#  On the Impact of an IRS on the Out-of-Band Performance in Sub-6 GHz &
mmWave Frequencies

L. Yashvanth, , and Chandra R. Murthy This work was financially supported by
the Qualcomm Innovation Fellowship, Qualcomm UR $6$G India Grant, a research
grant from MeitY, Govt. of India, and the Prime Minister’s Research
Fellowship, Govt. of India. A preliminary version of this work was presented
at the 24th IEEE International Workshop on Signal Processing Advances in
Wireless Communications (SPAWC), Sep. 2023 [DOI:
10.1109/SPAWC53906.2023.10304535] [1].The authors are with the Dept. of ECE,
Indian Institute of Science, Bengaluru, India 560 012. (e-mails: {yashvanthl,
cmurthy}@iisc.ac.in).

###### Abstract

Intelligent reflecting surfaces (IRSs) were introduced to enhance the
performance of wireless communication systems. However, from a service
provider’s viewpoint, a concern with the use of an IRS is its effect on out-
of-band (OOB) quality of service. Specifically, if two operators, say X and Y,
provide services in a given geographical area using non-overlapping frequency
bands, and if operator X uses an IRS to enhance the spectral efficiency (SE)
of its users (UEs), does it degrade the performance of UEs served by operator
Y? We answer this by analyzing the average and instantaneous performances of
the OOB operator considering both sub-6 GHz and mmWave bands. Specifically, we
derive the ergodic sum-SE achieved by the operators under round-robin
scheduling. We also derive the outage probability and analyze the change in
the SNR caused by the IRS at an OOB UE, using stochastic dominance theory.
Surprisingly, even though the IRS is randomly configured from operator Y’s
point of view, the OOB operator still benefits from the presence of the IRS,
witnessing a performance enhancement _for free_ in both sub-6 GHz and mmWave
bands. This is because the IRS introduces additional paths between the
transmitter and receiver, increasing the overall signal power arriving at the
UE and providing diversity benefits. Finally, we show that the use of
opportunistic scheduling schemes can further enhance the benefit of the
uncontrolled IRS at OOB UEs. We numerically illustrate our findings and
conclude that an IRS is always beneficial to every operator, even when the IRS
is deployed & controlled by only one operator.

###### Index Terms:

Intelligent reflecting surfaces, out-of-band performance, sub-6 GHz bands,
mmWave communications.

##  I Introduction

Intelligent reflecting surfaces (IRSs) have been extensively studied to
enhance the performance of beyond $5$G and $6$G communications [2, 3]. They
are passive reflecting surfaces with many elements that can introduce
carefully optimized phase shifts while reflecting an incoming radio-frequency
(RF) signal, thereby steering the signal towards a desired user equipment
(UE). Although a recent topic of research, several use-cases elucidating the
benefits of IRS-aided communications have already been investigated [3, 4, 5,
6, 7, 8]. All these studies implicitly assume that only _one operator_ deploys
and controls single or multiple IRS(s) to provide wireless services to their
subscribed UEs. However, in practice, multiple wireless network operators co-
exist in a given geographical area, each operating in non-overlapping
frequency bands. Then, if an IRS is deployed and optimized by one of the
operators to cater to its subscribed UEs’ needs, it is unclear whether the IRS
will boost or degrade the performance of the UEs served by the other
operators. This point is pertinent because the IRS elements are passive and
have no band-pass filters. They reflect all the RF signals that impinge upon
them across a wide range of frequency bands, including signals from _out-of-
band (OOB)_ operators intended to be received by OOB UEs. Thus, this paper
focuses on understanding how an IRS controlled by one operator affects the
performance of other OOB operators.

###  I-A Related Literature and Novelty of This Work

The benefits of using an optimized IRS has been illustrated in several use-
cases, see [3, 4, 5, 6, 7, 8]. For e.g., in [4], outage analysis with randomly
distributed IRSs is investigated. In [5, 9], coverage enhancement due to an
IRS is studied, and in [10], the effect of IRSs on inter base station (BS)
interferences is explored. Contrary to this, a few works also use randomly
configured IRSs and obtain benefits from them. For instance, [11, 12, 13] use
opportunistic communications using randomly configured IRSs. Similarly, blind
beam forming approaches and diversity order analysis are reported in [14], and
[15, 16, 17], respectively. Also, in [18], random IRSs are used to protect
against wireless jammers. However, none of these works consider the scenario
where multiple network operators coexist in an area. Very few studies have
considered using an IRS in multi-band, multi-operator systems. In [19], the
effect of pilot contamination on channel estimation in a multiple-operator
setting is reported. In [20, 21], the authors jointly optimize the IRS
configurations across multiple frequency bands via coordination among the BSs,
which is impractical and incurs high signal processing overhead.111The works
that assume inter-BS coordination to optimize the IRS configuration jointly
consider a single operator deploying many BSs. Coordination among BSs may be
feasible in this scenario. However, it is impractical for the BSs owned by two
different operators to coordinate with each other. Further, the solutions and
analyses provided therein are not scalable with the number of operators (or
frequency bands). More fundamentally, none of these works consider the effect
on the OOB performance, even in the scenario of two operators providing
services in non-overlapping bands when the IRS is optimized for only one
operator. In this work, we investigate whether an IRS degrades the performance
of other OOB operators. If not, can the mere presence of an IRS in the
vicinity provide _free gains_ to OOB operators? Below, we explain our
contributions in this context.

###  I-B Contributions

We consider a system with two network operators, X and Y, providing service in
different frequency bands in the same geographical area. The IRS is optimized
to serve the UEs subscribed to the _in-band_ operator X, and we are interested
in analyzing the spectral efficiency (SE) achieved and outage probability
witnessed by the UEs subscribed to the _OOB_ operator Y who does not control
the IRS. Specifically, we (separately) evaluate the IRS-assisted performance
in both sub-6 GHz and the mmWave bands, which are provisioned as the FR$1$ and
FR$2$ bands in $5$G, respectively [22]. Further, in the mmWave bands, inspired
by [23], we study two scenarios: (a) LoS (line-of-sight) and (b) (L+)NLoS (LoS
and Non-LoS.) In the LoS scenario, the IRS is optimized or aligned to the
dominant cascaded path (called the _virtual_ LoS path) of the in-band UE’s
channel. This is also considered in [24, 25], where the in-band UEs’ channels
are approximated by the dominant LoS path to reduce the signaling overhead
required for the base station (BS) to program the IRS: the phase of the second
IRS element relative to the first element determines the entire phase
configuration. Contrarily, in the (L+)NLoS case, the IRS optimally combines
all the spatial paths to maximize the signal-to-noise ratio (SNR) at the
receiver, for which the overhead scales linearly with the number of IRS
elements. Using tools from high-dimensional statistics, stochastic-dominance
theory, and array processing theory, we make the following contributions.

####  I-B1 OOB Performance in sub-6 GHz Bands (See Section III)

Here, the operators serve their UEs over the sub-6 GHz frequency band where
the channels are rich-scattering. In this context, our key findings are as
follows.

  1. 1-a)

We derive the ergodic sum-SEs of the two operators as a function of the system
parameters, under round-robin (RR) scheduling of the UEs served by both
operators. We show that the sum-SE scales log-quadratically and log-linearly
with the number of IRS elements for the in-band and OOB networks,
respectively, even though the OOB operator does not control the IRS (see
Theorem 1.)

  2. 1-b)

We show that the outage probability at an arbitrary OOB UE decreases
monotonically with the number of IRS elements. Further, via the complementary
cumulative distribution function (CCDF) of the difference in the OOB channel
gain with and without the IRS, we prove that the OOB channel gain with an IRS
_stochastically dominates_ the gain without the IRS, with the difference
increasing with the number of IRS elements. Thus, an OOB UE gets instantaneous
benefits that monotonically increase with the number of IRS elements (see
Theorem 2 and Proposition 1.)

####  I-B2 OOB Performance in mmWave Bands (See Section IV)

In the mmWave bands, the channels are directional, with only a few propagation
paths. In this context, using novel probabilistic approaches based on the
resolvable criteria of the mmWave spatial beams, our key findings are as
follows.

  1. 2-a)

In LoS scenarios, where the IRS is optimized to match the dominant path of the
in-band UE’s channel, we derive the ergodic sum-SEs of the two operators under
RR scheduling of the UEs. The SE at the in-band UE scales log-quadratically in
the number of IRS elements, whereas the SE gain at an OOB UE depends on the
number of spatial paths in the OOB UE’s channel. If there are a sufficient
number of paths in the cascaded channel, the SE improvement due to the IRS
scales log-linearly in the number of IRS elements. Otherwise, the OOB UE’s SE
improves only marginally compared to that in the absence of the IRS (see
Theorem 3.)

  2. 2-b)

We evaluate the outage probability and CCDF of an OOB UE’s channel gain
with/without an IRS in LoS scenarios and prove that the channel gain in the
presence of IRS _stochastically dominates_ the gain in its absence. Thus, even
in mmWave bands, the IRS provides positive instantaneous gains to all the OOB
UEs (see Theorem 4.)

  3. 2-c)

We next consider the (L+)NLoS scenario where the IRS is jointly optimized
considering all the spatial paths. We first evaluate the directional energy
response of the IRS and show that it exhibits peaks only at the channel angles
to which the IRS is optimized. This is a fundamental and new characterization
of the IRS response when it is aligned to multiple paths in an mmWave system.
(see Lemma 1.)

  4. 2-d)

We derive the ergodic sum-SE of both operators in (L+)NLoS scenarios. We find
that the OOB performance is even better than the LoS scenario (and hence
better than the system without an IRS.) This is because the odds that an OOB
UE benefits improve when the IRS has a nonzero response in multiple
directions. Thus, the OOB performance in mmWave bands does not degrade even
when the OOB operator serves its UEs while remaining oblivious to the presence
of the IRS (see Theorem 5.)

####  I-B3 Opportunistic Enhancement of OOB Performance (See Section V)

Having shown that an IRS positively benefits OOB UEs, we next suggest ways to
exploit the uncontrolled IRS to enhance the performance of OOB operators
further. In particular, by using opportunistic selection techniques, we
leverage multi-user diversity and show that a significant boost in the OOB
performance can be obtained compared to RR scheduling. Specifically, we
demonstrate the following.

  1. 3-a)

By using a proportional-fair scheduler over a large number of OOB UEs, the
sum-SE of operator Y converges to the so-called _beamforming SE_ , which is
the SE obtained when the IRS is optimized for an OOB UE in every time slot
(see Lemma 2.)

  2. 3-b)

By using a max-rate scheduler, the ergodic sum-SE of operator Y monotonically
increases with both the number of IRS elements and OOB UEs in the system (see
Lemma 3.)

Next, we highlight the practical utility of our results.

###  I-C Practical Implications and Useful Insights

Our results offer several interesting insights into the performance of IRS-
aided wireless systems where an IRS is deployed and controlled only by one
operator. First, in all the scenarios mentioned in Sec. I-B, we present novel,
compact, and insightful analytical expressions that uncover the dependence of
the performance on system parameters such as the number of IRS elements, SNR,
number of channel paths, etc. As shown both analytically and through
simulations, an IRS is beneficial to OOB users even though the IRS phase
configuration is chosen randomly from the OOB operator’s viewpoint, and this
holds both in terms of the average and instantaneous SE. In particular, in
rich scattering environments, the average SE at any OOB UE scales log-linearly
in the number of IRS elements. On the other hand, in mmWave channels, the
average SE improvement at the OOB UE due to the IRS is an increasing function
of the number of spatial paths in the OOB UE’s channel. Thus, deploying an IRS
enriches the overall wireless channels (in both sub-6 GHz and mmWave bands)
and can only benefit all wireless operators in the area. Our study also
reveals that there exists an interesting trade-off between reducing the
signaling overhead to program the IRS at the in-band operator (as in LoS
scenarios) versus boosting the gain at the OOB operators (as in (L+)NLoS
scenarios) in the mmWave bands. Finally, our results and insights derived for
$2$ operators directly extend to any number of OOB operators and to other
settings like planar array IRS or multiple antenna BSs.

_Notation:_ $|\cdot|,\angle\cdot$ denote the magnitude and phase of a complex
number (vector); $(\cdot)^{*}$ stands for complex conjugation;
$\mathbbm{1}_{\\{\cdot\\}}$ is the indicator function; $|\mathcal{A}|$ is the
cardinality of set $\mathcal{A}$; $A\stackrel{{\scriptstyle d}}{{=}}B$ means
$A$ and $B$ have the same distribution; $\odot$ is the Hadamard product;
$\Re(\cdot)$ and $\mathbb{R}^{+}$ are the real part and the set of positive
real numbers; ${\sf{Pr}}(\cdot)$ and $\mathbb{E}[\cdot]$ refer to the
probability measure and expectation operator.
$\mathcal{O}(\cdot),\Omega(\cdot)$, and $o(\cdot)$ denote Landau’s Big-O, Big-
Omega, and little-o functions.

##  II System Model

We consider a single-cell system with two mobile network operators, X and Y,
who provide service to $K$ and $Q$ UEs, respectively, on different frequency
bands and in the same geographical area. The BSs of operators X and Y
(referred to as BS-X and BS-Y, respectively) and UEs are equipped with a
single antenna, and all the channels undergo frequency-flat fading[26].222For
simplicity of exposition and to focus on the _effect of the IRS on OOB users_
, we consider single antennas at the BSs and UEs, similar to [27, 6, 7, 8,
28]. The extension to the multiple antenna case does not change our broad
conclusions, but we relegate this to future work. An $N$-element IRS is
deployed by operator X to enhance the SNR at the UEs it serves. So, operator X
configures the IRS with the SNR-optimal phase configuration for a UE scheduled
by BS-X in every time slot. On the other hand, operator Y does not deploy any
IRS and is oblivious to the presence of operator X’s IRS.

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

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

$y_{k}=\left(h_{d,k}+\mathbf{g}_{k}^{T}\boldsymbol{\Theta}\mathbf{f}^{X}\right)x_{k}+n_{k},$
(1)

where $\mathbf{g}_{k}\in\mathbb{C}^{N\times 1}$ is the channel from IRS to the
$k$th UE, $\mathbf{f}^{X}\in\mathbb{C}^{N\times 1}$ is the channel from BS-X
to the IRS, $\boldsymbol{\Theta}\in\mathbb{C}^{N\times N}$ is a diagonal
matrix containing the IRS reflection coefficients of the form $e^{j\theta}$,
and $h_{d,k}$ is the direct (non-IRS) path from BS to the UE-$k$. Also,
$x_{k}$ is the data symbol for UE-$k$ with average power
$\mathbb{E}[|x_{k}|^{2}]=P$, and $n_{k}$ is the additive noise
$\sim\mathcal{CN}(0,\sigma^{2})$ at UE-$k$. Similarly, at UE-$q$ served by
BS-Y, we have

$y_{q}=\left(h_{d,q}+\mathbf{g}_{q}^{T}\boldsymbol{\Theta}\mathbf{f}^{Y}\right)x_{q}+n_{q}.$
(2)

In Figure 1, we pictorially illustrate the considered network.

###  II-A Channel Model in sub-6 GHz Frequency Bands

Similar to [26], in the sub-6 GHz band, we consider that all the fading
channels are statistically independent333We consider that IRS elements are
placed sufficiently far apart that the spatial correlation in the channels
to/from the IRS is negligible. and identically distributed (i.i.d.) following
the Rayleigh distribution.444Our results can be easily extended to other
sub-$6$ GHz channel fading models also. From the approach used to show the
results here, it is easy to see that the scaling law of the SE as a function
of the IRS parameters will remain the same, although the scaling constants
depend on the fading model. Thus, the conclusions on the impact of the IRS on
OOB performance reported in this paper will continue to hold. Specifically,
$h_{d,k}=\sqrt{\beta_{d,k}}\tilde{h}_{d,k},h_{d,q}=\sqrt{\beta_{d,q}}\tilde{h}_{d,q};\tilde{h}_{d,k},\tilde{h}_{d,q}\mathrel{\overset{\text{i.i.d.}}{\scalebox{2.0}[1.0]{$\sim$}}}\mathcal{CN}(0,1)$;
$\mathbf{g}_{k}=\sqrt{\beta_{\mathbf{g},k}}\tilde{\mathbf{g}}_{k},\mathbf{g}_{q}=\sqrt{\beta_{\mathbf{g},q}}\tilde{\mathbf{g}}_{q};\tilde{\mathbf{g}}_{k},\tilde{\mathbf{g}}_{q}\mathrel{\overset{\text{i.i.d.}}{\scalebox{2.0}[1.0]{$\sim$}}}\mathcal{CN}(\mathbf{0},\mathbf{I}_{N})$;
$\mathbf{f}^{X}=\sqrt{\beta_{\mathbf{f}^{X}}}\tilde{\mathbf{f}}^{X},\mathbf{f}^{Y}=\sqrt{\beta_{\mathbf{f}^{Y}}}\tilde{\mathbf{f}}^{Y};\tilde{\mathbf{f}}^{X},\tilde{\mathbf{f}}^{Y}\mathrel{\overset{\text{i.i.d.}}{\scalebox{2.0}[1.0]{$\sim$}}}\mathcal{CN}(\mathbf{0},\mathbf{I}_{N})$.
All terms of the form $\beta_{x}$ represent the path losses.

###  II-B Channel Model in mmWave Frequency Bands

In the mmWave band, we consider a Saleh-Valenzuela type model for all the
channels [29]. At UE-$\ell$, the channels are

$\mathbf{f}^{p}\\!=\\!\\!\sqrt{\\!\frac{N}{L_{1,p}}}\\!\sum_{i=1}^{L_{1,p}}\\!\\!\gamma^{(1)}_{i,p}\\!\mathbf{a}^{*}_{N}(\phi_{i,p});\\!\text{
}\mathbf{g}_{\ell}\\!=\\!\\!\sqrt{\\!\frac{N}{L_{2,\ell}}}\\!\sum_{j=1}^{L_{2,\ell}}\\!\\!\gamma^{(2)}_{j,\ell}\\!\mathbf{a}^{*}_{N}(\psi_{j,\ell}),\\!\\!\\!$
(3)

where $p\in\\{X,Y\\}$, $L_{1,p}$ and $L_{2,\ell}$ are the number of spatial
paths in the BS-$p$ to IRS, and IRS to UE-$\ell$ links, respectively. Since
the in-band and OOB BSs/UEs are distributed arbitrarily with respect to the
IRS, for notational simplicity, we let $L_{1,X}=L_{1,Y}\triangleq L_{1}$, and
$L_{2,\ell}=L_{2}$ $\forall\ell$ (including in-band and OOB UEs.) Also,
$\phi_{i,p}$ and $\psi_{j,\ell}$ denote the sine of the angle of arrival of
the signal from BS-$p$ to the IRS via the $i$th path, and the sine of the
angle of departure from the IRS to the $\ell$th UE via the $j$th path, where
sine of an angle ($\phi_{x}$) is related to the physical spatial angle
($\chi$) by555In the sequel, the term “angle” will denote the sine of a
physical angle. $\phi_{x}=(2d/\lambda)\sin(\chi)$, with $d$, $\lambda$ being
the inter-elemental distance and signal wavelength, respectively. The sine
terms are sampled from a distribution $\mathcal{U}_{\mathcal{A}}$, which
depends on the beam resolution capability of the IRS (elaborated next.) The
fading coefficients, $\gamma^{(1)}_{i,p}$ and $\gamma^{(2)}_{j,\ell}$, are
independently sampled from $\mathcal{CN}(0,\beta_{\mathbf{f}^{p}})$, and
$\mathcal{CN}(0,\beta_{\mathbf{g},\ell})$, respectively. Finally,
$\mathbf{a}_{N}(\phi)$ is an array steering vector of a uniform linear array
(ULA)666All our main conclusions remain unchanged even if other array
geometries (e.g., a uniform planar array) are used at the IRS, similar to [30,
31, 32, 33]. based IRS oriented at the angle $\phi$, given by

$\mathbf{a}_{N}(\phi)=\frac{1}{\sqrt{N}}[1,e^{-j\pi\phi},\ldots,e^{-j(N-1)\pi\phi}]^{T}.$
(4)

####  II-B1 Beam Resolution Capability of the IRS

We now describe the beam resolution capability of an IRS, which is important
to account for when the array contains a finite number of elements. Beam
resolution measures the degree to which two closely located beams are
distinguishable (or resolvable.) Spatial paths in (3) will become non-
resolvable if the angles associated with the paths are spaced less the beam
resolution and can be modeled as being clustered into a single path with an
appropriate channel gain. To measure the beam resolution of the ULA-based IRS
model [30, 31, 32, 33], we enumerate all possible $D$ beams as

$\mathbf{A}=\left[\mathbf{a}_{N}(\phi_{1}),\mathbf{a}_{N}(\phi_{2}),\ldots,\mathbf{a}_{N}(\phi_{D})\right]\in\mathbb{C}^{N\times
D}.$ (5)

We can verify that $\mathbf{A}$ is a Vandermonde matrix and hence its column
vectors $\left\\{\mathbf{a}_{N}(\phi_{i})\right\\}^{D}_{i=1}$ are linearly
independent when $D\\!\leq\\!N$ provided $\\{\phi_{i}\\}_{i=1}^{D}$ are
distinct [34, Pg. 185]. On the other hand, when $D>N$, the vectors are always
linearly dependent and hence are non-resolvable. So, the total number of
independent (or resolvable) beams that can be formed by an $N$-element IRS is
at most $N$. This fact has been observed in the literature [29, 35, 36], and
also in the current works on IRS-aided mmWave wireless systems [23, 37]. Thus,
the complete set of resolvable beams $\mathcal{A}$ (called the _resolvable
beambook_) at the IRS is777For analytical tractability, we consider a _flat-
top_ RF directivity pattern of the IRS [38, Eq. 5]. This has been shown to be
a good approximation to practical array systems and becomes accurate as $N$
gets large [38, 29].

$\mathcal{A}\triangleq\left\\{\mathbf{a}_{N}(\phi),\phi\in\mathbf{\Phi}\right\\};\mathbf{\Phi}\triangleq\\!\left\\{\\!\left(\\!-1+\\!\dfrac{2i}{N}\right)\\!\bigg{\rvert}i=0,\ldots,N-1\\!\right\\}\\!.\\!$
(6)

Here $\mathbf{\Phi}$ is the _resolvable anglebook_ of the IRS, and, without
loss in generality, we model its distribution $\mathcal{U}_{\mathcal{A}}$ by

$\mathcal{U}_{\mathcal{A}}(\phi)=\frac{1}{\left|\mathbf{\Phi}\right|}\mathbbm{1}_{\left\\{\phi\in\mathbf{\Phi}\right\\}}=\frac{1}{N}\mathbbm{1}_{\left\\{\phi\in\mathbf{\Phi}\right\\}}.$
(7)

Also, for large $N$, $\mathcal{A}$ constitutes a set of orthonormal vectors:

$\mathbf{a}^{H}_{N}(\phi_{1})\mathbf{a}_{N}(\phi_{2})\rightarrow\delta_{\\{\phi_{1},\phi_{2}\\}}\hskip
14.22636pt\forall\hskip
5.69046pt\mathbf{a}_{N}(\phi_{1}),\mathbf{a}_{N}(\phi_{2})\in\mathcal{A},$ (8)

where $\delta_{\\{x,y\\}}$ is the usual dirac-delta function.

###  II-C Problem Statement

We are now ready to state the problem mathematically. Suppose BS-X tunes the
IRS with the SNR-optimal vector
$\boldsymbol{\theta}^{\mathrm{opt}}\triangleq\mathrm{diag}(\boldsymbol{\Theta}^{\mathrm{opt}})$
to serve UE-$k$ by solving the problem:

$\boldsymbol{\Theta}^{\mathrm{opt}}=\operatorname*{arg\,max}_{\boldsymbol{\Theta}\in\mathbb{C}^{N\times
N}}\quad\left|h_{d,k}+\mathbf{g}_{k}^{T}\boldsymbol{\Theta}\mathbf{f}^{X}\right|^{2},$
(9)

subject to $\boldsymbol{\Theta}$ being a diagonal matrix with unit-magnitude
diagonal entries. Then, we wish to characterize the impact of the IRS on the
channel gain,
$\left|h_{d,q}\\!+\mathbf{g}_{q}^{T}\boldsymbol{\Theta}^{\mathrm{opt}}\mathbf{f}^{Y}\right|^{2}$,
of OOB UE-$q$ scheduled by BS-Y. In particular, we wish to answer:

  1. 1.

Does the IRS degrade OOB performance? That is, when or how often will
$\left|h_{d,q}+\mathbf{g}_{q}^{T}\boldsymbol{\Theta}^{\mathrm{opt}}\mathbf{f}^{Y}\right|^{2}\\!<\\!\left|h_{d,q}\right|^{2}$
hold?

  2. 2.

How does the OOB channel gain
$\left|h_{d,q}+\mathbf{g}_{q}^{T}\boldsymbol{\Theta}^{\mathrm{opt}}\mathbf{f}^{Y}\right|^{2}$
depend on $N$?

Clearly, these aspects of an IRS are fundamental to understanding the overall
impact of an IRS in practical systems when many operators co-exist. In the
sequel, we answer these by analyzing the ergodic and instantaneous
characteristics of the OOB UE’s channel in both sub-6 GHz and mmWave bands in
the presence of an IRS controlled by a different operator.

######  Remark 1.

The next two sections focus on cases where both operators provide services in
the sub-$6$ GHz band (Sec. III) or both provide services in the mmWave band
(Sec. IV), using non-overlapping frequency allocations. We do not consider the
case where one operator uses the sub-$6$ GHz band while the other uses the
mmWave band, as it is unclear whether an IRS can efficiently reflect signals
in both these frequency bands.

##  III OOB Performance: sub-6 GHz bands

Suppose an operator X deploys and controls an IRS to enhance the SE of the UEs
being served by it in a sub-6 GHz band. We wish to characterize the effect of
the IRS on operator Y, which is operating in a different sub-6 GHz frequency
band with no control over the IRS. Thus, to serve the $k$th UE, BS-X
configures the IRS with the SNR/SE-optimal phases [2, 3, 12]

$\theta^{\mathrm{opt}}_{n,k}=e^{j\left(\angle
h_{d,k}-\left(\angle{f}^{X}_{n}+\angle{g}_{k,n}\right)\right)},\hskip
14.22636ptn=1,{\dots},N,$ (10)

which results in the coherent addition of the signals along the direct and IRS
paths, leading to the maximum possible received SNR. Then, the SE achieved by
the $k$th UE is

$R_{k}^{BF}=\log_{2}\\!\left(\\!1+\frac{P}{\sigma^{2}}\left||h_{d,k}|+\sum\nolimits_{n=1}^{N}|f^{X}_{n}{g_{k,n}}|\right|^{2}\right).$
(11)

Due to the independence of the channels of the UEs served by operators X and
Y, the IRS phase configuration used by operator X to serve its own UEs appears
as a _random_ phase configuration of the IRS for any UE served by operator Y.

We consider RR scheduling of UEs at both BS-X and BS-Y.888The extension of our
results to the proportional fair and max-rate scheduling schemes is provided
in Sec. V. Since the BSs are equipped with a single antenna, one UE from each
network is scheduled in every time slot. A summary of the protocol is given in
Fig. 2.

We characterize the average OOB performance by deriving the ergodic sum-SE of
both networks and then infer the degree of degradation/enhancement of the OOB
performance caused by the IRS. The ergodic SE at UE-$k$ is

$\\!\langle
R_{k}^{(X)}\rangle=\mathbb{E}\\!\left[\log_{2}\left(\\!1\\!+\left|\sum\nolimits_{n=1}^{N}\left|f^{X}_{n}{g_{k,n}}\right|\\!+\\!\left|h_{d,k}\right|\right|^{2}\\!\\!\\!\dfrac{P}{\sigma^{2}}\right)\\!\\!\right]\\!,\\!\\!\\!\\!$
(12)

since the IRS is configured with the optimal phase configuration for
(scheduled) UE-$k$. On the other hand, the ergodic SE for (scheduled) UE-$q$
of operator Y is

$\langle
R_{q}^{(Y)}\rangle=\mathbb{E}\\!\left[\log_{2}\\!\left(\\!1\\!+\left|\sum\nolimits_{n=1}^{N}\\!f^{Y}_{n}{g_{q,n}}\\!+h_{d,q}\right|^{2}\\!\\!\dfrac{P}{\sigma^{2}}\right)\right],\\!\\!$
(13)

where we used the fact that the channels are circularly symmetric random
variables, i.e., $f^{Y}_{n}g_{q,n}e^{j\theta}\stackrel{{\scriptstyle
d}}{{=}}f^{Y}_{n}g_{q,n}$ for any $\theta$. Here, the expectations are taken
with respect to the distribution of the channels to the respective UEs. With
RR scheduling, the ergodic sum-SEs of two operators are given by

$\bar{R}^{(X)}\triangleq\frac{1}{K}\sum_{k=1}^{K}\langle
R_{k}^{(X)}\rangle,\text{ and
}\bar{R}^{(Y)}\triangleq\frac{1}{Q}\sum_{q=1}^{Q}\langle R_{q}^{(Y)}\rangle.$
(14)

We note that closed-form expressions for the ergodic sum-SE are difficult to
obtain due to the complicated distribution of the SNR and SE terms (e.g., in
(12) and (13).) Although we can use the approach in [39] to obtain the exact
ergodic sum-SE in terms of standard integrals, the resulting expressions do
not provide insights into how the sum-SE scales with the system parameters.
Instead, we rely on applying tight approximations to obtain insightful
results. However, we note that it is important to choose approximations such
that the analysis across different scenarios can be unified in a single
framework and the corresponding results are comparable. To that end, we apply
Jensen’s inequality, and after careful simplification, we arrive at elegant
and interpretable expressions for the ergodic sum-SEs in all scenarios. In
this view, we have the following theorem for the sub-6 GHz band of
communication.

Figure 2: Flowchart of the round-robin scheduling based protocol.

######  Theorem 1.

Under independent Rayleigh fading channels in the sub-6 GHz bands, with RR
scheduling, and when the IRS is optimized to serve the UEs of operator X, the
ergodic sum-SEs of operators X and Y scale as

$\bar{R}^{(X)}\approx\frac{1}{K}\\!\sum_{k=1}^{K}\log_{2}\left(1+\left[N^{2}\\!\left(\frac{\pi^{2}}{16}\beta_{r,k}\\!\right)\\!+\right.\right.\\\
\left.\left.\\!N\\!\left(\\!\beta_{r,k}\\!-\frac{\pi^{2}}{16}\beta_{r,k}+\frac{\pi^{3/2}}{4}\sqrt{\beta_{d,k}\beta_{r,k}}\right)\\!+\\!\beta_{d,k}\right]\\!\frac{P}{\sigma^{2}}\right)\\!,\\!$
(15)

where $\beta_{r,k}\triangleq\beta_{\mathbf{f}^{X}}\beta_{\mathbf{g},k}$, and

$\bar{R}^{(Y)}\approx\frac{1}{Q}\sum_{q=1}^{Q}\log_{2}\left(1+\left[N\beta_{r,q}+\beta_{d,q}\right]\frac{P}{\sigma^{2}}\right),$
(16)

where $\beta_{r,q}\triangleq\beta_{\mathbf{f}^{Y}}\beta_{\mathbf{g},q}$.

###### Proof.

We compute the sum-SEs for both operators below.

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

We first compute $\langle R_{k}^{(X)}\rangle$ for a given $k$. By Jensen’s
inequality, we obtain

$\\!\\!\langle
R_{k}^{(X)}\rangle\leq\log_{2}\\!\left(\\!\\!1\\!+\\!\mathbb{E}\left[\left|\sum\nolimits_{n=1}^{N}|f^{X}_{n}{g_{k,n}}|+|h_{d,k}|\right|^{2}\right]\\!\frac{P}{\sigma^{2}}\\!\right)\\!.\\!$
(17)

We expand the expectation term as follows.

$\\!\\!\\!\\!\left|\sum_{n=1}^{N}|f^{X}_{n}{g_{k,n}}|+|h_{d,k}|\right|^{2}\\!\\!=\sum_{n,m=1}^{N}|f^{X}_{n}||g_{k,n}||f^{X}_{m}||g_{k,m}|+|h_{d,k}|^{2}\\\
+2\sum\nolimits_{n=1}^{N}|f^{X}_{n}||g_{k,n}||h_{d,k}|\ \
=\sum\nolimits_{n=1}^{N}|f^{X}_{n}|^{2}|g_{k,n}|^{2}+|h_{d,k}|^{2}+\\\
\mathop{\sum\nolimits_{n,m=1}^{N}}_{n\neq
m}|f^{X}_{n}||g_{k,n}||f^{X}_{m}||g_{k,m}|+2\sum_{n=1}^{N}|f^{X}_{n}||g_{k,n}||h_{d,k}|.$
(18)

Under Rayleigh fading,
$\mathbb{E}[|f_{n}^{X}|^{2}]=\beta_{\mathbf{f}^{X}},\mathbb{E}[|g_{k,n}|^{2}]=\beta_{\mathbf{g},k},\mathbb{E}[|h_{d,k}|^{2}]=\beta_{d,k}$,
$\mathbb{E}[|f_{n}^{X}|]=\sqrt{\dfrac{\pi}{4}\beta_{\mathbf{f}^{X}}},$
$\mathbb{E}[|g_{k,n}|]=\sqrt{\dfrac{\pi}{4}\beta_{\mathbf{g},k}},\mathbb{E}[|h_{d,k}|]=\\!\sqrt{\dfrac{\pi}{4}\beta_{d,k}},\forall
k=1,\ldots,K;n=1,\ldots,N$. Further, all the random variables are independent.
Taking the expectation in (18), and substituting for these values, we get

$\mathbb{E}\left[\left|\sum\nolimits_{n=1}^{N}|f^{X}_{n}{g_{k,n}}|+|h_{d,k}|\right|^{2}\right]=N^{2}\left(\frac{\pi^{2}}{16}\beta_{r,k}\right)\\\
+N\left(\beta_{r,k}-\frac{\pi^{2}}{16}\beta_{r,k}+\frac{\pi^{3/2}}{4}\sqrt{\beta_{d,k}\beta_{r,k}}\right)+\beta_{d,k},$
(19)

where $\beta_{r,k}$ is as defined in the statement of the theorem.
Substituting (19) in (17), and plugging in the resulting expression in (14)
yields (15), as desired.

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

As above, from Jensen’s inequality, we have

$\\!\\!\langle
R_{q}^{(Y)}\rangle\leq\log_{2}\left(\\!1+\mathbb{E}\left[\left|\sum\nolimits_{n=1}^{N}f^{Y}_{n}{g_{q,n}}+h_{d,q}\right|^{2}\right]\\!\frac{P}{\sigma^{2}}\right)\\!.\\!\\!$
(20)

Proceeding along the same lines, we get

$\left|\sum\nolimits_{n=1}^{N}f^{Y}_{n}{g_{q,n}}+h_{d,q}\right|^{2}=\sum\nolimits_{n=1}^{N}|f^{Y}_{n}|^{2}|g_{q,n}|^{2}+|h_{d,q}|^{2}+\\\
\\!\mathop{\sum\nolimits_{n,m=1}^{N}}_{n\neq
m}f^{Y}_{n}g_{q,n}f^{Y*}_{m}g^{*}_{q,m}\\!+\\!2\Re\left(\\!\sum\nolimits_{n=1}^{N}f^{Y}_{n}g_{q,n}h^{*}_{d,q}\right).$
(21)

Taking the expectation and simplifying,

$\mathbb{E}\left[\left|\sum\nolimits_{n=1}^{N}f^{Y}_{n}{g_{q,n}}+h_{d,q}\right|^{2}\right]=N\beta_{r,q}+\beta_{d,k}.$
(22)

Substituting (22) in (20), and plugging in the resulting expression in (14)
yields (16). ∎ From the above theorem, we infer the following:

  * •

The IRS enhances the average received SNR by a factor of $N^{2}$ at any
scheduled (in-band) UE of operator X when BS-X optimizes the IRS. This is the
benefit that operator X obtains by using an optimized $N$-element IRS.

  * •

Operator Y, who does not control the IRS, also witnesses an enhancement of
average SNR by a factor of $N$ for free, i.e., without any coordination with
the IRS. This happens because the IRS makes the wireless environment more
rich-scattering on average, and facilitates the reception of multiple copies
of the signals at the (OOB) UEs.

Next, we prove that even the instantaneous characteristics of the OOB channel
are favorable due to the IRS. We recognize that the instantaneous channel at
an OOB UE-$q$ is given by $|h_{q}|^{2}$, where
$h_{q}\triangleq\sum_{n=1}^{N}f^{Y}_{n}{g_{q,n}}+h_{d,q}$. In the sequel, we
provide two kinds of results: first, we compute the outage probability
experienced by UE-$q$ and then provide stronger results on the OOB channels
via stochastic dominance theory.

The outage probability of UE-$q$ is given by

$P^{\rho}_{q,\text{out}}=\mathsf{Pr}(|h_{q}|^{2}<\rho),$ (23)

where $\rho$ is a constant that depends on the receiver sensitivity threshold.
Although works like [16] provide closed-form expressions for the outage
probabilities of randomly configured IRSs, these expressions do not provide
insight into the system performance. So, using [12, Proposition $1$], we
approximate $h_{q}\sim\mathcal{CN}(0,N\beta_{r,q}+\beta_{d,q})$ which becomes
accurate as $N$ gets large.999This approximation works well even with $N=8$
[12]. Hence, $|h_{q}|^{2}$ is a exponential random variable with mean
$N\beta_{r,q}+\beta_{d,q}$, and the outage probability in (23) can be easily
obtained as shown in Theorem 2 below.

Next, we characterize the stochastic behavior of the channel gains witnessed
by UE-$q$ with/without an IRS. In this view, define the following random
variables.

$\left|h_{1,q}\right|^{2}\triangleq\left|\sum_{n=1}^{N}f^{Y}_{n}{g_{q,n}}+h_{d,q}\right|^{2};\left|h_{2,q}\right|^{2}\triangleq\left|h_{d,q}\right|^{2}.$
(24)

Note that $\left|h_{1,q}\right|^{2}$ and $\left|h_{2,q}\right|^{2}$ represent
the channel power gain of UE-$q$ in the presence and absence of the IRS,
respectively. We now characterize the _change_ in the channel gain at UE-$q$
served by BS-Y in the presence and absence of the IRS:

$Z^{(Y)}_{N}\triangleq\left|h_{1,q}\right|^{2}-\left|h_{2,q}\right|^{2}{\mathbbm{1}_{\\{N\neq
0\\}}}.$ (25)

The event $\\{Z^{(Y)}_{N}<0\\}$ indicates an SNR degradation at the OOB UE due
to the IRS. In the following theorem, we show that, almost surely,
$Z^{(Y)}_{N}$ is non-negative by deriving the CCDF of $Z^{(Y)}_{N}$,
$\bar{F}_{Z^{(Y)}_{N}}(z)\triangleq{\sf{Pr}}(Z^{(Y)}_{N}\geq z).$

######  Theorem 2.

The probability of outage at UE-$q$ served by an (OOB) operator Y in the sub-6
GHz band, when an IRS is optimized to serve the UEs of operator X, is given by

$P^{\rho}_{q,\text{out}}=1-e^{-\dfrac{\rho}{N\beta_{r,q}+\beta_{d,q}}}.$ (26)

Further, for reasonably large $N$,101010We will later numerically show that
the result holds even for $N\geq 4$. the CCDF $\bar{F}_{Z^{(Y)}_{N}}(z)$ of
$Z^{(Y)}_{N}$is given by

$\\!\bar{F}_{Z^{(Y)}_{N}}(z)=\begin{cases}1-\dfrac{1}{N\tilde{\beta}+2}\times\\!e^{\left(\dfrac{z}{\beta_{d,q}}\right)},\hskip
2.84544pt&\\!\\!\\!\\!\mathrm{if}\ z<0,\\\
\left(\dfrac{N\tilde{\beta}+1}{N\tilde{\beta}+2}\right)\times\\!e^{-\left(\dfrac{z}{\beta_{d,q}\left(1+N\tilde{\beta}\right)}\right)},&\\!\\!\\!\\!\mathrm{if}\
z\geq 0.\\\ \end{cases}$ (27)

where $\tilde{\beta}\triangleq\beta_{r,q}/\beta_{d,q}$.

###### Proof.

We can obtain $P^{\rho}_{q,\text{out}}$ using the CCDF of an exponential
random variable. For (27), see Appendix A. ∎

From Theorem 2, we see that the outage probability is strictly monotonically
decreasing with $N$, which shows that the IRS only improves the OOB
performance. In fact, from a first-order Taylor series approximation, we can
show that $P^{\rho}_{q,\text{out}}\approx{\rho}/{(N\beta_{r,q}+\beta_{d,q})}$,
i.e., the outage probability decreases linearly in $N$. Further, since
$\bar{F}_{Z^{(Y)}_{N}}(0)=1-{1}/\left({2+N\tilde{\beta}}\right),$ for a given
$\tilde{\beta}$, the probability that the SNR/gain offset in (25) is negative
decays as $\mathcal{O}\left({1}/{N}\right)$. Also,
$\bar{F}_{Z^{(Y)}_{N^{\prime}}}(z)\geq\bar{F}_{Z^{(Y)}_{N^{\prime\prime}}}(z)$
for all $z$ and $N^{\prime}\\!>\\!N^{\prime\prime}$. Consequently, we have the
following proposition.

######  Proposition 1.

For any $M,N\in\mathbb{N}$ with $M>N$, the random variable $Z^{(Y)}_{M}$
_stochastically dominates_ 111111A real-valued random variable $X$ is
stochastically larger than, or dominates, the real-valued random variable $Y$,
written $X>_{st}Y$, if  ${\sf{Pr}}(X>a)\geq{\sf{Pr}}(Y>a),\hskip
5.69046pt\text{for all }a.$ (28) Note that, if the random variables $X$ and
$Y$ have CCDFs $\bar{F}$ and $\bar{G}$, respectively, then
$X>_{st}Y\Longleftrightarrow\bar{F}(a)\geq\bar{G}(a)$ $\forall$
$a\in\mathbb{R}$ [40]. $Z^{(Y)}_{N}$. In particular, the channel gain in the
presence of the IRS stochastically dominates the channel gain in its absence.

The above proposition states that the random variables
$\left\\{Z^{(Y)}_{n}\right\\}_{n\in\mathbb{N}}$ form a sequence of
_stochastically larger_ random variables as a function of the number of IRS
elements, where $\mathbb{N}$ is the set of natural numbers. Thus, the SNR
offset increases with the number of IRS elements even at an OOB UE, i.e., the
IRS only enhances the channel quality at an OOB UE at any point in time, with
high probability. Therefore, the performance of OOB operators _does not
degrade_ even when the operator is entirely oblivious to the presence of the
IRS. Note that this holds true for any operator in the area; hence, no
operator will be at a disadvantage due to the presence of an IRS being
controlled by only one operator.

######  Remark 2.

The outage probability and the CCDF of the SNR offset as in Theorem 2 at the
in-band UE (for whom the IRS is optimized) decay to zero as
$\mathcal{O}(e^{-N})$ (see Appendix B.) This improvement in decay is the
benefit of using an optimized IRS.

Having asserted that IRS can only benefit every OOB operator in the sub-6 GHz
bands, we next move on to understand the effect of IRS on the OOB performance
in the mmWave bands of communications, where the IRSs can significantly boost
the in-band operators’ performance in establishing link connectivity,
improving coverage, etc.

##  IV OOB Performance: mmWave bands

In this section, we evaluate the in-band and OOB performance of an IRS-aided
mmWave system under RR scheduling of the UEs. The wireless channels in the
mmWave bands are typically spatially sparse and directional, with a few
propagation paths, as against the rich-scattering sub-6 GHz channels.
Consequently, the analysis in the sub-6 GHz bands does not extend to mmWave
bands, and it is necessary to evaluate the OOB performance in the mmWave bands
independently. We consider two different scenarios inspired by [23], namely,
the LoS and (L+)NLoS scenarios, as explained in Sec. I-B. In both cases, we
assume that the channel (see (3)) to the OOB UEs is a mixture of LoS and NLoS
paths.

###  IV-A IRS Optimized for LoS Scenarios

The dominant LoS channel of the in-band UE is [24, 25]

$\displaystyle h_{k}$
$\displaystyle=N\gamma^{(1)}_{1,X}\gamma^{(2)}_{1,k}\mathbf{a}_{N}^{H}(\psi_{1,k})\boldsymbol{\Theta}\mathbf{a}_{N}^{*}(\phi_{1,X})+h_{d,k}$
(29)
$\displaystyle\stackrel{{\scriptstyle(a)}}{{=}}N\left(\\!\gamma^{(1)}_{1,X}\gamma^{(2)}_{1,k}\left(\mathbf{a}_{N}^{H}(\phi_{1,X})\odot\mathbf{a}_{N}^{H}(\psi_{1,k})\right)\right)\\!\boldsymbol{\theta}+h_{d,k},$
(30)

where $\boldsymbol{\theta}=\text{diag}(\boldsymbol{\Theta})\in\mathbb{C}^{N}$,
and $(a)$ is obtained using the properties of the Hadamard product. Since the
Hadamard product of two array response vectors is also an array vector but
aligned in a different direction [30], we simplify (30) as

$h_{k}=N\gamma_{X,k}\mathbf{\dot{a}}_{N}^{H}(\omega^{1}_{X,k})\boldsymbol{\theta}+h_{d,k},$
(31)

where
$\omega^{1}_{X,k}\triangleq\sin_{(p)}^{-1}\left(\sin(\phi_{1,X})+\sin(\psi_{1,k})\right)$,121212We
define $\sin_{(p)}^{-1}(x)$ so that $x$ is in the principal argument $[-1,1)$
as  $\\!\\!\\!\sin_{(p)}^{-1}(x)=\begin{cases}\\!\sin^{-1}(x-2),&\mathrm{if}\
x\geq 1,\\\ \\!\sin^{-1}(x),&\mathrm{if}\ x\in[-1,1),\\\
\\!\sin^{-1}(x+2),&\mathrm{if}\ x<-1.\end{cases}$ (32) and
$\gamma_{X,k}\triangleq\gamma^{(1)}_{1,X}\gamma^{(2)}_{1,k}$. Here,
$\mathbf{\dot{a}}_{N}$ is an array vector that is normalized by $N$ instead of
$\sqrt{N}$ (see (4)), and so
$\mathbf{\dot{a}}_{N}(\cdot)=\frac{1}{\sqrt{N}}\mathbf{a}_{N}(\cdot)$. Then by
the Cauchy-Schwarz (CS) inequality, the $n$th entry of the optimal IRS
configuration vector $\boldsymbol{\theta}^{\mathrm{opt}}$ which maximizes the
channel gain $|h_{k}|^{2}$ is $\theta_{n}=e^{j\left(\angle
h_{d,k}-\pi(n-1)\omega^{1}_{X,k}-\angle\gamma_{X,k}\right)}$, and hence the
optimal IRS vector is

$\boldsymbol{\theta}^{\mathrm{opt}}=\dfrac{h_{d,k}\gamma^{*}_{X,k}}{\left|h_{d,k}\gamma_{X,k}\right|}\times
N\mathbf{\dot{a}}_{N}(\omega^{1}_{X,k}).$ (33)

From (33), it is clear that the IRS vector is _directional_ ; it is aligned in
the direction of the in-band UE’s channel. To illustrate this, we plot the
correlation function
$\mathbb{E}\left[\left|\mathbf{\dot{a}}^{H}(\nu)\boldsymbol{\theta}^{\mathrm{opt}}\right|^{2}\right]$
versus $\nu\in\boldsymbol{\Phi}$ in Fig. 3, where $\boldsymbol{\theta}$ is set
to align with an arbitrary in-band UE with $\omega^{1}_{X,k}=0.52$. The
expectation is with respect to the channel fading coefficients. Clearly, when
$\nu=0.52=\omega^{1}_{X,k}$, the function takes its maximum value of $1$. This
confirms that the IRS aligns with the in-band channel to which it is
optimized. Contrariwise, the IRS phase shift as optimized by BS-X will be
aligned at a _random_ direction from any OOB UE’s view with the distribution
given by $\mathcal{U}_{\mathcal{A}}$ in (7). We now characterize the ergodic
sum-SEs of both operators.

######  Theorem 3.

Under the Saleh-Valenzuela LoS model in the mmWave channels, with RR
scheduling, and when the IRS is optimized to serve the UEs of operator X, the
ergodic sum-SEs of operators X and Y scale as

$\bar{S}_{1}^{(X)}\approx\frac{1}{K}\sum_{k=1}^{K}\log_{2}\bigg{(}1+\bigg{[}N^{2}\beta_{r,k}\\\
\left.\left.+\
N\left(\frac{\pi^{3/2}}{4}\sqrt{\beta_{d,k}\beta_{r,k}}\right)+\beta_{d,k}\right]\frac{P}{\sigma^{2}}\right),$
(34)

and

$\bar{S}_{1}^{(Y)}\approx\frac{1}{Q}\sum_{q=1}^{Q}\left(\dfrac{\bar{L}}{N}\log_{2}\left(1+\left[\dfrac{N^{2}}{\bar{L}}\beta_{r,q}+\beta_{d,q}\right]\frac{P}{\sigma^{2}}\right)\right.\\\
\left.+\left(1-\dfrac{\bar{L}}{N}\right)\log_{2}\left(1+\beta_{d,q}\frac{P}{\sigma^{2}}\right)\right),$
(35)

respectively, where $\bar{L}\triangleq\min\left\\{L,N\right\\}$, and
$L\triangleq L_{1}L_{2}$.

###### Proof.

We derive the SEs of both operators separately below.

####  IV-A1 Ergodic sum-SE of operator X

Let the average SE seen by UE-$k$ be $\langle S_{k,1}^{(X)}\rangle$. Then the
ergodic sum-SE of operator X is
$\bar{S}_{1}^{(X)}=\frac{1}{K}\sum_{k=1}^{K}\langle S_{k,1}^{(X)}\rangle$. By
Jensen’s inequality,

$\\!\\!\langle
S_{k,1}^{(X)}\rangle\leq\log_{2}\\!\left(\\!1+\mathbb{E}\left[|h_{k}|^{2}\right]\\!\frac{P}{\sigma^{2}}\right).$
(36)

We recognize that

$\displaystyle\\!\\!\left|h_{k}\right|^{2}$
$\displaystyle=\left|N\gamma_{X,k}\mathbf{\dot{a}}_{N}^{H}(\omega^{1}_{X,k})\boldsymbol{\theta}^{\mathrm{opt}}+h_{d,k}\right|^{2}$
(37)
$\displaystyle\stackrel{{\scriptstyle(a)}}{{=}}{\left|h_{d,k}\\!+\\!N^{2}\dfrac{h_{d,k}\left|\gamma_{X,k}\right|^{2}}{\left|h_{d,k}\gamma_{X,k}\right|}\\!\times\\!\mathbf{\dot{a}}_{N}^{H}(\omega^{1}_{X,k})\mathbf{\dot{a}}_{N}(\omega^{1}_{X,k})\right|}^{2}\\!,\\!\\!$
(38)

where $(a)$ is due to (33). Further, using
$\mathbf{\dot{a}}_{N}^{H}(\omega^{1}_{X,k})\mathbf{\dot{a}}_{N}(\omega^{1}_{X,k})=\frac{1}{N}$,
we can compute
$\mathbb{E}\left[\left|h_{k}\right|^{2}\right]=\mathbb{E}\left[\left|\left|h_{d,k}\right|+N\left|\gamma_{X,k}\right|\right|^{2}\right]=\mathbb{E}\left[\left|h_{d,k}\right|^{2}\right]+N^{2}\mathbb{E}\left[\left|\gamma_{X,k}\right|^{2}\right]+2N\mathbb{E}\left[\left|h_{d,k}\right|\left|\gamma_{X,k}\right|\right]$.
Using the results from Sec. II-B, this can be simplified as

$\mathbb{E}\left[\left|h_{k}\right|^{2}\right]=N^{2}\beta_{r,k}+N\left(\dfrac{\pi^{3/2}}{4}\right)\sqrt{\beta_{r,k}\beta_{d,k}}+\beta_{d,k}.$
(39)

Substituting the above in (36), and plugging it into the expression for
$\bar{S}_{1}^{(X)}$ yields (34).

####  IV-A2 Ergodic sum-SE of operator Y

Recall that the ergodic sum-SE of operator Y is
$\bar{S}_{1}^{(Y)}=\frac{1}{Q}\sum_{q=1}^{Q}\langle S_{q,1}^{(Y)}\rangle$.
Then, the channel seen by an arbitrary OOB UE, say UE-$q$, can be derived in a
similar manner as (31) and is given by

$h_{q}=h_{d,q}+\dfrac{N}{\sqrt{L}}\sum\nolimits_{l=1}^{L}\gamma^{(1)}_{l,Y}\gamma^{(2)}_{l,q}\mathbf{\dot{a}}_{N}^{H}(\omega^{l}_{Y,q})\boldsymbol{\theta}^{\mathrm{opt}},$
(40)

where $\boldsymbol{\theta}^{\mathrm{opt}}$ is given by (33), and
$\gamma^{(1)}_{l,Y},\gamma^{(2)}_{l,q}$ denote the channel gains of the paths
between BS-IRS, and IRS-UE, respectively, corresponding to the $l$th cascaded
path. Then we have

$\displaystyle h_{q}$
$\displaystyle=h_{d,q}+\dfrac{N^{2}}{\sqrt{L}}\sum\nolimits_{l=1}^{L}\\!\\!\\!\gamma^{(1)}_{l,Y}\gamma^{(2)}_{l,q}\dfrac{h_{d,k}\gamma^{*}_{X,k}}{\left|h_{d,k}\gamma_{X,k}\right|}\mathbf{\dot{a}}_{N}^{H}(\omega^{l}_{Y,q})\mathbf{\dot{a}}_{N}(\omega^{1}_{X,k})$
$\displaystyle=h_{d,q}+\dfrac{N^{2}}{\sqrt{L}}\sum\nolimits_{l=1}^{L}\gamma^{(1)}_{l,Y}\gamma^{(2)}_{l,q}\frac{e^{j\angle
h_{d,k}}}{e^{j\angle\gamma_{X,k}}}\mathbf{\dot{a}}_{N}^{H}(\omega^{l}_{Y,q})\mathbf{\dot{a}}_{N}(\omega^{1}_{X,k})$
$\displaystyle\stackrel{{\scriptstyle
d}}{{=}}h_{d,q}+\dfrac{N^{2}}{\sqrt{L}}\sum\nolimits_{l=1}^{L}\gamma^{(1)}_{l,Y}\gamma^{(2)}_{l,q}\mathbf{\dot{a}}_{N}^{H}(\omega^{l}_{Y,q})\mathbf{\dot{a}}_{N}(\omega^{1}_{X,k}),$
(41)

where, in the last step, we used the fact that the product term
$\gamma^{(1)}_{l,Y}\gamma^{(2)}_{l,q}$ is circularly symmetric. We also have

$\langle
S_{q,1}^{(Y)}\rangle\triangleq\mathbb{E}\left[S_{q,1}^{(Y)}\right]=\mathbb{E}\left[\log_{2}\left(1+|h_{q}|^{2}\frac{P}{\sigma^{2}}\right)\right].$
(42)

Consider $L<N$. Now, let $\mathcal{E}_{1}$ be the event that one of the $L$
angles of the OOB channel “matches” with the IRS angle, and $\mathcal{E}_{0}$
be the event that none of the $L$ angles of the OOB channel match with the IRS
angle. Then, we can write (42) as

$\\!\\!\\!\\!\langle
S_{q,1}^{(Y)}\rangle\stackrel{{\scriptstyle(a)}}{{=}}\sum_{i=0}^{1}\mathbb{E}\left[S_{q,1}^{(Y)}\mathbbm{1}_{\\{\mathcal{E}_{i}\\}}\right]\stackrel{{\scriptstyle(b)}}{{=}}\sum_{i=0}^{1}\mathbb{E}\left[S_{q,1}^{(Y)}\bigg{\rvert}\mathcal{E}_{i}\right]{\sf{Pr}}(\mathcal{E}_{i}),\\!\\!$
(43)

where $\mathbbm{1}_{\\{\mathcal{E}_{i}\\}}$ is the indicator for the
occurrence of event $\mathcal{E}_{i}$, $(a)$ is because $\mathcal{E}_{0}$ and
$\mathcal{E}_{1}$ are mutually exclusive and exhaustive; $(b)$ is due to the
law of iterated expectations. Also, from (7), and flat-top directivity of the
IRS, ${\sf{Pr}}(\mathcal{E}_{1})=L/N$, and
${\sf{Pr}}(\mathcal{E}_{0})=\left(1-L/N\right)$. Then from (8), and Jensen’s
inequality, we have

$\\!\\!\\!\\!\mathbb{E}\left[S_{q,1}^{(Y)}\bigg{\rvert}\mathcal{E}_{1}\right]\\!\leq\log_{2}\\!\left(\\!\\!1\\!+\\!\mathbb{E}\left[\left|h_{d,q}+\dfrac{N}{\sqrt{L}}\gamma^{(1)}_{l^{*},Y}\gamma^{(2)}_{l^{*},q}\right|^{2}\right]\\!\\!\frac{P}{\sigma^{2}}\\!\right)\\!,\\!\\!\\!$
(44) $\vspace{-0.05cm}\text{and, \hskip 21.33955pt
}\mathbb{E}\left[S_{q,1}^{(Y)}\bigg{\rvert}\mathcal{E}_{0}\right]\leq\log_{2}\left(1+\mathbb{E}\left[\left|h_{d,q}\right|^{2}\right]\frac{P}{\sigma^{2}}\right),$
(45)

where
$l^{*}=\arg_{l}\left\\{\mathbf{\dot{a}}_{N}^{H}(\omega^{l}_{Y,q})\mathbf{\dot{a}}_{N}(\omega^{1}_{X,k})=1/N\right\\}$.
We can show that (similar to Sec. III-2),
$\mathbb{E}\left[|h_{q}|^{2}\bigg{\rvert}\mathcal{E}_{1}\right]=\left(\dfrac{N^{2}}{L}\beta_{r,q}+\beta_{d,q}\right)$,
and
$\mathbb{E}\left[|h_{q}|^{2}\bigg{\rvert}\mathcal{E}_{0}\right]=\beta_{d,q}$.
Collecting all the terms and substituting in (42) and using the expression for
$\bar{S}_{1}^{(Y)}$ yields (35). Now, when $L\geq N$, since an $N$-element IRS
can steer at most $N$ resolvable beams, some of the paths will be clustered
together, and the above analysis holds by replacing $L$ with $N$. We cover
both cases by $\bar{L}=\min\\{L,N\\}$ in (35). ∎ From the above theorem, it is
clear that an IRS can never degrade the OOB performance in LoS scenarios of
the mmWave bands. Even when $L$ is small and $N$ is large, the second term in
(35) remains, which is the achievable SE in the absence of IRS. Thus, almost
surely, the IRS results in an OOB-SE that is at least as high as the SE seen
in its absence.

Figure 3: Correlation response of the IRS vector and array steering vectors
pointing at different spatial angles, $\nu$, for (a) $N=50$ and (b) $N=500$.
When $\nu=\omega^{1}_{X,k}$, the response attains its maximum value of $1$.

######  Remark 3 (OOB performance as a function of $L$ for large $N$).

From (35), we can show that when $N\rightarrow\infty$ for a fixed and finite
$L$, the SE is a unimodal function of $N$. Hence, there exists a $N=N_{0}$
which provides the maximum gain and distinguishes the _favorable_ region
$(N\leq N_{0})$ from the _non-favorable_ region $(N\geq N_{0})$.131313By non-
favorable, we mean that the SE does not grow linearly with $N$ beyond
$N=N_{0}$. However, the SE never drops below the achievable SE in the absence
of the IRS. Here, $N_{0}$ itself increases with $L$ because the probability of
establishing connectivity via the IRS, i.e., ${\sf{Pr}}(\mathcal{E}_{1})$
increases.

######  Remark 4 (Distributed IRSs always benefit).

As noted in Remark 3, the OOB UEs benefit more if the number of channel paths
at these UEs is large. A natural way to obtain more paths is to deploy
multiple distributed IRSs so that each IRS can provide an additional path to
the UEs. This, in turn, increases the overall probability of benefitting the
OOB UEs. We refer the reader to our follow-up work in [41] for more details.

Similar to Theorem 2, the outage probability and CCDF of an OOB UE
with/without IRS in mmWave bands is analyzed next.

######  Theorem 4.

The probability of outage at UE-$q$ served by an OOB operator Y in the mmWave
bands, when an IRS is optimized to serve the UEs of operator X is

$\\!\\!\\!{F}_{|h_{q}|^{2}}(\rho)\triangleq{\sf{Pr}}(|h_{q}|^{2}<\rho)=1-e^{-\rho/\beta_{d,q}}\\\
\\!\\!\\!-\dfrac{\bar{L}}{N}\left(\dfrac{\bar{L}e^{\frac{\bar{L}\beta_{d,q}}{N^{2}\beta_{r,q}}}}{N^{2}\beta_{r,q}}\mathcal{I}_{0}\left(\rho;\beta_{d,q},\dfrac{N^{2}}{\bar{L}}\beta_{r,q}\right)\\!\\!-e^{-\rho/\beta_{d,q}}\\!\right)\\!,\\!\\!\\!$
(46)

where $\bar{L}\triangleq\min\\{L,N\\}$, and $\
\mathcal{I}_{0}(x;c_{1},c_{2})\\!\triangleq\\!\vbox{\hbox{\stretchto[440]{\displaystyle\int}{4.5ex}}}_{\\!\\!\\!c_{1}}^{\infty}\\!e^{\\!-\left(\\!\dfrac{x}{t}+\dfrac{t}{c_{2}}\right)}\\!\text{d}t.$
Further, define the random variables $G_{1}=|h_{q}|^{2}$, and
$G_{0}=|h_{d,q}|^{2}$ (the channel gain in the _presence_ and _absence_ of the
IRS, respectively.) Then,

${\sf{Pr}}(G_{1}>\rho)\geq{\sf{Pr}}(G_{0}>\rho)\hskip
8.5359pt\forall\rho\in\mathbb{R}^{+},$ (47)

i.e., the channel gain of the OOB-UE in the presence of an IRS _stochastically
dominates_ the channel gain in its absence.

###### Proof.

See Appendix C. ∎

The above theorem characterizes the instantaneous behavior of the OOB UEs’
channels in the LoS scenarios and shows that the IRS never degrades the
performance of OOB UEs.

###  IV-B IRS Optimized for (L+)NLoS Scenarios

In the previous section, we studied the OOB performance when the IRS is only
optimized to the dominant path in the cascaded channels of the scheduled in-
band UEs and concluded that the OOB performance does not degrade due to the
IRS. However, when the IRS is programmed to align jointly along all the paths
of the in-band UE’s channel, it remains unclear whether an IRS can still
benefit an OOB operator due to the unit-modulus preserving property of IRS
coefficients. In fact, the directional response of the IRS is not easy to
characterize since the IRS does not align completely along a single-channel
direction. To that end, we first determine the directional response of the IRS
when it is optimized to a channel comprising multiple paths. We can write the
channel to the in-band UE-$k$ with $L$ resolvable spatial paths as in the
following equation.141414For the sake of exposition, we consider that all
cascaded paths through the IRS have the same average energy, similar to [42].
However, our results can be directly extended to scenarios where the paths
have unequal energies.

$h_{k}=h_{d,k}+\frac{N}{\sqrt{L}}\sum\nolimits_{l=1}^{L}\gamma^{(1)}_{l,X}\gamma^{(2)}_{l,k}\mathbf{\dot{a}}_{N}^{H}(\omega^{l}_{X,k})\boldsymbol{\theta}.$
(48)

Then, we can show that $n$th element of the optimal IRS configuration
$\boldsymbol{\theta}^{\mathrm{opt}}$ is given by (using CS inequality)

$\theta^{\mathrm{opt}}_{n}=e^{j\angle
h_{d,k}}\dfrac{\sum_{l=1}^{L}\gamma^{(1)*}_{l,X}\gamma^{(2)*}_{l,k}e^{-j(n-1)\pi\omega^{l}_{X,k}}}{\left|\sum_{l=1}^{L}\gamma^{(1)*}_{l,X}\gamma^{(2)*}_{l,k}e^{-j(n-1)\pi\omega^{l}_{X,k}}\right|}.$
(49)

Hence, the optimal IRS vector $\boldsymbol{\theta}^{\mathrm{opt}}$ is

$\dfrac{h_{d,k}}{|h_{d,k}|}\left(\sum_{l=1}^{L}\gamma^{(1)*}_{l,X}\gamma^{(2)*}_{l,k}\mathbf{\dot{a}}_{N}(\omega^{l}_{X,k})\right)\odot\dfrac{1}{\left|\sum\limits_{l=1}^{L}\gamma^{(1)}_{l,X}\gamma^{(2)}_{l,k}\mathbf{\dot{a}}_{N}(\omega^{l}_{X,k})\right|},$
(50)

where $|\mathbf{x}|$ is the vector containing the magnitudes of the entries of
$\mathbf{x}$ and $1/|\mathbf{x}|$ is an entry-wise inverse.
_Directional Response of the IRS:_ To determine the directional response of an
IRS optimized for (L+)NLoS scenarios, similar to Sec. IV-A, we evaluate the
correlation function: $\rho_{\nu,\theta}\triangleq
N\mathbb{E}\left[\left|\mathbf{\dot{a}}^{H}(\nu)\boldsymbol{\theta}^{\mathrm{opt}}\right|\right]$
at various angles of $\nu$. We then focus on the values of $\nu$ that match
the channel angles of UE-$k$ and ascertain the distribution of the IRS-
reflected energy along these directions. We have the following lemma.

######  Lemma 1.

The optimal IRS configuration as in (50) has the following spatial amplitude
response $\rho_{\nu,\theta}$ as defined above:

$\rho_{\nu,\theta}=\begin{cases}\Omega\left(\dfrac{N}{\sqrt{L}}\right)+o(N),\hfill\mathrm{if}\
\nu\in\left\\{\omega^{1}_{X,k},\\!\\!\ldots,\omega^{L}_{X,k}\right\\},\\\
o(N),\hfill\mathrm{if}\
\nu\in\mathbf{\Phi}\setminus\left\\{\omega^{1}_{X,k},\ldots,\omega^{L}_{X,k}\right\\}.\end{cases}$
(51)

###### Proof.

See Appendix D. ∎

The above Lemma shows that when the IRS is optimized to align jointly along
$L$ directions, then it has a spatial energy response of $1/L$ in each of
these $L$ directions and negligible response in other directions. We perform a
similar experiment as in the LoS scenario to numerically verify Lemma 1. In
Fig. 4, we plot the normalized correlation response $\rho_{\nu,\theta}/N$ as
function of $\nu\in\boldsymbol{\Phi}$, for $L=2$ and $L=3$. We consider that
the IRS is optimized to an in-band UE whose equal-gain paths have channel
angles drawn from $\mathcal{L}_{2}=\\{-0.23,0.54\\}$, and
$\mathcal{L}_{3}=\\{-0.23,0.06,0.54\\}$, for $L=2$, and $L=3$, respectively.
We see that whenever $\nu\in\mathcal{L}_{2}$ for $L=2$ ($\mathcal{L}_{3}$ for
$L=3$), the response is maximum, and is nearly $1/\sqrt{L}$, and is $0$,
otherwise.

(a) $L=2.$

(b) $L=3.$

Figure 4: Normalized Correlation response of the IRS (in (L+)NLoS scenarios)
and array vectors at different $\nu$ for $N=500$ with (a) $L=2$, and (b)
$L=3$. When $\nu\in\mathcal{L}_{2}\text{ (or) }\mathcal{L}_{3}$, the response,
$\rho_{\nu,\theta}/N$ peaks and $\approx 1/\sqrt{L}$.

We now provide the ergodic sum-SEs of the system when the IRS is optimized for
(L+)NLoS scenarios.

######  Theorem 5.

Under the Saleh-Valenzuela (L+)NLoS model in the mmWave channels, with RR
scheduling, and when the IRS is optimized to serve the UEs of operator X, the
ergodic sum-SEs of operators X and Y scale as

$\\!\\!\bar{S}_{L}^{(X)}\\!\approx\\!\frac{1}{K}\\!\sum_{k=1}^{K}\log_{2}\\!\left(\\!1\\!+\\!\left(N^{2}\beta_{r,k}\\!+\\!N\\!\sqrt{\pi\beta_{d,k}\beta_{r,k}}\\!+\beta_{d,k}\right)\\!\dfrac{P}{\sigma^{2}}\right)\\!,\\!\\!$
(52)

and as (53) on the top of next page, respectively, where
$i_{0}\triangleq\max\\{0,2L-N\\}$.

$\displaystyle\\!\\!\\!\\!\bar{S}_{L}^{(Y)}\approx\begin{cases}\dfrac{1}{Q}\sum\limits_{q=1}^{Q}\sum\limits_{i=i_{0}}^{L}\dfrac{\scalebox{0.9}{$\dbinom{L}{i}$}\scalebox{0.9}{$\dbinom{N-L}{L-i}$}}{\scalebox{0.9}{$\dbinom{N}{L}$}}\times\log_{2}\left(1+\left(\beta_{d,q}+i\dfrac{N^{2}}{L^{2}}\beta_{r,q}\right)\dfrac{P}{\sigma^{2}}\right),&\mathrm{if}\
\ L<N,\\\
\dfrac{1}{Q}\sum\limits_{q=1}^{Q}\log_{2}\left(1+\left(\beta_{d,q}+N\beta_{r,q}\right)\dfrac{P}{\sigma^{2}}\right),&\mathrm{if}\
\ L\geq N.\end{cases}$ (53)

###### Proof.

We derive the SEs of both operators separately below.

####  IV-B1 Ergodic sum-SE of operator X

As before, we first compute the ergodic SE of UE-$k$, $\langle
S^{(X)}_{k,L}\rangle$, and then evaluate the ergodic sum-SE of operator X.
Using (48) and (50), the ergodic SE of UE-$k$ is bounded by $\langle
S^{(X)}_{k,L}\rangle\leq\\!\log_{2}\\!\left(\\!1\\!+\\!\mathbb{E}\\!\left[\left||h_{d,k}|+\frac{1}{\sqrt{L}}\\!\sum\limits_{n=1}^{N}\\!\left|\sum\limits_{l=1}^{L}\\!\\!\gamma^{(1)}_{l,X}\gamma^{(2)}_{l,K}e^{j\pi(n-1)\omega^{l}_{X,k}}\right|\right|^{2}\right]\\!\dfrac{P}{\sigma^{2}}\right)\\!.\\!\\!$
Using the fact that the distribution of $\gamma^{(1)}_{l,X}\gamma^{(2)}_{l,K}$
is circularly symmetric, we can rewrite the inner term as
$\mathbb{E}\left[\left||h_{d,k}|+\frac{N}{\sqrt{L}}\left|\sum\limits_{l=1}^{L}\\!\gamma^{(1)}_{l,X}\gamma^{(2)}_{l,K}\right|\right|^{2}\right]=\mathbb{E}\left[|h_{d,k}|^{2}+\dfrac{N^{2}}{L}\left|\sum\limits_{l=1}^{L}\gamma^{(1)}_{l,X}\gamma^{(2)}_{l,K}\right|^{2}+\dfrac{2N}{\sqrt{L}}|h_{d,k}|\left|\sum\limits_{l=1}^{L}\gamma^{(1)}_{l,X}\gamma^{(2)}_{l,K}\right|\\!\right].$
_Term I_ : It is straightforward to see that
$\mathbb{E}[|h_{d,k}|^{2}]=\beta_{d,k}$.
_Term II_ : We can write
$\\!\\!\mathbb{E}\\!\left[\left|\sum\nolimits_{l=1}^{L}\\!\gamma^{(1)}_{l,X}\gamma^{(2)}_{l,K}\right|^{2}\right]=\mathbb{E}\left[\mathop{\sum\limits_{l=1}^{L}}\left|\gamma^{(1)}_{l,X}\gamma^{(2)}_{l,K}\right|^{2}\\!\right]+\mathbb{E}\left[\mathop{\sum\limits_{l,\ell=1;{l\neq\ell}}^{L}}\gamma^{(1)}_{l,X}\gamma^{(2)}_{l,K}\gamma^{(1)*}_{\ell,X}\gamma^{(2)*}_{\ell,K}\right]\stackrel{{\scriptstyle(a)}}{{=}}\\!L\beta_{r,k},\\!$
where $(a)$ is due to the zero-mean and independence of
$\left\\{\gamma^{(1)}_{l,X},\gamma^{(2)}_{l,K}\right\\}_{l}$ across the
spatial paths.
_Term III_ : We have the following sequence of relations
$\mathbb{E}\left\\{|h_{d,k}|\left|\sum\limits_{l=1}^{L}\gamma^{(1)}_{l,X}\gamma^{(2)}_{l,K}\right|\right\\}\stackrel{{\scriptstyle(a)}}{{\leq}}\mathbb{E}|h_{d,k}|\sqrt{\mathbb{E}\left[\left|\sum\limits_{l=1}^{L}\gamma^{(1)}_{l,X}\gamma^{(2)}_{l,K}\right|^{2}\right]}=\dfrac{1}{2}\sqrt{\pi\beta_{d,k}}\cdot\sqrt{L\beta_{r,k}},$
where $(a)$ follows by using the independence of $h_{d,k}$ and
$\sum_{l=1}^{L}\gamma^{(1)}_{l,X}\gamma^{(2)}_{l,K}$, followed by the Jensen’s
inequality. Collecting all the terms yields (52).

####  IV-B2 Ergodic sum-SE of operator Y

Consider the case $L<N$. As above, the channel seen at the UE-$q$ is

$h_{q}=h_{d,q}+\frac{N}{\sqrt{L}}\sum\nolimits_{l=1}^{L}\gamma^{(1)}_{l,Y}\gamma^{(2)}_{l,q}\mathbf{\dot{a}}_{N}^{H}(\omega^{l}_{Y,q})\boldsymbol{\theta},$
(54)

where $\boldsymbol{\theta}$ is optimized by operator-X and is given by (50).
The IRS vector aligns along $L$ directions of $\boldsymbol{\Phi}$, while the
OOB UE’s channel is oriented along $L$ directions independent of the IRS. We
let the random variable $X$ denote the instantaneous SE at UE-$q$, and $Y$
denotes the number of matching (spatial) paths between the UE-$q$’s channel
and the IRS response. Clearly, the possible support of $Y$ is the set of
integers from $0$ to $L$. We now compute $\langle
S^{(Y)}_{q,L}\rangle=\mathbb{E}[X]$ as

$\\!\mathbb{E}[X]=\mathbb{E}_{Y}[\mathbb{E}_{X}[X|Y]]=\sum\nolimits_{i}\mathbb{E}_{X}[X|Y=i]{\sf{Pr}}(Y=i).\\!$
(55)

When $L<N$, we can show that the probability mass function (pmf) of $Y$ is
given by

${\sf{Pr}}(Y=i)=\dfrac{\binom{L}{i}\binom{N-L}{L-i}}{\binom{N}{L}},\hskip
5.69046pti=i_{0},i_{0}+1,\ldots,L,$ (56)

and $0$ otherwise, where $\binom{x}{y}$ is the usual binomial coefficient, and
$i_{0}=\max\\{0,2L-N\\}$. This is because, if $N-L\leq L$, not more than $N-L$
spatial paths of the OOB channel can be misaligned with the IRS, and hence,
the support of the above pmf begins at $2L-N$. Next, we compute the term
$\mathbb{E}_{X}[X|Y=i]$ which is the average SE seen by UE-$q$ when exactly
$i$ of the spatial paths are common between the IRS response and scheduled
UE-$q$’s channel angles. Then, $\mathbb{E}_{X}[X|Y=i]$

$\displaystyle\stackrel{{\scriptstyle(a)}}{{\leq}}\log_{2}\left(\\!1\\!+\mathbb{E}\left[\left|h_{d,q}\\!+\frac{N}{\sqrt{L}}\sum\nolimits_{{l^{\prime}}\in\mathcal{I}_{i}}\\!\gamma^{{l^{\prime}}}_{Y,q}\mathbf{\dot{a}}_{N}^{H}(\omega^{l^{\prime}}_{Y,q})\boldsymbol{\theta}\right|^{2}\right]\\!\frac{P}{\sigma^{2}}\right)\\!\\!$
$\displaystyle\stackrel{{\scriptstyle(b)}}{{=}}\log_{2}\left(1+\\!\mathbb{E}\left[\left|h_{d,q}+\frac{N}{L}\sum\nolimits_{{l^{\prime}}\in\mathcal{I}_{i}}\gamma^{{l^{\prime}}}_{Y,q}\right|^{2}\right]\frac{P}{\sigma^{2}}\\!\right),$
(57)

where $(a)$ is due to Jensen’s inequality,
$\gamma^{{l^{\prime}}}_{Y,q}\triangleq\gamma^{(1)}_{{l^{\prime}},Y}\gamma^{(2)}_{{l^{\prime}},q}$,
and $\mathcal{I}_{i}$ denotes the index set of the common path indices between
the IRS and UE-$q$’s channel such that $|\mathcal{I}_{i}|=i$, and $(b)$ is due
to Lemma 1. By expanding the square term and using the statistics of the
random variables, we can show that the above expectation becomes
$\beta_{d,q}+i\frac{N^{2}}{L^{2}}\beta_{r,q}$. Plugging this into (55) and
evaluating the ergodic sum-SE yields (53) when $L<N$. Now, for $L\geq N$ (and
hence $\bar{L}\triangleq\min\\{L,N\\}=N$), since the IRS can orient to at most
$N$ beams, with probability $1$, every channel path of the UE is aligned to
one of directions to which the IRS is steered. Thus, we have $\langle
S^{(Y)}_{q,L}\rangle$

$\displaystyle\leq\log_{2}\left(1+\mathbb{E}\left[\left|h_{d,q}+{\sqrt{N}}\sum\nolimits_{{l}=1}^{N}\gamma^{{l}}_{Y,q}\mathbf{\dot{a}}_{N}^{H}(\omega^{l}_{Y,q})\boldsymbol{\theta}\right|^{2}\right]\dfrac{P}{\sigma^{2}}\right)$
$\displaystyle=\log_{2}\left(1+\mathbb{E}\left[\left|h_{d,q}+\sum\nolimits_{{l}=1}^{N}\gamma^{{l}}_{Y,q}\right|^{2}\right]\vspace{-0.2cm}\dfrac{P}{\sigma^{2}}\right)$
(58)
$\displaystyle=\log_{2}\left(1+\left(\beta_{d,q}+N\beta_{r,q}\right)\frac{P}{\sigma^{2}}\right).$
(59)

The proof of (53) for $L\geq N$ now easily follows. ∎ Theorem 5 shows that the
IRS provides an $\mathcal{O}(\log_{2}(N))$ scaling of the SE at OOB UEs when
$L\geq N$. In fact, in Sec. VI, we numerically show that this bound can be
improved to $L\geq\sqrt{N}$ while preserving the $\mathcal{O}(\log_{2}(N))$
growth of SE. In that case, for $L<\sqrt{N}$, the OOB-SE is shown to be log-
sub-linear in $N$, which is still better than the SE in the absence of
IRS.151515We reiterate that we need $L\geq\sqrt{N}$ only for
$\mathcal{O}(\log_{2}(N))$ scaling of SE. More generally, the OOB-SE in mmWave
bands scales as $\mathcal{O}(\log_{2}(N^{\delta})),\delta\\!>\\!0$. Thus, an
IRS strictly benefits the OOB system under all conditions. Subsequently, we
show that the OOB sum-SE in (L+)NLoS scenarios is at least as good as in LoS
scenarios and present further insights on the OOB performance in the presence
of an uncontrolled IRS.

##  V Enhancement of OOB Performance Using Opportunistic Scheduling

In the previous sections, we saw that an IRS that is optimally configured to
serve the UEs of operator X also benefits operator Y, but to a lesser extent
than the benefit to the UEs of operator X. In this section, we show that the
benefit of the uncontrolled IRS to the UEs of operatory Y can be further
enhanced using opportunistic user selection. Specifically, since the IRS is
_randomly_ configured from an OOB UE’s view, if there are sufficiently many
OOB UEs, at least one of the UEs will experience an SNR that is close to the
SNR when the IRS is optimized for that UE (i.e., when the IRS is in
_beamforming configuration_ for that UE.) Then, opportunistically scheduling
the UE for which the IRS is in beamforming configuration in every slot
extracts multi-user diversity in the system and enhances the OOB performance
better than RR-based UE scheduling [43]. We now analyze the OOB performance
for two such opportunistic schedulers: the proportional-fair (PF) and max-rate
(MR) schedulers.

###  V-A Multi-user Diversity for Operator Y using PF Scheduler

The PF scheduler serves UE-$q^{*}$ at time $t$, where [43]

$\\!\\!\\!\\!q^{*}(t)=\operatorname*{arg\,max}_{q\in\\{1,2,\ldots,Q\\}}\frac{\log_{2}\left(1+|h_{q}(t)|^{2}P/\sigma^{2}\right)}{T_{q}(t)}\triangleq\frac{R_{q}(t)}{T_{q}(t)},$
(60)

where $T_{q}(t)$ is the exponential moving average SE seen by UE-$q$ till time
$t$ which is updated as

$T_{q}(t+1)=\begin{cases}\left(1-\frac{1}{\tau}\right)T_{q}(t)+\frac{1}{\tau}R_{q}(t),&\mathrm{if}\
\ q=q^{*}(t),\\\ \left(1-\frac{1}{\tau}\right)T_{q}(t),&\mathrm{if}\ \ q\neq
q^{*}(t).\end{cases}$ (61)

Here, the parameter $\tau$ controls the trade-off between fairness and
throughput [12, Sec.II]. The following Lemma shows the achievable sum-SE of
operator Y under PF scheduling.

######  Lemma 2.

([12, 11]) Under independent Rayleigh fading channels in the sub-$6$ GHz
bands, when the IRS is optimized to serve the UEs of operator X, and operator
Y uses the PF scheduler, the sum-SE of operator Y, $R^{(Y)}_{\mathrm{PF}}$,
obeys

$R^{(Y)}_{\mathrm{PF}}-\frac{1}{Q}\sum_{q=1}^{Q}\log_{2}\Bigg{(}\\!\\!1+\left|\sum_{n=1}^{N}|f_{n}^{Y}g_{q,n}|+|h_{d,q}|\right|^{2}\\!\\!\\!\frac{P}{\sigma^{2}}\\!\\!\Bigg{)}\longrightarrow
0,$ (62)

as $Q,\tau\longrightarrow\infty$.

From (62) and (11), we see that UE-$q$ achieves the optimal SNR from the IRS
even though the IRS is not explicitly programmed to be in beamforming
configuration to this UE. Thus, operator $Y$ also experiences
$\mathcal{O}(N^{2})$ gain in the SNR, free of cost, provided the number of UEs
it serves is very large. However, one drawback of using the PF scheduler with
large $\tau$ is that the latency in UE scheduling also becomes large [43].
Thus, an OOB operator has to judiciously choose $\tau$ to balance the
achievable sum rate with the latency in UE scheduling.

###  V-B Multi-user Diversity for Operator Y using MR Scheduler

The MR scheduler serves UE-$q^{*}$ at time $t$, where

$q^{*}(t)=\operatorname*{arg\,max}_{q\in\\{1,2,\ldots,Q\\}}\log_{2}\left(1+\left|h_{q}(t)\right|^{2}P/\sigma^{2}\right).$
(63)

Unlike the PF scheduler, an MR scheduler maximizes the overall system
throughput, disregarding fairness across the UEs. As a result, the achievable
benefit from the IRS when using an MR scheduler has a different flavor than
the PF scheduler. Specifically, the random IRS configurations are used to
enhance the dynamic range of fluctuations in the SNR at OOB UEs. Then, the
ergodic sum-SE of the operator Y is

$\bar{R}^{(Y)}_{\mathrm{MR}}=\mathbb{E}\left[\log_{2}\left(1+\max_{q\in\\{1,\ldots,Q\\}}|h_{q}|^{2}P/\sigma^{2}\right)\right].$
(64)

A closed-form characterization of (64) is given below under a special case
where $\\{h_{q}\\}_{q}$ form a set of i.i.d. random variables.

######  Lemma 3.

(Theorem $1$,[44]) Under i.i.d. Rayleigh fading channels in the sub-$6$ GHz
bands, when the IRS is optimized to serve the UEs of operator X, and operator
Y uses the MR scheduler, the ergodic sum-SE of operator Y,
$\bar{R}^{(Y)}_{\mathrm{MR}}$, scales as

$\bar{R}^{(Y)}_{\mathrm{MR}}\xrightarrow{Q\rightarrow\infty}\log_{2}\left(1+\log_{e}(Q)\left(N\beta_{r}+\beta_{d}\right)\frac{P}{\sigma^{2}}\right),$
(65)

where $\beta_{r}$, $\beta_{d}$ are the common direct & cascaded path losses.

Thus, the MR scheduler leverages multi-user diversity to obtain a
$\log_{e}(Q)$-fold improvement in SNR and focuses less on achieving the
beamforming SE at the selected OOB UE.

We note that practical aspects, such as the number of UEs required to achieve
a target multi-user diversity, efficient feedback schemes to identify the best
UE for scheduling, etc., are not discussed here due to space constraints. We
refer the reader to [44, Sec. II] and [12, Sec. III-A] for more details.

The results in Lemmas 2, 3 pertain to the sub-$6$ GHz bands. Similar results
can also be derived for operator Y in mmWave bands using PF and MR schedulers,
e.g., using the opportunistic scheme presented in [12, Sec.III-C] and [44,
Sec. III].

##  VI Numerical Results and Discussion

In this section, we demonstrate all our results via Monte Carlo simulations.
Let BS-X and BS-Y be located at $(0,50)$, and $(50,0)$ (in meters), the IRS is
at $(1025,1025)$, and the UEs are randomly and uniformly located in a
rectangular region with diagonally opposite corners $(950,950)$ and
$(1100,1100)$. The path loss in each link is modeled as
$\beta=C_{0}\left(d_{0}/d\right)^{\alpha}$, where $C_{0}$ is the path loss at
the reference distance $d_{0}$, $d$ is the distance of the link, and $\alpha$
is the path loss exponent. We let $d_{0}=1$ meter, and $C_{0}\\!=\\!-30$ dB
and $\\!-60$ dB for sub-6 GHz and mmWave bands, respectively. We use
$\alpha=2,2$, and $4.5$ in the BS X/Y-IRS, IRS-UE, and BS X/Y-UE (direct)
links, respectively. Finally, we consider $\\!K\\!=\\!Q\\!=\\!10$ UEs served
over $5000$ time slots.

###  VI-A OOB Performance in sub-6 GHz Bands using RR Scheduler

In Fig. 5, we plot the empirical ergodic sum-SE vs. the transmit SNR
$\left(\gamma\triangleq P/\sigma^{2}\right)$ for both the operators as a
function of $N$, the number of IRS elements, and in the sub-6 GHz frequency
band.161616We set the range of $\gamma$ in $110-160$ dB for sub-6 GHz bands.
For e.g., when $N=8$ and $\gamma=160$ dB, the _received_ SNR is $\approx
5,-10$ dB with and without an IRS, respectively, at an OOB UE at
$(1000,1000)$. We also plot the sum-SE obtained from the analytical
expressions in Theorem 1. We see that the IRS enhances the sum-SEs of both
operators, although operator X benefits more from the IRS. We also see that
the improvement in SE with $N$ is log-quadratic for operator X, while it is
log-linear for operator Y, as expected. For example, the gap between the
$N=64$ and $N=256$ curves for operator X is about $4$ bits
($\approx\log_{2}(4^{2})=4$), the gap is about $2$ bits ($\approx\log_{2}(4)$)
for operator Y. Also, the analytical expressions tightly match the simulated
values, i.e., the approximation error introduced using Jensen’s inequality is
very small.
Next, in Fig. 6, we examine the effect of the number of IRS elements, $N$, by
plotting the ergodic sum-SE vs. $\log_{2}(N)$ for transmit SNRs of $130$ dB
and $150$ dB to validate the scaling of the received SNR as a function of $N$.
On the plot, we mark the slope of the different curves, and as expected from
Theorem 1, it is clear that while received SNR for a UE served by operator X
scales as $N^{2}$, it also scales as $N$ for a UE served by operator Y.

Figure 5: Ergodic sum-SE vs. transmit SNR. Figure 6: Ergodic sum-SE vs.
$\log_{2}(N)$.

(a) CCDF of $Z^{(Y)}_{N}$ as a function of $N$.

(b) Decay of outage prob. at in-band UEs vs. OOB UEs.

(c) CCDF of the SNR offset at in-band UEs.

Figure 7: Instantaneous channel quality of the in-band/OOB UEs in the sub-6
GHz bands of communication. Figure 8: Ergodic sum-SE vs. $N$ in LoS scenarios
at $C_{0}\gamma=90$ dB. Figure 9: Ergodic sum-SE vs. $N$ in LoS scenarios at
$C_{0}\gamma=150$ dB.

(a) $L=5$.

(b) $L=20$.

(c) $L=50$.

Figure 10: CCDF of OOB UE’s channel gain $|h_{q}|^{2}$ as a function of $N$ in
LoS scenarios of the mmWave bands at $C_{0}\gamma=150$ dB. Figure 11: Ergodic
sum-SE vs. $N$ in (L+)NLoS scenarios. Figure 12: Comparison of the sum-SE in
LoS and (L+)NLoS scenarios. Figure 13: OOB SE versus $Q$ for different
schedulers. Figure 14: OOB SE versus $\log_{2}(N)$ for different schedulers.

Next, we study the effect of the IRS on the OOB operator Y by considering the
behavior of the random variable $Z^{(Y)}_{N}$ (see (25)), which represents the
difference in the SNR/channel gain at a UE $q$ served by BS-Y (which does not
control the IRS) with and without the IRS in the environment. In Fig. 7(a), we
plot the empirical and theoretical CCDF of $Z^{(Y)}_{N}$ (at $\gamma=130$ dB.)
The analytical expression derived in Theorem 2 matches well with the
simulations. Also, almost surely, $Z^{(Y)}_{N}$ is a non-negative random
variable for any $N>0$, which confirms that the channel gain at an OOB UE
_with_ an IRS is at least as good as the channel gain at the same UE _without_
an IRS with probability $1$. The CCDF shifts to the right as the $N$ is
increased, as expected. The left-most curve in the figure is the CCDF of
received SNR in the absence of the IRS, which shows that the probability that
an operator benefits from the presence of a randomly configured IRS increase
with $N$ for operators who do not control the IRS. Thus, the instantaneous
SNRs witnessed at an arbitrary UE of an OOB operator stochastically dominate
the SNR seen by the same UE in the absence of the IRS, in line with
Proposition 1.
Finally, we compare the outage probabilities of the in-band and OOB UEs in
Fig. 7(b) as a function of $N$. While the probability decreases as
$\mathcal{O}(e^{-N})$ at the in-band UEs (in-line with Remark 2), it also
uniformly decreases at OOB UEs, but at the rate of $\mathcal{O}(1/N)$ as per
Theorem 2. Thus, the outage probability of the OOB UEs monotonically decreases
with the IRS elements _free of cost._ For completeness, in Fig. 7(c), we also
validate that the CCDF of the SNR offset as in Remark 2 for the in-band UEs
grows to $1$ at the rate of $1-\mathcal{O}(e^{-N})$.

###  VI-B OOB Performance in mmWave Bands using RR Scheduler

We now numerically illustrate our findings in the mmWave band. First, we focus
on the OOB performance when the IRS is optimized to in-band UEs in the LoS
scenarios. In Figs. 8 and 9, we plot the ergodic sum-SEs vs. $N$, of both
operators, at low and high (received) SNRs, respectively. At low SNR, we
observe that $1$) The ergodic sum-SE scales linearly with $N$, and $2$) The
OOB performance is insensitive to $L$. This can be explained by analyzing the
behavior of the SE in (35) at low SNRs. Namely $\log(c\gamma)\approx c\gamma$
for small $c\gamma$. Thus, an OOB UE witnesses an SE which scales as $N^{2}/L$
for an $L/N$ fraction of time, leading to an effective scaling of SE as
$\mathcal{O}(N)$. Thus, if the OOB system is designed to operate at low SNRs,
then even an OOB UE obtains benefits from the IRS that monotonically increase
with $N$. On the other hand, at high SNRs, for any given $L$, the SE is an
unimodal function of $N$, as described in Remark 3. As long as $L\geq N$,
i.e., the number of paths is sufficiently large (at least as many as the
number of resolvable beams the IRS can form), almost surely, the IRS aligns
with one of the OOB UE’s channel angles and provides benefits which linearly
increases with $N$. However, when $L\\!<\\!N$, the IRS beam does not always
align with the OOB UE; hence, the performance starts to decline as $N$ grows
with the peak SE obtained when $L=N$. In any case, the OOB SE is higher than
the SE obtained without the IRS. This confirms that the IRS _never_ degrades
the average OOB performance even in mmWave bands. Also, the simulations and
analytical expressions match very well, which validates the correctness of
Theorem 3. Finally, we note that the SNR of the in-band operator monotonically
grows as $N^{2}$, as expected.

Next, we validate the instantaneous OOB performance by plotting the CCDFs of
the channel gains of an arbitrary OOB UE in Figs. 10(a), 10(b), and 10(c), for
$L=5$, $20$, and $50$, respectively. Clearly, the effective channel in the
presence of the IRS stochastically dominates the effective channel in its
absence, as found in Theorem 4. Further, with high probability, the channel
gains of the OOB UEs achieve their maximum value when $N=L$, which is in line
with our previous observations.

Next, we focus on the case when the IRS is optimized for (L+)NLoS mmWave
scenarios. Specifically, we study the operators’ ergodic sum-SE behavior in a
high SNR regime. In Fig. 11, we plot the ergodic sum-SE vs. $N$ when the IRS
is optimized for (L+)NLoS scenarios. The OOB performance in this case is very
similar to the LoS scenarios, where the enhancement in the OOB SE due to the
IRS is contingent on the number of spatial paths available in the OOB UEs’
channel. However, the main difference lies in the number of paths needed in
the OOB UE’s channel to obtain performance enhancement by the IRS. While the
LoS scenarios require $L$ to scale with $N$, it is sufficient for $L$ to scale
with $\sqrt{N}$ in the (L+)NLoS scenarios to obtain an OOB performance boost
which is monotonic in $N$. Intuitively, this is because, in addition to the
probability that one of the $L$ paths of the OOB channel aligns with the IRS
(which scales as $L/N$), the IRS provides an additional opportunity for
alignment by having a directional response in $L$ directions. This leads to
the probability of alignment improving to $L^{2}/N$ in the OOB system. We
confirm this in Fig. 11: the peak performance is attained roughly at $N=L^{2}$
for every $L$. Recall that the probability that $i$ of the IRS beams align
with the OOB channel is given by (56). Consider the scenario $L\ll N$. Using
Stirling’s approximation $\scalebox{1.0}{$\dbinom{n}{i}$}\approx n^{i}/i!$ we
can simplify (56) as
${\sf{Pr}}(Y=i)\approx\frac{1}{i!}\left(\frac{L^{2}}{N}\right)^{i}$. Then,
letting $i=1$ (to compare against the LoS scenario), the probability is
$L^{2}/N$, supporting the fact that $\sqrt{N}$ paths are sufficient to obtain
an OOB SE that log-linearly grows with the IRS elements. Finally, Fig. 12
compares the OOB performance in LoS and (L+)NLoS scenarios. For a fixed $N$,
$L$, the performance is enhanced much faster in (L+)NLoS than in LoS
scenarios, in line with our discussions.

###  VI-C OOB performance with PF and MR schedulers

We now numerically illustrate the OOB performance enhancement that can be
obtained using the PF and MR schedulers described in Sec. V. We plot the OOB
SE versus the number of OOB UEs $Q$ in Fig. 13. Here, we compare the SE for
RR, PF, and MR schedulers for different $N$. Using an MR scheduler, the OOB SE
increases monotonically as $\mathcal{O}\left(\log(\log(Q))\right)$, in line
with Lemma 3. Next, we consider the OOB SE using a PF scheduler. For $N=4$,
the SE increases with $Q$, and it converges to the SE that is achievable as
though the IRS is also optimal to the OOB UEs under RR scheduling, in line
with Lemma 2. Thus, for large $Q$, OOB UEs enjoy optimal beamforming IRS
benefits without explicitly optimizing the IRS to them when the OOB operator
uses a PF scheduler. Even with $N=16$ using a PF scheduler, a similar trend as
in $N=4$ is observed, except that the convergence rate to the optimal SE is
now reduced. We observe this from the unequal gaps between the optimal SE and
SE obtained by PF schedulers for $N=4$ and $16$, namely:
$\Delta_{16}>\Delta_{4}$. This is because the degree of randomness increases
with $N$, which requires the number of UEs to scale with $N$. We refer the
reader to [44, Prop. $1$] for more details on the required number of UEs for a
target gap in the SE. Finally, the OOB SE using the RR scheduler is invariant
to $Q$ because it does not leverage multi-user diversity in the system.
In the final study of this section, we plot the OOB SE as a function of
$\log_{2}(N)$ in Fig. 14 for different schedulers. First, we observe that the
slope of the curve using MR scheduler for both $Q=10,100$ is $1$, which
validates the $\mathcal{O}(\log(N))$ dependence of the SE as in Lemma 3.
Similarly, the RR scheduler also scales as $\mathcal{O}(\log_{2}(N))$, in line
with Theorem 1. On the other hand, the performance of the PF scheduler as a
function of $N$ crucially depends on the relative value of $Q$ with respect to
$N$. For smaller $N$, the OOB SNR scales with $N^{2}$, and as $N$ increases,
the OOB SNR scales as $N^{\epsilon}$, where $\epsilon\leq 2$ (notice this in
the decreasing values of slopes.) This happens because as $N$ increases, $Q$
is not large enough for the random IRS configuration to be near-optimal to at
least one OOB UE at every point in time.

##  VII Conclusions and Future Work

In this paper, we studied a fundamental issue in an IRS-aided system, namely,
the effect of deploying an IRS on the average and instantaneous performance of
an OOB operator who has no control over the IRS. Surprisingly, we found that
while the IRS optimally serves the in-band UEs, it simultaneously, and at no
additional cost, enhances the quality of the channels of the OOB UEs compared
to the system without any IRS. This performance enhancement is due to the
reception of multiple copies of the signal at the OOB UEs through the IRS.
Although this is always true in the sub-6 GHz bands, the degree of enhancement
in the mmWave communications is determined by the number of spatial paths in
the OOB UE’s channel. When the number of paths in the channel is at least the
number of resolvable beams at the IRS, the gain is monotonic in the number of
IRS elements similar to sub-6 GHz bands and is marginal otherwise. Further, we
showed that the systems where IRS is optimized to all the spatial paths of the
in-band UE’s channel outperform the ones where IRS is aligned only to the
dominant path. This is due to the additional degrees of freedom offered by the
IRS in the former, although it incurs additional BS-IRS signaling overhead at
the in-band operator to configure the IRS optimally. Therefore, the deployment
of an IRS benefits all the co-existing network operators, albeit to a lesser
extent than the operator that has control over the IRS phase configuration.
Further, it is possible to obtain much better benefits at the OOB UEs by using
an opportunistic scheduler at the OOB operator.
Future work can include the effect of multiple antennas at the BSs/UEs and
study the diversity-multiplexing gain tradeoff at OOB UEs. Other practical
considerations, like the non-availability of perfect channel state
information, quantized IRS phase shift levels, frequency selectivity of
channels, time & frequency offsets, near-field effects, etc., can be accounted
for to provide more insights. Another interesting line of study is to analyze
the impact of IRSs on the OOB performance in interference-limited scenarios
such as multi-cell environments.

##  Appendix A Proof of Theorem 2

We recognize that
$\tilde{h}_{1,q}\triangleq|{h}_{1,q}|^{2}\sim\exp(1/\left(N\beta_{r,q}+\beta_{d,q}\right))$,
and $\tilde{h}_{2,q}\triangleq|{h}_{2,q}|^{2}\sim\exp(1/\beta_{d,q})$. Now,
$Z^{(Y)}_{N}$ is the difference between two exponential random variables. We
first show that the correlation coefficient between $\tilde{h}_{1,q}$ and
$\tilde{h}_{2,q}$ decays inversely with $N$. Recall that the correlation
coefficient is defined
$\rho_{12}\triangleq\mathbb{E}\left[(\tilde{h}_{1,q}-\mathbb{E}[\tilde{h}_{1,q}])(\tilde{h}_{2,q}-\mathbb{E}[\tilde{h}_{2,q}])\right]\bigg{/}\sigma_{1}\sigma_{2}$,
where $\sigma_{1}^{2}$ and $\sigma_{2}^{2}$ are the variances of
$\tilde{h}_{1,q}$ and $\tilde{h}_{2,q}$, respectively. We can verify that $1)$
$\mu_{1}\triangleq\mathbb{E}[\tilde{h}_{1,q}]=N\beta_{r,q}+\beta_{d,q}$, and
$\mu_{2}\triangleq\mathbb{E}[\tilde{h}_{2,q}]=\beta_{d,q}$ $2)$
$\sigma_{1}^{2}={\left(N\beta_{r,q}+\beta_{d,q}\right)}^{2}$, and
$\sigma_{2}^{2}=\beta_{d,q}^{2}$. Thus, we get
$\rho_{12}=\left(\mathbb{E}\left[\tilde{h}_{1,q}\tilde{h}_{2,q}\right]-\left(N\beta_{r,q}+\beta_{d,q}\right)\beta_{d,q}\right)\bigg{/}\left(N\beta_{r,q}+\beta_{d,q}\right)\beta_{d,q}.$
Using (24), it is easy to verify that
$\mathbb{E}\left[\tilde{h}_{1,q}\tilde{h}_{2,q}\right]=N\beta_{r,q}\beta_{d,q}+2\beta_{d,q}^{2}$.
After simplification, we have

$\rho_{12}=1/\Big{(}1+N\left(\beta_{r,q}/\beta_{d,q}\right)\Big{)},$ (66)

which decays inversely with $N$. We now use a result from [45, Eq. 4.24] which
characterizes the distribution of the difference of two dependent and non-
identically distributed chi-square random variables and obtain the CDF of
$Z^{(Y)}_{N}$ as

$F_{Z^{(Y)}_{N}}(z)=\left\\{\begin{array}[]{lr}\dfrac{8}{\mu_{1}\mu_{2}(1-\rho_{12}^{2})\gamma\alpha^{-}}e^{\left(\frac{\alpha^{-}z}{4}\right)},&\text{if
}z<0,\\\
1-\dfrac{8}{\mu_{1}\mu_{2}(1-\rho_{12}^{2})\gamma\alpha^{+}}e^{-\left(\frac{\alpha^{+}z}{4}\right)},&\text{if
}z\geq 0,\end{array}\right.$ (67)

where

$\gamma\\!=\\!\dfrac{2\sqrt{{(\mu_{2}\\!-\\!\mu_{1})}^{2}\\!+4\mu_{1}\mu_{2}(1-\\!\rho_{12}^{2})}}{\mu_{1}\mu_{2}(1-\rho_{12}^{2})};\alpha^{\pm}\\!=\\!\gamma\pm\dfrac{2\left(\mu_{2}\\!-\\!\mu_{1}\right)}{\mu_{1}\mu_{2}(1-\\!\rho_{12}^{2})}.$
(68)

For large $N$, using (66) to let $\rho_{12}\rightarrow 0$ in (67), (68), and
recognizing that $\forall z$, $\bar{F}_{Z^{(Y)}_{N}}(z)=1-F_{Z^{(Y)}_{N}}(z)$,
we obtain

$\bar{F}_{Z^{(Y)}_{N}}(z)=\left\\{\begin{array}[]{lr}1-\dfrac{\mu_{2}}{\mu_{1}+\mu_{2}}e^{\frac{z}{\mu_{2}}},&\text{if
}z<0,\\\ \dfrac{\mu_{1}}{\mu_{1}+\mu_{2}}e^{-\frac{z}{\mu_{1}}},&\text{if
}z\geq 0,\end{array}\right.$ (69)

Substituting for $\mu_{1}$ and $\mu_{2}$ into (69) completes the proof. ∎

##  Appendix B Proof of the decay rate of outage probability and SNR offset at
the in-band UEs

We compute the decay rate of the outage probability, and SNR offset for the
in-band UEs (as in Theorem 2) below.

###  B-A Outage Probability

Let $X\\!\triangleq\\!\left||h_{d}|+\sum_{n}|f_{n}||g_{n}|\right|^{2}$ be the
channel gain at an arbitrary in-band UE (dropping index $k$). Then, the outage
probability is

$P^{\rho}_{\mathrm{out}}=\mathrm{Pr}(X\leq\rho).$ (70)

Since the true distribution of $X$ is complicated due to multiple products and
summations, we find an upper bound on $P^{\rho}_{\mathrm{out}}$ as

$X=\left||h_{d}|+\sum_{n}|f_{n}||g_{n}|\right|^{2}\\!\geq\\!\left|\sum_{n}|f_{n}||g_{n}|\right|^{2}\triangleq
Y,\ \ \text{almost surely}.$

As a result, we have

$P^{\rho}_{\mathrm{out}}=\mathrm{Pr}(X\leq\rho)\leq\mathrm{Pr}\left(\left|\sum\nolimits_{n}|f_{n}||g_{n}|\right|^{2}\leq\rho\right).$
(71)

Now by Central Limit Theorem (CLT) [46], we have

$\\!\\!\\!Z\triangleq\sum_{n}|f_{n}||g_{n}|\ {\longrightarrow}\
\mathcal{N}\left(N\dfrac{\pi}{4}\sqrt{\beta_{r}},N\left(1-\dfrac{\pi^{2}}{16}\right)\beta_{r}\right).$
(72)

Using (72), we can simplify (71) as

$\displaystyle P^{\rho}_{\mathrm{out}}$
$\displaystyle\stackrel{{\scriptstyle(a)}}{{\leq}}1-Q\left(\frac{\sqrt{\rho}}{c_{1}\sqrt{N}}-c_{2}\sqrt{N}\right)-Q\left(\frac{\sqrt{\rho}}{c_{1}\sqrt{N}}+c_{2}\sqrt{N}\right)$
$\displaystyle\stackrel{{\scriptstyle(b)}}{{\approx}}2Q\left(c_{2}\sqrt{N}\right),$
(73)

where, in $(a)$,
$c_{1}\triangleq\sqrt{\left(1-\frac{\pi^{2}}{16}\right)\beta_{r}}$ and
$c_{2}\triangleq\frac{\pi}{\sqrt{16-\pi^{2}}}$, and $(b)$ follows as for fixed
$\rho$, $\frac{\sqrt{\rho}}{c_{1}\sqrt{N}}\rightarrow 0$ for large $N$. Now,
by applying the Chernoff bound to $Q(\cdot)$ in (73), we see that
$Q(c_{2}\sqrt{N})\rightarrow 0$ at the rate of $\mathcal{O}(e^{-N})$. Hence,
the outage probability in (70) also decays at least as fast as
$\mathcal{O}(e^{-N})$.

###  B-B CCDF of the SNR offset

Let the channel gain with and without an IRS be
$h_{1}\triangleq\left||h_{d}|+\sum_{n}|f_{n}||g_{n}|\right|^{2}$ and
$h_{2}\triangleq|h_{d}|^{2}$. To compute the CCDF of $O\triangleq
h_{1}-h_{2}$, using (72), we notice

$\\!\\!\\!\bar{F}_{O}(\rho)\triangleq\mathrm{Pr}\left(h_{1}-h_{2}\geq\rho\right)\geq\mathrm{Pr}\Bigg{(}\underbrace{\left|Z\right|^{2}-|h_{d}|^{2}}_{\triangleq
O^{\prime}}\geq\rho\Bigg{)}.\vspace{-0.2cm}$ (74)

Now, we can simplify the lower bound of (74) as

$\displaystyle\\!\\!\\!\\!\mathrm{Pr}\left(|Z|^{2}-|h_{d}|^{2}\leq\rho\right)\\!=\\!\mathbb{E}_{Z}\left[\mathrm{Pr}\left(|z|^{2}-|h_{d}|^{2}\leq\rho\big{|}Z=z\right)\right]$
(75)
$\displaystyle\\!\\!\\!\\!=\mathbb{E}_{Z}\left[\mathrm{Pr}\left(|h_{d}|^{2}\geq|z|^{2}-\rho\big{|}Z=z\right)\right]\stackrel{{\scriptstyle(a)}}{{=}}\mathbb{E}_{Z}\Big{[}e^{-\left(\frac{|z|^{2}-\rho}{\beta_{d}}\right)}\Big{]}$
(76)
$\displaystyle\\!\\!\\!\\!=\sqrt{\frac{8}{N\pi(16-\pi^{2})\beta_{r}}}\\!\int_{-\infty}^{\infty}\\!\\!e^{-\left(\frac{|z|^{2}-\rho}{\beta_{d}}+\frac{\left(z-N\frac{\pi}{4}\sqrt{\beta_{r}}\right)^{2}}{2N\left(1-\frac{\pi^{2}}{16}\right)\beta_{r}}\right)}\text{d}z,\\!\\!\\!$
(77)

where in $(a)$, we used the property of the exponentially distributed random
variable $|h_{d}|^{2}$. To compute the above integral, we define
$\alpha\triangleq 2N\left(1-\frac{\pi^{2}}{16}\right)\beta_{r}$,
$\eta\triangleq N\frac{\pi}{4}\sqrt{\beta_{r}}$.

Then, by using standard integral solvers, we can show that

$\mathrm{Pr}\left(|Z|^{2}-|h_{d}|^{2}\geq\rho\right)=1-\dfrac{\sqrt{\beta_{d}}}{\sqrt{\beta_{d}+\alpha}}e^{-\left(\\!\\!\frac{\beta_{d}\eta^{2}-(\beta_{d}+\alpha)\rho}{\beta_{d}^{2}+\alpha\beta_{d}}\\!\\!\right)}.$
(78)

The above probability can be shown to be proportional to

$1-(1/\sqrt{N})\times e^{-N}\times e^{\left(\frac{1/N}{1/N}\right)}\sim
1-\mathcal{O}\left(e^{-N}\right).$ (79)

Thus, the lower bound grows to $1$ exponentially in $N$, and hence the CCDF of
the random variable $O$ also grows to $1$ at least as fast as the decay rate
of $e^{-N}$ to $0$. ∎

##  Appendix C Proof of Theorem 4

Assume $L<N$. From (41), we compute probability $P=$

$\\!\\!{\sf{Pr}}\left(\left|h_{d,q}+\dfrac{N^{2}}{\sqrt{L}}\sum_{l=1}^{L}\gamma^{(1)}_{l,Y}\gamma^{(2)}_{l,q}\mathbf{\dot{a}}_{N}^{H}(\omega^{l}_{Y,q})\mathbf{\dot{a}}_{N}(\omega^{1}_{X,k})\right|^{2}\\!\\!<\rho\right)\\!\\!.\\!\\!$
(80)

By the total law of probability, we obtain
$P={\sf{Pr}}(|h_{q}|^{2}<\rho\rvert\mathcal{E}_{1}){\sf{Pr}}(\mathcal{E}_{1})+{\sf{Pr}}(|h_{q}|^{2}<\rho\rvert\mathcal{E}_{0}){\sf{Pr}}(\mathcal{E}_{0}),$
where $\mathcal{E}_{0}$ and $\mathcal{E}_{1}$ are as defined in Sec. IV-A2.
Thus, we can simplify $P$ as

$\\!\\!\\!{\sf{Pr}}\\!\left(\left|h_{d,q}\\!+\\!\dfrac{N}{\sqrt{L}}\gamma^{(1)}_{l^{*},Y}\gamma^{(2)}_{l^{*},q}\right|^{2}\\!\\!<\\!\rho\\!\right)\\!{\sf{Pr}}(\mathcal{E}_{1})+{\sf{Pr}}\\!\left(\left|h_{d,q}\right|^{2}\\!<\\!\rho\\!\right)\\!{\sf{Pr}}(\mathcal{E}_{0}).\\!\\!$
(81)

Let
$\\!\left\\{\\!\tilde{\gamma}^{(1)}_{l^{*},Y},{\tilde{\gamma}}^{(2)}_{l^{*},q}\\!\right\\}\\!\triangleq\\!\sqrt{\\!\frac{N}{\sqrt{L}}}\\!\left\\{\\!\gamma^{(1)}_{l^{*},Y},\\!\gamma^{(2)}_{l^{*},q}\\!\right\\}$.
Then, by [47, Eq. 17],

$\\!\\!{\sf{Pr}}\\!\left(\\!\left|h_{d,q}\\!+\\!\tilde{\gamma}^{(1)}_{l^{*},Y}\tilde{\gamma}^{(2)}_{l^{*},q}\right|^{2}\\!\\!<\\!\rho\\!\right)\\!\\!=1-\dfrac{Le^{\frac{L\beta_{d,q}}{N^{2}\beta_{r,q}}}}{N^{2}\beta_{r,q}}\hskip
2.84544pt\mathcal{I}_{0}\left(\rho;\beta_{d,q},\dfrac{N^{2}}{L}\beta_{r,q}\right)\\!.\\!\\!\\!$
(82)

Also, ${\sf{Pr}}(\left|h_{d,q}\right|^{2}<\rho)=1-e^{-\rho/\beta_{d,q}}$.
Collecting these together completes the proof of (46) when $L<N$. Similarly,
we can prove for $L\geq N$. To obtain (47), it suffices to show that

$\forall L\lessgtr N,\ \ {\sf{Pr}}(G_{1}>\\!\rho)-{\sf{Pr}}(G_{0}>\\!\rho)\geq
0\stackrel{{\scriptstyle(a)}}{{\iff}}\\!\\!\\\
\frac{\bar{L}}{N}\\!\left(\\!\frac{\bar{L}e^{\frac{\bar{L}\beta_{d,q}}{N^{2}\beta_{r,q}}}}{N^{2}\beta_{r,q}}\mathcal{I}_{0}\left(\\!\rho;\beta_{d,q},\frac{N^{2}}{\bar{L}}\beta_{r,q}\\!\right)\\!-\\!e^{-\rho/\beta_{d,q}}\\!\right)\geq
0,$ (83)

where $(a)$ is due to (46). From [48], we know that $\mathcal{I}_{0}(\cdot)$
is related to the generalized upper incomplete Gamma function
$\Gamma(\alpha,x;b)$ for appropriate $\alpha,x,b$, and hence we can simplify

$\\!\\!\\!\mathcal{I}_{0}\\!\left(\\!\rho;\beta_{d,q},\frac{N^{2}}{\bar{L}}\beta_{r,q}\right)\\!=\frac{N^{2}\beta_{r,q}}{\bar{L}}\hskip
1.42271pt\Gamma\left(1,\frac{\bar{L}\beta_{d,q}}{N^{2}\beta_{r,q}};\frac{\bar{L}\rho}{N^{2}\beta_{r,q}}\\!\right)\\!.\\!\\!$
(84)

Then, it suffices to show that
$f(\rho)\triangleq\underbrace{e^{\frac{\bar{L}\beta_{d,q}}{N^{2}\beta_{r,q}}}\hskip
5.69046pt\Gamma\left(1,\frac{\bar{L}\beta_{d,q}}{N^{2}\beta_{r,q}};\frac{\bar{L}\rho}{N^{2}\beta_{r,q}}\right)}_{\triangleq
f_{1}(\rho)}-\underbrace{e^{-\rho/\beta_{d,q}}}_{\triangleq
f_{2}(\rho)}\\!\geq\\!0$. We first prove a few properties of $f(\rho)$. _1)
$f(0)=0$_: Clearly, $f_{2}(0)=1$. Now,
$f_{1}(0)=e^{\frac{\bar{L}\beta_{d,q}}{N^{2}\beta_{r,q}}}\hskip
5.69046pt\Gamma\left(1,\frac{\bar{L}\beta_{d,q}}{N^{2}\beta_{r,q}};0\right)$.
But, $\Gamma\left(1,x;0\right)=\Gamma(1,x)$ where $\Gamma(\alpha,x)$ is the
upper incomplete Gamma function. Hence,
$f_{1}(0)=e^{\frac{\bar{L}\beta_{d,q}}{N^{2}\beta_{r,q}}}\hskip
5.69046pt\\!\Gamma\left(1,\frac{\bar{L}\beta_{d,q}}{N^{2}\beta_{r,q}}\right)$.
But from [49, Eq. 8.4.5], $\Gamma\left(1,x\right)\\!=\\!e^{-x}$. Thus,
$f_{1}(0)\\!=\\!1$ and hence $f(0)\\!=\\!f_{1}(0)\\!-\\!f_{2}(0)=0$.

Figure 15: Graph of $f^{\prime}(\rho)$ vs. $\rho$ with $N=128$.

_2) $f(\rho)$ is non-decreasing_: From [50],
$\partial\Gamma(\alpha,x;b)/\partial b\\!=-\Gamma(\alpha\\!-1,x;b)$ and thus,
$f_{1}^{\prime}(\rho)=-e^{\frac{\bar{L}\beta_{d,q}}{N^{2}\beta_{r,q}}}\hskip
5.69046pt\Gamma\left(0,\frac{\bar{L}\beta_{d,q}}{N^{2}\beta_{r,q}};\frac{\bar{L}\rho}{N^{2}\beta_{r,q}}\right)$.
Further, for practical values of the system parameters,
$\bar{L}\beta_{d,q}/N^{2}\beta_{r,q}\\!\leq\\!1$, and thus we can approximate
$f_{1}^{\prime}(\rho)\approx-2e^{\frac{\bar{L}\beta_{d,q}}{N^{2}\beta_{r,q}}}K_{0}\left(\frac{2}{N}\sqrt{\frac{\bar{L}\rho}{\beta_{r,q}}}\right)$,
where $K_{0}(x)$ is the zeroth-order modified Bessel function of the second
kind. Further, $f_{2}^{\prime}(\rho)=-\left(1/d\right)e^{-\frac{\rho}{d}}$.
Hence, we can show that
$f^{\prime}(\rho)=f^{\prime}_{1}(\rho)-f^{\prime}_{2}(\rho)\geq 0$
$\forall\rho\in\mathbb{R}^{+}$ (see Fig. 15). Finally, since $f(\rho)$ is non-
decreasing, $f(\rho)\geq f(0)=0$. ∎

##  Appendix D Proof of Lemma 1

Using (50), we simplify
$\rho_{\nu,\theta}=N\mathbb{E}\left[\left|\mathbf{\dot{a}}^{H}(\nu)\boldsymbol{\theta}^{\mathrm{opt}}\right|\right]$
as

$\displaystyle\\!\\!\\!=\mathbb{E}\left[\left|\sum\nolimits_{n=1}^{N}e^{j\pi(n-1)\nu}\dfrac{\sum\nolimits_{l=1}^{L}\gamma^{(1)*}_{l,X}\gamma^{(2)*}_{l,k}e^{-j(n-1)\pi\omega^{l}_{X,k}}}{\left|\sum\nolimits_{l=1}^{L}\gamma^{(1)*}_{l,X}\gamma^{(2)*}_{l,k}e^{-j(n-1)\pi\omega^{l}_{X,k}}\right|}\right|\right]\\!\\!$
$\displaystyle\\!\\!\\!\\!\stackrel{{\scriptstyle(a)}}{{\geq}}\\!\dfrac{1}{\sqrt{L}}\mathbb{E}\left[\\!\dfrac{1}{\|\mathbf{h}_{c}\|_{2}}\\!\left|\sum_{n=1}^{N}\\!e^{j\pi(n-1)\nu}\sum\limits_{l=1}^{L}\\!\gamma^{(1)*}_{l,X}\\!\gamma^{(2)*}_{l,k}\\!e^{-j(n-1)\pi\omega^{l}_{X,k}}\\!\right|\\!\right]\\!\\!\\!$
$\displaystyle\\!\\!\\!\\!\stackrel{{\scriptstyle(b)}}{{=}}\dfrac{1}{\sqrt{L}}\hskip
2.84544pt\mathbb{E}\left[\dfrac{1}{\|\mathbf{h}_{c}\|_{2}}\left|\sum\limits_{l=1}^{L}\gamma^{(1)*}_{l,X}\gamma^{(2)*}_{l,k}\sum\limits_{n=1}^{N}e^{j\pi(n-1)(\nu-\omega^{l}_{X,k})}\right|\right]$
$\displaystyle\\!\\!\\!=\dfrac{1}{\sqrt{L}}\hskip
2.84544pt\mathbb{E}\left[\dfrac{1}{\|\mathbf{h}_{c}\|_{2}}\left|\sum\nolimits_{l=1}^{L}\gamma^{(1)*}_{l,X}\gamma^{(2)*}_{l,k}\hskip
2.84544pt\dfrac{e^{jN\frac{\pi}{2}(\nu-\omega^{l}_{X,k})}}{e^{j\frac{\pi}{2}(\nu-\omega^{l}_{X,k})}}\right.\right.$
$\displaystyle\hskip
93.89418pt\left.\left.\times\dfrac{\sin\left(0.5N\pi(\nu-\omega^{l}_{X,k})\right)}{\sin\left(0.5\pi(\nu-\omega^{l}_{X,k})\right)}\right|\right],$

where
$\mathbf{h}_{c}\triangleq[\gamma^{(1)}_{1,X}\gamma^{(2)}_{1,k},\ldots,\gamma^{(1)}_{L,X}\gamma^{(2)}_{L,k}]^{T}$.
Also, $(a)$ is because
$\left|\sum_{l=1}^{L}\gamma^{(1)*}_{l,X}\gamma^{(2)*}_{l,k}e^{-j(n-1)\pi\omega^{l}_{X,k}}\right|\leq\sqrt{L\sum_{l=1}^{L}|\gamma^{(1)}_{l,X}\gamma^{(2)}_{l,k}|^{2}}$
(the Cauchy-Schwarz inequality), and $(b)$ is obtained by changing the order
of summation. We recognize that
$\\!\sin\left(\\!0.5N\pi(\nu\\!-\\!\omega^{l}_{X,k})\\!\right)\\!/\sin\left(\\!0.5\pi(\nu\\!-\\!\omega^{l}_{X,k})\\!\right)\\!=\\!F_{N}(\nu\\!-\\!\omega^{l}_{X,k})$
where $F_{N}(\cdot)$ is the _Fejér Kernel_ given by

$F_{N}(x)=\begin{cases}N+o(N),&\mathrm{if}\ \ x=0,\\\ o(N),&\mathrm{if}\ \
x=\pm\frac{k}{N},k\in\mathbb{N}.\end{cases}$ (86)

As a consequence, we have

$\\!\\!\\!F_{N}(\nu-\omega^{l}_{X,k})=\begin{cases}\\!N+o(N),&\mathrm{if}\
\nu\in\left\\{\omega^{1}_{X,k},\ldots,\omega^{L}_{X,k}\right\\},\\\
\\!o(N),&\mathrm{if}\
\nu\in\mathbf{\Phi}\setminus\left\\{\omega^{1}_{X,k},\ldots,\omega^{L}_{X,k}\right\\}.\end{cases}$
(87)

Thus, when
$\nu\in\mathbf{\Phi}\setminus\left\\{\omega^{1}_{X,k},\omega^{2}_{X,k},\ldots,\omega^{L}_{X,k}\right\\},\rho_{\nu,\theta}\stackrel{{\scriptstyle
N\rightarrow\infty}}{{\longrightarrow}}0$; and when
$\nu\in\left\\{\omega^{1}_{X,k},\omega^{2}_{X,k},\ldots,\omega^{L}_{X,k}\right\\}$,
we have
$\rho_{\nu,\theta}\geq\dfrac{N}{\sqrt{L}}\times\mathbb{E}\left[\dfrac{|h_{l^{*}}|}{\|\mathbf{h}_{c}\|_{2}}\right]$,
implying $\rho_{\nu,\theta}=\Omega\left(N/\sqrt{L}\right)+o(N)$. ∎

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