# Distributed IRSs Always Benefit Every Mobile Operator

L. Yashvanth and Chandra R. Murthy The authors are with the Dept. of ECE,
Indian Institute of Science, Bengaluru, India 560 012. (e-mails: {yashvanthl,
cmurthy}@iisc.ac.in).

###### Abstract

We investigate the impact of multiple distributed intelligent reflecting
surfaces (IRSs), which are deployed and optimized by a mobile operator (MO),
on the performance of user equipments (UEs) served by other co-existing out-
of-band (OOB) MOs that do not control the IRSs. We show that, under round-
robin scheduling, in mmWave frequencies, the ergodic sum spectral efficiency
(SE) of an OOB MO is monotonic in the total number of IRS elements with a pre-
log factor that depends on the channel properties of the OOB UE. We further
show that the maximum achievable SE of OOB MO scales log-linearly in IRS
elements. Then, by specifying the minimum number of IRSs as a function of the
channel parameters, we design a distributed IRS system in which an OOB MO
almost surely obtains the maximum SE. Finally, we prove that the outage
probability at an OOB UE decreases exponentially in the number of IRSs, even
though they are randomly configured from the UE’s viewpoint. We numerically
verify our theory and conclude that distributed IRSs always help every MO, but
the MO controlling the IRSs benefits the most.

###### Index Terms:

Intelligent reflecting surfaces, out-of-band performance, mmWave frequency
bands, distributed systems.

##  I Introduction

Intelligent reflecting surfaces (IRSs) have been envisioned to meet the high
data requirements for next-generation wireless systems [1, 2, 3]. They are
made up of several passive elements that can be independently tuned to reflect
signals in desired directions. Further, in a distributed version of it,
multiple IRSs are uniformly spread out in the environment, which provides
additional diversity benefits to the system. Also, in practice, multiple
mobile operators (MOs) coexist in a geographical area and independently serve
multiple user equipments (UEs) over different frequency bands. Then, an
important problem within the paradigm of IRSs is how a distributed IRS,
deployed and optimized by only one of the MOs on a given frequency band,
affects the performance of other out-of-band (OOB) MOs. This aspect is
pertinent because an IRS does not have a band-pass filter and reflects every
signal that impinges it over a wide bandwidth. This paper addresses this
problem wherein every MO uses a different mmWave frequency band for
communication.

A few works in the literature have illustrated that distributed IRSs improve
the multiplexing gains in multiple antenna systems [2, 3]. Similarly, multiple
IRSs are leveraged to reduce link blockages for URLLC applications [4]. In
another work, the authors demonstrate the use of multiple IRSs to improve
coverage in cellular systems [5]. Also, distributed IRSs have been leveraged
to efficiently mitigate multi-user interference in multi-user scenarios [6].
Further, all of these merits can be obtained with minimal channel estimation
overheads that do not scale with the number of IRSs and their elements [7].

The above-existing works implicitly assume that a single MO controls the IRSs.
However, if multiple MOs co-exist and only one of them deploys multiple IRSs
to optimally serve its UEs, then the impact of the IRSs on the performances of
the OOB MOs (which do not control the IRSs) is unexplored, but is an important
aspect to be understood in real-world deployment scenarios. Although our prior
work studies this aspect in single IRS cases [8], the overall channel
properties change in distributed IRS settings due to multiple independent
links offered by the IRSs. This potentially leads to different results and
insights, and hence, it merits an independent study in the multiple IRS
scenario.

To set the context, we consider two MOs, X and Y, operating over non-
overlapping mmWave bands. The (in-band) MO X deploys and optimizes multiple
distributed IRSs to serve a UE scheduled by it in every time slot. Contrarily,
the (OOB) MO Y does not deploy any IRS and is oblivious to MO X’s IRSs.111The
broad conclusions of this paper can be shown to hold for any number of OOB MOs
and also when every MO has its own set of IRSs. Our key contributions are as
follows.

  1. 1.

Under round-robin (RR) scheduling of UEs, we derive the ergodic sum spectral
efficiencies (SE) of the MOs as a function of the number of IRS elements, $N$.
We show that the SE of MO X uniformly grows as $\\!\mathcal{O}(2\log_{2}(N))$,
and the SE of MO Y scales as $\mathcal{O}(\tau\log_{2}(N))$, where the pre-log
factor $\tau\\!\in\\![0,1]$ depends on the number of IRSs and the channel
properties of OOB UE. (See Theorem 1.)

  2. 2.

We next design a distributed IRS system by specifying the minimum number of
IRSs so that the MO Y _almost surely_ achieves the maximum SE (i.e.,
$\tau=1$.) (See Prop. 1.)

  3. 3.

Finally, we evaluate the outage probability at an arbitrary OOB UE and show
that it exponentially decreases in the number of IRSs deployed by MO X. (See
Theorem 2.)

Through numerical simulations, we affirm that distributed IRSs enhance the
performance of not only the UEs for which it is optimized but also of other
OOB UEs, at no additional signal processing costs, both instantaneously and on
average.
_Notation:_ $\mathcal{CN}(0,\sigma^{2}),\text{Ber}(p),\text{Bin}(n,p)$ are
circular symmetric Gaussian, Bernoulli, and Binomial distributions with
parameters $\sigma^{2}$, $p$, and $n,p$, respectively. $\odot$ is the Hadamard
product. $\mathsf{Pr}(\cdot)$ and $\mathbb{E}[\cdot]$ stand for probability
and expectation, $\mathcal{O}$ is the Landau’s Big-O notation, $x!$,
$\lceil\cdot\rceil$, $\mathbbm{1}_{\\{\cdot\\}}$, and $\cup$ refer to the
factorial of $x$, ceil and indicator functions, and set unions.

##  II System Model and Problem Statement

We consider a wireless system where two MOs, X and Y, provide services to $K$
and $Q$ UEs on non-overlapping mmWave frequency bands using their respective
base stations (BS-X and BS-Y). For the sake of exposition, the BSs are
equipped with a single antenna [9], and we emphasize that all the results in
this paper directly extend to multiple antenna BSs without changing the main
conclusions. MO X deploys and optimizes $S>1$ distributed IRSs with $M$
elements each (so, that total number of IRS elements is $N=SM$) to optimally
serve its (in-band) UEs in every time slot. On the other hand, MO Y does not
have any IRS and is oblivious to the presence of MO X’s IRS.

Figure 1: Network scenario of a distributed IRS-aided two-operator system.

Then, the downlink signal received at the $k$th (in-band) UE served by BS-X
is

$\vspace{-0.1cm}\\!\\!y_{k}=\left(h_{d,k}+\sum\nolimits_{s=1}^{S}\mathbf{g}_{s,k}^{T}\boldsymbol{\Theta}_{s}\mathbf{f}_{s}^{X}\right)x_{k}+n_{k},$
(1)

where $\\{\mathbf{g}_{s,k},\mathbf{f}_{s}^{X}\\}\in\mathbb{C}^{M\times 1}$ are
the channels from $s$th IRS to the $k$th UE and BS-X to the $s$th IRS,
respectively; the diagonal matrix
$\boldsymbol{\Theta}_{s}\in\mathbb{C}^{M\times M}$ contains the phase shifts
at the $s$th IRS, and $h_{d,k}$ is the direct channel from BS to UE-$k$. Also,
$x_{k}$ is UE-$k$’s data symbol with $\mathbb{E}[|x_{k}|^{2}]=P$, and $n_{k}$
is the AWGN $\sim\mathcal{CN}(0,\sigma^{2})$. Similarly, at OOB UE-$q$ served
by BS-Y,

$\vspace{-0.1cm}y_{q}=\left(h_{d,q}+\sum\nolimits_{s=1}^{S}\mathbf{g}_{s,q}^{T}\boldsymbol{\Theta}_{s}\mathbf{f}_{s}^{Y}\right)x_{q}+n_{q}.$
(2)

Figure 1 pictorially illustrates the network. We consider the Saleh-Valenzuela
channel model [10],

$\vspace{-0.1cm}\mathbf{f}_{s}^{p}\\!=\\!\\!\sqrt{\\!\frac{M}{L^{(1)}_{s,p}}}\sum_{i=1}^{L^{(1)}_{s,p}}\\!\\!\gamma^{(1)}_{i,s,p}\mathbf{a}^{*}_{M}(\phi^{i}_{s,p});\\!\text{
}\mathbf{g}_{s,t}\\!=\\!\\!\sqrt{\\!\frac{M}{L^{(2)}_{s,t}}}\sum_{j=1}^{L^{(2)}_{s,t}}\\!\\!\gamma^{(2)}_{j,s,t}\mathbf{a}^{*}_{M}(\psi^{j}_{s,t}),\\!\\!\\!$
(3)

where $p\in\\{X,Y\\}$, $L^{(1)}_{s,p}$ and $L^{(2)}_{s,t}$ are the number of
spatial paths in the BS-$p$ to $s$th IRS, and $s$th IRS to UE-$t$ links,
respectively. Since the in-band/OOB BSs/UEs are arbitrarily located in the
area, we let $L^{(1)}_{s,X}=L^{(1)}_{s,Y}\triangleq L_{1}$, and
$L^{(2)}_{s,t}=L_{2}$ $\forall t,s$. Further, $\phi^{i}_{s,p}$ and
$\psi^{j}_{s,t}$ denote the sines of the angle of arrival of the signal from
BS-$p$ at $s$th IRS in the $i$th path, and the departure from $s$th IRS to the
$t$th UE in the $j$th path, respectively, where sine of an angle
($\phi_{\chi}$) is related to the physical angle ($\chi$) by222In the sequel,
the term “angle” will denote the sine of a physical angle.
$\phi_{\chi}\\!=\\!(2d/\lambda)\sin(\chi).$ The fading coefficients,
$h_{d,t},\gamma^{(1)}_{i,s,p}$ and $\gamma^{(2)}_{j,s,t}$, are independently
sampled from
$\mathcal{CN}(0,\beta_{d,t}),\mathcal{CN}(0,\beta_{\mathbf{f}^{p}})$, and
$\mathcal{CN}(0,\beta_{\mathbf{g},t})$, respectively.333We consider equal path
loss in each cascaded link through the IRSs[11]. A precise choice of the path
losses of these links depends on the exact IRS locations, and optimizing these
values is beyond the scope of this paper. Further, $\mathbf{a}_{M}(\phi)$ is
the array response vector of a uniform linear array (ULA) based IRS aligned at
the angle $\phi$, given by444We note that the results in the paper can be
directly extended to any other types of IRS geometries. We consider ULA at the
IRS for simplicity [12].

$\vspace{-0.1cm}\mathbf{a}_{M}(\phi)=\dfrac{1}{\sqrt{M}}[1,e^{-j\pi\phi},\ldots,e^{-j(M-1)\pi\phi}]^{T}.$
(4)

Recall that an $M$-element IRS forms at most $M$ resolvable beams [10, 13].
So, the path angles in (3) are drawn independently and uniformly from the
angle-book,
$\mathbf{\Phi}\triangleq\\!\left\\{\\!\left(\\!-1\\!+\\!\frac{2i}{M}\right)\\!\big{\rvert}i=0,\ldots,M-1\\!\right\\}$.
Further, since only one of the $L\triangleq L_{1}L_{2}$ cascaded paths
contains most of the energy, aligning the IRS along this path procures near-
optimal benefits at the in-band UEs. So, the overall channels to the in-band
UEs can be simplified by their dominant paths [14]. Thus, for UE-$k$, using
(3) in (1) with $L_{1}=L_{2}=1$, we can write $h_{k}$

$\displaystyle\\!\\!=h_{d,k}\\!+\\!M\sum\nolimits_{s=1}^{S}\gamma^{(1)}_{1,s,X}\gamma^{(2)}_{1,s,k}\mathbf{a}_{M}^{H}(\psi^{1}_{s,k})\boldsymbol{\Theta}_{s}\mathbf{a}_{M}^{*}(\phi^{1}_{s,X})\\!\\!\\!$
(5)
$\displaystyle\stackrel{{\scriptstyle(a)}}{{=}}h_{d,k}\\!+\\!M\sum_{s=1}^{S}\left(\\!\gamma^{(1)}_{1,s,X}\gamma^{(2)}_{1,s,k}\left(\mathbf{a}_{M}^{H}(\phi^{1}_{s,X})\\!\odot\mathbf{a}_{M}^{H}(\psi^{1}_{s,k})\right)\right)\\!\boldsymbol{\theta}_{s}$
$\displaystyle\stackrel{{\scriptstyle(b)}}{{=}}h_{d,k}+M\sum\nolimits_{s=1}^{S}\gamma_{X,s,k}\mathbf{\dot{a}}_{M}^{H}(\omega^{1}_{X,s,k})\boldsymbol{\theta}_{s},$
(6)

where
$\boldsymbol{\theta}_{s}\\!\triangleq\\!\text{diag}(\boldsymbol{\Theta}_{s})$,
$(a)$ is due to the properties of Hadamard products. In $(b)$,
$\omega^{1}_{X,s,k}\triangleq\sin^{-1}\left(\sin(\phi^{1}_{s,X})+\sin(\psi^{1}_{s,k})\right)$,
$\gamma_{X,s,k}\triangleq\gamma^{(1)}_{1,s,X}\gamma^{(2)}_{1,s,k}$, and
$\mathbf{\dot{a}}_{M}$ is an array vector that is normalized by $M$ (see (4)),
i.e., $\mathbf{\dot{a}}_{M}(\cdot)=\frac{1}{\sqrt{M}}\mathbf{a}_{M}(\cdot)$.
Contrarily, since the IRS is not optimized to the OOB UEs, we retain the
channel at an OOB UE-$q$ with all the paths and simplified as

$h_{q}=h_{d,q}+\frac{M}{\sqrt{L}}\sum\nolimits_{s=1}^{S}\sum\nolimits_{\ell=1}^{L}\gamma^{\ell}_{Y,s,q}\mathbf{\dot{a}}_{M}^{H}(\omega^{\ell}_{Y,s,q})\boldsymbol{\theta}_{s},\vspace{-0.1cm}$
(7)

where $L\\!\triangleq\\!L_{s,Y}^{(1)}L_{s,q}^{(2)}$,
$\\{\gamma^{\ell}_{Y,s,q}\\}_{\ell=1}^{L}\\!\triangleq\\!\\{\gamma^{(1)}_{i,s,Y}\gamma^{(2)}_{j,s,q}\\}_{i=1,j=1}^{L^{(1)}_{s,Y},L^{(2)}_{s,q}}\
\forall s$.

We can now mathematically state our problem. Suppose the BS-X configures all
the $S$ IRSs to maximize the SE at UE-$k$ by solving the joint optimization
problem (across the IRSs):

$\\!\\!\\!\\!\\!\left\\{\boldsymbol{\Theta}_{s}^{\mathrm{opt}}\right\\}_{s=1}^{S}\\!\\!=\\!\operatorname*{arg\,max}_{\left\\{\boldsymbol{\Theta}_{s}\right\\}_{s=1}^{S}}\log_{2}\\!\\!\left(\\!\\!1\\!\\!+\\!\left|h_{d,k}\\!+\\!\\!\sum\limits_{s=1}^{S}\mathbf{g}_{s,k}^{T}\boldsymbol{\Theta}_{s}\mathbf{f}_{s}^{X}\right|^{2}\\!\\!\\!\\!\frac{P}{\sigma^{2}}\\!\\!\right)\\!\\!,\\!\\!\\!$
(P1)

subject to $\boldsymbol{\Theta}_{s}\\!\in\\!\mathbb{C}^{M\times M}$ being
diagonal matrices with unit modulus entries. Then, these phases are randomly
tuned from an OOB UE’s viewpoint. This is because different MOs do not
coordinate with each other, and so it is not feasible to configure the IRSs
which are jointly optimal to UEs served by both BS-X and BS-Y. In this
context, we address the following:

  1. 1.

What is the effect of the randomly configured IRSs on the ergodic sum-SE of
the user served by the OOB MO Y?

  2. 2.

What is the best ergodic sum-SE that the OOB MO Y can obtain, and when is it
achievable?

  3. 3.

How does the outage probability of UE-$q$ scale with $S$, $M$, and the channel
parameters?

We answer the above questions in the following sections.

##  III Ergodic Sum Spectral Efficiency Analysis

We begin by noting that the optimal IRS configuration at the $s$th IRS as the
solution to (P1) can be obtained as

$\boldsymbol{\theta}_{s}^{\mathrm{opt}}=\dfrac{h_{d,k}\gamma^{*}_{X,s,k}}{\left|h_{d,k}\gamma_{X,s,k}\right|}\times
M\mathbf{\dot{a}}_{M}(\omega^{1}_{X,s,k}),$ (8)

which is a function of the channel parameters of the in-band UE-$k$ only. The
IRS vector in (8) bears a directional response and has non-zero energy only in
the direction of the channel to which it is optimized. However, this direction
is randomly configured from the OOB UE-$q$’s view. So, out of the $L$ cascaded
paths at the OOB UE-$q$ through an IRS which can form $M$ possible beams, with
probability $L/M$, the IRS aligns with UE-$q$’s channel, and with the
probability $1\\!-\\!L/M$, the IRS does not contribute to the channel at
UE-$q$ [8]. Further, this event occurs independently at each IRS. This leads
to our result below, when the BSs use round-robin (RR) based UE scheduling in
every time slot.

######  Theorem 1.

Consider a distributed IRS-aided mmWave system consisting of $S$ IRSs, which
are non-colocated, each with $M$ elements. Then, if the IRSs are optimized (as
per (8)) to serve the UEs of operator X in every time slot, the ergodic sum-
SEs of MOs X and Y, $\bar{S}_{M}^{(X)}$ and $\bar{S}_{M}^{(Y)}$ under RR
scheduling, scale as (9) and (10) at the top of this page, respectively, where
$N$ is the total IRS elements, i.e., $N=SM$, and $M=\eta N$.

$\displaystyle\bar{S}_{M}^{(X)}\\!\approx\\!\frac{1}{K}\sum_{k=1}^{K}\log_{2}\left(1+\left[N^{2}\left(\dfrac{\pi^{2}}{16}+\eta\left(1-\dfrac{\pi^{2}}{16}\right)\right)\beta_{r,k}+N\frac{\pi^{3/2}}{4}\sqrt{\beta_{d,k}\beta_{r,k}}+\beta_{d,k}\right]\frac{P}{\sigma^{2}}\right),$
(9)

$\displaystyle\\!\\!\\!\\!\bar{S}_{M}^{(Y)}\approx\begin{cases}\dfrac{1}{Q}\sum\limits_{q=1}^{Q}\sum\limits_{\mathfrak{s}=0}^{S}\dfrac{S!}{\left(S-\mathfrak{s}\right)!\mathfrak{s}!}{\left(\dfrac{L}{M}\right)}^{\mathfrak{s}}{\left(1-\dfrac{L}{M}\right)}^{S-\mathfrak{s}}\log_{2}\left(1+\left[\dfrac{\mathfrak{s}{M}^{2}}{L}\beta_{r,q}+\beta_{d,q}\right]\dfrac{P}{\sigma^{2}}\right),&\mathrm{if}\
\ L<M,\\\
\dfrac{1}{Q}\sum\limits_{q=1}^{Q}\log_{2}\left(1+\left(\beta_{d,q}+N\beta_{r,q}\right)\dfrac{P}{\sigma^{2}}\right),&\mathrm{if}\
\ L\geq M.\end{cases}$ (10)

###### Proof.

We prove the theorem for both MOs separately.

####  III-1 Ergodic sum-SE of MO-X

Let the ergodic SE at an arbitrary UE served by operator X, say $k$, be
$\langle S_{M}^{k,X}\rangle$. Then, the ergodic sum-SE of MO X under RR
scheduling is $\bar{S}_{M}^{(X)}=\frac{1}{K}\sum_{k=1}^{K}\langle
S_{M}^{k,X}\rangle$. Computing the exact expression for $\langle
S_{M}^{k,X}\rangle$ leads to intractable forms and is not insightful. So, we
apply the Jensen’s approximation to $\langle S_{M}^{k,X}\rangle$ and obtain

$\\!\\!\langle
S_{M}^{k,X}\rangle\\!=\\!\mathbb{E}\\!\left[\log_{2}\\!\\!\left(\\!1\\!+\\!|h_{k}|^{2}\\!\frac{P}{\sigma^{2}}\\!\right)\\!\right]\\!\approx\\!\log_{2}\\!\\!\left(\\!1\\!+\\!\mathbb{E}\\!\left[|h_{k}|^{2}\right]\\!\frac{P}{\sigma^{2}}\\!\right),\\!\\!$
(11)

where the expectations are taken with respect to the channels of UE-$k$. To
evaluate $|h_{k}|^{2}$, we use (8) in (6), and obtain the overall channel gain
at UE-$k$ in (12).

$\displaystyle|h_{k}|^{2}=\left||h_{d,k}|+M\sum\nolimits_{s=1}^{S}|\gamma_{X,s,k}|\right|^{2}=|h_{d,k}|^{2}+M^{2}{\left(\sum\nolimits_{s=1}^{S}|\gamma_{X,s,k}|\right)}^{2}+2M|h_{d,k}|\sum\nolimits_{s=1}^{S}|\gamma_{X,s,k}|.$
(12)

Next, we evaluate the expected values of the terms in (12).
_Term I:_ It is clear that $\mathbb{E}[|h_{d,k}|^{2}]=\beta_{d,k}$. Let
$\beta_{r,k}\triangleq\beta_{\mathbf{f}^{X}}\beta_{\mathbf{g},k}$.
_Term II:_
$\mathbb{E}\left[\left(\sum\limits_{s=1}^{S}|\gamma_{X,s,k}|\right)^{2}\right]=\mathbb{E}\left[\mathop{\sum\limits_{s,p=1}^{S}}_{s\neq
p}|\gamma_{X,s,k}||\gamma_{X,p,k}|\right.$
$\left.+\sum\nolimits_{s=1}^{S}|\gamma_{X,s,k}|^{2}\right]=S(S-1)\left(\frac{\pi}{4}\sqrt{\beta_{r,k}}\right)^{2}+S\beta_{r,k}$.
Then, collecting and re-arranging the factors, the expected value of
$M^{2}\\!{\left(\sum\nolimits_{s=1}^{S}|\gamma_{X,s,k}|\\!\right)}^{\\!2}\\!$
is
$M^{2}\beta_{r,k}\left[S^{2}\frac{\pi^{2}}{16}+S\left(1-\frac{\pi^{2}}{16}\right)\right].$
_Term III:_ Finally, we can show
$2M\mathbb{E}\left[|h_{d,k}|\sum_{s=1}^{S}|\gamma_{X,s,k}|\right]=MS\frac{\pi^{3/2}}{4}\sqrt{\beta_{r,k}\beta_{d,k}}$,
due to independence of $h_{d,k}$ and $\gamma_{X,s,k}$. Now, summing up the
three terms and using the expression for $\bar{S}_{M}^{(X)}$, we obtain the
desired expression in (9).

####  III-2 Ergodic sum-SE of MO-Y

Let the ergodic SE at an arbitrary OOB UE, say $q$, be $\langle
S_{M}^{q,Y}\rangle$. Then, under RR scheduling, the ergodic sum-SE is
$\bar{S}_{M}^{(Y)}=\frac{1}{Q}\sum_{q=1}^{Q}\langle S_{M}^{q,Y}\rangle$.
First, we prove (10) for $L<M$. Define the random variable $A_{\mathrm{s}}$ to
denote whether $\mathrm{s}$th IRS aligns with the channel to OOB UE-$q$ or
not; then $A_{\mathrm{s}}\\!\sim\\!\text{Ber}(L/M)$ for all $\mathrm{s}$.
Further, let $B$ count the number of IRSs aligning to the channel of UE-$q$;
then, $B=\sum_{\mathrm{s}=1}^{S}A_{\mathrm{s}}$. So,
$B\\!\sim\\!\text{Bin}(S,L/M)$. Now, $\langle S_{M}^{q,Y}\rangle=$

$\displaystyle=\mathbb{E}_{|h_{q}|^{2},B}\\!\left[\log_{2}\left(1+\frac{|h_{q}|^{2}P}{\sigma^{2}}\right)\mathbbm{1}_{\left\\{B\in\\{0,1,2,\ldots,S\\}\\!\right\\}}\right]\\!$
(13)
$\displaystyle\stackrel{{\scriptstyle(a)}}{{=}}\sum_{\mathfrak{s}=0}^{S}\mathbb{E}_{|h_{q}|^{2}}\\!\\!\left[\log_{2}\\!\\!\left(1+\frac{|h_{q}|^{2}P}{\sigma^{2}}\right)\\!\\!\Bigg{|}B=\mathfrak{s}\right]\\!{\sf{Pr}}(B=\mathfrak{s})$
(14)
$\displaystyle\stackrel{{\scriptstyle(b)}}{{\approx}}\sum_{\mathfrak{s}=0}^{S}\log_{2}\left(1+\mathbb{E}\left[|h_{q}|^{2}\Big{|}B=\mathfrak{s}\right]\frac{P}{\sigma^{2}}\right){\sf{Pr}}(B=\mathfrak{s}),$
(15)

where $(a)$ is due to the law of iterated expectations, and (b) is due to
Jensen’s approximation, which is known to be numerically tight. Note that
${\sf{Pr}}(B=\mathfrak{s})=\frac{S!}{(S-\mathfrak{s})!\mathfrak{s}!}{\left(\frac{L}{M}\right)}^{\mathfrak{s}}{\left(1\\!-\\!\frac{L}{M}\right)}^{S-\mathfrak{s}}$.
Further, $\mathbb{E}\left[|h_{q}|^{2}\big{/}B=\mathfrak{s}\right]$ is the
average channel gain at UE-$q$ when the beams from exactly $\mathfrak{s}$ of
the IRSs align with one of the $L$ paths to the UE-$q$ through these IRSs.
Then, the channel gain due to these _contributing_ IRSs is computed as (see
(7))

$\vspace{-0.1cm}|h_{q}|^{2}\Big{|}\left\\{B=\mathfrak{s}\right\\}=\left|h_{d,q}+\frac{M}{\sqrt{L}}\sum\nolimits_{s=1}^{\mathfrak{s}}\gamma^{\ell^{*}}_{Y,s,q}\right|^{2},$
(16)

where $\gamma^{\ell^{*}}_{Y,s,q}$ is the channel gain of the cascaded path
that aligns with the $s$th IRS. We can now show that
$\mathbb{E}\left[|h_{q}|^{2}\Big{|}B=\mathfrak{s}\right]=(\mathfrak{s}{M}^{2}/L)\beta_{r,q}+\beta_{d,q}$,
where $\beta_{r,q}\triangleq\beta_{\mathbf{f}^{Y}}\beta_{\mathbf{g},q}$.
Collecting all the terms for (15) and plugging into $\bar{S}_{M}^{(Y)}$ yields
(10) for $L<M$. Now, for $L\geq M$, the probability term $L/M=1$. So, every
IRS almost surely aligns with the OOB UE. Thus, the overall channel gain and
its mean are

$\vspace{-0.1cm}|h_{q}|^{2}=\left|h_{d,q}+\sqrt{M}\sum\nolimits_{s=1}^{S}\gamma^{\ell^{*}}_{Y,s,q}\right|^{2},\text{
and }$ (17)

$N\beta_{r,q}\\!+\\!\beta_{d,q}$, respectively. The proof can now be finished.
∎ We observe from Theorem 1 that not only is the ergodic sum-SE of MO Y better
than a system without IRSs, but also monotonically grows in $N$, with the peak
scaling of $\mathcal{O}(\log_{2}(N))$ for $L\geq M$. For $L\\!<\\!M$, as we
show in Sec. V, the OOB SE scales as $\mathcal{O}(\tau\log_{2}(N))$ for
$\tau\in[0,1)$. Here, the exact value of $\tau$ depends on how $L$ compares
with $M$. The primary reason for this improvement in the OOB SE is that the
IRSs create more paths that enrich the channel at the OOB UEs. We note that
the above results generalize the work of [8], which considers $S=1$. Further,
the SE at the (in-band) MO still scales as $\mathcal{O}(2\log_{2}(N))$,
similar to single IRS scenarios. Thus, deploying multiple IRSs in a
distributed manner not only retains optimal benefits at the in-band MO but
also helps other MOs simultaneously at no significant system overheads.

###  III-A Design for achieving $\mathcal{O}(\log_{2}(N))$ growth in the OOB
SE

From (10), it is clear that the OOB SE maximally scales log-linearly in the
number of IRS elements and is achieved when the number of paths $L$ is
sufficiently large. Specifically, if $L\geq M$, then every IRS contributes to
the signal at an OOB UE. We leverage this fact to design new specifications
for distributed IRS systems that almost surely procure
$\mathcal{O}(\log_{2}(N))$ growth in the OOB SE for any $L,N$. We now have the
following result.

######  Proposition 1.

Consider a system where an MO deploys and controls multiple IRSs in a
distributed fashion, using a total of $N$ IRS elements. If the number of IRSs
$S$ each with $M$ elements providing $L$ cascaded paths at an OOB UE, satisfy

$M\leq M^{*}\triangleq N^{\delta^{*}},\text{ and, }S\geq
S^{*}\triangleq\left\lceil N^{1-\delta^{*}}\right\rceil,$ (18)

where $\delta^{*}=\min\\{1,\log_{N}L\\}$, then almost surely, the OOB SE
attains the maximum scaling of $\mathcal{O}(\log_{2}(N))$ for any $N,L$.

###### Proof.

If $S$ and $M$ satisfy (18), we have $L/M\geq 1$ under any scenario. Thus,
almost surely, every IRS contributes to the signal at an OOB UE. So, the
overall channel at the OOB UE is (17), for which the ergodic SE scales as
$\mathcal{O}(\log_{2}(N))$. ∎

Prop. 1 says that, if the number of paths is $L=N^{\delta}$ for
$0\\!\leq\\!\delta\\!\leq\\!1$, then a distributed system designed as per (18)
gives maximal benefits at the OOB MO _under all scenarios_. Further, this does
not affect the optimal growth of the SE for MO X.

We next analyze the instantaneous channel characteristics at an OOB UE due to
these arbitrarily configured IRSs.

##  IV Outage Probability Evaluation

In this section, we examine the instantaneous channel quality of an OOB UE-$q$
by evaluating its outage probability as a function of the IRS and channel
parameters. The outage probability
$P_{\text{out},q}^{\rho}\triangleq\mathsf{Pr}(|h_{q}|^{2}\leq\rho)$, which
from (7) is

$\\!\\!\\!P_{\text{out},q}^{\rho}\\!=\\!\mathsf{Pr}\\!\left(\\!\left|h_{d,q}\\!+\\!\frac{M}{\sqrt{L}}\sum\limits_{s=1}^{S}\\!\sum\limits_{\ell=1}^{L}\\!\\!\gamma^{\ell}_{Y,s,q}\mathbf{\dot{a}}_{M}^{H}(\omega^{\ell}_{Y,s,q})\boldsymbol{\theta}_{s}\right|^{2}\\!\\!\\!\\!\leq\\!\rho\\!\right)\\!\\!.\\!\\!\vspace{-0.1cm}$
(19)

Intuitively, each IRS provides an independent link at the OOB UE, so multiple
IRSs should provide better diversity gains than single or no IRS scenarios. We
have the following result.

######  Theorem 2.

The outage probability at an OOB UE-$q$ due to $S$ randomly configured and
distributed IRSs, each with $M$ elements and contributing to $L$ paths, is
given by

$\vspace{-0.2cm}P_{\emph{out},q}^{\rho}=\left(1-e^{-\rho/\beta_{d,q}}\right)\mathrm{P}^{S},$
(20)

where

$\vspace{-0.1cm}\\!\\!\mathrm{P}=1-\frac{\bar{L}}{M}\\!\left(\\!\\!\frac{\bar{L}e^{\frac{\bar{L}\beta_{d,q}}{M^{2}\beta_{r,q}}}}{M^{2}\beta_{r,q}}\mathcal{I}_{0}\\!\left(\\!\\!\rho;\beta_{d,q},\\!\frac{M^{2}}{\bar{L}}\beta_{r,q}\\!\right)\\!\\!-\\!e^{-\dfrac{\rho}{\beta_{d,q}}}\\!\\!\right)-e^{-\dfrac{\rho}{\beta_{d,q}}},\vspace{-0.1cm}$

$\bar{L}\\!\triangleq\\!\min\\{L,M\\}$, and
$\mathcal{I}_{0}(x;c_{1},c_{2})\triangleq\vbox{\hbox{\stretchto[440]{\displaystyle\int}{5.2ex}}}_{\\!\\!\\!\\!c_{1}}^{\infty}e^{-\left(\dfrac{x}{t}+\dfrac{t}{c_{2}}\right)}\text{d}t.$

###### Proof.

Define $\mathcal{E}_{s}$, which denotes the event that $s$th IRS aligns with
the OOB UE’s channel. Then $\mathcal{E}_{0}$ is the event that no IRS aligns
with the OOB channel. So, we can write

$\displaystyle\\!\\!\\!P_{\text{out},q}^{\rho}$
$\displaystyle\stackrel{{\scriptstyle(a)}}{{=}}\mathsf{Pr}\left(|h_{q}|^{2}\mathbbm{1}_{\\{\mathcal{E}_{0}\bigcup\mathcal{E}_{1}\bigcup\ldots\bigcup\mathcal{E}_{S}\\}}\leq\rho\right)$
(21)
$\displaystyle=\mathbb{E}_{\mathbbm{1}_{\\{\\!\mathcal{E}_{0}\\!\\}},\ldots,\mathbbm{1}_{\\{\\!\mathcal{E}_{S}\\!\\}}}\\!\\!\left[\mathsf{Pr}\left(|h_{q}|^{2}\leq\rho\Bigg{|}\mathbbm{1}_{\\{\\!\mathcal{E}_{0}\\!\\}},\ldots,\mathbbm{1}_{\\{\\!\mathcal{E}_{S}\\!\\}}\right)\right]\\!\\!\\!$
(22)
$\displaystyle\stackrel{{\scriptstyle(b)}}{{=}}\prod_{\mathfrak{s}=0}^{S}\mathbb{E}_{\mathbbm{1}_{\\{\mathcal{E}_{\mathfrak{s}}\\}}}\left[\mathsf{Pr}\left(|h_{q}|^{2}\leq\rho\Bigg{|}\mathbbm{1}_{\\{\mathcal{E}_{\mathfrak{s}}\\}}\right)\right]$
(23)
$\displaystyle\stackrel{{\scriptstyle(c)}}{{=}}\mathsf{Pr}(|h_{d,q}|^{2}\leq\rho)\\!\left(\mathbb{E}_{\mathbbm{1}_{\\{\\!\mathcal{E}_{1}\\!\\}}}\\!\\!\left[\mathsf{Pr}\\!\left(|h_{q}|^{2}\leq\rho\Big{|}\mathbbm{1}_{\\{\\!\mathcal{E}_{1}\\!\\}}\right)\right]\right)^{S}\\!\\!,\\!\\!$
(24)

where $(a)$ is because $\\{\mathcal{E}_{\mathfrak{s}}\\}_{\mathfrak{s}=0}^{S}$
is a set of mutually exhaustive events, $(b)$ is because these events are
independent, and in $(c)$, the first product term is because under
$\mathcal{E}_{0}$, the channel at UE-$q$ is only due to the direct link from
BS-Y, and the second term is because
$\\{h_{q}^{\mathfrak{s}}\\}_{\mathfrak{s}=1}^{S}$ and
$\\{\mathbbm{1}_{\\{\\!\mathcal{E}_{\mathfrak{s}}\\!\\}}\\}_{\mathfrak{s}=1}^{S}$
are sets of i.i.d. random variables. Here, $h_{q}^{\mathfrak{s}}$ is the
component of $h_{q}$ via the $\mathfrak{s}$th IRS. We can now evaluate the
expectation term in (24) as

$\vspace{-0.1cm}\mathbb{E}_{\mathbbm{1}_{\\{\\!\mathcal{E}_{1}\\!\\}}}\\!\\!\left[\mathsf{Pr}(|h_{q}|^{2}\leq\rho\Big{|}\mathbbm{1}_{\\{\mathcal{E}_{1}\\}})\right]=(1\\!-\\!\bar{L}/M)(\mathsf{Pr}(|h_{d,q}|^{2}\leq\rho))\\\
+\bar{L}/M(\mathsf{Pr}(|h_{d,q}+M/\sqrt{\bar{L}}\gamma^{\ell^{*}}_{Y,1,q}|^{2}\leq\rho)).\vspace{-0.1cm}$
(25)

While it is easy to show that
$\mathsf{Pr}(|h_{d,q}|^{2}\leq\\!\rho)\\!=\\!1\\!-e^{-\rho/\beta_{d,q}}$, the
second term in the above can be computed using [8, Appendix B]. Finally,
plugging all these values in (24) yields (20). ∎

From Theorem 2, it is evident that the outage probability at any OOB UE
decreases exponentially in $S$, even though the IRSs are randomly configured
from UE-$q$’s viewpoint. Thus, a distributed IRS-aided system benefits an OOB
MO at any instant in time, without incurring any optimization costs. We next
illustrate our findings via numerical simulations.

##  V Numerical Results

We verify our results through Monte Carlo simulations. We consider a single-
cell setup with the path loss modeled as $\beta=C_{0}(d_{0}/d)^{\alpha}$,
where $C_{0}$ is the path loss at the reference distance $d_{0}$, and $\alpha$
is the path loss exponent. We set $\alpha=2,2.2$ and $4.5$ in the BS-IRS, IRS-
UE, and BS-UE links. Further, BS-X and BS-Y serve a total of $K=10$ and $Q=10$
UEs using an RR scheduler over $10,000$ time slots.
In Fig. 2, we plot the ergodic sum-SE of both MOs versus the total number of
IRS elements, $N$, for different values for the number of IRSs, $S$. We use
$C_{0}\left(P/\sigma^{2}\right)=150$ dB and $L=2$ [8]. We first observe that
the SE of the MO X is invariant to the number of IRSs in the system. Thus, the
optimal SE scaling of $\mathcal{O}(2\log_{2}(N))$ is retained for any $S$.
However, the OOB SE significantly changes with $S$. For e.g., when $S=4$,
until some point, the OOB SE log-linearly increases in $N$ and then exhibits
log-sub-linear growth for larger values of $N$ (in the regime where $L<M$),
i.e., the pre-log factor of OOB SE lies in $[0,1)$. The latter is because, for
large $N$ and fixed small $L$, unless $S$ is reasonably large, the OOB UE does
not benefit much. We then plot the OOB SE for $S=S^{*}$, where $S^{*}$ is as
per Proposition 1. In this case, as indicated in the data marked inside the
figure, a $4\times$ increase in $N$ leads to a boost of SE by $2$ bps/Hz. Note
that $2+\log_{2}(N)=1\cdot\log_{2}\left((4N)\right)$. In other words, the OOB
SE achieves the (maximum) scaling of $\log_{2}(N)$ even for large $N$ when
sufficiently many IRSs are present. We also compare our results with that in
[8], which considers $S=1$ on the same plot. Clearly, the OOB SE obtained
using distributed IRSs is better than the single IRS case. Also, the
simulations match well with Theorem 1, illustrating the accuracy of the
derived rate scaling laws.
In Fig. 3, we illustrate the tightness of Proposition 1. Specifically, we plot
$\tau\triangleq\bar{S}_{M}^{(X)/(Y)}/\log_{2}(N)$ as a function of $S$, which
measures the pre-log factor of the in-band/OOB SE. Each sub-plot represents a
system with different $L$, through $\delta\triangleq\log_{N}L$ with $N=128$.
In all the cases, we observe that the result in Prop. 1 is very tight because
for any $S<S^{*}$, $\tau\in[0,1)$, i.e., the OOB SE grows only log-sub-
linearly in $N$. Contrarily, for $S\geq S^{*}$, the OOB-SE scales as
$\mathcal{O}(\log_{2}(N))$ for any $N,L$. Also, for the in-band SE, almost
surely $\tau=2$, in line with Fig. 2. Further, since the OOB SE is independent
of $N$ when there is no IRS, almost surely $\tau=0$ in this case.
Finally, in Fig. 4, we study the outage performance at an arbitrary OOB UE
after normalizing the channel path losses. We plot the outage probability with
$\rho=0.5$ as a function of $S$ for different $L$ through $\delta=\log_{N}L$,
and $N=512$. For a given $\delta$ (or $L$), the outage probability decreases
exponentially in the number of IRSs in the system, in line with Theorem 2.
Further, for a given $S$, the outage probability decreases with $\delta$ (or
$L$) because the likelihood that the randomly configured IRSs help the OOB UE
increases when there are many paths.

Figure 2: Ergodic sum-SE of the operators X and Y vs. $N$. Figure 3: Pre-log
factor of OOB SE, $\tau$ vs. $S$ as a function of $\delta$ (or $L$). Figure 4:
Outage probability vs. $S$ at an OOB UE.

##  VI Conclusions

We studied the impact of distributed IRSs on the SE and outage performance of
an OOB MO that does not have control over the IRSs. The SE of the OOB MO is
strictly better and scales as high as log-linear in the number of IRS elements
compared to having no IRSs. Primarily, this is because each IRS potentially
creates new paths at the OOB UEs. Then, we developed novel system
specifications that almost surely achieve the log-linear growth of OOB SE for
any number of IRS elements. Lastly, we proved that the outage probability at
an OOB UE decreases exponentially in the number of IRSs, although the IRSs are
randomly configured from the OOB UEs’ viewpoint. Also, since the choice of OOB
MO was arbitrary, our results identically apply to any OOB MO. Thus,
distributed IRSs help every MO in the system.

## References

  * [1] Q. Wu, S. Zhang, B. Zheng, C. You, and R. Zhang, “Intelligent reflecting surface-aided wireless communications: A tutorial,” _IEEE Trans. Commun._ , vol. 69, no. 5, pp. 3313–3351, May 2021.
  * [2] J. Choi, G. Kwon, and H. Park, “Multiple intelligent reflecting surfaces for capacity maximization in LOS MIMO systems,” _IEEE Wireless Commun. Lett._ , vol. 10, no. 8, pp. 1727–1731, Aug. 2021.
  * [3] Z. Chen, M.-M. Zhao, M. Li, M. Lei, and M.-J. Zhao, “IRS-aided joint spatial division and multiplexing for mmwave multiuser MISO systems,” _IEEE Trans. Wireless Commun._ , Nov. 2023.
  * [4] Y. Zhang, S. R. Khosravirad, X. Chu, and M. A. Uusitalo, “On the IRS deployment in smart factories considering blockage effects: Collocated or distributed?” _arXiv preprint arXiv:2311.07843_ , 2023.
  * [5] X. Shi, N. Deng, N. Zhao, and D. Niyato, “Coverage enhancement in millimeter-wave cellular networks via distributed IRSs,” _IEEE Trans. Commun._ , vol. 71, no. 2, pp. 1153–1167, Feb. 2023.
  * [6] X. Peng, X. Hu, and C. Zhong, “Distributed intelligent reflecting surfaces-aided communication system: Analysis and design,” _IEEE Trans. Green Commun. Netw._ , vol. 6, no. 4, pp. 1932–1944, Dec. 2022.
  * [7] L. Yashvanth and C. R. Murthy, “Cascaded channel estimation for distributed IRS aided mmwave massive MIMO systems,” in _Proc. IEEE Glob. Commun. Conf._ , Dec. 2022, pp. 717–723.
  * [8] ——, “On the impact of an IRS on the out-of-band performance in sub-6 GHz & mmwave frequencies,” _arXiv preprint arXiv:2308.14033_ , 2023.
  * [9] Y. Zhao _et al._ , “Cooperative reflection and synchronization design for distributed multiple-RIS communications,” _IEEE J. Sel. Topics Signal Process._ , vol. 16, no. 5, pp. 980–994, Aug. 2022.
  * [10] P. Wang, J. Fang, W. Zhang, and H. Li, “Fast beam training and alignment for IRS-assisted millimeter wave/terahertz systems,” _IEEE Trans. Wireless Commun._ , vol. 21, no. 4, pp. 2710–2724, Apr. 2022.
  * [11] L. Bai, T. Li, Z. Xiao, and J. Choi, “Performance analysis for SDMA mmwave systems: Using an approximate closed-form solution of downlink sum-rate,” _IEEE Access_ , vol. 5, pp. 15 641–15 649, Aug. 2017.
  * [12] F. Zhao _et al._ , “Joint beamforming optimization for IRS-aided THz communication with time delays,” _IEEE Wireless Commun. Lett._ , vol. 13, no. 1, pp. 49–53, Jan. 2024.
  * [13] C. You, B. Zheng, and R. Zhang, “Fast beam training for IRS-assisted multiuser communications,” _IEEE Wireless Commun. Lett._ , vol. 9, no. 11, pp. 1845–1849, Nov. 2020.
  * [14] J. Wang _et al._ , “Hierarchical codebook-based beam training for RIS-assisted mmwave communication systems,” _IEEE Trans. Wireless Commun._ , vol. 71, no. 6, pp. 3650–3662, Jun. 2023.