# Differential Chaos Shift Keying-based
Wireless Power Transfer with Nonlinearities

Authors  Priyadarshi Mukherjee, Member, IEEE, Constantinos Psomas, Senior
Member, IEEE, and Ioannis Krikidis, Fellow, IEEE P. Mukherjee, C. Psomas, and
I. Krikidis are with the Department of Electrical and Computer Engineering,
University of Cyprus, Nicosia 1678 (E-mail: {mukherjee.priyadarshi, psomas,
krikidis}@ucy.ac.cy). Preliminary results of this work have been presented at
the IEEE International Conference on Acoustics, Speech, and Signal Processing
$2021$, Toronto, Canada [1]. This work was co-funded by the European Regional
Development Fund and the Republic of Cyprus through the Research and
Innovation Foundation, under the projects EXCELLENCE/0918/0377 (PRIME),
INFRASTRUCTURES/1216/0017 (IRIDA) and POST-DOC/0916/0256 (IMPULSE). It has
also received funding from the European Research Council (ERC) under the
European Union’s Horizon 2020 research and innovation programme (Grant
agreement No. 819819).

###### Abstract

In this paper, we investigate conventional communication-based chaotic
waveforms in the context of wireless power transfer (WPT). Particularly, we
present a differential chaos shift keying (DCSK)-based WPT architecture, that
employs an analog correlator at the receiver, in order to boost the energy
harvesting (EH) performance. We take into account the nonlinearities of the EH
process and derive closed-form analytical expressions for the harvested direct
current (DC) under a generalized Nakagami-$m$ block fading model. We show
that, in this framework, both the peak-to-average-power-ratio of the received
signal and the harvested DC, depend on the parameters of the transmitted
waveform. Furthermore, we investigate the case of deterministic unmodulated
chaotic waveforms and demonstrate that, in the absence of a correlator,
modulation does not affect the achieved harvested DC. On the other hand, it is
shown that for scenarios with a correlator-aided receiver, DCSK significantly
outperforms the unmodulated case. Based on this observation, we propose a
novel DCSK-based signal design, which further enhances the WPT capability of
the proposed architecture; corresponding analytical expressions for the
harvested DC are also derived. Our results demonstrate that the proposed
architecture and the associated signal design, can achieve significant EH
gains in DCSK-based WPT systems. Furthermore, we also show that, even by
taking into account the nonlinearities at the transmitter amplifier, the
proposed chaotic waveform performs significantly better in terms of EH, when
compared with the existing multisine signals.

###### Index Terms:

Differential chaos shift keying, wireless power transfer, nonlinear energy
harvesting, Nakagami-$m$ fading channel.

##  I Introduction

The wireless traffic has been growing at an explosive rate in recent years;
according to Ericsson, it is expected to increase more than five times between
$2019$ and $2025$ [2]. For applications like the Internet of Things and
massive machine-type communications, where a large number of devices are
deployed, the overall network lifetime is often affected due to limited
battery constraints. Thus, powering or charging these devices becomes critical
as well as costly. As a result, low-powered and self-sustainable next
generation wireless communication networks is an important and relevant topic
of research, for both industry and academia. In this context, based on the
advances made in recent years, wireless power transfer (WPT) can be considered
as a suitable candidate, where the devices are wirelessly powered by
harvesting energy from ambient/dedicated radio-frequency (RF) signals. This is
achieved by employing a rectifying antenna (rectenna) at the receiver that
converts the received RF signals to direct current (DC) [3]. One of the main
advantages of this process is that it can be controlled, unlike conventional
energy sources, where the available power for harvesting is itself erratic in
nature [3].

The design of efficient WPT architectures fundamentally relies on accurate
mathematical models of the harvesting circuit. While some works propose
simplified linear energy harvesting (EH) models [4], the work in [5] proposes
a piece-wise linear model based on the sensitivity and saturation levels of
the circuit. On the other hand, several nonlinear EH models have also been
proposed. The authors in [6] propose a tractable logistic saturation nonlinear
model that originates from the saturation of the output power beyond a certain
input RF power due to the diode breakdown. As this model is obtained by
fitting measurements from practical RF-based EH circuits for a given
excitation signal, it is a significantly improved version of its over-
simplified piece-wise linear counterpart. However, all these models fail to
characterize the actual working principle of the harvesting circuit. As a
result, unlike the linear, piece-wise linear, and the logistic saturation
nonlinear model, the work in [7] proposes a realistic circuit-based nonlinear
model of the harvester circuit, which relies on the EH circuit characteristics
and also enables the design of waveforms that maximize the WPT efficiency.

This model has triggered recent interests in the area of waveform design for
WPT, with an objective of making the best use of the harvester to deliver a
maximum amount of DC power. The authors in [8] show that the nonlinearity of
the rectification process causes certain waveforms, with high peak-to-average-
power-ratio (PAPR), to provide a higher DC output compared to conventional
constant-envelop sinusoidal signals. Based on this observation, some works,
e.g. [9, 10, 11, 12, 13], investigate the effect of transmitted waveforms and
modulations on WPT. The authors in [9] propose the use of multisine waveforms
for WPT due to their high PAPR. The work in [10] proposes a novel simultaneous
wireless information and power transfer (SWIPT) architecture based on the
superposition of multi-carrier unmodulated and modulated waveforms at the
transmitter. The authors in [11] provide insights on how fading and diversity
are beneficial for boosting the RF-to-DC conversion efficiency; they develop a
new form of signal design for WPT, which relies on multiple dumb antennas at
the transmitter to induce fast fluctuations of the channel. The work in [12]
proposes an asymmetric modulation scheme specifically for SWIPT, that
significantly enhances the rate-energy region as compared to its existing
symmetric counterpart. Apart from the multisine waveforms, experimental
studies demonstrate that due to their high PAPR, chaotic waveforms outperform
conventional single-tone signals in terms of WPT efficiency [14].

Chaotic signals have been used in the past decades for wireless information
security and privacy [15, 16]. However, due to their properties such as
aperiodicity and sensitivity to initial data, chaotic waveforms have been
extensively used to improve the performance of wireless digital communication
systems [17]. In this context, differential chaos shift keying (DCSK), is one
of the most widely studied chaotic signal-based communication systems [18].
DCSK is a prominent benchmark in the class of non-coherent transmitted
reference modulation techniques, which comprises of a reference and an
identical/inverted replica of the reference depending on the data transmitted.
The majority of the related works focus on the error performance of DCSK-based
systems for various scenarios [19, 20, 21]. The authors in [19] propose an
$M$-ary DCSK system, in which successive bits are converted to a symbol and
then transmitted by using the same modulation scheme. The work in [20]
investigates a general set-up for evaluating the performance of DCSK system
over various communication channels and the performance of a cooperative
diversity-aided DCSK-based system is analysed in [21].

To exploit the benefits of both DCSK and WPT, there are a some works [22, 23,
24, 25] in the literature that investigate SWIPT in a chaotic framework. In
[22], a non-coherent short-reference DCSK SWIPT architecture is proposed by
using the time switching architecture to achieve higher data rate than
conventional systems. The authors in [23] investigate SWIPT with a power-
splitting receiver for a multi-carrier (MC) chaotic framework and propose two
WPT protocols. In the first protocol, a fraction of the reference sub-carrier
power is utilized for EH and in the second protocol, a fraction of the total
sub-carrier power is used for EH. Based on the proposed MC architecture, a
chaotic carrier-index (CI) system is investigated for a basic SWIPT set-up to
further reduce the energy consumption [24]. In particular, based on the
transmission characteristics of index modulation, the proposed SWIPT scheme
exploits the inactive carriers of CI-DCSK to deliver energy by transmitting
random noise-like signals. In [25], an adaptive link selection for buffer-
aided relaying is investigated in a decode-and-forward relay-based DCSK-SWIPT
architecture. Furthermore, by taking into account the decoding cost at the
relays, two link-selection schemes based on the harvested energy, and not
channel state information (CSI), are proposed.

The above studies focus on the performance of SWIPT in a DCSK-based scenario.
Furthermore, they consider a simplified linear model for harvesting, which is
impractical. However, the specific gains in EH from chaotic signals have not
been explored. Motivated by this, in this paper, we investigate chaotic
signal-based waveform designs for WPT. To the best of our knowledge, this is
the first work that presents a complete analytical framework of DCSK-based
WPT, by also taking into account the nonlinearities of the EH process.
Specifically, the contribution of this paper is threefold.

  * •

We propose a DCSK-based WPT architecture, where an analog correlator-aided EH
circuit is employed at the receiver. Although WPT is the main focus of this
work, we consider DCSK, i.e. an information-based waveform. We analytically
derive the PAPR of the signal at the harvester as a function of the
transmitted waveform parameters. We demonstrate that the analog correlator
prior to the harvester allows us to control the PAPR of the signal at the
input of the harvester and therefore the EH performance.

  * •

Analytical expressions of the harvested DC are derived for both cases, i.e.
with and without the analog correlator at the receiver, under a Nakagami-$m$
block fading scenario. The derived closed-form expressions are a function of
the spreading factor and the fading parameter $m$ and are verified by
extensive Monte Carlo simulations. They provide a quick and convenient
methodology of evaluating the system’s performance and obtaining insights into
how key system parameters affect the performance. In addition, our results
show that similar to [11], a Rayleigh fading scenario results in a higher
harvested DC performance, compared to a no-fading scenario, which shows that
fading is beneficial for DCSK-based WPT.

  * •

We apply the proposed WPT architecture to unmodulated chaotic waveforms and
investigate how it performs against its modulated DCSK counterpart. We
demonstrate that modulation does not affect the EH performance without the
correlator at the receiver. On the contrary, with a correlator-aided receiver,
DCSK significantly outperforms the unmodulated case. Based on this
observation, we propose a novel short reference DCSK-based signal design,
which results in further EH performance gains. Finally, we compare the
proposed waveform with the existing $N$-tone multisine signals, by taking into
account the imperfections of the high power amplifier (HPA) at the
transmitter.

The rest of this paper is organized as follows: Section II introduces the
proposed system architecture. Section III presents the complete analytical
framework of chaotic signal-based WPT by considering a nonlinear rectenna
model. Section IV proposes a novel WPT-optimal chaotic waveform. Numerical
results are presented in Section V, followed by our conclusions in Section VI.

##  II A Chaotic Signal-based WPT System Architecture

Figure 1: Proposed architecture for chaotic signal-based WPT; “D” corresponds
to the delay blocks and the symbol “$\int$” corresponds to the integrator.

###  II-A System model

We consider a point-to-point WPT set-up, where the transmitter employs a DCSK
generator and the receiver consists of an analog correlator (discussed in
Section II-C) followed by an EH circuit, as depicted in Fig. 1. Note that even
though DCSK is primarily a communication-based signal, in this work, we focus
on its effects on WPT111This assumption refers to communication networks that
operate based on DCSK modulation and EH receivers are added to operate by the
ambient radiation..

We assume that the wireless link suffers from both large-scale path-loss
effects and small-scale block fading. Specifically, the received power is
proportional to $r^{-\alpha}$, where $r$ is the transmitter-receiver (Tx-Rx)
distance and $\alpha$ denotes the path-loss exponent. Moreover, a Nakagami-$m$
distributed block fading channel $h$ is considered with unit mean power, where
$m\geq 1$.

###  II-B Chaotic signals

Assume a DCSK signal, where the current symbol is dependent on the previous
symbol [17] and different sets of chaotic sequences can be generated by using
different initial conditions. Each transmitted bit is represented by two sets
of chaotic signal samples, with the first set representing the reference, and
the other conveying information. If $+1$ is to be transmitted, the data sample
will be identical to the reference sample. Otherwise, an inverted version of
the reference sample will be used as the data sample [16]. During the $l$-th
transmission interval, the output of the transmitter is

$\displaystyle
s_{l,k}=\begin{cases}x_{l,k},&k=2(l-1)\beta+1,\dots,(2l-1)\beta,\\\
d_{l}x_{l,k-\beta},&k=(2l-1)\beta+1,\dots,2l\beta,\end{cases}$ (1)

where $d_{l}=\pm 1$ is the information bit, $x_{l,k}$ is the chaotic sequence
used as the reference signal, and $x_{l,k-\beta}$ is its delayed version. Let
$\beta$ be a non-negative integer, defined as the spreading factor. Then,
$2\beta$ chaotic samples are used to spread each information bit. Furthermore,
$x_{l,k}$ can be generated according to various existing chaotic maps. Due to
its good correlation properties, we consider the Chebyshev map of degree $\xi$
for chaotic signal generation, which is defined as [17]

$\displaystyle x_{k+1}=\cos(\xi\cos^{-1}(x_{k})),\forall\>\>|x_{k}|\leq 1.$
(2)

###  II-C Analog correlator

To further boost the EH performance, we propose a WPT receiver architecture,
where an analog correlator [26] preceeds the EH rectifier circuit (see Fig.
1). The motivation behind the employment of an analog correlator222It is to be
noted that the analog correlator is an active device, but with a nominal power
consumption. Specifically, it has been observed, that a CMOS-implemented
11-bit analog correlator consumes 40 nW of power [26]. As a result, we have
not considered this energy consumption in the context of applying the
correlator. block is two fold: i) it enables controlling the signal at the
input of the harvester, and ii) it enhances the PAPR of the signal. An analog
correlator primarily consists of a series of delay blocks, which result in
signal integration over a specified period of time; an ideal $\psi$-bit analog
correlator consists of $(\psi-1)$ number of delay blocks [26]. In what
follows, for the sake of simplicity, we consider $\psi$ as the transmitted
DCSK symbol length, i.e. $\psi=2\beta$. Accordingly, the analog correlator
output $y_{l}(t)$ for the $l$-th transmitted symbol is

$y_{l}(t)=\sqrt{P_{t}}h_{l}\sum\limits_{k=1}^{2\beta}s_{l,k}(t),$ (3)

where $P_{t}$ is the transmission power. Note that we propose the use of a
correlator and not a summing circuit in Rx. A summing circuit generates an
output, which is the sum of multiple input signals [27] whereas the analog
correlator allows an effective integration of the same signal over a certain
time interval [26], i.e. it generates an output, which is the sum of the
delayed versions of the same signal. In other words, the input signal is
correlated with a delayed version of itself. Note that the value $\psi=1$
corresponds to the conventional case without a correlator.

In the following proposition, we show the effect of the analog correlator on
the signal’s PAPR.

######  Proposition 1.

The signal PAPR at the harvester input is

$\displaystyle\mathrm{PAPR}=\begin{cases}2,&\text{without
correlator}\>\>(\psi=1),\\\ 4\beta,&\text{with
correlator}\>\>(\psi=2\beta).\end{cases}$ (4)

###### Proof.

See Appendix A. ∎

Therefore, since high PAPR signals are desirable for WPT [8], the correlator
significantly enhances the WPT performance.

###  II-D Energy transfer model

The WPT receiver is equipped with an antenna followed by a rectifier. The
rectifier, which generally consists of a diode (e.g., a Schottky diode) and a
passive low pass filter, acts as an envelope detector [28] and therefore
neglects the phase of the received signal $y(t)$. Based on the nonlinearity of
this circuit, the output DC current is approximated in terms of $y(t)$ as [7]

$z_{\rm
DC}=k_{2}R_{ant}\mathbb{E}\\{|y(t)|^{2}\\}+k_{4}R_{ant}^{2}\mathbb{E}\\{|y(t)|^{4}\\},$
(5)

where the parameters $k_{2},k_{4},$ and $R_{ant}$ are constants determined by
the characteristics of the circuit. Note that the conventional linear model is
a special case of this nonlinear model and can be obtained by considering only
the first term in (5). For the sake of presentation, we will use
$\varepsilon_{1}=r^{-\alpha}k_{2}R_{ant}P_{t}$ and
$\varepsilon_{2}=r^{-2\alpha}k_{4}R_{ant}^{2}P_{t}^{2}$. Hence, from (3) and
(5), we obtain

$\displaystyle z_{\rm MC}$
$\displaystyle=\varepsilon_{1}\mathbb{E}\left\\{\left(|h|\sum\limits_{k=1}^{2\beta}s_{k}\right)^{2}\right\\}+\varepsilon_{2}\mathbb{E}\left\\{\left(|h|\sum\limits_{k=1}^{2\beta}s_{k}\right)^{4}\right\\},$
(6)

for the scenario with an analog correlator at the receiver, where the
expectation is taken over both $|h|$ and $s_{k}$. On the other hand, the
output DC without a correlator is

$\displaystyle z_{\rm MNC}$
$\displaystyle=\varepsilon_{1}\mathbb{E}\left\\{|h|^{2}\sum\limits_{k=1}^{2\beta}s_{k}^{2}\right\\}+\varepsilon_{2}\mathbb{E}\left\\{|h|^{4}\sum\limits_{k=1}^{2\beta}s_{k}^{4}\right\\},$
(7)

where $2\beta$ samples are considered for fair comparison. This implies that
$\beta$ will have a considerable impact on the harvested energy. Finally, note
that the subscript MC and MNC refers to the scenario of modulated waveform
transmission (DCSK in this case), with and without the correlator at the
receiver, respectively.

##  III Chaotic Signal-based Wireless
Power Transfer

In this section, we investigate the effect of DCSK signals on WPT. We analyze
the performance of the proposed receiver architecture in terms of the
harvested energy, when DCSK waveforms are transmitted. Towards this direction,
we provide the following theorem.

######  Theorem 1.

For a WPT receiver with a correlator, the harvested DC is given by

$z_{\rm
MC}=\varepsilon_{1}\beta+\varepsilon_{2}\frac{3(1+m)}{m}\beta(2\beta-1).$ (8)

###### Proof.

See Appendix B. ∎

Theorem 1 provides a generalized closed-form expression for $z_{\rm MC}$ in
terms of $m$. Note that $z_{\rm MC}$ corresponding to a no-fading scenario
($m\to\infty$) can also be obtained as a special case, given in the following
corollary.

######  Corollary 1.

In a no-fading scenario, $z_{\rm MC}$ is given by

$\lim\limits_{m\rightarrow\infty}z_{\rm
MC}=\varepsilon_{1}\beta+3\varepsilon_{2}\beta(2\beta-1).$ (9)

The above corollary follows directly from Theorem 1, as $(1+m)/m\to 1$ for
$m\to\infty$.

######  Remark 1.

Note that in Theorem 1, $m=1$ results in an enhanced $z_{\rm MC}$ compared to
the $m\rightarrow\infty$ scenario, i.e. wireless fading enhances WPT. This
observation corroborates the claims made in [11] regarding the beneficial role
of fading in WPT systems.

Next, we consider the case of a conventional WPT receiver without the analog
correlator, which is given by the following theorem.

######  Theorem 2.

Without a correlator at the WPT receiver, the harvested DC is

$z_{\rm MNC}=\varepsilon_{1}\beta+\varepsilon_{2}\frac{3(1+m)}{4m}\beta.$ (10)

###### Proof.

See Appendix C. ∎

######  Remark 2.

From Theorems 1 and 2, we observe that $z_{\rm MC}$ and $z_{\rm MNC}$ is a
quadratic and linear function of $\beta$, respectively. This highlights the
benefits of the proposed architecture based on the nonlinearity of the EH
process. Note that, if a linear EH model is considered, which only accounts
for the second-order term in (5), we obtain $z_{\rm MC}=z_{\rm MNC}$.

We observe that, both $z_{\rm MC}$ and $z_{\rm MNC}$ are inversely
proportional to $m$. We also note that the analytical expressions obtained can
be extended to Rician fading scenarios (as special cases), by replacing
$m=\frac{(K+1)^{2}}{2K+1}$, where $K$ is the Rice factor [29].

For the sake of completeness, we also provide $z_{\rm MNC}$ corresponding to a
no-fading scenario.

######  Corollary 2.

The harvested DC $z_{\rm MNC}$ corresponding to a no-fading scenario is given
by

$\lim\limits_{m\rightarrow\infty}z_{\rm
MNC}=\varepsilon_{1}\beta+\frac{3}{4}\varepsilon_{2}\beta.$ (11)

##  IV DCSK-based Waveform Design for WPT

We now extend the analytical model, initially proposed for DCSK signals, to
unmodulated (deterministic) chaotic waveforms. Then, we modify the
conventional DCSK waveform at the transmitter, in a way that further enhances
the WPT performance.

Figure 2: (a) DCSK frame, and (b) SR-DCSK frame.

###  IV-A Unmodulated chaotic signals

We consider the case of using a deterministic, unmodulated chaotic waveform
for WPT and compare its performance with the conventional DCSK signal. For
this case, the $k$-th bit of the $l$-th transmitted unmodulated chaotic symbol
is given by

$s_{l,k}=x_{l,k},\quad k=2(l-1)\beta+1,\dots,2l\beta.$ (12)

It is worth noting that while it is essential to have a symbol length of
$2\beta$ in DCSK, such a constraint is not required in the unmodulated chaotic
case. However, we consider a symbol length $2\beta$ for a fair comparison of
the EH performance. Now, we state the following proposition, where we obtain
the harvested DC for unmodulated chaotic signals, with/without a correlator at
the WPT receiver.

######  Proposition 2.

Based on the symbol structure as defined in (12), we obtain

$z_{\rm
UM,C}=\varepsilon_{1}\beta+\varepsilon_{2}\frac{3(1+m)}{m}\beta\left(\beta-\frac{1}{4}\right),$
(13)

with a correlator and,

$z_{\rm UM,NC}=\varepsilon_{1}\beta+\varepsilon_{2}\frac{3(1+m)}{4m}\beta,$
(14)

without a correlator.

###### Proof.

See Appendix D. ∎

######  Remark 3.

Without a correlator at the receiver, we have $z_{\rm MNC}=z_{\rm UM,NC}$, for
a given set of system parameters (Theorem 2 and Proposition 2). On the other
hand, when a correlator is employed, we have $z_{\rm MC}>z_{\rm UM,C}$, with a
performance gain equal to
$\varepsilon_{2}\frac{3(1+m)}{m}\beta(\beta-\frac{3}{4})$ (Theorem 1 and
Proposition 2). In other words, a modulated DCSK signal positively affects the
EH performance in the presence of a correlator.

The above remark can be explained as follows. In contrast to the unmodulated
waveform, DCSK also carries information due to its inherent randomness, i.e.
the first $\beta$ symbols are chaotic, followed by their replica or inverted
replica. In case of a correlator-aided receiver, this randomness of the DCSK
waveform induces fluctuations of the transmit signal amplitude, which boosts
the harvested DC. The difference in performance, given in Remark 3, is in line
with the claims made in [10] regarding the importance of using modulated
multisine waveforms for efficient WPT.

Note that Remark 3 considers identical system parameters and similar channel
conditions between the two cases. However, this claim does not hold for all
possible scenarios. Indeed, an interesting insight can be obtained when we
analyze the energy harvested in the modulated case under the deterministic
fading scenario. Specifically, let $m_{1}$ and $m_{2}$ be the fading
parameters of the modulated and unmodulated case, respectively. Then, the
performance gap $\Delta=z_{\rm MC}-z_{\rm UM,C}$ can be written as

$\Delta=\varepsilon_{2}\frac{3(1+m_{1})}{m_{1}}\beta(2\beta-1)-\varepsilon_{2}\frac{3(1+m_{2})}{m_{2}}\beta\left(\beta-\frac{1}{4}\right).$
(15)

Observe that for the special case of $m_{1}=m_{2}$, we always have $\Delta>0$,
i.e. $z_{\rm MC}>z_{\rm UM,C}$ (Remark 3). On the other hand, $z_{\rm MC}$ in
a no-fading environment ($m_{1}\rightarrow\infty$) and $z_{\rm UM,C}$ in a
Rayleigh fading scenario ($m_{2}=1$) results in
$\Delta=-\frac{3}{2}\varepsilon_{2}\beta<0$. This implies that, with a
correlator-aided receiver, an unmodulated waveform in a Rayleigh fading
environment results in more harvested energy as compared to its modulated
counterpart in a no-fading environment.

Furthermore, we also observe from (15) that for finite $m_{1}$ and $m_{2}$,
$\Delta>0$ holds if and only if

$\displaystyle\frac{(1+m_{1})}{m_{1}}(2\beta-1)-\frac{(1+m_{2})}{m_{2}}\left(\beta-\frac{1}{4}\right)>0.$
(16)

Note that $m_{1}$ and $m_{2}$ are necessarily channel parameters, i.e. they
cannot be controlled. However, it is possible to tune $\beta$ accordingly to
satisfy (16). After some trivial algebraic manipulations, we obtain
$\beta>\beta_{\rm opt}^{+}$, where

$\beta_{\rm
opt}=\frac{\left(\frac{1+m_{1}}{m_{1}}\right)-\frac{1}{4}\left(\frac{1+m_{2}}{m_{2}}\right)}{2\left(\frac{1+m_{1}}{m_{1}}\right)-\left(\frac{1+m_{2}}{m_{2}}\right)}=\frac{1}{4}\left(\frac{4m_{2}+3m_{1}m_{2}-m_{1}}{2m_{2}+m_{1}m_{2}-m_{1}}\right)$
(17)

and $x^{+}=\max\\{x,0\\}$.

Thus, it is important to note that $\Delta>0$ does not hold for all values of
$\beta$ and in some scenarios, we have $\Delta<0$, i.e. it is possible to
obtain more harvested energy from an unmodulated waveform as compared to its
modulated counterpart.

###  IV-B WPT optimal DCSK-based waveform

In Section III, we showed that a communication-based chaotic waveform with the
proposed architecture boosts the WPT performance. The drawback of this
approach is a long symbol duration, which also implies that a large number of
chaotic chips are generated for every single symbol transmission. To this end,
a shorter symbol duration may address both these problems and accordingly, a
short reference DCSK (SR-DCSK) is proposed in [30].

An unmodulated chaotic component of length $\beta_{r}<\beta$ is considered in
SR-DCSK, followed by $\zeta$ copies of its replica, multiplied with the
information such that $\beta=\zeta\beta_{r}$. Hence, the symbol duration is
$\beta_{r}(1+\zeta)=\beta_{r}+\beta<2\beta$, i.e. the transmitted symbol is
characterized by a sequence of $\beta_{r}+\beta$ samples of the chaotic basis
signal, which is represented as

$\displaystyle s_{k}=\begin{cases}x_{k},&0<k\leq\beta_{r},\\\
dx_{k-\beta_{r}},&\beta_{r}<k\leq\beta_{r}+\beta,\end{cases}$ (18)

where $d$ is the transmitted information bit. An illustrative comparison of
DCSK and SR-DCSK is demonstrated in Fig. 2. While Fig. 2(a) shows the
conventional DCSK symbol structure, the short reference SR-DCSK is depicted in
Fig. 2(b). It is important to note here that the SR-DCSK is a technique
proposed mainly for the purpose of efficient information transfer, such that
its error performance always remains above a certain acceptable threshold. In
contrast, we investigate the impact of having a shorter symbol duration on WPT
and accordingly we propose a WPT optimal DCSK-based waveform.

$\displaystyle z_{\rm SR}=$
$\displaystyle\frac{1}{2}\varepsilon_{1}\beta^{2}+\frac{3(1+m)}{8m}\varepsilon_{2}\beta^{4},$
$\beta_{r}=0,$ (19a) $\displaystyle z_{\rm SR}=$
$\displaystyle\varepsilon_{1}\frac{\beta_{r}^{2}+\beta^{2}}{2\beta_{r}}+\varepsilon_{2}\frac{3(1+m)}{8m}\left(1+6\zeta^{2}+\zeta^{4}\right)\left(2\beta_{r}^{2}-\beta_{r}\right),$
$\beta_{r}>0.$ (19b)

Given that $\zeta$ replicas of the $\beta_{r}$ length reference signal are
concatenated in SR-DCSK, this implies that the reference signal is partially
correlated over $\zeta$ consecutive samples of the frame, i.e. the degree of
‘randomness’ increases within a single frame, as compared to DCSK. In other
words, this results in an increased correlation within the frame, which leads
to enhanced WPT. This is in line with the claim made in [31] regarding the
effect of correlation in transmitted signals with respect to WPT.

Hence, we investigate SR-DCSK to obtain a WPT-optimal frame structure. As the
symbol structure in (18) is dependent on $\beta_{r}$, the two extreme cases
are $\beta_{r}=0$ and $\beta_{r}=\beta$. Accordingly, the WPT performance for
all the possible values of $\beta_{r}$ is given by the following theorem.

######  Theorem 3.

A $\beta_{r}$-SR-DCSK results in $z_{\rm SR}$ given by (19b).

###### Proof.

See Appendix E. ∎

It is interesting to note from (19b), that $\beta_{r}$-SR-DCSK converges to
DCSK for $\beta_{r}=\beta.$ We observe that the increased degree of
‘randomness’ in $\beta_{r}$-SR-DCSK results in a significant performance
enhancement in terms of WPT, with an increase of $z_{\rm SR}$ proportional to
$\beta^{4}$, unlike $z_{\rm MC}$ in Theorem 1.

######  Proposition 3.

For a given $\beta$, the optimal $z_{\rm SR}$ is achieved at $\beta_{r}=1$.

###### Proof.

We know that $\beta=\zeta\beta_{r}$ and $\beta\in\mathbb{Z}^{+}$. Hence, by
comparing the second and fourth order terms of the EH process separately in
Theorem 3, we can state that $z_{\rm SR}$ is maximum for $\beta_{r}=1$, with
all other system parameters remaining constant. ∎

Hence, the WPT-optimal transmitted symbol of $(\beta+1)$ duration is defined
as

$\displaystyle s_{k}=\begin{cases}x_{k},&k=1,\\\
dx_{1},&1<k\leq\beta+1.\end{cases}$ (20)

The corresponding harvested DC

$z_{\rm SR}^{\rm
opt}=\frac{1}{2}\varepsilon_{1}(1+\beta^{2})+\frac{3(1+m)}{8m}\varepsilon_{2}(1+6\beta^{2}+\beta^{4}),$
(21)

is obtained by replacing $\beta_{r}=1$ in (19b). It is further interesting to
observe that having a single chip of chaotic signal followed by $\beta$ chips
of information modulated chaos in a symbol of $\beta+1$ duration results in a
maximum possible correlation; this is reflected in the WPT gain of the
proposed DCSK-based waveform. Therefore for large $\beta$, the reduction in
symbol duration is approximately $2\beta-(\beta+1)\approx\beta$ per symbol
duration; this also suggests that one chaotic chip needs to be generated per
symbol compared to $\beta$ chaotic chips per symbol in DCSK. Finally, a
shorter symbol duration also signifies the reduced receiver’s complexity.

Now, we compare the proposed chaotic waveform with the existing multisine
signal-based EH framework [9]. We know that for a $N$-tone multisine signal,
the PAPR at the harvester input is $2N$ [8]. On the contrary, we have proved
in Proposition $1$, that the PAPR of a DCSK-based signal is $4\beta$. As a
high PAPR translates to a better EH performance [8], we can intuitively say
that the proposed waveform will result in a higher harvested energy, even for
a set of comparable $N$ and $\beta$. Furthermore, for an $N$-tone multisine
signal, the linear term of the harvested DC is independent of $N$ and the
nonlinear term is linearly dependent on $N$ [9]. On the contrary, in the case
of the proposed $\beta_{r}$-SR-DCSK waveform, the linear and nonlinear terms
of $z_{\rm SR}^{\rm opt}$ (Eq. (21)) are proportional to $\beta^{2}$ and
$\beta^{4}$, respectively. As a result, the proposed architecture for
$\beta_{r}$-SR-DCSK significantly outperforms multisine waveforms, in terms of
WPT.

TABLE I: Summary of results.

Waveform | Modulation | Correlator | Harvested DC
---|---|---|---
$z_{\rm MC}$ | DCSK | $\checkmark$ | $\displaystyle\varepsilon_{1}\beta+\varepsilon_{2}\frac{3(1+m)}{m}\beta(2\beta-1)$
$z_{\rm MNC}$ | DCSK | $\times$ | $\displaystyle\varepsilon_{1}\beta+\varepsilon_{2}\frac{3(1+m)}{4m}\beta$
$z_{\rm UM,C}$ | $\times$ | $\checkmark$ | $\displaystyle\varepsilon_{1}\beta+\varepsilon_{2}\frac{3(1+m)}{m}\beta\left(\beta-\frac{1}{4}\right)$
$z_{\rm UM,NC}$ | $\times$ | $\times$ | $\displaystyle\varepsilon_{1}\beta+\varepsilon_{2}\frac{3(1+m)}{4m}\beta$
$z_{\rm SR}$ |  $\beta_{r}$-SR-DCSK | $\checkmark$ | $\begin{aligned} &\displaystyle\frac{1}{2}\varepsilon_{1}\beta^{2}+\frac{3(1+m)}{8m}\varepsilon_{2}\beta^{4},\qquad\qquad\qquad\qquad\qquad\qquad\quad\beta_{r}=0,\\\ &\displaystyle\varepsilon_{1}\frac{\beta_{r}^{2}+\beta^{2}}{2\beta_{r}}+\varepsilon_{2}\frac{3(1+m)}{8m}\left(1+6\zeta^{2}+\zeta^{4}\right)\left(2\beta_{r}^{2}-\beta_{r}\right),\quad\beta_{r}>0\end{aligned}$

Finally, Table I presents a summary of the main analytical results derived in
Section III and IV.

######  Remark 4.

It may appear from the analytical results in Table I that the limiting case of
$\beta\rightarrow\infty$ will result in an infinite amount of harvested
energy. However, note that the energy transfer model considered in this work
is based on the assumption that the harvester operates in the nonlinear region
[9]. If $\beta$ becomes too large, the diode inside the harvester will be
forced into the saturation region of operation, making the derived analytical
results inapplicable.

##  V Numerical Results

We validate our theoretical analysis through extensive Monte-Carlo
simulations. Without any loss of generality, we consider a transmission power
of $P_{t}=30$ dBm, a Tx-Rx distance $r=20$ m, and a pathloss exponent
$\alpha=4$. The parameters considered for the EH model are taken as
$k_{2}=0.0034,k_{4}=0.3829,$ and $R_{ant}=50$ $\Omega$ [10]. Also recall that,
$\psi=2\beta$ and $\psi=1$ corresponds to a WPT receiver with and without the
analog correlator, respectively.

Figure 3: Effect of the spreading factor on $z_{\text{DC}}$; lines correspond
to analysis and markers correspond to simulation results.

###  V-A Performance evaluation of the proposed architecture

Fig. 3 depicts the WPT performance of DCSK with respect to the spreading
factor, with and without the correlator at the receiver. Firstly, we observe
the significant gains in WPT performance achieved with the employment of the
correlator; the theoretical results (lines) match very closely with the
simulation results (markers); this verifies our proposed analytical framework.
This is related to the high PAPR, which is a function of the spreading factor
$\beta$ as stated in Proposition 1. Moreover, we observe that, the harvested
energy decreases with increasing $m$ for a given $\beta$. This observation is
inline with (8) and (10), where we note that, the harvested DC varies with the
quantity $\frac{1+m}{m}$, while the other system parameters remain constant.
Specifically, due to this ratio, the best performance is achieved with $m=1$,
whereas the worst is obtained with $m\to\infty$. Moreover, the loss in
performance between consecutive values of $m$ decreases as $m$ increases; for
example, observe the performance gap between $m=1$, $m=4$, and $m=20$.

###  V-B Effects of data modulation on WPT

Fig. 4 illustrates the impact of using a modulated waveforms for WPT and
compares the performance with unmodulated chaotic signals in terms of WPT. The
figure demonstrates that data modulation does not affect the WPT performance
without the correlator at the receiver, i.e. both DCSK and unmodulated chaotic
symbols result in an identical harvested DC. The only difference is the
degradation in WPT performance with increasing $m$, which has been discussed
earlier in Fig. 3. On the other hand, we observe that modulating the chaotic
symbols, it has a significant impact on the WPT performance. Finally, it is
interesting to note that the harvested DC corresponding to Rayleigh fading,
i.e. $m=1$, with unmodulated chaotic symbols is approximately identical to the
harvested DC obtained with modulated symbols and $m=10$. This verifies (15),
where in this case, the performance gap can be obtained by letting $m_{1}=10$
and $m_{2}=1$.

Figure 4: Effect of the modulation on $z_{\text{DC}}$.

###  V-C Effects of symbol length on WPT

Figure 5: Effect of the symbol length on $z_{\text{DC}}$.

Fig. 5 depicts the importance of the choice of $\beta$ as a function of the
channel parameter $m$. We consider the WPT performance of DCSK as a function
of $\beta$, for various fading scenarios $m_{1}=40,80,100,150$ and compare it
with its unmodulated counterpart in a Rayleigh fading scenario $(m_{2}=1)$.
Recall that a positive $\Delta$ implies that DCSK performs better in terms of
WPT and vice-versa. Observe that for all values of $m_{1}$, the parameter
$\Delta$ attains a positive value only beyond a certain $\beta$, i.e. the
unmodulated waveform performs better till that point and this value of $\beta$
increases with increasing $m_{1}$; this verifies our analysis to obtain (17).
Hence, we can rightly state that modulating the chaotic waveform does not
guarantee enhanced EH. Even in the proposed architecture, i.e. with a
correlator at the receiver, the choice of $\beta$ should be based on both the
channel condition and the modulation, to guarantee a better EH performance.

###  V-D Effects of a shorter reference on WPT

Figure 6: Effect of a short reference on WPT; lines correspond to analysis and
markers correspond to simulation results.

Fig. 6 demonstrates the effect of an SR symbol length on the WPT performance
of the proposed correlator-based architecture. The effect of having a
reference length $\beta_{r}<\beta$, with $\zeta$ copies of its data modulated
replica concatenated together such that $\beta=\zeta\beta_{r}$, is
demonstrated here, where we consider two cases of $\beta=60$ and $\beta=90$,
respectively. This figure illustrates the impact of correlation in enhancing
chaotic signal-based WPT. We observe that for a given $\beta$, increasing
$\beta_{r}$ results in obtaining the maximum harvested DC at $\beta_{r}=1$,
followed by a monotonically decreasing performance. This is justified by the
fact that as we have $\beta=\zeta\beta_{r}$, a higher $\beta_{r}$ implies a
lower $\zeta$ for a given $\beta$, i.e. the correlation decreases, which
results in deteriorating WPT performance. Finally, we observe that, for the
special case of $\beta_{r}=\beta$, the WPT performance coincides with that of
the DCSK-based transmission.

###  V-E Evaluation of WPT-optimal SR-DCSK

Fig. 7 demonstrates the performance of the proposed WPT-optimal SR-DCSK. The
figure exhibits the effect of the spreading factor for two scenarios of Tx-Rx
distance $20$ m and $30$ m, respectively. While the analytically obtained
values match closely with that of Monte Carlo simulation, we also note that
the harvested DC with Tx-Rx distance $30$ m is less compared to the harvested
DC with Tx-Rx distance $20$ m; this is intuitive due to the path-loss factor.

Figure 7: Performance of WPT-optimal SR-DCSK; lines correspond to analysis and
markers correspond to simulation results.

Fig. 8 illustrates the joint effect of fading parameter $m$ and spreading
factor $\beta$ on the WPT performance of the proposed symbol structure. We
observe that the performance of WPT-optimal SR-DCSK improves with increasing
$\beta$ and decreasing $m$ for a given Tx-Rx distance, where the harvested DC
is maximum at $m=1$ and $\beta=30$, i.e. minimum $m$ and maximum $\beta$. We
also note that even though the harvested energy increases monotonically with
$\beta$ for a given $m$, its variation against $m$ for any particular $\beta$
is not the same. The harvested energy saturates with increasing $m$ for a
given $\beta$; this observation can be justified by the fact that apart from
$\beta$, it is also a function of $\frac{1+m}{m}$.

Figure 8: Joint effect of the spreading factor $\beta$ and the fading
parameter $m$ on WPT-optimal SR-DCSK.

###  V-F Comparison of WPT-optimal SR-DCSK and DCSK

Fig. 9 compares the performance of the proposed WPT-optimal SR-DCSK and DCSK
against the transmit symbol length for several values of the fading parameter
$m$. As expected, the proposed waveform results in a significantly higher
harvested DC at approximately half the corresponding DCSK symbol length. A
small symbol length implies less number of chaotic signals generated per
symbol transmission and also a smaller number of delay blocks inside the
analog correlator, i.e. reduction in both receiver’s complexity and cost.
Besides having a short frame duration, the proposed frame structure
successfully exploits the effect of correlation in the transmitted signal.

Figure 9: Performance comparison of the proposed WPT-optimal SR-DCSK and DCSK.

###  V-G Performance comparison with multisine waveforms

Fig. 10 compares the WPT performance of the proposed chaotic and existing
different $N$-tone multisine waveforms [9]. Furthermore, in this figure, we
also consider the HPA non-linearities at the transmitter [32]. The figure
corroborates our claim that the proposed waveform significantly outperforms
the multitone signals in terms of WPT. Moreover, it also demonstrates the
effect of the transmission power $P_{t}$, as we observe that the harvested
energy saturates after a certain $P_{t}$ is reached. The reason for this
observation is attributed to the limited capability of a practical HPA at the
transmitter. A typical HPA cannot transmit with any arbitrary amount of power
and beyond a certain limit, it can only transmit with a fixed $P_{t}$,
irrespective of the power of its input signal [32]. In this figure, we observe
this phenomenon around $P_{t}=25$ dBm and hence the saturating effect in the
$P_{t}>25$ dBm range.

##  VI Conclusion

In this paper, we investigated the effects of conventional communication-based
chaotic waveforms in WPT. In particular, we considered a point-to-point set-
up, where a transmitter uses a DCSK generator and the receiver employs an
analog correlator followed by an EH circuit. By taking into account the
nonlinearities of the EH process, we proposed a novel WPT architecture and
analyzed it in terms of the received signal’s PAPR as well as the achieved
harvested DC. We showed that both of these performance metrics are dependent
on the parameters of the transmitted waveform. In addition, we demonstrated
that DCSK waveforms outperform their unmodulated counterparts, in terms of EH
performance, when a correlator is employed at the receiver. Based on this
observation, a novel SR-DCSK-based waveform design has been investigated,
which further enhances the WPT performance. Finally, we have shown that the
proposed chaotic waveform outperforms the conventional multisine signals in
terms of its WPT capability. An immediate extension of this work is to
investigate multi-dimensional chaotic signals for WPT, by considering a
generalized frequency selective fading scenario.

Figure 10: Performance comparison of the proposed WPT-optimal SR-DCSK and
multisine waveforms, considering HPA non-linearity; $m=4$.

##  Appendix A Proof of Proposition 1

Without a correlator at the receiver, i.e. for $\psi=1$, the $\rm PAPR$
corresponding to the $l$-th transmitted symbol is

$\mathrm{PAPR}=\frac{\max\limits_{l}\left\\{\sum\limits_{k=2(l-1)\beta+1}^{2l\beta}|h_{l}|^{2}s_{l,k}^{2}\right\\}}{\mathbb{E}\left\\{\sum\limits_{k=2(l-1)\beta+1}^{2l\beta}|h_{l}|^{2}s_{l,k}^{2}\right\\}}.$
(22)

By considering a given channel instance $h_{l}$ and the invariant probability
density function (PDF) of $x_{k}$, as [17]

$\displaystyle f_{X}(x)=\begin{cases}\frac{1}{\pi\sqrt{1-x^{2}}},&|x|<1,\\\
0,&\text{otherwise},\end{cases}$ (23)

we obtain
$\max\limits_{l}\left\\{\sum\limits_{k=2(l-1)\beta+1}^{2l\beta}|h_{l}|^{2}s_{l,k}^{2}\right\\}=2|h_{l}|^{2}\beta$
and

$\displaystyle\mathbb{E}\left\\{\sum\limits_{k=2(l-1)\beta+1}^{2l\beta}|h_{l}|^{2}s_{l,k}^{2}\right\\}$
$\displaystyle=|h_{l}|^{2}\sum\limits_{k=2(l-1)\beta+1}^{2l\beta}\mathbb{E}\\{s_{l,k}^{2}\\}$
$\displaystyle=|h_{l}|^{2}\sum\limits_{k=2(l-1)\beta+1}^{2l\beta}\int_{-\infty}^{\infty}x^{2}f_{X}(x)dx=|h_{l}|^{2}\beta.$
(24)

Hence, for $\psi=1$, we have

$\displaystyle{\rm PAPR}=\frac{2|h_{l}|^{2}\beta}{|h_{l}|^{2}\beta}=2.$ (25)

On the other hand, for $\psi=2\beta$, i.e. with a correlator at the receiver,

$\mathrm{PAPR}=\frac{\max\limits_{l}\left\\{\left(\sum\limits_{k=2(l-1)\beta+1}^{2l\beta}|h_{l}|s_{l,k}\right)^{2}\right\\}}{\mathbb{E}\left\\{\left(\sum\limits_{k=2(l-1)\beta+1}^{2l\beta}|h_{l}|s_{l,k}\right)^{2}\right\\}},$
(26)

where from (23), we get

$\displaystyle\max\limits_{l}\left\\{\sum\limits_{k=2(l-1)\beta+1}^{2l\beta}|h_{l}|s_{l,k}\right\\}^{2}=4|h_{l}|^{2}\beta^{2}.$
(27)

Note that for chaotic sequences generated by the Chebyshev map, we have
$\mathbb{E}\left\\{s_{l,i}s_{l,j}\right\\}=0$ for $i\neq j$ [33, Eq. 55]. As
such,

$\displaystyle\mathbb{E}\left\\{\left(\sum\limits_{k=2(l-1)\beta+1}^{2l\beta}|h_{l}|s_{l,k}\right)^{2}\right\\}$
$\displaystyle=|h_{l}|^{2}\sum\limits_{k=2(l-1)\beta+1}^{2l\beta}\mathbb{E}\\{s_{l,k}^{2}\\}$
$\displaystyle=|h_{l}|^{2}\beta,$ (28)

which follows from (A). Therefore, we have

$\displaystyle\mathrm{PAPR}=\frac{4|h_{l}|^{2}\beta^{2}}{|h_{l}|^{2}\beta}=4\beta.$
(29)

##  Appendix B Proof of Theorem 1

As we consider a unit mean power Nakagami-$m$ distributed block fading
channel, the PDF of $|h|$ is

$f_{|h|}(z)=\frac{2m^{m}z^{2m-1}e^{-mz^{2}}}{\Gamma(m)},\>\>\forall\>\>z\geq
0,$ (30)

where $\Gamma(\cdot)$ denotes the complete Gamma function and $m\geq 1$
controls the severity of the amplitude fading. The quantity
$\sum\limits_{k=1}^{2\beta}s_{k}$ in (6) can be alternatively written as

$\displaystyle\sum\limits_{k=1}^{2\beta}s_{k}$
$\displaystyle=(x_{1}+\dots+x_{\beta}+d(x_{1}+\dots+x_{\beta}))=p\sum\limits_{k=1}^{\beta}x_{k},$
(31)

where we have $p=1+d$. Then, by assuming equally likely transmissions of
$d=\pm 1$, we have the PDF $f_{P}(p)=\frac{1}{2},$ $\forall$ $p\in\\{0,2\\}$.
Hence, from (6) we have

$\displaystyle z_{\rm MC}$
$\displaystyle=\varepsilon_{1}\mathbb{E}\left\\{\left(|h|\sum\limits_{k=1}^{2\beta}s_{k}\right)^{2}\right\\}+\varepsilon_{2}\mathbb{E}\left\\{\left(|h|\sum\limits_{k=1}^{2\beta}s_{k}\right)^{4}\right\\}$
$\displaystyle=\varepsilon_{1}\mathbb{E}\left\\{|h|^{2}\right\\}\mathbb{E}\left\\{\left(p\sum\limits_{k=1}^{\beta}x_{k}\right)^{2}\right\\}$
$\displaystyle\qquad+\varepsilon_{2}\mathbb{E}\left\\{|h|^{4}\right\\}\mathbb{E}\left\\{\left(p\sum\limits_{k=1}^{\beta}x_{k}\right)^{4}\right\\}.$
(32)

Note that as we are considering a unit power Nakagami-$m$ fading scenario, we
have $\mathbb{E}\left\\{|h|^{2}\right\\}=1$ and

$\displaystyle\mathbb{E}\left\\{|h|^{4}\right\\}$
$\displaystyle=\frac{2m^{m}}{\Gamma(m)}\int_{0}^{\infty}z^{2m+3}e^{-mz^{2}}dz$
$\displaystyle=\frac{1}{m^{2}\Gamma(m)}\int_{0}^{\infty}v^{m+1}e^{-v}dv=\frac{(1+m)}{m},$
(33)

which follows from the transformation $mz^{2}\to v$. Then, the first term of
(B) can be evaluated as

$\displaystyle\mathbb{E}\left\\{\left(p\sum\limits_{k=1}^{\beta}x_{k}\right)^{2}\right\\}$
$\displaystyle=\mathbb{E}\left\\{p^{2}\right\\}\mathbb{E}\left\\{\left(\sum\limits_{k=1}^{\beta}x_{k}\right)^{2}\right\\}$
$\displaystyle\overset{(a)}{=}2\mathbb{E}\left\\{\sum\limits_{k=1}^{\beta}x_{k}^{2}+2\sum_{\begin{subarray}{c}l_{1},l_{2}=1\\\
l_{1}\neq l_{2}\end{subarray}}^{\beta}x_{l_{1}}x_{l_{2}}\right\\}$
$\displaystyle=2\sum\limits_{k=1}^{\beta}\mathbb{E}\left\\{x_{k}^{2}\right\\}+4\sum_{\begin{subarray}{c}l_{1},l_{2}=1\\\
l_{1}\neq
l_{2}\end{subarray}}^{\beta}\mathbb{E}\left\\{x_{l_{1}}x_{l_{2}}\right\\}$
$\displaystyle=\beta,$ (34)

where (a) follows from $\mathbb{E}\\{p^{2}\\}=2$ and the multinomial theorem,
and the final result follows from

$\mathbb{E}\left\\{x_{k}^{2}\right\\}=\int_{-1}^{1}\frac{x_{k}^{2}dx}{\pi\sqrt{1-x_{k}^{2}}}=\frac{1}{2}$
(35)

and by using [33, Eq. 55]. Similarly, by using the multinomial theorem, the
second term of (B) can be expanded as

$\displaystyle\mathbb{E}\left\\{\left(p\sum\limits_{k=1}^{\beta}x_{k}\right)^{4}\right\\}$
$\displaystyle=\mathbb{E}\left\\{p^{4}\right\\}\mathbb{E}\left\\{\left(\sum\limits_{k=1}^{\beta}x_{k}\right)^{4}\right\\}$
$\displaystyle\overset{(b)}{=}\mathbb{E}\left\\{p^{4}\right\\}\mathbb{E}\left\\{\sum\limits_{k_{1}+k_{2}+\cdots+k_{\beta}=4}\frac{4!}{k_{1}!\>k_{2}!\>\cdots\>k_{\beta}!}\prod\limits_{i=1}^{\beta}x_{i}^{k_{i}}\right\\}$
$\displaystyle\overset{(c)}{=}8\left(\sum\limits_{k=1}^{\beta}\mathbb{E}\left\\{x_{k}^{4}\right\\}+3\mathbb{E}^{2}\left\\{x_{k}^{2}\right\\}\beta(\beta-1)\right)$
$\displaystyle\overset{(d)}{=}8\left(\frac{3\beta}{8}+\frac{3}{4}\beta(\beta-1)\right)=3\beta(2\beta-1),$
(36)

where $(b)$ follows from the multinomial theorem. Furthermore, $(c)$ follows
from $\mathbb{E}\\{p^{4}\\}=0\times\frac{1}{2}+2^{4}\times\frac{1}{2}=8$,

$\mathbb{E}\left\\{x_{k}\right\\}=\int_{-1}^{1}\frac{x_{k}dx}{\pi\sqrt{1-x_{k}^{2}}}=0,$
(37)

and $(d)$ follows from (35) and

$\mathbb{E}\left\\{x_{k}^{4}\right\\}=\int_{-1}^{1}\frac{x_{k}^{4}dx}{\pi\sqrt{1-x_{k}^{2}}}=\frac{3}{8}.$
(38)

By combining (B), (B), (B), and (B), we obtain

$z_{\rm
MC}=\varepsilon_{1}\beta+\varepsilon_{2}\frac{3(1+m)}{m}\beta(2\beta-1).$ (39)

##  Appendix C Proof of Theorem 2

From the expression (7), we have

$\displaystyle z_{\rm MNC}$
$\displaystyle=\varepsilon_{1}\mathbb{E}\left\\{|h|^{2}\sum\limits_{k=1}^{2\beta}s_{k}^{2}\right\\}+\varepsilon_{2}\mathbb{E}\left\\{|h|^{4}\sum\limits_{k=1}^{2\beta}s_{k}^{4}\right\\}$
$\displaystyle=\varepsilon_{1}\mathbb{E}\left\\{|h|^{2}\right\\}\mathbb{E}\left\\{\sum\limits_{k=1}^{2\beta}s_{k}^{2}\right\\}+\varepsilon_{2}\mathbb{E}\left\\{|h|^{4}\right\\}\mathbb{E}\left\\{\sum\limits_{k=1}^{2\beta}s_{k}^{4}\right\\}$
$\displaystyle=\varepsilon_{1}\left(1+\sum\limits_{k=1}^{\beta}\mathbb{E}\left\\{d_{k}^{2}\right\\}\right)\left(\sum\limits_{k=1}^{\beta}\mathbb{E}\left\\{x_{k}^{2}\right\\}\right)$
$\displaystyle\qquad+\varepsilon_{2}\left(1+\sum\limits_{k=1}^{\beta}\mathbb{E}\left\\{d_{k}^{4}\right\\}\right)\left(\sum\limits_{k=1}^{\beta}\mathbb{E}\left\\{x_{k}^{4}\right\\}\right)$
$\displaystyle\overset{(a)}{=}\varepsilon_{1}\beta+\varepsilon_{2}\frac{3(1+m)}{4m}\beta,$
(40)

where $(a)$ follows from (35), (38) and by assuming equally likely
transmissions of $d_{k}=\pm 1,$ $\forall$ $k$.

##  Appendix D Proof of Proposition 2

We prove the proposition in the form of two separate sub-cases: i) with, and
ii) without the correlator.

####  D-1 With correlator

Based on (6) and (12), we obtain the harvested DC as

$z_{\rm
UM,C}=\varepsilon_{1}\mathbb{E}\left\\{\left(|h|\sum\limits_{k=1}^{2\beta}x_{k}\right)^{2}\right\\}+\varepsilon_{2}\mathbb{E}\left\\{\left(|h|\sum\limits_{k=1}^{2\beta}x_{k}\right)^{4}\right\\}.$
(41)

Hence,

$\displaystyle z_{\rm UM,C}$
$\displaystyle=\varepsilon_{1}\mathbb{E}\left\\{|h|^{2}\right\\}\mathbb{E}\left\\{\left(\sum\limits_{k=1}^{2\beta}x_{k}\right)^{2}\right\\}$
$\displaystyle\qquad+\varepsilon_{2}\mathbb{E}\left\\{|h|^{4}\right\\}\mathbb{E}\left\\{\left(\sum\limits_{k=1}^{2\beta}x_{k}\right)^{4}\right\\}$
$\displaystyle=\varepsilon_{1}\mathbb{E}\left\\{\sum\limits_{k=1}^{2\beta}x_{k}^{2}+2\sum_{\begin{subarray}{c}l_{1},l_{2}=1\\\
l_{1}\neq l_{2}\end{subarray}}^{2\beta}x_{l_{1}}x_{l_{2}}\right\\}$
$\displaystyle\qquad+\varepsilon_{2}\frac{(1+m)}{m}\mathbb{E}\left\\{\sum\limits_{k=1}^{2\beta}x_{k}^{4}+6\sum_{\begin{subarray}{c}l_{1},l_{2}=1\\\
l_{1}\neq
l_{2}\end{subarray}}^{\beta(2\beta-1)}x_{l_{1}}^{2}x_{l_{2}}^{2}\right\\}$
$\displaystyle=\varepsilon_{1}\sum\limits_{k=1}^{2\beta}\mathbb{E}\left\\{x_{k}^{2}\right\\}$
$\displaystyle+\varepsilon_{2}\frac{(1+m)}{m}\left(\sum\limits_{k=1}^{2\beta}\mathbb{E}\left\\{x_{k}^{4}\right\\}+6\beta(2\beta-1)\mathbb{E}^{2}\left\\{x_{k}^{2}\right\\}\right)$
$\displaystyle\overset{(a)}{=}\varepsilon_{1}\beta+\varepsilon_{2}\frac{3(1+m)}{m}\beta(\beta-\frac{1}{4}),$
(42)

where $(a)$ follows from (35) and (38).

####  D-2 Without correlator

Based on (7) and (12), we have

$\displaystyle z_{\rm UM,NC}$
$\displaystyle=\varepsilon_{1}\mathbb{E}\left\\{|h|^{2}\sum\limits_{k=1}^{2\beta}x_{k}^{2}\right\\}+\varepsilon_{2}\mathbb{E}\left\\{|h|^{4}\sum\limits_{k=1}^{2\beta}x_{k}^{4}\right\\}$
$\displaystyle\overset{(b)}{=}\varepsilon_{1}\times
2\beta\times\frac{1}{2}+\varepsilon_{2}\times\frac{(1+m)}{m}\times
2\beta\times\frac{3}{8}$
$\displaystyle=\varepsilon_{1}\beta+\varepsilon_{2}\frac{3(1+m)}{4m}\beta,$
(43)

where $(b)$ follows from (35) and (38).

##  Appendix E Proof of Theorem 3

We prove the theorem in the form of two separate sub-cases with $\beta_{r}=0$
and $\beta_{r}>0.$

###  E-A Case $\beta_{r}=0$

In this case, we obtain the harvested energy $z_{\rm SR}$ as

$\displaystyle z_{\rm SR}$
$\displaystyle=\varepsilon_{1}\mathbb{E}\left\\{\left(d\sum\limits_{k=1}^{\beta}x_{k}\right)^{2}\right\\}$
$\displaystyle\qquad+\varepsilon_{2}\frac{(1+m)}{m}\mathbb{E}\left\\{\left(d\sum\limits_{k=1}^{\beta}x_{k}\right)^{4}\right\\}.$
(44)

From Appendix B, we assume an equally likely transmission of $d=\pm 1$ and by
following a similar framework, we obtain

$\displaystyle z_{\rm SR}$
$\displaystyle=\varepsilon_{1}\beta^{2}\mathbb{E}\left\\{d^{2}\right\\}\mathbb{E}\left\\{x_{k}^{2}\right\\}+\varepsilon_{2}\beta^{4}\mathbb{E}\left\\{d^{4}\right\\}\mathbb{E}\left\\{x_{k}^{4}\right\\}$
$\displaystyle=\frac{1}{2}\varepsilon_{1}\beta^{2}+\frac{3(1+m)}{8m}\varepsilon_{2}\beta^{4}.$
(45)

###  E-B Case $\beta_{r}>0$

Based on (6) and (18), we obtain

$\displaystyle z_{\rm SR}$
$\displaystyle=\varepsilon_{1}\mathbb{E}\left\\{\left(\sum\limits_{k=1}^{\beta+\beta_{r}}s_{k}\right)^{2}\right\\}$
$\displaystyle\qquad+\varepsilon_{2}\frac{(1+m)}{m}\mathbb{E}\left\\{\left(\sum\limits_{k=1}^{\beta+\beta_{r}}s_{k}\right)^{4}\right\\},$
(46)

where from (18), we have

$\sum\limits_{k=1}^{\beta+\beta_{r}}s_{k}=(1+\zeta
d)\sum\limits_{k=1}^{\beta_{r}}x_{k}.$ (47)

Then, the first term of (E-B) can be evaluated as

$\displaystyle\mathbb{E}\left\\{\left(\sum\limits_{k=1}^{\beta+\beta_{r}}s_{k}\right)^{2}\right\\}$
$\displaystyle=\mathbb{E}\left\\{\left(1+\zeta
d\right)^{2}\right\\}\mathbb{E}\left\\{\left(\sum\limits_{k=1}^{\beta_{r}}x_{k}\right)^{2}\right\\},$
(48)

where

$\mathbb{E}\left\\{\left(1+\zeta
d\right)^{2}\right\\}=\frac{1}{2}\left\\{\left(1+\zeta\right)^{2}+\left(1-\zeta\right)^{2}\right\\}=1+\zeta^{2},$
(49)

and

$\mathbb{E}\left\\{\left(\sum\limits_{k=1}^{\beta_{r}}x_{k}\right)^{2}\right\\}=\sum\limits_{k=1}^{\beta_{r}}\mathbb{E}\left\\{x_{k}^{2}\right\\}=\frac{\beta_{r}}{2}.$
(50)

By combining (49) and (50), we obtain

$\mathbb{E}\left\\{\left(\sum\limits_{k=1}^{\beta+\beta_{r}}s_{k}\right)^{2}\right\\}=\frac{\beta_{r}}{2}\left(1+\zeta^{2}\right)=\frac{\beta_{r}^{2}+\beta^{2}}{2\beta_{r}}.$
(51)

Similarly, by using the multinomial theorem, the second term in (E-B) can be
expanded as

$\displaystyle\mathbb{E}\left\\{\left(\sum\limits_{k=1}^{\beta+\beta_{r}}s_{k}\right)^{4}\right\\}$
$\displaystyle=\mathbb{E}\left\\{\left(1+\zeta
d\right)^{4}\right\\}\mathbb{E}\left\\{\left(\sum\limits_{k=1}^{\beta_{r}}x_{k}\right)^{4}\right\\},$
(52)

where

$\displaystyle\mathbb{E}\left\\{\left(1+\zeta d\right)^{4}\right\\}$
$\displaystyle=\frac{1}{2}\left\\{\left(1+\zeta\right)^{4}+\left(1-\zeta\right)^{4}\right\\}$
$\displaystyle=1+6\zeta^{2}+\zeta^{4}$ (53)

and

$\displaystyle\mathbb{E}\left\\{\left(\sum\limits_{k=1}^{\beta_{r}}x_{k}\right)^{4}\right\\}$
$\displaystyle=\sum\limits_{k=1}^{\beta_{r}}\mathbb{E}\left\\{x_{k}^{4}\right\\}+3\mathbb{E}^{2}\left\\{x_{k}^{2}\right\\}\beta(\beta-1)$
$\displaystyle=\frac{3\beta_{r}}{8}+\frac{3}{4}\beta_{r}(\beta_{r}-1).$ (54)

By combining (E-B) and (E-B), we obtain

$\mathbb{E}\left\\{\left(\sum\limits_{k=1}^{\beta+\beta_{r}}s_{k}\right)^{4}\right\\}=\frac{3}{8}\left(1+6\zeta^{2}+\zeta^{4}\right)\left(2\beta_{r}^{2}-\beta_{r}\right).$
(55)

Hence, we have $z_{\rm SR}$ equal to

$\displaystyle z_{\rm SR}\\!$
$\displaystyle=\\!\varepsilon_{1}\mathbb{E}\left\\{\left(\sum\limits_{k=1}^{\beta+\beta_{r}}s_{k}\right)^{2}\right\\}+\varepsilon_{2}\frac{(1+m)}{m}\mathbb{E}\left\\{\left(\sum\limits_{k=1}^{\beta+\beta_{r}}s_{k}\right)^{4}\right\\}$
$\displaystyle=\varepsilon_{1}\frac{\beta_{r}^{2}+\beta^{2}}{2\beta_{r}}+\varepsilon_{2}\frac{3(1+m)}{8m}\left(1+6\zeta^{2}+\zeta^{4}\right)\left(2\beta_{r}^{2}-\beta_{r}\right).$
(56)

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