS

By introducing
${\mathbf{G}}_{1}\triangleq[\frac{\partial\boldsymbol{\mu}}{\partial\theta_{1}},\frac{\partial\boldsymbol{\mu}}{\partial\phi_{1}},\frac{\partial\boldsymbol{\mu}}{\partial
d_{1}},\frac{\partial\boldsymbol{\mu}}{\partial\theta_{2}},\frac{\partial\boldsymbol{\mu}}{\partial\phi_{2}},\frac{\partial\boldsymbol{\mu}}{\partial
d_{2}}]$,
$\hat{{\mathbf{G}}}_{1}\triangleq{\mathbf{G}}_{1}{\mathbf{T}}_{1}^{\mathsf{H}}$,
${\mathbf{G}}_{2}\triangleq[\frac{\partial\boldsymbol{\mu}}{\partial\theta_{3}},\frac{\partial\boldsymbol{\mu}}{\partial\phi_{3}},\frac{\partial\boldsymbol{\mu}}{\partial
d_{3}}]$, and
$\hat{{\mathbf{G}}}_{2}\triangleq{\mathbf{G}}_{2}{\mathbf{T}}_{2}^{\mathsf{H}}$,
the expression of ${\mathbf{J}}^{-1}(\boldsymbol{\kappa})$ in (36) can be
expressed as [13, 14]

${\mathbf{J}}^{-1}\\!(\boldsymbol{\kappa})\\!=\\!\frac{\sigma^{2}}{P}\begin{bmatrix}(\hat{{\mathbf{G}}}_{1}^{\mathsf{H}}({\mathbf{I}}\\!-\\!{\mathbf{P}}_{\hat{{\mathbf{G}}}_{2}})\hat{{\mathbf{G}}}_{1})^{-1}&*\\\
*&(\hat{{\mathbf{G}}}_{2}^{\mathsf{H}}({\mathbf{I}}\\!-\\!{\mathbf{P}}_{\hat{{\mathbf{G}}}_{1}})\hat{{\mathbf{G}}}_{2})^{-1}\end{bmatrix}\\!\\!,$
(52)

where
${\mathbf{P}}_{\hat{{\mathbf{G}}}_{i}}=\hat{{\mathbf{G}}}_{i}(\hat{{\mathbf{G}}}_{i}^{\mathsf{H}}\hat{{\mathbf{G}}}_{i})^{-1}\hat{{\mathbf{G}}}_{i}^{\mathsf{H}}$
is the orthogonal projection onto the column space of $\hat{{\mathbf{G}}}_{i}$
for $i=1$ and $2$.

In order to simplify the diagonal terms in (52), we rewrite ${\mathbf{G}}_{1}$
and ${\mathbf{G}}_{2}$ as
${\mathbf{G}}_{1}=[\eta_{1}{\mathbf{A}}_{1}{\mathbf{H}}_{1},\;\eta_{1}\epsilon_{2}{\mathbf{A}}_{2}{\mathbf{H}}_{2}]$
and ${\mathbf{G}}_{2}=\eta_{2}\epsilon_{1}{\mathbf{A}}_{3}{\mathbf{H}}_{3}$,
where
${\mathbf{H}}_{1}=[({\mathbf{D}}_{11}\otimes{\mathbf{I}}_{M_{x}}){\mathbf{h}}_{1},\;({\mathbf{D}}_{12}\otimes{\mathbf{I}}_{M_{x}}+{\mathbf{I}}_{M_{z}}\otimes{\mathbf{D}}_{13}){\mathbf{h}}_{1},\;\frac{-j2\pi
d_{1}/\lambda-1}{d_{1}}{\mathbf{h}}_{1}]$,
${\mathbf{H}}_{2}=[({\mathbf{D}}_{21}\otimes{\mathbf{I}}_{N_{x}}){\mathbf{h}}_{2},\;({\mathbf{D}}_{22}\otimes{\mathbf{I}}_{N_{x}}+{\mathbf{I}}_{N_{z}}\otimes{\mathbf{D}}_{23}){\mathbf{h}}_{2},\;\frac{-j2\pi
d_{2}/\lambda-1}{d_{2}}{\mathbf{h}}_{2}]$, and
${\mathbf{H}}_{3}=[({\mathbf{D}}_{31}\otimes{\mathbf{I}}_{N_{x}}){\mathbf{h}}_{3},\;({\mathbf{D}}_{32}\otimes{\mathbf{I}}_{N_{x}}+{\mathbf{I}}_{N_{z}}\otimes{\mathbf{D}}_{33}){\mathbf{h}}_{3},\;\frac{-j2\pi
d_{3}/\lambda-1}{d_{3}}{\mathbf{h}}_{3}]$. It can be seen that
${\mathbf{A}}_{2}$ is a function of $\bar{\boldsymbol{\Omega}}_{2}$ and
${\mathbf{A}}_{3}$ is a function of $\bar{\boldsymbol{\Omega}}_{1}$, while
${\mathbf{A}}_{1}$ is independent of $\bar{\boldsymbol{\Omega}}_{1}$ and
$\bar{\boldsymbol{\Omega}}_{2}$. By dividing ${\mathbf{T}}_{1}$ into two
submatrices, i.e., as
${\mathbf{T}}_{1}=[\tilde{{\mathbf{T}}}_{1}\in\mathbb{R}^{3\times
3},\bar{{\mathbf{T}}}_{1}\in\mathbb{R}^{3\times 3}]$, we can derive the
expressions
$\hat{{\mathbf{G}}}_{1}=\eta_{1}{\mathbf{A}}_{1}{\mathbf{H}}_{1}\tilde{{\mathbf{T}}}_{1}^{\mathsf{H}}+\eta_{1}\epsilon_{2}{\mathbf{A}}_{2}{\mathbf{H}}_{2}\bar{{\mathbf{T}}}_{1}^{\mathsf{H}}$
and
$\hat{{\mathbf{G}}}_{2}=\eta_{2}\epsilon_{1}{\mathbf{A}}_{3}{\mathbf{H}}_{3}{\mathbf{T}}_{2}^{\mathsf{H}}$.

For the sake of tractability for the STAR-RIS optimization (i.e.,
$\bar{\boldsymbol{\Omega}}_{1}$ and $\bar{\boldsymbol{\Omega}}_{2}$), we
maximize the principal angle (within $[0,\pi/2]$) between the subspaces of
$\hat{{\mathbf{G}}}_{1}$ and $\hat{{\mathbf{G}}}_{2}$ by following [13]. The
optimal solutions are founded when
$[\mathbf{1},\bar{\boldsymbol{\Omega}}_{2}^{\mathsf{T}}]$ is orthogonal to
$\bar{\boldsymbol{\Omega}}_{1}^{\mathsf{T}}$. In this regard, the largest
principal angle, e.g., $\pi/2$, is obtained [15]. The choices for the
$\bar{\boldsymbol{\Omega}}_{1}$ and $\bar{\boldsymbol{\Omega}}_{2}$ can be the
non-overlapping parts of a DFT or a Hadamard matrix.333Hadamard matrices
possess good properties, since they only contain $\\{-1,1\\}$. Therefore, only
$1$-bit quantization is needed for both the reflection and refraction matrices
at the STAR-RIS. Note that we need to assume that $K\geq 2N$ here in order to
guarantee that both $\bar{\boldsymbol{\Omega}}_{1}$ and
$\bar{\boldsymbol{\Omega}}_{2}$ are optimal.

##  IV Numerical Results

In this section’s numerical investigation, we have set the system parameters
as follows: ${\mathbf{p}}_{\text{B}}=(0,0,8)^{\mathsf{T}}$,
${\mathbf{p}}_{\text{R}}=(2,2,5)^{\mathsf{T}}$,
${\mathbf{p}}_{\text{U},1}=(5,1,2)^{\mathsf{T}}$, and
${\mathbf{p}}_{\text{U},2}=(1,5,2)^{\mathsf{T}}$. The numbers of BS antennas,
STAR-RIS elements, and SRSs from each MS were set as $M=16$, $N=64$, and
$K=128$. The signal-to-noise ratio (SNR) is defined as $P/\sigma^{2}$.

###  IV-A Results on the Channel Parameters’ Estimation

We evaluate the estimation of the intermediate parameters with different
setups for the STAR-RIS energy splitting coefficient $\epsilon_{1}$ and the
power allocation coefficient $\eta_{1}$, i.e.,
$\epsilon_{1}=\eta_{1}=\sqrt{0.5}$,
$\epsilon_{1}=\sqrt{0.9},\eta_{1}=\sqrt{0.5}$, and
$\epsilon_{1}=\sqrt{0.5},\eta_{1}=\sqrt{0.9}$. The simulation results on
estimation of $\theta_{i}$’s and $d_{i}$’s are shown in Figs. 2 and 3 in terms
of the mean square error (MSE). Due to the similar estimation performance for
$\theta_{i}$’s, the results for $\phi_{i}$’s are omitted here. From the
simulation results, we can see that when $\epsilon_{1}$ increases with fixed
$\eta_{1}$, better performance on the estimation of parameters in
${\mathbf{h}}_{3}$ can be achieved. Meanwhile, the performance degrades for
the parameter estimation in ${\mathbf{h}}_{1}$ and ${\mathbf{h}}_{2}$ when
$\eta_{1}$ decreases (less transmission power at the outdoor MS). Note that
when fixing $\eta_{1}$, the performance for the parameter estimation in
${\mathbf{h}}_{1}$ will not change with $\epsilon_{1}$.

Figure 2: CRLB on the estimation of $\\{\theta_{i}\\}$ for $i=1,2$ and $3$
with different pairs of $\epsilon_{1}$ and $\eta_{1}$.  Figure 3: CRLB on the
estimation of $\\{d_{i}\\}$ for $i=1,2$ and $3$ with different pairs of
$\epsilon_{1}$ and $\eta_{1}$.

###  IV-B CRLBs on 3D Localization

Based on the expressions (48) and (49), we can calculate the CRLBs for the
estimation of the 3D Cartesian coordinates of the indoor and outdoor MSs. The
simulation results are shown in Fig. 4 in terms of RMSE. The performance of 3D
localization has been controlled by the two parameters $\epsilon_{1}$ and
$\eta_{1}$. When $\epsilon_{1}=\eta_{1}=\sqrt{0.5}$, the performance gap
between the two MSs’ position estimation is small. However, in the other two
cases, the gap is obvious. In this sense, well-selected $\epsilon_{1}$ and
$\eta_{1}$ can satisfy the quality of services (QoSs) and user fairness for
both MSs simultaneously.

Figure 4: CRLB on the estimation of the 3D Cartesian coordinates of the two
MSs with different pairs of $\epsilon_{1}$ and $\eta_{1}$.

More comprehensive results on $\epsilon_{1}$ and $\eta_{1}$ on the 3D
localization are shown in Fig. 5. In general, a small $\epsilon_{1}$ and a
large $\eta_{1}$ offer poor performance for the indoor MS. On the contrary, a
large $\epsilon_{1}$ and a small $\eta_{1}$ results in poor performance for
the outdoor MS. However, for most of the cases, where it holds
$\eta_{1},\epsilon_{1}>0.3$, we can achieve promising 3D localization in term
of the RMSE for both MSs.

Figure 5: The effect of $\eta_{1}$ and $\epsilon_{1}$ on the 3D localization,
when the SNR is fixed to $15$ dB.

###  IV-C Effect of the STAR-RIS Design

We evaluate the effect of the STAR-RIS design by considering three different
cases: i) $\bar{\boldsymbol{\Omega}}_{1}$ and $\bar{\boldsymbol{\Omega}}_{2}$
are part of a DFT matrix; ii) $\bar{\boldsymbol{\Omega}}_{1}$ and
$\bar{\boldsymbol{\Omega}}_{2}$ are part of a Hadamard matrix; and iii) the
phases of $\bar{\boldsymbol{\Omega}}_{1}$ and $\bar{\boldsymbol{\Omega}}_{2}$
are randomly generated. The simulation results are shown in Fig. 6. We observe
that the first two cases have the same performance and outperform the third
one, since according to Section III-C they are optimal. The performance gain
is not obvious due to the fact that, for large $K$ and $N$, case iii)
approximates the optimal solutions (i.e., the subspaces of
$\hat{{\mathbf{G}}}_{1}$ and $\hat{{\mathbf{G}}}_{2}$ are nearly orthogonal).

Figure 6: The effect of the STAR-RIS design on the 3D localization.

##  V Conclusion and Future Work

In this paper, we studied the fundamental 3D localization performance limits
of STAR-RIS-aided mmWave MIMO systems for simultaneously serving one indoor MS
and one outdoor MS. We also investigated the effect of energy splitting at the
STAR-RIS and the power allocation between the two MSs to offer some useful
insights for practical implementations. Moreover, the optimal design of the
STAR-RIS reflection and refraction configuration matrices was derived by
maximizing the principal angle of two associated subspaces.

In the future, we will focus on the design of practical localization
algorithms, which attain this paper’s theoretical performance limits. We will
also extend the CRLB analysis to general multipath scenarios and consider
arbitrary orientations between the UPAs of the BS and the STAR-RIS.

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