# New Inner and Outer Bounds for Gaussian Broadcast Channels with
Heterogeneous Blocklength Constraints

Marcel Mross, Pin-Hsun Lin and Eduard A. Jorswieck  Institute for
Communications Technology, Technische Universität Braunschweig, Germany
E-mail: {Mross, Lin<EMAIL_ADDRESS>
###### Abstract

We investigate novel inner and outer bounds on the rate region of a 2-user
Gaussian broadcast channel with finite, heterogeneous blocklength constraints
(HB-GBC). In particular, we introduce a new, modified Sato-type outer bound
that can be applied in the finite blocklength regime and which does not
require the same marginal property. We then develop and analyze composite
shell codes, which are suitable for the HB-GBC. Especially, to achieve a lower
decoding latency for the user with a shorter blocklength constraint when
successive interference cancellation is used, we derive the number of symbols
needed to successfully early decode the other user’s message. We numerically
compare our derived outer bound to the best known achievable rate regions.
Numerical results show that the new early decoding performance in terms of
latency reduction is significantly improved compared to the state of the art,
and it performs very close to the asymptotic limit.

##  I Introduction

As one of the application areas of 5G and beyond, Ultra-Reliable Low Latency
Communication (URLLC) has attracted intensive attention. To achieve low
latency, asymptotic assumptions on the codeword sizes are no longer valid,
which motivates the finite blocklength analysis. Since the seminal work in
[1], there have been several extensions to multi-user channels, like the
multiple access channel (MAC) [2], the MAC with degraded message sets [3], the
broadcast channel [4, 5, 6] and the interference channel [7].

Due to different demands on Quality of Service for the wide range of available
applications in modern communication systems, the latency requirements can
differ among users. Therefore, in a realistic model, the two users may be
required to decode after having received differing numbers of symbols $n_{1}$
and $n_{2}$, which leads to the model of the Gaussian broadcast channel with
heterogeneous blocklength constraints (HB-GBC) [8, 9, 10, 11, 12]. The work
[10] shows the possibility to perform superposition coding at the encoder and
successive interference cancellation (SIC) at the decoder. The sufficient
conditions for a successful SIC are: 1) the channel gain of the user with the
stricter blocklength constraint is larger than that of the other user and 2)
the shorter blocklength between the two codewords is still sufficiently large.
This technique is called early decoding, since the longer codeword is decoded
before it is entirely received at the non-intended receiver. How early the
decoder can successfully decode was derived for asymptotic cases [13] and as a
second-order finite blocklength result [10].

However, it is unclear how the heterogeneous blocklength assumption affects
the capacity and, thus, how much the rates derived in [11, 12] can be
improved. Sato’s outer bound [14], a well-known asymptotic outer bound for
broadcast channels, requires the same marginal property (SMP), which is
invalid in the finite blocklength regime. Also, the authors of [10, 11, 12]
only consider independent and identically distributed (i.i.d.) Gaussian
codebooks, because shell codebooks bring unique challenges to the analysis in
the heterogeneous blocklength scenario. Since shell codes are known to have a
smaller dispersion than i.i.d. Gaussian codebooks, it is highly desirable to
apply them in early decoding.

The main contributions of this paper are as follows.

  * •

First, we present a novel scheme of applying a Sato-type outer bound technique
in the finite blocklength regime, where the SMP is invalid in general. Our
scheme still allows the minimization of the outer bound with respect to the
joint distributions having the same marginals. We then apply this idea to the
HB-GBC, leading to the first existing outer bound for this channel model. We
combine the bounding technique with the information spectrum converse method
[15] to derive the outer bounds.

  * •

Second, we improve the achievable rate and the latency of the ED scheme
proposed in [10] for the considered model. For this purpose, we introduce a
new class of shell codes, namely, composite shell codes, which is suitable for
the heterogeneous blocklength scenario and leads to a second-order latency
performance close to the asymptotic limit. The derived outer bounds are
numerically compared to the best known achievable rate regions.

  * •

Additionally, we investigate some other fundamental properties of the GBC in
the finite blocklength regime, which are different to those in the asymptotic
case.

This paper is organized as follows. In Section II, we introduce the system
model and some preliminaries. In Section III, we present our new outer bound
technique and the application to the GBC with heterogeneous blocklength
constraints. In Section IV, we present the improved achievability results. The
properties of the GBC in the finite blocklength regime are discussed in
Section V. Section VI shows some numerical calculations of the results from
the previous sections.

##  II Preliminaries and System Model

###  II-A Notation

Real constants are denoted by uppercase sans serif letters like $\mathsf{X}$
and $\mathsf{Y}$. Vectors are written with a superscript that indicates the
dimension of that vector, e.g. $x^{n}=[x_{1},...,x_{n}]$. The individual
elements of that vector are indexed by subscripts, i.e., $x_{i}$. If the
vector itself already has a subscript, e.g. $x_{1}^{n}$, then the index
follows that subscript, separated by a comma: $x_{1,i}$. The concatenation of
two vectors $x^{n}$ and $y^{m}$ is written as $[x^{n},y^{m}]$ and the vector
that consists solely of zeros is denoted by $\mathbf{0}^{n}$. Sets are denoted
by calligraphic letters like $\mathcal{X}$ and $\mathcal{Y}$.

Random variables are denoted by uppercase letters like $X$ and $Y$,
realizations of random variables are written in lowercase, like $x$ and $y$.
We write $X\sim P_{X}$ to indicate that the random variable $X$ follows the
distribution $P_{X}$, where $P_{X}$ denotes a probability measure. The
expectation and the variance of a random variable $X$ are denoted by
$\mathbb{E}[X]$ and $\mathrm{Var}[X]$, respectively, and the covariance of $X$
and $Y$ is denoted by $\mathrm{Cov}[X,Y]$. If two random variables $X$ and $Y$
are independent, we write
$X\mathrel{\bot}\joinrel\mathrel{\mkern-5.0mu}\joinrel\mathrel{\bot}Y$. A
random vector is indicated by a superscript like a deterministic vector. We
will denote the Gaussian distribution with mean $\mu$ and variance
$\sigma^{2}$ by $\mathcal{N}(\mu,\sigma^{2})$ and the $Q$-function by
$Q(\cdot)$. We will also use the indicator function $\mathbbm{1}(\cdot)$ and
the Landau symbols $o(\cdot)$ and $O(\cdot)$. We also denote the Gaussian
capacity function by $\mathsf{C}(x)=\frac{1}{2}\cdot\log(1+x)$, the shell
dispersion by $\mathsf{V}(x):=\frac{\log^{2}e}{2}\frac{x(x+2)}{(x+1)^{2}}$,
and the i.i.d. Gaussian dispersion by
$\mathsf{V}_{\mathrm{G}}(x):=\log^{2}e\frac{x}{x+1}$. All logarithms are taken
to the base 2.

###  II-B System Model

We consider a two-user Gaussian broadcast channel with heterogeneous
blocklength constraints (HB-GBC) and quasi-static block flat-fading. The
received signal at user $k$ at time $i\in\\{1,...,n_{k}\\}$ is

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

where $k=1,2$ and $Z_{1,i}\sim\mathcal{N}(0,1)$ and
$Z_{2,i}\sim\mathcal{N}(0,1)$ are i.i.d. and mutually independent additive
white Gaussian noise random variables. The transmitter as well as the
receivers have perfect knowledge of the channel gains $h_{k}$. Without loss of
generality, in this paper we will always assume $n_{1}\geq n_{2}$.
Furthermore, we will consider the case that the shorter blocklength constraint
belongs to the user with the larger channel gain, i.e., $h_{2}\geq h_{1}$.

######  Definition 1.

An
$(n_{1},n_{2},\mathsf{M}_{1},\mathsf{M}_{2},\epsilon,\mathcal{F}^{n_{1}})$-code
for an HB-GBC $W$ consists of:

  * •

two message sets $\mathcal{M}_{k}=\\{1,...,\mathsf{M}_{k}\\}$, $k=1,2$,

  * •

an encoder
$f_{n_{1}}:\mathcal{M}_{1}\times\mathcal{M}_{2}\to\mathcal{F}^{n_{1}}$, where
$\mathcal{F}^{n_{1}}\subseteq\mathbb{R}^{n_{1}}$ is some pre-defined set of
feasible codewords,

  * •

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

such that the average system error probability satisfies

$\displaystyle\epsilon^{({n_{1},n_{2}})}$ $\displaystyle:=$
$\displaystyle\Pr\left[\hat{M}_{1}\neq M_{1}\text{ or }\hat{M}_{2}\neq
M_{2}\right]\leq\epsilon.$ (2)

We denote by $\mathcal{C}_{W}(n_{1},n_{2},\mathcal{F}^{n_{1}},\epsilon)$ the
set of all second-order achievable message size pairs for an HB-GBC $W$, i.e.,
pairs of the form
$\log(\mathsf{M}_{k})=n_{k}\mathsf{C}_{k}+\sqrt{n_{k}\mathsf{V}_{k}}+o\left(\sqrt{n_{k}}\right),\quad
k=1,2,$ with some positive constants $\mathsf{C}_{k}$ and $\mathsf{V}_{k}$ for
which an
$(n_{1},n_{2},\mathsf{M}_{1},\mathsf{M}_{2},\epsilon,\mathcal{F}^{n_{1}})$-code
exists.

It is common to assume a power constraint $\mathsf{P}$ on the codewords. In
that case, the feasible set of channel inputs is

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

When superposition coding is used at the encoder, the codewords
$x_{1}^{n_{1}}$ and $x_{2}^{n_{2}}$ are generated independently for user 1 and
user 2 and then superimposed to get the channel input
$x^{n_{1}}=x_{1}^{n_{1}}+[x_{2}^{n_{2}},\mathbf{0}^{n_{1}-n_{2}}]$, where
$\mathbf{0}^{n_{1}-n_{2}}$ is an $(n_{1}-n_{2})$-dimensional vector consisting
solely of zeros. In that case, we also call (3) sum power constraint (SPC).
Alternatively, we can also impose a power constraint on the individual
codewords $x_{1}^{n_{1}}$ and $x_{2}^{n_{2}}$. In that case, which we call the
individual power constraint (IPC), the feasible set of codewords is defined as

$\displaystyle\mathcal{F}^{n_{1}}=\mathcal{F}_{\mathrm{max}}^{n_{1},n_{2}}(\mathsf{P}_{1},\mathsf{P}_{2}):=$
$\displaystyle\left\\{x^{n_{1}}=x_{1}^{n_{1}}+[x_{2}^{n_{2}},\mathbf{0}^{n_{1}-n_{2}}]:\|x_{k}^{n_{k}}\|^{2}\leq
n_{k}\mathsf{P}_{k},\;k=1,2\right\\}.$ (4)

In this paper, we will use individual error probabilities $\epsilon_{1}$ at
user 1 and $\epsilon_{\mathrm{SIC},1}$ and $\epsilon_{\mathrm{SIC},2}$ for the
different decoding steps of SIC at user 2. In order to conform with the
constraint in (2), we have to choose these individual error probabilities
properly s.t.

$\displaystyle\epsilon_{1}+\epsilon_{\mathrm{SIC},1}+\epsilon_{\mathrm{SIC},2}$
$\displaystyle\leq$ $\displaystyle\epsilon.$ (5)

If user 2 can decode user 1’s message $m_{1}$ from the first $n_{2}$ received
symbols $Y_{2,1},...,Y_{2,n_{2}}$ while fulfilling (2), we call that a
_successful early decoding_ [10].

Throughout this paper, we will use the symbol $p:=n_{2}/n_{1}\in(0,1]$ for the
blocklength ratio and $\bar{p}:=1-p$.

##  III A Modified Sato-Type Outer Bound

In the following, we use $\mathcal{W}_{\mathrm{SM}}(W)$ to denote the set of
channels having the same conditional marginals $P_{Y_{1}|X}(y_{1}|x)$ and
$P_{Y_{2}|X}(y_{2}|x)$ as a GBC $W=P_{Y_{1}Y_{2}|X}(y_{1},y_{2}|x)$.

###  III-A The Same Marginal Property

Sato’s outer bound [14] relies on the SMP, i.e., the property that the
capacity regions of Gaussian broadcast channels depend only on the marginal
distributions of the channels [16, Lemma 5.1]. The SMP only holds as long as
the decoding error probabilities at all the receivers are vanishing. In the
finite blocklength regime, however, we deal with non-vanishing error
probabilities. For the HB-GBC, the average system error probability in (2)
decomposes into

$\displaystyle\epsilon^{({n_{1},n_{2}})}$ $\displaystyle=$
$\displaystyle\epsilon_{1}^{({n_{1}})}+\epsilon_{2}^{({n_{2}})}-\epsilon_{1\land
2}^{({n_{1},n_{2}})},$ (6)

where $\epsilon_{k}^{({n_{k}})}:=\Pr[\hat{M}_{k}\neq M_{k}]$, $k=1,2$, and
$\epsilon_{1\land 2}^{({n_{1},n_{2}})}:=\Pr[\hat{M}_{1}\neq M_{1}\text{ and
}\hat{M}_{2}\neq M_{2}]$. Since a channel $V\in\mathcal{W}_{\mathrm{SM}}(W)$
can have different $\epsilon_{1\land 2}^{({n_{1},n_{2}})}$ from $W$ under
otherwise fixed conditions,
$\mathcal{C}_{W}(n_{1},n_{2},\mathcal{F}^{n_{1}},\epsilon)$ and
$\mathcal{C}_{V}(n_{1},n_{2},\mathcal{F}^{n_{1}},\epsilon)$ can also differ.
For example, consider a channel $W$ that has independent noise terms and
another channel $V\in\mathcal{W}_{\mathrm{SM}}(W)$ that has highly correlated
noise terms. Then we expect $V$ to have a larger $\epsilon_{1\land
2}^{({n_{1},n_{2}})}$ than $W$ and as a consequence, a smaller
$\epsilon^{({n_{1},n_{2}})}$. As a result, we cannot use the SMP in the finite
blocklength regime and as a consequence, neither the original Sato-type outer
bound.

###  III-B A Modified Sato-Type Outer Bound

Even though the SMP is not valid in the finite blocklength regime,
$\mathcal{C}_{V}(n_{1},n_{2},\mathcal{F}^{n_{1}},\epsilon)$ for a channel
$V\in\mathcal{W}_{\mathrm{SM}}(W)$ can serve as an outer bound of that of $W$
if the error probability constraint for $V$ is relaxed from $\epsilon$ to
$2\epsilon$. This is the statement of the following result, which can be used
instead of the SMP for finite blocklength outer bounds.

######  Lemma 1.

Let $W$ be an HB-GBC with blocklength constraints $n_{1}$ and $n_{2}$,
$n_{1}\geq n_{2}$, a set of feasible codewords $\mathcal{F}^{n_{1}}$, and
average system error probability constraint $\epsilon<0.25$. Then for all
$V\in\mathcal{W}_{\mathrm{SM}}(W)$, we have

$\displaystyle\mathcal{C}_{W}(n_{1},n_{2},\mathcal{F}^{n_{1}},\epsilon)$
$\displaystyle\subseteq$
$\displaystyle\mathcal{C}_{V}(n_{1},n_{2},\mathcal{F}^{n_{1}},2\epsilon).$ (7)

Please refer to Appendix -A for the proof. We now apply Lemma 1 to the HB-GBC
with SPC to derive our main result as follows.

######  Theorem 1 (Sato-type Outer Bound with Heterogeneous Blocklengths).

For a two-user HB-GBC with feasible set
$\mathcal{F}_{\mathrm{max}}^{n_{1}}(\mathsf{P})$ and channel gains $h_{2}\geq
h_{1}$, the message sizes of any coding scheme have to fulfill the following
inequalities:

$\displaystyle\log\mathsf{M}_{k}$ $\displaystyle\leq$ $\displaystyle
n_{k}\mathsf{C}(h_{k}\mathsf{P})-\sqrt{n_{k}\mathsf{V}(h_{k}\mathsf{P})}Q^{-1}(\epsilon)+\frac{1}{2}\log
n_{k}+O\left(1\right),\quad k=1,2,$ (8)
$\displaystyle\log\mathsf{M}_{1}+\log\mathsf{M}_{2}$ $\displaystyle\leq$
$\displaystyle
n_{1}\mathsf{C}_{\mathrm{s}}^{*}(h_{1},h_{2},p,\mathsf{P})-\sqrt{n_{1}\mathsf{V}_{\mathrm{s}}^{*}(h_{1},h_{2},p,\mathsf{P})}Q^{-1}(2\epsilon)+\frac{1}{2}\log
n_{1}+O\left(1\right),$ (9)

where

$\displaystyle\mathsf{C}_{\mathrm{s}}^{*}(h_{1},h_{2},p,\mathsf{P})$
$\displaystyle=$ $\displaystyle
p\mathsf{C}\left(h_{2}\mathsf{P}\right)+\bar{p}\mathsf{C}\left(h_{1}\mathsf{P}\right)+\log
e\frac{\bar{p}}{2}\left(\frac{h_{2}}{1+h_{2}\mathsf{P}}-\frac{h_{1}}{1+h_{1}\mathsf{P}}\right)\mathsf{P},$
(10) $\displaystyle\mathsf{V}_{\mathrm{s}}^{*}(h_{1},h_{2},p,\mathsf{P})$
$\displaystyle=$ $\displaystyle\frac{\log^{2}e}{4}\left(\frac{p\cdot
2h_{2}^{2}\mathsf{P}^{2}+4h_{2}\mathsf{P}}{(1+h_{2}\mathsf{P})^{2}}\\!+\\!\bar{p}\frac{2h_{1}^{2}\mathsf{P}^{2}}{(1+h_{1}\mathsf{P})^{2}}+\bar{p}\,\mathsf{V}_{\mathrm{s,a}}^{*}(h_{1},h_{2},\mathsf{P})\right),$
(11) $\displaystyle\mathsf{V}_{\mathrm{s,a}}^{*}(h_{1},h_{2},\mathsf{P})$
$\displaystyle=$
$\displaystyle\begin{cases}\frac{4h_{1}\mathsf{P}}{(1+h_{1}\mathsf{P})^{2}}-\frac{4h_{2}\mathsf{P}}{(1+h_{2}\mathsf{P})^{2}}&\text{if
}\mathsf{P}^{2}<\frac{1}{h_{1}h_{2}},\\\ 0&\text{otherwise}.\end{cases}$ (12)

###### Proof.

Appendix -B. ∎

We can specialize Theorem 1 to the homogeneous blocklength case by increasing
$n_{2}$ to $n_{1}$, i.e., inserting $p=1$ into (9). Since increasing $n_{2}$
gives the cooperative receiver used in the Sato-type outer bound an additional
advantage, the result is still an outer bound to the heterogeneous blocklength
case.

######  Corollary 1 (Sato-type outer bound with Homogeneous Blocklengths).

For a two-user HB-GBC with feasible set
$\mathcal{F}_{\mathrm{max}}^{n_{1}}(\mathsf{P})$ and channel gains $h_{2}\geq
h_{1}$, the message sizes of any coding scheme have to fulfill the
inequalities (8) and

$\displaystyle\log\mathsf{M}_{1}+\log\mathsf{M}_{2}$ $\displaystyle\leq$
$\displaystyle
n_{1}\mathsf{C}\left(h_{2}\mathsf{P}\right)-\sqrt{n_{1}\mathsf{V}\left(h_{2}\mathsf{P}\right)}Q^{-1}(2\epsilon)+\frac{1}{2}\log
n_{1}+O\left(1\right).$ (13)

Comparing only the first-order terms of Theorem 1 and Corollary 1 we find that
to get from $\mathsf{C}(h_{2}\mathsf{P})$ to
$\mathsf{C}_{\mathrm{s}}^{*}(h_{1},h_{2},p,\mathsf{P})$, we have to substitute
a portion $\frac{1-p}{2}\log\left[1+h_{2}\mathsf{P}\right]$ of
$\mathsf{C}(h_{2}\mathsf{P})$ by
$\frac{1-p}{2}\log\left[1+h_{1}\mathsf{P}\right]$ and add an additional
contribution $\log
e\frac{1-p}{2}\left(\frac{h_{2}}{1+h_{2}\mathsf{P}}-\frac{h_{1}}{1+h_{1}\mathsf{P}}\right)\mathsf{P}$.
This is an improvement if

$\displaystyle\log\left[1+h_{2}\mathsf{P}\right]$
$\displaystyle>\log\left[1+h_{1}\mathsf{P}\right]+\log
e\left(\frac{h_{2}}{1+h_{2}\mathsf{P}}-\frac{h_{1}}{1+h_{1}\mathsf{P}}\right)\mathsf{P}$
(14) $\displaystyle\Leftrightarrow\log\left[1+h_{2}\mathsf{P}\right]-\log
e\frac{h_{2}\mathsf{P}}{1+h_{2}\mathsf{P}}$
$\displaystyle>\log\left[1+h_{1}\mathsf{P}\right]-\log
e\frac{h_{1}\mathsf{P}}{1+h_{1}\mathsf{P}}.$ (15)

Since the function $f(x)=\log(1+x)-\log e\frac{x}{1+x}$ is monotonically
increasing for $x\geq 0$, this is equivalent to saying that the first-order
term of (9) is always smaller than that of (13), since by assumption we have
$h_{2}\geq h_{1}$. The first-order advantage of (9) over (13) then gets larger
with increasing $h_{2}$. However, note that the different second-order terms
make a full comparison more difficult. This shows that the bound from Theorem
1 takes the disadvantage of user 2 due to the shorter blocklength constraint
into account, since it is smaller than the more naive homogeneous blocklength
approach.

##  IV Improved Achievability Scheme

In this section, we will introduce a new achievability result for the
individual power constraint using composite shell codes, which are a new
extension of the shell code concept to the early decoding technique of [10].
We derive the number of necessary symbols for a successful early decoding and
the rate region using composite shell codes.

###  IV-A Composite Shell Codes

The authors of [10] derived an achievable rate region using superposition
coding with IPC by the early decoding technique with i.i.d. Gaussian
codebooks. Since the dispersion of i.i.d. Gaussian codebooks is suboptimal, it
is desirable to apply shell codewords to the setup. We define the set of shell
codewords as

$\displaystyle\mathcal{S}^{(n)}(\mathsf{P}):=\left\\{x^{n}:\|x^{n}\|^{2}=n\mathsf{P}\right\\}.$
(16)

They are characterized by the fact that they fulfill the power constraint
$\|x^{n}\|^{2}\leq n\mathsf{P}$ with equality (equal-power property). The
challenge is that, for early decoding, the codeword is not received
completely, so the received codeword does not necessarily fulfill the equal-
power property. Therefore, we introduce a modified class of shell codes to
solve this problem.

We define the set of composite shell codewords (CSC) as the set of all
codewords composed of two sub-codewords that both fulfill the equal-power
property, so the cost violation probability is equal to zero:

$\displaystyle\mathcal{S}^{(n_{2},n_{1})}(\mathsf{P})$ $\displaystyle:=$
$\displaystyle\left\\{x^{n_{1}}:\|[x_{1},...,x_{n_{2}}]\|^{2}=n_{2}\mathsf{P},\;\|[x_{n_{2}+1},...,x_{n_{1}}]\|^{2}=(n_{1}-n_{2})\mathsf{P}\right\\}.$
(17)

We will choose codewords according to the uniform distribution over
$\mathcal{S}^{(n_{2},n_{1})}(\mathsf{P})$, which will be shown to be obtained
by uniformly and independently selecting the two concatenated sub-codewords
from their respective power shells.

###  IV-B Improved Early Decoding

The following result is an improvement of [10, Theorem 1] by using composite
shell codes instead of an i.i.d. Gaussian codebook for user 1.

######  Theorem 2.

Consider an HB-GBC with channel gains $h_{1}\leq h_{2}$, blocklength
constraints $n_{1}\geq n_{2}$, feasible set
$\mathcal{F}_{\mathrm{max}}^{n_{1},n_{2}}(\mathsf{P}_{1},\mathsf{P}_{2})$ and
let $n_{1}$ be sufficiently large. Furthermore, let $\epsilon_{1}$,
$\epsilon_{\mathrm{SIC},1}$ and $\epsilon_{\mathrm{SIC},2}$ be the target
error probabilities at user 1 and at the two SIC steps at user 2,
respectively, s.t. they fulfill (5). Then successful early decoding can be
performed if the following inequality holds:

$\displaystyle
n_{2}\geq\left(\frac{\sqrt{\mathsf{V}(g_{2}\mathsf{P}_{1})}Q^{-1}(\epsilon_{\mathrm{SIC},1})}{2\mathsf{C}(g_{2}\mathsf{P}_{1})}+\sqrt{\frac{\mathsf{V}(g_{2}\mathsf{P}_{1})(Q^{-1}(\epsilon_{\mathrm{SIC},1}))^{2}}{4\mathsf{C}(g_{2}\mathsf{P}_{1})^{2}}+\frac{\log(\mathsf{M}_{1})}{\mathsf{C}(g_{2}\mathsf{P}_{1})}}\right)^{2},$
(18)

where $g_{2}:=\frac{h_{2}}{1+h_{2}\bar{\mathsf{P}}_{2}}$ and
$\bar{\mathsf{P}}_{2}:=\mathsf{P}_{2}-\delta,\delta>0$. In that case, all
message size pairs fulfilling the following inequalities:

$\displaystyle\log\mathsf{M}_{1}$ $\displaystyle\leq
n_{1}\bar{\mathsf{C}}_{1}-\sqrt{n_{1}\bar{\mathsf{V}}_{1}}Q^{-1}(\epsilon_{1})+O\left(1\right),$
(19) $\displaystyle\log\mathsf{M}_{2}$ $\displaystyle\leq
n_{2}\mathsf{C}(h_{2}\bar{\mathsf{P}}_{2})-\sqrt{n_{2}\mathsf{V}_{\mathrm{G}}(h_{2}\bar{\mathsf{P}}_{2})}Q^{-1}(\epsilon_{\mathrm{SIC},2})+O\left(1\right)$
(20)

are achievable, where $g_{1}:=\frac{h_{1}}{1+h_{1}\bar{\mathsf{P}}_{2}}$ and

$\displaystyle\bar{\mathsf{C}}_{1}$
$\displaystyle:=p\mathsf{C}(g_{1}\mathsf{P}_{1})+\bar{p}\mathsf{C}(h_{1}\mathsf{P}_{1}),$
(21) $\displaystyle\bar{\mathsf{V}}_{1}$
$\displaystyle:=p\mathsf{V}(g_{1}\mathsf{P}_{1})+\bar{p}\mathsf{V}(h_{1}\mathsf{P}_{1}).$
(22)

For the proof, please refer to Appendix -C. The main proof steps can be
summarized as follows:

  * •

We use the proposed composite shell codes for user 1 i, s.t. the first $n_{2}$
symbols also fulfill the equal-power property. User 2 still uses i.i.d.
Gaussian codewords, s.t. the interference he produces is still Gaussian.

  * •

We evaluate the information density at user 2 for the first SIC step. User 2
receives a shell codeword from user 1 of length $n_{2}$ with additive white
Gaussian noise. The expectation and variance of the information density are
$\mathsf{C}(g_{2}\mathsf{P}_{1})$ and $\mathsf{V}(g_{2}\mathsf{P}_{1})$,
respectively. The expectation as well as the variance are now independent of
the realization of $x^{n_{1}}$, which is an important difference to [10]. The
reason for the independence of $x^{n_{1}}$ is that the equal-power property
allows the useful simplification $\|x^{n_{2}}\|^{2}=n_{2}\mathsf{P}$ in the
derivation of the information density, just like in the point-to-point case
with shell codes. This simplification is not available when dealing with
i.i.d. Gaussian codebooks, where the expectation and variance of the
information density therefore depend on the specific realization of
$x^{n_{1}}$.

  * •

Then we proceed with similar steps as in the proof of [10, Theorem 1], but
with the difference that we do not need to bound $0\leq\|x^{n_{2}}\|^{2}\leq
n_{1}\mathsf{P}_{1}$ in several steps, since the expressions are already
independent of $x^{n_{2}}$.

  * •

Also, we do not have to bound $n_{1}>n_{2}$ as in [10], and we can solve for
$n_{2}$ using the quadratic formula.

Compared to [11], we now achieve the shell dispersion $\mathsf{V}(\cdot)$ in
the second-order term, instead of the i.i.d. Gaussian dispersion. The use of
CSC does not lead to a decreased second-order performance compared to standard
shell codes. The penalty from using CSC is captured in the constant
$\tilde{\mathsf{K}}$ (Lemma 2 in Appendix -C), which belongs to the third-
order term.

######  Remark 1.

As becomes clear from the proof of Theorem 2, the improved second-order
performance compared to [11] comes at the cost of introducing a constant
$\tilde{\mathsf{K}}$ which makes the third-order term larger. This will make
the second-order approach more inaccurate at smaller blocklengths.

Additionally, it is possible to extend this approach by using a concatenation
of more than two codewords, for example for the $m$-user HB-GBC. If $m$
codewords are concatenated (where $m$ may be any constant, but not a function
of $n$), the Radon-Nikodym derivative from Lemma 2 in Appendix -C will be
bounded by

$\displaystyle\frac{dP_{Y^{n}}(y^{n})}{dQ_{Y^{n}}(y^{n})}\leq\mathsf{K}^{m},$
(23)

which will make the third-order term larger and larger with increasing $m$.
Since it only considers the first- and second-order term, this behavior is not
captured by the Gaussian approximation.

##  V Properties of the GBC in the finite blocklength regime

In this section, we investigate the stochastical degradedness property of the
broadcast channel and the time sharing technique in the finite blocklength
regime. It will be shown that there exist important differences to the
asymptotic case.

###  V-A Stochastical Degradedness

The notion of stochastical degradedness is commonly used in the asymptotic
analysis of broadcast channels. A broadcast channel is called stochastically
degraded if it has the same condition marginal distributions as a physically
degraded channel [16, p. 112]. The SMP (Section III-A) states that two
channels having the same marginals is sufficient to guarantee that they also
have the same asymptotic capacity region. However, as we have shown, the SMP
is not valid for non-vanishing error probabilities. Therefore, even though the
definition of stochastical degradedness is still valid, we can no longer use
it to transform a broadcast channel in an equivalent, physically degraded
channel.

###  V-B Time Sharing

The time sharing technique is commonly used in achievability schemes to
convexify rate regions. Time sharing between two first-order achievable rate
pairs $(\tilde{\mathsf{R}}_{1},\tilde{\mathsf{R}}_{2})$ and
$(\hat{\mathsf{R}}_{1},\hat{\mathsf{R}}_{2})$ is a way to achieve all rates

$\displaystyle\left(\alpha\tilde{\mathsf{R}}_{1}+\bar{\alpha}\hat{\mathsf{R}}_{1},\alpha\tilde{\mathsf{R}}_{2}+\bar{\alpha}\hat{\mathsf{R}}_{2}\right),\qquad\alpha\in[0,1],$
(24)

where $\bar{\alpha}=1-\alpha$. These rates are obtained by dividing the total
blocklength into two sub-blocks of lengths $\alpha n$ and $(1-\alpha)n$,
assuming they both are integers, and using the scheme that achieves
$(\tilde{\mathsf{R}}_{1},\tilde{\mathsf{R}}_{2})$ in the first block and the
scheme that achieves $(\hat{\mathsf{R}}_{1},\hat{\mathsf{R}}_{2})$ in the
second block [16, p. 85].

When applying this concept in the finite blocklength regime, this means that
the two codewords are even shorter than the given blocklength constraint,
which was already noted by [4] for the multiple access channel. However, while
the authors of [4] only state that time sharing is not available because of
the blocklength constraint, we will now investigate what happens if we allow
the two codewords between which time sharing is applied to be shorter than the
original codeword. It turns out that the result (24) is not valid for finite
blocklengths111For the sake of simplicity, we consider homogeneous
blocklenghts. The results directly transfer to the heterogeneous blocklength
case.:

######  Theorem 3.

Time sharing in the finite blocklength regime results in a non-convex rate
region.

###### Proof.

Consider two second-order rate pairs

$\displaystyle(\tilde{\mathsf{R}}_{1},\tilde{\mathsf{R}}_{2})=\left(\tilde{\mathsf{R}}_{\mathrm{FO},1}-\frac{\tilde{\mathsf{R}}_{\mathrm{SO},1}}{\sqrt{n}},\tilde{\mathsf{R}}_{\mathrm{FO},2}-\frac{\tilde{\mathsf{R}}_{\mathrm{SO},2}}{\sqrt{n}}\right)$
(25)

and

$\displaystyle(\hat{\mathsf{R}}_{1},\hat{\mathsf{R}}_{2})=\left(\hat{\mathsf{R}}_{\mathrm{FO},1}-\frac{\hat{\mathsf{R}}_{\mathrm{SO},1}}{\sqrt{n}},\hat{\mathsf{R}}_{\mathrm{FO},2}-\frac{\hat{\mathsf{R}}_{\mathrm{SO},2}}{\sqrt{n}}\right),$
(26)

where we use the subscripts “FO” and “SO” to denote the first-order term and
the part of the second-order term which is independent of $n$. By using time
sharing as described above, i.e., using two sub-blocks of lengths $\alpha n$
and $\bar{\alpha}n$, we can achieve the rates

$\displaystyle\left(\alpha\tilde{\mathsf{R}}_{\mathrm{FO},1}+\bar{\alpha}\hat{\mathsf{R}}_{\mathrm{FO},1}-\frac{\alpha\tilde{\mathsf{R}}_{\mathrm{SO},1}}{\sqrt{\alpha
n}}-\frac{\bar{\alpha}\hat{\mathsf{R}}_{\mathrm{SO},1}}{\sqrt{\bar{\alpha}n}},\alpha\tilde{\mathsf{R}}_{\mathrm{FO},2}+\bar{\alpha}\hat{\mathsf{R}}_{\mathrm{FO},2}-\frac{\alpha\tilde{\mathsf{R}}_{\mathrm{SO},2}}{\sqrt{\alpha
n}}-\frac{\bar{\alpha}\hat{\mathsf{R}}_{\mathrm{SO},2}}{\sqrt{\bar{\alpha}n}}\right)$
(27)

Note that the scaling by $\alpha$ and $\bar{\alpha}$ also affects the
denominators of the second-order terms, as they are dependent on the
blocklength, unlike the first-order term. Consider the rates for user 1:

$\displaystyle\alpha\tilde{\mathsf{R}}_{\mathrm{FO},1}+\bar{\alpha}\hat{\mathsf{R}}_{\mathrm{FO},1}-\frac{\alpha\tilde{\mathsf{R}}_{\mathrm{SO},1}}{\sqrt{\alpha
n}}-\frac{\bar{\alpha}\hat{\mathsf{R}}_{\mathrm{SO},1}}{\sqrt{\bar{\alpha}n}}$
$\displaystyle=\alpha\tilde{\mathsf{R}}_{\mathrm{FO},1}+\bar{\alpha}\hat{\mathsf{R}}_{\mathrm{FO},1}-\frac{\sqrt{\alpha}\tilde{\mathsf{R}}_{\mathrm{SO},1}}{\sqrt{n}}-\frac{\sqrt{\bar{\alpha}}\hat{\mathsf{R}}_{\mathrm{SO},1}}{\sqrt{n}}$
(28)
$\displaystyle\leq\alpha\tilde{\mathsf{R}}_{\mathrm{FO},1}+\bar{\alpha}\hat{\mathsf{R}}_{\mathrm{FO},1}-\frac{\alpha\tilde{\mathsf{R}}_{\mathrm{SO},1}}{\sqrt{n}}-\frac{\bar{\alpha}\hat{\mathsf{R}}_{\mathrm{SO},1}}{\sqrt{n}}$
(29)
$\displaystyle=\alpha\tilde{\mathsf{R}}_{1}+\bar{\alpha}\hat{\mathsf{R}}_{1},$
(30)

where the inequality comes from the fact that $\alpha\leq 1$ and
$\bar{\alpha}\leq 1$. The same holds for user 2. Therefore, the achievable
rates by time sharing for both users are smaller than the convex combination
$\alpha\tilde{\mathsf{R}}_{k}+\bar{\alpha}\hat{\mathsf{R}}_{k}$ of the two
second-order rates, which leads to a non-convex rate region. ∎

Theorem 3 states that, in contrast to the asymptotic case, time sharing cannot
guarantee that all convex combinations of two achievable second-order rate
pairs are also achievable. Fig. 1 shows what time sharing between the single
user second-order rates

$\displaystyle
R_{k,0}:=\mathsf{C}(\mathsf{P})-\sqrt{\frac{\mathsf{V}_{\mathrm{G}}(\mathsf{P})}{n}},\qquad
k=1,2$ (31)

looks like for different blocklengths in a symmetric GBC with $h_{1}=h_{2}=1$,
$\mathsf{P}=10$ and $\epsilon=10^{-6}$. All rates are normalized w.r.t. these
single user rates, such that the rates for the different blocklengths are
comparable more easily. We vary $\alpha$ from zero to one with step size 0.05,
each point on the curves represents one value of $\alpha$. It can be observed
that for finite blocklengths, time sharing results in a non-convex rate
region, unlike in the asymptotic case, where the convex combination of all
achievable rate pairs can be achieved. The curves deviate more from the
asymptotic the smaller the blocklength is, since for smaller blocklengths, the
second-order term is becoming more dominant.

$\displaystyle 0.0$$\displaystyle 0.2$$\displaystyle 0.4$$\displaystyle
0.6$$\displaystyle 0.8$$\displaystyle 1.0$$\displaystyle
R_{1}/R_{1,0}$$\displaystyle 0.0$$\displaystyle 0.2$$\displaystyle
0.4$$\displaystyle 0.6$$\displaystyle 0.8$$\displaystyle 1.0$$\displaystyle
R_{2}/R_{2,0}$achievable rates, $\displaystyle n\to\infty$achievable rates,
$\displaystyle n=$ 1024achievable rates, $\displaystyle n=$ 512achievable
rates, $\displaystyle n=$ 256achievable rates, $\displaystyle n=$ 128 Figure
1: Achievable rate region through time sharing between the single-user rates
$R_{1,0}$ and $R_{2,0}$, normalized to the single user rates, at different
blocklengths, for $h_{1}=h_{2}=1$, $\mathsf{P}=10$ and $\epsilon=10^{-6}$.

Note that even the second-order approach to time sharing presented in Fig. 1
has its limitations. For $\alpha$ close to 1 or 0, the lengths of the
considered sub-blocks are very small and can be in the order of magnitude of
only a few bits. For these blocklengths, the Gaussian approximation is not
very accurate and therefore, our second-order time sharing results are not
very reliable for these values of $\alpha$. The most reliable results are from
the region where $\alpha$ is close to 0.5, where the two sub-blocks still have
a considerable blocklength.

##  VI Numerical Results

###  VI-A Outer Bound

In this section, we compare the sum rate upper bounds from Theorem 1 and
Corollary 1 to the achievable sum rates under the SPC that were derived in
[11, 12]. Therefore, in Fig. 2, we consider a HB-GBC with sum power constraint
$\mathsf{P}=10$, $\epsilon=2\cdot 10^{-6}$, channel gains $h_{1}=1$ and
$h_{2}\in\\{1.5,10\\}$, fixed blocklength ratio $p=n_{2}/n_{1}=0.9$ and we
vary $n_{1}\in[128,2048]$. We assume $\epsilon_{1}=10^{-6}$,
$\epsilon_{\mathrm{SIC},1}=\epsilon_{\mathrm{SIC},2}=5\cdot 10^{-7}$ for the
achievable rate expressions from [11].

$\displaystyle{250}$$\displaystyle{500}$$\displaystyle{750}$$\displaystyle{1000}$$\displaystyle{1250}$$\displaystyle{1500}$$\displaystyle{1750}$$\displaystyle{2000}$$\displaystyle
n_{1}$$\displaystyle{1.25}$$\displaystyle{1.50}$$\displaystyle{1.75}$$\displaystyle{2.00}$$\displaystyle{2.25}$$\displaystyle{2.50}$$\displaystyle{2.75}$$\displaystyle{3.00}$$\displaystyle{3.25}$$\displaystyle(\log\mathsf{M}_{1}+\log\mathsf{M}_{2})/n_{1}$UB
(hom), $\displaystyle h_{2}=10$UB (het), $\displaystyle h_{2}=10$sum rate,
$\displaystyle h_{2}=10$UB (hom), $\displaystyle h_{2}=1.5$UB (het),
$\displaystyle h_{2}=1.5$sum rate, $\displaystyle h_{2}=1.5$

Figure 2: Comparison of the sum rate upper bounds from Theorem 1 (het) and
Corollary 1 (hom) with the achievable sum rates based on [11] for two channels
with $h_{2}=1.5$ and $h_{2}=10$.

We can observe that for smaller $h_{2}$, the homogeneous upper bound and the
heterogeneous upper bound are closer to each other and the heterogeneous outer
bound always outperforms the homogeneous outer bound, which confirms the
observation in the end of Section III. The gap between achievable rates and
outer bounds is decreasing for larger $n_{1}$, but does not vanish, since
Sato’s outer bound is not tight even in the asymptotic regime.

In Fig. 3, we compare the complete outer bounds from Theorem 1 and Corollary 1
to the achievable rate regions from the achievability schemes early decoding
and hybrid NOMA from [11]. We keep most of the system parameters the same as
those in Figure 2, except $h_{2}=50$, $n_{1}=1024$ and $n_{2}=840$. The
maximal sum rate is achieved at $(0.22,3.48)$ because of the channel advantage
of user 2. Note that it is not possible to simply combine the two achievable
schemes into a convex rate region because time sharing is not available as a
tool for convexification in the finite blocklength regime (Section V-B).
Again, the heterogeneous outer bound is tighter than the homogeneous outer
bound. The gap between the achievable sum rates and the outer bound is smaller
when they are close to the single user rates. The sum rate upper bound is
looser than the single-user rate upper bounds due to the assumption of a
cooperative receiver for the sum rate upper bound.

$\displaystyle{0.0}$$\displaystyle{0.2}$$\displaystyle{0.4}$$\displaystyle{0.6}$$\displaystyle{0.8}$$\displaystyle{1.0}$$\displaystyle{1.2}$$\displaystyle{1.4}$$\displaystyle{1.6}$$\displaystyle\log\mathsf{M}_{1}/n_{1}$$\displaystyle{0.0}$$\displaystyle{0.5}$$\displaystyle{1.0}$$\displaystyle{1.5}$$\displaystyle{2.0}$$\displaystyle{2.5}$$\displaystyle{3.0}$$\displaystyle{3.5}$$\displaystyle\log\mathsf{M}_{2}/n_{1}$homogeneous
upper boundheterogeneous upper boundachievable rates (ED)achievable rates
(HNOMA)

Figure 3: Achievable rate regions using Early Decoding (ED) and Hybrid NOMA
(HNOMA) as in [11] compared to our outer bounds under the SPC.

###  VI-B Latency Reduction

We now turn to the HB-GBC with IPC and compare the number of necessary symbols
$n_{2}$ for a successful early decoding from Theorem 2 (“Shell”) to that from
[10] (“i.i.d.”) and the asymptotic result from [13], which is also considered
in [10]. The considered system parameters are
$\epsilon_{1}=\epsilon_{2}=10^{-6}$, $h_{1}=1$, $\mathsf{P}_{1}=8$ and
$\mathsf{P}_{2}=0.2$ and $n_{1}$ is varied. We numerically search for the best
value of $\epsilon_{\mathrm{SIC},1}\in(0,\epsilon_{2}]$.

$\displaystyle{10}$$\displaystyle{20}$$\displaystyle{30}$$\displaystyle{40}$$\displaystyle{50}$$\displaystyle{60}$$\displaystyle{70}$$\displaystyle{80}$$\displaystyle{90}$$\displaystyle
h_{2}$$\displaystyle{400}$$\displaystyle{600}$$\displaystyle{800}$$\displaystyle{1000}$$\displaystyle{1200}$$\displaystyle{1400}$$\displaystyle{1600}$$\displaystyle{1800}$Number
of Early Decoding Symbols $\displaystyle{n}_{2}$$\displaystyle
n_{1}=2048$$\displaystyle n_{1}=1024$$\displaystyle
n_{1}=512$i.i.dShellAsymptotic

Figure 4: Number of necessary symbols for a successful early decoding at
different blocklengths for the i.i.d. Gaussian inputs [10], shell inputs for
user 1 (Theorem 2) and the asymptotic analysis [13].

In Fig. 4, we observe that the improvement of Theorem 2 over [10] is
significant. Our results are very close to the asymptotic limit for all
considered blocklengths. From these results, we can conclude that early
decoding is a promising technique even in the finite blocklength regime,
especially when shell codes are used.

##  VII Conclusion

In this paper, we have developed a novel approach of deriving Sato-type outer
bounds in the finite blocklength regime. We have applied this technique to the
Gaussian broadcast channel with heterogeneous blocklength constraints using
two different bounding approaches. We have also improved previous
achievability results based on a technique called early decoding by using a
composite shell code for the user with the looser blocklength constraint.
Numerical results show a significant improvement in terms of latency reduction
over the previous early decoding result, and our results are now very close to
the asymptotic limit.

###  -A Proof of Lemma 1

According to (6), joint distributions with the same marginals affect the
probability of a simultaneous decoding error $\epsilon_{1\land
2}^{({n_{1},n_{2}})}$, but not the individual error probabilities
$\epsilon_{1}^{({n_{1}})}$ and $\epsilon_{2}^{({n_{2}})}$. Therefore, if we
consider a new channel $V\in\mathcal{W}_{\mathrm{SM}}(W)$ with the same
average system error probability constraint $\epsilon$,
$\mathcal{C}_{V}(n_{1},n_{2},\mathcal{F}^{n_{1}},\epsilon)$ may be larger or
smaller than $\mathcal{C}_{W}(n_{1},n_{2},\mathcal{F}^{n_{1}},\epsilon)$. If,
e.g., the joint error probability $\tilde{\epsilon}_{1\land
2}^{({n_{1},n_{2}})}$ of the new channel is smaller than $\epsilon_{1\land
2}^{({n_{1},n_{2}})}$, then by (6), we have to reduce
$\epsilon_{1}^{({n_{1}})}+\epsilon_{2}^{({n_{2}})}$ in order to fulfill the
same error probability constraint $\epsilon$. However, reducing the individual
error probabilities makes the
$\mathcal{C}_{V}(n_{1},n_{2},\mathcal{F}^{n_{1}},\epsilon)$ smaller and
therefore, we cannot simply use it as an outer bound on
$\mathcal{C}_{W}(n_{1},n_{2},\mathcal{F}^{n_{1}},\epsilon)$. As a result, to
derive an outer bound on
$\mathcal{C}_{W}(n_{1},n_{2},\mathcal{F}^{n_{1}},\epsilon)$, we have to ensure
that both the individual and total error probabilities are at least as large
as those in the original channel $W$.

When transmitting through $W$, the individual error probabilities of the
capacity-achieving code cannot be larger than $\epsilon$, since otherwise, the
error probability constraint (2) would be violated. Therefore, we now fix the
individual error probabilities of the new channel as
$\tilde{\epsilon}_{1}^{({n_{1}})}=\tilde{\epsilon}_{2}^{({n_{2}})}=\epsilon$,
and by (6), we get

$\displaystyle\tilde{\epsilon}^{({n_{1},n_{2}})}$
$\displaystyle=\tilde{\epsilon}_{1}^{({n_{1}})}+\tilde{\epsilon}_{2}^{({n_{2}})}-\tilde{\epsilon}_{1\land
2}^{({n_{1},n_{2}})}\geq\epsilon,$ (32)

since $\tilde{\epsilon}_{1\land
2}^{({n_{1},n_{2}})}\leq\min\\{\tilde{\epsilon}_{1}^{({n_{1}})},\tilde{\epsilon}_{2}^{({n_{2}})}\\}=\epsilon$.
Thus, we guarantee that the average system error probability of the new
channel is at least as large as that of the original channel, which results in
an outer bound on $\mathcal{C}_{W}(n_{1},n_{2},\mathcal{F}^{n_{1}},\epsilon)$.

On the other side, since $\tilde{\epsilon}_{1\land 2}^{({n_{1},n_{2}})}\geq
0$, we have $\tilde{\epsilon}^{({n_{1},n_{2}})}\leq 2\epsilon.$ Taking
$\tilde{\epsilon}^{({n_{1},n_{2}})}=2\epsilon$ gives us a uniform upper bound
of $\tilde{\epsilon}^{({n_{1},n_{2}})}$ w.r.t. all possible
$\tilde{\epsilon}_{1\land 2}^{({n_{1},n_{2}})}$, so we can guarantee that

$\displaystyle\mathcal{C}_{W}(n_{1},n_{2},\mathcal{F}^{n_{1}},\epsilon)\subseteq\mathcal{C}_{V}(n_{1},n_{2},\mathcal{F}^{n_{1}},2\epsilon).$
(33)

###  -B Proof of Theorem 1

The upper bounds on the individual rates directly follow from the single-user
rate upper bounds [1, Theorem 65].

For the sum rate upper bound, we use the modified information spectrum
approach presented in [17]. We consider a GBC where the two noise terms are
correlated via a correlation $\rho\in[0,1]$. The modified information density
for the cooperative receiver in this scenario is

$\displaystyle\tilde{i}(x^{n_{1}};Y_{1}^{n_{1}},Y_{2}^{n_{2}})$
$\displaystyle=$
$\displaystyle\sum_{i=1}^{n_{2}}\log\frac{dP_{Y_{1}Y_{2}|X}(Y_{1,i},Y_{2,i}|x_{i})}{dQ_{Y_{1}Y_{2}}(Y_{1,i},Y_{2,i})}+\sum_{i=n_{2}+1}^{n_{1}}\log\frac{dP_{Y_{1}|X}(Y_{1,i}|x_{i})}{dQ_{Y_{1}}(Y_{1,i})},$
(34)

where $P_{Y_{1}Y_{2}|X}$ is the channel distribution and $Q_{Y_{1}Y_{2}}$ is
the output distribution induced by i.i.d. Gaussian inputs. Its normalized
expectation is

$\displaystyle\mathbb{E}\left[\frac{1}{n_{1}}\tilde{i}(x^{n_{1}};Y_{1}^{n_{1}},Y_{2}^{n_{2}})\right]=\mathsf{C}_{\rho,1}+\frac{\mathsf{C}_{\rho,2}}{n_{1}}\cdot\sum_{i=n_{2}+1}^{n_{1}}x_{i}^{2},$
(35)

where

$\displaystyle\mathsf{C}_{\rho,1}:=p\mathsf{C}\left(h_{\rho}\mathsf{P}\right)+\bar{p}\mathsf{C}\left(h_{1}\mathsf{P}\right)+\frac{\bar{p}\log
e}{2}\left(\frac{h_{\rho}}{1+h_{\rho}\mathsf{P}}-\frac{h_{1}}{1+h_{1}\mathsf{P}}\right)\mathsf{P}$
(36)

and

$\displaystyle\mathsf{C}_{\rho,2}$ $\displaystyle:=\frac{\log
e}{2}\left(\frac{h_{1}}{1+h_{1}\mathsf{P}}-\frac{h_{\rho}}{1+h_{\rho}\mathsf{P}}\right).$
(37)

Here, we have introduced the abbreviation
$h_{\rho}:=(h_{1}+h_{2}-2\rho\sqrt{h_{1}h_{2}})/(1-\rho^{2})$. The variance of
(34) is

$\displaystyle\mathrm{Var}\left[\frac{1}{n_{1}}\tilde{i}(x^{n_{1}};Y_{1}^{n_{1}},Y_{2}^{n_{2}})\right]$
$\displaystyle=\frac{\mathsf{V}_{\rho,1}}{n_{1}}+\frac{\mathsf{V}_{\rho,2}}{n_{1}^{2}}\cdot\sum_{i=n_{2}+1}^{n_{1}}x_{i}^{2},$
(38)

where

$\displaystyle\mathsf{V}_{\rho,1}:=$
$\displaystyle\frac{\log^{2}e}{4}\left(\frac{p\cdot
2h_{\rho}^{2}\mathsf{P}^{2}+4h_{\rho}\mathsf{P}}{(1+h_{\rho}\mathsf{P})^{2}}+\bar{p}\frac{2h_{1}^{2}\mathsf{P}^{2}}{(1+h_{1}\mathsf{P})^{2}}\right)$
(39)

and

$\displaystyle\mathsf{V}_{\rho,2}$ $\displaystyle:=$
$\displaystyle\frac{\log^{2}e}{4}\left(\frac{4h_{1}}{(1+h_{1}\mathsf{P})^{2}}-\frac{4h_{\rho}}{(1+h_{\rho}\mathsf{P})^{2}}\right).$
(40)

We now employ the information spectrum converse [15, 17] to our setup, which
states that every $(n_{1},\mathsf{M},\epsilon,\mathcal{F}^{n_{1}})$-code has
to satisfy

$\displaystyle\epsilon\geq\mathrm{Pr}\left[\tilde{i}(X^{n_{1}};Y_{1}^{n_{1}},Y_{2}^{n_{2}})\leq\log\gamma_{n_{1}}\right]-\frac{\gamma_{n_{1}}}{\mathsf{M}},$
(41)

where for the cooperative receiver, we have
$\mathsf{M}=\mathsf{M}_{1}\mathsf{M}_{2}$. We set
$\gamma_{n_{1}}=\mathsf{M}/\sqrt{n_{1}}$ and lower bound the probability
expression in (41) using the Berry-Esseen Theorem [1, Theorem 44] by

$\displaystyle\mathrm{Pr}\left[\frac{1}{n_{1}}\tilde{i}(x^{n_{1}};Y_{1}^{n_{1}},Y_{2}^{n_{2}})\leq\frac{\mathsf{M}/\sqrt{n_{1}}}{n_{1}}\right]\geq
Q\Bigg{(}\underbrace{\frac{\mathsf{C}_{\rho,1}+\frac{\mathsf{C}_{\rho,2}}{n_{1}}\cdot\sum_{i=n_{2}+1}^{n_{1}}x_{i}^{2}-\log\left(\frac{\mathsf{M}}{\sqrt{n_{1}}}\right)/{n_{1}}}{\sqrt{\mathsf{V}_{\rho,1}+\frac{\mathsf{V}_{\rho,2}}{n_{1}}\cdot\sum_{i=n_{2}+1}^{n_{1}}x_{i}^{2}}/{n_{1}}}}_{=:r_{m,\rho}(n_{1})}\Bigg{)}-\frac{\mathsf{B}_{1}}{\sqrt{n_{1}}}.$
(42)

We observe that $\mathsf{C}_{\rho,2}$ is negative and $\mathsf{V}_{\rho,2}$ is
positive if $\mathsf{P}^{2}\geq\frac{1}{h_{1}h_{\rho}}$. We can now upper
bound $r_{\rho,m}(n_{1})$ by using $\sum_{i=n_{2}+1}^{n_{1}}x_{i}^{2}\geq 0$
in the numerator and
$0\leq\sum_{i=n_{2}+1}^{n_{1}}x_{i}^{2}\leq(n_{1}-n_{2})\mathsf{P}$ in the
denominator:

$\displaystyle r_{\rho,m}(n_{1})$ $\displaystyle\leq$
$\displaystyle\frac{\mathsf{C}_{\rho,1}-\log(\mathsf{M}/\sqrt{n_{1}})/{n_{1}}}{\sqrt{\mathsf{V}_{\rho,1}+\mathsf{V}_{\rho,2}\cdot(1-p)\mathsf{P}\cdot\mathbbm{1}\left(\mathsf{P}^{2}<\frac{1}{h_{1}h_{\rho}}\right)}},$
(43)

where $\mathbbm{1}(\cdot)$ is the indicator function. Inserting (43) into (42)
and using (42) to upper bound (41), we get

$\displaystyle 2\epsilon\geq
Q\left(\frac{\mathsf{C}_{\rho,1}-\log(\mathsf{M}/\sqrt{n_{1}})/{n_{1}}}{\sqrt{\mathsf{V}_{\rho,1}+\mathsf{V}_{\rho,2}\cdot(1-p)\mathsf{P}\cdot\mathbbm{1}\left(\mathsf{P}^{2}<\frac{1}{h_{1}h_{\rho}}\right)}}\right)-\frac{\bar{\mathsf{B}}}{\sqrt{n_{1}}},$
(44)

where $\bar{\mathsf{B}}:=\mathsf{B}_{1}+1$. Note that we have increased the
error probability constraint from $\epsilon$ to $2\epsilon$, because this will
later enable us to use Lemma 1. By solving for the number of messages, we find

$\displaystyle\log\mathsf{M}_{1}+\log\mathsf{M}_{2}$ $\displaystyle\leq
n_{1}\mathsf{C}_{\rho,1}-\sqrt{n_{1}\left(\mathsf{V}_{\rho,1}+\mathsf{V}_{\rho,2}\cdot(1-p)\mathsf{P}\cdot\mathbbm{1}\left(\mathsf{P}^{2}<\frac{1}{h_{1}h_{\rho}}\right)\right)}Q^{-1}\left(\epsilon+\frac{\bar{\mathsf{B}}}{\sqrt{n_{1}}}\right)+\frac{1}{2}\log
n_{1}$ (45)
$\displaystyle=n_{1}\mathsf{C}_{\rho,1}-\sqrt{n_{1}\left(\mathsf{V}_{\rho,1}+\mathsf{V}_{\rho,2}\cdot(1-p)\mathsf{P}\cdot\mathbbm{1}\left(\mathsf{P}^{2}<\frac{1}{h_{1}h_{\rho}}\right)\right)}Q^{-1}(\epsilon)+\frac{1}{2}\log
n_{1}+O\left(1\right),$ (46)

where the last step follows from the Taylor expansion of $Q^{-1}$.

Lemma 1 now enables us to use the channel distribution (or, equivalently, the
correlation $\rho$) that minimizes the first-order term (36) of (46). We
consider the individual summands separately.

  1. 1.

Since $\mathsf{C}(\cdot)$ is a monotonically increasing function, we can
minimize the argument of $\mathsf{C}(h_{\rho}\mathsf{P})$ as follows:

$\displaystyle\min_{\rho\in[0,1)}h_{\rho}\mathsf{P}=\min_{\rho\in[0,1)}\frac{h_{1}+h_{2}-2\rho\sqrt{h_{1}h_{2}}}{1-\rho^{2}}\mathsf{P}.$
(47)

Its derivative is

$\displaystyle\frac{d}{d\rho}h_{\rho}=\frac{-2\sqrt{h_{1}h_{2}}\rho^{2}+2(h_{1}+h_{2})\rho-2\sqrt{h_{1}h_{2}}}{(1-\rho^{2})^{2}}$
(48)

and the zeros of the derivatives are

$\displaystyle\rho_{1}=\sqrt{\frac{h_{1}}{h_{2}}}\qquad\text{and}\qquad\rho_{2}=\sqrt{\frac{h_{2}}{h_{1}}}.$
(49)

Since we know that $h_{2}\geq h_{1}$, we have $\rho_{2}\geq 1$, which is
outside of the feasible domain for $\rho$. It can be verified that
$h_{\rho_{1}}=h_{2}$ and

$\displaystyle\left.\frac{d^{2}}{d\rho^{2}}h_{\rho}\right|_{\rho=\rho_{1}}>0,$
(50)

i.e., $\rho_{1}$ is indeed the minimum.

  2. 2.

the term

$\displaystyle\frac{1-p}{2}\log\left[1+h_{1}\mathsf{P}\right]$ (51)

is independent of $\rho$,

  3. 3.

for the third term

$\displaystyle\log
e\frac{1-p}{2}\left(\frac{h_{1}+h_{2}-2\rho\sqrt{h_{1}h_{2}}}{1-\rho^{2}+\mathsf{P}(h_{1}+h_{2}-2\rho\sqrt{h_{1}h_{2}})}-\frac{h_{1}}{h_{1}\mathsf{P}+1}\right)\mathsf{P}=0,$
(52)

we define
$h_{12,\mathsf{P}}(\rho):=\frac{h_{1}+h_{2}-2\rho\sqrt{h_{1}h_{2}}}{1-\rho^{2}+\mathsf{P}(h_{1}+h_{2}-2\rho\sqrt{h_{1}h_{2}})}$,
which is the only part of (52) which is dependent on $\rho$, and find

$\displaystyle\frac{d}{d\rho}h_{12,\mathsf{P}}(\rho)=\frac{-2\sqrt{h_{1}h_{2}}\rho^{2}+2(h_{1}+h_{2})\rho-2\sqrt{h_{1}h_{2}}}{\left(1-\rho^{2}+\mathsf{P}(h_{1}+h_{2}-2\rho\sqrt{h_{1}h_{2}})\right)^{2}}$
(53)

and the zeros of that derivative are

$\displaystyle\rho_{1}=\sqrt{\frac{h_{1}}{h_{2}}}\qquad\text{and}\qquad\rho_{2}=\sqrt{\frac{h_{2}}{h_{1}}}.$
(54)

Again, only $\rho_{1}$ is a feasible value for $\rho$ and since
$\left.\frac{d^{2}}{d\rho^{2}}h_{12,\mathsf{P}}(\rho)\right|_{\rho=\rho_{1}}>0$,
it is a minimum.

Since $\rho=\sqrt{\frac{h_{1}}{h_{2}}}$ minimizes all three terms
individually, it also minimizes the first-order term (36) as a whole.
Inserting $\rho=\sqrt{\frac{h_{1}}{h_{2}}}$ into (46), we obtain the values
$\mathsf{C}_{\mathrm{s}}^{*}(h_{1},h_{2},p,\mathsf{P})$ and
$\mathsf{V}_{\mathrm{s}}^{*}(h_{1},h_{2},p,\mathsf{P})$ from (10) and (11).

###  -C Proof of Theorem 2

####  -C1 Number of symbols for a successful early decoding

The proof is based on the modified Dependence Testing bound introduced in [2,
Theorem 3], which we restate in the following for the single-user case:

######  Theorem 4.

For a general point-to-point channel, any input distribution $P_{X^{n}}$ and
any output distribution $Q_{Y^{n}}$, there exists an
$(n,\mathsf{M},\epsilon,\mathcal{F}^{n})$-code that satisfies

$\displaystyle\epsilon\leq$
$\displaystyle\Pr\left[\tilde{i}(X^{n};Y^{n})\leq\log\gamma_{n}\right]+\mathsf{K}_{n}\mathsf{M}\Pr\left[\tilde{i}(X^{n};\bar{Y}^{n})>\log\gamma_{n}\right]+\Pr[X^{n}\notin\mathcal{F}^{n}]$
(55)

where
$P_{X^{n}Y^{n}\bar{Y}^{n}}(a,b,c)=P_{X^{n}}(a)P_{Y^{n}|X^{n}}(b|a)Q_{Y^{n}}(c)$,
$\gamma_{n}$ is a positive threshold and the coefficient $\mathsf{K}_{n}$ is
defined over the Radon-Nikodym derivative

$\displaystyle\mathsf{K}_{n}:=\sup_{y^{n}\in\mathcal{Y}^{n}}\frac{dP_{Y^{n}}(y^{n})}{dQ_{Y^{n}}(y^{n})}.$
(56)

First, we derive the uniform distribution on the composite power shell
$\mathcal{S}^{(n_{2},n_{1})}(\mathsf{P})$. Choosing $x^{n_{1}}$ uniformly from
$\mathcal{S}^{(n_{2},n_{1})}(\mathsf{P})$ is equivalent to first choosing the
first $n_{2}$ symbols uniformly from $\mathcal{S}^{(n_{2})}(\mathsf{P})$ and
then independently choosing the remaining symbols uniformly from
$\mathcal{S}^{(n_{1}-n_{2})}(\mathsf{P})$. In the following, we use the
notation $x^{n_{2}}:=[x_{1},...,x_{n_{2}}]$ and
$x^{n_{1}-n_{2}}:=[x_{n_{2}+1},...,x_{n_{1}}]$. The distribution of the first
$n_{2}$ symbols is

$\displaystyle
P_{X^{n_{2}}}(x^{n_{2}})=\frac{\mathbbm{1}(x^{n_{2}}\in\mathcal{S}^{(n_{2})}(\mathsf{P}))}{\mathsf{S}_{n_{2}}(\sqrt{n_{2}\mathsf{P}})},$
(57)

where
$\mathsf{S}_{n}(r):=\frac{2\pi^{\frac{n}{2}}}{\Gamma\left(\frac{n}{2}\right)}r^{n-1}$,
and the distribution of the last $n_{1}-n_{2}$ symbols is

$\displaystyle
P_{X^{n_{1}-n_{2}}}(x^{n_{1}-n_{2}})=\frac{\mathbbm{1}(x^{n_{1}-n_{2}}\in\mathcal{S}^{(n_{1}-n_{2})}(\mathsf{P}))}{\mathsf{S}_{n_{1}-n_{2}}(\sqrt{(n_{1}-n_{2})\mathsf{P}})}.$
(58)

Since the choice of the last $n_{1}-n_{2}$ symbols is independent of the
choice of the first $n_{2}$ symbols, the total distribution of $x^{n_{1}}$ is

$\displaystyle
P_{X^{n}}(x^{n_{1}})=P^{\mathrm{Unif}}_{\mathcal{S}^{(n_{2},n_{1})}(\mathsf{P})}(x^{n_{1}}):=\frac{\mathbbm{1}(x^{n_{2}}\in\mathcal{S}^{(n_{2})}(\mathsf{P}))\cdot\mathbbm{1}(x^{n_{1}-n_{2}}\in\mathcal{S}^{(n_{1}-n_{2})}(\mathsf{P}))}{\mathsf{S}_{n_{2}}(\sqrt{n_{2}\mathsf{P}})\mathsf{S}_{n_{1}-n_{2}}(\sqrt{(n_{1}-n_{2})\mathsf{P}})},$
(59)

By definition of $\mathcal{S}^{(n_{2},n_{1})}(\mathsf{P})$, we have

$\displaystyle\mathbbm{1}(x^{n_{1}}\in\mathcal{S}^{(n_{2},n_{1})}(\mathsf{P}))=\mathbbm{1}(x^{n_{2}}\in\mathcal{S}^{(n_{2})}(\mathsf{P}))\cdot\mathbbm{1}(x^{n_{1}-n_{2}}\in\mathcal{S}^{(n_{1}-n_{2})}(\mathsf{P})),$
(60)

so we can simplify the expression (59) as

$\displaystyle
P^{\mathrm{Unif}}_{\mathcal{S}^{(n_{2},n_{1})}(\mathsf{P})}(x^{n_{1}})=\frac{\mathbbm{1}(x^{n_{1}}\in\mathcal{S}^{(n_{2},n_{1})}(\mathsf{P}))}{\mathsf{S}_{n_{2}}(\sqrt{n_{2}\mathsf{P}})\mathsf{S}_{n_{1}-n_{2}}(\sqrt{(n_{1}-n_{2})\mathsf{P}})}.$
(61)

Now, we can bound the Radon-Nikodym derivative that results from the change of
measure from
$P^{\mathrm{Unif}}_{\mathcal{S}^{(n_{2},n_{1})}(\mathsf{P})}(x^{n_{1}})$ to
the i.i.d. Gaussian output distribution.

######  Lemma 2.

Let $P_{Y^{n_{1}}}(y^{n_{1}})$ by the output distribution induced by
$P^{\mathrm{Unif}}_{\mathcal{S}^{(n_{2},n_{1})}(\mathsf{P})}(x^{n_{1}})$
defined in (61) and let
$Q_{Y^{n_{1}}}(y^{n_{1}})=\mathcal{N}(y^{n_{1}};\mathbf{0}^{n_{1}},(1+\mathsf{P})\cdot\mathbf{\mathrm{I}}^{n_{1}\times
n_{1}})$. Then, for $n_{1}$ sufficiently large,

$\displaystyle\frac{dP_{Y^{n_{1}}}(y^{n_{1}})}{dQ_{Y^{n_{1}}}(y^{n_{1}})}\leq\tilde{\mathsf{K}}:=729\cdot\frac{\pi}{8}\cdot\frac{(1+\mathsf{P})^{2}}{1+2\mathsf{P}}.$
(62)

For the proof of Lemma 2, please refer to Appendix -D. We can now analyze the
performance of the coding scheme:

  1. 1.

Codebook generation: For every message $m_{1}\in\mathcal{M}_{1}$, generate a
codeword according to the uniform distribution on the composite power shell:

$\displaystyle X_{1}^{n_{1}}(m_{1})\sim
P^{\mathrm{Unif}}_{\mathcal{S}^{(n_{2},n_{1})}(\mathsf{P}_{1})}.$ (63)

For every message $m_{2}\in\mathcal{M}_{2}$, generate a codeword according to
the i.i.d. Gaussian distribution:

$\displaystyle X_{2,i}(m_{2})\sim\mathcal{N}(0,\bar{\mathsf{P}}_{2}),\qquad
i=1,...,n_{2},$ (64)

where $\bar{\mathsf{P}}_{2}=\mathsf{P}_{2}-\delta$.

  2. 2.

Encoding: Superposition coding: to send the message pair $(m_{1},m_{2})$,
transmit
$x^{n_{1}}=x_{1}^{n_{1}}(m_{1})+[x_{2}^{n_{2}}(m_{2}),\mathbf{0}^{n_{1}-n_{2}}]$.

  3. 3.

Decoding: In the following, we use the notation
$x_{1}^{n_{2}}=[x_{1,1},...,x_{1,n_{2}}]$ and
$x_{2}^{n_{1}}=[x_{2}^{n_{2}},\mathbf{0}^{n_{1}-n_{2}}]$. User $k$ receives
the signal

$\displaystyle y_{k}^{n_{k}}$ $\displaystyle=$
$\displaystyle\sqrt{h_{k}}x_{1}^{n_{k}}+\sqrt{h_{k}}x_{2}^{n_{k}}+z_{k}^{n_{k}}.$
(65)

User 2 performs successive interference cancellation (SIC), while user 1
treats the interference as noise. Both decoders use the threshold decoding
technique from the modified DT bound (Theorem 4), where we use the threshold
$\gamma_{n_{1}}=\mathsf{K}_{n_{1}}\mathsf{M}_{1}$ for user 1,
$\gamma_{n_{2}}=\mathsf{K}_{n_{2}}\mathsf{M}_{1}$ for user 2 in the first SIC
phase, and $\gamma_{n_{2}}=\mathsf{K}_{n_{2}}\mathsf{M}_{2}$ for user 2 in the
second SIC phase.

     * •

User 1 treats user 2’s codeword as noise and decodes $\hat{m}_{1}$ as the
smallest message s.t.
$i(x_{1}^{n_{1}}(\hat{m}_{1}),y_{1}^{n_{1}})\geq\log(\mathsf{K}_{n_{1}}\mathsf{M}_{1})$.
If there is no such message, an error is declared.

     * •

User 2 uses successive interference cancellation (SIC) with early decoding.

       1. (a)

User 2 treats his own codeword as noise and decodes $\hat{m}_{1}$ as the
smallest message s.t.
$i\left(x_{1}^{n_{2}}(\hat{m}_{1}),y_{2}^{n_{2}}\right)\geq\log(\mathsf{K}_{n_{2}}\mathsf{M}_{1})$
using early decoding. If there is no such message, an error is declared.

       2. (b)

User 2 subtracts $x_{1}^{n_{2}}(\hat{m}_{1})$ from $y_{2}^{n_{2}}$ to obtain

$\displaystyle\tilde{y}_{2}^{n_{2}}$ $\displaystyle=$
$\displaystyle\sqrt{h_{2}}x_{2}^{n_{2}}+z_{2}^{n_{2}}.$ (66)

Then, he decodes $\hat{m}_{2}$ as the smallest message s.t.
$i\left(x_{2}^{n_{2}}(\hat{m}_{2}),\tilde{y}_{2}^{n_{2}}\right)\geq\log(\mathsf{K}_{n_{2}}\mathsf{M}_{2})$.

  4. 4.

Error analysis: For Theorem 2, only the first SIC step at user 2 is of
interest. Since we treat user 2’s codeword as noise, we have the effective
channel

$\displaystyle Y_{2}^{n_{2}}$ $\displaystyle=$
$\displaystyle\sqrt{h_{2}}X_{1}^{n_{2}}+(\sqrt{h_{2}}X_{2}^{n_{2}}+Z_{2}^{n_{2}}),$
(67)

with the noise

$\displaystyle\sqrt{h_{2}}X_{2,i}+Z_{2,i}$ $\displaystyle\sim$
$\displaystyle\mathcal{N}(0,1+h_{2}\bar{\mathsf{P}}_{2}),\qquad i=1,...n_{2}.$
(68)

By normalizing the noise power to 1, we get

$\displaystyle Y_{2}^{n_{2}}$ $\displaystyle=$
$\displaystyle\sqrt{g_{2}}X_{1}^{n_{2}}+\tilde{Z}_{2}^{n_{2}},$ (69)

where $\tilde{Z}_{2,i}\sim\mathcal{N}(0,1)$ is i.i.d.,
$\tilde{Z}_{2}^{n_{2}}\mathrel{\bot}\joinrel\mathrel{\mkern-5.0mu}\joinrel\mathrel{\bot}X^{n_{2}}$
and we define

$\displaystyle g_{2}$ $\displaystyle:=$
$\displaystyle\frac{h_{2}}{1+h_{2}\bar{\mathsf{P}}_{2}}.$ (70)

Since $X_{1}^{n_{1}}$ is a composite shell codeword, $X_{1}^{n_{2}}$ fulfills
the equal-power property. From Lemma 2, we can conclude that
$\mathsf{K}_{n_{2}}=\tilde{\mathsf{K}}$. We use the modified information
density

$\displaystyle\tilde{i}(X^{n};Y^{n}):=\log\frac{dP_{Y^{n}|X^{n}}(Y^{n}|X^{n})}{dQ_{Y^{n}}(Y^{n})},$
(71)

where $Q_{Y^{n}}(Y^{n})$ is the i.i.d. Gaussian output distribution. The
expectation and variance of the conditional information density are as in the
standard single-user case:

$\displaystyle\mathbb{E}\left[\frac{1}{n_{2}}\tilde{i}(x_{1}^{n_{2}};Y_{2}^{n_{2}})\right]$
$\displaystyle=$ $\displaystyle\mathsf{C}(g_{2}\mathsf{P}_{1}),$ (72)
$\displaystyle\mathrm{Var}\left[\frac{1}{n_{2}}\tilde{i}(x_{1}^{n_{2}};Y_{2}^{n_{2}})\right]$
$\displaystyle=$ $\displaystyle\frac{\mathsf{V}(g_{2}\mathsf{P}_{1})}{n_{2}}.$
(73)

Then, we can bound the outage and confusion probabilities of the DT bound
(Theorem 4):

$\displaystyle\Pr\left[\tilde{i}(x^{n_{2}};Y^{n_{2}})\leq\log\gamma_{n_{2}}\right]$
$\displaystyle\leq$ $\displaystyle
Q\left(\frac{n\mathsf{C}(g_{2}\mathsf{P}_{1})-\log\gamma_{n_{2}}}{\sqrt{n_{2}\mathsf{V}(g_{2}\mathsf{P}_{1})}}\right)+6\frac{\mathsf{B}_{1}(g_{2}\mathsf{P}_{1})}{\sqrt{n_{2}}},$
(74)
$\displaystyle\Pr\left[\tilde{i}(x^{n_{2}};\bar{Y}^{n_{2}})>\log\gamma_{n_{2}}\right]$
$\displaystyle\leq$
$\displaystyle\frac{\mathsf{B}_{2}(g_{2}\mathsf{P}_{1})}{\sqrt{n_{2}}\gamma_{n_{2}}},$
(75)

and therefore, we get with $\gamma_{n_{2}}=\tilde{\mathsf{K}}\mathsf{M}$,

$\displaystyle\epsilon_{\mathrm{SIC},1}^{(n_{2})}$ $\displaystyle\leq$
$\displaystyle
Q\left(\frac{n_{2}\mathsf{C}(g_{2}\mathsf{P}_{1})-\log(\tilde{\mathsf{K}}\mathsf{M}_{1})}{\sqrt{n_{2}\mathsf{V}(\mathsf{P}_{1})}}\right)+6\frac{\mathsf{B}_{1}(g_{2}\mathsf{P}_{1})}{\sqrt{n_{2}}}+\frac{\mathsf{B}_{2}(g_{2}\mathsf{P}_{1})}{\sqrt{n_{2}}}$
(76) $\displaystyle=$ $\displaystyle
Q\left(\frac{n_{2}\mathsf{C}(g_{2}\mathsf{P}_{1})-\log(\tilde{\mathsf{K}}\mathsf{M}_{1})}{\sqrt{n_{2}\mathsf{V}(g_{2}\mathsf{P}_{1})}}\right)+\frac{\mathsf{B}(\mathsf{P}_{1})}{\sqrt{n_{2}}},$
(77)

where we have defined
$\mathsf{B}(g_{2}\mathsf{P}_{1})=6\mathsf{B}_{1}(g_{2}\mathsf{P}_{1})+\mathsf{B}_{2}(g_{2}\mathsf{P}_{1}).$
By rearranging, we find

$\displaystyle\frac{n_{2}\mathsf{C}(g_{2}\mathsf{P}_{1})-\log(\mathsf{M}_{1})}{\sqrt{n_{2}\mathsf{V}(g_{2}\mathsf{P}_{1})}}$
$\displaystyle\geq$ $\displaystyle
Q^{-1}\left(\epsilon_{\mathrm{SIC},1}^{(n_{2})}-\frac{\mathsf{B}(g_{2}\mathsf{P}_{1})}{\sqrt{n_{2}}}\right)+\frac{\log(\tilde{\mathsf{K}})}{\sqrt{n_{2}\mathsf{V}(g_{2}\mathsf{P}_{1})}}$
(78) $\displaystyle=$ $\displaystyle
Q^{-1}\left(\epsilon_{\mathrm{SIC},1}^{(n_{2})}\right)+O\left(\frac{1}{\sqrt{n_{2}}}\right),$
(79)

where the last step follows from the Taylor expansion of the $Q$-function. By
dropping the $O(1/\sqrt{n_{2}})$ term and substituting $m=\sqrt{n_{2}}$, we
get the quadratic equation

$\displaystyle
m^{2}-\frac{\sqrt{\mathsf{V}(g_{2}\mathsf{P}_{1})}Q^{-1}\left(\epsilon_{\mathrm{SIC},1}^{(n_{2})}\right)}{\mathsf{C}(g_{2}\mathsf{P}_{1})}m-\frac{\log(\mathsf{M}_{1})}{{\mathsf{C}(g_{2}\mathsf{P}_{1})}}$
$\displaystyle\geq$ $\displaystyle 0,$ (80)

which has the roots

$\displaystyle m_{1,2}$ $\displaystyle=$
$\displaystyle\frac{\sqrt{\mathsf{V}(g_{2}\mathsf{P}_{1})}Q^{-1}\left(\epsilon_{\mathrm{SIC},1}^{(n_{2})}\right)}{2\mathsf{C}(g_{2}\mathsf{P}_{1})}\pm\sqrt{\frac{\mathsf{V}(g_{2}\mathsf{P}_{1})Q^{-1}\left(\epsilon_{\mathrm{SIC},1}^{(n_{2})}\right)^{2}}{4\mathsf{C}^{2}(g_{2}\mathsf{P}_{1})}+\frac{\log(\mathsf{M}_{1})}{{\mathsf{C}(g_{2}\mathsf{P}_{1})}}},$
(81)

where only the solution with the positive sign in front of the square root is
positive. Also, note that the polynomial in (80) has a positive sign in front
of the leading term, which means that the inequality (80) is fulfilled for all
$n_{2}$ greater than the second root. Therefore, early decoding is possible
for

$\displaystyle n_{2}$ $\displaystyle\geq$
$\displaystyle\left(\frac{\sqrt{\mathsf{V}(g_{2}\mathsf{P}_{1})}Q^{-1}\left(\epsilon_{\mathrm{SIC},1}^{(n_{2})}\right)}{2\mathsf{C}(g_{2}\mathsf{P}_{1})}+\sqrt{\frac{\mathsf{V}(\mathsf{P}_{1})Q^{-1}\left(\epsilon_{\mathrm{SIC},1}^{(n_{2})}\right)^{2}}{4\mathsf{C}^{2}(g_{2}\mathsf{P}_{1})}+\frac{\log(\mathsf{M}_{1})}{{\mathsf{C}(g_{2}\mathsf{P}_{1})}}}\right)^{2}.$
(82)

####  -C2 Achievable Rates

First, we analyze the rates at receiver 1. When treating user 2’s signal as
noise, the effective channel is

$\displaystyle Y_{1}^{n_{1}}$ $\displaystyle=$
$\displaystyle\sqrt{h_{1}}X_{1}^{n_{1}}+(\sqrt{h_{1}}X_{2}^{n_{1}}+Z_{1}^{n_{1}})=\sqrt{h_{1}}X_{1}^{n_{1}}+\tilde{Z}_{1}^{n_{1}},$
(83)

where

$\displaystyle\tilde{Z}_{1,i}$ $\displaystyle\sim$
$\displaystyle\mathcal{N}(0,\sigma_{i}),$ (84) $\displaystyle\sigma_{i}$
$\displaystyle:=$
$\displaystyle\begin{cases}1+h_{1}\bar{\mathsf{P}}_{2},&i=1,...,n_{2},\\\
1,&i=n_{2}+1,...,n_{1}.\end{cases}$ (85)

As the reference in the modified information density, we use the Gaussian
output distribution

$\displaystyle Y_{1,i}$ $\displaystyle\sim$
$\displaystyle\mathcal{N}\left(0,h_{1}\mathsf{P}_{1}+\sigma_{i}\right).$ (86)

In [11, eq. (109)], the conditional modified information density for this
scenario has been derived as

$\displaystyle\tilde{i}\left(x^{n_{1}},Y^{n_{1}}\right)$ $\displaystyle=$
$\displaystyle n_{1}\bar{\mathsf{C}}_{1}$ \+ loge2(1+h1(P1+¯P2))[∑_i =
1^n_2(h_1x_1,i^2 + 2 h_1x_1,i~Z_1,i) - g_1P_1∑_i=1^n_2~Z_1,i^2]

where $g_{1}=\frac{h_{1}}{1+h_{1}\bar{\mathsf{P}}_{2}}$ and

$\displaystyle\bar{\mathsf{C}}_{1}$ $\displaystyle:=$ $\displaystyle
p\mathsf{C}(g_{1}\bar{\mathsf{P}}_{1})+(1-p)\mathsf{C}(h_{1}\bar{\mathsf{P}}_{1}).$
(88)

In contrast to [11], we use composite shell codewords instead of i.i.d.
Gaussian codewords, so we can further simplify (LABEL:eq:info_density_rx_1) as

$\displaystyle\tilde{i}\left(x^{n_{1}},Y^{n_{1}}\right)$ $\displaystyle=$
$\displaystyle n_{1}\bar{\mathsf{C}}_{1}$ \+ loge2(1+h1(P1+¯P2))∑_i =
1^n_2[h_1P_1+ 2 h_1x_1,i~Z_1,i - g_1P_1~Z_1,i^2]

The expectation of (LABEL:eq:rx1_info_density) is

$\displaystyle\mathbb{E}\left[\tilde{i}\left(x^{n_{1}},Y^{n_{1}}\right)\right]$
$\displaystyle=$ $\displaystyle n_{1}\bar{\mathsf{C}}_{1}$ (90)

and the variance is

$\displaystyle\mathrm{Var}\left[\tilde{i}\left(x^{n_{1}},Y^{n_{1}}\right)\right]$
$\displaystyle=$ $\displaystyle
n_{2}\mathsf{V}(g_{1}\mathsf{P}_{1})+(n_{1}-n_{2})\mathsf{V}(h_{1}\mathsf{P}_{1}).$
(91)

Now, we can use the usual steps to bound the outage and confusion probability
in the DT bound and get the rates from Theorem 2.

The rate for user 2 is the standard single-user rate using i.i.d. Gaussian
codebooks, since after the successful SIC, user 2 is interference free.

###  -D Proof of Lemma 2

In [2, Appendix B], it is shown that the output distribution induced by the
uniform input distribution on the power shell can be upper bound by

$\displaystyle P_{Y^{n}}(y^{n})$ $\displaystyle\leq$ $\displaystyle
P_{\mathrm{out,shell}}(n,t):=\frac{c}{2}\frac{1+\sqrt{1+4\mathsf{P}t}}{\sqrt[4]{1+4\mathsf{P}t}}\pi^{-\frac{n}{2}}e^{-n\frac{(1+\mathsf{P})}{2}}e^{-n\frac{t}{2}}e^{n\frac{\sqrt{1+4\mathsf{P}t}}{2}}\left(1+\sqrt{1+4\mathsf{P}t}\right)^{-\frac{n}{2}},$

where $t:=\frac{\|y^{n}\|^{2}}{n}$ and $c=27\sqrt{\frac{\pi}{8}}$. Since in
our case, the input consists of the concatenation of two independent shell
codewords, and since the channel is memoryless, the output distribution in our
case will be the product of the output distributions of the two individual
power shell codewords. We define

$\displaystyle
t_{\mathrm{I}}:=\frac{\|[y_{1},...,y_{n_{2}}]\|^{2}}{n_{2}},\qquad
t_{\mathrm{II}}:=\frac{\|[y_{n_{2}+1},...,y_{n_{1}}]\|^{2}}{n_{1}-n_{2}},\qquad\text{and}\qquad
t:=\frac{\|y^{n_{1}}\|^{2}}{n_{1}}.$ (93)

Note that $pt_{\mathrm{I}}+(1-p)t_{\mathrm{II}}=t$, where
$p=\frac{n_{2}}{n_{1}}$. The output distribution for the composite shell code
is

$\displaystyle P_{Y^{n_{1}}}(y^{n_{1}})$ $\displaystyle\leq$ $\displaystyle
P_{\mathrm{out,shell}}(n_{2},t_{\mathrm{I}})\cdot
P_{\mathrm{out,shell}}(n_{1}-n_{2},t_{\mathrm{II}})$ (96) $\displaystyle=$
$\displaystyle\frac{c^{2}}{4}\frac{1+\sqrt{1+4\mathsf{P}t_{\mathrm{I}}}}{\sqrt[4]{1+4\mathsf{P}t_{\mathrm{I}}}}\frac{1+\sqrt{1+4\mathsf{P}t_{\mathrm{II}}}}{\sqrt[4]{1+4\mathsf{P}t_{\mathrm{II}}}}\pi^{-\frac{n_{2}}{2}}\pi^{-\frac{n_{1}-n_{2}}{2}}e^{-n_{2}\frac{(1+\mathsf{P})}{2}}e^{-(n_{1}-n_{2})\frac{(1+\mathsf{P})}{2}}$
$\displaystyle\cdot
e^{-n_{2}\frac{t_{\mathrm{I}}}{2}}e^{-(n_{1}-n_{2})\frac{t_{\mathrm{II}}}{2}}e^{n_{2}\frac{\sqrt{1+4\mathsf{P}t_{\mathrm{I}}}}{2}}e^{(n_{1}-n_{2})\frac{\sqrt{1+4\mathsf{P}t_{\mathrm{II}}}}{2}}$
$\displaystyle\cdot\left(1+\sqrt{1+4\mathsf{P}t_{\mathrm{I}}}\right)^{-\frac{n_{2}}{2}}\left(1+\sqrt{1+4\mathsf{P}t_{\mathrm{II}}}\right)^{-\frac{n_{1}-n_{2}}{2}}$
$\displaystyle=$
$\displaystyle\mathsf{K}_{1}(\mathsf{P},t_{\mathrm{I}},t_{\mathrm{II}})\cdot\pi^{-\frac{n_{1}}{2}}e^{-n_{1}\frac{(1+\mathsf{P})}{2}}e^{-n_{1}\frac{pt_{\mathrm{I}}+(1-p)t_{\mathrm{II}}}{2}}e^{n_{1}\left(p\frac{\sqrt{1+4\mathsf{P}t_{\mathrm{I}}}}{2}+(1-p)\frac{\sqrt{1+4\mathsf{P}t_{\mathrm{II}}}}{2}\right)}$
$\displaystyle\cdot
e^{-\frac{n_{1}}{2}\left(p\cdot\ln\left(1+\sqrt{1+4\mathsf{P}t_{\mathrm{I}}}\right)+(1-p)\cdot\ln\left(1+\sqrt{1+4\mathsf{P}t_{\mathrm{II}}}\right)\right)}$
$\displaystyle=$
$\displaystyle\mathsf{K}_{1}(\mathsf{P},t_{\mathrm{I}},t_{\mathrm{II}})\cdot\pi^{-\frac{n_{1}}{2}}e^{-\frac{n_{1}}{2}f_{1}(\mathsf{P},t_{\mathrm{I}},t_{\mathrm{II}},p)},$
(97)

where we have defined

$\displaystyle\mathsf{K}_{1}(\mathsf{P},t_{\mathrm{I}},t_{\mathrm{II}})$
$\displaystyle:=$
$\displaystyle\frac{c^{2}}{4}\frac{1+\sqrt{1+4\mathsf{P}t_{\mathrm{I}}}}{\sqrt[4]{1+4\mathsf{P}t_{\mathrm{I}}}}\frac{1+\sqrt{1+4\mathsf{P}t_{\mathrm{II}}}}{\sqrt[4]{1+4\mathsf{P}t_{\mathrm{II}}}}$
(98)

and

$\displaystyle f_{1}(\mathsf{P},t_{\mathrm{I}},t_{\mathrm{II}},p)$
$\displaystyle:=$
$\displaystyle(1+\mathsf{P})+pt_{\mathrm{I}}+(1-p)t_{\mathrm{II}}-p\sqrt{1+4\mathsf{P}t_{\mathrm{I}}}-(1-p)\sqrt{1+4\mathsf{P}t_{\mathrm{II}}}$
(99)
$\displaystyle+p\cdot\ln\left(1+\sqrt{1+4\mathsf{P}t_{\mathrm{I}}}\right)+(1-p)\cdot\ln\left(1+\sqrt{1+4\mathsf{P}t_{\mathrm{II}}}\right).$

The output distribution induced by i.i.d. Gaussian inputs is

$\displaystyle Q_{Y^{n_{1}}}(y^{n_{1}})$ $\displaystyle=$
$\displaystyle(2\pi)^{-\frac{n_{1}}{2}}(1+\mathsf{P})^{-\frac{n_{1}}{2}}e^{-n_{1}\frac{t}{2(1+\mathsf{P})}}$
(100) $\displaystyle=$
$\displaystyle\pi^{-\frac{n_{1}}{2}}e^{-\frac{n_{1}}{2}f_{2}(\mathsf{P},t)},$

where

$\displaystyle f_{2}(\mathsf{P},t)$ $\displaystyle:=$
$\displaystyle\ln\left(2(1+\mathsf{P})\right)+\frac{t}{(1+\mathsf{P})}.$ (101)

We obtain the ratio of output distributions by dividing (97) by (100):

$\displaystyle\frac{dP_{Y^{n_{1}}}(y^{n_{1}})}{dQ_{Y^{n_{1}}}(y^{n_{1}})}$
$\displaystyle\leq$
$\displaystyle\mathsf{K}_{1}(\mathsf{P},t_{\mathrm{I}},t_{\mathrm{II}})\cdot
e^{-\frac{n_{1}}{2}\left(f_{1}(\mathsf{P},t_{\mathrm{I}},t_{\mathrm{II}},p)-f_{2}(\mathsf{P},t)\right)}.$
(102)

Since we are looking for an upper bound of the ratio (102) for the worst case,
we are finding the realization of $y^{n_{1}}$ (or, equivalently,
$t_{\mathrm{I}}$ and $t_{\mathrm{II}}$) that maximizes (102).

First, we investigate the exponent in (102), which we want to minimize in
order to maximize (102):

$\displaystyle f(\mathsf{P},t_{\mathrm{I}},t_{\mathrm{II}},p)$
$\displaystyle:=$ $\displaystyle
f_{1}(\mathsf{P},t_{\mathrm{I}},t_{\mathrm{II}},p)-f_{2}(\mathsf{P},t)$ (105)
$\displaystyle=$
$\displaystyle(1+\mathsf{P})+pt_{\mathrm{I}}+(1-p)t_{\mathrm{II}}-p\sqrt{1+4\mathsf{P}t_{\mathrm{I}}}-(1-p)\sqrt{1+4\mathsf{P}t_{\mathrm{II}}}$
$\displaystyle+p\cdot\ln\left(1+\sqrt{1+4\mathsf{P}t_{\mathrm{I}}}\right)+(1-p)\cdot\ln\left(1+\sqrt{1+4\mathsf{P}t_{\mathrm{II}}}\right)$
$\displaystyle-\ln\left(2(1+\mathsf{P})\right)-\frac{t}{(1+\mathsf{P})}$
$\displaystyle=$
$\displaystyle(1+\mathsf{P})-\ln\left(2(1+\mathsf{P})\right)+\frac{(pt_{\mathrm{I}}+(1-p)t_{\mathrm{II}})\mathsf{P}}{(1+\mathsf{P})}-p\sqrt{1+4\mathsf{P}t_{\mathrm{I}}}$
$\displaystyle-(1-p)\sqrt{1+4\mathsf{P}t_{\mathrm{II}}}+p\cdot\ln\left(1+\sqrt{1+4\mathsf{P}t_{\mathrm{I}}}\right)$
$\displaystyle+(1-p)\cdot\ln\left(1+\sqrt{1+4\mathsf{P}t_{\mathrm{II}}}\right),$

where the last step follows from $t=pt_{\mathrm{I}}+(1-p)t_{\mathrm{II}}$. Its
derivative w.r.t. $t_{\mathrm{I}}$ is

$\displaystyle\frac{df(\mathsf{P},t_{\mathrm{I}},t_{\mathrm{II}},p)}{dt_{\mathrm{I}}}$
$\displaystyle=$ $\displaystyle\frac{p\mathsf{P}}{(1+\mathsf{P})}-\frac{p\cdot
4\mathsf{P}}{2\sqrt{1+4\mathsf{P}t_{\mathrm{I}}}}+\frac{p\cdot
4\mathsf{P}}{(2\sqrt{1+4\mathsf{P}t_{\mathrm{I}}})(1+\sqrt{1+4\mathsf{P}t_{\mathrm{I}}})},$
(106)

which has exactly one zero at $t_{\mathrm{I}}=1+\mathsf{P}$. It can also be
shown that

$\displaystyle\left.\frac{d^{2}f(\mathsf{P},t_{\mathrm{I}},t_{\mathrm{II}},p)}{dt_{\mathrm{I}}^{2}}\right|_{t_{\mathrm{I}}=1+\mathsf{P}}$
$\displaystyle>$ $\displaystyle 0,$ (107)

which means that $t_{\mathrm{I}}=1+\mathsf{P}$ is the minimizes
$f(\mathsf{P},t_{\mathrm{I}},t_{\mathrm{II}},p)$. Since
$f(\mathsf{P},t_{\mathrm{I}},t_{\mathrm{II}},p)$ is symmetric in its arguments
$t_{\mathrm{I}}$ and $t_{\mathrm{II}}$, the global minimum w.r.t.
$t_{\mathrm{II}}$ is also $t_{\mathrm{II}}=1+\mathsf{P}$. Inserting these
values of $t_{\mathrm{I}}$ and $t_{\mathrm{II}}$ into (105) yields

$\displaystyle
f(\mathsf{P},t_{\mathrm{I}}=1+\mathsf{P},t_{\mathrm{II}}=1+\mathsf{P},p)$
$\displaystyle=$ $\displaystyle 0.$ (108)

For $n_{1}$ sufficiently large, this leads to

$\displaystyle\frac{dP_{Y^{n_{1}}}(y^{n_{1}})}{dQ_{Y^{n_{1}}}(y^{n_{1}})}$
$\displaystyle\leq$
$\displaystyle\mathsf{K}_{1}(\mathsf{P},t_{\mathrm{I}}=1+\mathsf{P},t_{\mathrm{II}}=1+\mathsf{P})$
(109) $\displaystyle=$ $\displaystyle
c^{2}\frac{(1+\mathsf{P})^{2}}{1+2\mathsf{P}}.$ (110)

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