# Energy-Efficient Optimization for HARQ Schemes over Time-Correlated Fading
Channels

Zheng Shi  Shaodan Ma  Guanghua Yang  and Mohamed-Slim Alouini Copyright (c)
2015 IEEE. Personal use of this material is permitted. However, permission to
use this material for any other purposes must be obtained from the IEEE by
sending a request to<EMAIL_ADDRESS>work was supported in part
by National Natural Science Foundation of China under grants 61601524 and
61671488, in part by the Special Fund for Science and Technology Development
in Guangdong Province under Grant No. 2016A050503025, in part by the Research
Committee of University of Macau under grants MYRG2014-00146-FST and
MYRG2016-00146-FST, and in part by the Macau Science and Technology
Development Fund under grants 091/2015/A3 and 020/2015/AMJ.Zheng Shi is with
the School of Electrical and Information Engineering and the Institute of
Physical Internet, Jinan University (Zhuhai campus), Zhuhai 519070, China
(e-mail:shizheng0124@gmail.com).Shaodan Ma is with the Department of
Electrical and Computer Engineering, University of Macau, Macao (e-mail:
shaodanma@umac.mo).Guanghua Yang is with Institute of Physical Internet, Jinan
University (Zhuhai Campus), Zhuhai, China (e-mail: ghyang@jnu.edu.cn).Mohamed-
Slim Alouini is with the Computer, Electrical, and Mathematical Science and
Engineering (CEMSE) Division, King Abdullah University of Science and
Technology (KAUST) Thuwal, Makkah Province, Saudi Arabia
(e-mail:slim.alouini@kaust.edu.sa).The corresponding author is Guanghua Yang.

###### Abstract

Energy efficiency of three common hybrid automatic repeat request (HARQ)
schemes including Type I HARQ, HARQ with chase combining (HARQ-CC) and HARQ
with incremental redundancy (HARQ-IR), is analyzed and joint power allocation
and rate selection to maximize the energy efficiency is investigated in this
paper. Unlike prior literature, time-correlated fading channels is considered
and two widely concerned quality of service (QoS) constraints, i.e., outage
and goodput constraints, are also considered in the optimization, which
further differentiates this work from prior ones. Using a unified expression
of asymptotic outage probabilities, optimal transmission powers and optimal
rate are derived in closed-forms to maximize the energy efficiency while
satisfying the QoS constraints. These closed-form solutions then enable a
thorough analysis of the maximal energy efficiencies of various HARQ schemes.
It is revealed that with low outage constraint, the maximal energy efficiency
achieved by Type I HARQ is $\frac{1}{4\ln 2}$ bits/J, while HARQ-CC and HARQ-
IR can achieve the same maximal energy efficiency as
$\frac{\kappa_{\infty}}{4\ln 2}$ bits/J where $\kappa_{\infty}=1.6617$.
Moreover, time correlation in the fading channels has a negative impact on the
energy efficiency, while large maximal allowable number of transmissions is
favorable for the improvement of energy efficiency. The effectiveness of the
energy-efficient optimization is verified by extensive simulations and the
results also show that HARQ-CC can achieve the best tradeoff between energy
efficiency and spectral efficiency among the three HARQ schemes.

###### Index Terms:

Energy efficiency, time-correlated fading channels, hybrid automatic repeat
request, power allocation and rate selection.

##  I Introduction

The past few years have witnessed an explosive growth in wireless data traffic
and the number of mobile terminals. We are therefore obliged to continuously
enhance the spectral/energy efficiency of wireless communication systems to
meet the ever-increasing demand. Towards this end, adaptive modulation and
coding (AMC) and hybrid automatic repeat request (HARQ) are recognized as two
promising techniques so far [1, 2]. In particular, AMC is an efficient
physical-layer technique that adapts the modulation type, the transmission
rate and even the transmission power to channel state information (CSI)
available at the transmitter [3]. However, with only statistical or limited
CSI at the transmitter, HARQ turns out to be more efficient to offer higher
throughput and/or lower outage probability with the aid of multiple
transmissions adaptive to channel conditions [4]. Specifically, HARQ enables
reliable and robust data transmissions through leveraging forward error
correction at the physical layer as well as automatic repeat request at the
link layer. Generally, HARQ techniques are classified into three types based
on the encoding and decoding operations at the transceivers, i.e., Type I
HARQ, HARQ with chase combining (HARQ-CC) and HARQ with incremental redundancy
(HARQ-IR) [5]. The key difference among them is that Type I HARQ discards the
erroneously received sub-codewords and performs memoryless decoding at the
receiver, whereas HARQ-CC and HARQ-IR utilize the erroneous sub-codewords for
subsequent decoding through chase combining and code combining, respectively.
They provide various tradeoffs between performance and complexity and have a
wide range of applications in wireless communications.

To further boost the performance of HARQ schemes when statistical/limited CSI
is available, optimal design of HARQ system has sparked of considerable
research interests lately. The majority of the optimal designs in the
literature aim to maximize the spectral efficiency which is commonly
quantified by long-term average throughput (LTAT) [6, 7, 8, 9, 10, 11, 12, 13,
14]. For example, cooperative HARQ scheme is investigated and optimal rate and
transmission powers are found to maximize the LTAT through numerical search in
[6]. Noticing no closed-form solution and heavy computational overhead in [6],
two suboptimal algorithms are then proposed for rate selection in [7] to
substantially reduce the computational complexity while guaranteeing a
comparable performance to the globally optimal solution. By deriving the
throughput using Laplace transform, a parameterization-based method is
developed to find the optimal rate in (semi-)closed-form to maximize the
throughput for various HARQ schemes in [10]. Moreover, when outdated CSI is
available at the transmitter, the optimal transmission rate to maximize the
LTAT of HARQ-IR is found by using dynamic programming in [12]. Numerical
results demonstrate that a notable throughput gain can be achieved by this
optimal design.

Apart from spectral efficiency, there is another performance metric which
becomes of increasing concern in wireless communications, that is, energy
efficiency [15, 16, 14, 17, 18, 19, 20, 21, 22, 23]. It is particularly
important in energy limited networks, e.g., internet-of-things (IoT) networks.
However, there are very few optimal designs aiming at energy efficiency
maximization in the literature. Specifically, in [16], three optimum schemes
are proposed to achieve various tradeoffs between the energy efficiency and
the spectral efficiency for Type I HARQ under quasi-static fading channels. A
fixed power transmission scheme is proposed to maximize the energy efficiency
of HARQ-IR under independent fading channels and the optimal power is found
numerically in [14]. On the other hand, it is revealed in [17, 18] that the
maximal achievable energy efficiency of Type I HARQ is $\frac{1}{{\rm e}\ln
2}$ bits/J, while HARQ-CC and HARQ-IR can achieve the same maximal energy
efficiency as $\frac{1}{\ln 2}$ bits/J. Unfortunately, these prior analyses
and optimal designs rarely consider quality of service (QoS) constraints
(e.g., outage probability and throughput constraints) while maximizing the
energy efficiency. Such QoS constraints are practical and usually should be
satisfied for practical applications. They would definitely deteriorate the
energy efficiency due to the shrinking of the feasible region in the
optimization. Moreover, all the energy-efficient designs are applicable to
either quasi-static fading channels or independent fading channels, but not
optimal to time-correlated fading channels which usually occur in low-to-
medium mobility environments [24, 25]. Time-correlation among fading channels
usually causes negative impacts on the system performance, and it has been
intensively studied for various systems in the literature. Specifically, the
performance of HARQ-IR over time-correlated Rayleigh fading channels is
investigated in [26] through polynomial fitting technique. The analytical
results are further extended to cooperative HARQ-IR over time-correlated
Nakagami-m fading channels in [27]. Most of prior works are conducted for the
analysis of either outage probability or spectral efficiency. To our best
knowledge, analysis of energy-efficiency and its optimization for HARQ schemes
over time-correlated fading channels has not been discussed and solved yet.

Here we thus take a step further to investigate energy-efficient optimization
for various HARQ schemes over time-correlated fading channels with QoS
constraints, particularly, a tolerable outage constraint and a minimum goodput
constraint. A joint power allocation and rate selection scheme is developed to
maximize the energy efficiency while satisfying the QoS constraints. With a
unified expression of asymptotic outage probabilities, the optimal
transmission powers and rates for the three common HARQ schemes including Type
I HARQ, HARQ-CC and HARQ-IR are derived in closed-forms. The closed-form
optimal solutions then enable a thorough analysis of the maximum energy
efficiency when time-correlated fading channels and QoS constraints are
considered. It is found that the channel time correlation has a negative
impact on the energy efficiency, while large maximal allowable number of
transmissions is beneficial to the energy efficiency. More importantly, it is
revealed that with low outage constraint, the maximal energy efficiency of
Type I HARQ is $\frac{1}{4\ln 2}$ bits/J, while both HARQ-CC and HARQ-IR can
achieve a maximal energy efficiency of $\frac{\kappa_{\infty}}{4\ln 2}$ bits/J
where $\kappa_{\infty}=1.6617$. Noticing that the maximization of the energy
efficiency and the spectral efficiency are generally two conflicting goals,
the tradeoff between the spectral efficiency and the energy efficiency is also
discussed. Numerical results finally demonstrate the effectiveness of our
energy-efficient optimization and show that HARQ-CC in fact can achieve a
better tradeoff between the energy efficiency and the spectral efficiency than
Type I HARQ and HARQ-IR.

The rest of this paper is organized as follows. Section II introduces the
system model. An energy-efficient optimization is then proposed for the three
HARQ schemes and the optimal transmission powers are found in closed-form in
Section III. In Section IV, the optimal rates are derived in closed-form for
three HARQ schemes and their corresponding optimal energy efficiencies are
thoroughly analyzed. Section V presents the numerical results for verification
and investigates the tradeoff between the energy efficiency and the spectral
efficiency of our design. Finally, Section VI concludes this paper.

##  II System Model

This paper considers a point-to-point wireless communication system with HARQ
transmissions. Following HARQ protocol, the source first encodes $N_{b}$
information bits into a mother codeword of length $L{N_{s}}$, where $L$
denotes the maximal allowable number of transmissions. The codeword is then
punctured into $L$ sub-codewords, each with the same length $N_{s}$. These
sub-codewords are sequentially transmitted to the destination through multiple
rounds. Notice that in Type I HARQ and HARQ-CC, the same sub-codeword is
transmitted in each HARQ round, while in HARQ-IR a different sub-codeword with
new parity bits is transmitted in each HARQ round [8]. For all the three HARQ
schemes, the received signal ${\bf y}_{l}$ in the $l$th HARQ round at the
destination can be unified as

${\bf y}_{l}=\sqrt{P_{l}}h_{l}{\bf x}_{l}+{\bf z}_{l},\ \ l=1,2,\cdots,L,$ (1)

where ${\bf x}_{l}$ denotes a sub-codeword with length of $N_{s}$ and each
symbol of ${\bf x}_{l}$ follows Gaussian distribution with unit average
energy, i.e., ${\rm E}\\{{{\bf x}_{l}}^{\rm T}{{\bf x}_{l}}\\}=N_{s}$; $P_{l}$
stands for the transmission power in the $l$th HARQ round; ${\bf z}_{l}$
denotes complex Gaussian white noise vector whose elements have zero mean and
unit variance; and $h_{l}$ represents Rayleigh fading channel coefficient in
the $l$th HARQ round. Under low-to-medium mobility environments, wireless
communications usually experiences time-correlated block-fading channels. To
accommodate this channel time correlation, a widely used time-correlated
Rayleigh fading channel model given in [24] is adopted here

${h_{l}}={\sigma_{l}}\left({\sqrt{1-{\rho^{2\left({l-1}\right)}}}{\mathfrak{h}_{l}}+{\rho^{l-1}}{\mathfrak{h}_{0}}}\right),\
\ l=1,2,\cdots,L,$ (2)

where $\rho$ and ${\sigma_{l}}^{2}$ denote time correlation coefficient and
the variance of ${h_{l}}$, respectively; and ${\mathfrak{h}_{0}}$,
${\mathfrak{h}_{1}},\cdots,{\mathfrak{h}_{L}}$ follow independent circularly-
symmetric complex Gaussian distribution with zero mean and unit variance,
i.e.,
${\mathfrak{h}_{0}},{\mathfrak{h}_{l}}\sim\mathcal{CN}\left({0,1}\right)$.
This model is general and covers quasi-static fading channels (i.e., fully
correlated fading channels where $h_{1}=h_{2}=\cdots=h_{L}$ ) and fast fading
channels (i.e., independent fading channels where $h_{1}$, $h_{2}$, $\cdots$,
$h_{L}$ are independent) as special cases with $\rho=1$ and $\rho=0$,
respectively. In particular, it is worth highlighting that the fixed-rate
HARQ-IR is considered in this paper for the sake of simplicity and fair
comparison, that is, the length of each sub-codeword delivered in HARQ-IR
scheme keeps fixed throughout HARQ rounds. Hereby, as the length of each sub-
codeword is $N_{s}$ for all the HARQ schemes, the initial transmission rate
that denotes the data transmission rate in the first HARQ round is
$R=\frac{N_{b}}{N_{s}}$.

It is assumed that perfect instantaneous CSI is available at the receiver, but
only the statistical CSI is available at the transmitter. To boost the
communication performance, the transmission powers $P_{1},\cdots,P_{L}$ and
the transmission rate $R$ should be properly designed by using the statistical
CSI at the transmitter. In this paper, energy efficiency is concerned and our
design objective is to maximize the energy efficiency under QoS constraints by
jointly optimizing transmission powers and rate. Since the energy efficiency
and QoS constraints are generally defined based on the most fundamental
performance metric, i.e., outage probability, it is discussed first in the
following. For HARQ schemes, an outage event happens when the destination
fails to decode the message after $L$ transmissions. Here three common HARQ
schemes are particularly discussed: Type I HARQ, HARQ-CC and HARQ-IR. They are
differentiated by their encoding and decoding operations at the transceiver.
Specifically, Type I HARQ only takes the received signal in the latest round
for decoding, HARQ-CC adopts maximum ratio combining of the received signals
in multiple rounds for joint decoding, while HARQ-IR adopts code combining of
the received signals in multiple rounds for joint decoding. Therefore the
outage probabilities of the three HARQ schemes are different and given
respectively as (3) at the top of this page [5].

${p_{out,L}}=\left\\{{\begin{array}[]{*{20}{l}}{\Pr\left({{\log}_{2}}\left(1+\max\left({{P_{1}}{\left|{{h_{1}}}\right|^{2}}},{{P_{2}}{\left|{{h_{2}}}\right|^{2}}},\cdots,{{P_{L}}{\left|{{h_{L}}}\right|^{2}}}\right)\right)\leq
R\right),}&{{\rm{Type}}\;{\rm{I}}-{\rm HARQ}}\\\
{\Pr\left({{\log}_{2}}\left(1+\sum\nolimits_{l=1}^{L}{{P_{l}}{\left|{{h_{l}}}\right|^{2}}}\right)\leq
R\right),}&{{\rm{HARQ-CC}}}\\\
{\Pr\left(\sum\nolimits_{l=1}^{L}{{{\log}_{2}}\left({1+{{P_{l}}{\left|{{h_{l}}}\right|^{2}}}}\right)}\leq
R\right),}&{{\rm{HARQ-IR}}.}\end{array}}\right.$ (3)

The outage probabilities of the three HARQ schemes under time-correlated
fading channels have been derived in [28, 29, 30]. However, their expressions
are too complex and involve complicated special functions, which provide
little insights and hinder the optimal design based on them. Fortunately, by
assuming Gaussian codes and typical set decoding [5], the asymptotic outage
probabilities of Type I HARQ and HARQ-CC under time-correlated Rayleigh fading
channels have been exactly derived in simple forms in [28, 29], respectively,
while the asymptotic outage probability of HARQ-IR has been exactly derived in
closed-form in [30]. It has been shown that the asymptotic outage
probabilities match well with the exact outage probabilities under low outage
region or high SNR. They can be unified as

${p_{out,L}}\simeq{\phi_{L}}{\left({\prod\limits_{k=1}^{L}{{P_{k}}}}\right)^{-1}},$
(4)

where ${p_{out,0}}=1$ if $R>0$ and ${p_{out,0}}=0$ otherwise, “$\simeq$”
stands for the asymptotically equal operator, and ${\phi_{L}}$ changes among
various HARQ schemes and is explicitly given by

${\phi_{L}}=\left\\{{\begin{array}[]{*{20}{l}}{{{\varsigma_{L}}}{{\left({{2^{R}}-1}\right)}^{L}},}&{{\rm{Type}}\;{\rm{I}}-{\rm
HARQ}}\\\
{\frac{{{{\varsigma_{L}}}{{\left({{2^{R}}-1}\right)}^{L}}}}{{\Gamma\left({L+1}\right)}},}&{{\rm{HARQ-
CC}}}\\\ {{\varsigma_{L}}}{{{g_{L}}\left(R\right)},}&{{\rm{HARQ-
IR}},}\end{array}}\right.$ (5)

where $\phi_{0}=1$ if $R>0$ and $\phi_{0}=0$ otherwise. In (5),
$\Gamma(\cdot)$ denotes Gamma function, ${\varsigma_{L}}$ quantifies the
impact of time-correlated fading channels, i.e.,
${\varsigma_{L}}=\frac{{{{\left({\ell\left({L,\rho}\right)}\right)}^{-1}}}}{{\prod\nolimits_{k=1}^{L}{{{\sigma_{k}}^{2}}}}}$
in which $\ell\left({L,\rho}\right)$ particularly quantifies the impact of
time correlation with

$\ell\left({L,\rho}\right)=\left({1+\sum\limits_{k=1}^{L}{\frac{{{\rho^{2(k-1)}}}}{{1-{\rho^{2(k-1)}}}}}}\right)\prod\limits_{k=1}^{L}{\left({1-{\rho^{2(k-1)}}}\right)},$
(6)

and $\ell\left({0,\rho}\right)=1$. Hence ${\varsigma_{0}}=1$. Moreover, in
(5), if $R=0$ and $L=0$, $(2^{R}-1)^{L}=0$ by convention, and
${{g_{L}}\left(R\right)}$ is given by

$\displaystyle{{g_{L}}\left(R\right)}$
$\displaystyle=\frac{1}{{2\pi{\rm{i}}}}\int\nolimits_{{a}-{\rm{i}}\infty}^{{a}+{\rm{i}}\infty}{\frac{{{2^{Rs}}}}{{s{{\left({s-1}\right)}^{L}}}}ds}$
$\displaystyle={\left({-1}\right)^{L}}+{2^{R}}\sum\limits_{k=0}^{L-1}{{{\left({-1}\right)}^{k}}\frac{{{{\left({R\ln
2}\right)}^{L-k-1}}}}{{(L-k-1)!}}},$ (7)

with ${\rm i}=\sqrt{-1}$, $a>1$ and ${{g_{0}}\left(0\right)}=0$. The unified
asymptotic outage probability in (4) not only provides clear insights but also
enables optimal design of the transmission powers and transmission rate
analytically to achieve various objectives. It will lead to closed-form
optimal solutions and is adopted here for our optimal design. From the
simulation results shown later, the optimal design based on the asymptotic
outage probabilities can achieve similar performance to that based on the
exact outage probabilities through numerical search, but with much lower
complexity.

##  III Energy-Efficient Optimization

As defined in [15, 14], energy efficiency quantifies the amount of
successfully delivered information per unit energy. Specifically, for HARQ
schemes with maximum $L$ transmissions, based on the _renewal-reward theorem_
[31], the energy efficiency ${\eta}_{L}$ can be written as the ratio of the
average number of correctly received bits
$\bar{N}_{b}={{N_{b}}\left({1-{p_{out,L}}}\right)}$ to the average amount of
energy used
$\bar{\mathcal{E}}=\sum\nolimits_{l=1}^{L}{{p_{out,l-1}}{P_{l}}{\rm E}\\{{{\bf
x}_{l}}^{\rm T}{{\bf
x}_{l}}\\}}=N_{s}\sum\nolimits_{l=1}^{L}{{p_{out,l-1}}{P_{l}}}$, i.e.,

${\eta_{L}}=\frac{{{{\bar{N}}_{b}}}}{{\bar{\mathcal{E}}}}=\frac{{{N_{b}}\left({1-{p_{out,L}}}\right)}}{{{N_{s}}\bar{P}}}=\frac{\mathcal{T}_{g}}{{\bar{P}}},$
(8)

where $\bar{P}=\sum\nolimits_{l=1}^{L}{{p_{out,l-1}}{P_{l}}}$ is the average
total transmission power and $\mathcal{T}_{g}=R(1-{p_{out,L}})$. It is worth
noting that $\mathcal{T}_{g}$ is frequently termed as the goodput/effecitve
rate, which is an important performance metric to evaluate the throughput of
HARQ schemes [32, 33]. As pointed out in [32], the goodput denotes the number
of bits successfully delivered per packet transmission. It is asymptotically
equivalent to the spectral efficiency under high SNR. Notice that the spectral
efficiency denotes the average number of successfully delivered bits per
channel use and its definition will be given later.

With statistical CSI at the transmitter, the transmission powers
$P_{1},\cdots,P_{L}$ and the transmission rate $R$ could be optimally designed
to maximize the energy efficiency ${\eta_{L}}$. In the literature, some
optimal/sub-optimal designs have been proposed for HARQ schemes [16, 14].
Unfortunately, most of them are applicable for independent fading channels and
rarely consider QoS constraints. Considering the wide occurrences of time-
correlated fading channels and QoS requirements in practice, we take a step
forward to incorporate both of them in the energy-efficient optimization. Two
widely concerned QoS constraints are particularly considered here. One is the
outage constraint, i.e., ${{p_{out,L}}\leq\varepsilon}$, and the other is
minimum goodput constraint 111The goodput constraint is asymptotically
equivalent to the spectral efficiency constraint[32]. Since the goodput
expression is simpler than the spectral efficiency, the consideration of
goodput constraint will simplify the optimization and lead to close-form
solution without loss of the practical significance., i.e.,
${\mathcal{T}_{g}\geq\mathcal{T}_{0}}$. With these QoS constraints, the
optimum design of transmission powers and transmission rate to maximize the
energy efficiency can be formulated as

$\begin{array}[]{*{20}{l}}{\mathop{\rm\max}\limits_{{P_{1}},\cdots{P_{L}},R}}&{{\eta_{L}}}\\\
{{\rm{subject}}\,{\rm{to}}}&{{p_{out,L}}\leq\varepsilon}\\\
{}&{{\mathcal{T}_{g}}\geq\mathcal{T}_{0}}\\\ {}&{{P_{l}}\geq 0,\quad 1\leq
l\leq L}\\\ {}&{R\geq 0},\end{array}$ (9)

where $\varepsilon$ and $\mathcal{T}_{0}$ denote the maximal allowable outage
probability and the minimum required goodput, respectively. It is clear that
(9) is a fractional programming problem. Due to the complicated expressions of
the energy efficiency and outage probability under time-correlated fading
channels, the optimal design is very challenging and it is difficult to find
the optimal solution directly. But by introducing an auxiliary variable
$\alpha={p_{out,L}}$ (_target outage probability_), the fractional programming
problem can be reformulated as

$\begin{array}[]{*{20}{l}}{\mathop{{\rm{\max}}}\limits_{{P_{1}},\cdots{P_{L}},R,\alpha}}&{\frac{R(1-\alpha)}{{\bar{P}}}}\\\
{{\rm{subject}}\,{\rm{to}}}&{{p_{out,L}}=\alpha}\\\
{}&{0\leq\alpha\leq\varepsilon}\\\ {}&{R(1-\alpha)\geq{\mathcal{T}}_{0}}\\\
{}&{{P_{l}}\geq 0,\quad 1\leq l\leq L}\\\ {}&{R\geq 0},\end{array}$ (10)

which can be further decomposed into three subproblems equivalently with
regard to transmission powers $P_{1},\cdots,P_{L}$, target outage probability
$\alpha$ and transmission rate $R$, respectively [34, Eqs.11-13]. It should be
noted that this equivalent decomposition holds without the necessity of any
conditions. They can be solved individually in closed-forms in the following.

###  III-A Optimal Power Allocation

Given the transmission rate $R$ and the target outage probability $\alpha$,
the problem in (10) reduces to the design of the transmission powers as

$\begin{array}[]{*{20}{l}}{\mathop{\rm\min}\limits_{{P_{1}},\cdots{P_{L}}}}&{\bar{P}}\\\
{{\rm{subject}}\,{\rm{to}}}&{{p_{out,L}}=\alpha}\\\ {}&{{P_{l}}\geq 0,1\leq
l\leq L}.\end{array}$ (11)

This optimization problem is similar to the power allocation problem in [29],
except that the equality outage constraint ${{p_{out,L}}=\alpha}$ is changed
as inequality outage constraint ${{p_{out,L}}\leq\alpha}$. It has been proved
in [29] that the minimal average total transmission power with inequality
outage constraint is achieved at the outage boundary, i.e.,
${{p_{out,L}}=\alpha}$. Therefore, the closed-form optimal power solution in
[29] is applicable to our power design problem in (11). Specifically, as shown
in [29], the optimal powers are given as functions of the transmission rate
$R$ and target outage probability $\alpha$ as

$\displaystyle{P_{L}^{*}}$
$\displaystyle{={{\left({\frac{{{\phi_{L}}\prod\limits_{k=2}^{L}{{{\left({\frac{{2{\phi_{k-1}}}}{{{\phi_{k-2}}}}}\right)}^{{2^{1-k}}}}}}}{{{2^{L-1}}\alpha{\phi_{L-1}}}}}\right)}^{\frac{{{2^{L-1}}}}{{{2^{L}}-1}}}}},$
(12)
$P_{l}^{*}=\prod\limits_{k=l+1}^{L}{{{\left({\frac{{2{\phi_{k-1}}}}{{{\phi_{k-2}}}}}\right)}^{{2^{l-k}}}}}{P_{L}^{*}}^{{2^{l-L}}},\quad
1\leq l\leq L-1,$ (13)

and the minimal average total transmission power $\bar{P}^{*}$ can be obtained
in a simple form as

${{\bar{P}}^{*}}=\frac{{\left({{2^{L}}-1}\right){\alpha^{-\frac{1}{{{2^{L}}-1}}}}}}{{{2^{\frac{L}{{1-{2^{-L}}}}-2}}}}{\left({\prod\limits_{k=1}^{L}{{{\left({\frac{{{\phi_{k}}}}{{{\phi_{k-1}}}}}\right)}^{{2^{-k}}}}}}\right)^{\frac{1}{{1-{2^{-L}}}}}}.$
(14)

Clearly from (12), (13) and (14), the decrease of the target outage
probability $\alpha$ will lead to the increase of transmission powers
$P^{*}_{l}$ and then the increase of the minimal average total transmission
power $\bar{P}^{*}$. In addition, it can be found that the minimal average
total transmission power $\bar{P}^{*}$ becomes irrelevant to the target outage
probability $\alpha$ when $L\to\infty$.

###  III-B Optimal Outage Probability

Now putting the optimal powers (12) and (13) into the original optimization
problem (10), the original problem can be reduced to the optimization of two
variables of the transmission rate $R$ and target outage probability $\alpha$
as

$\begin{array}[]{*{20}{l}}{\mathop{\rm\max}\limits_{R,\alpha}}&\frac{R(1-\alpha)}{{{\bar{P}}^{*}}}\\\
{{\rm{subject}}\,{\rm{to}}}&{0\leq\alpha\leq\varepsilon}\\\
{}&{R(1-\alpha)\geq\mathcal{T}_{0}}\\\ {}&{R\geq 0}.\end{array}$ (15)

With (14), the objective function of (15) can be rewritten as

$\frac{{R(1-\alpha)}}{{{{\bar{P}}^{*}}}}=\underbrace{\frac{{{2^{\frac{L}{{1-{2^{-L}}}}-2}}}}{{\left({{2^{L}}-1}\right)}}}_{\triangleq\psi}\underbrace{(1-\alpha){\alpha^{\frac{1}{{{2^{L}}-1}}}}}_{\triangleq
f\left(\alpha\right)}{\frac{R}{{{{\left({\prod\limits_{k=1}^{L}{{{\left({\frac{{{\phi_{k}}}}{{{\phi_{k-1}}}}}\right)}^{{2^{-k}}}}}}\right)}^{\frac{1}{{1-{2^{-L}}}}}}}}}.$
(16)

It is clear that the target outage probability is only involved in the term
$f\left(\alpha\right)$ in (16). When given the transmission rate $R$, the
problem in (15) is equivalent to the following target outage probability
optimization as

$\begin{array}[]{*{20}{l}}{\mathop{{\rm{\max}}}\limits_{\alpha}}&{f\left(\alpha\right)}\\\
{{\rm{subject\,to}}}&{0\leq\alpha\leq\varepsilon}\\\
{}&{R(1-\alpha)\geq{\mathcal{T}_{0}}},\end{array}$ (17)

whose feasibility and optimal solution can be found in the following theorem.

######  Theorem 1.

The optimization problem (17) is infeasible when $R\leq\mathcal{T}_{0}$. When
$R>\mathcal{T}_{0}$, the optimal target outage probability is
${\alpha^{*}}={\min\left\\{{\varepsilon,1-\frac{{{{\mathcal{T}_{0}}}}}{R}},2^{-L}\right\\}}$
and the optimal $f\left({{\alpha^{*}}}\right)$ can be written as

$\displaystyle f\left({{\alpha^{*}}}\right)$
$\displaystyle={{\frac{{{{\mathcal{T}_{0}}}}}{R}{{\left({1-\frac{{{{\mathcal{T}_{0}}}}}{R}}\right)}^{c}}}}\left({{\chi}\left({R-{{\mathcal{T}_{0}}}}\right)-{\chi}\left({R-\frac{{{{\mathcal{T}_{0}}}}}{{1-\Delta}}}\right)}\right)$
$\displaystyle\quad+{{(1-\Delta){\Delta^{c}}}}{\chi}\left({R-\frac{{{{\mathcal{T}_{0}}}}}{{1-\Delta}}}\right),$
(18)

where $c=\frac{1}{{{2^{L}}-1}}$,
$\Delta=\min\left\\{{\varepsilon,{2^{-L}}}\right\\}$, and $\chi(t)$ is an
indicator function as

$\chi(t){\rm{=}}\left\\{{\begin{array}[]{*{20}{c}}{\rm{0}}&{t<0}\\\
{\rm{1}}&{t\geq 0}\end{array}}\right..$ (19)

###### Proof.

Please see Appendix A. ∎

###  III-C Optimal Rate Selection

After determining the optimal target outage probability $\alpha^{*}$ and
considering the feasible region of $R>\mathcal{T}_{0}$ in Theorem 1, the
energy-efficient optimization in (15) is finally reduced to the optimal rate
selection as

$\begin{array}[]{*{20}{l}}{\mathop{{\rm{\max}}}\limits_{R}}&{{\eta_{L}}=\psi
f\left({{\alpha^{*}}}\right){\frac{R}{{{{\left({\prod\limits_{k=1}^{L}{{{\left({\frac{{{\phi_{k}}}}{{{\phi_{k-1}}}}}\right)}^{{2^{-k}}}}}}\right)}^{\frac{1}{{1-{2^{-L}}}}}}}}}}\\\
{{\rm{subject\,to}}}&{R>{{\mathcal{T}_{0}}}.}\end{array}$ (20)

Plugging (1) into (20), although the optimal transmission rate $R^{*}$ can be
computed numerically through one dimensional search, it is lacking of clear
insights. In this paper, we aim to derive the optimal rate in closed-form and
extract clear insights for energy-efficient optimization. Since $\phi_{k}$ in
the objective function depends on the transmission rate $R$ as shown in (5)
and changes among different HARQ schemes, the optimal rate selection for
various HARQ schemes should be investigated individually and will be discussed
one by one in the next section.

##  IV Optimal Rate and Optimal Energy Efficiency

###  IV-A Type I HARQ

####  IV-A1 Optimal Rate

Putting (5) into the objective function in (20), the energy efficiency of Type
I HARQ can be written as

${\eta_{I,L}}=\psi{\theta_{L}}\frac{f\left({{\alpha^{*}}}\right)R}{{{2^{R}}-1}},$
(21)

where

$\displaystyle{\theta_{L}}$
$\displaystyle={\left({\prod\limits_{k=1}^{L}{{{\left({\frac{{{\varsigma_{k-1}}}}{{{\varsigma_{k}}}}}\right)}^{{2^{-k}}}}}}\right)^{\frac{1}{{1-{2^{-L}}}}}}$
$\displaystyle={\left({\prod\limits_{k=1}^{L}{{{\left({\frac{{\ell\left({k,\rho}\right){{\sigma_{k}}^{2}}}}{{\ell\left({k-1,\rho}\right)}}}\right)}^{{2^{-k}}}}}}\right)^{\frac{1}{{1-{2^{-L}}}}}}.$
(22)

Substituting (1) into (21) yields

$\displaystyle{\eta_{I,L}}$
$\displaystyle={{\psi{\theta_{L}}}}{{{{\mathcal{T}_{0}}}}}\frac{{{{\left({1-\frac{{{{\mathcal{T}_{0}}}}}{R}}\right)}^{c}}}}{{{2^{R}}-1}}\left({{\chi}\left({R-{{\mathcal{T}_{0}}}}\right)-{\chi}\left({R-\frac{{{{\mathcal{T}_{0}}}}}{{1-\Delta}}}\right)}\right)$
$\displaystyle\quad+{{\psi{\theta_{L}}}}{{(1-\Delta){\Delta^{c}}}}\frac{R}{{{2^{R}}-1}}{\chi}\left({R-\frac{{{{\mathcal{T}_{0}}}}}{{1-\Delta}}}\right).$
(23)

From (IV-A1), we can see that when $R>\frac{\mathcal{T}_{0}}{1-\Delta}$, the
first term is zero and the energy efficiency reduces to
${\eta_{I,L}}={{\psi{\theta_{L}}}}{{(1-\Delta){\Delta^{c}}}}\frac{R}{{{2^{R}}-1}}$
which is a decreasing function of $R$. Moreover, the energy efficiency
${\eta_{I,L}}$ is continuous in the domain of $R>\mathcal{T}_{0}$. Therefore,
considering the constraint of $R>\mathcal{T}_{0}$ in (20), we can conclude
that the optimal rate ${R^{*}}$ to achieve the maximal energy efficiency
should lie within the range of
$\left({\mathcal{T}_{0}},\frac{\mathcal{T}_{0}}{1-\Delta}\right]$. In this
range, the second term in (IV-A1) is zero and the energy efficiency can be
simplified as

$\displaystyle{\eta_{I,L}}$
$\displaystyle={{\psi{\theta_{L}}}}{{{{\mathcal{T}_{0}}}}}\frac{{{{\left({1-\frac{{{{\mathcal{T}_{0}}}}}{R}}\right)}^{c}}}}{{{2^{R}}-1}}.$
(24)

Then the problem of rate selection in (20) is equivalent to a minimization
problem as

$\begin{array}[]{*{20}{l}}{\mathop{{\rm{\min}}}\limits_{R}}&{\Phi\left(R\right)=\left({{2^{R}}-1}\right){\left({1-\frac{{{{\mathcal{T}_{0}}}}}{R}}\right)^{-c}}}\\\
{{\rm{subject}}{\mkern
1.0mu}\,{\rm{to}}}&{{{\mathcal{T}_{0}}}<R\leq\frac{{{{\mathcal{T}_{0}}}}}{{1-\Delta}}}.\end{array}$
(25)

The optimal solution to (25) can be found in closed-form and is shown in the
following theorem.

######  Theorem 2.

The optimal rate to maximize the energy efficiency while guaranteeing the
outage and goodput performance for Type I HARQ is

${R^{*}}=\min\left\\{{\frac{{{{\cal
T}_{0}}}}{{1-\Delta}}},{{\varphi^{-1}}(0)}\right\\},$ (26)

where $\varphi^{-1}$ denotes the inverse function with respect to
$\varphi\left(R\right)=\ln\left(2\right)R\left({R-{{\mathcal{T}_{0}}}}\right){2^{R}}-c{{\mathcal{T}_{0}}}\left({{2^{R}}-1}\right)$
and $\varphi^{-1}(0)$ refers to the zero point of $\varphi(R)$. When
$\varepsilon\geq 2^{-L}$, the optimal transmission rate reduces to
$R^{*}=\varphi^{-1}(0)$.

###### Proof.

Please see Appendix B. ∎

It is noteworthy that the zero point $\varphi^{-1}(0)$ can be easily computed
since $\varphi\left(R\right)$ is an increasing function of $R$ within the
range of $(\mathcal{T}_{0},\infty)$.

####  IV-A2 Optimal Energy Efficiency

Putting the optimal rate (26) into (24), the optimal energy efficiency of Type
I HARQ can be obtained as

${{\eta}_{I,L}^{*}}={{\psi{\theta_{L}}}}{{{{\mathcal{T}_{0}}}}}\frac{{{{\left({1-\frac{{{{\mathcal{T}_{0}}}}}{{{R^{*}}}}}\right)}^{c}}}}{{{2^{{R^{*}}}}-1}}={{\psi{\theta_{L}}}}{{{{\mathcal{T}_{0}}}}}\frac{{{{\left({{R^{*}}-{{\mathcal{T}_{0}}}}\right)}^{c}}}}{{{R^{*}}^{c}\left({{2^{{R^{*}}}}-1}\right)}}.$
(27)

As aforementioned, the maximal energy efficiency of HARQ schemes over
independent Rayleigh fading channels without QoS constraints has been studied
in [18]. It has been proved that the maximal energy efficiency of Type I HARQ
operating over _independent Rayleigh fading channels without QoS constraints_
is $\frac{1}{{\rm e}\ln(2)}$. From (27), we can see that channel time
correlation will affect the optimal energy efficiency through the term
$\theta_{L}$. As proved in Appendix C, $\theta_{L}$ is a decreasing function
of time correlation coefficient $\rho$. In other words, channel time
correlation has a negative impact on the optimal energy efficiency and time-
correlated fading channels provide lower energy efficiency than time
independent fading channels with $\rho=0$. We thus can expect that the maximal
energy efficiency of Type I HARQ over time-correlated fading channels with QoS
constraints will be lower than $\frac{1}{{\rm e}\ln(2)}$ and will be
investigated here.

To proceed with the investigation, we first analyze the monotonic property of
the optimal energy efficiency with respect to the maximal number of
transmissions $L$. From the original energy efficiency maximization problem in
(9), we can find the following property.

######  Property 1.

The optimal energy efficiencies of all the three HARQ schemes are non-
decreasing functions of the maximal number of transmissions $L$ and
${\eta_{L}^{*}}\leq\mathop{\lim}\limits_{L\to\infty}{\eta_{L}^{*}}\triangleq{\eta_{\infty}^{*}}$.

###### Proof.

Please see Appendix D. ∎

Notice that this property is applicable to all the three HARQ schemes. For
Type I HARQ scheme, the maximal energy efficiency is thus achieved when
$L\to\infty$ and is denoted as ${\eta_{I,\infty}^{*}}$ which can be found as
shown in the following theorem.

######  Theorem 3.

Under time-correlated Rayleigh fading channels, the optimal energy efficiency
of Type I HARQ with outage and goodput constraints is upper bounded by

${\eta_{I,L}^{*}}\leq{\eta_{I,\infty}^{*}}=\frac{{{\theta_{\infty}}}{{\mathcal{T}_{0}}}}{{4\left({{2^{{{\mathcal{T}_{0}}}}}-1}\right)}},$
(28)

where
$\theta_{\infty}\triangleq{\mathop{\lim}\limits_{L\to\infty}{\theta_{L}}}$
exists. The maximal energy efficiency ${\eta_{I,\infty}^{*}}$ is a decreasing
function of the goodput threshold $\mathcal{T}_{0}$. Particularly, for
Rayleigh fading channels with unit channel gains as
${\sigma_{1}}^{2}=\cdots={\sigma_{L}}^{2}=1$, $\theta_{\infty}\leq 1$ and the
maximal energy efficiency of Type I HARQ with outage and goodput constraints
satisfies

${\eta_{I,\infty}^{*}}=\frac{{{\theta_{\infty}}}{{\mathcal{T}_{0}}}}{{4\left({{2^{{{\mathcal{T}_{0}}}}}-1}\right)}}\leq\mathop{\lim}\limits_{{\mathcal{T}_{0}}\to
0}\frac{{{{\mathcal{T}_{0}}}}}{{4\left({{2^{{{\mathcal{T}_{0}}}}}-1}\right)}}=\frac{1}{4\ln(2)}\triangleq{\eta_{I,\infty}^{\max}}.$
(29)

###### Proof.

Please see Appendix E. ∎

It means that when QoS constraints are considered, the maximal energy
efficiency which can be achieved by Type I HARQ is $\frac{1}{4\ln(2)}$.

###  IV-B HARQ-CC

####  IV-B1 Optimal Rate

Similarly to Section IV-A, with the optimal powers
$P_{1}^{*},\cdots,P_{L}^{*}$, the optimal target outage probability
$\alpha^{*}$ and the definition in (5), the energy efficiency of HARQ-CC in
(20) can be rewritten as

${\eta_{CC,L}}={\kappa_{L}}^{\frac{1}{{1-{2^{-L}}}}}\psi{\theta_{L}}\frac{f\left({{\alpha^{*}}}\right)R}{{{2^{R}}-1}},$
(30)

where ${\kappa_{L}}=\prod\nolimits_{k=1}^{L}{{{{{k}}}^{{2^{-k}}}}}$. Noticing
that the only difference between (21) and (30) is at an additional term
${\kappa_{L}}^{\frac{1}{{1-{2^{-L}}}}}$ involved in (30) and this term is
irrelevant to the transmission rate, the optimal transmission rate $R^{*}$ of
HARQ-CC can thus be derived as the same as that for Type I HARQ shown in (26)
in Theorem 2.

####  IV-B2 Optimal Energy Efficiency

Accordingly, the optimal energy efficiency of HARQ-CC can be written as

${\eta_{CC,L}^{*}}={\kappa_{L}}^{\frac{1}{{1-{2^{-L}}}}}{\eta_{I,L}^{*}}.$
(31)

Since $\kappa_{L}>1$, it thus reveals that HARQ-CC surpasses Type I HARQ in
terms of the optimal energy efficiency, i.e.,
${\eta_{CC,L}^{*}}>{\eta_{I,L}^{*}}$.

Moreover, based on Property 1 and Theorem 3, it is easy to get the following
result of maximal energy efficiency of HARQ-CC.

######  Theorem 4.

Under time-correlated Rayleigh fading channels, the optimal energy efficiency
of HARQ-CC with outage and goodput constraints is upper bounded by

${\eta_{CC,L}^{*}}\leq{\eta_{CC,\infty}^{*}}=\frac{{{\kappa_{\infty}\theta_{\infty}}}{{\mathcal{T}_{0}}}}{{4\left({{2^{{{\mathcal{T}_{0}}}}}-1}\right)}},$
(32)

where ${\kappa_{\infty}}=\mathop{\lim}\limits_{{L}\to\infty}\kappa_{L}\approx
1.6617$ as proved in Appendix F. Particularly, for Rayleigh fading channels
with unit channel gains as ${\sigma_{1}}^{2}=\cdots={\sigma_{L}}^{2}=1$,
$\theta_{\infty}\leq 1$ and the maximal energy efficiency of HARQ-CC with
outage and goodput constraints satisfies

${\eta_{CC,\infty}^{*}}=\frac{{{\kappa_{\infty}\theta_{\infty}}}{{\mathcal{T}_{0}}}}{{4\left({{2^{{{\mathcal{T}_{0}}}}}-1}\right)}}\leq\mathop{\lim}\limits_{{\mathcal{T}_{0}}\to
0}\frac{{{{\kappa_{\infty}\mathcal{T}_{0}}}}}{{4\left({{2^{{{\mathcal{T}_{0}}}}}-1}\right)}}=\frac{\kappa_{\infty}}{4\ln(2)}\triangleq{\eta_{CC,\infty}^{\max}}.$
(33)

In other words, when QoS constraints are considered, the maximal energy
efficiency of HARQ-CC is $\frac{\kappa_{\infty}}{4\ln(2)}$ which is higher
than that of Type I HARQ.

###  IV-C HARQ-IR

####  IV-C1 Optimal Rate

Putting (5) into (20) yields the energy efficiency of HARQ-IR as

$\displaystyle{\eta_{IR,L}}$
$\displaystyle=\psi{\theta_{L}}\frac{f\left({{\alpha^{*}}}\right)R}{{{{\left({\prod\limits_{k=1}^{L}{{{\left({\frac{{{g_{k}}\left(R\right)}}{{{g_{k-1}}\left(R\right)}}}\right)}^{{2^{-k}}}}}}\right)}^{\frac{1}{{1-{2^{-L}}}}}}}}.$
(34)

Substituting (1) into (34), it becomes

${\eta_{IR,L}}=\frac{{{\psi{\theta_{L}}}}{{{{\mathcal{T}_{0}}}}}}{\Lambda\left(R\right)}\left({{\chi}\left({R-{{\mathcal{T}_{0}}}}\right)-{\chi}\left({R-\frac{{{{\mathcal{T}_{0}}}}}{{1-\Delta}}}\right)}\right)\\\
+\frac{{{\psi{\theta_{L}}}}{{(1-\Delta){\Delta^{c}}}}}{\left({\prod\limits_{k=1}^{L}{{{\left({\cal
G}_{k}\left(R\right)\right)}^{{2^{-k}}}}}}\right)^{\frac{1}{{1-{2^{-L}}}}}}{\chi}\left({R-\frac{{{{\mathcal{T}_{0}}}}}{{1-\Delta}}}\right),$
(35)

where
${\Lambda\left(R\right)\triangleq{{\left({1-\frac{{{{\mathcal{T}_{0}}}}}{R}}\right)}^{-c}}{{\left({\prod\limits_{k=1}^{L}{{{\left({\frac{{{g_{k}}\left(R\right)}}{{{g_{k-1}}\left(R\right)}}}\right)}^{{2^{-k}}}}}}\right)}^{\frac{1}{{1-{2^{-L}}}}}}}$
and ${\cal
G}_{k}\left(R\right)\triangleq\frac{{{g_{k}}\left(R\right)}}{{R{g_{k-1}}\left(R\right)}}$.
When $R>\frac{\mathcal{T}_{0}}{1-\Delta}$, the first term in (35) is zero and
the energy efficiency reduces as
${\eta_{IR,L}}=\frac{{{\psi{\theta_{L}}}}{{(1-\Delta){\Delta^{c}}}}}{\left({\prod\limits_{k=1}^{L}{{{\left({\cal
G}_{k}\left(R\right)\right)}^{{2^{-k}}}}}}\right)^{\frac{1}{{1-{2^{-L}}}}}}$.
As proved in Appendix G that ${\cal G}_{k}\left(R\right)$ is a monotonically
increasing function of $R$. Thus when $R>\frac{\mathcal{T}_{0}}{1-\Delta}$,
${\eta_{IR,L}}$ is a decreasing function of $R$. Together with the continuity
of ${\eta_{IR,L}}$ at the point $R={\frac{{{{\mathcal{T}_{0}}}}}{{1-\Delta}}}$
and the constraint of $R>\mathcal{T}_{0}$ in (20), we can conclude that the
optimal rate to achieve the maximal energy efficiency of HARQ-IR is located
within the range of
$\left({\mathcal{T}_{0}},\frac{\mathcal{T}_{0}}{1-\Delta}\right]$.
Accordingly, when ${\mathcal{T}_{0}}<R\leq\frac{\mathcal{T}_{0}}{1-\Delta}$,
the energy efficiency ${\eta_{IR,L}}$ can be simplified as

${\eta_{IR,L}}=\frac{{{\psi{\theta_{L}}}}{{{{\mathcal{T}_{0}}}}}}{\Lambda\left(R\right)}.$
(36)

Then the rate selection problem in (20) is equivalent to

$\begin{array}[]{*{20}{l}}{\mathop{{\rm{\min}}}\limits_{R}}&{\Lambda\left(R\right)}\\\
{{\rm{subject\,to}}}&{{{\mathcal{T}_{0}}}<R\leq\frac{{{{\mathcal{T}_{0}}}}}{{1-\Delta}}}.\end{array}$
(37)

Due to the complicated form of ${\Lambda\left(R\right)}$, it is difficult to
derive a closed-form solution for the optimal transmission rate. Fortunately,
after analyzing the function ${\Lambda\left(R\right)}$, we find a special
property of ${\Lambda\left(R\right)}$ in the following.

######  Property 2.

The function ${\Lambda\left(R\right)}$ is bounded by

${\left({\ln\left(2\right)}\right)^{\frac{{{2^{-1}}-{2^{-L}}}}{{1-{2^{-L}}}}}}{\kappa_{L}}^{-\frac{1}{{1-{2^{-L}}}}}{\left({\varpi\left(R\right)}\right)^{\frac{{{2^{-1}}}}{{1-{2^{-L}}}}}}\leq\Lambda\left(R\right)\\\
\leq{\left({\ln\left(2\right)}\right)^{\frac{{{2^{-1}}-{2^{-L}}}}{{1-{2^{-L}}}}}}{\kappa_{L-1}}^{-\frac{{{2^{-1}}}}{{1-{2^{-L}}}}}{\left({\varpi\left(R\right)}\right)^{\frac{2^{-1}}{{1-{2^{-L}}}}}},$
(38)

where $\varpi\left(R\right)={\left({R-{{\cal
T}_{0}}}\right)^{-{2^{-L+1}}}}\left({{2^{R}}-1}\right)R$.

###### Proof.

Please see Appendix H. ∎

With these bounds and Intermediate Value Theorem [35, Theorem 4.23],
$\Lambda\left(R\right)$ can be rewritten as

$\Lambda\left(R\right)={\left({\ln\left(2\right)}\right)^{\frac{{{2^{-1}}-{2^{-L}}}}{{1-{2^{-L}}}}}}\zeta{\left({\varpi\left(R\right)}\right)^{\frac{{{2^{-1}}}}{{1-{2^{-L}}}}}},$
(39)

where $\zeta$ is bounded as

${\kappa_{L}}^{-\frac{1}{{1-{2^{-L}}}}}\leq\zeta\leq{\kappa_{L-1}}^{-\frac{{{2^{-1}}}}{{1-{2^{-L}}}}}.$
(40)

Based on (39), the optimization problem (37) can be rewritten as

$\begin{array}[]{*{20}{l}}{\mathop{{\rm{\min}}}\limits_{R}}&{\varpi\left(R\right)}\\\
{{\rm{subject\,to}}}&{{{\mathcal{T}_{0}}}<R\leq\frac{{{{\mathcal{T}_{0}}}}}{{1-\Delta}}}.\end{array}$
(41)

Following a similar proof as that for Theorem 2, the optimal solution to (41)
can be derived by using KKT conditions, as given in the following theorem.

######  Theorem 5.

The optimal rate to the problem in (41) is

${R^{*}}=\min\left\\{{\frac{{{{\mathcal{T}_{0}}}}}{{1-\Delta}}},\Upsilon^{-1}(0)\right\\}.$
(42)

where
$\Upsilon\left(R\right)=\left({R-{{\mathcal{T}_{0}}}}\right)\left({{2^{R}}\ln\left(2\right)R+{2^{R}}-1}\right)-{2^{-L+1}}\left({{2^{R}}-1}\right)R$
and $\Upsilon^{-1}(0)$ refers to the zero point of $\Upsilon(R)$. When
$\varepsilon\geq 2^{-L}$, the optimal transmission rate reduces to
$R^{*}=\Upsilon^{-1}(0)$.

The proof is omitted here to avoid redundancy. Notice that the zero point
$\Upsilon^{-1}(0)$ can be easily found since $\Upsilon(R)$ is an increasing
function with respect to $R$.

####  IV-C2 Optimal Energy Efficiency

With the definition of ${\Lambda\left(R\right)}$, the energy efficiency of
HARQ-IR in (36) can be written as

${\eta_{IR,L}}=\frac{{{\psi{\theta_{L}}}}{{{{\mathcal{T}_{0}}}}}}{{{{\left({1-\frac{{{{\mathcal{T}_{0}}}}}{R}}\right)}^{-c}}{{\left({\prod\limits_{k=1}^{L}{{{\left({\frac{{{g_{k}}\left(R\right)}}{{{g_{k-1}}\left(R\right)}}}\right)}^{{2^{-k}}}}}}\right)}^{\frac{1}{{1-{2^{-L}}}}}}}}$
(43)

As proved in Appendix I, we have the following inequality

$\frac{{{g_{k}}\left(R\right)}}{{{g_{k-1}}\left(R\right)}}\leq\frac{{{2^{R}}-1}}{k}.$
(44)

Applying this inequality to (43) yields

$\displaystyle{\eta_{IR,L}}$
$\displaystyle\geq\frac{{\left(\prod\limits_{k=1}^{L}k^{2^{-k}}\right)^{\frac{1}{{1-{2^{-L}}}}}}{{\psi{\theta_{L}}}}{{{{\mathcal{T}_{0}}}}}{{\left({1-\frac{{{{\mathcal{T}_{0}}}}}{R}}\right)}^{c}}}{{{{\left({\prod\limits_{k=1}^{L}{{{\left({{2^{R}}-1}\right)}^{{2^{-k}}}}}}\right)}^{\frac{1}{{1-{2^{-L}}}}}}}}$
$\displaystyle={\kappa_{L}}^{\frac{1}{{1-{2^{-L}}}}}{{\psi{\theta_{L}}}}{{{{\mathcal{T}_{0}}}}}\frac{{{{\left({1-\frac{{{{\mathcal{T}_{0}}}}}{R}}\right)}^{c}}}}{{{2^{R}}-1}}={\eta_{CC,L}}.$
(45)

It means that HARQ-IR outperforms HARQ-CC in terms of energy efficiency.
Moreover, the optimal energy efficiency of HARQ-IR is also higher than or
equal to that of HARQ-CC, i.e., ${\eta_{IR,L}^{*}}\geq{\eta_{CC,L}^{*}}$.

Now putting the optimal rate (42) into (39) and then (36) together with the
bounds of $\zeta$ in (40), the optimal energy efficiency of HARQ-IR is bounded
as

${\kappa_{L-1}}^{\frac{{{2^{-1}}}}{{1-{2^{-L}}}}}\tilde{\eta}_{IR,L}^{*}\leq\eta_{IR,L}^{*}\leq{\kappa_{L}}^{\frac{1}{{1-{2^{-L}}}}}\tilde{\eta}_{IR,L}^{*}.$
(46)

where

$\tilde{\eta}_{IR,L}^{*}={{\psi{\theta_{L}}}}{{{{\mathcal{T}_{0}}}}}{\left({\ln\left(2\right)}\right)^{-\frac{{{2^{-1}}-{2^{-L}}}}{{1-{2^{-L}}}}}}{\left({\varpi\left(R^{*}\right)}\right)^{-\frac{2^{-1}}{{1-{2^{-L}}}}}}.$
(47)

Combining (46) with the inequality ${\eta_{IR,L}^{*}}\geq{\eta_{CC,L}^{*}}$,
the optimal energy efficiency of HARQ-IR is consequently found to be bounded
by

$\max\left\\{{\kappa_{L-1}}^{\frac{{{2^{-1}}}}{{1-{2^{-L}}}}}\tilde{\eta}_{IR,L}^{*},{\eta_{CC,L}^{*}}\right\\}\leq\eta_{IR,L}^{*}\leq{\kappa_{L}}^{\frac{1}{{1-{2^{-L}}}}}\tilde{\eta}_{IR,L}^{*}.$
(48)

When the number of transmissions approaches infinity, i.e., $L\to\infty$, the
following bounds also hold

$\max\left\\{\sqrt{\kappa_{\infty}}\tilde{\eta}_{IR,\infty}^{*},{\eta_{CC,\infty}^{*}}\right\\}\leq\eta_{IR,\infty}^{*}\leq{\kappa_{\infty}}\tilde{\eta}_{IR,\infty}^{*},$
(49)

where ${\eta_{CC,\infty}^{*}}$ has been given in Theorem 4, and
$\tilde{\eta}_{IR,\infty}^{*}$ is defined as
$\tilde{\eta}_{IR,\infty}^{*}\triangleq\mathop{\lim}\limits_{L\to\infty}\tilde{\eta}_{IR,L}^{*}$
and can be further derived from (47) as

$\displaystyle\tilde{\eta}_{IR,\infty}^{*}$
$\displaystyle=\mathop{\lim}\limits_{L\to\infty}{{\psi{\theta_{L}}}}{{{{\mathcal{T}_{0}}}}}{\left({\ln\left(2\right)}\right)^{-\frac{{{2^{-1}}-{2^{-L}}}}{{1-{2^{-L}}}}}}$
$\displaystyle\quad\times{\left({{{\left({{{R}^{*}}-{{\mathcal{T}_{0}}}}\right)}^{-{2^{-L+1}}}}\left({{2^{{{R}^{*}}}}-1}\right){{R}^{*}}}\right)^{-\frac{{{2^{-1}}}}{{1-{2^{-L}}}}}}$
$\displaystyle=\frac{{\theta_{\infty}}{\mathcal{T}_{0}}}{4\sqrt{\ln(2)}}\mathop{\lim}\limits_{L\to\infty}{\left({{{\left({{{R}^{*}}-{{\mathcal{T}_{0}}}}\right)}^{-{2^{-L+1}}}}\left({{2^{{{R}^{*}}}}-1}\right){{R}^{*}}}\right)^{-\frac{{{2^{-1}}}}{{1-{2^{-L}}}}}}.$
(50)

In (IV-C2), the last equality follows based on the limit result
$\mathop{\lim}\limits_{L\to\infty}\psi=\frac{1}{4}$.

When $L\to\infty$, the inequality $\varepsilon\geq 2^{-L}$ would hold. Based
on Theorem 5, the optimal transmission rate is $R^{*}=\Upsilon^{-1}(0)$ and
thus $\Upsilon(R^{*})=0$ which can be rewritten as

${{R}^{*}}={{\mathcal{T}_{0}}}+\frac{{{2^{-L+1}}\left({{2^{{{R}^{*}}}}-1}\right){{R}^{*}}}}{{{2^{{{R}^{*}}}}\ln\left(2\right){{R}^{*}}+{2^{{{R}^{*}}}}-1}}.$
(51)

Meanwhile, noticing that the optimal rate ${{R}^{*}}$ is bounded as
${\mathcal{T}_{0}}<{{R}^{*}}\leq\frac{\mathcal{T}_{0}}{1-\Delta}$,
$\Delta=\min\left\\{\varepsilon,2^{-L}\right\\}$ and
$\mathop{\lim}\limits_{L\to\infty}\Delta=0$, it follows by using squeeze
theorem that $\mathop{\lim}\limits_{L\to\infty}{{R}^{*}}={\mathcal{T}_{0}}$.
With this limit and plugging (51) into (IV-C2), it yields

$\displaystyle\tilde{\eta}_{IR,\infty}^{*}$
$\displaystyle=\frac{{{\theta_{\infty}}{{\cal
T}_{0}}}}{{4\sqrt{\ln(2)}}}\mathop{\lim}\limits_{L\to\infty}{\left({\left({{2^{{{R}^{*}}}}-1}\right){{R}^{*}}}\right)^{-\frac{{{2^{-1}}}}{{1-{2^{-L}}}}}}\mathop{\lim}\limits_{L\to\infty}{\left({{{R}^{*}}-{{\cal
T}_{0}}}\right)^{\frac{{{2^{-L}}}}{{1-{2^{-L}}}}}}$
$\displaystyle=\frac{{{\theta_{\infty}}}}{4}\sqrt{\frac{{{\mathcal{T}_{0}}}}{{\ln(2)\left({{2^{{\mathcal{T}_{0}}}}-1}\right)}}}\mathop{\lim}\limits_{L\to\infty}{\left({\frac{{{2^{-L+1}}\left({{2^{{{R}^{*}}}}-1}\right){{R}^{*}}}}{{{2^{{{R}^{*}}}}\ln\left(2\right){{R}^{*}}+{2^{{{R}^{*}}}}-1}}}\right)^{\frac{{{2^{-L}}}}{{1-{2^{-L}}}}}}$
$\displaystyle=\frac{{{\theta_{\infty}}}}{4}\sqrt{\frac{{{{\cal
T}_{0}}}}{{\ln(2)\left({{2^{{{\cal T}_{0}}}}-1}\right)}}}.$ (52)

Plugging (32) and (IV-C2) into (49), it then follows that

$\max\left\\{{\frac{{{\theta_{\infty}}}}{4}\sqrt{\frac{{{\kappa_{\infty}}{{\cal
T}_{0}}}}{{\ln(2)\left({{2^{{{\cal
T}_{0}}}}-1}\right)}}},\frac{{{\kappa_{\infty}}{\theta_{\infty}}{{\cal
T}_{0}}}}{{4\left({{2^{{{\cal
T}_{0}}}}-1}\right)}}}\right\\}\leq\eta_{IR,\infty}^{*}\\\
\leq\frac{{{\kappa_{\infty}}{\theta_{\infty}}}}{4}\sqrt{\frac{{{{\cal
T}_{0}}}}{{\ln(2)\left({{2^{{{\cal T}_{0}}}}-1}\right)}}}.$ (53)

Clearly, both the lower and upper bounds of $\eta_{IR,\infty}^{*}$ in (53) are
decreasing functions of the the goodput threshold ${\cal T}_{0}$. In addition,
the following inequality of their limits holds

$\max\left\\{{\frac{{\theta_{\infty}}}{4}\sqrt{\frac{{{\kappa_{\infty}}}}{{\ln(2)}}}\sqrt{\mathop{\lim}\limits_{{{\mathcal{T}}_{0}}\to
0}\frac{{{\mathcal{T}_{0}}}}{{{2^{{\mathcal{T}_{0}}}}-1}}},\frac{{{\kappa_{\infty}}}{\theta_{\infty}}}{4}\mathop{\lim}\limits_{{{\cal
T}_{0}}\to
0}\frac{{{\mathcal{T}_{0}}}}{{{2^{{\mathcal{T}_{0}}}}-1}}}\right\\}\\\
\leq\mathop{\lim}\limits_{{{\cal T}_{0}}\to
0}\eta_{IR,\infty}^{*}\leq\frac{{{\kappa_{\infty}}}{\theta_{\infty}}}{{4\sqrt{\ln(2)}}}\sqrt{\mathop{\lim}\limits_{{{\cal
T}_{0}}\to 0}\frac{{{\mathcal{T}_{0}}}}{{{2^{{\mathcal{T}_{0}}}}-1}}}.$ (54)

Noticing that $\mathop{\lim}\limits_{{\mathcal{T}_{0}}\to
0}\frac{{{\mathcal{T}_{0}}}}{{{2^{{\mathcal{T}_{0}}}}-1}}=\frac{1}{\ln\left(2\right)}$,
we have

$\max\left\\{{\frac{{\sqrt{{\kappa_{\infty}}}}{\theta_{\infty}}}{{4\ln(2)}},\frac{{{\kappa_{\infty}}}{\theta_{\infty}}}{{4\ln(2)}}}\right\\}\leq\mathop{\lim}\limits_{{{\cal
T}_{0}}\to
0}\eta_{IR,\infty}^{*}\leq\frac{{{\kappa_{\infty}}}{\theta_{\infty}}}{{4\ln(2)}}.$
(55)

Since ${\kappa_{\infty}}>1$ and using squeeze theorem, it follows

$\mathop{\lim}\limits_{{{\cal T}_{0}}\to
0}\eta_{IR,\infty}^{*}=\frac{{{\kappa_{\infty}}}{\theta_{\infty}}}{{4\ln(2)}}.$
(56)

Summarizing the results in (53), (56) and Property 1 finally leads to the
following result of the optimal energy efficiency of HARQ-IR.

######  Theorem 6.

Under time-correlated Rayleigh fading channels, the optimal energy efficiency
of HARQ-IR with outage and goodput constraints is upper bounded by

${\eta_{IR,L}^{*}}\leq{\eta_{IR,\infty}^{*}}\leq\frac{{{\kappa_{\infty}}{\theta_{\infty}}}}{4}\sqrt{\frac{{{{\cal
T}_{0}}}}{{\ln(2)\left({{2^{{{\cal T}_{0}}}}-1}\right)}}}.$ (57)

Particularly, for Rayleigh fading channels with unit channel gains
${\sigma_{1}}^{2}=\cdots={\sigma_{L}}^{2}=1$, $\theta_{\infty}\leq 1$ and the
maximal energy efficiency of HARQ-IR with outage and goodput constraints
follows

${\eta_{IR,\infty}^{*}}\leq\frac{{{\kappa_{\infty}}{\theta_{\infty}}}}{4}\sqrt{\frac{{{{\cal
T}_{0}}}}{{\ln(2)\left({{2^{{{\cal
T}_{0}}}}-1}\right)}}}\leq\frac{\kappa_{\infty}}{4\ln(2)}\triangleq{\eta_{IR,\infty}^{\max}}.$
(58)

Comparing the results in Theorems 4 and 6, we can conclude that HARQ-CC and
HARQ-IR can reach the same maximal energy efficiency of
$\frac{\kappa_{\infty}}{4\ln(2)}$.

##  V Numerical Results and Discussions

The performance of our proposed energy-efficient optimization is tested and
the impact of various parameters is discussed in this section. Unless
otherwise stated, the results are provided over time-correlated Rayleigh
fading channels with time correlation $\rho=0.5$ and unit channel gains
${\sigma_{1}}^{2}=\cdots={\sigma_{L}}^{2}=1$.

###  V-A Numerical Verification

In our design, asymptotic outage probability is adopted to enable the
derivation of closed-form solutions for the optimal transmission powers and
the optimal rate. To validate the correctness of our design, the optimal
energy efficiency achieved by our design is compared with that achieved
through numerical exhaustive search based on the exact outage probabilities
derived in [28, 29, 30]. The results versus the outage tolerance $\varepsilon$
and the minimum goodput requirement $\mathcal{T}_{0}$ are shown in Figs. 1 and
2, respectively. It is readily observed from Fig. 1 that there is an excellent
match between the optimal energy efficiencies based on the asymptotic and the
exact outage probabilities when $\varepsilon\leq 10^{-2}$, because the outage
probability can be well approximated by the asymptotic outage probability (4)
under a _low outage_ (or _high SNR_). This result demonstrates that the
effectiveness of the proposed close-form solution holds true when outage
constraint is strict, i.e., the allowable outage probability is small, which
is usually true in practical applications since the high QoS is generally
required in practice. The effectiveness of our design is further demonstrated
by the results in Fig. 2 where the optimal energy efficiency achieved by our
design coincides well with that achieved based on the exact outage probability
no matter how large the minimum goodput. Moreover, it is also shown in Fig. 1
that the optimal energy efficiency $\eta_{L}^{*}$ is an increasing function of
$\varepsilon$. For instance, $\eta_{L}^{*}$ of HARQ-IR scheme is increased by
about $0.15$ bits/J when the outage tolerance is relaxed from $10^{-3}$ to
$10^{-1}$. However, as shown in Fig. 2, the optimal energy efficiency
decreases with the increase of the minimum goodput requirement
$\mathcal{T}_{0}$, which verifies our result that the maximal energy
efficiency is achieved when no goodput constraint is considered, i.e.,
$\mathcal{T}_{0}\to 0$. In addition, both Figs. 1 and 2 show that HARQ-IR
performs the best in terms of optimal energy efficiency, while Type I HARQ
provides the worst performance without exploiting the erroneously received
sub-codewords.

Figure 1: The optimal energy efficiency versus the outage tolerance with $L=2$
and $\mathcal{T}_{0}=2$ bps/Hz. Figure 2: The optimal energy efficiency versus
the minimum goodput requirement with $L=2$ and $\varepsilon=10^{-2}$.

To further demonstrate the superiority of our design, our energy-efficient
design is compared with uniform power design and their optimal energy
efficiencies are shown in Fig. 3. Notice that in the uniform power design, the
transmission powers at different HARQ rounds are set equal, i.e.,
$P_{1}=\cdots=P_{L}=P$, and the transmission power $P$ and the rate are then
optimally chosen to maximize the energy efficiency under the same QoS
constraints as our design. Clearly, our design can achieve a significant
enhancement of energy efficiency over the uniform power design under a
stringent outage constraint. Interestingly, for Type I HARQ, the energy
efficiency achieved by the proposed design converges to that of the uniform
power design under loose outage constraint, i.e., $\varepsilon\to 1$. However,
there is a non-negligible gap between the energy efficiencies of the proposed
design and the uniform power design for both HARQ-CC and HARQ-IR. This is due
to the fact that the erroneously received packets are directly discarded by
Type I HARQ and thus the previously consumed resources such as transmission
powers are not exploited. To overcome this problem, both HARQ-CC and HARQ-IR
combine the currently received packet with the erroneously received packets
for reutilizing these resources. Particularly for HARQ-CC and HARQ-IR, the
concluded results are totally different from [36], where the uniform power
allocation can offer similar performance as the optimal solution for spectral
efficiency maximization. This essentially stems from the difference between
the energy and the spectral efficiencies. The energy efficiency is in fact a
ratio of spectral efficiency to the long term average power [37], so it would
be significantly affected by transmission powers. As opposed to [18, Fig. 3],
[36] where the spectral efficiency maximization is targeted and only a slight
spectral efficiency improvement can be achieved via optimal power allocation,
the adaptation of transmission powers appears to be very crucial to the energy
efficiency maximization.

Figure 3: Comparison of the proposed design with uniform power design for
various HARQ schemes by setting $\mathcal{T}_{0}=2$ bps/Hz and $L=5$.

###  V-B Impact of System Parameters

To investigate the impact of various system parameters, the optimal energy
efficiency versus the maximal number of transmissions is shown in Fig. 4. It
is clear that the optimal energy efficiency $\eta_{L}^{*}$ is an increase
function of $L$ and is upper bounded. The maximal energy efficiency is
achieved when $L\to\infty$, which is named as lossless HARQ for convenience.
This is consistent with our analytical results in Theorems 3, 4 and 6. Since
the energy efficiency is upper bounded, when $L$ is large, further increase of
the maximal number of transmission can only contribute a very limited
improvement on the energy efficiency, but it would definitely lead to the
reduction of spectral efficiency since more transmissions are conducted to
convey the same information. Thus there would exist a tradeoff between the
energy efficiency and the spectral efficiency, which will be particularly
discussed later. Herein, it should be pointed out that $\theta_{\infty}$ is a
function of $\rho$, and can be approximated as $\theta_{L}$ with high accuracy
by choosing $L=20$.

Figure 4: The optimal energy efficiency versus the maximal number of
transmissions with $\mathcal{T}_{0}=2$ bps/Hz and $\varepsilon=10^{-4}$.

To test the effect of the outage tolerance $\varepsilon$, the optimal energy
efficiency $\eta_{L}^{*}$ versus $\varepsilon$ is plotted in Fig. 5. It is
found that the optimal energy efficiency first increases as $\varepsilon$
increases up to $0.06$, while it remains constant when $\varepsilon$ increases
further. This result can be well explained by Theorem 2 and Theorem 5. More
precisely, when $\varepsilon>2^{-L}$, the optimal rate is independent of the
outage tolerance $\varepsilon$ and thus the optimal energy efficiency becomes
irrelevant to $\varepsilon$. In the case of $L=4$, we have
$2^{-L}=2^{-4}\approx 0.06$. Therefore, when $\varepsilon>0.06$, the optimal
energy efficiency would become constant in this case.

Figure 5: The optimal energy efficiency versus the outage tolerance with
$\mathcal{T}_{0}=2$ bps/Hz and $L=4$.

The effect of the minimal goodput requirement $\mathcal{T}_{0}$ is then shown
in Fig. 6. It can be seen that the optimal energy efficiency $\eta_{L}^{*}$
decreases with the increase of $\mathcal{T}_{0}$. For all the three HARQ
schemes, the maximal energy efficiency is achieved when $\mathcal{T}_{0}\to
0$. Moreover, HARQ-CC and HARQ-IR can achieve the same maximal energy
efficiency of $\frac{\kappa_{\infty}}{4\ln(2)}$ when $\mathcal{T}_{0}\to 0$.
However, the superiority of HARQ-IR over HARQ-CC becomes more significant as
$\mathcal{T}_{0}$ increases.

Figure 6: The optimal energy efficiency versus the minimum goodput requirement
with $L=10$ and $\varepsilon=10^{-2}$.

The impact of the time correlation $\rho$ on the optimal energy efficiency
$\eta_{L}^{*}$ is finally investigated and the results are shown in Fig. 7. As
proved in Appendix C, $\theta_{L}$ is a decreasing function of the time
correlation $\rho$ and thus the time correlation has a detrimental impact on
the energy efficiency. This can be verified by the results in Fig. 7. It can
also been seen that there is a significant drop of the energy efficiency when
$\rho>0.5$, which is consistent with the result in [38] that time correlation
lower than $0.5$ would not lead to a significant performance degradation.

Figure 7: Effect of time correlation with $L=5$, $\mathcal{T}_{0}=2$ bps/Hz
and $\varepsilon=10^{-4}$.

###  V-C Spectral Efficiency

As defined in [16, 14], spectral efficiency of HARQ schemes signifies the
average number of successfully delivered bits per channel use (bps/Hz) and is
defined as

${\xi_{L}}=\frac{{R(1-{p_{out,L}})}}{{\sum\nolimits_{l=0}^{L-1}{{p_{out,l}}}}}.$
(59)

Generally, the optimizations of the energy efficiency and the spectral
efficiency are two conflicting goals. This conflict can be clearly observed
from the results of energy and spectral efficiencies achieved by our design as
shown in Fig. 8. It is shown that the spectral efficiency ${\xi_{L}}$
decreases while the optimal energy efficiency $\eta_{L}^{*}$ increases with
the increase of $L$ and higher energy efficiency is achieved at the cost of
spectral efficiency degradation. In addition, with our energy-efficient
optimization, the HARQ-IR scheme can achieve the highest energy efficiency but
with the lowest spectral efficiency since our design is targeting at energy
efficiency maximization. Notice that this result is different from that in [9,
10, 5] where the design objective is to maximize the spectral efficiency. To
further illustrate the tradeoff between the energy and spectral efficiencies
for the three HARQ schemes, the minimum goodput requirement $\mathcal{T}_{0}$
is varied from $0.5$bps/Hz to $10$bps/Hz and the corresponding efficiencies
are shown in Fig. 9. It is clear from Figs. 8 and 9 that HARQ-CC can always
achieve spectral and energy efficiencies in between those of Type I HARQ and
HARQ-IR given the same objective and constraints for optimization. In other
words, HARQ-CC can achieve a better tradeoff between the energy and spectral
efficiencies than the other schemes given the same objective and constraints
for optimization.

Figure 8: Spectral and energy efficiencies of various HARQ schemes with
$\mathcal{T}_{0}=2$ bps/Hz and $\varepsilon=10^{-4}$. Figure 9: Tradeoff
between the optimal energy efficiency and spectral efficiency with
$\varepsilon=10^{-4}$ and $L=5$.

To demonstrate the generality of the above results, a practical long term
evolution (LTE) system with a coding rate of $1/2$ and a modulation scheme of
16QAM is also taken as an example for simulations. The system level simulation
is conducted by using LTE system toolbox for MATLAB. Note that since
${{{\mathcal{T}_{0}}}<R\leq\frac{{{{\mathcal{T}_{0}}}}}{{1-\Delta}}}$ and
$\Delta$ is small, ${\mathcal{T}_{0}}$ can be approximated as
${\mathcal{T}_{0}}\approx R=1/2\times\log_{2}16=2$ bps/Hz. Hereby, by using
the closed-form solution of problem (9) with ${\mathcal{T}_{0}}=2$ bps/Hz,
Fig. 10 shows the spectral and the energy efficiencies of the three HARQ
schemes against the outage constraint under LTE system settings. Similar
results can be observed as Fig. 8, that is, HARQ-CC can strike the best
balance between the energy and the spectral efficiencies.

Figure 10: Spectral and energy efficiencies for the three HARQ schemes in LTE
system with a coding rate of $1/2$, a modulation scheme of 16QAM,
$N_{b}=12960$ bits and $L=4$.

##  VI Conclusion

Energy-efficient optimization for HARQ schemes has been proposed in this
paper. Different from the prior designs, widely occurred time-correlated
fading channels and practical QoS constraints are considered in the optimal
design to maximize the energy efficiency. The optimal transmission powers and
the optimal rate have been derived in closed-forms and have further enabled
the analysis of the maximal energy efficiencies of various HARQ schemes. It
has been found that with low outage constraint, the maximal energy
efficiencies of Type I HARQ and HARQ-CC/IR are $\frac{1}{4\ln 2}$ bits/J and
$\frac{\kappa_{\infty}}{4\ln 2}$ bits/J, respectively. Our numerical results
have also shown that the energy efficiency improvement is achieved in
sacrifice of the spectral efficiency, because the spectral efficiency and the
energy efficiency are two conflicting objectives. In addition, HARQ-CC can
achieve a better tradeoff between the energy efficiency and the spectral
efficiency than Type I HARQ and HARQ-IR.

##  Appendix A Proof of Theorem 1

It is clear from (12) that the target outage probability $\alpha$ should be
nonzero, i.e., $\alpha\neq 0$. Together with the first constraint in (17), we
have $0<\alpha\leq\varepsilon$. Further considering the second constraint
$R(1-\alpha)\geq\mathcal{T}_{0}$, the transmission rate should satisfy
$R>\mathcal{T}_{0}$, otherwise the second constraint cannot be satisfied and
the problem (17) is then infeasible. Then the problem (17) can be reformulated
as

$\begin{array}[]{*{20}{l}}{\mathop{{\rm{max}}}\limits_{\alpha}}&{f\left(\alpha\right)}\\\
{{\rm{subject\,to}}}&{0<\alpha\leq\min\left\\{\varepsilon,1-\frac{{{\mathcal{T}_{0}}}}{R}\right\\}}.\end{array}$
(60)

The Lagrangian associated with (60) can be written as

${\mathcal{L}_{1}}\left({\alpha,u}\right)=f(\alpha)+u\left({\alpha-\min\left\\{\varepsilon,1-\frac{{{\mathcal{T}_{0}}}}{R}\right\\}}\right)-\nu\alpha,$
(61)

where $u$ and $\nu$ refer to Lagrange multipliers. The optimal solution to
(60) should satisfy the Karush-Khun-Tucker (KKT) conditions as

${\left.{\frac{{\partial{{\cal
L}_{1}}}}{{\partial\alpha}}}\right|_{\left({{\alpha^{*}},{u^{*}},\nu^{*}}\right)}}=f^{\prime}({\alpha^{*}})+{u^{*}}-\nu^{*}=0,$
(62)
${u^{*}}\left({{\alpha^{*}}-\min\left\\{{\varepsilon,1-\frac{{{\mathcal{T}_{0}}}}{R}}\right\\}}\right)=0,$
(63) $\nu^{*}\alpha^{*}=0,$ (64)

$0<{\alpha^{*}}\leq\min\left\\{{\varepsilon,1-\frac{{{\mathcal{T}_{0}}}}{R}}\right\\}$
and ${u^{*}},\nu^{*}\geq 0$. Since $\alpha^{*}>0$, (64) implies $\nu^{*}=0$.
Substituting $\nu^{*}=0$ into (62) yields

$\displaystyle{u^{*}}$
$\displaystyle=-f^{\prime}({\alpha^{*}})=\frac{{{{\alpha^{*}}^{\frac{1}{{{2^{L}}-1}}}}}}{{{2^{L}}-1}}\left({{{{\alpha^{*}}^{-1}}-{2^{L}}}}\right).$
(65)

Note that the Lagrange multiplier ${{u^{*}}}$ is either larger than or equal
to $0$. Suppose that ${{u^{*}}}=0$, it follows from (65) that
$\alpha^{*}={2^{-L}}$. Combining this with (63) and the constraint
$0<{\alpha^{*}}\leq\min\left\\{{\varepsilon,1-\frac{{{\mathcal{T}_{0}}}}{R}}\right\\}$,
we have
$\alpha^{*}={2^{-L}}\leq\min\left\\{{\varepsilon,1-\frac{{{{\mathcal{T}_{0}}}}}{R}}\right\\}$.
If ${{u^{*}}}>0$, on the other hand, (63) implies that
${{\alpha^{*}}=\min\left\\{{\varepsilon,1-\frac{{{{\mathcal{T}_{0}}}}}{R}}\right\\}}$.
Besides, combining ${u^{*}}>0$ with (65) gives ${{\alpha^{*}}^{-1}}-{2^{L}}>0$
so that
${{\alpha^{*}}=\min\left\\{{\varepsilon,1-\frac{{{{\mathcal{T}_{0}}}}}{R}}\right\\}}<2^{-L}$.
To summarize, the optimal target outage probability is therefore given by
${{\alpha^{*}}=\min\left\\{{\varepsilon,1-\frac{{{{\mathcal{T}_{0}}}}}{R},2^{-L}}\right\\}}$
for $R>\mathcal{T}_{0}$.

By defining $\Delta=\min\left\\{{\varepsilon,{2^{-L}}}\right\\}$, the optimal
target outage probability ${\alpha^{*}}$ can be rewritten as

${\alpha^{*}}=\min\left\\{{\Delta,1-\frac{{{{\mathcal{T}_{0}}}}}{R}}\right\\}=\left({1-\frac{{{{\mathcal{T}_{0}}}}}{R}}\right)\times\\\
\left({\chi\left({R-{{\mathcal{T}_{0}}}}\right)-\chi\left({R-\frac{{{{\mathcal{T}_{0}}}}}{{1-\Delta}}}\right)}\right)+\Delta\chi\left({R-\frac{{{{\mathcal{T}_{0}}}}}{{1-\Delta}}}\right),$
(66)

where $\chi(\cdot)$ denotes the indicator function as (19). Plugging (66) into
$f(\alpha)={{(1-{\alpha}){\alpha}^{\frac{1}{{{2^{L}}-1}}}}}$ finally leads to
(1).

##  Appendix B Proof of Theorem 2

The Lagrangian associated with (25) is written as

${\mathcal{L}_{2}}\left({R,v,w}\right)=\Phi\left(R\right)+v\left({{{\mathcal{T}_{0}}}-R}\right)+w\left({R-\frac{{{{\mathcal{T}_{0}}}}}{{1-\Delta}}}\right),$
(67)

where $v$ and $w$ represent the Lagrange multipliers. The optimal solution to
(25) should satisfy the KKT conditions as

${\left.{\frac{{\partial{\mathcal{L}_{2}}}}{{\partial
R}}}\right|_{\left({{R^{*}},{v^{*}},{w^{*}}}\right)}}=\Phi^{\prime}\left(R^{*}\right)-v^{*}+w^{*}=0,$
(68) $v^{*}\left({{{\mathcal{T}_{0}}}-R^{*}}\right)=0,$ (69)
$w^{*}\left({R^{*}-\frac{{{{\mathcal{T}_{0}}}}}{{1-\Delta}}}\right)=0,$ (70)

${{{\mathcal{T}_{0}}}<R^{*}\leq\frac{{{{\mathcal{T}_{0}}}}}{{1-\Delta}}}$ and
$v^{*},w^{*}\geq 0$. Herein, the first derivative of $\Phi\left(R\right)$ with
respect to $R$ is

$\displaystyle\Phi^{\prime}\left(R\right)$
$\displaystyle=\frac{1}{{{R^{1-c}}{{\left({R-{{\mathcal{T}_{0}}}}\right)}^{c+1}}}}\varphi\left(R\right),$
(71)

where
$\varphi\left(R\right)=\ln\left(2\right)R\left({R-{{\mathcal{T}_{0}}}}\right){2^{R}}-c{{\mathcal{T}_{0}}}\left({{2^{R}}-1}\right)$.
Noticing ${{{\mathcal{T}_{0}}}}<R^{*}$, it follows from (69) that $v^{*}=0$.
Clearly from (68), $w^{*}$ is thus given by

$w^{*}=-\Phi^{\prime}\left(R^{*}\right)=-\frac{1}{{{{R^{*}}^{1-c}}{{\left({R^{*}-{{\mathcal{T}_{0}}}}\right)}^{c+1}}}}\varphi\left(R^{*}\right).$
(72)

Note that $w^{*}$ is either larger than or equal to $0$. If $w^{*}>0$, it
follows from (70) that ${R^{*}}=\frac{{{{\mathcal{T}_{0}}}}}{{1-\Delta}}$.
Together with (72), we have
$\varphi\left(R^{*}\right)=\varphi\left({\frac{{{{\mathcal{T}_{0}}}}}{{1-\Delta}}}\right)<0$.
On the other hand, if $w^{*}=0$, following from (72), we have
$\varphi\left(R^{*}\right)=0$. By defining $\varphi^{-1}\left(y\right)$ as the
inverse function of $\varphi$, we have ${R^{*}}=\varphi^{-1}(0)$. Since
$R>{\mathcal{T}_{0}}$, the following inequality holds

$\varphi^{\prime}\left(R\right)=\ln\left(2\right){2^{R}}\left({\ln\left(2\right)R\left({R-{{\mathcal{T}_{0}}}}\right)+2R-\left({1+c}\right){{\mathcal{T}_{0}}}}\right)>0.$
(73)

It means that $\varphi\left(R\right)$ is a monotonically increasing function
of the rate $R$. Together with
$R^{*}\leq\frac{{{{\mathcal{T}_{0}}}}}{{1-\Delta}}$, we have
$\varphi\left(\frac{{{{\mathcal{T}_{0}}}}}{{1-\Delta}}\right)\geq\varphi\left(R^{*}\right)=0$.
Then the optimal solution $R^{*}$ under the above two cases can be summarized
as

${R^{*}}=\left\\{{\begin{array}[]{*{20}{c}}{\frac{{{{\mathcal{T}_{0}}}}}{{1-\Delta}},}&{\varphi\left({\frac{{{{\mathcal{T}_{0}}}}}{{1-\Delta}}}\right)<0}\\\
{{\varphi^{-1}(0)},}&{\varphi\left({\frac{{{{\mathcal{T}_{0}}}}}{{1-\Delta}}}\right)\geq
0.}\end{array}}\right.$ (74)

Noticing that $\varphi\left(R\right)$ is an increasing function of $R$, (74)
can be rewritten as (26).

Clearly from (74), the optimal rate $R^{*}$ is determined by the sign of
$\varphi\left({\frac{{{{\mathcal{T}_{0}}}}}{{1-\Delta}}}\right)$. Based on its
definition, $\varphi\left({\frac{{{{\mathcal{T}_{0}}}}}{{1-\Delta}}}\right)$
can be explicitly expressed as

$\displaystyle\varphi\left({\frac{{{{\mathcal{T}_{0}}}}}{{1-\Delta}}}\right)$
$\displaystyle={2^{\frac{{{{\mathcal{T}_{0}}}}}{{1-\Delta}}}}{{\mathcal{T}_{0}}}\left({{\ln\left(2\right)\frac{{\Delta{{\mathcal{T}_{0}}}}}{{{{\left({1-\Delta}\right)}^{2}}}}+c\left({{2^{-\frac{{{{\mathcal{T}_{0}}}}}{{1-\Delta}}}}-1}\right)}}\right).$
(75)

By using the inequality ${\rm 2}^{-x}\geq 1-x\ln 2$, we have

$\varphi\left({\frac{{{{\cal
T}_{0}}}}{{1-\Delta}}}\right)\geq{2^{\frac{{{{\cal
T}_{0}}}}{{1-\Delta}}}}{{\cal T}_{0}}\ln\left(2\right)\frac{{{{\cal
T}_{0}}}}{{1-\Delta}}\left({\frac{\Delta}{{1-\Delta}}-c}\right).$ (76)

When $\varepsilon\geq 2^{-L}$, $\Delta=\min\\{\varepsilon,2^{-L}\\}=2^{-L}$
and the right hand side of (76) equals to 0 since $c=\frac{1}{{{2^{L}}-1}}$.
In other words, when $\varepsilon\geq 2^{-L}$, we have
$\varphi\left({\frac{{{{\cal T}_{0}}}}{{1-\Delta}}}\right)\geq 0$. From (74),
the optimal transmission rate is thus given by $R^{*}=\varphi^{-1}(0)$. The
proof is then completed.

##  Appendix C Decreasing Monotonicity of $\theta_{L}$

From (27), the optimal energy efficiency can be rewritten as

${{\eta}_{I,L}^{*}}=\mathcal{K}{\theta_{L}}.$ (77)

where
$\mathcal{K}={{\psi}}{{{{\mathcal{T}_{0}}}}}\frac{{{{\left({{R^{*}}-{{\mathcal{T}_{0}}}}\right)}^{c}}}}{{{R^{*}}^{c}\left({{2^{{R^{*}}}}-1}\right)}}$
and is independent of $\rho$. Accordingly, the monotonicity of $\theta_{L}$
with respect to $\rho$ is the same as that of ${\eta_{I,L}^{*}}$. With the
same monotonicity, we can prove the decreasing monotonicity of $\theta_{L}$
through the analysis of the monotonicity of ${\eta_{I,L}^{*}}$ as follows.

Specifically, as proved in [30, Lemma 3], the asymptotic outage probability is
an increasing function of $\rho$. Thus it follows from (8) that the energy
efficiency $\eta_{I,L}$ is a decreasing function of $\rho$. To proceed with
the proof, we assume two distinct time correlation coefficients $\rho_{1}$ and
$\rho_{2}$ with $\rho_{1}>\rho_{2}$. Denote the optimal solution to the
original problem (9) under the time correlation of $\rho_{1}$ and the
corresponding optimal energy efficiency as
$({P_{1}}^{*},\cdots,{P_{L}}^{*},R^{*})$ and
$\eta_{I,L,\rho_{1}}^{*}=\eta_{I,L,\rho_{1}}({P_{1}}^{*},\cdots,{P_{L}}^{*},R^{*})$,
respectively. When the channel time correlation is reduced to $\rho_{2}$, the
solution $({P_{1}}^{*},\cdots,{P_{L}}^{*},R^{*})$ is still feasible to the
problem (9) since it satisfies all the constraints in (9). With the decreasing
monotonicity of $\eta_{I,L}$ with respect to $\rho$, we have
$\eta_{I,L,\rho_{1}}({P_{1}}^{*},\cdots,{P_{L}}^{*},R^{*})<\eta_{I,L,\rho_{2}}({P_{1}}^{*},\cdots,{P_{L}}^{*},R^{*})$.
Moreover, when the time correlation of $\rho_{2}$ is considered, the
optimization in (9) will definitely lead to the optimal energy efficiency
$\eta_{I,L,\rho_{2}}^{*}$ not lower than
$\eta_{I,L,\rho_{2}}({P_{1}}^{*},\cdots,{P_{L}}^{*},R^{*})$, i.e.,
$\eta_{I,L,\rho_{2}}^{*}\geq\eta_{I,L,\rho_{2}}({P_{1}}^{*},\cdots,{P_{L}}^{*},R^{*})$.
Therefore, we have $\eta_{I,L,\rho_{2}}^{*}>\eta_{I,L,\rho_{1}}^{*}$. It means
that the optimal energy efficiency ${{\eta}_{I,L}^{*}}$ is a monotonically
decreasing function of the time correlation $\rho$. Combining this
monotonicity with (77), we can conclude that $\theta_{L}$ is a decreasing
functions of $\rho$ and the proof is finally completed.

##  Appendix D Proof of Property 1

Consider two different maximal numbers of transmissions $L_{1}$ and $L_{2}$
with $L_{1}\leq L_{2}$. We denote the optimal transmission powers and rate and
the corresponding optimal energy efficiency under the case with maximal
$L_{1}$ transmissions as
$\mathcal{S}_{1}^{*}=(P_{1}^{*},\cdots,P_{L_{1}}^{*},R^{*})$ and
$\eta_{L_{1}}^{*}$, respectively. Now when the maximal number of transmissions
increases to $L_{2}$, we construct a solution of transmission powers and rate
as
$\mathcal{S}_{2}=(P_{1}^{*},\cdots,P_{L_{1}}^{*},P_{L_{1}+1},\cdots,P_{L_{2}},R^{*})$
where $P_{L_{1}+1}=\cdots=P_{L_{2}}=0$. Clearly from the definition of outage
probability in (3), we have ${p_{out,{L_{1}}}}=\cdots={p_{out,{L_{2}}}}$. When
the maximal number of transmissions is $L_{2}$, $\mathcal{S}_{2}$ constitues a
feasible point of the problem (9), since $\mathcal{S}_{2}$ satisfies both
outage and goodput constraints. Denote its corresponding energy efficiency as
$\eta_{L_{2}}(\mathcal{S}_{2})$. We directly have
$\eta_{L_{2}}(\mathcal{S}_{2})=\eta_{L_{1}}^{*}$. Through the optimization in
(9), when the maximal number of transmissions is $L_{2}$, we can definitily
find the optimal energy efficiency $\eta_{L_{2}}^{*}$ not lower than
$\eta_{L_{2}}(\mathcal{S}_{2})$, i.e.,
$\eta_{L_{2}}^{*}\geq\eta_{L_{2}}(\mathcal{S}_{2})$. It then follows
$\eta_{L_{2}}^{*}\geq\eta_{L_{1}}^{*}$, which means that the optimal energy
efficiency is non-decreasing function of the maximum number of transmissions
and thus completes the proof.

##  Appendix E Proof of Theorem 3

###  E-A Proof of (28)

Taking the limit $L\to\infty$ of both sides of (27) yields

$\displaystyle{\eta_{I,\infty}^{*}}$
$\displaystyle=\mathop{\lim}\limits_{L\to\infty}{{\psi{\theta_{L}}}}{{{{\mathcal{T}_{0}}}}}\frac{{{{\left({{R^{*}}-{{\mathcal{T}_{0}}}}\right)}^{c}}}}{{{R^{*}}^{c}\left({{2^{{R^{*}}}}-1}\right)}}$
$\displaystyle=\mathop{\lim}\limits_{L\to\infty}\frac{{\left({2^{\frac{L}{{1-{2^{-L}}}}-2}}\right){\theta_{L}}{{\mathcal{T}_{0}}}}}{{{{2^{L}}-1}}}\mathop{\lim}\limits_{L\to\infty}\frac{{{{\left({{R^{*}}-{{\mathcal{T}_{0}}}}\right)}^{c}}}}{{{R^{*}}^{c}\left({{2^{{R^{*}}}}-1}\right)}}$
$\displaystyle=\frac{{{\theta_{\infty}}{{\cal
T}_{0}}}}{4}\mathop{\lim}\limits_{L\to\infty}\frac{{{{\left({{R^{*}}-{{\cal
T}_{0}}}\right)}^{c}}}}{{{R^{*}}^{c}\left({{2^{{R^{*}}}}-1}\right)}},$ (78)

where
$\theta_{\infty}\triangleq{\mathop{\lim}\limits_{L\to\infty}{\theta_{L}}}$ and
the existence of $\theta_{\infty}$ will be proved in Appendix E-B. As
$L\to\infty$, we surely have $\Delta=\min\\{\varepsilon,2^{-L}\\}=2^{-L}$.
Recalling
${{{\mathcal{T}_{0}}}<R^{*}\leq\frac{{{{\mathcal{T}_{0}}}}}{{1-\Delta}}}$,
$\mathop{\lim}\limits_{L\to\infty}{R^{*}}={{\mathcal{T}_{0}}}$ follows by
applying squeeze theorem. Plugging this into (E-A) along with
$\mathop{\lim}\limits_{L\to\infty}{c}={0}$ leads to

$\displaystyle{\eta_{I,\infty}^{*}}$
$\displaystyle=\frac{\theta_{\infty}{{{\mathcal{T}_{0}}}}}{{4\left({{2^{{{\mathcal{T}_{0}}}}}-1}\right)}}{{\mathop{\lim}\limits_{L\to\infty}{{\left({{R^{*}}-{{\mathcal{T}_{0}}}}\right)}^{c}}}}.$
(79)

Noticing that both ${{R^{*}}-{{\mathcal{T}_{0}}}}$ and $c$ in (79) converge to
zero, the occurrence of the limit $0^{0}$ complicates the derivations.
Fortunately, based on Theorem 2, we have $R^{*}=\varphi^{-1}(0)$ when
$L\to\infty$ as $\varepsilon$ becomes greater than $2^{-L}$. It implies that
$R^{*}$ is a zero point of $\varphi\left({{R}}\right)$, i.e.,
$\varphi\left({{R^{*}}}\right)=0$. Following the definition of
$\varphi\left({{R}}\right)$, we then have

${R^{*}}={{\mathcal{T}_{0}}}+c{{\mathcal{T}_{0}}}\Xi(R^{*}),$ (80)

where $\Xi(R)=\frac{{{2^{R}}-1}}{{\ln\left(2\right)R{2^{R}}}}$. Now putting
(80) into (79) yields

$\displaystyle{\eta_{I,\infty}^{*}}$
$\displaystyle=\frac{{{{{\theta_{\infty}}}}{{\cal
T}_{0}}}}{{4\left({{2^{{{\cal
T}_{0}}}}-1}\right)}}{{\mathop{\lim}\limits_{L\to\infty}{{\left({c{{\cal
T}_{0}}\Xi({R^{*}})}\right)}^{c}}}}$
$\displaystyle=\frac{{{{{\theta_{\infty}}}}{{\cal
T}_{0}}}}{{4\left({{2^{{{\cal
T}_{0}}}}-1}\right)}}{{\mathop{\lim}\limits_{L\to\infty}{c^{c}}{{\left({\mathop{\lim}\limits_{L\to\infty}{{\cal
T}_{0}}\Xi({R^{*}})}\right)}^{\mathop{\lim}\limits_{L\to\infty}c}}}}$
$\displaystyle=\frac{{{{{\theta_{\infty}}}}{{\cal
T}_{0}}}}{{4\left({{2^{{{\cal T}_{0}}}}-1}\right)}}.$ (81)

The result in (28) then directly follows by combining (E-A) with Property 1.

Clearly, ${\eta_{I,\infty}^{*}}$ is a decreasing function of the goodput
threshold ${{\cal T}_{0}}$. Moreover, when
${\sigma_{1}}^{2}=\cdots={\sigma_{L}}^{2}=1$ and $\rho=0$, we have
$\theta_{L}=1$ based on its definition in (IV-A1). As shown in Appendix C,
$\theta_{L}$ is a decreasing function of the time correlation coefficient
$\rho$. Therefore, when ${\sigma_{1}}^{2}=\cdots={\sigma_{L}}^{2}=1$ with
$L\to\infty$, ${\theta_{\infty}}$ would be generally lower than or equal to
that for the case with $\rho=0$. That is, ${\theta_{\infty}}\leq 1$. Applying
this inequality into the maximal energy efficiency (E-A) and using the
decreasing monotonicity of ${\eta_{I,\infty}^{*}}$ with respect to ${{\cal
T}_{0}}$, it is easy to get (29).

###  E-B Existence of $\theta_{\infty}$

Based on the definition in (IV-A1), $\theta_{\infty}$ can be written as

${\theta_{\infty}}=\mathop{\lim}\limits_{L\to\infty}\underbrace{\prod\limits_{k=1}^{L}{{{\left({{\sigma_{k}}^{2}}\right)}^{{2^{-k}}}}}}_{\omega_{L}}\mathop{\lim}\limits_{L\to\infty}\underbrace{\prod\limits_{k=1}^{L}{{{\left({\frac{{\ell\left({k,\rho}\right)}}{{\ell\left({k-1,\rho}\right)}}}\right)}^{{2^{-k}}}}}}_{{\vartheta_{L}}}.$
(82)

Here $\omega_{L}$ can be rewritten as
$\omega_{L}=\exp\left({\sum\nolimits_{k=1}^{L}{{2^{-k}}\ln{\sigma_{k}}^{2}}}\right)$.
Since the channel power gain ${\sigma_{1}}^{2},\cdots,{\sigma_{\infty}}^{2}$
are generally bounded in practice and every absolutely convergent series is
convergent [35], the exponent
${\sum\nolimits_{k=1}^{L}{{2^{-k}}\ln{\sigma_{k}}^{2}}}$ absolutely converges
as $L\to\infty$. It follows that $\omega_{L}$ converges as $L\to\infty$, i.e.,
the limit of $\omega_{L}$ for $L\to\infty$ exists.

With respect to the term $\vartheta_{L}$, following the definition of
$\ell\left({k,\rho}\right)$ in (6), we have

$\displaystyle\frac{{\ell\left({k,\rho}\right)}}{{\ell\left({k-1,\rho}\right)}}$
$\displaystyle=\frac{{1+\left({1-{\rho^{2(k-1)}}}\right)\sum\limits_{t=1}^{k-1}{\frac{{{\rho^{2(t-1)}}}}{{1-{\rho^{2(t-1)}}}}}}}{{1+\sum\limits_{t=1}^{k-1}{\frac{{{\rho^{2(t-1)}}}}{{1-{\rho^{2(t-1)}}}}}}}\leq
1.$ (83)

Accordingly, the sequence $\\{\vartheta_{L}\\}$ decreases with $L$ and is
lower bounded as $\vartheta_{L}\geq 0$. Therefore, the limit
$\mathop{\lim}\limits_{L\to\infty}{\vartheta_{L}}$ exists. The existence of
both the limits $\mathop{\lim}\limits_{L\to\infty}{\vartheta_{L}}$ and
$\mathop{\lim}\limits_{L\to\infty}{\omega_{L}}$ then guarantees the existence
$\theta_{\infty}$ and completes the proof.

##  Appendix F Derivation of $\kappa_{\infty}$

Noticing ${{2^{-k}}}>0$ together with
${\kappa_{L}}=\prod\nolimits_{k=1}^{L}{{{{{k}}}^{{2^{-k}}}}}$, $\kappa_{L}$ is
readily found to be an increasing sequence, such that
${\kappa_{1}}<{\kappa_{2}}<\cdots<{\kappa_{\infty}}$. To prove the convergence
of the sequence ${\kappa_{L}}$, it suffices to prove that ${\kappa_{\infty}}$
is upper bounded. To this end, ${\kappa_{\infty}}$ is rewritten as
${\kappa_{\infty}}={e^{\sum\nolimits_{k=1}^{\infty}{{2^{-k}}\ln\left(k\right)}}}$.
Then applying Jensen’s inequality yields

${\kappa_{\infty}}\leq{e^{\ln\left({\sum\limits_{k=1}^{\infty}{k{2^{-k}}}}\right)}}={{\sum\limits_{k=1}^{\infty}{k{2^{-k}}}}}.$
(84)

By using [39, Eq.0.231], we have ${\kappa_{\infty}}\leq{2}$. Accordingly, the
sequence ${\kappa_{L}}$ is convergent, and the limit
${\kappa_{\infty}}=\mathop{\lim}\limits_{{L}\to\infty}\kappa_{L}$ exists. It
is difficult to evaluate ${\kappa_{\infty}}$ exactly, however
${\kappa_{\infty}}$ can be approximated as ${\kappa_{L}}$ with a large $L$. To
guarantee the approximate accuracy, $L$ should be properly chosen to achieve a
sufficiently low approximation error which is defined as

${e_{L}}\triangleq\frac{{{\kappa_{\infty}}-{\kappa_{L}}}}{{{\kappa_{\infty}}}}=1-\prod\limits_{k=L+1}^{\infty}{{k^{{2^{-k}}}}}.$
(85)

Clearly, the approximation error ${e_{L}}$ decreases with $L$, and is upper
bounded as

$\displaystyle{e_{L}}$
$\displaystyle=1-{e^{{-2^{-L}}\sum\limits_{k=L+1}^{\infty}{\frac{{{2^{-k}}}}{{{2^{-L}}}}\ln\left(k\right)}}}$
$\displaystyle\leq
1-{e^{{-2^{-L}}\ln\left({\sum\limits_{k=L+1}^{\infty}{\frac{{{2^{-k}}}}{{{2^{-L}}}}k}}\right)}}$
$\displaystyle=1-{e^{{-2^{-L}}\ln\left({{\sum\limits_{k=1}^{\infty}{{2^{-k}}k}+L\sum\limits_{k=1}^{\infty}{{2^{-k}}}}}\right)}}$
$\displaystyle=1-{\left({{L+2}}\right)^{{-2^{-L}}}},$ (86)

wherein the inequality holds by using Jensen’s inequality. Based on (F), by
setting $L=20$, the approximation error is upper bounded as ${e_{L}}\leq
2.95*10^{-6}$. Thus ${\kappa_{\infty}}$ can be approximated as
$\kappa_{\infty}\approx\kappa_{20}=1.6617$ with approximation error less than
$2.95*10^{-6}$ which is sufficiently low for practical applications.

##  Appendix G Proof of Increasing Monotonicity of ${\cal
G}_{k}\left(R\right)$

To prove the increasing monotonicity of ${\cal G}_{k}\left(R\right)$, we
resort to analyze its first order derivative of ${\cal G}_{k}\left(R\right)$
with respect to $R$, given by

$\displaystyle{\cal G}_{k}^{\prime}\left(R\right)=\frac{{{\cal
U}_{k}\left(R\right)}}{{{{\left({R{g_{k-1}}\left(R\right)}\right)}^{2}}}},$
(87)

where
$\mathcal{U}_{k}\left(R\right)=R{g_{k}}^{\prime}\left(R\right){g_{k-1}}\left(R\right)-R{g_{k-1}}^{\prime}\left(R\right){g_{k}}\left(R\right)-{g_{k-1}}\left(R\right){g_{k}}\left(R\right)$
and ${g_{k}}^{\prime}(R)$ can be further written by using Residue theorem as

$\displaystyle{g_{k}}^{\prime}\left(R\right)$
$\displaystyle=\frac{{\ln\left(2\right)}}{{2\pi{\rm{i}}}}\int\nolimits_{{a}-{\rm{i}}\infty}^{{a}+{\rm{i}}\infty}{\frac{{{2^{Rs}}}}{{{{\left({s-1}\right)}^{k}}}}ds}$
$\displaystyle=\frac{{\ln\left(2\right){{\left({R\ln\left(2\right)}\right)}^{k-1}}}}{{\left({k-1}\right)!}}{2^{R}},\,k\geq
1.$ (88)

Specifically when $k=1$, it follows from (II) that
$\mathcal{U}_{k}\left(R\right)$ reduces to
$\mathcal{U}_{1}\left(R\right)=R\ln(2)2^{R}-2^{R}+1={2^{R}}\left({R\ln(2)-1+{e^{-\ln\left(2\right)R}}}\right)$.
Clearly by using the inequality $e^{-x}\geq 1-x$, we have
$\mathcal{U}_{1}\left(R\right)\geq 0$. It means that ${\cal
G}_{k}\left(R\right)$ is an increasing function when $k=1$. On the other hand,
when $k\geq 2$, it becomes very difficult to determine the sign of
$\mathcal{U}_{k}\left(R\right)$. To proceed, we turn to analyze the first
order derivative of $\mathcal{U}_{k}\left(R\right)$ with respect to $R$ given
as

$\mathcal{U}_{k}^{\prime}\left(R\right)=R{g_{k}}^{\prime\prime}\left(R\right){g_{k-1}}\left(R\right)\\\
-\left({2{g_{k-1}}^{\prime}\left(R\right)+R{g_{k-1}}^{\prime\prime}\left(R\right)}\right){g_{k}}\left(R\right),$
(89)

where similarly to (G), the second order derivative of ${g_{k}}(R)$ can also
be written by using Residue theorem as

${g_{k}}^{\prime\prime}\left(R\right)=\frac{\left({\ln 2}\right)^{2}}{2\pi\rm
i}\int\nolimits_{{a}-{\rm{i}}\infty}^{{a}+{\rm{i}}\infty}{\frac{{s{2^{Rs}}}}{{{{\left({s-1}\right)}^{k}}}}ds}=\frac{{{{\left({\ln
2}\right)}^{2}}}}{{\left({k-1}\right)!}}\times\\\ \left({{2^{R}}{{\left({R\ln
2}\right)}^{k-1}}+\left({k-1}\right){2^{R}}{{\left({R\ln
2}\right)}^{k-2}}}\right),k\geq 2.$ (90)

Substituting (G) and (90) into (89), it follows that

$\displaystyle\mathcal{U}_{k}^{\prime}\left(R\right)$
$\displaystyle=\frac{{\ln\left(2\right){{\left({R\ln
2}\right)}^{k-2}}{2^{R}}}}{{\left({k-2}\right)!}}$
$\displaystyle\times\underbrace{\left(\begin{array}[]{l}\frac{{R\ln\left(2\right)}}{{\left({k-1}\right)}}\left({R\ln\left(2\right)+\left({k-1}\right)}\right){g_{k-1}}\left(R\right)\\\
-\left({R\ln\left(2\right)+k}\right){g_{k}}\left(R\right)\end{array}\right)}_{{\cal
J}\left(R\right)}.$ (93)

It can be proved that $\mathcal{J}(R)$ is an increasing and convex function,
since its first and second order derivatives of $\mathcal{J}(R)$ with respect
to $R$ satisfy

$\displaystyle\mathcal{J}^{\prime}\left(R\right)$
$\displaystyle=\left({\frac{{2R{{\left({\ln\left(2\right)}\right)}^{2}}}}{{\left({k-1}\right)}}+\ln\left(2\right)}\right){g_{k-1}}\left(R\right)$
$\displaystyle\quad-\ln\left(2\right){g_{k}}\left(R\right)-\frac{{\ln\left(2\right){{\left({R\ln\left(2\right)}\right)}^{k-1}}}}{{\left({k-1}\right)!}}{2^{R}},$
(94) $\displaystyle\mathcal{J}^{\prime\prime}\left(R\right)$
$\displaystyle=\frac{{2{{\left({\ln\left(2\right)}\right)}^{2}}}}{{\left({k-1}\right)}}{g_{k-1}}\left(R\right)\geq
0.$ (95)

Noticing that $\mathcal{J}^{\prime\prime}\left(R\right)\geq 0$ and
$g_{k}(0)=0$, thus $\mathcal{J}^{\prime}(R)\geq\mathcal{J}^{\prime}(0)=0$ for
$k\geq 2$. Analogously, we can sequentially prove
$\mathcal{J}(R)\geq\mathcal{J}(0)=0$,
$\mathcal{U}_{k}^{\prime}(R)\geq\mathcal{U}_{k}^{\prime}(0)=0$,
$\mathcal{U}_{k}(R)\geq\mathcal{U}_{k}(0)=0$, then we have
$\mathcal{G}_{k}^{\prime}(R)\geq 0$. Thus the increasing monotonicity of
$\mathcal{G}_{k}(R)$ follows as well for $k\geq 2$. The proof is eventually
completed.

##  Appendix H Proof of Property 2

Prior to proving (38), the following upper and lower bounds associated with
$\frac{{{g_{k}}\left(R\right)}}{{{g_{k-1}}\left(R\right)}}$ are obtained
first.

###  H-A
$\frac{{{g_{k}}\left(R\right)}}{{{g_{k-1}}\left(R\right)}}\leq\frac{{R\ln\left(2\right)}}{{k-1}}$

To prove this inequality, it suffices to show that
${\mathfrak{A}}(R)\triangleq\frac{{R\ln\left(2\right)}}{{k-1}}{g_{k-1}}\left(R\right)-{g_{k}}\left(R\right)\geq
0$. By taking the first order derivative of ${\mathfrak{A}}(R)$ with respect
to $R$, it follows that

$\displaystyle\mathfrak{A}^{\prime}\left(R\right)$
$\displaystyle=\frac{{\ln\left(2\right)}}{{k-1}}{g_{k-1}}\left(R\right)+\frac{{R\ln\left(2\right)}}{{k-1}}{g_{k-1}}^{\prime}\left(R\right)-{g_{k}}^{\prime}\left(R\right)$
$\displaystyle=\frac{{\ln\left(2\right)}}{{k-1}}{g_{k-1}}\left(R\right)\geq
0.$ (96)

where the second equality holds by using (G). Accordingly, ${\mathfrak{A}}(R)$
is an increasing function, which implies
${\mathfrak{A}}(R)\geq{\mathfrak{A}}(0)=0$. Then the upper bound of
$\frac{{{g_{k}}\left(R\right)}}{{{g_{k-1}}\left(R\right)}}$ holds.

###  H-B
$\frac{{{g_{k}}\left(R\right)}}{{{g_{k-1}}\left(R\right)}}\geq\frac{{R\ln\left(2\right)}}{k}$

To prove this lower bound, it is equivalent to prove
$\mathfrak{B}(R)\triangleq\frac{{R\ln\left(2\right)}}{k}{g_{k-1}}\left(R\right)-{g_{k}}\left(R\right)\leq
0$. To this end, taking the first order derivative of $\mathfrak{B}(R)$ with
respect to $R$, we have

$\displaystyle\mathfrak{B}^{\prime}\left(R\right)$
$\displaystyle=\frac{{\ln\left(2\right)}}{k}{g_{k-1}}\left(R\right)-\frac{{\ln\left(2\right){{\left({R\ln\left(2\right)}\right)}^{k-1}}{2^{R}}}}{{k!}}.$
(97)

Then taking the second order derivative of $\mathfrak{B}(R)$ with respect to
$R$ yields

$\displaystyle\mathfrak{B}^{\prime\prime}\left(R\right)$
$\displaystyle=-\frac{{{{\left({\ln\left(2\right)}\right)}^{2}}{{\left({R\ln\left(2\right)}\right)}^{k-1}}{2^{R}}}}{{k!}}\leq
0.$ (98)

Combining (97) and (98), it is clearly found that $\mathfrak{B}\left(R\right)$
is a decreasing function, which indicates
$\mathfrak{B}(R)\leq\mathfrak{B}(0)=0$. Thus the lower bound of
$\frac{{{g_{k}}\left(R\right)}}{{{g_{k-1}}\left(R\right)}}$ is proved.

As a consequence, substituting the upper and lower bounds of
$\frac{{{g_{k}}\left(R\right)}}{{{g_{k-1}}\left(R\right)}}$ into
$\Lambda\left(R\right)$ directly yields the upper and lower bounds of
$\Lambda\left(R\right)$, respectively, as shown in (38).

##  Appendix I Proof of (44)

For the proof, it suffices to show that
$\mathfrak{C}(R)\triangleq{g_{k}}\left(R\right)-\frac{{{2^{R}}-1}}{k}{g_{k-1}}\left(R\right)\leq
0$. Taking the first order derivative of $\mathfrak{C}(R)$ with respect to $R$
gives

$\mathfrak{C}^{\prime}\left(R\right)={2^{R}}\ln\left(2\right)\left({\frac{{{{\left({R\ln\left(2\right)}\right)}^{k-1}}}}{{\left({k-1}\right)!}}-}\right.\\\
\left.{\frac{{k-1}}{k}\frac{{\left({{2^{R}}-1}\right){{\left({R\ln\left(2\right)}\right)}^{k-2}}}}{{\left({k-1}\right)!}}-\frac{{{g_{k-1}}\left(R\right)}}{k}}\right).$
(99)

By using the lower bound of
$\frac{{{g_{k}}\left(R\right)}}{{{g_{k-1}}\left(R\right)}}$ in Appendix H-B,
namely
$\frac{{{g_{k}}\left(R\right)}}{{{g_{k-1}}\left(R\right)}}\geq\frac{{R\ln\left(2\right)}}{k}$,
it follows that

${g_{k-1}}\left(R\right)\geq\frac{{R\ln\left(2\right)}}{{k-1}}{g_{k-2}}\left(R\right)\geq\cdots\\\
\geq\frac{{{{\left({R\ln\left(2\right)}\right)}^{k-2}}}}{{\left({k-1}\right)!}}{g_{1}}\left(R\right)=\frac{{{{\left({R\ln\left(2\right)}\right)}^{k-2}}\left({{2^{R}}-1}\right)}}{{\left({k-1}\right)!}}.$
(100)

Substituting (100) into (99) yields

$\displaystyle\mathfrak{C}^{\prime}\left(R\right)$
$\displaystyle\leq\frac{{{2^{R}}\ln\left(2\right){{\left({R\ln\left(2\right)}\right)}^{k-2}}}}{{\left({k-1}\right)!}}\left({R\ln\left(2\right)-\left({{2^{R}}-1}\right)}\right)\leq
0.$ (101)

Thus $\mathfrak{C}(R)$ turns out to be a decreasing function of $R$, which
implies $\mathfrak{C}(R)\leq\mathfrak{C}(0)=0$. The proof is then completed.

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