# Channel-Aware Ordered Successive Relaying with Finite-Blocklength Coding

Lingrui Zhang, Yuxing Han, Qiong Wang, and Wei Chen, Lingrui Zhang and Wei
Chen are with the Department of Electronic Engineering and Beijing National
Research Center for Information Science and Technology, Tsinghua University,
Beijing 100084, China, e-mail<EMAIL_ADDRESS>;
<EMAIL_ADDRESS>Han is with the Shenzhen International Graduate
School, Tsinghua University, e-mail<EMAIL_ADDRESS>Wang
is with the State Grid Beijing Electric Power Company, e-mail:
<EMAIL_ADDRESS>
###### Abstract

Successive relaying premises to improve the transmission rate by allowing the
source and relays to transmit messages simultaneously, but it may cause severe
inter-relay interference (IRI). To mitigate IRI, IRI cancellation schemes are
proposed. However, the interference cancellation method has a high risk of
error propagation resulting in a severe transmission rate loss in the Finite
Blocklength Regime. Thus, how to jointly decode for successive relaying with
Finite-Blocklength Coding (FBC) remains open. In this paper, we present an
optimized channel aware ordered successive relaying protocol with finite-
blocklength coding (CAO-SIR-FBC), which can recover the rate loss by carefully
adapting relays’ transmission order and rate. We analyse the average
throughput of CAO-SIR-FBC, based on which the closed-form expression of it in
the high signal-to-noise regime (SNR) is presented. The average throughput
analysis and simulations show that CAO-SIR-FBC outperforms the conventional
two-timeslot half-duplex relaying in terms of spectral efficiency.

###### Index Terms:

Successive relaying, finite-blocklength coding, interference cancellation,
decode-and-forward, rate adaptation

##  I Introduction

With the rapid development of 5G and 6G wireless networks, ultra-reliable low-
latency communications (URLLC) has attracted considerable attention for its
potential applications in telesurgery, smart cities and autonomous driving.
3GPP has researchers from academia and industry have begun to look beyond 5G
and start the research of 6G systems for multiple use cases such as artificial
intelligence, augmented/virtual reality (AR/VR), and automatic driving
[yang20196g]. An ultra-reliable low-latency communications (URLLC) has
attracted considerable attention, because it support the stringent
requirements of 6G system. Specifically, URLLC has strict standards requiring
$10^{-10}$ error probability and $1$ ms latency time [20196gwirelessnetwork].
To achieve this strict requirements, how to support a low delay violation
probability has attracted significant attention[2019effective].

As one of three main usage scenarios for the 5G system, URLLC still relies on
orthogonal frequency division multiplexing (OFDM) as 4G does [20173GPP170379].
Different from the 4G system, the subcarrier spacing is allowed to be enlarged
[20173GPP38.913]. Due to the enlarged subcarrier spacing, the duration of one
timeslot is diminished [20183GPP38.300]. Accordingly, the latency of the time-
slotted scheduling strategy is also diminished. However, the submillisecond
duration of a timeslot results in a too short blocklength of a data packet
[2016toward]. The Shannon’s capacity given in [1948mathematical] is no longer
achievable with the short packet. Consequently, short-packet transmissions are
adopted in URLLC with the FBC coding approaches. In the finite blocklength
regime, the maximal transmission rate over Additive White Gaussian Noise
(AWGN) channels was a decrease function of error probability $\varepsilon$ and
an increase function of the blocklength $n$. Moreover, The transmission rate
analysis has been extended to other channel models, such as multiple-antenna
fading channels [2014quasi], multiple-antenna Rayleigh-fading channels
[2015short], block-fading channels [2013block], and hybrid automatic repeat
request (HARQ) [2018low].

Satisfying the reliable requirement of URLLC with FBC is a challenge because
there is a fundamental tradeoff between latency and reliability [2020joint].
More specifically, short blocklength causes a severe loss in coding gain
[2018short]. Because of larger subcarrier spacing, the number of subcarriers
is declined under the same bandwidth, which decreases the maximum frequency
diversity gain in OFDM systems. Hence, there is a need to reuse the spatial
domain to ensure reliability in URLLC. This needs to be done both at each
network node and by densifying the whole network [20185gdesign]. At the node
level, MIMO (Multiple-Input Multiple-Output) systems are adopted in URLLC
because they achieve huge capacity increases and Antenna diversity gains over
the harsh link [1999capacity]. Combined with OFDM, the MIMO-OFDM system can
achieve higher capacity and reliability [2006MIMOOFDM]. However, the use of
MIMO technology may not be practical if nodes in a network are small,
inexpensive, or have typically severe energy constraints [2008unified].
Cooperative communication is a solution to overcome this limitation by which
the high transmission rate is achieved over the harsh wireless links, without
causing high implementation complexity. There are also numerous advantages of
cooperative communication, such as substantial diversity gains and a low
occupancy rate of additional frequency resources. Cooperative communication
was first proposed by Sendonaris $et\ al.$ for Code Division Multiple Access
systems [sendonaris2003user, sendonaris2003user2]. From then on, much research
focused on cooperative communication. The space diversity and diversity-
multiplexing tradeoff of various cooperative relaying schemes, including
fixed, selection, and incremental relaying, were studied by Laneman $et\ al.$
[laneman2004cooperative]. These studies showed that the amplify-and-forward
(AF) and the selective decode-and-forward (DF) protocols can achieve the
maximal diversity gain over fading channels [nabar2004fading]. The maximal
diversity gain of these protocols is equal to the employed nodes’ number in
cooperative transmission. An opportunistic relaying protocol was proposed
based on network path selection [bletsas2006simple]. Most of the research
mentioned above is mainly focused on two-timeslot relaying protocols, in which
relays receive the signals from the source in the first timeslot and then
forward the signals to the destination node in the second timeslot. However,
these relaying protocols have to suffer from a half-duplex constraint, i.e. a
node cannot receive and forward simultaneously in the same frequency band. As
a result, there is a significant multiplexing loss. The multiplexing gains of
two-timeslot relaying protocols are upper-bounded by $\frac{1}{2}$. To
overcome the multiplexing loss, some cooperative relaying networks with full-
duplex relays were studied in recent researches [li2016multi]. The full-duplex
relays can receive and forward signals at the same time. However, the self-
interference caused at full-duplex relays still makes it challenging to
implement in practice.

Successive relaying is also a feasible method of recovering the multiplexing
loss besides using full-duplex relays [yang2007towards]. Its basic idea relies
on the current transmission of the source and relays. More specifically, the
source transmits to one relay while another relay transmits to the destination
synchronously. In this protocol, the cost of sending one message is less than
two timeslots. Therefore, the multiplexing gain of successive relaying is
greater than 1/2, while the half-duplex constraint is satisfied.
Unfortunately, successive relaying may cause severe IRI, resulting in a low
transmission rate and poor reliability. Therefore, mitigating IRI becomes a
vital issue for successive relaying. The interference cancellation method for
AF successive relaying was proposed in [ji2014partial, ren2018successive]. At
the destination node, full interference cancellation and partial interference
cancellation were proposed to suppress the IRI for two-path successive
relaying systems in [ji2014partial]. More recently, a generalized self-
interference method to mitigate IRI was proposed in [ren2018successive]. For
DF-based successive relaying, IRI causes severe decoding errors and error
propagation. Hence, IRI cancellation is a more critical issue. Relay
scheduling or selection was applied to improve the throughput of the DF-based
two-path successive relaying in [nomikos2012successive]. Hu $et\ al.$
investigated an IRI mitigation method, in which two signals transmitted by
source and relay are jointly decoded, in a multiple-relay cooperative
network[hu2012efficient]. In [wei2020successive], a DF relaying protocol was
proposed based on an analog network interference cancellation method with
linear processing without decoding the signals from the source relay. A
channel-aware successive relaying protocol (CAO-SIR) is proposed in [2014CAO],
where the decode-and-cancel protocol is extended to cancel the IRI completely.

Recently, researchers have also investigated relaying under the finite
blocklength regime. The authors in [2019finite] derived closed-form
expressions for the coding rates of relay communications under the Nakagami-m
fading channel in the finite blocklength regime. In [2016finite], Du $et\ al.$
studied the relaying throughput performance of Multi-Hop Relaying Networks
with FBC under the quasistatic Rayleigh channels, found the optimal distance
and blocklength of the network and compared that to the infinite blocklength
regime. However, successive relaying protocol in the finite blocklength regime
suffers from a severe loss of transmission rate because of the high risk of
error propagation caused by the interference cancellation method of the
network. Thus, achieving low-latency short-packet communications is still an
open problem for successive relaying protocol.

In this paper, we investigate a DF-based successive relaying protocol, also
referred to as CAO-SIR-FBC. The IRI cancellation method is based on the CAO-
SIR [2014CAO]. More specifically, an interfered relay can obtain prior
knowledge transmitted from the source. The prior knowledge is used to mitigate
IRI without high computational complexity. We further analyze the influence of
FBC. The error probabilities of every decoding process are not ignored in this
context. The error probability at the destination, which usually is
constrained to guarantee the high-reliability requirement of the system, is
given by the error-propagation theory. This observation motivates us to
optimize the CAO-SIR-FBC scheme further. Specifically, the transmission rate
in every timeslots is optimized according to the error probabilities in the
decoding processes. However, it is not trivial to derive the close-form
expressions of the transmission rate in arbitrary SNR. Based on the optimized
CAO-SIR-FBC, we presented the average throughput of our scheme. Compared to
classic protocols, the average throughput is increased in our scheme.
Furthermore, CAO-SIR-FBC also outperforms conventional protocols in terms of
the multiplexing gain.

The rest of this paper is organized as follows. Section II presents the system
model. The CAO-SIR-FBC is presented along with its algorithm in Section III.
The performance analysis of CAO-SIR-FBC is carried out in Section IV. Finally,
the numerical results and conclusions are presented in Sections V and VI,
respectively.

##  II System Model

Figure 1: System model

We consider a cooperative communication system in which a source node $S$
communicates to a destination node $D$ with $N$ relays denoted by
$\mathcal{N}=\\{1,2,\cdots,N\\}$, as shown in Fig. 1. The channel coefficient
of links between nodes $a$ and $b$ are denoted by $h_{a,b}$, where
$a\in\\{S,1,\cdots,N\\}$ and $b\in\\{1,\cdots,N,D\\}$. We denote by $g_{a,b}$
the channel gain of links between nodes $a$ and $b$, namely,
$g_{a,b}=|h_{a,b}|^{2}$.

We consider a quasi-static or slow-fading channel, in which the channel state
remains constant in each successive relaying period. In addition, channel
estimation with a pilot and channel-state-information (CSI) feedback are
adopted. Specifically, the source and relay nodes send the pilots at the
beginning of each successive relaying period. After receiving the broadcast
pilots, a relay can then estimate the channel coefficients of the source-relay
and relay-relay links. Similarly, the destination node $D$ can estimate the
channel coefficients of the source-destination and relay-destination links.
Consequently, $D$ has the CSI of source-destination and relay-destination
channels. A relay has the CSI of links from $S$ and other relays to itself.
Finally, the source-relay, source-destination, and relay-destination channel
gains are fed back to the source $S$ through a signaling channel.

Half-duplex constraint is assumed throughout this paper. The relay node cannot
transmit and receive at the same time. Time-slotted scheduling is assumed in
this context. We use $X_{a}[k]$ to denote the signal transmitted by node $a$
in the $k$th timeslot. Node $b$, which does not transmit in the $k$th timeslot
because of the half-duplex constraint, receives a signal denoted by

$Y_{b}[k]=\sum_{a\in\mathfrak{A}[k]}h_{a,b}X_{a}[k]+Z_{b}[k],$ (1)

where $\mathfrak{A}[k]$ denote the set of transmitting nodes in the $k$th
timeslot. Notation $Z_{b}[k]$ denotes the Additive White Gaussian Noise (AWGN)
at node $b$ and is subject to normal distribution with zero mean and a
variance of $\sigma^{2}$, i.e., $Z_{b}[k]\sim\mathcal{CN}(0,\sigma^{2})$. The
transmission power of node $a$ is denoted by $P_{a}=E\\{|X_{a}[k]|^{2}\\}$. We
assume that each node has the same transmission power $P$, i.e.,
$P=E\\{|X_{a}[k]|^{2}\\}$ for all $a\in\\{S,1,2,\cdots,N\\}$. The transmitter-
side SNR of node $a$ is denoted by $\gamma=\frac{P}{\sigma^{2}}$. Hence, the
receiver-side SNR of the link between node $a$ and node $b$ is denoted by
$\rho_{a,b}=g_{a,b}\gamma$.

Because our research is based on CAO-SIR model, we follow the relay order and
scheduling schemes of basic CAO-SIR [2014CAO]. More specifically, the channel
gains of relay set satisfy that

$g_{S,1}\leq g_{S,2}\leq\cdots\leq g_{S,N}.$ (2)

The received signals of the relay $m$, $m=1,2\cdots,N$, are given by

$\displaystyle Y_{m}[1]=$ $\displaystyle h_{S,m}X_{S}[1]+Z_{m}[1],$ (3)
$\displaystyle\begin{split}Y_{m}[k]=&h_{S,m}X_{S}[k]+h_{k-1,m}X_{k-1}[k]+Z_{m}[k],\qquad
k=2,\cdots,m.\end{split}$ (4)

At the destination node $D$, the received signals are presented by

$\displaystyle Y_{D}[1]=$ $\displaystyle h_{S,D}X_{S}[1]+Z_{D}[1],$ (5)
$\displaystyle\begin{split}Y_{D}[k]=&h_{S,D}X_{S}[k]+h_{k-1,D}X_{k-1}[k]+Z_{D}[k],\qquad
k=2,\cdots,N,\end{split}$ (6) $\displaystyle Y_{D}[N+1]=$ $\displaystyle
h_{N,D}X_{N}[N+1]+Z_{D}[N+1].$ (7)

The IRI cancellation mechanism at relay $m$ is shown by

$Y_{m}[i+1]-h_{i,m}X_{i}[i+1]=h_{S,m}X_{S}[i+1]+Z_{m}[i+1].$ (8)

The right-hand side of Eq. (8) is equivalent to an AWGN channel without IRI.
Hence, relay $m$ can decode $W_{m}$. Similarly, node $D$ is capable of
thoroughly mitigating the interference of $S$ in Eq. (6) as

$Y_{D}[i]-h_{S,D}X_{S}[i]=h_{i-1,D}X_{i-1}[i]+Z_{D}[i].$ (9)

Finite-blocklength coding is assumed. Specifically, a packet consisting of $k$
payload bits is typically encoded into a complex symbol whose blocklength is
$n$. The ratio denoted by $R=k/n$, i.e., the number of information bits per
complex symbol represents the transmission rate. Let $R^{*}(\varepsilon)$
denote the largest transmission rate $k/n$ of a coding scheme whose packet
error probability does not exceed $\varepsilon$. Additionally, we assume that
the blocklength $n$ is a constant in the whole transmission. Therefore, a
normal approximate rate given in [polyanskiy2010channel] is presented by

$R^{*}(\varepsilon)=C-\sqrt{\frac{V}{n}}Q^{-1}(\varepsilon)+\frac{\log
n}{2n}.$ (10)

Here, $Q^{-1}(\cdot)$ denotes the inverse of the Gaussian $Q$ function (the
tail distribution function of the standard normal distribution).111 As usual,
$Q(x)=\int_{x}^{\infty}\frac{1}{\sqrt{2\pi}}\mathrm{e}^{-t^{2}/2}dt.$ Notation
$V$ is the channel dispersion [polyanskiy2010channel].

Regarding the case of a real AWGN channel with the receiver-side SNR $\rho$,
the capacity is given by [2016toward]

$\displaystyle C(\rho)$ $\displaystyle=\frac{1}{2}\log(1+\rho).$ (11)

The channel dispersion is expressed as

$V(\rho)=\frac{\rho}{2}\frac{(2+\rho)}{(1+\rho)^{2}}(\log\mathrm{e})^{2}.$
(12)

##  III Successive DF relaying protocol with FBC

In this section, we first propose the optimization problem of the channel-
aware rate adaptation of CAO-SIR-FBC. Furthermore, the algorithms to compute
the maximal transmission rate are expressed in the second part.

###  III-A Channel-aware rate adaptation

Because of FBC, the packet-error probability in every timeslot should be
considered. We denote by $\varepsilon_{m}[k]$ the supremum of the error
probability when relay $m$ decodes $W_{k}$ . When node $D$ decodes message
$W_{k}$, similarly, the supremum of the error probability is denoted by
$\varepsilon_{D}[k]$. These error probabilities of $W_{k}$ are dependent.
Based on the definition of rate, the rate of message $W_{k}$ remains constant
in the transmission process where the number of payload bits is constant. In
other words, the rates of transmitting message $W_{k}$ through $S$-relay $m$
and relay $k$-$D$ channels are constant for all $m=k,k+1,\cdots,N$. Therefore,
based on Eq. (10), the maximal rates of message $W_{k}$ in $S$-relay $m$
channels are presented by

$\displaystyle R^{*}_{k}=$ $\displaystyle
C_{S,m}-\sqrt{\frac{V_{S,m}}{n}}Q^{-1}\left(\varepsilon_{m}[k]\right)+\frac{\log
n}{2n}.$ (13)

The maximal rates of message $W_{k}$ in the relay $k$-$D$ channel is presented
by

$\displaystyle R^{*}_{k}=$ $\displaystyle
C_{k,D}-\sqrt{\frac{V_{k,D}}{n}}Q^{-1}(\varepsilon_{D}[k])+\frac{\log n}{2n}.$
(14)

Here, $C_{S,m}$ and $C_{k,D}$ denote the capacity of channels of $S$-relay $m$
and relay $k$-$D$, respectively.We use $V_{S,m}$ and $V_{k,D}$ to denote the
capacity of channels of $S$-relay $m$ and relay $k$-$D$, respectively.

Because all the messages $W_{k}$, $k=1,2\cdots,N$, are reliably transmitted in
$N+1$ timeslots, the average rate of DF-FC-SR is presented by

$\bar{R}^{*}=\frac{1}{N+1}\sum\limits_{k=1}^{N}R^{*}_{k}.$ (15)

Substituting Eq. (15) into Eq. (10), we have

$\begin{split}\bar{R}^{*}&=\frac{1}{N+1}\left(\sum\limits_{k=1}^{N}C_{S,k}-\sqrt{\frac{1}{n}}\sum\limits_{k=1}^{N}\sqrt{V_{S,k}}Q^{-1}(\varepsilon_{k}[k])+N\cdot\frac{\log
n}{2n}\right).\end{split}$ (16)

From Eq. (13), all of the error probabilities about $W_{k}$ are expressed as

$\displaystyle\begin{split}\varepsilon_{m}[k]=&Q\left(\sqrt{\frac{n}{V_{S,m}}}\left(C_{S,m}-C_{S,k}\right)+\sqrt{\frac{V_{S,k}}{V_{S,m}}}Q^{-1}\left(\varepsilon_{k}[k]\right)\right),\end{split}$
(17) $\displaystyle\varepsilon_{D}[k]=$ $\displaystyle
Q\left(\sqrt{\frac{n}{V_{k,D}}}(C_{k,D}-C_{S,k})+\sqrt{\frac{V_{S,k}}{V_{k,D}}}Q^{-1}(\varepsilon_{k}[k])\right).$
(18)

Next, the reliability of CAO-SIR-FBC is given by the following theorem.

###### Theorem 1

In CAO-SIR-FBC, the reliability ,denoted by $\zeta$ , is given by

$\zeta=\prod_{k=1}^{N}(1-\varepsilon_{D}[k])\prod_{k=1}^{N}\prod_{m=k}^{N}(1-\varepsilon_{m}[k]),$
(19)

where $m$ is the relay’s order.

###### Proof 1

See Appendix A.

In communication system, the error probability of the decoding process at node
$D$ is a constraint of the system. We suppose that the supremum of the error
probability of $D$ is denoted as $\varepsilon_{d}$. Hence, based on the
definition of $\varepsilon_{d}$, we have

$\zeta\geq 1-\varepsilon_{d}.$ (20)

Because only the second item on the right-hand side in Eq. (16) is denoted by
$\varepsilon$, we formulate the optimization problem presented by

$\begin{split}\min\quad&\sum\limits_{k=1}^{N}\sqrt{V_{S,k}}Q^{-1}(\varepsilon_{k}[k])\\\
s.t.\quad&\begin{cases}\prod\limits_{k=1}^{N}(1-\varepsilon_{D}[k])\prod\limits_{k=1}^{N}\prod\limits_{m=k}^{N}(1-\varepsilon_{m}[k])\geq
1-\varepsilon_{d}\\\ \begin{aligned}
\varepsilon_{m}[k]&=Q\left(\sqrt{\frac{n}{V_{S,m}}}(C_{S,m}-C_{S,k})+\sqrt{\frac{V_{S,k}}{V_{S,m}}}Q^{-1}(\varepsilon_{k}[k])\right)\\\
\varepsilon_{D}[k]&=Q\left(\sqrt{\frac{n}{V_{k,D}}}(C_{k,D}-C_{S,k})+\sqrt{\frac{V_{S,k}}{V_{k,D}}}Q^{-1}(\varepsilon_{k}[k])\right).\end{aligned}\end{cases}\end{split}$
(21)

###  III-B algorithms to compute the maximal transmission rate

Because the optimization problem (21) can not be solved directly, we need to
transform the optimization problem into an equivalent simpler problem. We re-
ordered the $\varepsilon$ according to their values’ magnitude. The re-ordered
error probabilities are given by

$\varepsilon_{1}\leq\varepsilon_{2}\leq\cdots\leq\varepsilon_{M}.$ (22)

Here, notation $M$, which is the number of $\varepsilon$, is given by

$\begin{split}M=\frac{N(N+3)}{2}.\end{split}$ (23)

With the re-ordered $\varepsilon$, Eq. (19) is expressed as

$\zeta=\prod_{i=1}^{M}(1-\varepsilon_{i}).$ (24)

Based on the research on linear approximation conditions for nonlinear models
[1980Relative], the constraint is a linear approximation at the stable
operating point of CAO-SIR-FBC. In other words, the constraint is simplified
to be

$1-\sum_{i=1}^{M}\varepsilon_{i}\geq 1-\varepsilon_{d}^{\prime}\ ,$ (25)

where $\varepsilon_{d}^{\prime}$ is a parameter in the neighborhood of
$\varepsilon_{d}$. As a result, the optimization problem (21) is expressed as

$\begin{split}\min\quad&\sum\limits_{k=1}^{N}\sqrt{V_{S,k}}Q^{-1}(\varepsilon_{k}[k])\\\
s.t.\quad&\begin{cases}\sum\limits_{k=1}^{N}\varepsilon_{D}[k]+\sum\limits_{k=1}^{N}\sum\limits_{m=k}^{N}\varepsilon_{m}[k]\leq\varepsilon_{d}^{\prime}\\\
\begin{aligned}
\varepsilon_{m}[k]&=Q\left(\sqrt{\frac{n}{V_{S,m}}}(C_{S,m}-C_{S,k})+\sqrt{\frac{V_{S,k}}{V_{S,m}}}Q^{-1}(\varepsilon_{k}[k])\right)\\\
\varepsilon_{D}[k]&=Q\left(\sqrt{\frac{n}{V_{k,D}}}(C_{k,D}-C_{S,k})+\sqrt{\frac{V_{S,k}}{V_{k,D}}}Q^{-1}(\varepsilon_{k}[k])\right).\end{aligned}\end{cases}\end{split}$
(26)

The range of $\varepsilon_{d}^{\prime}$ is given by the following lemma.

###### Lemma 1

The range of $\varepsilon_{d}^{\prime}$ is given by

$\begin{split}\varepsilon_{\min}\leq&\varepsilon_{d}^{\prime}\leq\varepsilon_{\max},\end{split}$
(27)

where $\varepsilon_{\min}=\varepsilon_{d}$ and
$\varepsilon_{\max}=\frac{M-\sqrt{M^{2}-2M(M-1)\varepsilon_{d}}}{M-1}$.

###### Proof 2

See Appendix B.

Hear notations $\varepsilon_{\min}$ and $\varepsilon_{\max}$ denote the
infimum and supremum of $\varepsilon_{d}^{\prime}$, respectively.

Next, we define a kind of parameter $x$ that satisfies

$x:=Q^{-1}(\varepsilon).$ (28)

By substituting Eq. (28) into Eqs. (18) and (17), we have a simple linear
relation of parameter $x$ . Consequently, after substituting parameter $x$ for
$\varepsilon$, the optimization problem (26) is simplified as

$\begin{split}\min\quad&\sum\limits_{k=1}^{N}\sqrt{V_{S,k}}\quad x_{k}[k]\\\
s.t.\quad&\begin{cases}\sum\limits_{k=1}^{N}\left(\sum\limits_{m=k}^{N}Q(x_{m}[k])+Q(x_{D}[k])\right)\leq\varepsilon_{d}^{\prime}\\\
x_{m}[k]=\sqrt{\frac{n}{V_{S,m}}}(C_{S,m}-C_{S,k})+\sqrt{\frac{V_{S,m}}{V_{S,k}}}x_{k}[k]\\\
x_{D}[k]=\sqrt{\frac{n}{V_{k,D}}}(C_{k,D}-C_{S,k})+\sqrt{\frac{V_{S,k}}{V_{k,D}}}x_{k}[k]\\\
x_{k}[k]>0,k=1,2,\cdots,N,\end{cases}\end{split}$ (29)

where the constraint of $x_{k}[k]$ are attributed to
$\varepsilon_{k}[k]<\varepsilon_{d}^{\prime}<0.5,k=1,2,\cdots,N$.

A careful reader may notice that the optimization problem (29) actually is a
convex optimization. However, the gradient of $Q(x)$ at the operating point
satisfies that

$\nabla Q(x)\ll 1.$ (30)

Hence the convergence is very slow and it is complicated to obtain the
solution through the traditional interior-point method [2004Convex] or CVX
toolkit [2008CVX]. A further simplification is necessary.

Based on the optimization problem (29), we propose the following lemma and
substitute the inequality constraints with an equality constraint.

###### Lemma 2

The solution to the optimization problem (29) satisfies that

$\sum\limits_{k=1}^{N}\left(\sum\limits_{m=k}^{N}Q(x_{m}[k])+Q(x_{D}[k])\right)=\varepsilon_{d}^{\prime}\
.$ (31)

###### Proof 3

See Appendix C.

We then transform the optimization problem (29) into an abstract optimization
problem to analyze its properties. The coefficient matrices and coefficient
vector of the optimization problem (29) are denoted by
$(a_{ij})_{N\times(N+1)},(b_{ij})_{N\times(N+1)}$ and $(c_{i})_{N\times 1}$,
respectively, in which the matrix element is denoted by

$\begin{split}&a_{ij}=\sqrt{\frac{V_{S,i}}{V_{S,j}}},\\\
&b_{ij}=\sqrt{\frac{n}{V_{S,j}}}(C_{S,j}-C_{S,i}),\qquad 1\leq j\leq N,\\\
&a_{i(N+1)}=\sqrt{\frac{V_{S,i}}{V_{i,D}}}\ ,\\\
&b_{i(N+1)}=\sqrt{\frac{n}{V_{i,D}}}(C_{i,D}-C_{S,i}),\\\
&c_{i}=\sqrt{V_{S,i}}.\end{split}$

Here, we regard the destination node $D$ as an $N+1$ relay because of the
similarity between its rate equation and that of other relays. Furthermore, an
optimization variable $\bm{x}=(x_{1},x_{2},\cdots,x_{N})$ is adapted, in which
the element $x_{k}$ denotes $x_{k}[k]$ for $k=1,2,\cdots,N$.

Substituting coefficient matrices and coefficient vector into the optimization
problem (29), we obtain the optimization problem expressed as

$\begin{split}\min\quad&f(\bm{x})=\sum\limits_{i=1}^{N}c_{i}x_{i}\\\
s.t.\quad&\begin{cases}&h(\bm{x})=\sum\limits_{i=1}^{N}\sum\limits_{j=i}^{N+1}Q(a_{i,j}x_{i}+b_{i,j})-\varepsilon_{d}^{\prime}=0\\\
&x_{i}\geq 0,i=1,2,\cdots,N.\end{cases}\end{split}$ (32)

It is worth noting that the optimization problem (32) is not a convex
optimization, because the equality constraint function $h(\bm{x})$ is not
affine. As a result, we propose Lemma 3 to exchange the equality constraint
function and objective functions.

###### Lemma 3

Let the solution of the optimization problem (32) be denoted as $\bm{x_{0}}$.
The value of the objective function is presented by parameter $v_{0}$. Then,
the optimization problem with the same solution $\bm{x_{0}}$ of the
optimization problem (32) is expressed as

$\begin{split}\min\quad&f(\bm{x})=\sum\limits_{i=1}^{N}\sum\limits_{j=i}^{N+1}Q(a_{i,j}x_{i}+b_{i,j})\\\
s.t.\quad&\begin{cases}&h(\bm{x})=\sum\limits_{i=1}^{N}c_{i}x_{i}-v_{0}=0\\\
&x_{i}\geq 0,i=1,2,\cdots,N.\end{cases}\end{split}$ (33)

###### Proof 4

See Appendix D.

In the optimization problem (33), the equality constraint function is affine,
and the objective function is convex because

$\nabla^{2}f(\bm{x})\geq 0,$

for all $x_{i}>0$. Consequently, problem (33) is a convex optimization
problem. The minimal $f(\bm{x})$ is obtained with a specific $v_{0}$.
Additionally, the range of $v_{0}$ can be obtained by the following lemma.

###### Lemma 4

The objective function value in the solution of the optimization problem (32)
satisfies that

$\displaystyle v_{1}\leq v_{0}\leq v_{2},$ (34)

where
$v_{1}=Q^{-1}\left(\varepsilon_{d}^{\prime}\right)\sum\limits_{i=1}^{N}c_{i}$
and $v_{2}=x_{1}\sum\limits_{i=1}^{N}c_{i}$. Here the parameter $x_{1}$ is the
solution to the equation expressed as

$\sum\limits_{i=1}^{N}\sum\limits_{j=i}^{N+1}Q(a_{i,j}x+b_{i,j})-\varepsilon_{d}^{\prime}=0.$
(35)

###### Proof 5

The vector $\bm{x_{1}}$ is equal to $x_{1}\cdot(1)_{1\times N}^{T}$ satisfies
the constraint of problem (32), so we have

$v_{0}\leq v_{2}.$ (36)

Furthermore, because every error probability satisfies
$\varepsilon_{m}[k]\leq\varepsilon_{d}^{\prime}$, the solution $v_{0}$
satisfies that

$v_{0}\geq v_{1}.$ (37)

Accordingly, the solution to the optimization problem (32) is obtained by
Bisection algorithm. More specifically, the midpoint of the interval
$v_{3}=\frac{v_{1}+v_{2}}{2}$ is substituted into the optimization problem
(33). The solution $\bm{x_{3}}$ of the optimization problem (33) with $v_{3}$
is calculated. Then the objective function $f(\bm{x_{3}})$ is compared with
$\varepsilon_{d}$. If $f(\bm{x_{3}})-\varepsilon_{d}$ is sufficiently small or
$\frac{v_{2}-v_{1}}{2}$ is smaller than tolerance, the convergence is
satisfactory. On the other hand, $v_{1}$ is replaced with $v_{3}$ if
$f(\bm{x_{3}})<\varepsilon_{d}$. When $f(\bm{x_{3}})>\varepsilon_{d}$, $v_{3}$
replaces $v_{2}$ so that there is a solution crossing within the new interval.
A detailed process is shown in Algorithm 1.

The complexity of Algorithm 1 is
$O\left(\log_{2}\left({Q^{-1}\left(\varepsilon_{d}\right)}\right)\right)$. In
contrast, the complexity of exhaustive method is
$O\left({Q^{-1}\left(\varepsilon_{d}\right)}\right)$. The proposed method
reduces the computational complexity significantly.

Input: Relay’s number $N$, error bound $\varepsilon$, maximum iterations
$k_{\max}$, tolerance $TOL$, objective error probability
$\varepsilon_{d}^{\prime}$, channel gain $g_{a,b}$, where
$a\in\\{S,1,\cdots,N\\}$ and $b\in\\{1,\cdots,N,D\\}$

Output: maximal average rate $\bar{R}^{*}$

initialize pre-existing parameter $P,n$, interval $[v_{1},v_{2}]$, iteration
number $k$, etc

Compute Shannon capacity $C_{a,b}$, channel dispersion $V_{a,b}$, parameter
matrices $(a_{ij})_{N\times(N+1)},(b_{ij})_{N\times(N+1)}$, coefficient vector
$(c_{i})_{N\times 1}$

while  _$k <k_{\max}$_ do

Compute solution $\bm{x_{0}}$ of optimization problem (33) through the
interior-point method

Compute objective functions $f\left(\bm{x_{0}}\right)$

if  _$\left\lvert f(\bm{x_{0}})-\varepsilon_{d}^{\prime}\right\rvert
<\varepsilon$ $\mathbf{or}$ $\frac{v_{2}-v_{1}}{2}<TOL$_ then

Compute and return $\bar{R}^{*}$

end if

if  _$f(\bm{x_{0}}) <\varepsilon_{d}^{\prime}$_ then

$v_{1}\leftarrow v_{3}$

else

$v_{2}\leftarrow v_{3}$

end if

end while

Return message “ $\bar{R}^{*}$ is not found”

Algorithm 1 Compute maximal average rate $\bar{R}^{*}$ with specific
$\varepsilon_{d}^{\prime}$ Figure 2: Computing process on $\bar{R}^{*}$ for
$i.i.d.$ cases with relay number $N=2$, blocklength $n=300$, transmission
power $P=1$, and error probabilities $\varepsilon_{d}=10^{-3}$. Channel gain
obeys index distribution and $\bar{g}_{a,b}=1$.

Based on Algorithm 1, we propose a method of computing the approximated
maximal rate of CAO-SIR-FBC denoted by $\hat{R}$. As shown in Fig. 2, the
difference of probabilities between the achievability bound and converse bound
is slight. Therefore, the rate $\hat{R}$ is given by Algorithm 1 with the
approximated error probability $\hat{\varepsilon}_{d}$ denoted by

$\hat{\varepsilon}_{d}=\frac{1}{2}\left(\varepsilon_{d}+\frac{M-\sqrt{M^{2}-2M(M-1)\varepsilon_{d}}}{M-1}\right).$
(38)

In the general case ($N\leq 10$), the percent error of $\hat{\varepsilon}_{d}$
is less than $0.024\%$. Consequently, a simple approximated maximal average
rate $\hat{R}$ is expressed as

$\hat{R}=f(\hat{\varepsilon}_{d}).$ (39)

In most cases, the approximated rate is reliable. Furthermore, if the percent
error of rate $\hat{R}$ is insufferable, we examine every
$\varepsilon_{d}^{\prime}$ in the range
[$\varepsilon_{\min},\varepsilon_{\max}$] with the step $\Delta\varepsilon$.
Then, we compute and cache every $\bm{x_{0}}$ of $\varepsilon_{d}^{\prime}$.
With the $\bm{x_{0}}$, the $\zeta$ is obtained by Eq. (19). The more accurate
approximated solution is obtained when the difference is less than the error
bound. The specific process is shown in Algorithm 2.

Input: error probability $\varepsilon_{d}$, iteration number $k$, significance
level $\alpha$, Relay’s number $N$, channel gain $g_{a,b}$, where
$a\in\\{S,1,\cdots,N\\}$ and $b\in\\{1,\cdots,N,D\\}$

Output: approximated maximal rate $\hat{R}$

Initialize pre-existing parameter of Algorithm 1, error bound $\varepsilon$,
etc

Compute $\varepsilon_{\min}$, $\varepsilon_{\max}$, approximated error
probability $\hat{\varepsilon_{d}}$, achievability bound $R_{a}$, converse
bound $R_{c}$, approximated value $\hat{R}$ and percent error of $\hat{R}$
expressed as  $\delta=\frac{\max\\{R_{c}-\hat{R},\hat{R}-R_{a}\\}}{\hat{R}}$
(40) if  _$\delta <\alpha$_ then

Return $\hat{R}$

else

Compute $\Delta\varepsilon=(\varepsilon_{\max}-\varepsilon_{\min})/{k}$

for
_$\varepsilon_{d}^{\prime}=\varepsilon_{\min}:\Delta\varepsilon:\varepsilon_{\max}$_
do

Compute solution $\\{R_{0},\bm{x_{0}}\\}$ by Algorithm 1

Compute $\zeta$ of $\bm{x_{0}}$

if  _$\left\lvert\varepsilon_{d}-\left(1-\zeta\right)\right\rvert
<\varepsilon$ _ then

Return $\hat{R}=R_{0}$

end if

end for

end if

Algorithm 2 Compute approximated maximal $\hat{R}$

Now, we obtain the specific method of computing the reliable $\hat{R}$. The
analysis of the protocol’s performance is presented in Section IV.

##  IV Performance Analysis

In this section, the performance of the CAO-SIR-FBC protocol is analyzed. We
are interested in the average throughput in fading channel. In this section,
the channel spanning between node $a$ and node $b$ is assumed to be an
independent distributed Rayleigh fading channel. The probability density
function $(p.d.f)$ of the channel gain $g_{a,b}$ is obtained by
$f_{g_{a,b}}(x)=\frac{1}{\bar{g}_{a,b}}\ \exp(-\frac{x}{\bar{g}_{a,b}})$. Here
$\bar{g}_{a,b}=\mathbb{E}g_{a,b}=\mathbb{E}|h_{a,b}|^{2}$ denote the average
channel gain of the link from node $a$ to $b$.

Because it is not a trivial to derive the close-form expression of the average
throughput of CAO-SIR-FBC, we focus on the approximation of the average
throughout in the high blocklength, low and high SNR. Consider the high
blocklength first. Because the rate $\bar{R}^{*}$ increases and trends to its
capacity when $n$ increases, we have

$\lim_{n\to\infty}\bar{R}^{*}(\varepsilon,n)=\bar{C}.$ (41)

Here, the capacity is presented by [2014CAO]

$\displaystyle\bar{C}=\frac{1}{N+1}\sum\limits_{i=1}^{N}C_{i},$ (42)

where $C_{i}=\min\left\\{C_{S,i},C_{i,D}\right\\}$. In other word, when the
blocklength of coding $n$ trends to infinity, the CAO-SIR-FBC protocol is
equivalent to the basic CAO-SIR. As a result, the average throughput is given
by

$\begin{split}\lim_{n\to\infty}\mathbb{E}\\{\bar{R}^{*}(\varepsilon,n)\\}=&\mathbb{E}\\{\bar{C}\\}\\\
=&\frac{2N}{N+1}\int\limits_{0}^{\infty}\log(1+x\gamma)\exp(-2x)\,\mathrm{d}x.\end{split}$
(43)

Next, we focus on the CAO-SIR-FBC protocol’s performance in the SNR domain. In
the low-SNR regime, due to the channel dispersion, the rate of CAO-SIR-FBC is
0, as shown in Fig. 3. In other words, it is impossible to transmit the
message with the desired error probability $\varepsilon_{d}$.

Figure 3: Maximal average rate $\bar{R}^{*}$ for $i.i.d.$ cases with relay
number $N=2$, blocklength $n=300$, and error probabilities
$\varepsilon_{d}=10^{-3}$. Channel gain obeys index distribution and
$\bar{g}_{a,b}=1$.

In the high SNR regime, it is worth noting that the notation $V$ trends to a
constant presented by

$\lim\limits_{\rho\to\infty}V\left(\rho\right)=\frac{(\log\mathrm{e})^{2}}{2}.$
(44)

Accordingly, problem (32) in the high SNR regime is obtained by

$\begin{split}\min\quad&f(\bm{x})=\sqrt{\frac{(\log\mathrm{e})^{2}}{2}}\sum\limits_{i=1}^{N}x_{i}\\\
s.t.\quad&\begin{cases}&h(\bm{x})=\sum\limits_{i=1}^{N}\sum\limits_{j=i}^{N+1}Q\left(x_{i}+\frac{\sqrt{2n}}{\log\mathrm{e}}\log\left(\frac{g_{S,j}}{g_{S,i}}\right)\right)-\varepsilon_{d}^{\prime}=0\\\
&x_{i}\geq 0.\end{cases}\end{split}$ (45)

It is not difficult to verify that this optimization problem is independent of
arbitrary SNR. The solution to Eq. (45) is denoted by a constant vector
$\bm{x_{0}}=\big{[}x_{0}\left[1\right],x_{0}[2],\cdots,x_{0}\left[N\right]\big{]}$
and we have

$f(\bm{x_{0}})=\sqrt{\frac{(\log\mathrm{e})^{2}}{2}}\sum\limits_{i=1}^{N}x_{0}[i].$
(46)

As a result, the average throughput of CAO-SIR-FBC in high SNR regime is given
by

$\begin{split}\mathbb{E}\\{\bar{R}^{*}\\}=&\mathbb{E}\\{\bar{C}\\}-\frac{1}{N+1}\left(\sqrt{\frac{(\log\mathrm{e})^{2}}{2n}}\sum\limits_{i=1}^{N}x_{0}[i]-N\frac{\log
n}{2n}\right)\\\
=&\frac{2N}{N+1}\int\limits_{0}^{\infty}\log(1+x\gamma)\exp(-2x)\,\mathrm{d}x-\frac{1}{N+1}\left(\sqrt{\frac{(\log\mathrm{e})^{2}}{2n}}\sum\limits_{i=1}^{N}x_{0}[i]-N\frac{\log
n}{2n}\right).\end{split}$ (47)

As shown in Eq. (47), the difference between $R^{*}$ and $C$ trends to a
constant in the high SNR regime. In addition, the slope of CAO-SIR-FBC is
equal to its capacity in the high SNR regime. Channel dispersion does not
influence the slope, also referred to as multiplexing gain, of the CAO-SIR-FBC
protocol.

##  V Numerical Results

Numerical results are presented in this section to verify the above analysis
and the advantage of CAO-SIR-FBC. Assume that there are three potential
relays. Both independent and identically distributed ($i.i.d.$) and
independent non-identical distributed ($i.ni.d.$) Rayleigh channels
[2006Fundamentals, 2013Outage] are considered. For the $i.i.d.$ case, we
suppose the channel gains $\frac{1}{\bar{g}_{a,b}}=1$ for every node pair
$(a,b)$. For the $i.ni.d.$ case, we assume
$\frac{1}{\bar{g}_{a,b}}=d_{a,b}^{-2}$, where $d_{a,b}$ denotes the distance
between node pair $(a,b)$, because of the pass-loss model. To determine
$d_{a,b}$, we assume the following network topology. Node $S$ and $D$ are
located in coordinates (0, 0) and (0, 1), respectively. Three relay’s
coordinates are given by ($\frac{\sqrt{3}}{2},\frac{1}{2}$),
($-\frac{\sqrt{3}}{2},\frac{1}{2}$), and ($0,\frac{1}{2}$), respectively. To
provide more insight, the CAO-SIR-FBC protocols are compared with the
conventional two-timeslot DF relay protocols (with and without the direct
$S-D$ link) with fixed target rate [laneman2003distributed,
laneman2004cooperative, bletsas2006simple], and two-timeslot DF relaying
without the direct $S-D$ link [bletsas2006simple]. The average throughput of
traditional protocols also considers the influence of finite-blocklength
coding. The fixed target rate is $r=1$. Different from Eq. (11), the
transmission rate is computed in the complex dimension to analyze the slope
and multiplexing gains easily. For other system parameters, blocklength
$n=300$ and the error probability at the destination node is
$\varepsilon_{d}=10^{-3}$.

Figure 4:  Average throughput in $i.i.d.$ case. Proposed CAO-SIR-FBC protocols
are compared with conventional two-timeslot relaying.  Figure 5:  Average
throughput in $i.ni.d.$ case. Proposed CAO-SIR-FBC protocols are compared with
conventional two-timeslot relaying.

The average throughput versus transmitter-side SNR curves for $i.i.d.$ and
$i.ni.d.$ are presented in Figs. 4 and 5, respectively. In the high SNR
regime, we focus on the slope of the throughput, which also means the
multiplexing gains. The multiplexing gain of CAO-SIR-FBC in both the $i.i.d.$
and $i.ni.d.$ cases is $\frac{3}{4}$. The multiplexing gain of CAO-SIR-FBC is
equal to that of the basic CAO-SIR [2014CAO], which verifies the theory that
the channel dispersion does not influence the multiplexing gain. The
multiplexing gain of two-timeslot DF relaying without the direct $S-D$ link is
$\frac{1}{2}$ in both the $i.i.d.$ and $i.ni.d.$ cases, much lower than that
of CAO-SIR-FBC because it spends two timeslots to transmit one message. Then
We focus on the required SNR when the average throughput is $4$ bit/s/Hz. In
the $i.i.d.$ case, CAO-SIR-FBC has SNR gains of $4.0$ dB over two-timeslot DF
relaying without the direct $S-D$ link. For the $i.ni.d.$ scenario, the SNR
gains of CAO-SIR-FBC over two-timeslot DF relaying without the direct $S-D$
link are $1.2$ dB. This shows that in the high SNR regime the employed relays
of CAO-SIR-FBC are beneficial to obtaining a high average throughput.

Next, we focus on comparing CAO-SIR-FBC and conventional relaying protocols
with fixed target rates. It should be noted that it might be unfair to set the
fixed target rate of conventional relaying protocols, because the average
throughput for fixed-rate relaying is constrained in the high SNR regime.
Specifically, we focus on a transmitter-side SNR of 20 dB. With the fixed
target rate $r=1$, the average throughput of two-timeslot DF relaying with and
without direct $S-D$ links is $1$ and $\frac{1}{2}$ bit/s/Hz, respectively,
regardless if in the $i.i.d.$ case or $i.ni.d.$ case. Although the upper bound
in the high SNR regime is increased with the fixed rate, it is worth noting
that a higher target rate also causes a lower average throughput in the low-
SNR regime. In addition, even without the fixed target rate, two-timeslot DF
relaying without the $S-D$ link is still only $2.9$ bit/s/Hz in the $i.i.d.$
case and $3.7$ bit/s/Hz in the $i.ni.d.$ case. In contrast, because CAO-SIR-
FBC is capable of adapting its transmission rate to the channel gains, always
achieves the optimal average throughput in the high SNR regime. In the
$i.i.d.$ case, CAO-SIR-FBC achieves throughput gains of $117\%$, $344\%$, and
$688\%$ over two-timeslot DF relaying as well as fixed-rate two-timeslot DF
relaying with and without direct $S-D$ links, respectively. In the $i.ni.d.$
case, the throughput gains of CAO-SIR-FBC over two-timeslot DF relaying as
well as fixed-rate two-timeslot DF relaying with and without direct $S-D$
links are $105\%$, $392\%$, and $784\%$, respectively. Hence, we can conclude
that the joint channel-aware rate-adaptation mechanism greatly improves the
efficiency of successive relaying.

##  VI Conclusions

This paper presents an optimized relaying protocol referred to as CAO-SIR-FBC
based on the CAO-SIR scheme. Faced with a short-packet transmission in which
the Shannon formula is not achievable, we analyze the optimization problems of
a successive relaying protocol with FBC in detail. We then solve the
optimization problem and clarify the adjustments of the successive relaying
protocol with FBC. To analyze the performance of CAO-SIR-FBC, we carried out a
series of simulation experiments and analyses. Experimental results show that
the CAO-SIR-FBC protocol has high reliability and excellent transmission
performance in short-packet transmission, and its performance is better than
those of conventional protocols with FBC. In the future, we hope to continue
studying the relay ranking, relay selection, power allocation, and other
mechanisms of the CAO-SIR-FBC protocol for improvement .

##  Appendix A PROOF OF THEOREM 1

The proof is based on the following observation. In CAO-SIR-FBC, the relay’s
protocol is successive decoding. More specifically, $W_{1}$ is decoded by each
relay in the first timeslot. In the second timeslot, $W_{2}$ is decoded
successfully only if $W_{1}$ is decoded successfully. Likewise, message
$W_{k}$ is decoded successfully only if messages $W_{1},W_{2},\cdots W_{k-1}$
are decoded successfully. As a result, error probabilities are increased in
each timeslot, also referred as error propagation.

Let $\hat{W}_{k}$ denote the message decoded by relay $k$ in the $k$th
timeslot. The message $W_{k}$ is decoded successfully by relay $k$ only if
$\hat{W}_{k}=W_{k}$. Consequently, for $2\leq k\leq N$, the probability of
decoding successful, denoted by
$\mathrm{Pr}\left\\{\hat{W}_{k}=W_{k}\right\\}$, is presented by

$\begin{split}\mathrm{Pr}\left\\{\hat{W}_{k}=W_{k}\right\\}=&\mathrm{Pr}\left\\{\hat{W}_{k}=W_{k}|\hat{W}_{k-1}=W_{k-1}\right\\}\times\mathrm{Pr}\left\\{\hat{W}_{k-1}=W_{k-1}\right\\}\\\
=&\mathrm{Pr}\left\\{\hat{W_{k}}=W_{k}|\hat{W}_{k-1}=W_{k-1}\right\\}\times\mathrm{Pr}\left\\{\hat{W}_{k-1}=W_{k-1}|\hat{W}_{k-2}=W_{k-2}\right\\}\\\
&\times\mathrm{Pr}\left\\{\hat{W}_{k-2}=W_{k-2}\right\\}\\\
=&\mathrm{Pr}\left\\{\hat{W}_{1}=W_{1}\right\\}\prod\limits_{i=1}^{k-1}\mathrm{Pr}\left\\{\hat{W}_{i+1}=W_{i+1}|\hat{W_{i}}=W_{i}\right\\}.\end{split}$
(48)

Additionally, based on the definition of $\varepsilon$, when $k=1$, we have

$\mathrm{Pr}\left\\{\hat{W}_{1}=W_{1}\right\\}\geq 1-\varepsilon_{1}[1].$ (49)

When $2\leq k\leq N$, because the event
$\left\\{\hat{W}_{k}=W_{k}|\hat{W}_{k-1}=W_{k-1}\right\\}$ means that the
relay $k$ decodes the message $W_{1},W_{2},\cdots W_{k}$ successfully, it is
obtained that

$\mathrm{Pr}\left\\{\hat{W}_{k}=W_{k}|\hat{W}_{k-1}=W_{k-1}\right\\}\geq\prod_{i=1}^{k}(1-\varepsilon_{k}[i]).$
(50)

Based on the multiplication theorem, the probability of relay $N$ decoding
message $W_{N}$ successfully satisfies that

$\mathrm{Pr}\left\\{\hat{W}_{N}=W_{N}\right\\}\geq\prod_{k=1}^{N}\prod_{i=1}^{k}(1-\varepsilon_{k}[i]).$
(51)

Rewritten according to relay’s number $m$, Eq. (51) is expressed as

$\mathrm{Pr}\left\\{\hat{W}_{N}=W_{N}\right\\}\geq\prod_{k=1}^{N}\prod_{m=k}^{N}(1-\varepsilon_{m}[k]).$
(52)

For node $D$, its protocol is reverse successive decoding. Specifically,
$W_{k}$ is decoded successfully at node $D$ only if
$W_{k+1},W_{k+2},\cdots,W_{N}$ are also decoded successfully. Let
$\hat{W}_{k}^{D}$ denote the estimate of $W_{k}$ at the destination node $D$.
The message $W_{k}$ is decoded successfully by $D$ only if
$\hat{W}_{k}^{D}=W_{k}$. For $1\leq k\leq N-1$, the probability of decoding
successful at node $D$, denoted by
$\mathrm{Pr}\left\\{\hat{W}_{k}^{D}=W_{k}\right\\}$, is expressed as

$\begin{split}\mathrm{Pr}\left\\{\hat{W}_{k}^{D}=W_{k}\right\\}=&\mathrm{Pr}\left\\{\hat{W}_{k}^{D}=W_{k}|\hat{W}_{k+1}^{D}=W_{k+1}\right\\}\times\mathrm{Pr}\left\\{\hat{W}_{k+1}^{D}=W_{k+1}\right\\}\\\
=&\mathrm{Pr}\left\\{\hat{W}_{k}^{D}=W_{k}|\hat{W}_{k+1}^{D}=W_{k+1}\right\\}\times\mathrm{Pr}\left\\{\hat{W}_{k+1}^{D}=W_{k+1}|\hat{W}_{k+2}^{D}=W_{k+2}\right\\}\\\
&\times\mathrm{Pr}\left\\{\hat{W}_{k+2}^{D}=W_{k+2}\right\\}\\\
=&\mathrm{Pr}\left\\{\hat{W}_{N}^{D}=W_{N}\right\\}\prod\limits_{i=k}^{N-1}\mathrm{Pr}\left\\{\hat{W}_{i}^{D}=W_{i}|\hat{W}_{i+1}^{D}=W_{i+1}\right\\}.\end{split}$
(53)

When $k=N$, message $W_{N}$ is decoded correctly at $D$ only if relay $N$
decodes the message successfully, so
$\mathrm{Pr}\left\\{\hat{W}_{N}^{D}=W_{N}\right\\}$ is given by

$\begin{split}\mathrm{Pr}\left\\{\hat{W}_{N}^{D}=W_{N}\right\\}=\mathrm{Pr}\left\\{\hat{W}_{N}^{D}=W_{N}|\hat{W}_{N}=W_{N}\right\\}\mathrm{Pr}\left\\{\hat{W}_{N}=W_{N}\right\\}.\end{split}$
(54)

Hence, $\mathrm{Pr}\left\\{\hat{W}_{N}^{D}=W_{N}\right\\}$ satisfies that

$\begin{split}\mathrm{Pr}\left\\{\hat{W}_{N}^{D}=W_{N}\right\\}\geq\mathrm{Pr}\left\\{\hat{W}_{N}=W_{N}\right\\}(1-\varepsilon_{D}[N]).\end{split}$
(55)

Additionally, when $1\leq k\leq N-1$, we have

$\mathrm{Pr}\left\\{\hat{W}_{k}^{D}=W_{k}|\hat{W}_{k+1}^{D}=W_{k+1}\right\\}\geq
1-\varepsilon_{D}[k].$ (56)

As a result, the minimal probability of
$\left\\{\hat{W}^{D}_{1}=W_{1}\right\\}$ satisfies that

$\mathrm{Pr}\left\\{\hat{W}^{D}_{1}=W_{1}\right\\}\geq\prod_{k=1}^{N}(1-\varepsilon_{D}[k])\prod_{k=1}^{N}\prod_{m=k}^{N}(1-\varepsilon_{m}[k]).$
(57)

We use the notation $\zeta$ to denote the right-hand side of Eq. (57), which
is also the lower limit of
$\mathrm{Pr}\left\\{\hat{W}^{D}_{1}=W_{1}\right\\}$, so we have

$\zeta=\prod_{k=1}^{N}(1-\varepsilon_{D}[k])\prod_{k=1}^{N}\prod_{m=k}^{N}(1-\varepsilon_{m}[k]).$
(58)

It is worth noting that $\mathrm{Pr}\left\\{\hat{W}^{D}_{1}=W_{1}\right\\}$ is
also the probabilities of $D$ decoding all messages without errors, which also
referred as the reliability of the communication system. Therefore, the
theorem 1 is proved.

##  Appendix B PROOF OF LEMMA 1

The proof is based on the following observation. The expansion of Eq. (19)
with the first and second remainder terms are expressed as

$\begin{split}\zeta=&1-\sum_{i=1}^{M}\varepsilon_{i}+o\left(\sum_{i=1}^{M}\varepsilon_{i}\right)\\\
=&1-\sum_{i=1}^{M}\varepsilon_{i}+\sum_{i=1}^{M-1}\sum_{j=i+1}^{M}\varepsilon_{i}\varepsilon_{j}-o\left(\sum_{i=1}^{M-1}\sum_{j=i+1}^{M}\varepsilon_{i}\varepsilon_{j}\right),\end{split}$
(59)

where the remainder term $o\left(\sum_{i=1}^{M}\varepsilon_{i}\right)$ means
that it is the higher order infinitesimal of $\sum_{i=1}^{M}\varepsilon_{i}$.
Similarly, remainder term
$o\left(\sum_{i=1}^{M-1}\sum_{j=i+1}^{M}\varepsilon_{i}\varepsilon_{j}\right)$
is the higher order infinitesimal of
$\sum_{i=1}^{M-1}\sum_{j=i+1}^{M}\varepsilon_{i}\varepsilon_{j}$. Because each
remainder term is positive definite, we have the following inequality
presented by

$\displaystyle\zeta$ $\displaystyle>1-\sum_{i=1}^{M}\varepsilon_{i},$ (60)
$\displaystyle\zeta$
$\displaystyle<1-\sum_{i=1}^{M}\varepsilon_{i}+\sum_{i=1}^{M-1}\sum_{j=i+1}^{M}\varepsilon_{i}\varepsilon_{j}.$
(61)

For the second-order term in the right-hand side of Eq. (61), because of the
inequality of arithmetic and geometric means (AM–GM inequality), we have

$\sum_{i=1}^{M-1}\sum_{j=i+1}^{M}\varepsilon_{i}\varepsilon_{j}\leq\frac{M-1}{2}\sum_{i=1}^{M}\varepsilon_{i}^{2}<\frac{M-1}{2}\varepsilon_{d}\sum_{i=1}^{M}\varepsilon_{i}.$
(62)

Here, the first term is equal to the second term if and only if every
$\varepsilon_{i}$ is equal to each other for $i=1,2,\cdots,M$. The second
inequality relation results from $\varepsilon_{i}<\varepsilon_{d}$ for
$i=1,2,\cdots,M$.

Additionally, We substitute Eq. (62) into the expansion of
$(\sum_{i=1}^{M}\varepsilon_{i})^{2}$. The intersection of inequality is
expressed as

$\begin{split}\left(\sum_{i=1}^{M}\varepsilon_{i}\right)^{2}&=\sum_{i=1}^{M}\varepsilon_{i}^{2}+2\sum_{i=1}^{M-1}\sum_{j=i+1}^{M}\varepsilon_{i}\varepsilon_{j}\\\
&\geq\frac{2M}{M-1}\sum_{i=1}^{M-1}\sum_{j=i+1}^{M}\varepsilon_{i}\varepsilon_{j}.\end{split}$
(63)

As a result, we have

$\sum_{i=1}^{M-1}\sum_{j=i+1}^{M}\varepsilon_{i}\varepsilon_{j}\leq\frac{M-1}{2M}\left(\sum_{i=1}^{M}\varepsilon_{i}\right)^{2}.$
(64)

Through substituting the inequality (61) into the right-hand side of
inequality (64) and the third term of inequalities (62), it follows that

$\begin{cases}\zeta<1-\frac{2-(M-1)\varepsilon_{d}}{2}\sum\limits_{i=1}^{M}\varepsilon_{i},\\\
\zeta<1-\sum\limits_{i=1}^{M}\varepsilon_{i}+\frac{M-1}{2M}\left(\sum\limits_{i=1}^{M}\varepsilon_{i}\right)^{2}.\end{cases}$
(65)

By substituting the right-hand side of inequality (60) into the item $\zeta$
in inequality (20), we have

$1-\sum_{i=1}^{M}\varepsilon_{i}\geq 1-\varepsilon_{d}^{\prime}.$ (66)

It is important to note that Eq. (66) is the equivalent condition of Eq. (20).
According to Eq. (60), the right-hand side of Eq.(20) should be greater than
the right-hand side of Eq. (66), so $\varepsilon_{d}^{\prime}$ satisfies that

$\varepsilon_{d}^{\prime}\geq\varepsilon_{d}.$ (67)

Likewise, we obtain two converse bounds of $\bar{R}^{*}$ by substituting
$\zeta$ with into the right-hand side of inequalities (62) and (64). The
inequalities are

$\begin{cases}\varepsilon_{d}\leq\frac{2-(M-1)\varepsilon_{d}}{2}\sum\limits_{i=1}^{M}\varepsilon_{i}\\\
\varepsilon_{d}\leq\sum\limits_{i=1}^{M}\varepsilon_{i}+\frac{M-1}{2M}\left(\sum\limits_{i=1}^{M}\varepsilon_{i}\right)^{2}.\end{cases}$
(68)

Choosing the intersection of inequalities, we obtain a reliable upper bound of
$\varepsilon_{d}^{\prime}$,

$\varepsilon_{d}^{\prime}\leq\frac{M-\sqrt{M^{2}-2M(M-1)\varepsilon_{d}}}{M-1}.$
(69)

Then, Lemma 1 is proved.

##  Appendix C PROOF OF LEMMA 2

We now prove this lemma by contradiction. First, we assume that there is a set
of solutions $\left\\{x_{1}[1],x_{2}[2],\cdots,x_{N}[N]\right\\}$ to the
optimization problem, and the solution satisfies

$\sum\limits_{k=1}^{N}\left(\sum\limits_{m=k}^{N}Q(x_{m}[k])+Q(x_{D}[k])\right)<\varepsilon_{d}^{\prime}.$
(70)

Then, because $Q(x)$ is a continuous function, we have $\Delta x>0$ satisfies
that

$\begin{split}&\sum\limits_{k=1}^{N-1}\left(\sum\limits_{m=k}^{N}Q(x_{m}[k])+Q(x_{D}[k])\right)+Q(x_{N}[N]-\Delta
x)+Q\Big{(}a\left(x_{N}[N]-\Delta
x\right)+b\Big{)}<\varepsilon_{d}^{\prime}.\end{split}$ (71)

Here, the parameter $a=\sqrt{V_{S,N}/V_{N,D}}$ and
$b=(C_{N,D}-C_{S,N})\sqrt{n/V_{N,D}}$. Therefore, we have another set of
solutions $\\{x_{1}[1],x_{2}[2],\cdots,x_{N}[N]-\Delta x\\}$ that satisfies
the constraint. Its equivalent is lesser than the value of the assumed
solution, which contradicts our supposition.

Additionally, we have

$\sum\limits_{k=1}^{N}\left(\sum\limits_{m=k}^{N}Q(0)+Q(0)\right)>\varepsilon_{d}^{\prime},$
(72)

Consequently, $\varepsilon_{d}^{\prime}$ is an achievable point, due to the
intermediate value theorem. The set of solutions must satisfy

$\sum\limits_{k=1}^{N}\left(\sum\limits_{m=k}^{N}Q(x_{m}[k])+Q(x_{D}[k])\right)=\varepsilon_{d}^{\prime},$
(73)

which completes the proof.

##  Appendix D PROOF OF LEMMA 3

The proof is based on the following observation. To easily describe it, we
denote the equality objective functions and constraint function of the
original optimization problem (32) by $f_{0}(\bm{x})$ and $h_{0}(\bm{x})$,
respectively. Similarly, the equality objective functions and constraint
function of the optimization problem (33) are named
$f_{1}(\bm{x}),h_{1}(\bm{x})$. The relationship between
$f_{0}(\bm{x}),h_{0}(\bm{x})$ and $f_{1}(\bm{x}),h_{1}(\bm{x})$ is expressed
as

$\displaystyle f_{1}(\bm{x})=h_{0}(\bm{x})+\varepsilon_{d}^{\prime},$ (74)
$\displaystyle h_{1}(\bm{x})=f_{0}(\bm{x})-v_{0}.$ (75)

Based on problem (32), we obtain the Karush–Kuhn–Tucker (KKT) conditions:

$\begin{cases}\nabla f_{0}(\bm{x_{0}})+\lambda\nabla h_{0}(\bm{x_{0}})=0,\\\
h_{0}(\bm{x_{0}})=0..\\\ \end{cases}$ (76)

Because $\nabla f_{0}(\bm{x})\neq 0,$ for all $\bm{x}$, it follows that
$\lambda\neq 0$. As a result, by substituting $\bm{x_{0}}$ and $1/\lambda$
into the changed optimization problem’s KKT condition, we have

$\displaystyle\begin{split}\nabla f_{1}(\bm{x_{0}})+\frac{1}{\lambda}\nabla
h_{1}(\bm{x_{0}})=&\nabla h_{0}(\bm{x_{0}})+\frac{1}{\lambda}(\nabla
f_{0}(\bm{x_{0}}))\\\ =&\frac{1}{\lambda}(\nabla
f_{0}(\bm{x_{0}})+\lambda\nabla h_{0}(\bm{x_{0}}))\\\ =&0,\end{split}$ (77)
$\displaystyle\begin{split}h_{1}(\bm{x_{0}})=&f_{0}(\bm{x_{0}})-v_{0}\\\
=&0.\end{split}$ (78)

Therefore, $(\bm{x_{0}},\frac{1}{\lambda})$ is a solution to the optimization
problem (32). Furthermore, because the KKT condition of the optimization
problem (32) consists of $N+1$ independent equations, there is only one
solution satisfying the KKT condition. Hence, the solution to the optimization
problem (32) is also $\bm{x_{0}}$, which completes the proof.