# Finite Blocklength Regime Performance of Downlink Large Scale Networks

Nourhan Hesham, Student Member, IEEE, Anas Chaaban, Senior Member, IEEE,
Hesham ElSawy, Senior Member, IEEE, Jahangir Hossain, Senior Member, IEEE
N. Hesham, A. Chaaban, and J. Hossain are with the School of Engineering,
University of British Columbia, Kelowna, BC V1V 1V7, Canada (e-mail:
{nourhan.soliman,anas.chaaban,jahangir.hossain}@ubc.ca), and N. Hesham is on
leave from the Department of Electronics and Electrical Engineering, Cairo
University, Cairo, Egypt. H. ElSawy is with the School of Computing, Queen’s
University, Kingston, ON K7L 2N8, Canada. (e-mail: hesham.elsawy@queensu.ca).
This publication is based upon work supported by King Abdullah University of
Science and Technology (KAUST) under Award No. OSR-2018-CRG7-3734. Part of
this work was presented in the IEEE International Conference on Communications
(ICC 2021) [1]

###### Abstract

Some emerging 5G and beyond use-cases impose stringent latency constraints,
which necessitates a paradigm shift towards finite blocklength performance
analysis. In contrast to Shannon capacity-achieving codes, the codeword length
in the finite blocklength regime (FBR) is a critical design parameter that
imposes an intricate tradeoff between delay, reliability, and information
coding rate. In this context, this paper presents a novel mathematical
analysis to characterize the performance of large-scale downlink networks
using short codewords. Theoretical achievable rates, outage probability, and
reliability expressions are derived using the finite blocklength coding theory
in conjunction with stochastic geometry, and compared to the performance in
the asymptotic regime (AR). Achievable rates under practical modulation
schemes as well as multilevel polar coded modulation (MLPCM) are investigated.
Numerical results provide theoretical performance benchmarks, highlight the
potential of MLPCM in achieving close to optimal performance with short
codewords, and confirm the discrepancy between the performance in the FBR and
that predicted by analysis in the AR. Finally, the meta distribution of the
coding rate is derived, providing the percentiles of users that achieve a
predefined target rate in a network.

###### Index Terms:

Average Coding Rate, Finite Blocklength, Meta Distribution, Multilevel Polar-
Coded Modulation, Stochastic Geometry.

##  I Introduction

The proliferating applications and diverse use-cases of beyond 5G (B5G)
networks enforce stringent key performance indicators that cannot be realized
via conventional coding schemes [2]. For instance, mission critical
applications require sub-$1\ \text{msec}$ latency that cannot be achieved with
conventional long codes [3]. Moreover, many Internet of Things (IoT) devices
have an extreme low-power consumption profile that cannot support the
complexity and long transmission intervals of conventional long codes [4].
Consequently, there is a paradigm shift towards using short codes to comply
with the stringent latency and power consumption constraints of the foreseen
B5G networks.111In a network, there are four dominant delays: transmission
delay, propagation delay, processing delay, and queuing delay. As the network
becomes denser, the transmission delay can dominate the propagation delay
which motivates using short codes. Such reduced code length comes at the
expense of inevitable errors and lower information coding rates. The tolerance
to the induced errors differs with the application, which can be lower than
$10^{-9}$ for ultra-reliable low-latency communications (URLLC) [5]. This
motivates the study of the finite blocklength coding theory [6, 7], which
characterizes the information coding rate as a function of the code length and
frame error rate (FER). Therefore, for a context-aware short code design in
B5G networks, it is of primary importance to extend finite blocklength coding
theory to interference-prone scenarios that account for the intrinsic large-
scale and dense deployments of future networks.

Performance of large-scale cellular networks is well investigated using the
classical Shannon capacity [8, 9, 10, 11, 12, 13], defined as the maximum
achievable rate such that the error probability vanishes as the code length
increases [14]. Such idealistic scenario is hereafter denoted as the
asymptotic regime (AR). The AR analysis in [8, 9, 10, 11, 12, 13] is not
applicable for networks operating in the finite blocklength regime (FBR).
Short codes (e.g., 128 symbols) violate the AR assumptions and lead to
inevitable errors as a cost for constraints on delay and/or power consumption.
Thus, the code length $n$ of the FBR is a critical design parameter that
imposes a delicate trade-off between FER ($\epsilon$) and information coding
rate. To characterize such trade-off, the work of Polyanskiy et al. in [6, 7]
finds the maximum achievable rate as function of the code length $n$ and FER
$\epsilon$ for a point-to-point link.

Motivated by the need to migrate towards the FBR, the work in [6, 7] is
extended for several use cases in [15, 16, 17, 18, 19, 20, 21, 22, 23, 24,
25]. For instance, the work in [15] extends the FBR analysis to relaying
channels. The achievable rate for non-orthogonal multiple access in the FBR is
studied in [16]. The impact of the FBR on rate splitting is characterized in
[17]. The performance of multi-user MIMO, massive-MIMO, and cell-free MIMO in
the FBR are investigated in [18], [19], and [20], respectively. The authors in
[21, 22] optimize the blocklength and other network parameters (e.g., transmit
power or transmitter location) to minimize the decoding error probability
subject to a latency constraint. Works was also done in [23, 24, 25] to
maximize the achievable rate, improve latency, enhance reliability, and/or
provide a secure communication in the FBR. However, the models in [15, 16, 17,
18, 19, 20, 21, 22, 23, 24, 25] are limited to small scale networks. To the
best of the authors’ knowledge, performance characterization of large-scale
networks in the FBR is still an open problem.

Using the coding theory in the FBR in conjunction with stochastic geometry
tools, this paper intends to contribute to the aforementioned research gap, as
an extension of [1]. In particular, this paper develops novel mathematical
analysis to characterize the trade-off between the code length $n$, the FER
$\epsilon$, and the information coding rate $R$ in large-scale downlink (DL)
networks using orthogonal multiple access (OMA). To this end, achievable rates
and outage in the FBR are characterized for Gaussian codebooks as well as
practical modulation schemes. Multi-level polar-coded modulation (MLPCM) [26,
27, 28, 29], is presented as a practical validation for the obtained
theoretical achievable rates.222Polar codes is standardized for 5G in 3GPP
Release 16 Specification $\\#$ 38.212[30], and MLPCM was proposed to be used
in the future generations in [31, 32] The contributions of this paper are
summarized as follows.

The paper analyzes the performance of a large-scale DL network in the FBR as a
function of $n$ and $\epsilon$ in terms of:

  * •

the average coding rate under a random distance between the user and its
serving BS, under Gaussian codebooks,

  * •

the average coding rate using QAM constellations,

  * •

the rate outage probability and reliability bounds and approximations, and

  * •

the approximation of the coding rate meta distribution.

Additionally, the paper simulates the average coding rate achieved by MLPCM
for different QAM modulation orders in comparison with the obtained
theoretical benchmarks. All results are validated via Monte Carlo simulations.
It is shown that MLPCM achieves rates close to the derived theoretical
benchmarks. It is also shown that the derived outage probability bounds are
tight and coincide with simulation results. Additionally, it is shown that the
proposed approximation of the coding rate meta-distribution is fairly tight
under moderate and high SINR. Throughout the paper, we conduct qualitative and
quantitative comparisons with the performance in the AR to motivate the
importance of analyzing the performance in the FBR and demonstrate the
discrepancy with AR results. We also investigate the effect of various network
parameters in the FBR.

This paper is organized as follows. In Sec. II, the system model and
assumptions of the analysis are presented. In Sec. III, the average coding
rate of a large-scale DL network in the FBR is derived, under Gaussian
codebooks and under a constraint of QAM constellations. Then, the theoretical
results are numerically evaluated and compared with the performance of MLPCM
as a practical scheme. In Sec. IV, bounds on the rate outage probability are
derived, followed by the characterization of the reliability, the meta
distribution of the coding rate, numerical evaluations, and a detailed
discussion. Finally, the paper is concluded in Sec. V.

##  II System Model

Consider a single-tier OMA large-scale DL network with universal frequency
reuse and no intra-cell interference. The base stations (BSs) are located
according to a Poisson point process (PPP) $\Psi\subset\mathbb{R}^{2}$ with
intensity $\lambda\ \text{BS}/\text{km}^{2}$. The network applies universal
frequency reuse with one user equipment (UE) served per BS at a given resource
block. Each UE is served by its geographically closest BS, where the intended
distance between a UE and its serving BS is denoted as $r_{0}$. The distances
to the interfering BSs ordered with respect to the intended UE are denoted as
$r_{1},\ r_{2},\dots,\ r_{i},\dots$ such that $r_{i+1}>r_{i}$. For the sake of
simple presentation, the set $\tilde{\Psi}\in\mathbb{R}$ is defined as the BSs
distances to the desired UE. Hence, the received signal at the desired UE is
given by

$y=\sqrt{\mathcal{P}}h_{0}r_{0}^{-\eta/2}s_{0}+\underset{I_{agg}}{\underbrace{\sum_{r_{i}\in\tilde{\Psi}\setminus\\{r_{0}\\}}\sqrt{\mathcal{P}}h_{i}r_{i}^{-\eta/2}s_{i}}}+w,$
(1)

where $\mathcal{P}$ is the transmit power of the BSs, $h_{0}$ (resp. $h_{i}$)
represents the channel fading of the intended (resp. $i^{th}$ interfering)
channel, $\eta$ is the path loss exponent, $s_{0}$ (resp. $s_{i}$) is a unit
average power codeword symbol transmitted by the serving (resp. $i^{th}$
interfering) BS, $I_{agg}$ is the aggregate interference from all other BSs,
and $w\sim\mathcal{CN}(0,\sigma_{w}^{2})$ is circularly symmetric complex
Gaussian noise with zero mean and variance $\sigma_{w}^{2}$. We model $h_{0}$
and $h_{i}$ to be circularly symmetric complex Gaussian with zero mean and
unit variance (modeling Rayleigh fading). An independent and identically
distributed block fading channel model is assumed where the channel
coefficients $h_{0}$ and $h_{i}$ remain constant for a duration of $L$
consecutive symbols and change to independent realizations at the end of each
symbol interval.

The serving BS wants to send information to the UE using codewords from a code
with the rate $R$ and length $n$ symbols, where $n$ can be defined following a
latency and/or energy consumption constraint. To ensure that the channel
remains constant during the transmission of $n$ symbols, we require $n=L/l$
for some integer $l$ [33]. This ensures that the channels remain constant
during the transmission, but makes consecutive transmission blocks have
identical channels. Nonetheless, by considering the sequence of blocks
consisting of the first block (of $n$ symbols) in each coherence interval (of
$L$ symbols), we have independent channels and we can derive the average
performance over all transmissions. This is because the remaining sequences of
blocks (such as the second frame in each coherence interval) have identical
statistics.

To this end, two modes of operation are considered. The first mode assumes
knowledge of channel-state information (CSI) at the transmitter and the
receiver in the form of SINR availability.333Assuming SINR knowledge
benchmarks the performance of the network. Under this mode, we investigate the
average coding rate $\mathbb{E}\\{R\\}$ as a function of the FER and
codelength (see Sec. III). The second mode assumes that the CSI is unknown at
the transmitter but known at the receiver. Under this scenario, we derive the
outage probability and utilize the coding-rate meta distribution to
characterize the percentile of users achieving a given transmission
reliability (See Sec. IV).

##  III Average Rate Analysis

Considering the first mode of operation, we assume that the SINR is available
at the transmitter and the receiver. Hence, the coding rate can be adapted
based on the SINR in each transmission block so that the transmission using
codelength $n$ is successful with a desired FER $\epsilon$. This rate
adaptation can be realized using the following lemma, which characterizes the
maximum achievable rate over an additive white Gaussian noise (AWGN) channel
in the FBR [6].

######  Lemma 1.

([6]) For an AWGN channel with signal to noise power ratio (SNR) $\alpha$,
blocklength $n$, and FER $\epsilon\in(0,0.5)$, the maximum coding rate is
approximated as444This approximation is shown to be tight at blocklengths
($>100$) and FER $>10^{-6}$ in [6].

$R_{n,\epsilon}(\alpha)=C_{\infty,0}(\alpha)-\frac{\sqrt{V(\alpha)}Q^{-1}(\epsilon)}{\sqrt{n}}{+\frac{1}{2n}\log_{2}n},$
(2)

where $C_{\infty,0}(\alpha)=\log_{2}(1+\alpha)$ is the AWGN channel capacity
in the AR, $V(\alpha)=\frac{\alpha(\alpha+2)}{(\alpha+1)^{2}}\log_{2}^{2}(e)$
is known as the channel dispersion, and $Q^{-1}(\cdot)$ is the inverse of the
Q-function.

In this mode, due to the random changes of the channels $h_{0}$ and $h_{i}$,
we characterize performance by the average coding rate subject to a target FER
$\epsilon$ and codelength $n$. However, Lemma 1 is derived for an AWGN
channel, and hence, cannot be directly applied to a large-scale network due to
the additional non-Gaussian interference term $I_{agg}$ in (1). To overcome
this, we utilize the equivalence in distribution (EiD) approach to express the
interference as a conditionally Gaussian random variable [34, 35, 12], as
described next.

###  III-A Conditional Gaussian Representation

Using the EiD approach, we represent the aggregate interference for a given
$r_{0}$ as follows

$\displaystyle I_{agg}{\overset{\text{eid}}{=}}\sqrt{\mathcal{B}}{g},$ (3)

where $\overset{\text{eid}}{=}$ denotes the EiD, ${g}\sim\mathcal{CN}(0,1)$
and $\mathcal{B}>0$ is a positive random variable independent of $g$ with a
probability density function (PDF) whose Laplace transform (LT) is given by
[12],

$\displaystyle{\mathcal{L}_{\mathcal{B}}(u)=\exp\left\\{\sum_{k=1}^{\infty}(-1)^{k}2\pi\lambda
r_{0}^{2}\left(\frac{\mathcal{P}}{r_{0}^{\eta}}\right)^{k}\frac{\mathbb{E}\big{\\{}|s|^{2k}\big{\\}}u^{k}}{(\eta
k-2)k!}\right\\}.}$ (4)

where $s$ is the transmitted symbol from an arbitrary codebook by an arbitrary
BS and follows the same distribution as $s_{0}$ and $s_{i}$.

The EiD in (3) is proved by showing that the product $\sqrt{\mathcal{B}}{g}$
has the same characteristic function as $I_{agg}$ [12]. Using (3), the
equivalent system model is expressed as

$\displaystyle
y=\sqrt{\mathcal{P}}h_{0}r_{0}^{-\eta/2}s_{0}+\sqrt{\mathcal{B}}{g}+w,$

where for a given $\mathcal{B}$, the lumped term $\sqrt{\mathcal{B}}{g}+w$ is
Gaussian with zero mean and variance $\mathcal{B}+\sigma_{w}^{2}$. Hence, the
resulting conditional SINR, given $\mathcal{B}$ and $h_{0}$, is given by

${\Upsilon}=\frac{|h_{0}|^{2}\mathcal{P}r_{0}^{-\eta}}{\mathcal{B}+\sigma_{w}^{2}}.$
(5)

The random variable $\mathcal{B}$ in (5) augments the noise power with the
aggregate network interference power, for any constellation of $s_{0}$ and
$s_{i}$. If $s_{0}$ and $s_{i}$ is Gaussian distributed (Gaussian codebooks),
it can be shown that (4) simplifies to [12]

$\displaystyle\mathcal{L}_{\mathcal{B}_{g}}(u)=\exp\left\\{\frac{-2\pi\lambda
u\mathcal{P}r_{0}^{2-\eta}}{\eta-2}{}_{2}F_{1}\left(1,1-\frac{2}{\eta};2-\frac{2}{\eta};-\frac{u\mathcal{P}}{r_{0}^{\eta}}\right)\right\\}.$
(6)

Since at this point, interference is modeled as conditionally Gaussian, and
since Lemma 1 provides a tight approximation when noise is Gaussian, then the
tightness of the approximation provided in Lemma 1 holds here too. Thus, from
this point onward, we focus on analyzing performance using $R_{n,\epsilon}$ as
a surrogate for the maximum achievable rate in the FBR given that this is a
tight approximation. Under this framework, the maximum average coding rate is
defined next.

######  Definition 1.

Given a DL large-scale network with a distance $r_{0}$ between the desired UE
and the serving BS as defined in (1), assuming the SINR knowledge is available
at the transmitter and the receiver, we define the maximum average coding rate
in the FBR with blocklength $n$ and FER $\epsilon$ with
$\alpha_{0}=\frac{\mathcal{P}r_{0}^{-\eta}}{\sigma_{w}^{2}}$, as
$R_{n,\epsilon}(\alpha_{0})=\mathbb{E}_{h_{0},\mathcal{B}}\\{R_{n,\epsilon}(\Upsilon)\\}$,
where $\Upsilon$ is the SINR defined in (5) and $R_{n,\epsilon}(\Upsilon)$ is
as defined in Lemma 1.

Next, we use Lemma 1 and Def. 1 to characterize the average coding rate for a
large-scale network in the FBR, first using Gaussian codebooks, and then under
finite constellations (QAM). Then, we present a practical MLPCM to validate
the achievability of the derived theoretical benchmarks.

###  III-B Average Coding Rate in the Finite Block-Length Regime

By virtue of the EiD, the AWGN results in Lemma 1 can be now extended to
large-scale networks. For a given intended UE distance $r_{0}$, the FBR
average coding rate is characterized in the following theorem.

######  Theorem 1.

The maximum average coding rate of the large-scale network modeled by (1) with
Gaussian codebooks, blocklength $n$, FER $\epsilon$, and distance $r_{0}$
between the UE and its serving BS is given by

$\displaystyle\mathcal{R}_{n,\epsilon}({\alpha_{0}})=\mathcal{C}_{\infty,0}({\alpha_{0}})-\frac{\mathcal{V}({\alpha_{0}})Q^{-1}(\epsilon)}{\sqrt{n}}{+\frac{1}{2n}\log_{2}n},$
(7)

where
$\mathcal{C}_{\infty,0}({\alpha_{0}})=\int_{0}^{\infty}\exp\left(-\frac{2^{c}-1}{{\alpha_{0}}}\right)\mathcal{L}_{\mathcal{B}_{g}}\left\\{\frac{2^{c}-1}{\mathcal{P}r_{0}^{-\eta}}\right\\}dc,$
$\mathcal{V}({\alpha_{0}})=\int_{0}^{\log_{2}(e)}e^{\frac{-z(v)}{{\alpha_{0}}}}\mathcal{L}_{\mathcal{B}_{g}}\left\\{\frac{r_{0}^{\eta}}{\mathcal{P}}z(v)\right\\}d{v}$,
$\alpha_{0}=\frac{\mathcal{P}r^{-\eta}_{0}}{\sigma_{w}^{2}}$,
$z(v)=\sqrt{\frac{1}{1-\frac{{v^{2}}}{\log_{2}^{2}(e)}}}-1$, and
$\mathcal{L}_{\mathcal{B}_{g}}\\{\cdot\\}$ is given in (6).

###### Proof.

By virtue of the EiD approach, Lemma 1 is applicable to large-scale networks
by replacing $\alpha$ by $\Upsilon$ where $\Upsilon$ is defined in (2), which
leads to a maximum coding rate $R_{n,\epsilon}(\Upsilon)$ under a given
$h_{0}$ and $\mathcal{B}$. The average rate is then
$\mathcal{R}_{n,\epsilon}({\alpha_{0}})=\mathbb{E}\\{R_{n,\epsilon}(\Upsilon)\\}$,
where the averaging is over $h_{0}$ and $\mathcal{B}$. Thus, we need to
average $C_{\infty,0}(\Upsilon)$ and $\sqrt{V(\Upsilon)}$ over $\mathcal{B}$
and $h_{0}$. The term
$\mathcal{C}_{\infty,0}({\alpha_{0}})=\mathbb{E}\\{C_{\infty,0}(\Upsilon)\\}$
is the average capacity in a large-scale network in the AR with CSI available
at the BSs, which was derived in [36]. It remains to find
$\mathbb{E}\\{\sqrt{{V}(\Upsilon)}\\}$ which is shown to be equal to
$\mathcal{V}(\alpha_{0})$ in App. A. ∎

Theorem 1 governs the maximum average rate at a fixed distance $r_{0}$ between
the UE and its serving BS, considering the stochastic locations of the
interfering BSs and the Rayleigh fading environment. The following corollary
generalizes Theorem 1 by accounting for the randomness of $r_{0}$.

######  Corollary 1.

The average coding rate of the large-scale network modeled by (1) with
blocklength $n$, FER $\epsilon$, and Gaussian codebooks is given by

$\displaystyle\mathcal{\bar{R}}_{n,\epsilon}=\mathcal{\bar{C}}_{\infty,0}-\frac{\mathcal{\bar{V}}Q^{-1}(\epsilon)}{\sqrt{n}}{+\frac{1}{2n}\log_{2}n},$

where
$\mathcal{\bar{C}}_{\infty,0}=\int_{0}^{\infty}\mathcal{{C}}_{\infty,0}({\alpha_{0}})f_{r_{0}}(r_{0})dr_{0}$
and
$\mathcal{\bar{V}}=\int_{0}^{\infty}\mathcal{{V}}({\alpha_{0}})f_{r_{0}}(r_{0})dr_{0}$.

###### Proof.

The average capacity and average channel dispersion are derived by directly
averaging (7) with respect to $r_{0}$ which has the following probability
density function[36]

$\displaystyle f_{r_{0}}(r_{0})=2\pi\lambda r_{0}e^{-\pi\lambda r_{0}^{2}},\ \
0<r_{0}<\infty.$ (8)

∎

As can be seen in Theorem 1 and Corollary 1, the average coding rate of the
large-scale network in the FBR is the average capacity in the AR, plus a
penalty term (loss) which is a function of the blocklength $n$, and FER
$\epsilon$, in addition to the average SNR $\alpha_{0}$ and the stochastic
geometry of the network, manifested in the Laplace transform term. This
penalty term implies a trade-off between the reliability and the average rate:
the higher the reliability requirement (low FER), the lower the rate. However,
for a given average achievable rate, the reliability can be increased
($\epsilon$ decreased) by increasing the blocklength, converging to the AR
performance as $n$ grows to infinity.

Theorem 1 expresses the average coding rate under Gaussian codebooks, which is
of theoretical relevance. Characterizing the performance in the FBR when using
a finite constellation is of practical interest and is discussed next.

###  III-C Average Rate Using Finite Constellations

This section focuses on achievable rates in the FRB under transmission using
standard finite constellations. The maximum achievable rate using an $M$-ary
constellation over an AWGN channel in the FBR was given in [37]. Following the
same methodology as in Sec. III-B, we use the EiD approach to extend the
results of [37] to large-scale networks.

Consider an encoder that maps messages from a message set into a length $n$
sequence of symbols chosen from an $M$-ary constellation consisting of $M$
complex-valued symbols $s_{1},s_{2},...,s_{M}$. Using an $M$-ary QAM
constellation, denote by
$\mathcal{C}^{\text{\tiny(M)}}_{\infty,0}(\alpha_{0})$ the average capacity
achieved in the AR, by ${\mathcal{V}}^{\text{\tiny{(M)}}}(\alpha_{0})$ the
average square root channel dispersion, and by
$\mathcal{R}^{\text{\tiny(M)}}_{n,\epsilon}(\alpha_{0})$ the average coding
rate under a blocklength $n$ and FER $\epsilon$. To characterize the average
capacity of the $M$-ary QAM under FBR, the conditional mutual information
$I(s_{0};y|h_{0},\mathcal{B},r_{0})$ has to be averaged with respect to the
aggregate interference power $\mathcal{B}$ and the channel gain $h_{0}$.
However, the distribution of the aggregate interference power $\mathcal{B}$ is
unknown which leads to an intractable expression. Although such an expression
can be evaluated using Monte Carlo simulations, an expression that is amenable
to numerical integration is preferable. A similar argument applies for
${\mathcal{V}}^{\text{\tiny{(M)}}}(\alpha_{0})$. Hence, to obtain tractable
integral expressions, we adopt an approximation for the distribution of the
interference power $\mathcal{B}$ proposed in [38, 39] which was shown to be a
tight approximation. In particular, in [38, 39], the distribution of the
interference power $\mathcal{B}$ was approximated by a Gamma distribution

$\displaystyle
f(x;q,\theta)=\frac{x^{q-1}e^{\frac{-x}{\theta}}}{\Gamma(q)\theta^{q}},\ \ \ \
\ \ \ \ \ x>0$ (9)

where $q=\frac{4\pi\lambda r_{0}^{2}(\eta-1)}{(\eta-2)^{2}}$ is the shape
parameter, and $\theta=\frac{(\eta-2)\mathcal{P}}{2(\eta-1)r_{0}^{\eta}}$ is
the scale parameter. The approximation is based on a moment-matching approach.
The first two moments of $\mathcal{B}$ can be obtained from (4) as

$\displaystyle\mathbb{E}\\{\mathcal{B}\\}$
$\displaystyle=\left.\frac{d\mathcal{L}_{\mathcal{B}}(u)}{du}\right|_{u=0}=\frac{2\pi\lambda
r_{0}^{2-\eta}\mathcal{P}}{(\eta-2)}$
$\displaystyle\mathbb{E}\\{\mathcal{B}^{2}\\}$
$\displaystyle=\left.\frac{d^{2}\mathcal{L}_{\mathcal{B}}(u)}{du^{2}}\right|_{u=0}=\left(\frac{2\pi\lambda
r_{0}^{2-\eta}\mathcal{P}}{(\eta-2)}\right)^{2}+\frac{\pi\lambda
r_{0}^{2-2\eta}\mathcal{P}^{2}}{(\eta-1)}.$

The scale and shape parameters of the gamma distribution are then obtained as
$q=\frac{\mathbb{E}\\{\mathcal{B}\\}^{2}}{\mathbb{V}ar\\{\mathcal{B}\\}}$ and
$\theta=\frac{\mathbb{V}ar\\{\mathcal{B}\\}}{\mathbb{E}\\{\mathcal{B}\\}}$,
where
$\mathbb{V}ar\\{\mathcal{B}\\}=\mathbb{E}\\{\mathcal{B}^{2}\\}-\mathbb{E}\\{\mathcal{B}\\}^{2}$.

Fig. 1 numerically demonstrates the accuracy of this approximation by plotting
the approximate CDF of $\mathcal{B}$ (using (9)) and its exact CDF obtained
from a numerical inversion of (4). The figure shows that the Gamma
approximation in (9) provides a fairly tight approximation for the CDF of
$\mathcal{B}$. Note that the Gamma approximation is crucial to maintain the
tractability of the analysis. Using this approximation, the average coding
rate of a large-scale network in the FBR, at a given distance $r_{0}$ and
using $M$-ary QAM constellation, is characterized.

Figure 1: CDF of the interference power $\mathcal{B}$ and its approximation at
different values of $\mathcal{P}$, $\lambda$ and $r_{0}$

The maximum achievable rate for a finite constellation is provided in [37].
However, the channel dispersion in [37] is defined as the variance of the
information density ($V=Var(i(x;y))$). This definition is considered to be
imprecise according to [6], where the channel dispersion is defined as the
conditional variance of the information density for finite constellation and
is defined as the unconditional variance of the information density in the
case of using Gaussian signals. Therefore, in this paper, we derive the
maximum achievable rate using this definition for large-scale networks in the
FBR in the following theorem.

######  Theorem 2.

For a DL large-scale network with BS density $\lambda\
\mathrm{BS}/\mathrm{km}^{2}$, blocklength $n$, FER $\epsilon$, distance
$r_{0}$ between the UE and the serving BS, and an $M$-ary constellation with
symbols $\\{\underline{s}_{m}\\}_{m=1}^{M}$ where
$\underline{s}_{m}=[\mathcal{R}e\\{s_{m}\\},\mathcal{I}m\\{s_{m}\\}]^{T}$ and
$\mathcal{R}e\\{s_{m}\\}$ and $\mathcal{I}m\\{s_{m}\\}$ are the real and
imaginary parts of $s_{m}$, respectively, if the interference power
$\mathcal{B}$ follows a Gamma distribution (9), then the average achievable
coding rate can be expressed by

$\displaystyle\mathcal{R}^{\text{\tiny(M)}}_{n,\epsilon}(\alpha_{0})=\mathcal{C}^{\text{\tiny(M)}}_{\infty,0}(\alpha_{0})-\frac{\mathcal{{V}}^{\text{\tiny{(M)}}}(\alpha_{0})Q^{-1}(\epsilon)}{\sqrt{n}}{+\frac{1}{2n}\log_{2}n},$
(10)

where $\alpha_{0}=\frac{\mathcal{P}r_{0}^{-\eta}}{\sigma_{w}^{2}}$,

$\displaystyle\mathcal{C}^{\text{\tiny(M)}}_{\infty,0}(\alpha_{0})$
$\displaystyle=\log_{2}(M)-\frac{1}{M\pi}\sum_{m=1}^{M}\int_{0}^{\infty}\int_{{\mathbb{R}^{2}}}e^{-\|\underline{t}\|^{2}}{g_{m}(\underline{t})}f_{{\Upsilon}}(\Upsilon|r_{0})d\underline{t}\
d\Upsilon$ (11) $\displaystyle{\mathcal{V}}^{\text{\tiny{(M)}}}(\alpha_{0})$
$\displaystyle=\int_{0}^{\infty}\sqrt{V_{\text{\tiny
M}}(\Upsilon)}f_{{\Upsilon}}(\Upsilon|r_{0})\ d\Upsilon$ (12)
$\displaystyle{V}_{\text{\tiny{M}}}(\Upsilon)$
$\displaystyle=\sum_{m=1}^{M}\left(\int_{{\mathbb{R}^{2}}}\frac{e^{-\|\underline{t}\|^{2}}}{M\pi}{g_{m}(\underline{t})^{2}}d\underline{t}-\left(\int_{\mathbb{R}^{2}}\frac{e^{-\|\underline{t}\|^{2}}}{M\pi}{g_{m}(\underline{t})}d\underline{t}\right)^{2}\right)$
(13)

where
$g_{m}(\underline{t})=\log_{2}\left(\sum_{l=1}^{M}e^{-2\sqrt{{\Upsilon}}\underline{t}^{T}(\underline{s}_{m}-\underline{s}_{l})-{\Upsilon}\|\underline{s}_{m}-\underline{s}_{l}\|^{2}}\right)$,
$\underline{t}=[t_{1},t_{2}]^{T}$ is a 2-D real-valued vector, $\Upsilon$ is
the SINR given in (5), and

$\displaystyle
f_{{\Upsilon}}(\Upsilon|r_{0})=\frac{\exp\left\\{\frac{-\Upsilon\sigma_{w}^{2}}{\mathcal{P}r_{0}^{-\eta}}\right\\}}{\mathcal{P}r_{0}^{-\eta}}\frac{(1+\theta
r_{0}^{\eta}\Upsilon/\mathcal{P})\sigma_{w}^{2}+q\theta}{(1+\theta
r_{0}^{\eta}\Upsilon/\mathcal{P})^{q+1}},0\leq\Upsilon\leq\infty.$ (14)

###### Proof.

The average capacity of an $M$-ary constellation
($\mathcal{C}^{\text{\tiny(M)}}_{\infty,0}(\alpha_{0})$) at a fixed $r_{0}=r$
is derived by averaging the conditional mutual information between the
transmitted symbol $s_{0}\in\\{s_{m}\\}_{m=1}^{M}$ and the received signal
$y$, i.e., $I(s_{0};y|\Upsilon,r_{0}=r)$, provided in [37], with respect to
the SINR $\Upsilon$ which is a function of interference power $\mathcal{B}$
and fading channel $h_{0}$ statistics. Similarly, the average square-root
channel dispersion (${\mathcal{V}}^{\text{\tiny{(M)}}}(\alpha_{0})$) is
derived by averaging the channel dispersion in (13) with respect to
$\Upsilon$. The derivation of (13) is provided in App. B. However, instead of
averaging over the distribution of $\Upsilon$ which is unknown, we average
over $f_{{\Upsilon}}(\Upsilon|r_{0})$ in (14) which is obtained using the
Gamma approximation provided in (9) (See App. C). ∎

The following corollary generalizes Theorem 2 by accounting for the randomness
of $r_{0}$.

######  Corollary 2.

The average achievable coding rate of the large-scale network modeled by (1)
with blocklength $n$, FER $\epsilon$, and using an $M$-ary constellation with
symbols $\\{s_{m}\\}_{m=1}^{M}$ is given by

$\displaystyle\bar{\mathcal{R}}^{\text{\tiny(M)}}_{n,\epsilon}\approx\bar{\mathcal{C}}^{\text{\tiny(M)}}_{\infty,0}-\frac{\mathcal{\bar{V}}^{\text{\tiny{(M)}}}Q^{-1}(\epsilon)}{\sqrt{n}}{+\frac{1}{2n}\log_{2}n},$
(15)

where
${\bar{\mathcal{C}}^{\text{\tiny(M)}}_{\infty,0}=\int_{0}^{\infty}{\mathcal{C}}^{\text{\tiny{(M)}}}_{\infty,0}(\alpha_{0})f_{r_{0}}(r_{0})dr_{0}}$,
${\bar{\mathcal{V}}^{\text{\tiny{(M)}}}=\int_{0}^{\infty}{\mathcal{V}}^{\text{\tiny{(M)}}}(\alpha_{0})f_{r_{0}}(r_{0})dr_{0}}$,
and $f_{r_{0}}(r_{0})$ as defined in (8).

###### Proof.

This is derived by averaging (10) with respect to $r_{0}$. ∎

The average achievable rate depends on the distances between symbols in the
constellation set, i.e.
$\|\underline{d}_{ml}\|=\|\underline{s}_{m}-\underline{s}_{l}\|$ for $m\neq
l$, where neighboring symbols contribute more to error than far ones. Such
distances are in turn affected by the constellation choice. Moreover, the
average achievable rate is affected by the network parameters $\lambda,\
r_{0}$ which will be investigated later in the numerical evaluations section.

The achievable rates provided so far are theoretical. In the next section, we
use MLPCM, which was proposed in [31, 32] to be used in future generations, to
validate the achievability of the obtained theoretical benchmarks.

###  III-D Multilevel Polar-Coded Modulation

$\mathbb{X}$

Demultiplexer

$x_{1}$$x_{2}$$x_{\log_{2}M}$EncoderEncoderEncoderEncoder

M-ary Modulator

$u_{1}$$u_{2}$$u_{\log_{2}M}$${s_{0}}$Channel$\mathbb{y}$Decoder$\widetilde{x}_{1}$DelayDecoder$\widetilde{x}_{2}$DelayDecoder$\widetilde{x}_{\log_{2}M-1}$$\widetilde{x}_{\log_{2}M-2}$$\widetilde{x}_{1}$DelayDecoder$\widetilde{x}_{\log_{2}M}$$\widetilde{x}_{1}$

Multiplexer

$\widetilde{\mathbf{X}}$ Figure 2: Multilevel Coded Modulation Block Diagram

To validate the theoretical results in Sec. III-B and III-C, we use the MLPCM
scheme shown in Fig. 2. The information bits (message)
$\mathbf{X}\in\mathbb{F}_{2}^{k\log_{2}(M)}$ are fed into a demultiplexer that
splits them into $\log_{2}(M)$ sub-messages $\\{x_{i}\\}_{i=1}^{\log_{2}(M)}$
where $x_{i}\in\mathbb{F}_{2}^{k}$. Each sub-message $x_{i}$ is encoded using
a polar encoder with rate $R_{c}=k/n$ to obtain codewords
$\mathbf{U}\in\mathbb{F}_{2}^{\log_{2}(M)\times n}$, where
$u_{i}\in\mathbb{F}_{2}^{n}$ is the $i^{th}$ row of $\mathbf{U}$ ($i^{th}$
codeword). Then, the transmitter collects one bit from each of the codewords
$u_{i}$ to form a $\log_{2}(M)$-bit symbol which is then mapped to a symbol
$s_{0}$ from an $M$-ary constellation such as the $M$-QAM constellation. The
symbols are then scaled according to the power constraint in Sec. II. The
result of repeating this process for all $n$ codeword symbols of all
$\log_{2}(M)$ sub-messages is one MLPCM codeword of length $n$, which is then
transmitted through the channel. At the receiver side, multistage decoding
takes place to efficiently recover the message. It breaks the decoding process
into $\log_{2}(M)$ stages.

In each stage $i=1,\dots,\log_{2}(M)$, the receiver feeds the demapper of the
$i^{th}$ stage with the received signal $y$ and the submessages recovered from
previous stages $(\tilde{x}_{1},\dots,\tilde{x}_{i-1})$, (which is defined as
$\emptyset$ for $i=1$), and the demapper demaps the symbols into bits to
obtain $\tilde{u}_{i}$, using the multilevel decoding criteria [26]. Then, the
demapped bits $\tilde{u}_{i}$ are decoded using a polar decoder to obtain
$\tilde{x}_{i}$ using the successive cancellation algorithm provided in [40].
Finally, all sub-messages are fed to a multiplexer to form the message
$\mathbf{\tilde{X}}$. This MLPCM will be incorporated in the large-scale DL
network Monte Carlo simulator to validate the achievability of the theoretical
rates obtained in Sec. III-B and III-C.

(a) The maximum average coding rate at $r_{0}=250\ m$ for different values of
$n$ and $\epsilon$

(b) The average coding rate for M-QAM at $r_{0}=150\ m$, $n=128$, and
$\epsilon=10^{-2}$.

Figure 3: Average coding rate analysis of a large-scale network with
$\lambda=1\ \mathrm{BS/km^{2}}$ for fixed $r_{0}$

###  III-E Average Rate Numerical Results

This subsection presents illustrative numerical results for the theoretical
rate analysis provided in Sec. III-B and III-C, which are also validated via
Monte Carlo simulation. The results presented investigate the effect of the
blocklength $n$ and the FER $\epsilon$, the tightness of the proposed
theoretical approximations, and the performance comparison between the FBR and
the AR. Then, the performance of the MLPCM is investigated in comparison with
the theoretical average coding rate.

Fig. 3 plots the maximum average coding rate and the average coding rate for
M-QAM versus the average SNR $\alpha_{0}$ for a fixed $r_{0}$. In Fig. 3(a),
the average coding rate is plotted for $n\in\\{128,2048\\}$, FER
$\epsilon\in\\{10^{-2},\ 10^{-5},\ 10^{-6}\\}$, and $r_{0}=250$ m. Given a FER
$\epsilon$, a gap exists between the average coding rate in the FBR and the
AR. The gap is around $1$ dB for $n=2048$ and is between $3$ and $5$ dB for
$n=128$ at $\epsilon=10^{-5}$ and $10^{-6}$, respectively, which shows the
severity of the SNR penalty in the FBR compared to the AR. The figure shows
the trade-off between reliability (represented by FER) and maximum average
coding rate at a given $n$, where the maximum average coding rate
decreases/increases as the FER decreases/increases. For instance, to achieve a
given maximum average coding rate at $n=128$ and FER $\epsilon=10^{-5}$, we
need around $2$ dB more power compared with the same blocklength $n$ under the
FER $\epsilon=10^{-2}$. Furthermore, the impact of small $n$ becomes more
severe when the FER $\epsilon$ is smaller, resulting in rate degradation.
Overall, this shows that it is impractical to use results from the AR in the
FBR.

Second, the maximum achievable coding rate for different modulation schemes,
presented in Sec. III-C, is simulated for $r_{0}=150\ m$, $\lambda=1\
\text{BS}/{\text{km}}^{2}$, blocklength $n=128$, and FER $\epsilon=10^{-2}$ in
Fig. 3(b). Fig. 3(b) demonstrates the approximated maximum achievable coding
rate using the Gamma approximation for various modulation orders,
${M}=\\{2,4,8,16\\}$, (Theorem 2), along with the maximum achievable rate
under Gaussian codebooks (Theorem 1) versus $\alpha_{0}$. The Monte Carlo
simulated maximum achievable coding rate for QAM is provided to show the
tightness of the approximation. The approximated maximum achievable rate is
very close to the simulated maximum achievable rate. Thus, the Gamma
approximation in (9) leads to a fairly tight approximation. It is also
observed that as the QAM modulation order increases, the performance improves
towards the maximum average coding rate (without Gaussian codebooks) in a
large-scale network in the FBR (Theorem 1).

The spatially averaged coding rate is depicted in Fig 4(a) versus the transmit
signal to noise power ratio $\frac{\mathcal{P}}{\sigma_{w}^{2}}$.555The figure
is plotted for a wide range of $\frac{\mathcal{P}}{\sigma_{w}^{2}}$ for
illustrative purposes, bearing in mind that some values of
$\frac{\mathcal{P}}{\sigma_{w}^{2}}$ in the plot may be impractical. The
figure shows that for high SNR (above $0$ dB), one cannot achieve the same
rate and FER achieved at a given $n$ by decreasing $n$ and paying an SNR
penalty. For instance, the rate achieved at $n=2048$, $\epsilon=10^{-5}$,
$\text{SNR}=10$ dB cannot be achieved at $n=128$ and $\epsilon=10^{-5}$. The
only way to achieve such a rate at $n=128$ is by paying a FER penalty, which
is impractical for applications that need high reliability. Hence, it is
important to select $n$ that balances the trade-off between reliability and
coding rate. Similar to Fig. 3(a), Fig. 4(a) shows the importance of the
results in this paper for characterizing and optimizing the performance of
large-scale networks with latency-sensitive applications.

(a) Maximum Average Coding Rate for different values of $n$ and $\epsilon$.

(b) Average coding rate for M-QAM at $n=512$, and $\epsilon=10^{-2}$.

Figure 4: Average coding rate analysis for random $r_{0}$ at $\lambda=1\
\mathrm{BS/km^{2}}$

Fig. 4(b) plots, the average achievable rate versus
$\frac{\mathcal{P}}{\sigma_{w}^{2}}$ under a Rayleigh distributed $r_{0}$, QAM
constellation with $M=2,4,8,$ and $16$, blocklenth $n=512$, and FER
$\epsilon=10^{-2}$, showing a similar trend as for fixed $r_{0}$. It is
observed that the average achievable rate saturates at a rate of $\approx 2\
\text{bits/transmission}$ for this choice of $\lambda$, $n$, and $\epsilon$,
which is low compared to the case $r_{0}=150\ m$. This performance is a
consequence of averaging the performance of nearby UEs which have relatively
strong received desired signals (high SINR) and far away UEs which suffer from
more interference (low SINR).

Finally, considering the same setting in Fig. 3(b), we simulate MLPCM with a
QAM constellation for modulation orders $M=2,4,8,$ and $16$ in Fig. 5. The
simulations are performed by fixing the SNR and increasing the coding rate
until reaching the FER $\epsilon$. The achievable rates are plotted along with
the theoretical expressions proposed in (7) and (10). The figure shows that
the achievable rates of the MLPCM scheme with 2-QAM or 4-QAM approaches the
theoretical rate of the corresponding QAM constellation. However, for 8-QAM
and 16-QAM, a gap of $\approx 2.5$ dB exists between the theoretical results
and the simulation results. Nonetheless, this SNR gap translates to a small
rate gap of fractions of bits/transmission due to the slow increase of the
rate versus SNR in the large-scale network which is caused by interference.
Fig. 5 thus validates that the theoretical performance under finite
constellations can be approached using MLPCM, which validates the potential of
MLPCM for future technologies.

Figure 5: The average coding rate versus $r_{0}^{-\eta}\alpha$ at $\lambda=1\
\mathrm{BS/km^{2}}$, $r_{0}=150\ m$, and $n=128$.

##  IV Rate Outage Probability & Meta Distribution

In the previous section, we studied the case where the SINR is known at the
transmitter. However, if the BS does not track the SINR, then the rate
adaptation investigated in the previous section is not possible. Considering
the second mode of operation, where the SINR is unknown at the transmitter but
known at the receiver, the transmitter encodes at a constant (target) rate
$R_{t}$ and blocklength $n$. As a result, the resulting (conditional) FER
conditioned on SINR $\Upsilon$ may or may not satisfy a desired design FER
$\epsilon$ due to the unknown stochastic variation of the intended channel
fading and aggregate interference. In this case, it is important to keep this
conditional FER below a threshold. This is especially relevant bearing in mind
that $n=L/l$, i.e., a large conditional FER means that there are coherence
intervals that produce large FERs for all blocks within these coherence
intervals. This can lead to bursts of frame errors when the channel undergoes
a period of coherence intervals with weak channels, which is undesired.

Thus, we define an FER threshold $\bar{\epsilon}$, and it is desired to
maintain the conditional FER below $\bar{\epsilon}$. If the conditional FER
exceeds $\bar{\epsilon}$, we say that we have an outage. This equivalently
means that rate $R_{n,\bar{\epsilon}}(\Upsilon)$ supported by the channel at
the given $n$ and FER threshold $\bar{\epsilon}$ decreases below $R_{t}$. This
is defined formally as follows.

######  Definition 2.

Given a target rate $R_{t}$ and a DL large-scale network with a distance
$r_{0}$ between the desired UE and the serving BS as defined in (1), we define
the outage probability in the FBR with blocklength $n$ and FER threshold
$\bar{\epsilon}$ as
$\mathcal{O}(r_{0},R_{t},n,\bar{\epsilon})=P(R_{n,\bar{\epsilon}}(\Upsilon)<R_{t})$,
where $R_{n,\bar{\epsilon}}(\Upsilon)$ is as defined in Lemma 1.

This outage probability is studied in this section.

###  IV-A DL Outage Analysis in the Finite Blocklength Regime

We start by reviewing the outage probability of a large-scale network in the
AR. In this case, the outage is defined as the probability that the channel
capacity (defined as the maximum rate such that the error probability vanishes
as $n\to\infty$) is lower than a target rate $R_{t}$. The rate outage
probability under a given $r_{0}$ is given as [41]

$\displaystyle\mathcal{O}(r_{0},R_{t})=1-e^{-\frac{(2^{R_{t}}-1)\sigma_{w}^{2}}{\mathcal{P}r_{0}^{-\eta}}}\mathcal{L}_{\mathcal{B}_{g}}\left\\{\frac{2^{R_{t}}-1}{\mathcal{P}r_{0}^{-\eta}}\right\\}.$
(16)

Studying the rate outage probability under blocklength $n$ and FER threshold
$\bar{\epsilon}$ is generally difficult to simplify as it is a function of the
fading distribution as well as the network geometry. Instead, we provide
bounds which are fairly tight in the following theorem.

######  Theorem 3.

For an average SNR $\alpha_{0}$, the rate outage probability defined in Def. 2
satisfies
$\mathcal{O}_{l}(r_{0},R_{t},n,\bar{\epsilon})\leq\mathcal{O}(r_{0},R_{t},n,\bar{\epsilon})\leq\mathcal{O}_{u}(r_{0},R_{t},n,\bar{\epsilon})$,
where

$\displaystyle\mathcal{O}_{l}(r_{0},R_{t},n,\bar{\epsilon})$
$\displaystyle=\mathcal{O}(r_{0},R_{t}),$ (17)
$\displaystyle\mathcal{O}_{u}(r_{0},R_{t},n,\bar{\epsilon})$
$\displaystyle=\mathcal{O}(r_{0},R_{t}+a_{n,\bar{\epsilon}}{-b_{n}}),$ (18)

where $\mathcal{O}(r_{0},R_{t})$ is defined in (16),
$a_{n,\bar{\epsilon}}=\sqrt{\frac{\log_{2}^{2}(e)}{n}}Q^{-1}(\bar{\epsilon})$,
and ${b_{n}=\frac{1}{2n}\log_{2}n}$.

###### Proof.

From Def. 2, we have

$\displaystyle\mathcal{O}(r_{0},R_{t},n,\bar{\epsilon})=\mathbb{P}(R_{n,\bar{\epsilon}}(\Upsilon)<R_{t})=\mathbb{P}\Bigg{(}\log_{2}(1+\Upsilon)-{a_{n,\bar{\epsilon}}}\sqrt{1-\frac{1}{(1+\Upsilon)^{2}}}{+b_{n}}<R_{t}\Bigg{)}.$
(19)

Noting that $\mathcal{K}(\Upsilon)=\sqrt{1-\frac{1}{(1+\Upsilon)^{2}}}$ is
monotonically increasing in $\Upsilon$ as shown in Fig. 6, satisfying
$\mathcal{K}(\Upsilon)\in[0,1]$, we conclude that
$\mathcal{O}_{l}(r_{0},R_{t},n,\bar{\epsilon})\leq\mathcal{O}(r_{0},R_{t},n,\bar{\epsilon})\leq\mathcal{O}_{u}(r_{0},R_{t},n,\bar{\epsilon})$,
where the lower and upper bounds are obtained by setting
$\mathcal{K}(\Upsilon)$ to $0$ and $1$, respectively. ∎

Figure 6: $\mathcal{K}(\Upsilon)$ versus $\Upsilon$ and the PDF of the SINR at
$r_{0}\in\\{150,250\\}\ m$

The outage probability lower bound confirms that the outage probability in the
FBR is larger than that in the AR and the upper bound provides a guaranteed
performance. By observing Fig. 6, it can be seen that $\mathcal{K}(\Upsilon)$
quickly approaches one as $\Upsilon$ grows. The figure also shows that the
density $f(\Upsilon|r_{0})$ in the interval of SINR, where the approximation
$\mathcal{K}(\Upsilon)\approx 1$ is tight ($\Upsilon>5$ e.g. ), increases as
$r_{0}$ decreases (which is the case for dense networks). Hence, the upper
bound $\mathcal{O}_{u}(r_{0},R_{t},n,\bar{\epsilon})$ can be used to evaluate
guaranteed outage performance in general and can be used as a tight
approximation for dense networks. This is verified in Fig. 7 & 8. Next, the
spatially averaged outage probability is presented.

######  Corollary 3.

The average rate outage probability defined in Def. 2 with blocklength $n$,
FER $\bar{\epsilon}$, and target rate $R_{t}$ can be upper bounded as

$\displaystyle\bar{\mathcal{O}}(R_{t},n,\bar{\epsilon})=1-\int_{0}^{\infty}2\pi\lambda
r_{0}$ $\displaystyle
e^{-\frac{2^{R_{t}+a_{n,\bar{\epsilon}}{-b_{n}}}-1}{\alpha_{0}}-\pi\lambda
r_{0}^{2}}\mathcal{L}_{\mathcal{B}_{g}}\left\\{\frac{2^{R_{t}+a_{n,\bar{\epsilon}}{-b_{n}}}-1}{\mathcal{P}r_{0}^{-\eta}}\right\\}dr_{0}$
(20)

###### Proof.

Using the law of iterated expectation, the average outage probability is the
average of (18) with respect to $r_{0}$, i.e.,
$\mathbb{P}(R>R_{t})=\mathbb{E}_{r_{0}}\\{\mathbb{P}(R>R_{t}|r_{0})\\}$. The
result then follows since $r_{0}$ follows a Rayleigh distribution. ∎

For a shadowed urban cellular network, the path-loss exponent $\eta$ lies in
the range $3-5$ [42], where $\eta=4$ is a common value of practical relevance
that is widely utilized in the literature. For the case $\eta=4$, the outage
probability in (20) can be simplified as given next.

######  Corollary 4.

The average rate outage probability of the large-scale network modeled by (1)
with blocklength $n$, FER $\bar{\epsilon}$, target rate $R_{t}$, and path loss
exponent $\eta=4$, can be upper bounded as

$\bar{\mathcal{O}}(R_{t},n,\bar{\epsilon})=1-\pi\lambda\sqrt{\frac{\pi\mathcal{P}}{\sigma_{w}^{2}\mu}}e^{\frac{\sigma_{w}^{2}\mu
z^{2}}{4\mathcal{P}}}Q\left(\sqrt{\frac{z^{2}\mu\sigma_{w}^{2}}{2\mathcal{P}}}\right)$
(21)

where $\mu=2^{R_{t}+a_{n,\bar{\epsilon}}{-b_{n}}}-1$ and
$z=\frac{\lambda\pi\mathcal{P}}{\sigma_{w}^{2}\mu}\left(\sqrt{\mu}\arctan(\sqrt{\mu})+1\right)$.

###### Proof:

A closed-form expression for the upper bounded outage probability at $\eta=4$
is obtained as follows

$\displaystyle\bar{\mathcal{O}}(R_{t},n,\bar{\epsilon})$
$\displaystyle=1-\int_{0}^{\infty}e^{-\frac{\mu
r_{0}^{4}\sigma_{w}^{2}}{\mathcal{P}}}\mathcal{L}_{\mathcal{B}_{g}}\left\\{\frac{\mu
r_{0}^{4}}{\mathcal{P}}\right\\}f_{r_{0}}(r_{0})\ dr_{0}$ (22)
$\displaystyle\overset{(i)}{=}1-\int_{0}^{\infty}2\pi\lambda
r_{0}e^{-\frac{\sigma_{w}^{2}\mu}{\mathcal{P}}\left(r_{0}^{4}+z\
r_{0}^{2}\right)}dr_{0}$ (23)
$\displaystyle\overset{(ii)}{=}1-\pi\lambda\sqrt{\frac{\pi\
\mathcal{P}}{\sigma_{w}^{2}\mu}}e^{\frac{z^{2}\mu\sigma_{w}^{2}}{4\mathcal{P}}}Q\left(\sqrt{\frac{z^{2}\mu\sigma_{w}^{2}}{2\mathcal{P}}}\right)$
(24)

where we used
${\mathcal{L}_{\mathcal{B}_{g}}\\{s\\}}_{\eta=4}=\exp\left\\{-\pi\lambda\sqrt{s\mathcal{P}}\text{tan}^{-1}\left(\sqrt{\frac{s\mathcal{P}}{r_{0}^{4}}}\right)\right\\}$,
$(i)$ is obtained using a change of variable
$z=\frac{\lambda\pi\mathcal{P}}{\mu\sigma_{w}^{2}}\left(\sqrt{\mu}\arctan(\sqrt{\mu})+1\right)$,
and $(ii)$ is obtained using the change of variables and the CDF of the
Gaussian distribution with mean $\left(\frac{z}{2}\right)$ and variance
$\left(\frac{\mathcal{P}}{2\mu\sigma_{w}^{2}}\right)$. ∎

The rate outage probability of a large-scale network in the FBR is upper
bounded, with guaranteed performance at high SINR, in a generic form in (18)
and (20) and in a closed-form expression for $\eta=4$ in (24), as a function
of BS density $\lambda$, FER $\bar{\epsilon}$, blocklength $n$, and average
SNR $\alpha_{0}$. Next, the reliability of the network under the FBR.

###  IV-B Reliability of a Large-Scale Network in the Finite Blocklength
Regime

We define a metric to evaluate the overall network reliability for a fixed
$r_{0}$, defined as the probability of correctly decoding a codeword. A
codeword is decoded correctly in two cases in the FBR. The first case is when
$\mathcal{R}_{n,\bar{\epsilon}}(\Upsilon)>R_{th}$, in which case the codeword
is decoded correctly with a probability larger than or equal to
$(1-\bar{\epsilon})$. The second case is when
$\mathcal{R}_{n,\bar{\epsilon}}(\Upsilon)<R_{th}$, in which case few codewords
may still be decoded correctly. Since in practice, it is desirable to operate
at low outage probability, hence the second case does not contribute much to
reliability. Thus, to simplify the analysis, we will lower bound the
probability of decoding a codeword correctly in the second case by $0$ (i.e.,
its contribution to reliability is neglected) and we will only consider the
first case. Hence, the reliability is lower bounded by the following
expression, providing a guaranteed performance for a given $r_{0}$,

$\mathcal{T}_{n,\bar{\epsilon}}(r_{0},R_{t})=(1-\mathcal{O}(r_{0},R_{t},n,\bar{\epsilon}))(1-\bar{\epsilon}).$
(25)

The guaranteed reliability averaged over $r_{0}$ is given by

$\bar{\mathcal{T}}_{n,\bar{\epsilon}}(R_{t})=(1-\bar{\mathcal{O}}(R_{t},n,\bar{\epsilon}))(1-\bar{\epsilon}),$
(26)

Also, for the sake of comparison, the reliability of the AR can be defined as

${\mathcal{T}}_{\infty,0}(r_{0},R_{t})=1-\mathcal{O}(r_{0},R_{t}),$ (27)

since the error probability is assumed to be vanishing as $n\to\infty$, and
hence failure occurs when there is an outage. The reliability will be assessed
numerically in Sec. IV-D.

Studying the rate outage probability and the overall system reliability gives
more insight into the performance of the network. Moreover, studying the
coding rate meta distribution, which provides the percentiles of users
achieving a rate $R_{t}$, will provide a complete picture of the network
performance. Hence, the meta distribution of the coding rate is derived next.

###  IV-C Coding Rate Meta Distribution in the Finite Blocklength Regime

The meta distribution provides fine-grained information about network
performance, in the form of the fraction of users that can achieve a minimum
data rate $R_{t}$ at a FER threshold $\bar{\epsilon}$ with probability at
least $p_{t}$. The definition and analysis of this quantity are provided next.

######  Definition 3.

Given a target rate $R_{t}$ and a DL large-scale network with a distance
$r_{0}$ between the desired UE and the serving BS as defined in (1), we define
the coding rate meta distribution in the FBR with blocklength $n$ and FER
threshold $\bar{\epsilon}$ as
$F_{R_{t}}(p_{t})={P}(P_{s}(R_{t},n,\bar{\epsilon})<p_{s})$ where
$P_{s}(R_{t},n,\bar{\epsilon})=P(R_{n,\bar{\epsilon}}(\Omega)>R_{t}|\tilde{\Psi})$
is the probability of success, $\Omega$ is the signal-to-interference ratio
(SIR) and $R_{n,\bar{\epsilon}}(\Omega)$ is as defined in Lemma 1.

For simplicity, we assume an interference-limited scenario where noise is
neglected and the SIR is given by
$\Omega=\frac{\mathcal{P}r_{o}^{-\eta}|h_{0}|^{2}}{\sum_{r_{i}\in\tilde{\Psi}/r_{o}}\mathcal{P}r_{i}^{-\eta}|h_{i}|^{2}}$.
Then, $P_{s}(R_{t},n,\bar{\epsilon})$ is given by

$\displaystyle
P_{s}(R_{t},n,\bar{\epsilon})=\mathbb{P}^{!}(R_{n,\bar{\epsilon}}(\Omega)>R_{t}|\tilde{\Psi})=\mathbb{P}^{!}\left(\log_{2}(1+\Omega)-\frac{\sqrt{V(\Omega)}Q^{-1}(\bar{\epsilon})}{\sqrt{n}}{+b_{n}}>R_{t}|\tilde{\Psi}\right)$
(28)

where $\mathbb{P}^{!}(\cdot)$ is the reduced Palm measure of the point process
[43],[44, Def. 8.8]. Note that $P_{s}(R_{t},n,\bar{\epsilon})$ is a random
variable which is a function of the stochasticity of the network. Its moments
are defined as

$\displaystyle\mathcal{M}_{d}=\mathbb{E}\left\\{P_{s}(R_{t},n,\bar{\epsilon})^{d}\right\\},$
(29)

and are used to calculate the meta distribution as follows [43]

$\displaystyle
F_{R_{t}}(p_{t})=\mathbb{P}(P_{s}(R_{t},n,\bar{\epsilon})>p_{t}){=\frac{1}{2}+\frac{1}{\pi}\int_{0}^{\infty}\frac{\mathcal{I}m(e^{-t\log(p_{t})}\mathcal{M}_{jt})}{t}dt},$
(30)

where $\mathcal{I}m(\cdot)$ is the imaginary part of a complex number and
$\mathcal{M}_{jt}$ is obtained at $d=jt$ in (29).

In [43], a simple yet tight approximating distribution of
$P_{s}(R_{t},n,\bar{\epsilon})$ is proposed for the AR, which is the beta
distribution given by

$\displaystyle
f(P_{s};R_{t})\approx\frac{P_{s}^{\frac{\vartheta(\beta+1)-1}{1-\vartheta}}(1-P_{s})^{\beta-1}}{\text{B}\left(\frac{\vartheta\beta}{1-\vartheta},\beta\right)},$
(31)

where $\text{B}(\cdot,\cdot)$ is the beta function,
$\vartheta=\mathcal{M}_{1}$ and the
$\beta=\frac{(\mathcal{M}_{1}-\mathcal{M}_{2})(1-\mathcal{M}_{1})}{\mathcal{M}_{2}-\mathcal{M}_{1}^{2}}$.
Hence, the meta distribution could be computed as the CCDF of the beta
distribution. However, $\mathcal{M}_{1}$ and $\mathcal{M}_{2}$ provided in
(29) are difficult to evaluate in the FBR, and hence, an approximation is
proposed in this paper. At high SIR, where the term
$\mathcal{K}(\Upsilon){=\sqrt{1-\frac{1}{(1+\Upsilon)^{2}}}}$ is approximately
equal to 1, the approximated probability of success is given by

(a) $n=128$

(b) $n=2048$

Figure 7: The outage probability versus $r_{0}$ at $\lambda\in\\{1,9\\}\
\mathrm{BS/km^{2}}$, $\bar{\epsilon}\in\\{10^{-2},10^{-6}\\}$, and $\alpha=0\
\text{dB}$.

$\displaystyle\hat{P}_{s}(R_{t},n,\bar{\epsilon})$
$\displaystyle=\mathbb{P}^{!}\left(\log_{2}(1+\Omega)-a_{n,\bar{\epsilon}}{+b_{n}}>R_{t}|\tilde{\Psi}\right).$

Therefore, the approximated probability of success can be expressed as a
function of the SIR instead of the coding rate, as follows,

$\displaystyle\hat{P}_{s}(R_{t},n,\bar{\epsilon})=\mathbb{P}^{!}\left(\Omega>2^{R_{t}+a_{n,\bar{\epsilon}}{-b_{n}}}-1|\tilde{\Psi}\right)=\prod_{r_{i}\in\tilde{\Psi}\setminus\\{r_{0}\\}}\frac{1}{1+(2^{R_{t}+a_{n,\bar{\epsilon}}{-b_{n}}}-1)\left(\frac{r_{0}}{r_{i}}\right)^{\eta}},$
(32)

The expression provided in (32) is obtained by averaging over the channel
gains. Then, its moments are given by

$\displaystyle\hat{\mathcal{M}}_{d}$
$\displaystyle=\mathbb{E}\left\\{\hat{P}_{s}(R_{t},n,\bar{\epsilon})^{d}\right\\}$
$\displaystyle=\mathbb{E}\left\\{\prod_{r_{i}\in\tilde{\Psi}\setminus\\{r_{0}\\}}\frac{1}{\left(1+(2^{R_{t}+a_{n,\bar{\epsilon}}{-b_{n}}}-1)\left(\frac{r_{0}}{r_{i}}\right)^{\eta}\right)^{d}}\right\\}$
$\displaystyle=\exp\left\\{-2\pi\lambda\int_{r_{0}}^{\infty}r\left(1-\frac{1}{(1+(2^{R_{t}+a_{n,\bar{\epsilon}}{-b_{n}}}-1)(r_{0}/r)^{\eta})^{d}}\right)dr\right\\},$
(33)

Thus, the meta distribution can be evaluated using the approximated moments in
(IV-C) as follows

$\displaystyle\hat{F}_{R_{t}}(p_{t})=\mathbb{P}(\hat{P}_{s}(R_{t},n,\bar{\epsilon})>p_{t})=\frac{1}{2}+\frac{1}{\pi}\int_{0}^{\infty}\frac{\mathcal{I}m(e^{-t\log(p_{t})}\hat{\mathcal{M}}_{jt})}{t}dt.$
(34)

Using the beta distribution approximation, the meta-distribution can be
approximated as

$\displaystyle\hat{F}_{R_{t}}(p_{t})\approx
1-I_{p_{t}}\left(\frac{\hat{\mathcal{M}}_{1}(\hat{\mathcal{M}}_{1}-\hat{\mathcal{M}}_{2})}{(\hat{\mathcal{M}}_{2}-\hat{\mathcal{M}}_{1}^{2})},\frac{(1-\hat{\mathcal{M}}_{1})(\hat{\mathcal{M}}_{1}-\hat{\mathcal{M}}_{2})}{(\hat{\mathcal{M}}_{2}-\hat{\mathcal{M}}_{1}^{2})}\right),$
(35)

where $I_{z}(x,y)=\frac{1}{\text{B}(x,y)}\int_{0}^{z}t^{x-1}(1-t)^{y-1}dt$ is
the regularized incomplete beta function.

(a) $\bar{\epsilon}=10^{-2}$

(b) $\bar{\epsilon}=10^{-6}$

Figure 8: The outage probability versus the rate threshold $R_{t}$ at
$\lambda=1\ \mathrm{BS/km^{2}}$, $n\in\\{128,\ 2048\\}$, and
$\frac{\mathcal{P}}{\sigma_{w}^{2}}=0\ \text{dB}$, averaged with respect to
Rayleigh distributed $r_{0}$.

###  IV-D Numerical Results

Now, we provide numerical results for the bounds and approximations of the
rate outage probability, reliability, and the coding rate meta-distribution.
We investigate the tightness of these approximations and compare them with
results from the literature for the AR.

First, in Fig. 7, the rate outage probability is plotted versus $r_{0}$ with
$\lambda\in\\{1,\ 9\\}\ \mathrm{BS/km^{2}}$, $\eta=4$,
$\bar{\epsilon}\in\\{10^{-2},\ 10^{-6}\\}$,
$\frac{\mathcal{P}}{\sigma_{w}^{2}}=0\ \text{dB}$, and $R_{t}=1$
bit/transmission. The PDF of $r_{0}$ is plotted to illustrate the probability
of occurrence of the $r_{0}$ values. The figure shows the Monte Carlo
simulated outage probability, in addition to the upper bound in (18) and the
outage probability in the AR (16) which serves as a lower bound (cf. Thm. 3).
It shows that the rate outage in the AR underestimates the rate outage
probability in the FBR and the gap to the actual outage probability increases
as the FER threshold decreases. This gap can be as large as $30\%$ in some
cases, such as with $r_{0}=200$ m, $\lambda=1$ BS/km2, $n=128$, and
$\bar{\epsilon}=10^{-6}$, where the actual outage probability is 0.13, but
(16) underestimates it as 0.1. Fig. 7 also shows that the upper bound of the
outage probability expression in (18) is fairly accurate to provide a
convenient approximation for a broad range of $n,\lambda$, and
$\bar{\epsilon}$. The outage probability is seen to increase as the FER
threshold $\bar{\epsilon}$ decreases because the term
$\sqrt{\frac{V(\Upsilon)}{n}}Q^{-1}(\bar{\epsilon})$ in (7) increases as
$\bar{\epsilon}$ decreases. However, at a specific $\bar{\epsilon}$, as $n$
increases, the outage probability approaches the outage probability in the AR
as expected. Thus, while the results in [12] can be a good representative of
the performance when $n$ is relatively large (Fig. 7(a)), this is not the case
when $n$ is small in which case our derived bounds and approximation are more
accurate. Moreover, in Fig. 8, the rate outage probability approximation for a
Rayleigh distributed $r_{0}$, provided in (24), is plotted versus the target
rate $R_{t}$ and compared to the simulated outage probability for BS density
$\lambda=1\ \text{BSs}/\text{km}^{2}$, blocklengths $n\in\\{128,\ 2048\\}$,
FER threshold $\bar{\epsilon}\in\\{10^{-2},\ 10^{-6}\\}$, and
$\frac{\mathcal{P}}{\sigma_{w}^{2}}=0\ \text{dB}$. This plot also proves the
tightness of the upper bound and the inaccuracy of the AR.

In Fig. 9(a), the rate outage probability under QAM constellations is shown
for various modulation orders versus $r_{0}$, where the rate outage for a
specific modulation scheme is defined as
$\mathcal{O}^{\text{\tiny(M)}}(r_{0},R_{t},n,\bar{\epsilon})=\mathbb{P}(\mathcal{R}_{n,\bar{\epsilon}}^{\text{\tiny(M)}}(\Upsilon)<R_{t}|r_{0})$.
This expression is computed using Monte Carlo simulation. The figure is
plotted for $n=128$, $\bar{\epsilon}=10^{-2}$, $\lambda=1$,
$\frac{\mathcal{P}}{\sigma_{w}^{2}}=0\ \text{dB}$, and
$R_{t}\in\\{0.825,1.85\\}$. Similar to the previous plots, the rate outage
probability $\mathcal{O}(r_{0},R_{t},n,\bar{\epsilon})$ acts as the lower
bound of the outage probability of all constellation sets. Also, the higher
order modulation (i.e. $M=16$) yields an outage probability lower than the
lower order modulation (i.e. $M=2$) at a fixed $R_{t}$ as the modulation order
limits the maximum transmission rate $(\log_{2}(M))$. However, at lower
$R_{t}$ as such $R_{t}=0.825$, most of the high order modulation schemes
provide a performance close to the theoretical lower bound
$\mathcal{O}(r_{0},R_{t},n,\bar{\epsilon})$. Fig. 9(b) shows the outage
probability for random $r_{0}$, i.e.,
$\bar{\mathcal{O}}^{\text{\tiny(M)}}(R_{t},n,\bar{\epsilon})=\mathbb{P}(\mathcal{R}_{n,\bar{\epsilon}}^{\text{\tiny(M)}}(\Upsilon)<R_{t})$
versus $R_{t}$. As shown, the rate outage probability increases as the rate
threshold $R_{t}$ increases, and it jumps to 1 at a certain $R_{t}$ for each
modulation order, which is when $R_{t}$ exceeds the maximum rate achievable by
the modulation order ($\log_{2}(M)$).

(a) Outage probability versus $r_{0}$, for $R_{t}\in\\{0.825,1.85\\}$

(b) Outage probability versus $R_{t}$

Figure 9: Outage probability using M-QAM at $\lambda=1\
\text{BS}/\text{km}^{2}$, $\bar{\epsilon}=10^{-2}$, $n=128$ and
$\frac{\mathcal{P}}{\sigma_{w}^{2}}=0\ \text{dB}$

The overall network reliability versus the FER threshold $\bar{\epsilon}$ is
evaluated in Fig. 10 for fixed and random $r_{0}$ for $n\in\\{128,2048\\}$ and
$R_{t}\in\\{0.1375,1,3.46\\}$. It can be observed that the reliability of the
network increases as the FER threshold increases until it reaches a maximum
around $\bar{\epsilon}=10^{-2}$ then it rapidly decays. This is because the
increase of $(1-\mathcal{O}(r_{0},R_{t},n,\bar{\epsilon}))$ with respect to
$\bar{\epsilon}$ is faster than the decrease of $(1-\bar{\epsilon})$ for small
$\bar{\epsilon}$. However, for large $\bar{\epsilon}$,
$(1-\mathcal{O}(r_{0},R_{t},n,\bar{\epsilon}))$ saturates to $1$ and
$(1-\bar{\epsilon})$ dominates and contributes to the decay of the reliability
at high $\bar{\epsilon}$. In other words, at $\bar{\epsilon}<10^{-2}$, the
rate of increase of $(1-\mathcal{O}(r_{0},R_{t},n,\bar{\epsilon}))$ is higher
than the rate of decay of $(1-\bar{\epsilon})$. However, at
$\bar{\epsilon}>10^{-2}$, the rate of increase of
$(1-\mathcal{O}(r_{0},R_{t},n,\bar{\epsilon}))$ is nearly zero and the rate of
decay of $(1-\bar{\epsilon})$ dominates. Therefore, the reliability rapidly
decays. It is shown that for a fixed $R_{t}$, a gap exists between curves
simulated under FBR and AR. This gap grows wider up to $0.15$ as FER threshold
$\bar{\epsilon}$ decreases because the short blocklengths (e.g. $n=128$) are
less reliable. Also, it is obvious that the reliability of the network at
random $r_{0}$ is lower than that at fixed $r_{0}$ as it averages the
performance of users which are close and those that are far away from the BS.
Moreover, as $R_{t}$ increases the approximated reliability becomes close to
the exact one. This is because the outage upper bound provided in Corollary 3
is tight for high SINR. Hence, from Def. 2, as the $R_{t}$ increases, only
high SINRs can achieve the desired rate and FER threshold at the given $n$,
which is where the approximation becomes tighter. Whereas, for low $R_{t}$,
low SINRs can also achieve the desired performance which loosens the
approximation.

(a) Fixed $r_{0}$: $\lambda=1$ $\text{BS}/\text{km}^{2}$,
$\frac{\mathcal{P}}{\sigma_{w}^{2}}=0\ \text{dB}$ and $r_{0}=250\ m$

(b) Random $r_{0}$: $\lambda=0.1$ $\text{BS}/\text{km}^{2}$ and
$\frac{\mathcal{P}}{\sigma_{w}^{2}}=10\ \text{dB}$

Figure 10: Reliability versus the FER threshold $\bar{\epsilon}$

Finally, we validate the coding rate meta distribution in Fig. 11(a). The
coding rate meta distribution provided in (30) is simulated via Monte-Carlo
simulations at $\lambda=1\ \text{BSs}/\text{km}^{2}$, $r_{0}=\ 150\ m$, an
area of $500\ \text{km}^{2}$, a blocklength $n=128$, FER threshold
$\bar{\epsilon}=10^{-5}$, and a rate threshold
$R_{t}=\\{0.1375,0.3964,1,2.0574,3.4594\\}$. It is observed that there is a
significant gap between the performance in the AR and the FBR, showing that
the meta distribution of the coding rate in the AR overestimates the
performance of the network in the FBR. This gap can reach up to $0.09$ at
$R_{t}=3.46$ and $p_{t}=0.9$, which means $9\%$ of the users predicted to
achieve the required performance in the AR will not achieve it in the FBR. It
can also be observed that the gap increases as $R_{t}$ or $p_{t}$ increases.
Thus, the AR analysis in [43, 45, 46] cannot be used to provide precise
results in the FBR. For the sake of preciseness of the results, we have
simulated the coding rate meta distribution including the noise term with
noise power $N_{0}=-90\ \rm dBm$ [47]. The results show that the network is an
interference-limited network and the noise can be neglected for the simplicity
of the analysis. In Fig. 11(b), the approximation proposed in (32) is shown to
be very tight compared to the exact coding rate meta distribution in (30) for
different $r_{0}$. The tightness of the approximation increases by decreasing
$r_{0}$ or alternatively increasing SIR as the term
$\sqrt{1-\frac{1}{(1+\Omega)^{2}}}$ converges to $1$ as the SIR increases.
Using this approximation it is easier to find an expression for the moments of
the success probability and use it to evaluate the meta distribution using the
beta distribution provided in (35). Hence, the beta approximation of the meta-
distribution provides a close performance to the exact and approximated meta-
distribution.

(a) The FBR versus the AR at $n=128$ and $\bar{\epsilon}=10^{-5}$

(b) Exact versus Approximation at $r_{0}\in\\{150,\ 250,\ 500\\}m$ at
$R_{t}=1$, $n=128$, and $\bar{\epsilon}=10^{-2}$

Figure 11: The meta distribution of the coding rate versus $p_{t}$

Thus, we saw that our proposed expressions provide an accurate representation
of the rate outage probability and the coding rate meta distribution in the
FBR, which is more accurate than using AR expressions from the literature.

##  V Conclusion

This paper investigates different aspects of the performance of a large-scale
DL network in the FBR using the theory of coding in the FBR in conjunction
with stochastic geometric tools. We start with rate analysis where we derive
the average coding rate using Gaussian codebooks and $M$-ary QAM
constellation, and investigate the practical scheme of MLPCM in the FBR
showing that its performance approaches the theoretical benchmarks. We also
study the rate outage probability, reliability, and the coding rate meta
distribution, where we provide bounds and approximations. From the results, we
conclude that the performance analysis provided under an AR provides a
misleading (overestimated) performance and cannot be used to characterize the
performance in the FBR. The performance in the FBR provides accurate rate and
FER characterization, and approaches the AR performance at sufficiently large
blocklengths $>2048$ and high FER $>10^{-2}$. The developed expressions for
the average coding rate, the rate outage probability, and the coding rate meta
distribution are fairly tight as shown by our numerical results. In
conclusion, the results are relevant for the theoretical analysis of large-
scale networks in the FBR, and the theoretical performance can be approached
using practical transmission schemes such as MLPCM. The results in this paper
can be extended to characterize the performance of networks under different
multiple access schemes as in [48, 18, 17].

##  Appendix A The Average Square Root Channel Dispersion
$\mathcal{V}(\alpha_{0})$

From the definition of channel dispersion provided in (2), the average square
root channel dispersion $\mathcal{V}(\alpha_{0})$ can be written as follows

$\displaystyle\mathcal{V}({\alpha_{0}})=\mathbb{E}\left\\{\sqrt{V(\Upsilon)}\right\\}$
$\displaystyle=\mathbb{E}\left\\{\log_{2}(e)\sqrt{\left(1-\frac{1}{(\Upsilon+1)^{2}}\right)}\right\\}$
$\displaystyle=\int_{0}^{\infty}(1-\mathbb{F}_{\sqrt{V}}({v}))d{v},$ (36)

where $\mathbb{F}_{\sqrt{V}}(v)$ is the cumulative distribution function (CDF)
of $\sqrt{V(\Upsilon)}$, and the last step follows as an application of
Fubini’s theorem [49]. The CDF of $\sqrt{V(\Upsilon)}$ for a given
$\mathcal{B}_{g}$ is given by

$\displaystyle\mathbb{F}_{\sqrt{V}}({v}|\mathcal{B}_{g})$
$\displaystyle=\mathbb{P}(\sqrt{V(\Upsilon)}<{v}\big{|}\mathcal{B}_{g})$
$\displaystyle=\mathbb{P}\left(|h_{0}|^{2}<\left(\frac{1}{{\alpha_{0}}}+\frac{\mathcal{B}_{g}}{\mathcal{P}r_{0}^{-\eta}}\right){z(v)}\bigg{|}\mathcal{B}_{g}\right)$
$\displaystyle=1-\exp\left(-\left(\frac{1}{{\alpha_{0}}}+\frac{\mathcal{B}_{g}}{\mathcal{P}r_{0}^{-\eta}}\right){z(v)}\right),$
(37)

where the last step follows from the exponential distribution of
$|h_{0}|^{2}$. Averaging with respect to the interference term
$\mathcal{B}_{g}$ yields
$\mathbb{E}_{\mathcal{B}_{g}}\\{\mathbb{F}_{\sqrt{V}}({v}|\mathcal{B}_{g})\\}=\mathbb{E}_{\mathcal{B}_{g}}\left\\{1-\exp\left(\frac{-{z(v)}}{{\alpha_{0}}}\right)\exp\left(-\frac{\mathcal{B}_{g}r_{0}^{\eta}}{\mathcal{P}}{z(v)}\right)\right\\}$.
Hence, the CDF of the square root of the channel dispersion is given as

$\displaystyle\mathbb{F}_{\sqrt{V}}({v})=1-\exp\left(\frac{-{z(v)}}{{\alpha_{0}}}\right)\mathcal{L}_{\mathcal{B}_{g}}\left\\{\frac{r_{0}^{\eta}}{\mathcal{P}}{z(v)}\right\\},\
\ \ \ \ \ \ \ 0\leq v<\log_{2}(e),$ (38)

which is obtained using the definition of Laplace transform
$\left(\mathcal{L}_{x}\\{u\\}=\mathbb{E}\left\\{e^{-ux}\right\\}\right)$. By
substituting (38) in (36), we obtain

$\displaystyle\mathcal{V}(\alpha_{0})=\int_{0}^{\log_{2}(e)}\exp\left(\frac{-{z(v)}}{{\alpha_{0}}}\right)\mathcal{L}_{\mathcal{B}_{g}}\left\\{\frac{r_{0}^{\eta}}{\mathcal{P}}{z(v)}\right\\}d{v},$

as defined in Theorem 1.

##  Appendix B The Channel Dispersion for A Large-Scale Network Under a Finite
Constellation

The channel dispersion is defined as the conditional variance of the
information density conditioned on the input distribution as follows

$\displaystyle V_{\text{\tiny M}}(\Upsilon)$
$\displaystyle=\mathbb{E}_{S}\left\\{\mathbb{V}ar\left\\{\log_{2}\frac{P_{Y|S}(y|s)}{P_{Y}(y)}\mid
S\right\\}\right\\}$
$\displaystyle=\frac{1}{M}\sum_{m=1}^{M}\mathbb{E}\left\\{\log_{2}^{2}\frac{P_{Y|S}(y|\underline{s}_{m})}{P_{Y}(y)}\right\\}-\frac{1}{M}\sum_{m=1}^{M}\mathbb{E}\left\\{\log_{2}\frac{P_{Y|S}(y|\underline{s}_{m})}{P_{Y}(y)}\right\\}^{2}.$

where
$\underline{s}_{m}=[\mathcal{R}e\\{s_{m}\\},\mathcal{I}m\\{s_{m}\\}]^{T}$ is
the 2-D real-valued vector form of the complex-valued symbol $s_{m}$ of an
$M$-ary constellation.

Define
$I_{1}=\mathbb{E}\left\\{\log_{2}^{2}\frac{P_{Y|S}(y|\underline{s}_{m})}{P_{Y}(y)}\right\\}$
and
$I_{2}=\mathbb{E}\left\\{\log_{2}\frac{P_{Y|S}(y|\underline{s}_{m})}{P_{Y}(y)}\right\\}^{2}$.
The first term $I_{1}$ is derived as follows

$\displaystyle
I_{1}=\mathbb{E}\left\\{\log_{2}^{2}P_{Y|S}(y|\underline{s}_{m})\right\\}-2\mathbb{E}\left\\{\log_{2}P_{Y|S}(y|\underline{s}_{m})\log_{2}P_{Y}(y)\right\\}+\mathbb{E}\left\\{\log_{2}^{2}P_{Y}(y)\right\\}=I_{11}-2I_{12}+I_{13}.$

The term $I_{11}$ is given by

$\displaystyle
I_{11}=\int_{{\mathbb{R}^{2}}}\frac{e^{-\frac{\|{\underline{y}}-h_{0}\sqrt{\mathcal{P}}r_{0}^{-\eta/2}{\underline{s}_{m}}\|^{2}}{N_{0}+\mathcal{B}}}}{{\pi(N_{0}+\mathcal{B})}}\log_{2}^{2}\frac{e^{-\frac{\|{\underline{y}}-h_{0}\sqrt{\mathcal{P}}r_{0}^{-\eta/2}{\underline{s}_{m}}\|^{2}}{N_{0}+\mathcal{B}}}}{{\pi(N_{0}+\mathcal{B})}}\
d{\underline{y}},$

where $\underline{y}=[y_{1},y_{2}]^{T}$. Define
${\underline{t}}=\frac{{\underline{y}}-h_{0}\sqrt{\mathcal{P}}r_{0}^{-\eta/2}{\underline{s}_{m}}}{\sqrt{N_{0}+\mathcal{B}}}$,
hence by substitution, $I_{11}$ is given by

$\displaystyle I_{11}$
$\displaystyle=\int_{{\mathbb{R}^{2}}}\frac{e^{-\|{\underline{t}}\|^{2}}}{\pi}\left(\log_{2}^{2}\frac{1}{\pi(N_{0}+\mathcal{B})}+2\|{\underline{t}}\|^{2}\log_{2}\left(e\right)\log_{2}\frac{1}{\pi(N_{0}+\mathcal{B})}+\log_{2}^{2}e^{\|{\underline{t}}\|^{2}}\right)d{\underline{t}}$
$\displaystyle=\log_{2}^{2}\frac{1}{\pi(N_{0}+\mathcal{B})}-2\log_{2}(e)\log_{2}\frac{1}{\pi(N_{0}+\mathcal{B})}+3\log_{2}^{2}(e).$

The term $I_{12}$ is given by

$\displaystyle
I_{12}=\int_{{\mathbb{R}^{2}}}\frac{e^{-\frac{\|{\underline{y}}-h_{0}\sqrt{\mathcal{P}}r_{0}^{\eta/2}{\underline{s}_{m}}\|^{2}}{(N_{0}+\mathcal{B})}}}{\pi(N_{0}+\mathcal{B})}\log_{2}\left(\frac{1}{M}\sum_{l=1}^{M}\frac{e^{-\frac{\|{\underline{y}}-h_{0}\sqrt{\mathcal{P}}r_{0}^{\eta/2}{\underline{s}_{l}}\|^{2}}{(N_{0}+\mathcal{B)}}}}{\pi(N_{0}+\mathcal{B})}\right)\log_{2}\left(\frac{e^{\frac{-\|{\underline{y}}-h_{0}\sqrt{\mathcal{P}}r_{0}^{\eta/2}{\underline{s}_{m}}\|^{2}}{(N_{0}+\mathcal{B)}}}}{\pi(N_{0}+\mathcal{B})}\right)d{\underline{y}},$

and is divided into four terms as
$I_{12}=I_{12}^{(1)}+I_{12}^{(2)}+I_{12}^{(3)}+I_{12}^{(4)}$, where

$\displaystyle I_{12}^{(1)}$
$\displaystyle=\int_{{\mathbb{R}^{2}}}\frac{e^{-\frac{\|{\underline{y}}-h_{0}\sqrt{\mathcal{P}}r_{0}^{\eta/2}{\underline{s}_{m}}\|^{2}}{(N_{0}+\mathcal{B})}}}{\pi(N_{0}+\mathcal{B})}\log_{2}\frac{1}{\pi
M(N_{0}+\mathcal{B})}\log_{2}\frac{1}{\pi(N_{0}+\mathcal{B})}d{\underline{y}}\hskip
170.71652pt$ $\displaystyle=\log_{2}\frac{1}{\pi
M(N_{0}+\mathcal{B})}\log_{2}\frac{1}{\pi(N_{0}+\mathcal{B})},$

$\displaystyle I_{12}^{(2)}$
$\displaystyle=\int_{{\mathbb{R}^{2}}}\frac{e^{-\frac{\|{\underline{y}}-h_{0}\sqrt{\mathcal{P}}r_{0}^{\eta/2}{\underline{s}_{m}}\|^{2}}{(N_{0}+\mathcal{B})}}}{\pi(N_{0}+\mathcal{B})}\log_{2}\frac{1}{\pi
M(N_{0}+\mathcal{B})}\log_{2}e^{\frac{-\|{\underline{y}}-h_{0}\sqrt{\mathcal{P}}r_{0}^{\eta/2}{\underline{s}_{m}}\|^{2}}{(N_{0}+\mathcal{B})}}d{\underline{y}}\hskip
56.9055pt\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ $
$\displaystyle=-\log_{2}(e)\log_{2}\frac{1}{\pi M(N_{0}+\mathcal{B})},$
$\displaystyle I_{12}^{(3)}$
$\displaystyle=\int_{{\mathbb{R}^{2}}}\frac{e^{-\frac{\|{\underline{y}}-h_{0}\sqrt{\mathcal{P}}r_{0}^{\eta/2}{\underline{s}_{m}}\|^{2}}{(N_{0}+\mathcal{B})}}}{\pi(N_{0}+\mathcal{B})}\log_{2}\frac{1}{\pi(N_{0}+\mathcal{B})}\log_{2}\sum_{l=1}^{M}e^{-\frac{\|{\underline{y}}-h_{0}\sqrt{\mathcal{P}}r_{0}^{\eta/2}{\underline{s}_{l}}\|^{2}}{(N_{0}+\mathcal{B})}}\
d{\underline{y}}\hskip 170.71652pt$
$\displaystyle\stackrel{{\scriptstyle\text{(i)}}}{{=}}\frac{1}{\pi}\int_{{\mathbb{R}^{2}}}e^{-\|\underline{t}\|^{2}}\log_{2}\frac{1}{\pi(N_{0}+\mathcal{B})}\log_{2}\sum_{l=1}^{M}e^{-\|\underline{t}\|^{2}-2\underline{t}^{T}(\underline{s}_{m}-\underline{s}_{l})\frac{h_{0}\sqrt{\mathcal{P}}r_{0}^{-\eta/2}}{\sqrt{N_{0}+\mathcal{B}}}-\frac{|h_{0}|^{2}{\mathcal{P}}r_{0}^{-\eta}}{{N_{0}+\mathcal{B}}}\left\|\underline{s}_{m}-\underline{s}_{l}\right\|^{2}}d\underline{t}$
$\displaystyle\stackrel{{\scriptstyle\text{(ii)}}}{{=}}\frac{1}{\pi}\int_{{\mathbb{R}^{2}}}e^{-\|\underline{t}\|^{2}}\log_{2}\frac{1}{\pi(N_{0}+\mathcal{B})}\log_{2}\sum_{l=1}^{M}e^{-\left(\|\underline{t}\|^{2}+2\sqrt{\Upsilon}\underline{t}^{T}\underline{d}_{ml}+\Upsilon\left\|\underline{d}_{ml}\right\|^{2}\right)}d\underline{t}$
$\displaystyle=-\log_{2}(e)\log_{2}\frac{1}{\pi(N_{0}+\mathcal{B})}+\mathbb{E}\left\\{\log_{2}\frac{1}{\pi(N_{0}+\mathcal{B})}\log_{2}\sum_{l=1}^{M}e^{-2\sqrt{\Upsilon}\underline{t}^{T}\underline{d}_{ml}-\Upsilon\left\|\underline{d}_{ml}\right\|^{2}}\right\\},$

where $(\rm i)$ follows from a change of variable
${\underline{t}}=\frac{{\underline{y}}-h_{0}\sqrt{\mathcal{P}}r_{0}^{-\eta/2}{\underline{s}_{m}}}{\sqrt{N_{0}+\mathcal{B}}}$,
and $(\rm ii)$ follows by using
$\underline{d}_{ml}=\underline{s}_{m}-\underline{s}_{l}$ and
$\Upsilon=\frac{|h_{0}|^{2}\mathcal{P}r_{0}^{-\eta}}{{N_{0}+\mathcal{B}}}$,
and $I_{12}^{(4)}$ is given by

$\displaystyle I_{12}^{(4)}$
$\displaystyle=\int_{{\mathbb{R}^{2}}}\frac{e^{-\frac{\|{\underline{y}}-h_{0}\sqrt{\mathcal{P}}r_{0}^{\eta/2}{\underline{s}_{m}}\|^{2}}{(N_{0}+\mathcal{B})}}}{\pi(N_{0}+\mathcal{B})}\log_{2}e^{\frac{-\|{\underline{y}}-h_{0}\sqrt{\mathcal{P}}r_{0}^{\eta/2}{\underline{s}_{m}}\|^{2}}{(N_{0}+\mathcal{B})}}\log_{2}\sum_{l=1}^{M}e^{-\frac{\|{\underline{y}}-h_{0}\sqrt{\mathcal{P}}r_{0}^{\eta/2}{\underline{s}_{l}}\|^{2}}{(N_{0}+\mathcal{B})}}d{\underline{y}}\hskip
85.35826pt$
$\displaystyle\stackrel{{\scriptstyle\text{(i)}}}{{=}}\int_{{\mathbb{R}^{2}}}\frac{e^{-\|\underline{t}\|^{2}}}{\pi}\log_{2}e^{-\|\underline{t}\|^{2}}\log_{2}\sum_{l=1}^{M}e^{-\left(\|\underline{t}\|^{2}+2\underline{t}^{T}\underline{d}_{ml}\sqrt{\Upsilon}+\Upsilon\left\|\underline{d}_{ml}\right\|^{2}\right)}d{\underline{t}}$
$\displaystyle=3\log_{2}^{2}(e)-\log_{2}(e)\mathbb{E}_{\underline{t}}\left\\{\|\underline{t}\|^{2}\log_{2}\sum_{l=1}^{M}e^{-2\sqrt{\Upsilon}\underline{t}^{T}\underline{d}_{ml}-\Upsilon\|\underline{d}_{ml}\|^{2}}\right\\},$

where $(\rm i)$ is obtained by the substitution of $\underline{t}$,
$\underline{d}_{ml}$, and $\Upsilon$.

The term $I_{13}$ is given by

$\displaystyle
I_{13}=\int_{{\mathbb{R}^{2}}}\frac{e^{-\frac{\|{\underline{y}}-h_{0}\sqrt{\mathcal{P}}r_{0}^{\eta/2}{\underline{s}_{m}}\|^{2}}{(N_{0}+\mathcal{B})}}}{\pi(N_{0}+\mathcal{B})}\left(\log_{2}\frac{1}{M\pi(N_{0}+\mathcal{B})}+\log_{2}\sum_{l=1}^{M}e^{\frac{-\|{\underline{y}}-h_{0}\sqrt{\mathcal{P}}r_{0}^{\eta/2}{\underline{s}_{l}}\|^{2}}{(N_{0}+\mathcal{B})}}\right)^{2}d{\underline{y}},$

and is divided into three terms as
$I_{13}=I_{13}^{(1)}+I_{13}^{(2)}+I_{13}^{(3)}$, where

$\displaystyle I_{13}^{(1)}$
$\displaystyle=\int_{{\mathbb{R}^{2}}}\frac{e^{-\frac{\|{\underline{y}}-h_{0}\sqrt{\mathcal{P}}r_{0}^{\eta/2}{\underline{s}_{m}}\|^{2}}{(N_{0}+\mathcal{B})}}}{\pi(N_{0}+\mathcal{B})}\log_{2}^{2}\frac{1}{\pi
M(N_{0}+\mathcal{B})}\ d{\underline{y}}\hskip 85.35826pt\ \ \ \ \ \ \ \ \ \ \
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ $
$\displaystyle=\log_{2}^{2}\frac{1}{\pi M(N_{0}+\mathcal{B})},$ $\displaystyle
I_{13}^{(2)}$
$\displaystyle={2}\int_{{\mathbb{R}^{2}}}\frac{e^{-\frac{\|{\underline{y}}-h_{0}\sqrt{\mathcal{P}}r_{0}^{\eta/2}{\underline{s}_{m}}\|^{2}}{(N_{0}+\mathcal{B})}}}{\pi(N_{0}+\mathcal{B})}\log_{2}\frac{1}{\pi
M(N_{0}+\mathcal{B})}\log_{2}\sum_{l=1}^{M}e^{\frac{-\|{\underline{y}}-h_{0}\sqrt{\mathcal{P}}r_{0}^{\eta/2}{\underline{s}_{l}}\|^{2}}{(N_{0}+\mathcal{B})}}\
d{\underline{y}}$ $\displaystyle=-2\log_{2}(e)\log_{2}\frac{1}{\pi
M(N_{0}+\mathcal{B})}+2\log_{2}\frac{1}{\pi
M(N_{0}+\mathcal{B})}\mathbb{E}\left\\{\log_{2}\sum_{l=1}^{M}e^{-2\sqrt{\Upsilon}{\underline{t}^{T}}\underline{d}_{ml}-\Upsilon\|\underline{d}_{ml}\|^{2}}\right\\},$
$\displaystyle I_{13}^{(3)}$
$\displaystyle=\int_{{\mathbb{R}^{2}}}\frac{1}{\pi(N_{0}+\mathcal{B})}e^{-\frac{\|{\underline{y}}-h_{0}\sqrt{\mathcal{P}}r_{0}^{\eta/2}{\underline{s}_{m}}\|^{2}}{(N_{0}+\mathcal{B})}}\log_{2}^{2}\sum_{l=1}^{M}e^{\frac{-\|{\underline{y}}-h_{0}\sqrt{\mathcal{P}}r_{0}^{\eta/2}{\underline{s}_{l}}\|^{2}}{(N_{0}+\mathcal{B})}}\
d{\underline{y}}$ $\displaystyle=3\log_{2}^{2}(e)-2\log_{2}(e)\
\mathbb{E}\left\\{\|{\underline{t}}\|^{2}\log_{2}\sum_{l=1}^{M}e^{-2\sqrt{\Upsilon}{\underline{t}^{T}}\underline{d}_{ml}-\Upsilon\|\underline{d}_{ml}\|^{2}}\right\\}+\mathbb{E}\left\\{\log_{2}\sum_{l=1}^{M}e^{-2\sqrt{\Upsilon}{\underline{t}^{T}}\underline{d}_{ml}-\Upsilon\|\underline{d}_{ml}\|^{2}}\right\\},$

By adding $I_{11}$, $I_{12}$, and $I_{13}$, $I_{1}$ is given by

$\displaystyle
I_{1}=\log_{2}^{2}\frac{1}{M}+2\log_{2}\frac{1}{M}\mathbb{E}\left\\{\log_{2}\sum_{l=1}^{M}e^{-2\sqrt{\Upsilon}{\underline{t}^{T}}{\underline{d}_{ml}}-\Upsilon\|{\underline{d}_{ml}}\|^{2}}\right\\}+\mathbb{E}\left\\{\log_{2}^{2}\sum_{l=1}^{M}e^{-2\sqrt{\Upsilon}{\underline{t}^{T}}{\underline{d}_{ml}}-\Upsilon\|{\underline{d}_{ml}}\|^{2}}\right\\},$

The second term $I_{2}$ is the square of the relative entropy given as

$\displaystyle
I_{2}=\log_{2}^{2}(M)-2\log_{2}(M)\mathbb{E}\left\\{\log_{2}\sum_{l=1}^{M}e^{-2\sqrt{\Upsilon}{\underline{t}^{T}}{\underline{d}_{ml}}-\Upsilon\|{\underline{d}_{ml}}\|^{2}}\right\\}+\mathbb{E}^{2}\left\\{\log_{2}\sum_{l=1}^{M}e^{-2\sqrt{\Upsilon}{\underline{t}^{T}}{\underline{d}_{ml}}-\Upsilon\|{\underline{d}_{ml}}\|^{2}}\right\\}.$

To compute the average square-root channel dispersion, $I_{2}$ is subtracted
from $I_{1}$ and then averaged over all constellation symbols, leading to the
following expression as indicated in (13)

$\displaystyle V_{\text{\tiny
M}}(\Upsilon)=\frac{1}{M}\sum_{m=1}^{M}\left\\{\mathbb{E}_{\underline{t}}\left\\{\log_{2}^{2}\sum_{l=1}^{M}e^{-2\sqrt{\Upsilon}{\underline{t}^{T}}{\underline{d}_{ml}}-\Upsilon\|{\underline{d}_{ml}}\|^{2}}\right\\}-\mathbb{E}_{\underline{t}}\left\\{\log_{2}\sum_{l=1}^{M}e^{-2\sqrt{\Upsilon}{\underline{t}^{T}}{\underline{d}_{ml}}-\Upsilon\|{\underline{d}_{ml}}\|^{2}}\right\\}^{2}\right\\}.$

##  Appendix C The SINR distribution at a Fixed $r_{0}$

The distribution of the SINR $\Upsilon$ at a fixed $r_{0}$ is derived by
defining the CDF as follows

$\displaystyle\mathbb{F}_{\Upsilon}(\Upsilon|r_{0})$
$\displaystyle=\mathbb{P}\left(|h_{0}|^{2}<\frac{\Upsilon(\mathcal{B}+\sigma_{w}^{2})}{\mathcal{P}r_{0}^{-\eta}}\right)$
$\displaystyle=1-\mathbb{E}\left\\{e^{\frac{-\Upsilon\mathcal{B}}{\mathcal{P}r_{0}^{-\eta}}}\right\\}e^{\frac{-\Upsilon\sigma_{w}^{2}}{\mathcal{P}r_{0}^{-\eta}}}$
$\displaystyle=1-\mathcal{L}_{{\mathcal{B}}}\left\\{\frac{\Upsilon}{\mathcal{P}r_{0}^{-\eta}}\right\\}e^{\frac{-\Upsilon\sigma_{w}^{2}}{\mathcal{P}r_{0}^{-\eta}}},$

using the Gamma approximation provided in (9). The PDF of the SINR $\Upsilon$
at a fixed $r_{0}$ is given by differentiating the CDF as follows

$\displaystyle
f(\Upsilon|r_{0})=\frac{e^{\frac{-\Upsilon\sigma_{w}^{2}}{\mathcal{P}r_{0}^{-\eta}}}}{\mathcal{P}r_{0}^{-\eta}}\frac{(1+\frac{\theta
r_{0}^{\eta}\Upsilon}{\mathcal{P}})\sigma_{w}^{2}+q\theta}{(1+\frac{\theta
r_{0}^{\eta}\Upsilon}{\mathcal{P}})^{q+1}},$ (39)

this proves (14).

## References

  * [1] N. Hesham and A. Chaaban, “On the performance of large-scale wireless networks in the finite block-length regime,” in _ICC 2021 - IEEE Int. Conf. Commun._ , 2021, pp. 1–6.
  * [2] Z. Ma, M. Xiao, Y. Xiao, Z. Pang, H. V. Poor, and B. Vucetic, “High-reliability and low-latency wireless communication for internet of things: Challenges, fundamentals, and enabling technologies,” _IEEE Internet of Things Journal_ , vol. 6, no. 5, pp. 7946–7970, 2019.
  * [3] H. Yu, H. Lee, and H. Jeon, “What is 5G? emerging 5G mobile services and network requirements,” _Sustainability_ , vol. 9, no. 10, p. 1848, 2017.
  * [4] A. J. Pinheiro, J. de M. Bezerra, C. A. Burgardt, and D. R. Campelo, “Identifying IoT devices and events based on packet length from encrypted traffic,” _Comput. Commun._ , vol. 144, pp. 8 – 17, 2019.
  * [5] W. Khalid, H. Yu, R. Ali, and R. Ullah, “Advanced physical-layer technologies for beyond 5G wireless communication networks,” _Sensors_ , vol. 21, no. 9, 2021. [Online]. Available: https://www.mdpi.com/1424-8220/21/9/3197
  * [6] Y. Polyanskiy, H. V. Poor, and S. Verdú, “Channel coding rate in the finite blocklength regime,” _IEEE Trans. Inf. Theory_ , vol. 56, no. 5, pp. 2307–2359, 2010.
  * [7] W. Yang, G. Durisi, T. Koch, and Y. Polyanskiy, “Block-fading channels at finite blocklength,” in _Int. Symp. Wireless Commun. Sys._ , 2013, pp. 1–4.
  * [8] H. Q. Nguyen, F. Baccelli, and D. Kofman, “A stochastic geometry analysis of dense IEEE 802.11 networks,” in _IEEE Int. Conf. Comput. Commun._ , 2007, pp. 1199–1207.
  * [9] M. Haenggi, J. G. Andrews, F. Baccelli, O. Dousse, and M. Franceschetti, “Stochastic geometry and random graphs for the analysis and design of wireless networks,” _IEEE J. Sel. Areas Commun._ , vol. 27, no. 7, pp. 1029–1046, 2009.
  * [10] C.-H. Lee, C.-Y. Shih, and Y.-S. Chen, “Stochastic geometry based models for modeling cellular networks in urban areas,” _Wireless Netw._ , vol. 19, no. 6, pp. 1063–1072, 2013.
  * [11] Y. Hmamouche, M. Benjillali, S. Saoudi, H. Yanikomeroglu, and M. D. Renzo, “New trends in stochastic geometry for wireless networks: A tutorial and survey,” _Proc. IEEE_ , vol. 109, no. 7, pp. 1200–1252, 2021.
  * [12] H. ElSawy, A. Sultan-Salem, M.-S. Alouini, and M. Z. Win, “Modeling and analysis of cellular networks using stochastic geometry: A tutorial,” _IEEE Commun. Surveys Tuts._ , vol. 19, no. 1, pp. 167–203, 2016.
  * [13] H. ElSawy, E. Hossain, and M. Haenggi, “Stochastic geometry for modeling, analysis, and design of multi-tier and cognitive cellular wireless networks: A survey,” _IEEE Commun. Surveys Tuts._ , vol. 15, no. 3, pp. 996–1019, 2013.
  * [14] C. E. Shannon, “Coding theorems for a discrete source with a fidelity criterion,” _IRE Nat. Conv. Rec_ , vol. 4, no. 142-163, p. 1, 1959.
  * [15] Y. Hu, J. Gross, and A. Schmeink, “On the capacity of relaying with finite blocklength,” _IEEE Trans. Veh. Technol._ , vol. 65, no. 3, pp. 1790–1794, 2015.
  * [16] E. Dosti, M. Shehab, H. Alves, and M. Latva-aho, “On the performance of non-orthogonal multiple access in the finite blocklength regime,” _Ad Hoc Networks_ , vol. 84, pp. 148 – 157, 2019. [Online]. Available: http://www.sciencedirect.com/science/article/pii/S157087051830708X
  * [17] Y. Xu, Y. Mao, O. Dizdar, and B. Clerckx, “Rate-splitting multiple access with finite blocklength for short-packet and low-latency downlink communications,” _arXiv preprint arXiv:2105.06198_ , 2021.
  * [18] J. Feng, H. Q. Ngo, and M. Matthaiou, “Coherent MU-MIMO in block fading channels: A finite blocklength analysis,” in _IEEE Int. Conf. Commun. Workshops (ICC Workshops)_ , 2020, pp. 1–6.
  * [19] J. Östman, A. Lancho, G. Durisi, and L. Sanguinetti, “URLLC with massive MIMO: Analysis and design at finite blocklength,” _IEEE Trans. Wireless Commun._ , vol. 20, no. 10, pp. 6387–6401, 2021.
  * [20] A. Lancho, G. Durisi, and L. Sanguinetti, “Cell-free massive MIMO with short packets,” in _IEEE Int. Workshop Signal Process. Adv. Wireless Commun. (SPAWC)_ , 2021, pp. 416–420.
  * [21] H. Ren, C. Pan, Y. Deng, M. Elkashlan, and A. Nallanathan, “Joint power and blocklength optimization for URLLC in a factory automation scenario,” _IEEE Trans. Wireless Commun._ , vol. 19, no. 3, pp. 1786–1801, 2020.
  * [22] W. J. Ryu and S. Y. Shin, “Power allocation for URLLC using finite blocklength regime in downlink NOMA systems,” in _Int. Conf. Inf. Commun. Technol. Convergence (ICTC)_ , 2019, pp. 770–773.
  * [23] C. Shen, T.-H. Chang, H. Xu, and Y. Zhao, “Joint uplink and downlink transmission design for URLLC using finite blocklength codes,” in _Int. Symp. Wireless Commun. Sys. (ISWCS)_ , 2018, pp. 1–5.
  * [24] Z. Wang, T. Lv, Z. Lin, J. Zeng, and P. T. Mathiopoulos, “Outage performance of URLLC NOMA systems with wireless power transfer,” _IEEE Wireless Commun. Lett._ , vol. 9, no. 3, pp. 380–384, 2020.
  * [25] H. Ren, C. Pan, Y. Deng, M. Elkashlan, and A. Nallanathan, “Resource allocation for secure URLLC in mission-critical IoT scenarios,” _IEEE Trans. Commun._ , vol. 68, no. 9, pp. 5793–5807, 2020.
  * [26] M. Seidl, A. Schenk, C. Stierstorfer, and J. B. Huber, “Polar-coded modulation,” _IEEE Trans. Commun._ , vol. 61, no. 10, pp. 4108–4119, 2013\.
  * [27] H. Khoshnevis, I. Marsland, and H. Yanikomeroglu, “Throughput-based design for polar-coded modulation,” _IEEE Trans. Commun._ , vol. 67, no. 3, pp. 1770–1782, 2019.
  * [28] T. Koike-Akino, Y. Wang, D. S. Millar, K. Kojima, and K. Parsons, “Polar coding for multilevel shaped constellations,” in _OSA Signal Process. Photonic Commun._ , 2018, p. SpW1G.1.
  * [29] H. Khoshnevis, “Multilevel polar coded-modulation for wireless communications,” Ph.D. dissertation, Carleton University, 2018.
  * [30] 3GPP, “Technical Specification Group Radio Access Network; NR; Multiplexing and channel coding,” 3rd Generation Partnership Project (3GPP), TS 38.212, Release 16, V16.8.0, Jan. 2022.
  * [31] J. Dai, D. Zhang, K. Niu, Z. Si, P. Zhang, S. Wang, Y. Yuan _et al._ , “Generalized polarization transform: A novel coded transmission paradigm,” _arXiv preprint arXiv:2110.12224_ , 2021.
  * [32] B. Tomasi, F. Gabry, V. Bioglio, I. Land, and J.-C. Belfiore, “Low-complexity receiver for multi-level polar coded modulation in non-orthogonal multiple access,” in _2017 IEEE Wireless Communications and Networking Conference Workshops (WCNCW)_ , 2017, pp. 1–6.
  * [33] E. Malkamaki and H. Leib, “Performance of truncated type-II hybrid ARQ schemes with noisy feedback over block fading channels,” _IEEE Trans. Commun._ , vol. 48, no. 9, pp. 1477–1487, 2000.
  * [34] M. D. Renzo and W. Lu, “The equivalent-in-distribution (EiD)-based approach: On the analysis of cellular networks using stochastic geometry,” _IEEE Commun. Lett._ , vol. 18, no. 5, pp. 761–764, 2014.
  * [35] P. C. Pinto and M. Z. Win, “Communication in a Poisson field of interferers–part I: Interference distribution and error probability,” _IEEE Transactions on Wireless Communications_ , vol. 9, no. 7, pp. 2176–2186, 2010.
  * [36] J. G. Andrews, F. Baccelli, and R. K. Ganti, “A tractable approach to coverage and rate in cellular networks,” _IEEE Trans. Commun._ , vol. 59, no. 11, pp. 3122–3134, 2011.
  * [37] C. Zhag, X. Mu, J. Yuan, Y. Zhou, and J. Shi, “The performance of coded modulation with Gallager mapping in the finite length regime,” in _Int. Conf. Wireless Commun. Signal Process. (WCSP)_ , 2019, pp. 1–6.
  * [38] R. K. Ganti and M. Haenggi, “Interference in ad hoc networks with general motion-invariant node distributions,” in _IEEE Int. Symp. Inf. Theory_. IEEE, 2008, pp. 1–5.
  * [39] M. Kountouris and N. Pappas, “Approximating the interference distribution in large wireless networks,” in _Int. Symp. Wireless Commun. Sys._ IEEE, 2014, pp. 80–84.
  * [40] H. Vangala, E. Viterbo, and Y. Hong, “Permuted successive cancellation decoder for polar codes,” in _Int. Symp. Inf. Theory Appl._ , 2014, pp. 438–442.
  * [41] J. G. Andrews, A. K. Gupta, and H. S. Dhillon, “A primer on cellular network analysis using stochastic geometry,” _arXiv preprint arXiv:1604.03183_ , 2016\.
  * [42] F. Palacio, R. Agustí, J. Pérez-Romero, M. López-Benítez, S. Grimoud, B. Sayrac, I. Dagres, A. Polydoros, J. Riihijärvi, J. Nasreddine, P. Mähönen, L. Gavrilovska, V. Atanasovski, and J. Beek, “Radio environmental maps: information models and reference model. document number D4.1,” _Deliverable D4.1 del projecte Europeu FARAMIR (248351)_ , 2011\.
  * [43] M. Haenggi, “The meta distribution of the SIR in Poisson bipolar and cellular networks,” _IEEE Trans. Wireless Commun._ , vol. 15, no. 4, pp. 2577–2589, 2016.
  * [44] ——, _Stochastic Geometry for Wireless Networks_. Cambridge University Press, 2012.
  * [45] H. ElSawy and M.-S. Alouini, “On the meta distribution of coverage probability in uplink cellular networks,” _IEEE Commun. Lett._ , vol. 21, no. 7, pp. 1625–1628, 2017.
  * [46] H. Ibrahim, H. Tabassum, and U. T. Nguyen, “The meta distributions of the SIR/SNR and data rate in coexisting sub-6GHz and millimeter-wave cellular networks,” _IEEE Open J. Commun. Soc._ , vol. 1, pp. 1213–1229, 2020.
  * [47] W. Hassan, H.-S. Jo, S. Ikki, and M. Nekovee, “Spectrum-sharing method for co-existence between 5G OFDM-based system and fixed service,” _IEEE Access_ , vol. 7, pp. 77 460–77 475, 01 2019.
  * [48] K. S. Ali, M. Haenggi, H. ElSawy, A. Chaaban, and M.-S. Alouini, “Downlink non-orthogonal multiple access (NOMA) in Poisson networks,” _IEEE Trans. Commun._ , vol. 67, no. 2, pp. 1613–1628, 2018.
  * [49] G. Fubini, “Sugli integrali multipli,” _Rend. Acc. Naz. Lincei_ , vol. 16, pp. 608–614, 1907.