# Gaussian Broadcast Channels under Heterogeneous Blocklength Constraints

Pin-Hsun Lin†, Shih-Chun Lin‡, Peng-Wei Chen§, Marcel Mross†, and
Eduard A. Jorswieck†
†Institute for Communication Technology,
Technische Universität Braunschweig, Germany
‡Department of Electrical Engineering, National Taiwan University, Taiwan,
§Department of Electrical and Computer Engineering,
National Taiwan University of Science and Technology, Taiwan,
Email:{Lin, Mross<EMAIL_ADDRESS>
<EMAIL_ADDRESS><EMAIL_ADDRESS>Part of the work is presented
in ISIT 2021 [1] and ICC 2022 [2].

###### Abstract

Future wireless access networks aim to simultaneously support a large number
of devices with heterogeneous service requirements, including data rates,
error rates, and latencies. While achievable rate and capacity results exist
for Gaussian broadcast channels in the asymptotic blocklength regime, the
characterization of second-order achievable rate regions for heterogeneous
blocklength constraints is not available. Therefore, we investigate a two-user
Gaussian broadcast channel (GBC) with heterogeneous blocklength constraints,
specified according to users’ channel output signal-to-noise ratios (SNRs). We
assume the user with higher output SNR has a shorter blocklength constraint.
We show that with sufficiently large output SNR, the stronger user can perform
the early decoding (ED) technique to decode and subtract the interference via
successive interference cancellation (SIC). To achieve it, we derive an
explicit lower bound on the necessary number of received symbols for a
successful ED, using an independent and identically distributed Gaussian
input. A second-order rate of the weaker user who suffers from an SNR change
due to the heterogeneous blocklength constraint, is also derived. Numerical
results show that ED can outperform the hybrid non-orthogonal multiple access
scheme when the stronger channel is sufficiently better than the weaker one.
Under the considered setting, about 7-dB SNR gain can be achieved. These
results shows that ED with SIC is a promising technique for the future
wireless networks.

##  I Introduction

Ultra-reliable and low-latency communication (URLLC) is one of the target
application scenarios in 5G and beyond [3, 4, 5], which has attracted many
research efforts. One important branch of research is finite blocklength
analysis [6], which has been extended to different multiuser cases such as the
multiple access channel [7] and [8], asymmetric broadcast channel [9],
Gaussian broadcast channel [10] (GBC), the strong converse for GBC [11], GBC
with hard deadline [12], and the channel with state [13]. For 5G, the
flexibility for resource allocation was obtained by fine numerology [14].
Within each service class, most studies considered homogeneous blocklength and
latency constraints among the users. In practice, heterogeneous blocklengths
among users should be considered due to different requirements of latency,
quality of service, and different channel conditions among users. Therefore,
we investigate heterogeneous blocklengths among users.

The GBC with heterogeneous blocklength constraints has been studied in [15],
[16]. In [15], hybrid non-orthogonal multiple access (HNOMA) is analyzed,
where treating interference as noise (TIN), normal superposition coding
(homogeneous blocklength at all users) with successive interference
cancellation (SIC), and time-division multiple access (TDMA) are used in this
case. In particular, the weaker user’s codeword is divided into two shorter
codewords, where one of them has the same blocklength as that of the stronger
user. Therefore, the normal superposition coding with SIC can be applied. In
[16], after TDMA transmission, a user is forced to decode both the intended
and interference codewords. Such decoding is similar to decoding a common
message in a GBC, and the rate is limited by the link with the lower output
SNR. Aside from the above, one unsolved but important issue is how to perform
superposition coding with SIC when the two users use codewords of different
lengths. More specifically, the performance of SIC is unclear for the user
with a higher output signal-to-noise ratio (SNR) but shorter blocklength in a
two-user GBC.

In contrast to [15], [16], we argue that the stronger user with sufficiently
large output SNR can still decode the interference to perform SIC based on the
partially received symbols. The key is the early decoding (ED) technique:
assume that a code is designed for a channel with a specific output SNR under
a specific error probability and blocklength. Via ED, the message can be
decoded using only a certain fraction of the codeword when it is transmitted
through a channel with a larger output SNR. This concept has been investigated
with traditional first-order asymptotic analysis in [17], which tells us that
in a GBC, a user with higher output SNR than the other can decode successfully
using fewer received symbols. The concept of ED has already been used in
several wireless scenarios, not only to improve the latency performance, but
also to increase the throughput of a network. A popular application of ED is
in cognitive radio (CR) [18]. In addition to [17], there are further works
about ED: In [19], ED is applied to short message noisy network coding under
the same asymptotic assumption as in [17]. In [20], the authors consider the
necessary number of symbols for ED for binary input channels via numerical
simulations under a finite blocklength assumption. Note that the analytical
result for the finite blocklength regime of ED is missing in the above
references, which motivates our work in this paper.

Our main contributions are as follows:

  * •

We consider a two-user GBC with heterogeneous blocklength constraints and two
private messages (no common message), while the SIC is applied at the user
having higher output SNR with a shorter (stricter) blocklength constraint,

  * •

We derive a second-order lower bound on the necessary number of received
symbols such that ED works as the first step of SIC, i.e., decoding the
interference, while fulfilling the input power and error probability
constraints. In particular, we analyze the dependence testing (DT) bound [6]
over a fixed code instead of using Shannon’s random coding scheme, to ensure
that a specific code can be decoded when being transmitted through two
different channels while fulfilling the aforementioned constraints. By
applying ED, the stronger user can avoid using TIN as [15] and the stronger
user’s rate can be improved. We also derive the second-order rate of the
weaker user, whose received symbols encounter an SNR change due to the
heterogeneous blocklengths.

  * •

Based on the derived second-order rates of the stronger and weaker users in a
GBC, we formulate the rate region problems for ED and HNOMA. In particular, we
start the investigation from individual power constraint (IPC) and generalize
it to sum power constraint (SPC), and solve the corresponding programming
problems numerically.

  * •

Numerical results show that ED can significantly reduce latency in the finite
blocklength regime. Under the considered setting, more than 10dB SNR gain can
be achieved. In addition, ED partly outperforms HNOMA regarding the sum rate
and the rate region, when the stronger user has a sufficiently better channel
than the weaker user. Therefore, a hybrid system combining ED and HNOMA is the
best-known achievable scheme.

This paper is organized as follows. Section II introduces the system model and
preliminaries. Section III shows our main result: the minimum number of
received symbols for a successful ED. In addition, we derive the weaker user’s
second-order rate, under both IPC and SPC. In Section IV we show the
performance improvements numerically. We conclude this paper in Section V.

_Notation_ $\colon$ Upper/lower case normal letters denote
random/deterministic variables. Upper case calligraphic letters denote sets.
The notation $a_{i}^{j}$ denotes a row vector
$[a_{i},\,a_{i+1},\,\ldots,\,a_{j}]$ while $a_{1}^{j}$ is simplified to
$a^{j}$. We denote the inner product of two vectors $a^{j}$ and $b^{j}$ by
$\langle a^{j},b^{j}\rangle$. The probability of event $\mathcal{A}$ is
denoted by Pr$(\mathcal{A})$. The expectation and variance are denoted by
$\mathds{E}[\cdot]$ and $\mbox{Var}[\cdot]$, respectively. We denote the
probability density function (PDF) and cumulative distribution function (CDF)
of a random variable $X$ by $f_{X}$ and $F_{X}$, respectively. The random
variable $X$ following the distribution with CDF $F$ is denoted by
$\,X\sim\,F$. $\mathrm{Unif}(a,b)$ denotes the uniform distribution between
$a\in\mathds{R}$ and $b\in\mathds{R}$. We use
$X\mathbin{\rotatebox[origin={c}]{90.0}{$\vDash$}}Y$ to denote that $X$ and
$Y$ are stochastically independent. The logarithms used in the paper are all
with respect to base 2. We define
${\textsf{C}}(x)\triangleq\frac{1}{2}\log(1+x)$. Real additive white Gaussian
noise (AWGN) with zero mean and variance $\sigma^{2}$ is denoted by
$\mathcal{N}(0,\sigma^{2})$. We denote the indicator function and identity
matrix with dimension $n$ by $\mathds{1}$ and $\bm{I}_{n}$, respectively. We
denote the inverse $Q$-function by $Q^{-1}(.)$ and the big-O and small-o by
$\mathcal{O}(.)$ and ${o}(.)$, respectively.

##  II System Model and Preliminaries

###  II-A System Model

We consider a two-user GBC, where only private messages, but no common
message, are transmitted to each user. Denote the blocklength of user $k$ by
$n_{k}\in\mathds{N}^{+}$, $k=1,\,2$. We assume $n_{1}\geq n_{2}$. The received
signal at user $k$ at time $i$ is expressed as follows:

$\displaystyle Y_{k,i}$
$\displaystyle=\sqrt{h_{k}}X_{i}+Z_{k,i},\,i\in\\{1,\ldots,n_{k}\\},\,k=1,\,2,\,$
(1)

where the channel input
$X_{i}\in\mathcal{F}_{\ell}^{n_{1}}\subseteq\mathds{R}^{n_{1}}$,
$\ell=\\{IPC,\,SPC\\}$, while $\mathcal{F}_{SPC}^{n_{1}}$ and
$\mathcal{F}_{IPC}^{n_{1}}$ are sets of feasible codewords satisfying the
upcoming power constraints in (3) and (5), respectively,
$Z_{1,i}\sim\mathcal{N}(0,1)$ and $Z_{2,i}\sim\mathcal{N}(0,1)$, are
independent and identically distributed (i.i.d.) and mutually independent
additive white Gaussian noises. Define two message sets
$\mathcal{M}_{k}:=\\{1,\ldots,\textsf{M}_{k}\\},\,k=1,\,2$ for each user.
Assume that the message tuple $(m_{1},m_{2})$ is uniformly selected from
$\mathcal{M}_{1}\times\mathcal{M}_{2}$. The considered code
$(\textsf{M}_{1},\,\textsf{M}_{2},\,n_{1},\,n_{2},\,\epsilon,\,\mathcal{F}_{\ell}^{n_{1}})$
consists of

  * •

two message sets $\mathcal{M}_{k}=\\{1,\ldots,\textsf{M}_{k}\\},\,k=1,\,2$,

  * •

one encoder $f$:
$\mathcal{M}_{1}\times\mathcal{M}_{2}\mapsto\mathcal{F}_{\ell}^{n_{1}},\,\ell\in\\{IPC,\,SPC\\}$,

  * •

two decoders $\phi_{k}$: $\mathds{R}^{n_{k}}\mapsto\mathcal{M}_{k}$,
$\,k=1,\,2$,

such that

$\displaystyle
P_{e}^{n_{1},n_{2}}:=\frac{1}{\textsf{M}_{1}\textsf{M}_{2}}\sum_{(m_{1},m_{2})=(1,1)}^{(\textsf{M}_{1},\textsf{M}_{2})}\mbox{Pr}(\hat{m}_{1}\neq
m_{1}\mbox{ or }\hat{m}_{2}\neq m_{2}|(m_{1},m_{2})\mbox{ is
sent})\leq\epsilon,$ (2)

where $\epsilon\in(0,1)$ is a error probability constraint. Given $n_{1}$,
$n_{2}$, and $\epsilon$, a message-size tuple
$\left(\textsf{M}_{1},\,\textsf{M}_{2}\right)$ is achievable, if an
$(\textsf{M}_{1},\,\textsf{M}_{2},\,n_{1},\,n_{2},\,\epsilon,\,\mathcal{F}_{\ell}^{n_{1}})$-code
satisfying (2) exists. We consider the maximal power constraint on the channel
input by defining the following set:

$\displaystyle\mathcal{F}_{SPC}^{n_{1}}:=\left\\{x^{n_{1}}:\,||x^{n_{1}}||^{2}\leq
n_{1}\textsf{P}\right\\}.$ (3)

An encoding error is declared if a generated codeword $x^{n_{1}}$ does not
belong to $\mathcal{F}^{n_{1}}_{SPC}$.

To implement the encoder and decoder for the above model, we consider the
superposition coding with successive interference cancellation (SIC),
summarized in Fig. 1. The received signals at the two receivers can be
respectively expressed as follows:

$\displaystyle
Y_{k,i}=\begin{cases}&\sqrt{h_{k}}(X_{1,i}+X_{2,i})+Z_{k,i},\,\,\,i\in\\{1,\ldots,n_{2}\\},\\\
&\sqrt{h_{k}}X_{1,i}+Z_{k,i},\,\hskip
42.67912pti\in\\{n_{2}+1,\ldots,n_{1}\\},\end{cases}$ (4)

where the codewords of the two codes
$\\{X_{1}^{n_{1}}(m_{1}):\,m_{1}\in\mathcal{M}_{1}\\}$ and
$\\{X_{2}^{n_{2}}(m_{2}):\,m_{2}\in\mathcal{M}_{2}\\}$ are generated according
to i.i.d. Gaussian distributions:
$X_{k,i}\sim\mathcal{N}(0,\textsf{P}_{k}),\,i\in\\{1,\ldots,n_{k}\\}$,
$k=1,\,2$, and $\\{X_{1}^{n_{1}}(m_{1}):\,m_{1}\in\mathcal{M}_{1}\\}$ and
$\\{X_{2}^{n_{2}}(m_{2}):\,m_{2}\in\mathcal{M}_{2}\\}$ are mutually
independent.

Figure 1: The considered 2-user GBC with heterogeneous blocklength
constraints.

In addition to (3), in the following analysis we also consider the maximal
power constraint on each user’s codewords, namely, the IPC, by defining the
following set:

$\displaystyle\mathcal{F}^{n_{1}}_{IPC}:=\left\\{x^{n_{1}}:\,x^{n_{1}}=x_{1}^{n_{1}}+[x_{2}^{n_{2}},\,{0}^{n_{1}-n_{2}}],\,||x_{k}^{n_{k}}||^{2}\leq
n_{k}\textsf{P}_{k}\right\\},\,\,k=1,\,2,$ (5)

where ${\textsf{P}}_{k}$ is the power constraint at user $k$. An encoding
error is declared if a generated codeword $x_{1}^{n_{1}}$ or $x_{2}^{n_{2}}$
does not belong to $\mathcal{F}^{n_{1}}_{IPC}$. In contrast to the IPC
described by (5), we also call (3) the SPC.

We consider the following decoding schemes. At the stronger user (user 2), the
decoder first finds a unique $m\in\mathcal{M}_{1}$, such that
$i(x_{1}^{n_{2}}(m);y_{2}^{n_{2}})>\log({\textsf{M}}_{1})$, where $i(.;.)$ is
the information density. If a unique index $m$ is found, set $\hat{m}_{1}=m$.
Otherwise, it declares an error. Based on $\hat{m}_{1}$, find a unique
$m\in\mathcal{M}_{2}$, such that
$i(x_{2}^{n_{2}}(m);\tilde{y}_{2}^{n_{2}})>\log({\textsf{M}}_{2})$, where
$\tilde{y}_{2}^{n_{2}}$ is the received signal without the signal
$x_{1}^{n_{2}}$. If a unique index $m$ is found, set $\hat{m}_{2}=m$.
Otherwise, it declares an error. At the weaker user (user 1), it finds a
unique $m\in\mathcal{M}_{1}$, such that
$i(x_{1}^{n_{1}}(m);y_{1}^{n_{1}})>\log({\textsf{M}}_{1})$. If a unique index
$m$ is found, set $\hat{m}_{1}=m$.

######  Remark 1.

The reason for considering the i.i.d. Gaussian codes instead of shell codes
[6] in this work is as follows. Assume $x_{1}^{n_{1}}$ and $x_{2}^{n_{2}}$ are
two shell codes. When user 2 tries to decode $m_{1}$ from the received signal
$y_{2}^{n_{2}}=\sqrt{h_{1}}(x_{1}^{n_{2}}+x_{2}^{n_{2}})+z_{2}^{n_{2}}$,
however, the truncated $x_{1}^{n_{2}}:=[x_{1,1},\ldots,\,x_{1,n_{2}}]$ is no
longer a shell code. Then results in [9] and [12], which are based on shell
codes, cannot be applied here. To avoid this issue, we apply i.i.d. Gaussian
codes as a starting point to investigate the ED.

###  II-B Preliminaries

When $n_{1}=n_{2}$, user 2 can perform the traditional two-step SIC. The first
step is decoding user 1’s codeword and removing its interference. Next, user 2
decodes his own codeword. However, with $n_{1}>n_{2}$, the performance of
using SIC is not clear. This motivates our investigation of using the ED as
the first step of SIC. We first define a successful ED as follows.

######  Definition 1.

A successful ED means that the user with a shorter blocklength constraint
(user 2) can decode message of the user with the longer blocklength constraint
(user 1) from the first $\tilde{n}_{1}$ received symbols:
$Y_{2,1},\,Y_{2,2},\ldots,Y_{2,\tilde{n}_{1}}$, where $\tilde{n}_{1}\leq
n_{2}<n_{1}$, while the resulting error probability fulfills the error
probability constraint.

Note that the channel output distribution of (4) is still jointly Gaussian
when $n_{2}<n_{1}$, if an i.i.d. Gaussian codebook is used. From [6] we know
that the second-order achievable number of messages of a point-to-point
channel with channel gain $h$, blocklength $n$, error probability constraint
$\epsilon$, and power constraint P with i.i.d. Gaussian input, can be
specialized from our model by nulling $m_{2}$. It is described as follows:

$\displaystyle\log{\textsf{M}}\leq
n{{\textsf{C}}}(h\bar{{\textsf{P}}})-\sqrt{n{\textsf{V}}_{G}(h\cdot\bar{{\textsf{P}}})}Q^{-1}(\epsilon)+o\left(1\right),$
(6)

where
${\textsf{V}}_{G}(\bar{{\textsf{P}}}):=\log^{2}e\cdot\frac{\bar{{\textsf{P}}}}{1+\bar{{\textsf{P}}}}$
and $\bar{{\textsf{P}}}:={\textsf{P}}-\delta,\,\delta>0$. Based on (6), we
define the parameterized second order achievable rate as follows:

$\displaystyle{\textsf{R}}(n,{\textsf{SNR}},\epsilon)$
$\displaystyle:={\textsf{C}}({\textsf{SNR}})-\sqrt{\frac{{\textsf{V}_{G}}({\textsf{SNR}})}{n}}Q^{-1}(\epsilon).$
(7)

##  III Main Results

In this section we introduce our main results: the necessary number of
received symbols for a successful ED at the stronger user and the second-order
achievable region $({\textsf{M}}_{1},{\textsf{M}}_{2})$ of the two users under
both IPC and SPC.

###  III-A Second Order Achievable Rate Region
$({\textsf{M}}_{1},{\textsf{M}}_{2})$ of ED with IPC

In the following, we apply the concept of ED to a two-user GBC with SIC.

######  Theorem 1.

Denote the necessary number of symbols to successfully early decode user 1’s
signal at user 2 by $\tilde{n}_{1}$. Assume $n_{2}\leq n_{1}$ and $h_{1}\leq
h_{2}$. If all the following conditions

$\displaystyle n_{2}$
$\displaystyle\geq\tilde{n}_{1}\geq\frac{\log\,{\textsf{M}_{1}}}{{\textsf{C}}(g_{2}{\bar{\textsf{P}}_{1}})-\log
e\cdot\frac{g_{2}{\bar{\textsf{P}}_{1}}}{2(1+g_{2}{\bar{\textsf{P}}_{1}})}}+\frac{\log
e\sqrt{4g_{2}{\bar{\textsf{P}}_{1}}+2(g_{2}{\bar{\textsf{P}}_{1}})^{2}}Q^{-1}({{\epsilon}_{SIC1}})}{2(1+g_{2}{\bar{\textsf{P}}_{1}}){\textsf{C}}(g_{2}{\bar{\textsf{P}}_{1}})-\log
e\cdot g_{2}{{\bar{\textsf{P}}_{1}}}}\cdot\sqrt{n_{1}},$ (8)

and

$\displaystyle\epsilon_{SIC1}+\epsilon_{SIC2}-\epsilon_{SIC1}\epsilon_{SIC2}+\epsilon_{1}\leq\epsilon,\,\,\quad
0<\epsilon_{1},\,\epsilon_{2},\,\epsilon_{SIC1},\,\epsilon_{SIC2}<1$ (9)

hold, then under IPC all tuples of message sizes
$({\textsf{M}}_{1},{\textsf{M}}_{2})$ in $\mathcal{M}_{IPC}$ are achievable,
where

$\displaystyle\mathcal{M}_{IPC}=\Bigg{\\{}({\textsf{M}}_{1},{\textsf{M}}_{2}):\,\,\log{\textsf{M}}_{1}$
$\displaystyle\leq
n_{1}\bar{{\textsf{C}}}_{1}-\sqrt{n_{1}\bar{{\textsf{V}}}_{1}}Q^{-1}(\epsilon_{1})+\mathcal{O}(1),$
(10) $\displaystyle\log{\textsf{M}}_{2}$ $\displaystyle\leq
n_{2}{\textsf{C}}(h_{2}{\bar{\textsf{P}}_{2}})-\sqrt{n_{2}{{\textsf{V}}_{G}(h_{2}\bar{\textsf{P}}_{2})}}Q^{-1}({\epsilon}_{SIC2})+\mathcal{O}(1)\Bigg{\\}},$
(11) $\displaystyle\bar{{\textsf{C}}}_{1}$
$\displaystyle:=p{\textsf{C}}(g_{1}\bar{{\textsf{P}}}_{1})+(1-p){\textsf{C}}(h_{1}\bar{{\textsf{P}}}_{1}),$
(12) $\displaystyle\bar{{\textsf{V}}}_{1}$
$\displaystyle:=\log^{2}e\cdot\left\\{p\frac{g_{1}\bar{{\textsf{P}}}_{1}}{1+g_{1}\bar{{\textsf{P}}}_{1}}+(1-p)\frac{h_{1}\bar{{\textsf{P}}}_{1}}{1+h_{1}\bar{{\textsf{P}}}_{1}}\right\\},$
(13) $\displaystyle g_{1}$
$\displaystyle:=\frac{h_{1}}{1+h_{1}\bar{{\textsf{P}}}_{2}},$ (14)

$p:=\frac{n_{2}}{n_{1}},\,\bar{\textsf{P}}_{k}:={\textsf{P}}_{k}-\delta,\,k=1,\,2,\,\delta>0$,
${\epsilon}_{SIC1}$ and ${\epsilon}_{SIC2}$ are the target decoding error
probabilities of $m_{1}$ and $m_{2}$ at user 2 at the 1st and 2nd steps of
SIC, respectively, $g_{2}:=\frac{h_{2}}{1+h_{2}\bar{\textsf{P}}_{2}}$, and
$g_{2}\bar{\textsf{P}}_{1}$ is the equivalent output SNR at user 2 in the
first step of SIC.

The proof of the lower bound of $\tilde{n}_{1}$ is relegated to Appendix I and
the proof of the region $\mathcal{M}_{IPC}$ is relegated to Appendix II.

In contrast to the pure private message case, the ED scheme can be naturally
applied to a two-user GBC with only common-message (or equivalently, a multi-
cast channel) and different decoding latencies. That is, a common message $m$
is sent, and the signal received at time $i$ at user $k$ is
$Y_{k,i}=\sqrt{h_{k}}X_{i}+Z_{k,i},\,i=1,\ldots,n_{k}$, where
$X_{i}\sim\mathcal{N}(0,\bar{{\textsf{P}}})$ is the transmitted symbol at time
$i$. User $k$ wants to decode $m$ through $n_{k}$ received symbols. We can
compare this specialization to the asymptotic case. Denote the feasible lower
bound of $\tilde{n}_{1}$ derived from the common-message specialization based
one Theorem 1 by $g(n_{1})$. Then we have the following

$\displaystyle\lim_{n_{1}\rightarrow\infty}\frac{\tilde{n}_{1}}{n_{1}}$
$\displaystyle\geq\lim_{n_{1}\rightarrow\infty}\frac{g(n_{1})}{n_{1}}=\frac{{\textsf{C}}(h_{1}\bar{{\textsf{P}}})}{{\textsf{C}}(h_{2}\bar{{\textsf{P}}})-\frac{\log
e\cdot h_{2}\bar{{\textsf{P}}}}{2(1+h_{2}\bar{{\textsf{P}}})}}$ (15)
$\displaystyle>\frac{{\textsf{C}}(h_{1}\bar{{\textsf{P}}})}{{\textsf{C}}(h_{2}\bar{{\textsf{P}}})},$
(16)

where (16) is the asymptotic result for the ED [17]. The strict inequality
comes from an strictly upper bounding during the error analysis shown in
Appendix I.

In contrast, without using ED or when (8) is violated, TIN can be used,
instead. Then user 2’s second-order achievable $\log{\textsf{M}}_{2}$ can be
derived from (6) as follows

$\displaystyle\log{\textsf{M}}_{2}\leq
n_{2}{\textsf{C}}\left(g_{2}\bar{{\textsf{P}}}_{2}\right)-\sqrt{n_{2}{\textsf{V}_{G}}\left(\tilde{h}_{2}\bar{{\textsf{P}}}_{2}\right)}Q^{-1}(\epsilon_{2})+\mathcal{O}(1),$
(17)

where $g_{2}:=\frac{h_{2}}{1+h_{2}{\textsf{P}}_{1}}$ and $\epsilon_{2}$ is
user 2’s target error probability.

###  III-B Rate region of ED with SPC

The SPC with heterogenous blocklength constraints is stated as follows

$\displaystyle\sum_{j=1}^{n_{2}}(x_{1,j}(m_{1})+x_{2,j}(m_{2}))^{2}+\sum_{j=n_{2}+1}^{n_{1}}x_{1,j}^{2}(m_{1})\leq
n_{1}{\textsf{P}}_{T},$ (18)

for all $m_{k}\in\mathcal{M}_{k},\,k=1,\,2.$ Because the cross-term in the
first term on the left hand side (LHS) of (18) complicates the power
allocation, we consider the following sum power constraint instead, for the
following derivation and simulation:

$\displaystyle\sum_{j=1}^{n_{2}}\left(x_{1,j}^{2}(m_{1})+x_{2,j}^{2}(m_{2})\right)+\sum_{j=n_{2}+1}^{n_{1}}x_{1,j}^{2}(m_{1})\leq
n_{1}{\textsf{P}}_{T},$ (19)

for all $m_{k}\in\mathcal{M}_{k},\,k=1,\,2$. The validity of considering (19)
instead of (18) is derived in Lemma 1 as shown below. We first assume

$\displaystyle\sum_{j=1}^{n_{2}}x_{1,j}^{2}(m_{1})\leq
n_{2}{\textsf{P}}_{11},\,\sum_{j=1}^{n_{2}}x_{2,j}^{2}(m_{2})\leq
n_{2}{\textsf{P}}_{2},\,\sum_{j=n_{2}+1}^{n_{1}}x_{1,j}^{2}(m_{1})\leq(n_{1}-n_{2}){\textsf{P}}_{12},$
(20)

for all $m_{k}\in\mathcal{M}_{k},\,k=1,\,2.$ Then we can consider the
following power constraint instead of (18)

$\displaystyle
n_{2}({\textsf{P}}_{11}+{\textsf{P}}_{2})+(n_{1}-n_{2}){\textsf{P}}_{12}\leq
n_{1}{\textsf{P}}_{T}.$ (21)

In short, if (20) and (21) are fulfilled, then the probability that the
constraint in (18) is violated, is upper bounded by $e^{-\mathcal{O}(n_{2})}$.
Note that without incurring confusion, in the following we omit the
parameterized $m_{1}$ and $m_{2}$ in codewords to simplify the notation.

######  Lemma 1.

Let $X_{1,j}\sim\mathcal{N}(0,\bar{{\textsf{P}}}_{11})\mbox{ and
}\,X_{2,j}\sim\mathcal{N}(0,\bar{{\textsf{P}}}_{2})\mbox{ be i.i.d. generated
and}\,\\{X_{1,j}\\}$ and $\\{X_{2,j}\\}$ are mutually independent,
$j=1,\cdots,n_{2}$, also let
$X_{1,j}\sim\mathcal{N}(0,\bar{{\textsf{P}}}_{12})\mbox{ be i.i.d.
generated},$ $j=n_{2}+1,\cdots,n_{1}$, where
$\bar{{\textsf{P}}}_{11}={\textsf{P}}_{11}-\delta$,
$\bar{{\textsf{P}}}_{12}={\textsf{P}}_{12}-\delta$, and
$\bar{{\textsf{P}}}_{2}={\textsf{P}}_{2}-\delta$, $\delta>0$. Then

$\displaystyle\Pr\left(\sum_{j=1}^{n_{2}}\left(X_{1,j}+X_{2,j}\right)^{2}+\sum_{j=n_{2}+1}^{n_{1}}X_{1,j}^{2}>n_{1}{\textsf{P}}_{T}\right)\leq
e^{-\mathcal{O}(n_{2})}$ (22)

and

$\displaystyle\Pr\left(\sum_{j=1}^{n_{2}}\left(X_{1,j}^{2}+X_{2,j}^{2}\right)+\sum_{j=n_{2}+1}^{n_{1}}X_{1,j}^{2}>n_{1}{\textsf{P}}_{T}\right)\leq
e^{-\mathcal{O}(n_{2})}.$ (23)

The proof is relegated to Appendix III. The main idea is to treat the event
$\left\\{\sum_{j=1}^{n_{2}}x_{1,j}x_{2,j}\geq n_{2}\delta\right\\}$ as an
outage and collect the probability of input violation
${\mbox{Pr}}(\sum_{j=1}^{n_{2}}X_{1,j}X_{2,j}\geq n_{2}\delta)$ into the big-O
term during the error analysis. After that, we use the concept of power
backoff to ensure that by selecting $2n_{2}\delta$ as the power backoff, the
total energy that is allocated to $\sum_{j=1}^{n_{1}}x_{1,j}^{2}$ and
$\sum_{j=1}^{n_{2}}x_{2,j}^{2}$ will be no larger than
$n_{1}P_{T}-2n_{2}\delta$ with probability close to 1. Based on Lemma 1, we
can extend the analysis in Theorem 1 to the case with SPC as the following
result.

######  Proposition 1.

Denote a power backoff for the sum power constraint (19) by $\delta$, such
that $\bar{{\textsf{P}}}_{11}={\textsf{P}}_{11}-\delta\geq 0$,
$\bar{{\textsf{P}}}_{12}={\textsf{P}}_{12}-\delta\geq 0$,
$\bar{{\textsf{P}}}_{2}={\textsf{P}}_{2}-\delta\geq 0$ fulfill

$\displaystyle
p\bar{{\textsf{P}}}_{11}+(1-p)\bar{{\textsf{P}}}_{12}\leq{\textsf{P}}_{T}-p\bar{{\textsf{P}}}_{2}.$
(24)

Assume the blocklengths $n_{1}>n_{2}$ and channel gains $h_{2}>h_{1}$. If (8),
(9), and (24) hold, then under SPC all tuples of message sizes
$({\textsf{M}}_{1}^{\prime},{\textsf{M}}_{2})$ in $\mathcal{M}_{SPC}$ are
achievable, where

$\displaystyle\mathcal{M}_{SPC}=\Bigg{\\{}({\textsf{M}}_{1}^{\prime},{\textsf{M}}_{2}):\,\,\log{\textsf{M}}_{1}^{\prime}$
$\displaystyle\leq
n_{1}\bar{{\textsf{C}}}^{\prime}_{1}-\sqrt{n_{1}\bar{{\textsf{V}}}_{1}^{\prime}}Q^{-1}(\epsilon_{1})+\mathcal{O}(1),$
(25) $\displaystyle\log{\textsf{M}}_{2}$ $\displaystyle\leq
n_{2}{\textsf{C}}(h_{2}{\bar{\textsf{P}}_{2}})-\sqrt{n_{2}{{\textsf{V}}_{G}(h_{2}\bar{\textsf{P}}_{2})}}Q^{-1}({\epsilon}_{SIC2})+\mathcal{O}(1)\Bigg{\\}},$
(26)

and

$\displaystyle\bar{{\textsf{C}}}_{1}^{\prime}$
$\displaystyle:=p{\textsf{C}}(g_{1}^{\prime}\bar{{\textsf{P}}}_{11})+(1-p){\textsf{C}}(h_{1}\bar{{\textsf{P}}}_{12}),$
(27) $\displaystyle\bar{{\textsf{V}}}_{1}^{\prime}$
$\displaystyle:=\log^{2}e\cdot\left\\{p\frac{g_{1}^{\prime}\bar{{\textsf{P}}}_{11}}{1+g_{1}^{\prime}\bar{{\textsf{P}}}_{11}}+(1-p)\frac{h_{1}\bar{{\textsf{P}}}_{12}}{1+h_{1}\bar{{\textsf{P}}}_{12}}\right\\},$
(28) $\displaystyle g_{1}^{\prime}$
$\displaystyle:=\frac{h_{1}}{1+h_{1}\bar{{\textsf{P}}}_{2}}.$ (29)

The proof is relegated in Appendix IV.

##  IV Numerical Results

In this section, we first formulate the achievable regions of the ED and HNOMA
schemes as optimization problems, where the latter case is used as a
comparison baseline. Based on these problem formulations, we then show the
latency reduction of ED compared to the normal decoding, followed by the
comparisons of achievable regions.

###  IV-A Programming Formulation of the Achievable Rate Region

Denote the target system error probability by $\epsilon$. Denote the rates at
users 1 and 2 by ${\textsf{R}}_{1}$ and ${\textsf{R}}_{2}$, respectively. Let
the intermediate variables $\epsilon_{SIC_{k}},\,k=1,\,2,$ denote the target
error probabilities of decoding user $k$’s messages at the $k$-th step of SIC
at user 2, respectively. Let $\epsilon_{1}^{ED}$ denote the target error
probability of decoding user 1’s messages at user 1 when ED is used at
receiver 2. Let $\epsilon_{1,j}^{HNOMA},\,j=1,\,2$ denote the target error
probabilities of the sub-blocks 1 and 2, respectively, when HNOMA is used
[15]. Note that for HNOMA, sub-blocks 1 and 2 are codewords with blocklengths
$n_{2}$ and $n_{1}-n_{2}$, respectively. Denote the weighting in the weighted
sum rate formulation by $\omega$, $0\leq\omega\leq 1$. Note that to optimize
the sum-rate and rate region, the inequalities in the error probability
constraints should be equalities, due to the tradeoff between the error
probability and the rate in finite blocklength analysis. Define

$\displaystyle{\textsf{SNR}}_{11}$
$\displaystyle:=\frac{h_{1}\bar{{\textsf{P}}}_{1}}{1+h_{1}\bar{{\textsf{P}}}_{2}},\quad{\textsf{SNR}}_{21}:=\frac{h_{2}\bar{{\textsf{P}}}_{1}}{1+h_{2}\bar{{\textsf{P}}}_{2}},\quad{\textsf{SNR}}_{12}:=h_{1}\bar{{\textsf{P}}}_{1},\quad\mbox{and
}{\textsf{SNR}}_{22}:=h_{2}\bar{{\textsf{P}}}_{2}.$ (30)

We then formulate an optimization problem for an enhanced HNOMA as follows:
$\mathbf{P_{1}^{IPC}}$ (enhanced weighted sum-rate of HNOMA with IPC):

$\displaystyle\max$
$\displaystyle\quad\omega{\textsf{R}}_{1}\left(\epsilon_{1,1}^{HNOMA},\,\epsilon_{1,2}^{HNOMA}\right)+(1-\omega){\textsf{R}}\left(n_{2},{\textsf{SNR}}_{22},\epsilon_{SIC2}\right)$
(31) s.t. $\displaystyle\quad
2-(1-\epsilon_{1,1}^{HNOMA})(1-\epsilon_{1,2}^{HNOMA})-(1-\epsilon_{SIC1})(1-\epsilon_{SIC2})\leq\epsilon$
(32) $\displaystyle\quad
0<\epsilon_{SIC1},\,\epsilon_{SIC2},\,\epsilon_{1,1}^{HNOMA},\,\epsilon_{1,2}^{HNOMA}<1,$
(33)

where

$\displaystyle{\textsf{R}}_{1}\left(\epsilon_{1,1}^{HNOMA},\,\epsilon_{1,2}^{HNOMA}\right)$
$\displaystyle:=p\cdot\min\left\\{{\textsf{R}}(n_{2},\,{\textsf{SNR}}_{11},\,\epsilon_{1,1}^{HNOMA}),\,{\textsf{R}}(n_{2},\,{\textsf{SNR}}_{21},\,\epsilon_{SIC1})\right\\}+\left(1-p\right){\textsf{R}}(n_{1}-n_{2},\,{\textsf{SNR}}_{12},\,\epsilon_{1,2}^{HNOMA}).$
(34)

Recall that $({\textsf{SNR}}_{11},{\textsf{SNR}}_{12})$ and $p$ are defined in
(30) and below (14), respectively and R on the RHS in (34) is defined in (7);
the minimum in (34) is to ensure that the first sub-block of the weaker user
can be decoded at the stronger user with an error probability
$\epsilon_{SIC1}$ within blocklength $n_{2}$. Note that the higher order terms
are neglected under the assumption of a sufficiently large blocklength.

To optimize the weighted sum-rate of SIC with ED, we formulate the following
problem:

$\mathbf{P_{2}^{IPC}}$ (weighted sum-rate of ED with IPC):

$\displaystyle\max$
$\displaystyle\quad\omega{\textsf{R}}_{1,ED}\left(\epsilon_{1}\right)+(1-\omega){\textsf{R}}\left(n_{2},{\textsf{SNR}}_{22},\epsilon_{SIC2}\right)$
(35) s.t.
$\displaystyle\quad\eqref{EQ_SIC_constraintED_ED_rate_improvement},\,\eqref{EQ_Pe_constr},$
$\displaystyle\quad
n_{2}\geq\frac{n_{1}{\textsf{R}}_{1,ED}(\epsilon_{1})}{{\textsf{C}}({\textsf{SNR}}_{21})-\log
e\cdot\frac{{\textsf{SNR}}_{21}}{2(1+{\textsf{SNR}}_{21})}}+\frac{\log
e\sqrt{4{\textsf{SNR}}_{21}+2{\textsf{SNR}}_{21}^{2}}Q^{-1}(\epsilon_{SIC1})}{2(1+{\textsf{SNR}}_{21}){\textsf{C}}({\textsf{SNR}}_{21})-\log
e\cdot{\textsf{SNR}}_{21}}\cdot\sqrt{n_{1}},$ (36)

where ${\textsf{R}}_{1,ED}$ is derived from (10) as follows:

$\displaystyle{\textsf{R}}_{1,ED}\left(\epsilon_{1}\right)$
$\displaystyle:={\bar{\textsf{C}}}_{1}-\sqrt{\frac{{\bar{\textsf{V}}}_{1}}{n_{1}}}Q^{-1}(\epsilon_{1}),$
(37)

and
$\bar{{\textsf{C}}}_{1}:=p{\textsf{C}}(g_{1}\bar{{\textsf{P}}}_{1})+(1-p){\textsf{C}}(h_{1}\bar{{\textsf{P}}}_{1}),\,\bar{{\textsf{V}}}_{1}:=\log
e^{2}\cdot\left\\{p\frac{g_{1}\bar{{\textsf{P}}}_{1}}{1+g_{1}\bar{{\textsf{P}}}_{1}}+(1-p)\frac{h_{1}\bar{{\textsf{P}}}_{1}}{1+h_{1}\bar{{\textsf{P}}}_{1}}\right\\},\,\mbox{
and }g_{1}:=\frac{h_{1}}{1+h_{1}\bar{{\textsf{P}}}_{2}}.$

Note that (36) ensures that the weaker user’s signal can be decoded with error
probability $\epsilon_{SIC1}$ within blocklength $n_{2}$ and the channel SNR
is as ${\textsf{SNR}}_{21}$, which plays a similar role to the minimum in (34)
of the HNOMA. Note also that we assume that the blocklength is sufficiently
large, such that we omit the big-O term in the programming formulation.

For the SPC, we first redefine
${\textsf{SNR}}_{11}:=\frac{h_{1}\bar{{\textsf{P}}}_{11}}{1+h_{1}\bar{{\textsf{P}}}_{2}},\,\quad{\textsf{SNR}}_{12}:=h_{1}\bar{{\textsf{P}}}_{12},\,\quad\mbox{and
}{\textsf{SNR}}_{21}:=\frac{h_{2}\bar{{\textsf{P}}}_{11}}{1+h_{2}\bar{{\textsf{P}}}_{2}}$.
Then we formulate the following optimization problem for the enhanced weighted
sum-rate of HNOMA:
$\mathbf{P_{1}^{SPC}}$ (enhanced weighted sum-rate of HNOMA with SPC):

$\displaystyle\max$
$\displaystyle\quad\omega{\textsf{R}}_{1}\left(\epsilon_{1,1}^{HNOMA},\,\epsilon_{1,2}^{HNOMA}\right)+(1-\omega){\textsf{R}}\left(n_{2},{\textsf{SNR}}_{22},\epsilon_{SIC2}\right)$
(38) s.t. $\displaystyle\quad\bar{{\textsf{P}}}_{11}\geq
0,\,\bar{{\textsf{P}}}_{12}\geq 0,\,\bar{{\textsf{P}}}_{2}\geq 0,$ (39)
$\displaystyle\quad\eqref{EQ_SPC_power_backoff},\,\eqref{EQ_P1p_1st_constraint},\,\eqref{EQ_P1P_IPC_eps}.$

The optimization problem of the ED with SPC is formulated as follows:

$\mathbf{P_{2}^{SPC}}$ (weighted sum-rate of ED with SPC):

$\displaystyle\max$
$\displaystyle\quad\omega{\textsf{R}}^{\prime}_{1,ED}\left(\epsilon_{1}\right)+(1-\omega){\textsf{R}}\left(n_{2},{\textsf{SNR}}_{22},\epsilon_{SIC2}\right)$
(40) s.t.
$\displaystyle\quad\eqref{EQ_SIC_constraintED_ED_rate_improvement},\,\eqref{EQ_Pe_constr},\,\eqref{EQ_SPC_power_backoff},\,\eqref{EQ_constraint_positive_power},$
(41)

where ${\textsf{R}}^{\prime}_{1}\left(\epsilon_{1}^{ED}\right)$ is derived
from (25).

###  IV-B Latency Reduction

We compare the latencies among three cases: 1) the number of received symbols
necessary for a successful ED under asymptotic analysis; 2) decoding after the
complete codeword is received (without ED); 3) the derived number of received
symbols necessary for a successful ED under finite blocklength analysis. We
consider the following setting: $\epsilon=2\cdot 10^{-6}$, $h_{1}=1$,
${\textsf{P}}_{1}=8\mbox{ and }{\textsf{P}}_{2}=0.2$ for IPC. We consider
three different blocklengths: $n_{1}=$512, 1024, and 2048. The three schemes
are compared in Fig. 2. Without ED, the stronger user can start to decode only
after receiving $n_{1}$ symbols. When ED is activated and successful, we can
observe that the improvement of the latency reduction increases with an
increasing $h_{2}$. We use (16), which is a latency lower bound, as the
baseline for comparison. When the channel gain is sufficiently large, the
early decoding gain in terms of latency reduction can be achieved around 450
symbols when $n_{1}=2048$ is considered, while the gap of ED to the asymptotic
scenario is around 350 symbols. Note that the gap between the results of the
early decodings in asymptotic and finite blocklength analyses is not only from
the channel dispersion but also from the bounding error when deriving (8).

Figure 2: Comparison of necessary numbers of received symbols for successfully
decoding a message under the constraints of average probability of error under
maximal channel input power constraint: ED with finite blocklength, ED with
infinite blocklength, and complete decoding.

Assume that we consider the case in which $n_{2}$ is set as the lower bound in
(8). Then we compare (26) and (17) in Fig. 3, where the rate gain by using the
ED over TIN is apparent. In particular, about 7-dB SNR gain can be achieved by
the ED under our setting. Please note that similar comparisons as Fig. 2 and
Fig. 3 for GBC with only common messages can be seen in [1].

Figure 3: Comparison of the stronger users’ rates with and without ED given a
latency constraint.

###  IV-C Comparison of Sum Rate/Rate Region

####  IV-C1 Sum-rate comparison

In the following, we use two examples to compare the sum-rates
($\omega=\frac{1}{2}$) of the ED in $\mathbf{P_{2}^{IPC}}$ and HNOMA in
$\mathbf{P_{1}^{IPC}}$, under individual power constraints. We use grid search
to find the optimal solution in the three problems with step sizes
$\epsilon/100$. We only show the range of $n_{2}$ where $\mathbf{P_{2}^{IPC}}$
is feasible. Therefore, the sum rates below a threshold of $n_{2}$ will be
zero. We consider the following setting:
$n_{1}=1024,\,h_{1}=1,{\textsf{P}}_{1}=8,\,{\textsf{P}}_{2}=0.2,\,\epsilon=2\cdot
10^{-6}$ under $h_{2}=10$ and $20$. We can find that ED outperforms HNOMA for
all $n_{2}$ feasible for ED, under individual power constraint, for
$h_{2}\in\\{10,\,20\\}$. In the same figure we can also observe that a larger
$h_{2}$ will not only enlarge the feasible region of operating the ED but also
enhance the sum-rate performance, which is consistent with the intuition.
Besides, we can interpret curves in Fig. 4 as a tradeoff between latency and
sum-rate. In particular, when a lower latency at the stronger user is
requested, we can use a smaller $n_{2}$ if the ED is feasible. However, due to
the error probability constraint, the stronger user’s rate will also be
reduced, which causes the sum-rate as an increasing function of $n_{2}$.

Figure 4: Comparison of the sum rates between ED and HNOMA under IPC with
different $h_{2}$.

####  IV-C2 Rate regions comparison

We now consider the rate regions under SPC from solving $\mathbf{P_{1}^{SPC}}$
and $\mathbf{P_{2}^{SPC}}$ with the following system parameters:
$h_{1}=1,\,h_{2}=50,\,P_{T}=10,\,n_{1}=1024,\,n_{2}=840,\,\epsilon=2\cdot
10^{-5}$ in Fig. 5. We can observe that both ED and HNOMA have their own
advantages. As for ED, it can be beneficial when transmitting a single
codeword with a longer blocklength compared to HNOMA, whose weaker user
decodes two concatenated shorter codewords. Therefore, when the weighting
$\omega$ is higher, ED outperforms HNOMA when $h_{2}$ is sufficiently large.
On the contrary, by HNOMA, the transmission consists of 2 segments, which
brings the flexibility of non-overlapping transmission, and therefore, it can
outperform ED when $h_{2}$ is smaller. In particular, when $h_{2}$ is smaller,
the feasibility constraint (8) is harder to fulfill.

Figure 5: Comparison of achievable rate regions:
$h_{1}=1,h_{2}=50,P_{T}=10,n_{1}=1024,n_{2}=840$.

##  V conclusion

We investigate a two-user Gaussian broadcast channel with heterogeneous
blocklength constraints. Unlike the traditional GBC where two users have the
same blocklength constraints, here, the user with higher output SNR has a
shorter blocklength constraint. We show that with sufficiently large output
SNR, the stronger user can perform early decoding to decode the interference,
followed by the successive interference cancellation, which is not yet
reported in the literature. To achieve this goal, we derive an explicit lower
bound on the necessary number of received symbols for a successful ED, using
an independent and identically distributed Gaussian input. A second-order rate
of the weaker user who suffers from an SNR change due to the heterogeneous
blocklength constraint, is also derived. Numerical results show that ED has a
larger sum rate and rate region than HNOMA, when the channel gain of the
better channel is sufficiently larger than the other one. Then ED with SIC is
a promising technique for future broadcast channels with heterogeneous
blocklength constraints. Our future works include deriving a second order
capacity outer bound and improving the second order performance by a better
input distribution than i.i.d. Gaussian input.

##  Appendix I Proof of Theorem 1

To derive the necessary number of received symbols, namely, $n_{2}$, for a
successful ED, we investigate the error analysis at the stronger user. We
consider the following average error probability (at the first step of SIC) at
user 2, from the dependence testing bound given a specific code $\mathcal{C}$
[6, Lemma 19]:

$\displaystyle\frac{1}{\textsf{M}_{1}}\sum_{m_{1}=1}^{\textsf{M}_{1}}\mbox{Pr}(\hat{m}_{1}\neq
m_{1}\mbox{ at user }2|\,\,m_{1}\mbox{ is sent, }\mathcal{C}\mbox{ is used})$
$\displaystyle\leq$
$\displaystyle\frac{1}{{\textsf{M}}_{1}}\sum_{m=1}^{{\textsf{M}}_{1}}\left\\{\mathds{1}_{x_{1}^{n_{1}}(m)\notin\mathcal{F}^{n_{1}}}+P_{Y_{2}^{{n_{2}}}|X_{1}^{{n_{2}}}=x_{1}^{n_{2}}(m)}\left(i(x_{1}^{n_{2}}(m);Y_{2}^{{n_{2}}})\leq\log{\textsf{M}}_{1}\right)\right.$
$\displaystyle\hskip
227.62204pt\left.+{\textsf{M}}_{1}P_{Y_{2}^{{n_{1}}}}\left(i(x_{1}^{n_{2}}(m);Y_{2}^{{n_{2}}})>\log{\textsf{M}}_{1}\right)\right\\}$
(42) $\displaystyle\leq$
$\displaystyle\frac{1}{{\textsf{M}}_{1}}\sum_{m=1}^{{\textsf{M}}_{1}}\left\\{\mathds{1}_{||x_{1}^{n_{1}}(m)||^{2}>n_{1}{\textsf{P}}_{1}}+\mathds{1}_{||x_{1}^{n_{1}}(m)||_{\infty}>{n_{1}}^{a}}+P_{Y_{2}^{n_{2}}|X_{1}^{{n_{2}}}=x_{1}^{n_{2}}(m)}\left(i(x_{1}^{n_{2}}(m);Y_{2}^{{n_{2}}})\leq\log{\textsf{M}}_{1}\right)\right.$
$\displaystyle\hskip
227.62204pt\left.+{\textsf{M}}_{1}P_{Y_{2}^{{n_{2}}}}\left(i(x_{1}^{n_{2}}(m);Y_{2}^{{n_{2}}})>\log{\textsf{M}}_{1}\right)\right\\}$
(43) $\displaystyle\leq$
$\displaystyle\frac{1}{{\textsf{M}}_{1}}\sum_{m=1}^{{\textsf{M}}_{1}}\left\\{P_{\tilde{Y}_{2}^{n_{2}}|X_{1}^{n_{2}}=x_{1}^{n_{2}}(m)}\left(i(x_{1}^{n_{2}}(m);\tilde{Y}_{2}^{n_{2}})\leq\log{\textsf{M}}_{1}\right)\right.$
$\displaystyle\hskip
113.81102pt\left.+{\textsf{M}}_{1}P_{\tilde{Y}_{2}^{n_{2}}}\left(i(x_{1}^{n_{2}}(m);\tilde{Y}_{2}^{n_{2}})>\log{\textsf{M}}_{1}\right)\right\\}+e^{-\frac{n_{2}\delta^{2}}{4}}+e^{\frac{-n_{1}^{2a}}{2{\textsf{P}}_{1}}+\ln(2n_{1})},$
(44)

where the code $\mathcal{C}$ is specified by (6). In particular, from random
coding analysis there must exist a code
$\mathcal{C}=\\{x_{1}^{n_{1}}(1),\,x_{1}^{n_{1}}(2),\,\ldots
x_{1}^{n_{1}}({\textsf{M}}_{1})\\}$ with blocklength ${n_{1}}$ achieving the
rate (6) while fulfilling the input power and error probability constraints,
where each codeword is i.i.d. generated according to
$\prod_{k=1}^{n_{1}}p_{X_{1}}(x_{1,k})$,
$x_{1,k}\in\mathcal{X}_{1},\,k=1,\cdots,n_{1}$. In (42), $\mathcal{F}^{n_{1}}$
is the channel input constraint, including the maximal power constraint (5)
and the peak constraint $||x_{1}^{n_{1}}(m)||_{\infty}\leq{n_{1}}^{a}$ ***The
peak constraint is only for the proof purpose, i.e., to ensure the vanishing
property of the Berry-Esseen ratio, in particular, when calculating the third
moment of the information density. Furthermore, with this additional
constraint, the derived rate is a lower bound of the achievable rate of the
original system., $0<a<1$. Then the first term in (42) (with the normalization
with respect to ${\textsf{M}}_{1}$) is the probability of input-constraint
violation, the third and the fourth terms are outage and confusion
probabilities, respectively, given a specific codeword [6, (75)]. With the
selected code $\mathcal{C}$, we can re-map the messages of all the codewords
violating the input constraint to one arbitrary vector which fulfills the
power constraint while the decoding region is kept unchanged by this
remapping. Under such a setting, the probability of the input power constraint
being violated is merged into the decoding error probability [6, Theorem 20].
In (44), we use concentration inequalities to upper bound the violation
probability of maximal power constraint and the peak constraint. A detailed
derivation can be seen in [21].

In the first step of SIC at user 2, the received signal can be equivalently
expressed as

$\displaystyle\tilde{Y}_{2}:=\sqrt{g_{2}}X_{1}+{\tilde{Z}}_{2},$ (45)

where
${\tilde{Z}}_{2}\sim\mathcal{N}(0,1),\,\tilde{Z}_{2}\mathbin{\rotatebox[origin={c}]{90.0}{$\vDash$}}X$,
and

$\displaystyle g_{2}:=\frac{h_{2}}{1+h_{2}{\textsf{P}}_{2}}.$ (46)

Fix any codeword $x_{1}^{n_{2}}(m),\,m\in\\{1,\ldots,\,{\textsf{M}}_{1}\\}$
from $\mathcal{C}$, the information density
$i(x_{1}^{n_{2}}(m);\tilde{Y}_{2}^{n_{2}})$ can be calculated as follows:

$\displaystyle{i(x_{1}^{n_{2}}(m);\tilde{Y}_{2}^{n_{2}})}=$
$\displaystyle\log\frac{(2\pi)^{-{n_{2}}/2}e^{-\frac{||\tilde{Y}_{2}^{n_{2}}-\sqrt{g_{2}}x_{1}^{n_{2}}(m)||^{2}}{2}}}{(2\pi(1+g_{2}{\textsf{P}}_{1}))^{-{n_{2}}/2}e^{-\frac{||\tilde{Y}_{2}^{n_{2}}||^{2}}{2(1+g_{2}{\textsf{P}}_{1})}}}=\sum_{j=1}^{n_{2}}W_{j},$
(47)

where the second equality in (47) is due to the memoryless channel, and we
define

$\displaystyle W_{j}:={\textsf{C}}(g_{2}{\textsf{P}}_{1})+\frac{\log e\cdot
g_{2}(x_{1,j}^{2}-{\textsf{P}}_{1}\tilde{Z}_{j}^{2})}{2(1+g_{2}{\textsf{P}}_{1})}+\frac{\log
e}{1+g_{2}{\textsf{P}}_{1}}\sqrt{g_{2}}x_{1,j}\tilde{Z}_{j}.$ (48)

The mean of $W_{j}$ conditioned on $x_{1,j}$ is as follows

$\displaystyle\mathds{E}_{{\tilde{Y}_{2,j}}|X_{1,j}=x_{1,j}}[W_{j}]={\textsf{C}}(g_{2}{\textsf{P}}_{1})+\frac{\log
e\cdot g_{2}(x_{1,j}^{2}-{\textsf{P}}_{1})}{2(1+g_{2}{\textsf{P}}_{1})}.$ (49)

Then the centralized information density of the $j$-th symbol conditioned on
$x_{1,j}$ is as follows:

$\displaystyle W_{j}-\mathds{E}_{{\tilde{Y}_{2,j}}|X_{1,j}=x_{1,j}}[W_{j}]=$
$\displaystyle\frac{\log
e}{1+g_{2}{\textsf{P}}_{1}}\left(\sqrt{g_{2}}x_{1,j}\tilde{Z}_{2,j}+g_{2}\frac{{\textsf{P}}_{1}}{2}(1-\tilde{Z}_{2,j}^{2})\right).$
(50)

To upper bound both the confusion (when the wrong codewords are treated as the
transmitted ones) and outage (when the correct codewords is treated not be
transmitted) probabilities, we derive the Berry-Esseen (B-E) ratio as follows.
First, the absolute centralized third moment of the information density given
$x_{1}^{n}$ can be upper bounded through standard analysis as follows

$\displaystyle\sum_{j=1}^{n_{2}}\mathds{E}_{{\tilde{Y}_{2,j}}|X_{1,j}=x_{1,j}}\left[\left|W_{j}-\mathds{E}_{{\tilde{Y}_{2,j}}|X_{1,j}=x_{1,j}}[W_{j}]\right|^{3}\right]\leq
4\sum_{j=1}^{n_{2}}\left(\frac{\log
e\cdot\sqrt{g_{2}}}{1+g_{2}{\textsf{P}}_{1}}\right)^{3}\left(8(\sqrt{g_{2}}{\textsf{P}}_{1})^{3}+2|x_{1,j}|^{3}\right).$
(51)

The variance of the information density given $x_{1}^{n_{2}}$ can be lower
bounded as follows:

$\displaystyle\sum_{j=1}^{n_{2}}\mbox{Var}_{\tilde{Y}_{2,j}|X_{1,j}=x_{1,j}}\left[W_{j}\right]$
$\displaystyle=\sum_{j=1}^{n_{2}}\mathds{E}_{\tilde{Y}_{2,j}|X_{1,j}=x_{1,j}}\left[\left(\frac{\log
e}{1+g_{2}{\textsf{P}}_{1}}\left(\sqrt{g_{2}}x_{1,j}\tilde{Z}_{2,j}+g_{2}\frac{{\textsf{P}}_{1}}{2}(1-\tilde{Z}_{2,j}^{2})\right)\right)^{2}\right]$
(52) $\displaystyle=\sum_{j=1}^{n_{2}}\left(\frac{\log
e}{1+g_{2}{\textsf{P}}_{1}}\right)^{2}g_{2}\left(x_{1,j}^{2}+g_{2}\frac{{\textsf{P}}_{1}^{2}}{2}\right)\geq{n_{2}}\left(\frac{\log
e\cdot g_{2}{\textsf{P}}_{1}}{\sqrt{2}(1+g_{2}{\textsf{P}}_{1})}\right)^{2},$
(53)

where (52) is from (50) and in (53), we remove the term related to
$x_{1,j}^{2}$.

After substituting (51) and (53) into Berry-Esseen theorem, we can upper bound
the Berry-Esseen ratio conditioned on $x_{1}^{n_{2}}$ as follows

$\displaystyle\frac{T}{V^{3/2}}$
$\displaystyle\leq\frac{4\sum_{j=1}^{n_{2}}\left(\frac{\sqrt{g_{2}}}{1+g_{2}{\textsf{P}}_{1}}\right)^{3}\left(8({\sqrt{g_{2}}}{\textsf{P}}_{1})^{3}+2|x_{1,j}|^{3}\right)}{\left(\frac{1}{2}{n_{2}}\left(\frac{\sqrt{g_{2}}{\textsf{P}}_{1}}{1+g_{2}{\textsf{P}}_{1}}\right)^{2}\right)^{\frac{3}{2}}}$
(54)
$\displaystyle:=\frac{4d_{1}\left(8n_{2}(\sqrt{g_{2}}{\textsf{P}}_{1})^{3}+2\sum_{j=1}^{n_{2}}|x_{1,j}|^{3}\right)}{\left(\frac{1}{2}d_{1}n_{2}\right)^{\frac{3}{2}}}$
(55) $\displaystyle\leq
c_{0}\cdot{n_{2}}^{-\frac{1}{2}}+c_{1}\cdot{n_{2}}^{a-\frac{1}{2}}:=B_{0}(n_{2}),$
(56)

where in (55), we define
$d_{1}:=\left(\frac{\sqrt{g_{2}}}{1+g_{2}{\textsf{P}}_{1}}\right)^{3}$, which
is a constant only dependent on ${\textsf{P}}_{1}$ and channel gains, but
independent of $n_{2}$; in (56), we define
$c_{0}:=\frac{62\sqrt{2}(\sqrt{g_{2}}{\textsf{P}}_{1})^{3}}{\sqrt{d_{1}}}$,
$c_{1}:=\frac{128\sqrt{2}{\textsf{P}}_{1}}{\sqrt{d_{1}}p^{1+a}}$ and the
inequality comes from the assumption of peak constraint
$|x_{1,j}|\leq{n_{1}}^{a}$, $a<\frac{1}{2}$, with the following upper bounding

$\displaystyle\sum_{j=1}^{n_{2}}|x_{1,j}|^{3}$
$\displaystyle\leq|\max\,x_{1,j}|\sum_{j=1}^{n_{2}}|x_{1,j}|^{2}\overset{(a)}{\leq}|\max\,x_{1,j}|{n_{1}}{\textsf{P}}_{1}\overset{(b)}{\leq}{n_{1}}^{a+1}{\textsf{P}}_{1}=\left(\frac{{n_{2}}}{p}\right)^{a+1}{\textsf{P}}_{1},$
(57)

where (a) comes from the maximal power constraint (5) and (b) comes from the
assumption of peak constraint.

Besides, the confusion probability conditioned on $x_{1}^{n_{2}}$ can be upper
bounded as

$\displaystyle
P_{\tilde{Y}_{2}^{n_{2}}}\left[\sum_{j=1}^{n_{2}}W_{j}>\log\gamma_{n_{2}}\right]$
$\displaystyle=\mathds{E}_{\tilde{Y}^{n_{2}}|X^{n}=x^{n}}\left[\exp\left(-\sum_{j=1}^{n_{2}}W_{j}\cdot\mathds{1}\left\\{\frac{\sum_{j=1}^{n_{2}}W_{j}}{n_{2}}>\frac{\log\gamma_{n_{2}}}{n_{2}}\right\\}\right)\right]$
(58) $\displaystyle\overset{(a)}{\leq}\frac{2}{\gamma_{n_{2}}}\left(\frac{\ln
2}{\sqrt{\pi d_{1}n_{2}}}+B_{0}(n_{2})\right)$ (59)
$\displaystyle:=\frac{B_{1}(n_{2})}{\gamma_{n_{2}}},$ (60)

where (58) is from the change of measure [6, (257)], (59) is from [6, Lemma
47] with (53), which is upper bounded by the same step used in (54). In
addition, we use (56) to bound the Berry-Esseen ratio. Note that
$B_{1}(n_{2})$ is a constant depending only on $n_{2}$, $h_{1}$, and
${\textsf{P}}_{2}$ but not on the realization $x^{n_{2}}$. Therefore, the
total confusion probability can be simply derived from (60) as

$\displaystyle{\textsf{M}}_{1}\cdot
P_{X^{n_{2}}}P_{\tilde{Y}_{2}^{n_{2}}}\left[\sum_{j=1}^{n_{2}}W_{j}>\log\gamma_{n_{2}}\right]$
$\displaystyle\leq\frac{{\textsf{M}}_{1}\cdot B_{1}(n_{2})}{\gamma_{n_{2}}}.$
(61)

Meanwhile, again by Berry-Esseen theorem, the outage probability conditioned
on ${x}^{n_{2}}$ can be expressed as follows

$\displaystyle
P_{\tilde{Y}_{2}^{n_{2}}|X_{1}^{n_{2}}=x_{1}^{n_{2}}}\left[\sum_{j=1}^{n_{2}}W_{j}\leq\log\gamma_{n_{2}}\right]$
$\displaystyle\leq Q\left(r_{m}({n_{2}})\right)+B_{0}(n_{2}),$ (62)

where

$\displaystyle
r_{m}({n_{2}}):=\frac{n_{2}\mu_{m}-\log{\textsf{M}}_{1}}{n_{2}\sigma_{m}},$
(63)

$\mu_{m}$ and $\sigma^{2}_{m}$ are defined as the RHS of (49) and the LHS of
(53), respectively, and $B_{0}(n_{2})$ is defined in (56).

By selecting $\gamma_{n_{2}}={\textsf{M}}_{1}$, we can then bound the
conditional confusion and outage probabilities in (61) and (62) respectively
as follows

$\displaystyle{\textsf{M}}_{1}\cdot
P_{X_{1}^{n_{2}}}P_{\tilde{Y}_{2}^{n_{2}}}\left[\sum_{j=1}^{n_{2}}W_{j}>\log\gamma_{n_{2}}\right]$
$\displaystyle\leq B_{1}(n_{2}),$ (64) $\displaystyle
P_{\tilde{Y}_{2}^{n_{2}}|X_{1}^{n_{2}}=x_{1}^{n_{2}}(m)}\left(\sum_{j=1}^{n_{2}}W_{j}\leq\log{\textsf{M}}_{1}\right)$
$\displaystyle=Q\left(r_{m}({n_{2}})\right)+B_{0}(n_{2}).$ (65)

Note that we need $r_{m}({n_{2}})>0$ since we consider the case in which
$\epsilon_{SIC1}<\frac{1}{2}$. Note also that $r_{m}({n_{2}})$ is a function
of $||x_{1}^{n_{2}}(m)||^{2}$. To derive an upper bound of (65), we resort to
finding a lower bound of $r_{m}({n_{2}})$ since $Q$-function is monotonically
decreasing. Furthermore, we aim to find a uniform lower bound of
$r_{m}({n_{2}})$, which will be independent of the given $x_{1}^{n_{2}}(m)$,
as shown as follows

$\displaystyle r_{m}({n_{2}})=$
$\displaystyle\frac{n_{2}{\textsf{C}}(g_{2}{\textsf{P}}_{1})+\frac{\log e\cdot
g_{2}}{2(1+g_{2}{\textsf{P}}_{1})}\left(||x_{1}^{n_{2}}(m)||^{2}-n_{2}{\textsf{P}}_{1}\right)-\log{\textsf{M}}_{1}}{\frac{\log
e}{2(1+g_{2}{\textsf{P}}_{1})}\sqrt{4g_{2}||x_{1}^{n_{2}}(m)||^{2}+2{n_{2}}g_{2}^{2}{\textsf{P}}_{1}^{2}}}$
(66) $\displaystyle\geq$
$\displaystyle\frac{2(1+g_{2}{\textsf{P}}_{1})\left({n_{2}}{\textsf{C}}(g_{2}{\textsf{P}}_{1})-\log{\textsf{M}}_{1}\right)-\log
e\cdot{n_{2}}g_{2}{\textsf{P}}_{1}}{\log
e\cdot\sqrt{4g_{2}||x_{1}^{n_{2}}(m)||^{2}+2{n_{2}}g_{2}^{2}{\textsf{P}}_{1}^{2}}}$
(67) $\displaystyle:=$ $\displaystyle r_{m,1}(n_{2}),$ (68)

where (66) is from (63).

We further lower bound $r_{m,1}(n_{2})$ by substituting the following upper
bound:

$\displaystyle||x_{1}^{n_{2}}(m)||^{2}\leq||x_{1}^{n_{1}}(m)||^{2}\leq
n_{1}{\textsf{P}}_{1},$ (69)

into the denominator of (67), while we need to ensure the numerator of (67) is
positive, i.e.,

$\displaystyle
n_{2}\geq\frac{\log\,{\textsf{M}}_{1}}{{\textsf{C}}(g_{2}{\textsf{P}}_{1})-\log
e\cdot\frac{g_{2}{\textsf{P}}_{1}}{2(1+g_{2}{\textsf{P}}_{1})}}.$ (70)

We will check the validity of the additional condition (70) at the end of the
proof by comparing it to our derived lower bound on $n_{2}$. Then we can lower
bound $r_{m,1}(n_{2})$ as follows

$\displaystyle{r}_{m,1}(n_{2})\geq$
$\displaystyle\frac{[2(1+g_{2}{\textsf{P}}_{1}){\textsf{C}}(g_{2}{\textsf{P}}_{1})-\log
e\cdot
g_{2}{\textsf{P}}_{1}]n_{2}-2(1+g_{2}{\textsf{P}}_{1})\log{\textsf{M}}_{1}}{\log
e\sqrt{4g_{2}n_{1}{\textsf{P}}_{1}+2n_{2}(g_{2})^{2}{\textsf{P}}_{1}^{2}}}$
$\displaystyle\geq$
$\displaystyle\frac{[2(1+g_{2}{\textsf{P}}_{1}){\textsf{C}}(g_{2}{\textsf{P}}_{1})-\log
e\cdot
g_{2}{\textsf{P}}_{1}]n_{2}-2(1+g_{2}{\textsf{P}}_{1})\log{\textsf{M}}_{1}}{\log
e\sqrt{4g_{2}{\textsf{P}}_{1}+2(g_{2})^{2}{\textsf{P}}_{1}^{2}}\sqrt{n_{1}}}$
(71)

for $m\in[1,\,{\textsf{M}}_{1}]$, where (71) is due to $n_{1}\geq n_{2}$.

After substituting (64), (65), (68), and (71) into (44), we can derive the
following result

$\displaystyle\frac{1}{\textsf{M}_{1}}\sum_{m_{1}=1}^{\textsf{M}_{1}}\mbox{Pr}(\hat{m}_{1}\neq
m_{1}\mbox{ at user }2|\,\,m_{1}\mbox{ is sent, }\mathcal{C}\mbox{ is used,
and use }n_{2}\mbox{ symbols to decode})$ $\displaystyle\leq$ $\displaystyle
Q\\!\left(\\!\\!\frac{[2(1\\!+\\!g_{2}{\textsf{P}}_{1}){\textsf{C}}(g_{2}{\textsf{P}}_{1})\\!-\\!\log
e\\!\cdot
g_{2}{\textsf{P}}_{1}]n_{2}\\!-\\!2(1\\!+\\!g_{2}{\textsf{P}}_{1})\log{\textsf{M}}_{1}}{\log
e\sqrt{4g_{2}{\textsf{P}}_{1}+2(g_{2})^{2}{\textsf{P}}_{1}^{2}}\sqrt{n_{1}}}\\!\\!\right)+c_{2}$
(72) $\displaystyle\leq$ $\displaystyle\epsilon_{SIC1},$ (73)

where

$\displaystyle c_{2}:=$ $\displaystyle
B_{0}(n_{2})+B_{1}(n_{2})+e^{-\frac{n_{2}}{2}}+e^{\frac{-n_{1}^{2a}}{2{\textsf{P}}_{1}}+\ln(2n_{1})}$
$\displaystyle=$ $\displaystyle\frac{2}{\sqrt{n_{2}}}\left(\frac{\ln
2}{\sqrt{\pi d_{1}}}+\frac{3}{2}(c_{0}+c_{1}\cdot
n_{2}^{a})\right)+e^{-\frac{n_{2}}{2}}+e^{\frac{-n_{1}^{2a}}{2{\textsf{P}}_{1}}+\ln(2n_{1})}$
(74) $\displaystyle\leq$
$\displaystyle\frac{1}{\sqrt{n_{2}d_{1}}}\left(\frac{2\ln
2}{\sqrt{\pi}}+64\sqrt{2}\left((\sqrt{g_{2}}{\textsf{P}}_{1})^{3}+2{\textsf{P}}_{1}\cdot
n_{2}^{a}\right)\right)+e^{-\frac{n_{2}}{2}}+e^{\frac{-n_{2}^{2a}}{2{\textsf{P}}_{1}}+\ln(2n_{2})},$
(75)

where (74) is from (56) and (60),
$d_{1}:=\left(\frac{\sqrt{g_{2}}}{1+g_{2}{\textsf{P}}_{1}}\right)^{3/2}$ and
$g_{2}:=\frac{h_{2}}{1+h_{2}{\textsf{P}}_{2}}$. Note that in (73) we enforce
the upper bound of the average error probability to be no larger than the
target value $\epsilon_{SIC1}$.

Now we further rearrange (72) and (73) by taking the inverse function of
$Q$-function as follows

$\displaystyle\frac{[2(1+g_{2}{\textsf{P}}_{1}){\textsf{C}}(g_{2}{\textsf{P}}_{1})-\log
e\cdot
g_{2}{\textsf{P}}_{1}]n_{2}-2(1+g_{2}{\textsf{P}}_{1})\log{\textsf{M}}_{1}}{\log
e\sqrt{4g_{2}{\textsf{P}}_{1}+2(g_{2})^{2}{\textsf{P}}_{1}^{2}}\sqrt{n_{1}}}\geq$
$\displaystyle Q^{-1}\left({\epsilon_{SIC1}}-c_{2}\right)$ (76)
$\displaystyle=$ $\displaystyle
Q^{-1}\left({\epsilon_{SIC1}}\right)+\mathcal{O}(c_{2}),$ (77)

where (76) is due to the fact that Q-function is monotonically decreasing and
(77) is due to the Q-function being continuous so we can apply the Taylor
expansion as [6, (267)]. By simple algebra, we can solve a lower bound of
$n_{2}$ shown as (8). Now compare (8) and (70), we find that (8) is stricter.
With a power backoff $\delta$, we complete the proof. $\square$

##  Appendix II Proof of the rate region in Theorem 1

To analyze the weaker user’s rate, we modify (44) by considering the channel
gain $g^{n_{1}}$ as part of the channel output and in contrast to the analysis
in Appendix I, here we use the random coding argument as follows

$\displaystyle\epsilon\leq$ $\displaystyle
P_{{X}_{1}^{n_{1}}}P_{\tilde{Y}^{n_{1}}G^{n_{1}}|{X}_{1}^{n_{1}}}\left(i({X}_{1}^{n_{1}};\tilde{Y}^{n_{1}})\leq\log{\textsf{M}}\right)+\textsf{M}\cdot\left[P_{{X}_{1}^{n_{1}}}P_{\tilde{Y}^{n_{1}}}\left(i({X}_{1}^{n_{1}};\tilde{Y}^{n_{1}})\geq\log{\textsf{M}}\right)\right]+P_{X_{1}^{n_{1}}}\left(X_{1}^{n_{1}}\notin\mathcal{F}_{IPC}^{n_{1}}\right),$
(78)

where

$\displaystyle
P_{\tilde{Y}^{n_{1}}}=\Pi_{j=1}^{n_{1}}P_{\tilde{Y}_{j}}\sim\mathcal{N}(\mathbf{0},{\bm{\Sigma}}),$
(79)

and

$\displaystyle\bm{\Sigma}:=\left(\begin{array}[]{cc}\bm{I}_{n_{2}}\cdot[1+h_{1}({\textsf{P}}_{1}+{\textsf{P}}_{2})]&\bm{0}\\\
\bm{0}&\bm{I}_{n_{1}-n_{2}}\cdot[1+h_{1}{\textsf{P}}_{1}]\\\
\end{array}\right).$ (82)

Based on (79), we have the following modified information density:

$\displaystyle i(X_{1}^{n_{1}};Y^{n_{1}})$ $\displaystyle=$
$\displaystyle\log\left(\frac{(2\pi)^{\frac{-n_{1}}{2}}(1+h_{1}{\textsf{P}}_{2})^{\frac{-n_{2}}{2}}1^{\frac{-(n_{1}-n_{2})}{2}}\exp\left(\frac{-\sum_{i=1}^{n_{2}}(Y_{1,i}-\sqrt{h_{1}}X_{1,i})^{2}}{2(1+h_{1}{\textsf{P}}_{2})}\right)\exp\left(\frac{-\sum_{i=n_{2}+1}^{n_{1}}(Y_{1,i}-\sqrt{h_{1}}X_{1,i})^{2}}{2}\right)}{(2\pi)^{\frac{-n_{1}}{2}}(1+h_{1}({\textsf{P}}_{1}+{\textsf{P}}_{2}))^{\frac{-n_{2}}{2}}(1+h_{1}{\textsf{P}}_{1})^{\frac{-(n_{1}-n_{2})}{2}}\exp\left(\frac{-\sum_{i=1}^{n_{2}}Y_{1,i}^{2}}{2(1+h_{1}({\textsf{P}}_{1}+{\textsf{P}}_{2}))}\right)\exp\left(\frac{-\sum_{i=n_{2}+1}^{n_{1}}Y_{1,i}^{2}}{2(1+h_{1}{\textsf{P}}_{1})}\right)}\right)$
(83) $\displaystyle=$ $\displaystyle n_{1}\bar{{\textsf{C}}}_{1}+\frac{\log
e}{2(1+h_{1}({\textsf{P}}_{1}+{\textsf{P}}_{2}))}\left[{h_{1}}\sum_{i=1}^{n_{2}}\left(X_{1,i}^{2}-\frac{{\textsf{P}}_{1}}{1+h_{1}{\textsf{P}}_{2}}\tilde{Z}_{1,i}^{2}\right)+2\langle\sqrt{h_{1}}X_{1}^{n_{2}},\tilde{Z}_{1}^{n_{2}}\rangle\right]+$
$\displaystyle\hskip 142.26378pt\frac{\log
e}{2(1+h_{1}{\textsf{P}}_{1})}\left[{h_{1}}\sum_{i=n_{2}+1}^{n_{1}}(X_{1,i}^{2}-{\textsf{P}}_{1}\tilde{Z}_{1,i}^{2})+2\langle\sqrt{h_{1}}X_{1,n_{2}+1}^{n_{1}},\tilde{Z}_{1,n_{2}+1}^{n_{1}}\rangle\right],$
(84)

where in (84) we define $\bar{{\textsf{C}}}_{1}$ and ${g}_{1}$ as in (12) and
(14), respectively. Based on (84) we can derive the variance of the modified
information density as follows. For $i\in[1,\,n_{2}]$, from (84) we can see
that

$\displaystyle\mathds{E}[i(X_{1,i};Y_{i})]={\textsf{C}}\left(\frac{h_{1}{\textsf{P}}_{1}}{1+h_{1}{\textsf{P}}_{2}}\right),$
(85)

and

$\displaystyle\mathop{\rm Var}(i(X_{1,i};Y_{i}))=$
$\displaystyle\frac{\log^{2}e}{4(1+h_{1}({\textsf{P}}_{1}+{\textsf{P}}_{2}))^{2}}\left\\{\mathds{E}\left[\left({h_{1}}\left(X_{1,i}^{2}-\frac{{\textsf{P}}_{1}}{1+h_{1}{\textsf{P}}_{2}}\tilde{Z}_{1,i}^{2}\right)+2\sqrt{h_{1}}X_{1,i},\tilde{Z}_{1,i}\right)^{2}\right]\right\\}$
(86) $\displaystyle=$ $\displaystyle\frac{\log^{2}e\cdot
h_{1}{\textsf{P}}_{1}}{1+h_{1}({\textsf{P}}_{1}+{\textsf{P}}_{2})},$ (87)

where in (87) we use the fact that $\mathds{E}[X_{1,i}^{2}]={\textsf{P}}_{1}$,
$\mathds{E}[X_{1,i}^{3}]=0$, $\mathds{E}[X_{1,i}^{4}]=3{\textsf{P}}_{1}^{2}$,
$\mathds{E}[\tilde{Z}_{1,i}^{2}]=1+h_{1}{\textsf{P}}_{2}$,
$\mathds{E}[\tilde{Z}_{1,i}^{3}]=0$, and
$\mathds{E}[\tilde{Z}_{1,i}^{4}]=3(1+h_{1}{\textsf{P}}_{2})^{2}$. On the other
hand, if $i\in[n_{2}+1,\,n_{1}]$, by setting ${\textsf{P}}_{2}=0$ in (87), we
have

$\displaystyle\mathds{E}[i(X_{1,i};Y_{i})]$
$\displaystyle={\textsf{C}}(h_{1}{\textsf{P}}_{1}),$ (88)
$\displaystyle\mathop{\rm Var}(i(X_{1,i};Y_{i}))$
$\displaystyle=\frac{\log^{2}e\cdot
h_{1}{\textsf{P}}_{1}}{1+h_{1}{\textsf{P}}_{1}}.$ (89)

Then from (87) and (89) we compare the dispersion is as follows

$\displaystyle\bar{{\textsf{V}}}_{1}$
$\displaystyle=\frac{1}{n_{1}}\mathop{\rm
Var}\left(i(X_{1}^{n_{1}};Y^{n_{1}})\right)$ (90)
$\displaystyle=\log^{2}e\cdot\left(p\frac{h_{1}{\textsf{P}}_{1}}{1+h_{1}({\textsf{P}}_{1}+{\textsf{P}}_{2})}+(1-p)\frac{h_{1}{\textsf{P}}_{1}}{1+h_{1}{\textsf{P}}_{1}}\right).$
(91)

To show the convergence of the Berry-Esseen ratio of the weaker user, in
addition to (91), we need to derive an upper bound of the absolute centralized
third moment of the information density, which is shown as follows. We denote
$W_{j}$ by the information density of the weaker user at the $j$-th symbol.
Then from the above we know that

$\displaystyle W_{j}-\mu_{W_{j}}=\left\\{\begin{array}[]{ll}\frac{\log
e}{2(1+h_{1}({\textsf{P}}_{1}+{\textsf{P}}_{2}))}\left({h_{1}}\left(X_{1,i}^{2}-\frac{{\textsf{P}}_{1}}{1+h_{1}{\textsf{P}}_{2}}\tilde{Z}_{1,i}^{2}\right)+2\sqrt{h_{1}}X_{1,i},\tilde{Z}_{1,i}\right),&1\leq
j\leq n_{2}\\\ \frac{\log
e}{2(1+h_{1}{\textsf{P}}_{1})}\left({h_{1}}\left(X_{1,i}^{2}-{\textsf{P}}_{1}\tilde{Z}_{1,i}^{2}\right)+2\sqrt{h_{1}}X_{1,i},\tilde{Z}_{1,i}\right),&n_{2}+1\leq
j\leq n_{1},\end{array}\right.$ (94)

and we can bound the absolute centralized third moment of the information
density as follows:

$\displaystyle\sum_{j=1}^{n_{1}}\mathds{E}_{XY}[|W_{j}-\mu_{W_{j}}|^{3}]\leq$
(95)
$\displaystyle\frac{9}{8}\log^{3}e\cdot\left\\{\frac{1}{(1+h_{1}({\textsf{P}}_{1}+{\textsf{P}}_{2}))^{3}}\sum_{j=1}^{n_{2}}\\!\\!\\!\\!\left(\\!\\!\\!\\!9\left(\mathds{E}\left[\left|h_{1}X_{1,j}^{2}\right|^{3}\right]\right)\\!\\!+\mathds{E}\left[\left|\frac{{\textsf{P}}_{1}}{1+h_{2}{\textsf{P}}_{2}}\tilde{Z}_{1,j}^{2}\right|^{3}\right]\right.\right.\\!\\!\\!\\!+\\!\\!\left.\mathds{E}\left[\left|2\sqrt{h_{1}}X_{1,j}\tilde{Z}_{1,j}\right|^{3}\right]\\!\\!\right)$
$\displaystyle\hskip
28.45274pt+\frac{1}{(1+h_{1}{\textsf{P}}_{1})^{3}}\sum_{j=n_{2}+1}^{n_{1}}\left(9\left(\mathds{E}\left[\left|h_{1}X_{1,j}^{2}\right|^{3}\right]+\mathds{E}\left[\left|{\textsf{P}}_{1}\tilde{Z}_{1,j}^{2}\right|^{3}\right]\right)+\left.\left.\mathds{E}\left[\left|2\sqrt{h_{1}}X_{1,j}\tilde{Z}_{1,j}\right|^{3}\right]\right)\right.\right\\}.$
(96)

We can easily see that each term of the summation in (96) is finite, which can
be calculated by the absolute moments of $X_{1,j}^{2}$ and
$\tilde{Z}_{1,j}^{2}$. Therefore, we can further express (96) as follows:

$\displaystyle\sum_{j=1}^{n_{1}}\mathds{E}_{XY}[|W_{j}-\mu_{W_{j}}|^{3}]$
$\displaystyle\leq a\cdot n_{2}+b(n_{1}-n_{2})$ (97)
$\displaystyle\leq\left\\{\begin{array}[]{ll}a\cdot n_{1},&\mbox{ if }a\geq
b\\\ b\cdot n_{1},&\mbox{ else },\end{array}\right.$ (100)

where in (97), we collect the coefficients of $n_{2}$ and $n_{1}-n_{2}$ as $a$
and $b$, respectively, where $a>0$ and $b>0$. Then (96) is a linear scale of
$n_{1}$. After dividing (96) by $(\mathop{\rm
Var}(X^{n_{1}};\tilde{Y}^{n_{1}}))^{\frac{3}{2}}$, which can be easily seen
from (91), we can observe that the Berry-Esseen ratio of the weaker user is
upper bounded by $\mathcal{O}(n^{\frac{1}{2}})$. Based on the above derived
results, we can follow the same steps in Appendix I to complete the proof.
$\square$

##  Appendix III Proof of
$\Pr\left(\sum_{j=1}^{n_{2}}\left(X_{1,j}+X_{2,j}\right)^{2}+\sum_{j=n_{2}+1}^{n_{1}}X_{1,j}^{2}>n_{1}{\textsf{P}}_{T}\right)\leq
e^{-\mathcal{O}(n_{2})}$

To simplify the expression, we define
$\tilde{X}_{1,j}:=X_{1,j}/\sqrt{\bar{{\textsf{P}}}_{11}}$,
$\tilde{X}_{2,j}:=X_{2,j}/\sqrt{\bar{{\textsf{P}}}_{2}}$, and
$Z_{j}:=\tilde{X}_{1,j}\tilde{X}_{2,j}$ where
$\tilde{X}_{1,j},\,\tilde{X}_{2,j}\in\mathcal{N}(0,1)$,
$\tilde{X}_{1,j}\mathbin{\rotatebox[origin={c}]{90.0}{$\vDash$}}\tilde{X}_{2,j}$,
$j=1,\cdots,n_{2}$. Then

$\displaystyle{\mbox{Pr}}\left(\sum_{j=1}^{n_{2}}X_{1,j}X_{2,j}\geq
n_{2}\delta\right)={\mbox{Pr}}\left(\sum_{j=1}^{n_{2}}Z_{j}\geq n_{2}\cdot
t\right),$ (101)

where
$t:=\frac{\delta}{\sqrt{\bar{{\textsf{P}}}_{11}\bar{{\textsf{P}}}_{2}}}.$ We
can derive the moment generating function of $Z_{j}$ as follows:

$\displaystyle\mathds{E}\left[e^{\lambda
Z_{j}}\right]=\frac{1}{\sqrt{1-\lambda^{2}}}.$ (102)

By Chernoff bound, we can derive the following:

$\displaystyle{\mbox{Pr}}\left(\sum_{j=1}^{n_{2}}Z_{j}\geq n_{2}\cdot
t\right)\leq\frac{(1-\lambda^{2})^{\frac{-n_{2}}{2}}}{e^{\lambda n_{2}t}}.$
(103)

By properly selecting an upper bound of the RHS of (103), e.g.,

$\displaystyle\frac{1}{\sqrt{1-\lambda^{2}}}\leq e^{2\lambda^{2}},$ (104)

where $|\lambda|<0.8$, the RHS of (104) fulfills the definition of sub-
exponential distribution [22, Def. 2.2] with the parameters
$(\nu,b)=(2,\,1/0.8)$. This is because $|\lambda|<\frac{1}{b}$ by definition,
which can be proved by simple calculus. Then we can invoke the sub-exponential
tail bound [22, (2.20)] to derive (22) as follows:

$\displaystyle{\mbox{Pr}}\left(\sum_{j=1}^{n_{2}}Z_{j}\geq n_{2}\cdot
t\right)\leq e^{\frac{-n_{2}\cdot
t^{2}}{2\nu^{2}}}=e^{\frac{-n_{2}\cdot\delta^{2}}{8\bar{{\textsf{P}}}_{11}\bar{{\textsf{P}}}_{2}}}.$
(105)

Now we consider the outage of (18) with the definition
$A_{11}:=\sum_{j=1}^{n_{2}}X_{1,j}^{2},\,A_{2}:=\sum_{j=1}^{n_{2}}X_{2,j}^{2},\,C^{\prime}:=2\sum_{j=1}^{n_{2}}X_{1,j}X_{2,j}-2n_{2}\delta,$
and $A_{12}:=\sum_{j=n_{2}+1}^{n_{1}}X_{1,j}^{2}$, as follows:

$\displaystyle\Pr\left(\sum_{j=1}^{n_{2}}\left(X_{1,j}+X_{2,j}\right)^{2}+\sum_{j=n_{2}+1}^{n_{1}}X_{1,j}^{2}>n_{1}{\textsf{P}}_{T}\right)$
$\displaystyle\hskip
14.22636pt:=\Pr\left(A_{11}+A_{2}+A_{12}+C^{\prime}>n_{1}{\textsf{P}}_{T}-2n_{2}\delta\right)$
(106) $\displaystyle\hskip
14.22636pt=\Pr\left(A_{11}+A_{2}+A_{12}+C^{\prime}>n_{1}{\textsf{P}}_{T}-2n_{2}\delta\mbox{
and }C^{\prime}>0\right)+$ $\displaystyle\hskip
170.71652pt\Pr\left(A_{11}+A_{2}+A_{12}+C^{\prime}>n_{1}{\textsf{P}}_{T}-2n_{2}\delta\mbox{
and }C^{\prime}\leq 0\right)$ (107) $\displaystyle\hskip
14.22636pt\leq\Pr\left(C^{\prime}>0\right)+\Pr\left(A_{11}+A_{2}+A_{12}+C^{\prime}>n_{1}{\textsf{P}}_{T}-2n_{2}\delta\mbox{
and }C^{\prime}\leq 0\right)$ (108) $\displaystyle\hskip
14.22636pt=\Pr\left(C^{\prime}>0\right)+\int_{c^{\prime}\in\mbox{supp}(C^{\prime})\mbox{
and }c^{\prime}\leq
0}\Pr\left(A_{11}+A_{2}+A_{12}>n_{1}{\textsf{P}}_{T}-2n_{2}\delta+|c^{\prime}|\right)dF_{C^{\prime}}(c^{\prime})$
(109) $\displaystyle\hskip 14.22636pt\leq
e^{\frac{-n_{2}\cdot\delta^{2}}{8\bar{{\textsf{P}}}_{11}}}+\int_{c^{\prime}\in\mbox{supp}(C^{\prime})\mbox{
and }c^{\prime}\leq
0}\Pr\left(A_{11}+A_{2}+A_{12}>n_{1}{\textsf{P}}_{T}-2n_{2}\delta+|c^{\prime}|\right)dF_{C^{\prime}}(c^{\prime}),$
(110)

where in (106) we define $C^{\prime}:=C-2n_{2}\delta$, in (107) less
conditions lead to larger probabilities, in (110) we use (105) to upper bound
the first term in (109). Now define
$A_{11}^{\prime}:=A_{11}-n_{2}{\textsf{P}}_{11}+\frac{2n_{2}\delta}{3}$,
$A_{12}^{\prime}:=A_{12}-(n_{1}-n_{2}){\textsf{P}}_{12}+\frac{2n_{2}\delta}{3}$,
$A_{2}^{\prime}:=A_{2}-n_{2}{\textsf{P}}_{2}+\frac{2n_{2}\delta}{3}$. Then we
can further express the integrand in (109) as follows:

$\displaystyle\Pr\left(A_{11}+A_{2}+A_{12}>n_{1}{\textsf{P}}_{T}-2n_{2}\delta+|c^{\prime}|\right)$
$\displaystyle\leq\Pr(A_{11}^{\prime}+A_{2}^{\prime}+A_{12}^{\prime}>|c^{\prime}|)$
(111) $\displaystyle\leq\Pr(A_{11}^{\prime}+A_{2}^{\prime}+A_{12}^{\prime}>0)$
(112)
$\displaystyle\leq\Pr(A_{11}^{\prime}>0)+\Pr(A_{2}^{\prime}>0)+\Pr(A_{12}^{\prime}>0),$
(113)

where (111) is due to the definitions of $A_{11},\,A_{12},\,A_{2}$ with the
inequality (21), in (113) we recursively use (107) and (108). Combine (110)
and (113) with the fact that with a proper power backoff, (113) can be upper
bounded by $e^{-\mathcal{O}(n_{2})}$ from the concentration inequality [23,
(4.3)], we have (22). To derive (23), we can follow the same steps to derive
(113) with a slight modification, which completes the proof. $\square$.

##  Appendix IV Proof of Proposition 1

We first rearrange (21) as follows

$\displaystyle
p{\textsf{P}}_{1,1}+(1-p){\textsf{P}}_{1,2}\leq{\textsf{P}}_{T}-p{\textsf{P}}_{2}.$
(114)

We then generalize the dispersion expression with IPC in (91) to the following
form:

$\displaystyle\bar{{\textsf{V}}}_{1}^{\prime}=\log^{2}e\cdot\left(p\frac{g_{1}^{\prime}{\textsf{P}}_{1,1}}{1+g_{1}^{\prime}{\textsf{P}}_{1,1}}+(1-p)\frac{g_{2}^{\prime}{\textsf{P}}_{1,2}}{1+g_{2}^{\prime}{\textsf{P}}_{1,2}}\right),$
(115)

where $g_{1}^{\prime}:=\frac{h_{1}}{1+h_{1}{\textsf{P}}_{2}}$ and
$g_{2}^{\prime}:=h_{1}$ are the equivalent channel gains when the indices of
code symbols are from 1 to $n_{2}$ and from $n_{2}+1$ to $n_{1}$,
respectively. With a power backoff $\delta$, we can derive
${\bar{\textsf{V}}}_{1}^{\prime}$ as (28). Similarly, we can derive
${\bar{\textsf{C}}}^{\prime}$ from (85) as (27), which completes the proof.
$\square$

## References

  * [1] P.-H. Lin, S.-C. Lin, and E. A. Jorswieck, “Early decoding for Gaussian broadcast channels with heterogeneous blocklength constraints,” in _Proc. IEEE Int. Symp. Inf. Theory (ISIT) 2021, Melbourne, Australia_ , 2021\.
  * [2] P.-H. Lin, S.-C. Lin, P.-W. Chen, M. Mross, and E. A. Jorswieck, “Rate region of Gaussian broadcast channels with heterogeneous blocklength constraints,” in _Proc. IEEE Int. Conf. on Communi. (ICC) 2022, Seoul, Korea_ , 2022.
  * [3] 3GPP, “Study on scenarios and requirements for next generation access technologies,” _Technical Report 38.913, Release 14_ , Oct. 2016.
  * [4] ——, “Summary of email discussion on the link level evaluation for LTE URLLC,” _Technical Report, TSG RAN WG1 Meeting No.92, R1—1801385_ , Mar. 2018.
  * [5] G. Durisi, T. Koch, and P. Popovski, “Toward massive, ultrareliable, and low-latency wireless communication with short packets,” _Proceedings of the IEEE_ , vol. 104, no. 9, pp. 1711–1726, 2016.
  * [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, May 2010.
  * [7] E. MolavianJazi and J. N. Laneman, “A second-order achievable rate region for Gaussian multi-access channels via a central limit theorem for functions,” _IEEE Trans. Inf. Theory_ , vol. 61, no. 12, pp. 6719–6733, 2015.
  * [8] J. Scarlett and V. Y. F. Tan, “Second-order asymptotics for the Gaussian MAC with degraded message sets,” _IEEE Trans. Inf. Theory_ , vol. 61, no. 12, pp. 6700–6718, 2015.
  * [9] V. Y. F. Tan and O. Kosut, “On the dispersions of three network information theory problems,” _IEEE Trans. Inf. Theory_ , vol. 60, no. 2, pp. 881–903, 2014.
  * [10] A. Ünsal and J. Gorce, “The dispersion of superposition coding for Gaussian broadcast channels,” in _Proc. IEEE Information Theory Workshop (ITW)_ , 2017, pp. 414–418.
  * [11] S. L. Fong and V. Y. F. Tan, “A proof of the strong converse theorem for Gaussian broadcast channels via the Gaussian Poincaré inequality,” _IEEE Trans. Inf. Theory_ , vol. 63, no. 12, pp. 7737–7746, 2017.
  * [12] P. Sheldon, D. Tuninetti, and B. Smida, “The Gaussian broadcast channels with a hard deadline and a global reliability constraint,” in _Proc. IEEE Int. Conf. on Communi. (ICC)_ , 2021, pp. 1–6.
  * [13] J. Scarlett, “On the dispersions of the Gel’fand-Pinsker channel and dirty paper coding,” _IEEE Trans. Inf. Theory_ , vol. 61, no. 9, pp. 4569–4586, Sept. 2015.
  * [14] P. K. Korrai, E. Lagunas, A. Bandi, S. K. Sharma, and S. Chatzinotas, “Joint power and resource block allocation for mixed-numerology-based 5G downlink under imperfect CSI,” _IEEE Open Journal of the Communications Society_ , vol. 1, pp. 1583–1601, 2020.
  * [15] Y. Xu, C. Shen, T. Chang, S. Lin, Y. Zhao, and G. Zhu, “Transmission energy minimization for heterogeneous low-latency NOMA downlink,” _IEEE Trans. Wireless Commun._ , vol. 19, no. 2, pp. 1054–1069, 2020.
  * [16] D. Tuninetti, B. Smida, N. Devroye, and H. Seferoglu, “Scheduling on the Gaussian broadcast channel with hard deadlines,” in _Proc. IEEE Int. Conf. on Communi. (ICC)_ , 2018, pp. 1–7.
  * [17] K. Azarian, H. E. Gamal, and P. Schniter, “On the achievable diversity-multiplexing tradeoff in half-duplex cooperative channels,” _IEEE Trans. Inf. Theory_ , vol. 51, no. 12, pp. 4152–4272, Dec. 2005.
  * [18] A. Jovicic and P. Viswanath, “Cognitive radio: an information-theoretic perspective,” _IEEE Trans. Inf. Theory_ , vol. 55, no. 9, pp. 3945–3958, Sep. 2009.
  * [19] J. Hou and G. Kramer, “Short message noisy network coding with a decode-forward option,” _IEEE Trans. Inf. Theory_ , vol. 62, no. 1, pp. 89–107, 2016.
  * [20] C. Sahin, L. Liu, and E. Perrins, “Early decoding for transmission over finite transport blocks,” in _Proc. IEEE International Symposium on Information Theory (ISIT)_ , 2014, pp. 1558–1562.
  * [21] P.-H. Lin, S.-C. Lin, P.-W. Chen, M. Mross, and E. A. Jorswieck, “Gaussian broadcast channels in heterogeneous blocklength constrained networks,” _arxiv:2109.07767_ , 2021.
  * [22] M. Wainwright, _High-Dimensional Statistics: A Non-Asymptotic Viewpoint_ , 1st ed. Cambridge university press, 2019\.
  * [23] B. Laurent and P. Massart, “Adaptive estimation of a quadratic functional by model selection,” _The Annals of Statistics_ , vol. 28, no. 5, pp. 1302 – 1338, 2000. [Online]. Available: https://doi.org/10.1214/aos/1015957395