# Information-Energy Regions in the Finite Block-Length Regime with Finite Channel Inputs

Sadaf Ul Zuhra, Samir M Perlaza, H. Vincent Poor, Eitan Altman, Mikael Skoglund

► To cite this version:

Sadaf Ul Zuhra, Samir M Perlaza, H. Vincent Poor, Eitan Altman, Mikael Skoglund. Information-Energy Regions in the Finite Block-Length Regime with Finite Channel Inputs. 2022. hal-03848708

HAL Id: hal-03848708

https://inria.hal.science/hal-03848708

Preprint submitted on 10 Nov 2022

HAL is a multi-disciplinary open access
archive for the deposit and dissemination of sci-
entific research documents, whether they are pub-
lished or not. The documents may come from
teaching and research institutions in France or
abroad, or from public or private research centers.

L'archive ouverte pluridisciplinaire HAL, est
destinée au dépôt et à la diffusion de documents
scientifiques de niveau recherche, publiés ou non,
émanant des établissements d'enseignement et de
recherche français ou étrangers, des laboratoires
publics ou privés.
# Simultaneous Information and Energy Transmission with Short Packets and Finite Constellations

Sadaf ul Zuhra, Member, IEEE, Samir M. Perlaza, Senior Member, IEEE, H. Vincent Poor, Fellow, IEEE, Mikael Skoglund, Fellow, IEEE

**Abstract**—This paper characterizes the trade-offs between information and energy transmission over an additive white Gaussian noise channel in the finite block-length regime with finite channel input symbols. These trade-offs are characterized in the form of inequalities involving the information transmission rate, energy transmission rate, decoding error probability (DEP) and energy outage probability (EOP) for a given finite block-length code. The first set of results identify a set of necessary conditions that a given code must satisfy for simultaneous information and energy transmission. Following this, a novel method for constructing a family of codes that can satisfy a target information rate, energy rate, DEP and EOP is proposed. Finally, achievability results identify the set of tuples of information rate, energy rate, DEP and EOP that can be simultaneously achieved by the constructed family of codes.

**Index Terms**—Simultaneous information and energy transmission, finite block-length, and finite channel inputs.

## 1. INTRODUCTION

Simultaneous information and energy transmission (SIET), also known as simultaneous wireless information and power transfer (SWIPT), is a technology that employs radio frequency (RF) signals to simultaneously transmit both information and energy to (possibly different) remote devices [4]–[6]. A central question in modern SIET is to characterize the information rate, energy rate, decoding error probability (DEP) and energy outage probability (EOP) that can be simultaneously achieved by any given transmission technique. Such question has been traditionally studied under at least one of two simplifying assumptions. The first assumption is that the transmission duration is infinitely long, as in [5], [7]–[11]. Under this assumption, the DEP and the EOP can be made

arbitrarily close to zero, and thus, the focus is exclusively on the information and energy transmission rates. If this assumption is dropped, as in the remainder of this work, the DEP and EOP are bounded away from zero, and thus, the performance analysis of SIET shall take into account four metrics: information rate, energy rate, DEP, and EOP. The second common assumption is that channel inputs symbols can be drawn from probability distributions whose support is uncountable, e.g., Gaussian distributions as in [12]. This presumes the ability to transmit with infinitely many channel input symbols, which is not the case in practice. In a departure from this assumption, this work considers the more practical case of finite channel input symbols, which provides insights for practical systems operating with modulation schemes built upon commercial constellations.

### A. State of the art

The existing body of work in SIET can be divided into three main categories. The first category is the study of the trade-offs between the information and energy rates that can be simultaneously transmitted by an RF signal. In the asymptotic block-length regime, the focus is on the notion of information-energy regions to characterize the set of all information and energy rate pairs that can be simultaneously achieved [8]. To capture the trade-off between the information and energy rates, a capacity-energy function is defined in [7] for various channels including the discrete memoryless channel, binary symmetric channel, and the additive white Gaussian noise (AWGN) channel. The information-energy capacity region of the Gaussian multiple access channel is characterized in [10], [13], whereas the information-energy capacity region of the Gaussian interference channel is approximated in [11]. In the finite block-length regime, [14] and [15] provide a characterization of the information-energy capacity region with binary antipodal channel inputs and fixed DEP and EOP. Converse and achievability bounds on these metrics for finite channel input symbols are presented in [1] and [3], respectively. The fundamental limits of SIET for the case of semantic communications are characterized in [16].

The second category of research studies the modeling of energy harvesting (EH) circuits. One of the key questions in this regard tackles the estimation of the amount of energy that can be harvested from an RF signal. This line of inquiry has revealed that, due to the presence of non-linear elements such as diodes in the EH circuits, the expected energy harvested from a signal is a function of the second and fourth powers

Sadaf ul Zuhra, Samir M. Perlaza, and H. Vincent Poor are with the Department of Electrical and Computer Engineering, Princeton University, 08540 Princeton, NJ, USA. {sadaf.zuhra, poor}@princeton.edu

Samir M. Perlaza is with INRIA, Centre Inria d'Université Côte d'Azur, 2004 Route des Lucioles, 06902 Sophia Antipolis, France; and also with the Laboratoire de Mathématiques GAATI, Université de la Polynésie Française, BP 6570, 98702 Faa, French Polynesia. {samir.perlaza}@inria.fr

Mikael Skoglund is with the School of Electrical Engineering and Computer Science, Malvinas Väg 10, KTH Royal Institute of Technology, 11428 Stockholm, Sweden. (skoglund@kth.se)

This research was supported in part by the U.S. National Science Foundation under grant ECCS-2335876; the European Commission through the H2020-MSCA-RISE-2019 project 872172; the French National Agency for Research (ANR) through the project ANR-21-CE25-0013 and the project ANR-22-PEFT-0010 of the France 2030 program PEPR Réseaux du Futur; and in part by the Agence de l'innovation de défense (AID) through the project UK-FR 2024352.

Parts of this work were presented at the IEEE Information Theory Workshop (ITW) in 2021 [1] and in 2022 [2], and IEEE International Symposium on Information Theory (ISIT) in 2022 [3].
# Information-Energy Regions in the Finite Block-Length Regime with Finite Channel Inputs

Sadaf ul Zuhra, Member, IEEE, Samir M. Perlaza, Senior Member, IEEE, H. Vincent Poor, Fellow, IEEE, Eitan Altman, Fellow, IEEE, Mikael Skoglund, Fellow, IEEE

## Abstract

This paper provides a complete characterization of the information-energy region of simultaneous information and energy transmission over an additive white Gaussian noise channel in the finite block-length regime with finite sets of channel input symbols. Given a set of channel input symbols, the converse characterizes the tuples of information rate, energy rate, decoding error probability (DEP) and energy outage probability (EOP) that cannot be achieved by any code built using the given set of channel inputs. A novel method for constructing a family of codes that respects the given information rate, energy rate, DEP and EOP requirements is proposed. The achievable region identifies the set of tuples of information rate, energy rate, DEP and EOP that can be achieved by the constructed family of codes. The proposed construction proves to be information rate, energy rate, and EOP optimal. The achieved DEP is, however, sub-optimal, owing to the choice of the decoding regions made during the construction.

## Index Terms

Simultaneous information and energy transmission, SIET, SWIPT, information-energy regions, achievability, converse, finite block-length, finite channel inputs

## 1. INTRODUCTION

Simultaneous information and energy transmission (SIET) (also known as simultaneous wireless information and power transfer (SWIPT)) employs radio frequency (RF) signals to simultaneously accomplish the tasks of conveying information and providing energy to (possibly different) devices. In the following, for the sake of correctness, the denomination “energy transmission” is preferred against “power transfer”, and thus, the acronym SIET is adopted in the remainder of this paper. A fundamental question of interest in SIET is that, given a certain code, what can and cannot be accomplished by such a code in terms of the information and energy transmission rates. It is particularly interesting to answer this question in the finite block-length regime with finite sets of channel input symbols. The trade-offs between information and energy transmission rates in SIET have previously been studied in the asymptotic regime [1]–[6] where the assumption of infinitely long transmissions guarantees that the decoding error probability (DEP) and the energy outage probability (EOP) can be made arbitrarily close to zero. Thus, the focus of the asymptotic results is only on the information and energy transmission rates. In the finite block-length regime, however, the DEP and EOP are bounded away from zero and pose additional constraints on the fundamental limits of SIET.

A common assumption in the study of SIET over an additive white Gaussian noise (AWGN) channel is that the channel inputs are derived independently from a Gaussian distribution. In a departure from this norm, this work considers the more meaningful, discrete set of channel input symbols that define the channel. This work characterizes the fundamental limits of SIET for such channels in the finite block-length regime in the presence of AWGN noise. Unlike Gaussian inputs, the fundamental limits obtained for finite discrete sets of channel input symbols provide insights for practical systems that operate with finite block-lengths and finite sets of channel inputs.

Sadaf ul Zuhra, Samir M. Perlaza and Eitan Altman are with INRIA, Centre de Recherche de Sophia Antipolis - Méditerranée, 2004 Route des Lucioles, 06902 Sophia Antipolis, France. {sadaf-ul.zuhra, samir.perlaza, eitan.altman}@inria.fr

H. Vincent Poor, Sadaf ul Zuhra, and Samir M. Perlaza are with the Department of Electrical and Computer Engineering, Princeton University, 08540 Princeton, NJ, USA. (poor@princeton.edu)

Mikael Skoglund is with the School of Electrical Engineering and Computer Science, Malvinas Väg 10, KTH Royal Institute of Technology, 11428 Stockholm, Sweden. (skoglund@kth.se)

Samir M. Perlaza is also with the Laboratoire de Mathématiques GAATI, Université de la Polynésie Française, BP 6570, 98702 Faaa, French Polynesia.

Eitan Altman is also with the Laboratoire d'Informatique d'Avignon, Université d'Avignon, 84911 Avignon, France; and with the Laboratory of Information, Network and Communication Sciences, 75013 Paris, France.

This research was supported in part by the European Commission through the H2020-MSCA-RISE-2019 program under grant 872172; in part by the Agence Nationale de la Recherche (ANR) through the project MAESTRO-5G (ANR-18-CE25-0012); in part by the U.S. National Science Foundation under Grant CCF-1908308; and in part by the French Government through the “Plan de Relance” and “Programme d’investissements d’avenir”.

This paper was presented in part at the IEEE Information Theory Workshop (ITW) 2021, IEEE International Symposium on Information Theory 2022 and IEEE ITW 2022.
of the signal magnitude [17]. Recent research on EH non-linearities [12], [18] has shown that energy models that do not account for these non-linearities result in inaccurate estimates of the harvested energy. The impact of the energy harvester non-linearities on the fundamental limits of SIET in the finite block-length regime has been studied in [2]. In [19], a learning based approach is proposed for modeling the EH circuits where the memory of non-linear elements in the EH circuit is modeled as a Markov decision process.

The third line of research covers the system design, implementation, and applications of SIET networks. Optimal waveform design for SIET from a multi-antenna transmitter to multiple single antenna receivers is studied in [20]. Signal and system design exclusively for wireless energy transmission has been studied in [18], [21]–[24] and [25]. In [26], the authors optimize resource allocation and beamforming for intelligent reflecting surfaces aided SIET. An algorithm for designing a circular quadrature amplitude modulation scheme for SIET that maximizes the peak-to-average power ratio has been proposed in [27].

More comprehensive overviews of the work on SIET in the second and third categories detailed above, can be found in [28], [29], [30], and [31]. A comparison of relevant aspects of the existing literature on SIET with this work is provided in Table I below. A tick mark indicates that the specific factor has been taken into consideration while a dash indicates the opposite.

## B. Contributions

This work provides a comprehensive characterization of the relationships between the parameters of short packet codes with finite channel input symbols (finite constellations, e.g., QPSK, 16-QAM, 64-QAM, among others) for simultaneous transmission of information and energy. The main contributions can be summarized as follows.

* The first set of results, presented in Section 3, provide bounds on the information transmission rate, energy transmission rate, DEP and EOP of any given code over a point-to-point AWGN channel. All these bounds emerge to be functions of common code parameters such as the block-length and the type. Together, these bounds form necessary conditions that any code must satisfy for accomplishing the tasks of information and energy transmission simultaneously. Through the dependence on common parameters of the code, these bounds also characterize the trade-offs between information rate, energy rate, DEP and EOP of a given code.

* A method for constructing codes for SIET over short packet communication is proposed. Using this method, a family of codes can be designed to achieve given rates of information and energy transmission while also ensuring that the DEP and EOP are kept within the required thresholds. The feasible tuples of information rate, energy rate, DEP and EOP that can be achieved by the codes thus constructed are also characterized.

* The performance of the constructed codes is evaluated by comparing achievability results with the identified necessary conditions. Comprehensive numerical analyses are also conducted to illustrate the trade-offs between the competing objectives of information and energy transmission identified by the results in this work.

## C. Notation

The sets of natural, real and complex numbers are denoted by $\mathbb{N}$, $\mathbb{R}$ and $\mathbb{C}$, respectively. In particular, $0 \notin \mathbb{N}$. Random variables and random vectors are denoted by uppercase letters and lowercase bold letters, respectively. Scalars are denoted by lowercase letters and vectors by lowercase bold letters. The real and imaginary parts of a complex number $c \in \mathbb{C}$ are denoted by $\Re(c)$ and $\Im(c)$, respectively. The complex conjugate of $c \in \mathbb{C}$ is denoted by $c^*$ and the magnitude of $c$ is denoted by $|c|$. The imaginary unit is denoted by $i$, i.e., $i^2 = -1$. The empty set is denoted by $\emptyset$. The sinc function is defined as follows

$$ \text{sinc}(t) \triangleq \frac{\sin(\pi t)}{\pi t}. \qquad (1) $$

The tail distribution function of the standard normal distribution is referred to as the $Q$-function where

$$ Q(x) = \int_{x}^{\infty} \frac{1}{\sqrt{2\pi}} \exp\left(-\frac{t^2}{2}\right) dt. \qquad (2) $$

## 2. SYSTEM MODEL

Consider a communication system formed by a transmitter, an information receiver (IR), and an energy harvester (EH). The objective of the transmitter is to simultaneously send information to the IR at a rate of $R$ bits per second while also delivering $B$ Joules of energy to the EH over an AWGN channel. The transmission takes place over a finite duration of $n \in \mathbb{N}$ channel uses. Thus, the rate of energy transmission is $B/n$ Joules per channel use. The transmitter uses $L$ symbols from the set of channel input symbols

$$ \mathcal{X} \triangleq \{x^{(1)}, x^{(2)}, \dots, x^{(L)}\} \subset \mathbb{C}. \qquad (3) $$

For all $m \in \{1, 2, \dots, n\}$, denote by $\nu_m \in \mathcal{X}$, the symbol to be transmitted during channel use $m$. Denote the vector of channel input symbols transmitted over $n$ channel uses by

$$ \boldsymbol{\nu} = (\nu_1, \nu_2, \dots, \nu_n)^{\mathrm{T}}. \qquad (4) $$

The baseband frequency of the transmitter in Hertz (Hz) is $f_w$. Denote by $T \triangleq \frac{1}{f_w}$, the duration of a channel use in time units. Hence, the transmission takes place over $nT$ time units. The complex baseband signal at time $t$, with $t \in [0, nT]$ is given by

$$ x(t) = \sum_{m=1}^{n} \nu_m \text{sinc}(f_w (t - (m-1)T)), \qquad (5) $$

where the sinc function is defined in (1). The signal $x(t)$ in (5) has a bandwidth of $\frac{f_w}{2} > 0$ Hz. Let $f_c > \frac{f_w}{2}$ denote the center frequency of the transmitter. The RF signal input to the channel at time $t$, denoted by $\tilde{x}(t)$, is obtained by the
The existing body of work in SIET can be studied under three main categories. The first is the study of the trade-offs between the information and energy rates that can be simultaneously transmitted by an RF signal. Earlier research in this area is focused primarily on the asymptotic regime [1]–[4]. In this case, the notion of information-energy region generalizes to the set of all information and energy rate tuples that can be simultaneously achieved in the asymptotic block-length regime [2]. To capture the trade-off between the information and energy rates, [1] defines a capacity-energy function for various channels including the discrete memoryless channel, binary symmetric channel, and the AWGN channel. In [4], the information-energy trade-off is studied for a coupled-inductor circuit that models a slow frequency-selective fading channel. The information-energy capacity region of the Gaussian multiple access channel is characterized in [5], whereas the information-energy capacity region of the Gaussian interference channel is approximated in [6]. Nonetheless, the information-energy trade-off is not the only trade-off involved in SIET. In the finite block-length regime, several other trade-offs appear which are taken into consideration in this paper.

Within the finite block-length regime, [7] and [8] provide a characterization of the information-energy capacity region with binary antipodal channel inputs. A converse and an achievable information-energy region of SIET with arbitrary number of channel inputs is presented in [9] and [10], respectively. It has been shown in [9] and [10] that, in the non-asymptotic regime, these limits are a function of the type induced by the code used for the transmission. A type is understood in the sense of the empirical frequency with which each channel input symbol appears in the codewords [11]. The impact of energy harvester non-linearities on the fundamental limits of SIET in the finite block-length regime has been studied in [12].

The second area of research that has received considerable attention in SIET is the modelling of the energy harvester (EH) circuits with the aim of providing accurate estimates of the energy gathered from an RF signal. This line of inquiry has revealed that, due to the presence of non-linear elements such as diodes in the EH circuits, the expected energy harvested from a signal is a function of the fourth power of the signal magnitude [13] in addition to the squared magnitude as was conventionally assumed (see [2], [5] and [8]). Recent research on EH non-linearities [14], [15] has shown that energy models that do not account for these non-linearities result in inaccurate estimates of the harvested energy. This work accounts for the EH non-linearities by adopting the models proposed in the literature for determining the energy harvested from the transmitted RF signals.

The third category of research deals with aspects related to the design and implementation of SIET such as signal and system design, resource allocation, receiver architectures, energy harvester circuits, and decoding strategies. Optimal waveform design for SIET from a multi-antenna transmitter to multiple single antenna receivers is studied in [16]. Signal and system design exclusively for wireless energy transmission has been studied in [15], [17]–[20] and [21]. In [22], the authors optimize resource allocation and beamforming for intelligent reflecting surfaces aided SIET. The memory of non-linear elements in the EH circuit is modeled as a Markov decision process in [23] and a learning based model is proposed for the EH circuit. An algorithm for designing a circular quadrature amplitude modulation scheme for SIET that maximizes the peak-to-average power ratio has been proposed in [24]. This paper contributes to this line of research by providing a method of code construction for SIET that proves to be information rate, energy rate and EOP optimal.

More comprehensive overviews of the work on SIET in the second and third categories detailed above, can be found in [25], [26], [27], and [28]. A comparison of relevant aspects of the existing literature on SIET with this work is provided in Table I below. A tick mark indicates that the specific factor has been taken into consideration in the paper while a dash indicates that the concerned feature has not been considered in that reference.

The main contributions of this paper are summarized below.

* The converse for finite block-length SIET over an AWGN channel with a peak-amplitude constraint is characterized for a given finite set of channel input symbols. The converse defines the set of information rate, energy rate, DEP and EOP tuples that cannot be achieved by any code that employs the given set of channel input symbols.

It is of interest to characterize the information-energy regions for a given set of channel input symbols because channel inputs are usually designed for very specific functions and are, therefore, fixed by design in a system. Ideally, the converse information-energy region would be characterized for the optimal set of channel inputs. However, determining the optimal set of channel inputs is known to be a very difficult problem [44]–[46], even without energy considerations. Interestingly, the process of characterizing the information-energy regions for given sets of channel input symbols results in guidelines for designing optimal channel input symbols for SIET. This characterization also reveals the dependence of the information rate, the energy rate, the DEP and the EOP on various parameters of the code which, in turn, provides insights into which parameters should be changed and how in order to achieve a certain performance objective in SIET.

* Based upon the insights provided by the converse information-energy region, a method of constructing sets of channel input symbols and codes for SIET over an AWGN channel in the finite block-length regime is proposed.

* The achievable information-energy region for the constructed family of codes is characterized. The achievable region defines the set of information rate, energy rate, DEP and EOP tuples that can be achieved by at least one code from the
<table><caption>TABLE I: Summary of the state-of-art</caption><thead><tr><th>Reference</th><th>Channel inputs</th><th>Block-length</th><th>EH non-linearities</th><th>Channel</th><th>DEP</th><th>EOP</th></tr></thead><tbody><tr><td>[7]</td><td>Infinite</td><td>Asymptotic</td><td>-</td><td>DMC, AWGN</td><td>-</td><td>-</td></tr><tr><td>[5]</td><td>Infinite</td><td>Asymptotic</td><td>-</td><td>AWGN + Frequency-selective fading</td><td>-</td><td>-</td></tr><tr><td>[9]</td><td>Infinite</td><td>Asymptotic</td><td>✓</td><td>Rayleigh fading</td><td>-</td><td>-</td></tr><tr><td>[32]</td><td>Infinite</td><td>Asymptotic</td><td>-</td><td>Flat fading</td><td>-</td><td>-</td></tr><tr><td>[19]</td><td>Infinite</td><td>Asymptotic</td><td>-</td><td>AWGN</td><td>-</td><td>-</td></tr><tr><td>[33]</td><td>Finite</td><td>Asymptotic</td><td>-</td><td>AWGN + Fading</td><td>-</td><td>-</td></tr><tr><td>[26]</td><td>Finite</td><td>Asymptotic</td><td>-</td><td>AWGN + Flat-fading</td><td>-</td><td>-</td></tr><tr><td>[34]</td><td>Finite</td><td>-</td><td>-</td><td>AWGN + Rayleigh fading</td><td>✓</td><td>-</td></tr><tr><td>[27]</td><td>Finite</td><td>-</td><td>-</td><td>AWGN</td><td>-</td><td>-</td></tr><tr><td>[17]</td><td>Infinite</td><td>Asymptotic</td><td>✓</td><td>AWGN</td><td>-</td><td>-</td></tr><tr><td>[12]</td><td>Infinite</td><td>Asymptotic</td><td>✓</td><td>Multi-path fading</td><td>-</td><td>-</td></tr><tr><td>[35]</td><td>Uncountable</td><td>Asymptotic</td><td>-</td><td>AWGN + Rayleigh fading</td><td>✓</td><td>-</td></tr><tr><td>[36]</td><td>Uncountable</td><td>Asymptotic</td><td>-</td><td>AWGN + Rayleigh and Rician fading</td><td>-</td><td>-</td></tr><tr><td>[37]</td><td>Uncountable</td><td>Asymptotic</td><td>-</td><td>AWGN</td><td>-</td><td>-</td></tr><tr><td>[38]</td><td>Uncountable</td><td>Asymptotic</td><td>✓</td><td>Multi-path fading</td><td>-</td><td>-</td></tr><tr><td>[39]</td><td>Uncountable</td><td>Asymptotic</td><td>-</td><td>Multi-path fading</td><td>-</td><td>-</td></tr><tr><td>[40]</td><td>Uncountable</td><td>Asymptotic</td><td>-</td><td>AWGN + Flat-fading</td><td>-</td><td>-</td></tr><tr><td>[41]</td><td>Finite</td><td>Finite</td><td>✓</td><td>AWGN</td><td>-</td><td>-</td></tr><tr><td>[14]</td><td>Finite</td><td>Finite</td><td>-</td><td>BSC</td><td>✓</td><td>✓</td></tr><tr><td>[15]</td><td>Finite</td><td>Finite</td><td>-</td><td>BSC</td><td>-</td><td>-</td></tr><tr><td><b>This work</b></td><td>Finite</td><td>Finite</td><td>✓</td><td>AWGN</td><td>✓</td><td>✓</td></tr></tbody></table>

frequency up-conversion of the baseband signal $x(t)$ in (5) as follows:

$$ \tilde{x}(t) = \Re(x(t)\sqrt{2}\exp(i2\pi f_c t)), \quad (6) $$

where $i$ is the complex unit. The RF outputs of the AWGN channel at time $t \in [0, nT]$ are the random variables

$$ Y(t) = \tilde{x}(t) + N_1(t), \quad \text{and} \quad (7a) $$

$$ Z(t) = \tilde{x}(t) + N_2(t), \quad (7b) $$

where, for all $t \in [0, nT]$, the random variables $N_1(t)$ and $N_2(t)$ represent real white Gaussian noise with zero mean and variance $\sigma^2$; and the random variables $Y(t)$ and $Z(t)$ are the inputs to the IR and the EH, respectively.

At the IR, the received signal $Y(t)$ in (7a) is first multiplied with $\sqrt{2}\exp(-i2\pi f_c t)$ to obtain the down-converted output. The down-converted output is then passed through a unit gain low pass filter with impulse response $f_w$ sinc($f_w t$) that has a cut-off frequency of $\frac{f_w}{2}$ Hz to obtain the complex baseband equivalent of $Y(t)$. This is followed by ideally sampling the complex baseband output at intervals of $1/f_w$. The resulting discrete time baseband output at the end of $n$ channel uses is given by the following random vector [42, Section 2.2.4]:

$$ Y = \nu + N, \quad (8) $$

where the vector $Y = (Y_1, Y_2, \dots, Y_n)^T \in \mathbb{C}^n$ is the input to the IR; the vector of channel input symbols $\nu$ is in (4); and the noise vector $N = (N_1, N_2, \dots, N_n)^T \in \mathbb{C}^n$ is such that, for all $m \in \{1, 2, \dots, n\}$, the random variable $N_m$ is a complex circularly symmetric Gaussian random variable whose real and imaginary parts have zero means and variances $\frac{1}{2}\sigma^2$. Moreover, the random variables $N_1, N_2, \dots, N_n$ are mutually independent (see [42, Section 2.2.4]). Therefore, for all $y = (y_1, y_2, \dots, y_n)^T \in \mathbb{C}^n$, and for all $\nu =$

$(\nu_1, \nu_2, \dots, \nu_n)^T \in \mathbb{C}^n$, the conditional probability density function of the channel output $Y$ in (8) is given by

$$ f_{Y|X}(y|\nu) = \prod_{m=1}^{n} f_{Y|X}(y_m|\nu_m), \quad (9a) $$

where, for all $m \in \{1, 2, \dots, n\}$,

$$ f_{Y|X}(y_m|\nu_m) = \frac{1}{\pi\sigma^2} \exp\left(-\frac{|y_m - \nu_m|^2}{\sigma^2}\right). \quad (9b) $$

At the EH, the RF signal $Z(t)$ in (7b) is not down-converted or filtered. See for instance [12], [18].

## A. Information and Energy Transmission

Let $M$ be the cardinality of the set of symbols at the output of the information source. To accomplish the tasks of information and energy transmission, the transmitter makes use of an $(n, M, \mathcal{X})$-code defined as follows.

**Definition 2.1** ((n, M, X)-code). An $(n, M, X)$-code, with $X$ in (3), is a set of pairs

$$ \{(u(1), D_1), (u(2), D_2), \dots, (u(M), D_M)\}, \quad (10) $$

such that for all $(i, j) \in \{1, 2, \dots, M\}^2$, with $i \neq j$,

$$ u(i) = (u_1(i), u_2(i), \dots, u_n(i)) \in \mathcal{X}^n, \quad (11a) $$

$$ D_i \subseteq \mathbb{C}^n, \text{ and} \quad (11b) $$

$$ D_i \cap D_j = \emptyset. \quad (11c) $$

Note that Definition 2.1 specifies the exact set of channel input symbols via the set $\mathcal{X}$. Thus, constraints on the average power or peak-amplitude power are vacuous as those are fixed by the definition of $\mathcal{X}$.

The associated encoding, decoding and energy harvesting operations for a given $(n, M, \mathcal{X})$-code are defined as follows.
<table><caption>TABLE I: Summary of the state-of-art</caption><thead><tr><th rowspan="2">Reference</th><th rowspan="2">Channel inputs</th><th rowspan="2">Block-length</th><th colspan="3">Point-to-point</th></tr><tr><th>EH non-linearities</th><th>Channel</th><th>DEP</th><th>EOP</th></tr></thead><tbody><tr><td>[1]</td><td>Infinite</td><td>Asymptotic</td><td>-</td><td>DMC, AWGN</td><td>-</td><td>-</td></tr><tr><td>[4]</td><td>Infinite</td><td>Asymptotic</td><td>-</td><td>AWGN + Frequency-selective fading</td><td>-</td><td>-</td></tr><tr><td>[3]</td><td>Infinite</td><td>Asymptotic</td><td>✓</td><td>Rayleigh fading</td><td>-</td><td>-</td></tr><tr><td>[29]</td><td>Infinite</td><td>Asymptotic</td><td>-</td><td>Flat fading</td><td>-</td><td>-</td></tr><tr><td>[23]</td><td>Infinite</td><td>Asymptotic</td><td>-</td><td>AWGN</td><td>-</td><td>-</td></tr><tr><td>[30]</td><td>Finite</td><td>Asymptotic</td><td>-</td><td>AWGN + Fading</td><td>-</td><td>-</td></tr><tr><td>[22]</td><td>Finite</td><td>Asymptotic</td><td>-</td><td>AWGN + Flat-fading</td><td>-</td><td>-</td></tr><tr><td>[31]</td><td>Finite</td><td>-</td><td>-</td><td>AWGN + Rayleigh fading</td><td>✓</td><td>-</td></tr><tr><td>[24]</td><td>Finite</td><td>-</td><td>-</td><td>AWGN</td><td>-</td><td>-</td></tr><tr><td>[13]</td><td>Infinite</td><td>Asymptotic</td><td>✓</td><td>AWGN</td><td>-</td><td>-</td></tr><tr><td>[14]</td><td>Infinite</td><td>Asymptotic</td><td>✓</td><td>Multi-path fading</td><td>-</td><td>-</td></tr><tr><td>[32]</td><td>Uncountable</td><td>Asymptotic</td><td>-</td><td>AWGN + Rayleigh fading</td><td>✓</td><td>-</td></tr><tr><td>[33]</td><td>Uncountable</td><td>Asymptotic</td><td>-</td><td>AWGN + Rayleigh and Rician fading</td><td>-</td><td>-</td></tr><tr><td>[34]</td><td>Uncountable</td><td>Asymptotic</td><td>-</td><td>AWGN</td><td>-</td><td>-</td></tr><tr><td>[35]</td><td>Uncountable</td><td>Asymptotic</td><td>✓</td><td>Multi-path fading</td><td>-</td><td>-</td></tr><tr><td>[36]</td><td>Uncountable</td><td>Asymptotic</td><td>-</td><td>Multi-path fading</td><td>-</td><td>-</td></tr><tr><td>[37]</td><td>Uncountable</td><td>Asymptotic</td><td>-</td><td>AWGN + Flat-fading</td><td>-</td><td>-</td></tr><tr><td>[38]</td><td>Finite</td><td>Finite</td><td>✓</td><td>AWGN</td><td>-</td><td>-</td></tr><tr><td>[7]</td><td>Finite</td><td>Finite</td><td>-</td><td>BSC</td><td>✓</td><td>✓</td></tr><tr><td>[8]</td><td>Finite</td><td>Finite</td><td>-</td><td>BSC</td><td>-</td><td>-</td></tr><tr><td><strong>This work</strong></td><td>Finite</td><td>Finite</td><td>✓</td><td>AWGN</td><td>✓</td><td>✓</td></tr></tbody></table><table><thead><tr><th></th><th colspan="3">MAC</th></tr></thead><tbody><tr><td>[5]</td><td>Finite</td><td>Asymptotic</td><td>AWGN</td></tr><tr><td>[6]</td><td>Finite</td><td>Asymptotic</td><td>Gaussian interference</td></tr><tr><td>[39]</td><td>Uncountable</td><td>Asymptotic</td><td>Path-loss, Rayleigh fading</td></tr><tr><td>[40]</td><td>Uncountable</td><td>Asymptotic</td><td>AWGN</td></tr><tr><td>[41]</td><td>Uncountable</td><td>Asymptotic</td><td>AWGN + flat fading</td></tr><tr><td>[42]</td><td>Uncountable</td><td>Asymptotic</td><td>AWGN + flat fading</td></tr><tr><td>[43]</td><td>Uncountable</td><td>Asymptotic</td><td>AWGN + flat fading</td></tr></tbody></table>

constructed family of codes.

* The trade-offs between the information rate, energy rate, DEP and EOP are illustrated through various instructive examples. The information-energy region is also illustrated for a given set of channel input symbols. The study of the gaps between the converse and achievability regions shows that the constructed codes are information rate, energy rate and EOP optimal. The DEP achieved by the constructed codes is, however, sub-optimal.

The rest of this paper is organized as follows. The notations used in this paper and the system model are presented in Sections 2 and 3, respectively. The converse region for finite block-length SIET is characterized in Section 4. Section 5 provides the construction of codes for finite block-length SIET followed by the characterization of an achievable region. The trade-offs between various parameters of finite block-length SIET are elucidated using a simple set of channel input symbols in Section 6. Section 7, illustrates the characterized converse and achievable regions and provides insights into the optimality of the constructed class of codes for SIET.

## 2. NOTATION

The sets of natural, real and complex numbers are denoted by $\mathbb{N}$, $\mathbb{R}$ and $\mathbb{C}$, respectively. In particular, $0 \notin \mathbb{N}$. The Borel $\sigma$-algebra on $\mathbb{R}$ is denoted by $\mathcal{B}(\mathbb{R})$. Random variables and random vectors are denoted by uppercase letters and uppercase bold letters, respectively. Scalars are denoted by lowercase letters and vectors by lowercase bold letters. The real and imaginary parts of a complex number $c \in \mathbb{C}$ are denoted by $\Re(c)$ and $\Im(c)$, respectively. The complex conjugate of $c \in \mathbb{C}$ is denoted by $c^*$ and the magnitude of $c$ is denoted by $|c|$. The imaginary unit is denoted by $i$, i.e., $i^2 = -1$. The Kullback-Leibler (KL) divergence between two measures $P$ and $R$, with $P$ absolutely continuous with $R$, is denoted by $D(P||R) = \int \log \frac{dP}{dR}dP$, where $\frac{dP}{dR}$ denotes the Radon-Nykodim derivative of $P$ with respect to $R$. The Fourier transform of an integrable signal $x(t)$ is denoted by $\hat{x}(t)$.

$$ \text{sinc}(t) \triangleq \frac{\sin(\pi t)}{\pi t}, \qquad (1) $$

and the $Q$ function is given by the following:

$$ Q(x) = \int_{x}^{\infty} \frac{1}{\sqrt{2\pi}} \exp \left( -\frac{t^2}{2} \right) dt. \qquad (2) $$
### 3. SYSTEM MODEL

Consider a communication system formed by a transmitter, an information receiver (IR), and an energy harvester (EH). The objective of the transmitter is to simultaneously send information to the IR at a rate of $R$ bits per second; and energy to the EH at a rate of $B$ Joules per second over an additive white Gaussian noise (AWGN) channel. The transmission takes place over a finite duration of $n \in \mathbb{N}$ channel uses. The transmitter uses $L$ symbols from the set

$$ \mathcal{X} \triangleq \{x^{(1)}, x^{(2)}, \dots, x^{(L)}\} \subset \mathbb{C} \qquad (3) $$

that contains all possible channel input symbols. That is,

$$ L \triangleq |\mathcal{X}|. \qquad (4) $$

For all $m \in \{1, 2, \dots, n\}$, denote by $\nu_m \in \mathcal{X}$, the symbol to be transmitted during channel use $m$. Denote the vector of channel input symbols over $n$ channel uses by

$$ \boldsymbol{\nu} = (\nu_1, \nu_2, \dots, \nu_n)^{\mathrm{T}}. \qquad (5) $$

The baseband frequency of the transmitter in Hertz (Hz) is denoted by $f_w$. Denote by $T = \frac{1}{f_w}$, the duration of a channel use in time units. Hence, the transmission takes place during $nT$ time units. The complex baseband signal at time $t$, with $t \in [0, nT]$ is given by

$$ x(t) = \sum_{m=1}^{n} \nu_m \mathrm{sinc}(f_w(t - (m-1)T)), \qquad (6) $$

where the sinc function is in (1). The signal $x(t)$ in (6) has a bandwidth of $f_w > 0$ Hz. Let $f_c > f_w$ denote the center frequency of the transmitter. The RF signal input to the channel at time $t$, denoted by $\tilde{x}(t)$, is obtained by the frequency up-conversion of the baseband signal $x(t)$ in (6) as follows:

$$ \tilde{x}(t) = \Re(x(t)\sqrt{2}\exp(i2\pi f_c t)), \qquad (7) $$

where $i$ is the complex unit. The RF outputs of the AWGN channel at time $t \in [0, nT]$ are the random variables

$$ Y(t) = \tilde{x}(t) + N_1(t), \quad \text{and} \qquad (8a) $$

$$ Z(t) = \tilde{x}(t) + N_2(t), \qquad (8b) $$

where, for all $t \in [0, nT]$, the random variables $N_1(t)$ and $N_2(t)$ represent real white Gaussian noise with zero mean and variance $\sigma^2$. $Y(t)$ and $Z(t)$ are the inputs to the IR and the EH, respectively.

At the IR, the received signal $Y(t)$ in (8a) is first multiplied with $\sqrt{2}\exp(-i2\pi f_c t)$ to obtain the down-converted output. The down-converted output is then passed through a unit gain low pass filter with impulse response $f_w \mathrm{sinc}(f_w t)$ that has a cut-off frequency of $\frac{f_w}{2}$ Hz to obtain the complex baseband equivalent of $Y(t)$. This is followed by ideally sampling the complex baseband output at intervals of $1/f_w$. The resulting discrete time baseband output at the end of $n$ channel uses is given by the following random vector [47, Section 2.2.4]:

$$ Y = \nu + N, \qquad (9) $$

where the vector $Y = (Y_1, Y_2, \dots, Y_n)^{\mathrm{T}} \in \mathbb{C}^n$ is the input to the IR; $\nu$ is the vector of channel input symbols in (5); and $N = (N_1, N_2, \dots, N_n)^{\mathrm{T}} \in \mathbb{C}^n$ is the noise vector such that, for all $m \in \{1, 2, \dots, n\}$, the random variable $N_m$ is a complex circularly symmetric Gaussian random variable whose real and imaginary parts have zero means and variances $\frac{1}{2}\sigma^2$. Moreover, the random variables $N_1, N_2, \dots, N_n$ are mutually independent (see [47, Section 2.2.4]). That is, for all $\mathbf{y} = (y_1, y_2, \dots, y_n)^{\mathrm{T}} \in \mathbb{C}^n$, and all $\boldsymbol{\nu} = (\nu_1, \nu_2, \dots, \nu_n)^{\mathrm{T}} \in \mathbb{C}^n$, the conditional probability density function of the channel output $\mathbf{Y}$ in (9) is given by

$$ f_{\mathbf{Y}|\mathbf{X}}(\mathbf{y}|\mathbf{x}) = \prod_{m=1}^{n} f_{\mathbf{Y}|\mathbf{X}}(y_m|\nu_m) \qquad (10) $$

and for all $m \in \{1, 2, \dots, n\}$,

$$ f_{\mathbf{Y}|\mathbf{X}}(y_m|\nu_m) = \frac{1}{\pi\sigma^2} \exp\left(-\frac{|y_m - \nu_m|^2}{\sigma^2}\right) \qquad (11) $$

$$ = \frac{1}{\pi\sigma^2} \exp\left(-\frac{(\Re(y_m) - \Re(\nu_m))^2 + (\Im(y_m) - \Im(\nu_m))^2}{\sigma^2}\right). \qquad (12) $$

The EH does not down-convert or filter the received input $Z(t)$ (see [15] and [14]). The RF signal $Z(t)$ in (8b) is used as is for harvesting the energy contained in it.

Within this framework, two tasks must be accomplished: information transmission and energy transmission.
ting a message $i \in \{1, 2, \dots, M\}$, such that $e_i$ in (21) satisfies $e_i < B$. That is,

$$ \theta(\mathcal{C}, B) \triangleq \sum_{i \in \{j \in \{1, 2, \dots, M\} : e_j < B\}} P_E(e_i) \quad (25) $$

$$ = \sum_{i \in \{j \in \{1, 2, \dots, M\} : e_j < B\}} \frac{1}{M} \sum_{i=1}^{M} 1_{\{e=e_i\}} \quad (26) $$

$$ = \frac{1}{M} \left| \{ i \in \{1, 2, \dots, M\} : e_i < B \} \right| \quad (27) $$

$$ = \frac{1}{M} \sum_{i=1}^{M} 1_{\{e_i < B\}}, \quad (28) $$

where the equality in (26) follows from (24).

Definition 2.1 can now be refined to include the DEP and the EOP as follows.

**Definition 2.2** ((n, M, X, ϵ, B, δ)-code). An (n, M, X)-code C is said to be an (n, M, X, ϵ, B, δ)-code for the random transformation in (9) if the following hold:

$$ \gamma(\mathcal{C}) \leq \epsilon, \text{ and} \quad (29) $$

$$ \theta(\mathcal{C}, B) \leq \delta, \quad (30) $$

where $γ(\mathcal{C})$ and $θ(\mathcal{C}, B)$ are defined in (16) and (25), respectively.

The (n, M, X, ϵ, B, δ)-codes thus defined can be employed for transmitting at most B energy units in n channel uses and at most $\frac{\log_2 M}{n}$ bits per channel use, while ensuring that the resulting DEP is smaller than ϵ and the EOP is smaller than δ.

The primary objective of what follows in Section 3 is a characterization of the trade-offs between the various information and energy parameters of (n, M, X, ϵ, B, δ)-codes. These trade-offs are presented in the form of inequalities involving the code parameters n, M, X, ϵ, B and δ. This characterization serves two main purposes. First, these inequalities define what tuples (n, M, X, ϵ, B, δ) are “possible”. In other words, these inequalities define necessary conditions that an (n, M, X, ϵ, B, δ)-code must satisfy. Second, the dependence of M, ϵ, B, δ on common parameters precisely quantifies the trade-offs between the objectives of information and energy transmission.

### 3. NECESSARY CONDITIONS

This section introduces some inequalities involving the information and energy rates, the DEP and the EOP that any (n, M, X, ϵ, B, δ)-code satisfies. From this perspective, these inequalities are recognized to be necessary conditions for SIET. Furthermore, these inequalities quantify the mutual dependence between the information rate, the energy rate, the DEP and the EOP of (n, M, X, ϵ, B, δ)-codes, which in turn, provides insights into the feasibility of SIET for specific reliability guarantees. For a specific (n, M, X, ϵ, B, δ)-code, e.g., the code C in (12), this mutual dependence is presented in terms of the type induced by each of the codewords of such a code. The type induced by the codeword $u(i)$, with

$i \in \{1, 2, \dots, M\}$, is a probability mass function, denoted by $P_{u(i)}$, such that for all $x \in X$,

$$ P_{u(i)}(x) \triangleq \frac{1}{n} \sum_{m=1}^{n} 1_{\{u_m(i)=x\}}. \quad (31) $$

The type induced by all the codewords in C is also a probability mass function on the set X in (3). This type is denoted by $P_C$ and for all $x \in X$,

$$ P_C(x) \triangleq \frac{1}{M} \sum_{i=1}^{M} P_{u(i)}(x). \quad (32) $$

Due to space constraints and for the ease of presentation, the main results are presented for a specific class of (n, M, X, ϵ, B, δ)-codes, referred to as constant composition codes. Nonetheless, the results presented in this section can be easily extended to the whole class of (n, M, X, ϵ, B, δ)-codes. For extensions to (n, M, X, ϵ, B, δ)-codes that are not constant composition codes, only the assumption in (31) must be relaxed. Generalizations of these results for all (n, M, X, ϵ, B, δ)-codes can be found in our research report [43].

**Definition 3.1** (Constant Composition Codes). An (n, M, X)-code C is said to be a constant composition code if for all $i \in \{1, 2, \dots, M\}$ and for all $x \in X$, it holds that

$$ P_{u(i)}(x) = P_C(x), \quad (33) $$

where, $P_{u(i)}$ and $P_C$ are the types defined in (31) and (32), respectively.

Constant composition codes are (n, M, X)-codes in which a given channel input symbol is used the same number of times in all the codewords. As a consequence, for a constant composition code C, the amount of energy $e_i$ in (21) harvested while transmitting the codeword $i \in \{1, 2, \dots, M\}$, is invariant with respect to i. That is, for all $i \in \{1, 2, \dots, M\}$,

$$ e_i = e_C, \quad (34) $$

and the positive real $e_C$ satisfies

$$ e_C \triangleq k_1 \sum_{x \in \mathcal{X}} n P_C(x) |x|^2 + k_2 \sum_{x \in \mathcal{X}} n P_C(x) |x|^4, \quad (35) $$

where the constants $k_1$ and $k_2$ are the same as in (21); X in (3); and $P_C$ in (32).

The following theorem introduces a first characterization of the dependences among the performance metrics of an (n, M, X, ϵ, B, δ)-code.

**Theorem 3.1.** If a given constant composition (n, M, X)-code, C, with X in (3), is an (n, M, X, ϵ, B, δ)-code for the
A. Information Transmission

Let $M$ be the resolution of the set of indices from which a message is transmitted within $n$ channel uses. That is,

$$M \le 2^{n \log L}, \qquad (13)$$

where $L$ is the number of channel input symbols in (4). To reliably transmit a message index, the transmitter uses an $(n, M)$-code defined as follows.

**Definition 3.1** ((n, M)-code). An $(n, M)$-code for the random transformation in (9) is a system:

$$\{(\mathbf{u}(1), \mathcal{D}_1), (\mathbf{u}(2), \mathcal{D}_2), \dots, (\mathbf{u}(M), \mathcal{D}_M)\}, \qquad (14)$$

where, for all $(i, j) \in \{1, 2, \dots, M\}^2$, $i \ne j$,

$$\mathbf{u}(i) = (u_1(i), u_2(i), \dots, u_n(i)) \in \mathcal{X}^n, \qquad (15a)$$

$$\mathcal{D}_i \cap \mathcal{D}_j = \phi, \qquad (15b)$$

$$\bigcup_{i=1}^{M} \mathcal{D}_i \subseteq \mathbb{C}^n, \text{ and} \qquad (15c)$$

$$|u_m(i)| \le P, \qquad (15d)$$

where the real $P > 0$ is the peak-amplitude constraint.

Assume that the transmitter uses the $(n, M)$-code

$$\mathcal{C} \triangleq \{(\mathbf{u}(1), \mathcal{D}_1), (\mathbf{u}(2), \mathcal{D}_2), \dots, (\mathbf{u}(M), \mathcal{D}_M)\}, \qquad (16)$$

that satisfies (15). The results in this paper are presented in terms of the types induced by the codewords of the given code $\mathcal{C}$ in (16). The type induced by the codeword $\mathbf{u}(i)$, with $i \in \{1, 2, \dots, M\}$, is a probability mass function (pmf) whose support is equal to or a subset of $\mathcal{X}$ in (3). This pmf is denoted by $P_{\mathbf{u}(i)}$ and for all $x \in \mathcal{X}$,

$$P_{\mathbf{u}(i)}(x) \triangleq \frac{1}{n} \sum_{m=1}^{n} 1_{\{u_m(i)=x\}}. \qquad (17)$$

The type induced by all the codewords in $\mathcal{C}$ is also a pmf on the set $\mathcal{X}$ in (3). This pmf is denoted by $P_{\mathcal{C}}$ and for all $x \in \mathcal{X}$,

$$P_{\mathcal{C}}(x) \triangleq \frac{1}{M} \sum_{i=1}^{M} P_{\mathbf{u}(i)}(x). \qquad (18)$$

The information rate of any $(n, M)$-code $\mathcal{C}$ is given by

$$R(\mathcal{C}) = \frac{\log M}{n} \qquad (19)$$

in bits per channel use. To transmit the message index $i$, with $i \in \{1, 2, \dots, M\}$, the transmitter uses the codeword $\mathbf{u}(i) = (u_1(i), u_2(i), \dots, u_n(i))$. That is, at channel use $m$, with $m \in \{1, 2, \dots, n\}$, the transmitter inputs the RF signal corresponding to symbol $u_m(i)$ into the channel. At the end of $n$ channel uses, the IR observes a realization of the random vector $\mathbf{Y} = (Y_1, Y_2, \dots, Y_n)^T$ in (9). The IR decides that message index $j$, with $j \in \{1, 2, \dots, M\}$, was transmitted, if the following event takes place:

$$\mathbf{Y} \in \mathcal{D}_j, \qquad (20)$$

with $\mathcal{D}_j$ in (16). That is, the set $\mathcal{D}_j \subseteq \mathbb{C}^n$ is the region of correct detection for message index $j$. Therefore, the DEP associated with the transmission of message index $i$ is given by

$$\gamma_i(\mathcal{C}) \triangleq 1 - \int_{\mathcal{D}_i} f_{\mathbf{Y}|\mathbf{X}}(\mathbf{y}|\mathbf{u}(i))d\mathbf{y}, \qquad (21)$$

and the average DEP for code $\mathcal{C}$ is given by

$$\gamma(\mathcal{C}) \triangleq \frac{1}{M} \sum_{i=1}^{M} \gamma_i(\mathcal{C}). \qquad (22)$$

Using this notation, Definition 3.1 can be refined as follows.

**Definition 3.2** ((n, M, $\epsilon$)-codes). An $(n, M)$-code $\mathcal{C}$ for the random transformation in (9), is said to be an $(n, M, \epsilon)$-code if

$$\gamma(\mathcal{C}) \le \epsilon. \qquad (23)$$
random transformation in (9), then the following conditions simultaneously hold:

$$ \epsilon \ge (M-1)Q \left( \sqrt{\frac{\sum_{\ell=1}^{L} n P_{\mathcal{C}}(x^{(\ell)}) |x^{(\ell)} - \bar{x}^{(\ell)}|^2}{2\sigma^2}} \right), \quad (36a) $$

$$ R(\mathcal{C}) \le \frac{1}{n} \log_2 \left( \frac{n!}{\prod_{\ell=1}^{L} (n P_{\mathcal{C}}(x^{(\ell)}))!} \right), \quad (36b) $$

$$ \delta \ge \mathbb{1}_{\{e_{\mathcal{C}} < B\}}, \quad (36c) $$

$$ B \le e_{\mathcal{C}}, \quad (36d) $$

where, the information rate $R(\mathcal{C})$ is defined in (36b); the function $Q$ is defined in (2); for all $\ell \in \{1, 2, \dots, L\}$, the complex $\bar{x}^{(\ell)} \in \mathcal{X}$ is

$$ \bar{x}^{(\ell)} \in \arg \max_{x \in \mathcal{X} \setminus \{x^{(\ell)}\}} |x^{(\ell)} - x|; \quad (37) $$

$P_{\mathcal{C}}$ is the type defined in (32); and $e_{\mathcal{C}}$ is defined in (35).

*Proof.* The proofs are provided in Appendix A in [44]. □

Theorem 3.1 introduces the fundamental limits on the main performance metrics, namely the information transmission rate, energy transmission rate, DEP, and EOP, of all constant composition $(n, M, \mathcal{X})$-codes. Essentially, if a given constant composition $(n, M, \mathcal{X})$-code $\mathcal{C}$ is an $(n, M, \mathcal{X}, \epsilon, B, \delta)$-code, then the DEP $\epsilon$ cannot be smaller than the right-hand side of (36a); the information transmission rate $R(\mathcal{C})$ in (17) cannot be bigger than the right-hand side of (36b); the EOP $\delta$ cannot be smaller than the right-hand side of (36c); and the energy transmitted cannot be bigger than the right-hand side of (36d). These fundamental upper and lower bounds on the performance metrics depend on the type induced by the code $\mathcal{C}$. This observation reveals a profound conclusion: Two different $(n, M, \mathcal{X})$-codes $\mathcal{C}_1$ and $\mathcal{C}_2$ of the form in (12), might exhibit different codewords and different decoding regions, nonetheless, if they induce the same types, i.e., $P_{\mathcal{C}_1} = P_{\mathcal{C}_2}$, then their performance metrics to simultaneously transmit information and energy are subject to identical constraints. This reveals the relevance of the type $P_{\mathcal{C}}$ in (36) as the key parameter in code design for SIET.

In the remainder of this section, the conditions of Theorem 3.1 on the information and energy transmission parameters of $(n, M, \mathcal{X}, \epsilon, B, \delta)$-codes are discussed in further detail.

### A. DEP

To minimize the DEP for the code $\mathcal{C}$, the decoding regions $\mathcal{D}_1, \mathcal{D}_2, \dots, \mathcal{D}_M$ are defined using the maximum a posteriori (MAP) decision rule [45, Chapter 21], i.e., for all $i \in \{1, 2, \dots, M\}$,

$$ \mathcal{D}_i = \left\{ \mathbf{y} \in \mathbb{C} : \forall j \in \{1, 2, \dots, M\}, \frac{f_{\mathbf{Y}|\mathbf{X}}(\mathbf{y}|\mathbf{u}(i))}{f_{\mathbf{Y}|\mathbf{X}}(\mathbf{y}|\mathbf{u}(j))} \ge 1 \right\}. \quad (38) $$

The lower bound on the DEP, $\epsilon$, in (36a), is independent of the decoding regions $\mathcal{D}_1, \mathcal{D}_2, \dots, \mathcal{D}_M$ of the code $\mathcal{C}$. This lower bound is the exact DEP of a code in which all the channel input symbols in $\mathcal{X}$ are equidistant from one another, such

as in the case of binary phase-shift keying. Hence, the lower bound on the DEP, $\epsilon$, in (36a) is tight for such a code.

### B. Energy Transmission Rate and EOP

The lower and upper bounds in (36c) and (36d), which might appear redundant with respect to each other in the case of constant composition codes, are more informative in the general case as discussed below. The following lemma provides a lower bound on $\delta$ for the general case of $(n, M, \mathcal{X}, \epsilon, B, \delta)$-codes.

**Lemma 3.2.** If a given $(n, M, \mathcal{X})$-code, $\mathcal{C}$, with $\mathcal{X}$ in (3), is an $(n, M, \mathcal{X}, \epsilon, B, \delta)$-code for the random transformation in (9), then the following holds for $\delta$ in (30):

$$ \delta \ge \frac{1}{M} \sum_{i=1}^{M} \mathbb{1}_{\{e_i < B\}}, \quad (39) $$

where, for all $i \in \{1, 2, \dots, M\}$, the real $e_i \in [0, \infty)$ is in (21).

*Proof.* The result follows from (28) and (30). □

In Lemma 3.2, the lower bound on the EOP, $\delta$, is the probability of the set of symbols that deliver energy that is smaller than the required value of $B$. That is, $\delta \ge P_W(\{i \in \{1, 2, \dots, M\} : e_i < B\})$, with $P_W$ in (13). Hence, if there exists at most one codeword $\mathbf{u}(i)$, for some $i \in \{1, 2, \dots, M\}$, such that $e_i < B$, then, the EOP is bounded away from zero. More specifically, $\delta \ge \frac{1}{M}$. If there are $\rho$ codewords $\mathbf{u}(\ell_1), \mathbf{u}(\ell_2), \dots, \mathbf{u}(\ell_{\rho})$, such that for all $j \in \{1, 2, \dots, \rho\}$, $e_{\ell_j} < B$, then, $\delta \ge \frac{\rho}{M}$. If all the codewords $\mathbf{u}(1), \mathbf{u}(2), \dots, \mathbf{u}(M)$ are such that $\max_i e_i < B$, then the EOP is equal to one. That is, it is impossible to guarantee that $B$ units of energy will be harvested at the EH. This reveals the entanglement between $B$ and $\delta$.

To quantify the relationship between $B$ and $\delta$, denote by $M' \le M$, the number of unique values in the vector $(e_1, e_2, \dots, e_M)^{\top}$ with $e_i$ in (21). These $M'$ unique energy levels are represented by $\{\bar{e}_1, \bar{e}_2, \dots, \bar{e}_{M'}\} \subset \mathbb{R}$. More specifically, for all $i \in \{1, 2, \dots, M\}$, there exists $j \in \{1, 2, \dots, M'\}$ such that $e_i = \bar{e}_j$.

Assume without loss of generality that the following holds:

$$ 0 < \bar{e}_1 < \bar{e}_2 < \dots < \bar{e}_{M'}. \quad (40) $$

Using this notation, the following lemma provides an upper bound on the energy requirement $B$ for any $(n, M, \mathcal{X}, \epsilon, B, \delta)$-code as a function of the EOP $\delta$.

**Lemma 3.3.** Given an $(n, M, \mathcal{X}, \epsilon, B, \delta)$-code $\mathcal{C}$ for the random transformation in (9), with $\mathcal{X}$ in (3), let $j^+ \in \mathbb{N}$ be defined as follows:

$$ j^+ \triangleq \min \left\{ j \in \{1, \dots, M'\} : \delta \le \frac{\sum_{k=1}^{j} \sum_{i=1}^{M} \mathbb{1}_{\{e_i = \bar{e}_k\}}}{M} \right\}. \quad (41) $$
Using (17) and (18), a special class of codes of particular interest in this study, called homogeneous codes, is defined hereunder.

**Definition 3.3 (Homogeneous Codes).** An $(n, M, \epsilon)$-code $\mathcal{C}$ for the random transformation in (9) of the form in (16) is said to be homogeneous if for all $i \in \{1, 2, \dots, M\}$ and for all $x \in \mathcal{X}$, with $\mathcal{X}$ in (3) it holds that

$$P_{u(i)}(x) = P_{\mathcal{C}}(x), \qquad (24)$$

where, $P_{u(i)}$ and $P_{\mathcal{C}}$ are the types defined in (17) and (18), respectively.

Homogeneous codes are essentially $(n, M, \epsilon)$-codes in which a given channel input symbol is used the same number of times in all codewords.

## B. Energy Transmission

While transmitting message $i \in \{1, 2, \dots, M\}$, the channel output observed at the EH is denoted by $Z_i(t)$, with $t \in [0, nT]$. From (7) and (8b), the channel output $Z_i(t)$ is given by

$$Z_i(t) = \Re(\sqrt{2} \sum_{m=1}^{n} u_m(i) \sinc(f_w(t - (m-1)T)) \exp(i2\pi f_c t)) + N_2(t) \qquad (25)$$

$$= x_i(t) + N_2(t), \qquad (26)$$

where, for all $m \in \{1, 2, \dots, n\}$, the complex $u_m(i)$ is the $m^{th}$ symbol of the codeword $u(i)$ in (15a); for all $t \in [0, nT]$, the signal $x_i(t)$ in (26) is

$$x_i(t) = \Re(\sqrt{2} \sum_{m=1}^{n} u_m(i) \sinc(f_w(t - (m-1)T)) \exp(i2\pi f_c t)); \qquad (27)$$

and the random variable $N_2(t)$ is a real Gaussian random variable with zero mean and variance $\sigma^2$ in (10). For all $t \in [0, nT]$, the channel output $Z_i(t)$ in (26) is a real Gaussian random variable with mean $x_i(t)$ and variance $\sigma^2$.

The non-linear energy model in [14] and [15] states that the energy harvested from a signal is proportional to the DC component of the second and fourth powers of the signal. Using this model, for all $i \in \{1, 2, \dots, M\}$, the expected energy harvested from the channel output $Z_i(t)$ in (25) during the time $t \in [0, nT]$ is given by the following:

$$e_i \triangleq k_1 \sum_{m=1}^{n} |u_m(i)|^2 + k_2 \sum_{m=1}^{n} |u_m(i)|^4 \qquad (28)$$

$$= k_1 \sum_{x \in \mathcal{X}} n P_{u(i)}(x) |x|^2 + k_2 \sum_{x \in \mathcal{X}} n P_{u(i)}(x) |x|^4, \qquad (29)$$

where $x \in \mathcal{X}$, with $\mathcal{X}$ in (3); $k_1$ and $k_2$ are positive constants, with $k_1 = 0.0034$ and $k_2 = 0.3829$ [14]; for all $m \in \{1, 2, \dots, n\}$, the complex $u_m(i)$ is in (15a) and $P_{u(i)}$ is the type defined in (17). From (29), for a homogeneous code $\mathcal{C}$ (Definition 3.3), the expected energy harvested from the channel output $Z_i(t)$ in (26) during the time $t \in [0, nT]$ is equal for all $i \in \{1, 2, \dots, M\}$ and is given by the following:

$$e_{\mathcal{C}} \triangleq k_1 \sum_{x \in \mathcal{X}} n P_{\mathcal{C}}(x) |x|^2 + k_2 \sum_{x \in \mathcal{X}} n P_{\mathcal{C}}(x) |x|^4, \qquad (30)$$

where $x \in \mathcal{X}$, with $\mathcal{X}$ in (3) and $P_{\mathcal{C}}$ is the type defined in (18).

Denote by $M' \le M$, the number of unique values in the vector $(e_1, e_2, \dots, e_M)^T$ with $e_i$ in (28). The $M'$ unique energy levels are given by $\{\bar{e}_1, \bar{e}_2, \dots, \bar{e}_{M'}\}$. Assume without loss of generality that the following holds:

$$0 < \bar{e}_1 < \bar{e}_2 < \dots < \bar{e}_{M'}. \qquad (31)$$

More specifically, for all $i \in \{1, 2, \dots, M\}$, there exists $j \in \{1, 2, \dots, M'\}$ such that $e_i = \bar{e}_j$ with $e_i$ in (28).

For all $j \in \{1, 2, \dots, M'\}$, define variables $y_j$ to be the number of codewords that carry energy $\bar{e}_j$. More precisely, for all $j \in \{1, 2, \dots, M'\}$, $y_j$ is given by

$$y_j = \sum_{i=1}^{M} 1_{\{e_i = \bar{e}_j\}}. \qquad (32)$$

It follows that $\sum_{j=1}^{M'} y_j = M$.
where, the positive integer $M'$ and the reals $\bar{e}_1, \bar{e}_2, \dots, \bar{e}_{M'}$ are in (40). Then, the following holds for the energy requirement $B$:

$$B \le \bar{e}_{j+} \qquad (42)$$

*Proof.* The proof is provided in Appendix C in [44]. □

Lemma 3.3 provides an upper bound on the value of $B$ for a given $(n, M, \mathcal{X}, \epsilon, B, \delta)$-code $\mathcal{C}$. From the discussion following Lemma 3.2, it is clear that $\delta$ can only take values from a discrete set that depends on the $M'$ unique energy levels $\{\bar{e}_1, \bar{e}_2, \dots, \bar{e}_{M'}\}$ defined in (40) as well as the number of codewords that possess these energy values. Furthermore, from (39), it follows that the value of $\delta$ defines the maximum number of values in the set $\{\bar{e}_1, \bar{e}_2, \dots, \bar{e}_{M'}\}$ that can be less than $B$ so that the energy and EOP requirements of the given code $\mathcal{C}$ are satisfied. This is precisely quantified by the variable $j^+$ defined in (41). Finally, from (41) it follows that, for the EOP to be less than or equal to $\delta$, the required energy $B$ can be no more than $\bar{e}_{j+}$. Thus, together with Lemma 3.2, Lemma 3.3 completely quantifies the relationship between $B$ and $\delta$ for a given $(n, M, \mathcal{X}, \epsilon, B, \delta)$-code.

### C. Information Transmission Rate

Though the bound on $R(\mathcal{C})$ in (36b) is tight, it can quickly become computationally infeasible as the block-length $n$ increases. The following theorem provides a tractable approximation of this bound.

**Theorem 3.4.** If a given constant composition $(n, M, \mathcal{X})$-code, $\mathcal{C}$, with $\mathcal{X}$ in (3), is an $(n, M, \mathcal{X}, \epsilon, B, \delta)$-code for the random transformation in (9), then the following holds for $R(\mathcal{C})$ in (17):

$$R(\mathcal{C}) \le H(P_{\mathcal{C}}) + \frac{1}{n^2} \left( \frac{1}{12} - \sum_{\ell=1}^{L} \frac{1}{12P_{\mathcal{C}}(x^{(\ell)}) + 1} \right) \quad (43) \\ + \frac{1}{n} \left( \log(\sqrt{2\pi}) - \sum_{\ell=1}^{L} \log \sqrt{2\pi P_{\mathcal{C}}(x^{(\ell)})} \right) - \frac{\log n}{n} \left( \frac{L-1}{2} \right),$$

where, $P_{\mathcal{C}}$ is the type defined in (32); and $L$ is the number of channel input symbols in (3).

*Proof.* The proof is provided in Appendix B in [44]. □

Note that all terms in (43), except the entropy $H(P_{\mathcal{C}})$, vanish with the block-length $n$. This implies the well known information theoretic result that information rate is essentially constrained by the entropy of the channel input symbols. In particular, note that $H(P_{\mathcal{C}}) \le \log_2 L$. Furthermore, this bound holds with equality for a code $\mathcal{C}$ when the type $P_{\mathcal{C}}$ is uniform, i.e., all the symbols from the set of channel inputs $\mathcal{X}$ are used with the same frequency in $\mathcal{C}$. It is important to note that, even though it may not be explicitly clear in Theorem 3.1 and Theorem 3.4, the inequalities on the information rate and the DEP are in fact dependent on each other. Each of the inequalities are a function of the type $P_{\mathcal{C}}$, which defines the dependence between these parameters.

**Remark 3.1.** The value $\frac{n!}{\prod_{\ell=1}^{L} (n P_{\mathcal{C}}(x^{(\ell)}))!}$ in (36b) is the exact number of different codewords of length $n$, while exhibiting an average type $P_{\mathcal{C}}$. Hence, codes such that every channel input symbol is used the same number of times among all the codewords are less constrained in terms of the information rate. This is the case in which $P_{\mathcal{C}}$ is a uniform distribution. Alternatively, using a uniform type $P_{\mathcal{C}}$ might reduce the energy transmission rate significantly. For instance, assume that the set of channel input symbols is such that for at least one pair $(x_1, x_2) \in \mathcal{X}^2$, with $\mathcal{X}$ in (3), it holds that $|x_1| < |x_2|$. Then, from (35), it is clear that using the symbol $x_1$ equally often as $x_2$ constraints the value of $e_{\mathcal{C}}$ and hence, the energy $B$. Alternatively, codes that exhibit the largest energy rates are those in which the symbols that have the largest magnitude are used most often. This clearly deviates from the uniform distribution and thus, constrains the information rate $R$.

### 4. CODE CONSTRUCTION FOR SIET

This section introduces a method for constructing codes for SIET that ensure certain reliability guarantees. The tuples of information rate, energy rate, DEP and EOP that can be achieved by the constructed codes are also characterized. This is accomplished in two steps. The first step is the construction of the constellation $\mathcal{X}$ defined in (3) and the construction of a particular $(n, M, \mathcal{X})$-code, whose codewords and decoding sets are subject to certain conditions such that such the code is an $(n, M, \mathcal{X}, \epsilon, B, \delta)$-code.

#### A. Constellation Design

Consider a constellation represented by the set $\mathcal{X}$ in (3) and assume that $\mathcal{X}$ is the union of $C \in \mathbb{N}$ sets of channel input symbols, which will be referred to as layers. A layer is a set of symbols that have the same magnitude. For all $c \in \{1, 2, \dots, C\}$, denote by $L_c \in \mathbb{N}$ the number of symbols and by $A_c \in \mathbb{R}^+$ the amplitude of the symbols in layer $c$. The underlying assumption is that symbols in layer $c$ are equally spaced along the circle of radius $A_c$. This is typically the case in phase-shift keying (PSK) modulations [45, Chapter 16] since the symbols being equally spaced is the most favorable in terms of the DEP.

Let $\alpha_c \in [0, 2\pi]$ denote the phase shift of the symbols in layer $c$. The layer $c$ is denoted as

$$\mathcal{U}(A_c, L_c, \alpha_c) \triangleq \{x_c^{(1)}, x_c^{(2)}, \dots, x_c^{(L_c)}\} \quad (44)$$

$$= \{x \in \mathbb{C} : x = A_c \exp \left( i \left( \frac{2\pi}{L_c} \ell + \alpha_c \right) \right),$$

$$\ell \in \{0, 1, 2, \dots, (L_c - 1)\}, \quad (45)$$

where $i$ is the complex unit. Hence, the set $\mathcal{X}$ is given by

$$\mathcal{X} = \bigcup_{c=1}^{C} \mathcal{U}(A_c, L_c, \alpha_c). \qquad (46)$$

The vector of the amplitudes $A_c$ in (46) is denoted by

$$\mathbf{A} = (A_1, A_2, \dots, A_C)^{\mathrm{T}}, \qquad (47)$$
Let W be a random variable that denotes the message index sent during time $t \in [0, nT]$. For all $i \in \{1, 2, \dots, M\}$, the probability of transmitting message index $i$ is given by

$$P_W(i) = \frac{1}{M}. \qquad (33)$$

The energy harvested during time $t \in [0, nT]$ is also a random variable denoted by $E$. The probability of harvesting energy $e$ given that message index $i$ was transmitted is given by

$$P_{E|W}(e|i) = \mathbb{1}_{\{e=e_i\}}, \qquad (34)$$

where $e_i$ is in (28). The pmf of the random variable $E$, denoted by $P_E$ is given by the following:

$$P_E(e) = \sum_{i=1}^{M} P_{E|W}(e|i) P_W(i) \qquad (35)$$

$$= \frac{1}{M} \sum_{i=1}^{M} \mathbb{1}_{\{e=e_i\}}. \qquad (36)$$

Using (35), the EOP associated with code $\mathcal{C}$ is defined as follows:

$$\theta(\mathcal{C}, B) \triangleq \Pr(E < B) = \sum_{i \in \{j \in \{1, 2, \dots, M\} : e_j < B\}} P_E(e_i) \qquad (37)$$

$$= \sum_{i \in \{j \in \{1, 2, \dots, M\} : e_j < B\}} \frac{1}{M} \sum_{i=1}^{M} \mathbb{1}_{\{e_i = e_j\}} \qquad (38)$$

$$= \frac{1}{M} |\{i \in \{1, 2, \dots, M\} : e_i < B\}| \qquad (39)$$

$$= \frac{1}{M} \sum_{i=1}^{M} \mathbb{1}_{\{e_i < B\}}, \qquad (40)$$

where, $P_E$ is the pmf defined in (35), $e_i$ is defined in (28) and, the equality in (38) follows from (36).

From (40), an interesting insight can be derived about $\theta(\mathcal{C}, B)$. The EOP $\theta(\mathcal{C}, B)$ can only take discrete values in the set $\{0, \frac{1}{M}, \frac{2}{M}, \dots, 1\}$. Moreover, for homogeneous codes [9, Definition 4], it holds that $\theta(\mathcal{C}, B) \in \{0, 1\}$. The following refinement of Definition 3.2 follows from (37).

**Definition 3.4** ((n, M, e, B, δ)-code). An $(n, M, e)$-code $\mathcal{C}$ for the random transformation in (9) is said to be an $(n, M, e, B, \delta)$-code if

$$\theta(\mathcal{C}, B) \le \delta. \qquad (41)$$

## 4. INFORMATION-ENERGY CONVERSE REGION

The main results in this section provide upper bounds on the information rate $R$ in (19), lower bounds on the average DEP $\epsilon$ in (23) and lower bounds on the EOP $\delta$ in (41) for all possible $(n, M, \epsilon, B, \delta)$-codes that employ a given set of channel input symbols $\mathcal{X}$ of the form in (3).

### A. Energy Transmission Rate and EOP

The following corollary introduces a lower bound on the EOP that holds for all possible $(n, M, \epsilon, B, \delta)$-codes.

**Corollary 4.1.** Given an $(n, M, \epsilon, B, \delta)$-code $\mathcal{C}$ for the random transformation in (9) of the form in (16), the following holds:

$$\delta \ge \frac{1}{M} \sum_{i=1}^{M} \mathbb{1}_{\{e_i < B\}}, \qquad (42)$$

where, for all $i \in \{1, 2, \dots, M\}$, the real $e_i \in [0, \infty)$ is in (28).

*Proof.* The result follows from (40) and (41). □

<figure><img src="image_1.png" /></figure>

The following lemma provides the bound on the EOP for the class of homogeneous codes.

**Lemma 4.2.** Given a homogeneous $(n, M, \epsilon, B, \delta)$-code $\mathcal{C}$ for the random transformation in (9) of the form in (16), the following holds:

$$\delta = \mathbb{1}_{\{e_{\epsilon} < B\}}, \qquad (43)$$
and without any loss of generality, assume that

$$A_1 > A_2 > \dots > A_C. \quad (48)$$

The vector of the number of symbols in each layer in (46) is denoted by

$$\mathbf{L} = (L_1, L_2, \dots, L_C)^{\mathrm{T}}; \quad (49)$$

and the vector of the phase shifts of the symbols in each layer in (46) is denoted by

$$\alpha = (\alpha_1, \alpha_2, \dots, \alpha_C)^{\mathrm{T}}. \quad (50)$$

The total number of symbols $L$ in (3) satisfies

$$L = \sum_{c=1}^{C} L_c. \quad (51)$$

Using the design in (46) for the set of channel input symbols, $\mathcal{X}$ in (3), the $(n, M, \mathcal{X}, \epsilon, B, \delta)$-code $\mathcal{C}$ is constructed in two steps. The first step consists of designing the codewords, while the second step designs the corresponding decoding regions.

## B. Codeword Design

Consider an $(n, M, \mathcal{X})$-code of the form in (12), with $\mathcal{X}$ in (46), whose codewords are all different and are constructed satisfying a constraint on the number of times the symbols in the different layers are used. For all $c \in \{1, 2, \dots, C\}$, let $p_c$ be the empirical probability of symbols of the $c^{th}$ layer. That is,

$$p_c = \frac{1}{Mn} \sum_{\ell=1}^{L_c} \sum_{i=1}^{M} \sum_{m=1}^{n} \mathbb{1}_{\{u_m(i)=x_c^{(\ell)}\}} \quad (52)$$

$$= \frac{1}{M} \sum_{\ell=1}^{L_c} \sum_{i=1}^{M} P_{u(i)}(x_c^{(\ell)}) \quad (53)$$

$$= \sum_{\ell=1}^{L_c} P_{\mathcal{C}}(x_c^{(\ell)}), \quad (54)$$

where, for all $i \in \{1, 2, \dots, M\}$ and for all $m \in \{1, 2, \dots, n\}$, the complex $u_m(i)$ is the $m^{th}$ channel input symbol of the codeword $\mathbf{u}(i)$ in (11a); the types $P_{u(i)}$ and $P_{\mathcal{C}}$ are defined in (31) and in (32), respectively; and $x_c^{(\ell)}$ is in (44). The resulting probability vector is denoted by

$$\mathbf{p} = (p_1, p_2, \dots, p_C)^{\mathrm{T}}, \quad (55)$$

which is a free parameter in the construction of code $\mathcal{C}$. The constraint on the codewords of code $\mathcal{C}$ is that for all $c \in \{1, 2, \dots, C\}$ and for all $\ell \in \{1, 2, \dots, L_c\}$, the frequency with which the symbol $x_c^{(\ell)}$ appears in $\mathcal{C}$ satisfies

$$P_{\mathcal{C}}(x_c^{(\ell)}) = \frac{p_c}{L_c}. \quad (56)$$

That is, symbols within the same layer are used with the same frequency in $\mathcal{C}$. The motivation for this assumption is that intuitively, while using two symbols within the same layer with different frequencies does not impact the energy transmission rate $B$ or EOP $\delta$, it does have an impact on the information transmission rate and DEP of $\mathcal{C}$.

## C. Decoding Regions

The focus is now on the design of the decoding regions $\mathcal{D}_1, \mathcal{D}_2, \dots, \mathcal{D}_M$ of the code $\mathcal{C}$, for the codewords $\mathbf{u}(1), \mathbf{u}(2), \dots, \mathbf{u}(M)$ constructed above. For all $c \in \{1, 2, \dots, C\}$ and $\ell \in \{1, 2, \dots, L_c\}$, let the set $\mathcal{G}_c^{(\ell)} \subseteq \mathbb{C}$ be a circle of radius $r_c \in \mathbb{R}^+$ centered at $x_c^{(\ell)}$, with $x_c^{(\ell)}$ in (44). That is,

$$\mathcal{G}_c^{(\ell)} = \{y \in \mathbb{C} : |y - x_c^{(\ell)}|^2 \le r_c^2\}. \quad (57)$$

For all $i \in \{1, 2, \dots, M\}$, the decoding region $\mathcal{D}_i$ for codeword $\mathbf{u}(i)$ is built as follows:

$$\mathcal{D}_i = \mathcal{D}_{i,1} \times \mathcal{D}_{i,2} \times \dots \times \mathcal{D}_{i,n}, \quad (58)$$

where, for all $m \in \{1, 2, \dots, n\}$,

$$\mathcal{D}_{i,m} = \mathcal{G}_c^{(\ell)}, \quad (59)$$

with $c$ and $\ell$ satisfying $u_m(i) = x_c^{(\ell)}$.

To ensure that the decoding regions $\mathcal{D}_1, \mathcal{D}_2, \dots, \mathcal{D}_M$ are mutually disjoint, for all $c \in \{1, 2, \dots, C-1\}$, the amplitudes $A_c$ in (44) and the radii $r_c$ in (57) satisfy the following:

$$A_{c+1} - A_c \ge r_{c+1} + r_c. \quad (60)$$

The vector of the radii in (57) is denoted by

$$\mathbf{r} = (r_1, r_2, \dots, r_C)^{\mathrm{T}}. \quad (61)$$

When a peak-power constraint is in place, changing the number of layers $C$ has varying impacts on the information and energy parameters. Firstly, if $C$ as well as the number of channel input symbols is increased, the information transmission rate increases. However, the DEP of the resulting code would also increase. This is because, adding more layers decreases the Euclidean distance between the channel input symbols, resulting in increased decoding errors. The impact of the number of layers on the energy transmission parameters is less obvious. Adding layers with amplitudes higher than the mean will result in an increase in the energy transmission rate $B$ and a decrease in the EOP $\delta$ and vice versa.

Note that there are multiple ways of choosing the parameters $\mathbf{A}, \alpha, \mathbf{p}$ and $\mathbf{r}$ in the proposed construction to satisfy the same information rate, energy rate, DEP and EOP requirements. Thus, the construction defines a family of codes

$$C(C, \mathbf{A}, \mathbf{L}, \alpha, \mathbf{p}, \mathbf{r}), \quad (62)$$

with the number of layers $C$ in (46); $\mathbf{A}$ in (47); $\mathbf{L}$ in (49); $\alpha$ in (50); $\mathbf{p}$ in (55); and $\mathbf{r}$ in (61). The codewords $\mathbf{u}(1), \mathbf{u}(2), \dots, \mathbf{u}(M)$ and their respective decoding regions $\mathcal{D}_1, \mathcal{D}_2, \dots, \mathcal{D}_M$ completely define an $(n, M, \mathcal{X})$-code within the family $C(C, \mathbf{A}, \mathbf{L}, \alpha, \mathbf{p}, \mathbf{r})$.

## D. Achievable Tuples

The following theorem characterizes the tuples of information rate, energy rate, DEP and EOP that can be achieved by codes within the family $C(C, \mathbf{A}, \mathbf{L}, \alpha, \mathbf{p}, \mathbf{r})$ in (62).
where $e_{\mathcal{C}} \in [0, \infty)$ as in (30).

*Proof.* From (40), the average EOP of code $\mathcal{C}$ is given by

$$ \theta(\mathcal{C}, B) = \frac{1}{M} \sum_{i=1}^{M} 1_{\{e_i < B\}}. \qquad (44) $$

Since $\mathcal{C}$ is homogeneous, from (44) and (30) it follows that

$$ \theta(\mathcal{C}, B) = \frac{1}{M} \sum_{i=1}^{M} 1_{\{e_{\mathcal{C}} < B\}} \qquad (45) $$

$$ = 1_{\{e_{\mathcal{C}} < B\}}. \qquad (46) $$

From (41) and (46), it follows that

$$ \delta = 1_{\{e_{\mathcal{C}} < B\}}. \qquad (47) $$

This completes the proof.

<figure><img src="image_2.png" /></figure>

Note that, while (41) would suggest that (47) should be an inequality, namely $\delta \ge 1_{\{e_{\mathcal{C}} < B\}}$. However, since $\delta \in \{0, 1\}$ for the homogeneous code $\mathcal{C}$ in Lemma 4.2, (47) holds with equality.

The derived lower bound on $\delta$ in Corollary 4.1 can now be used for determining the upper bound on the energy transmission rate $B$ as follows.

**Lemma 4.3.** Given an $(n, M, \epsilon, B, \delta)$-code $\mathcal{C}$ for the random transformation in (9) of the form in (16), the energy transmission rate $B$ satisfies,

$$ B \le \begin{cases} \bar{e}_1 & \text{if } 0 \le \delta < \frac{y_1}{M}, \\ \bar{e}_j & \text{if } \frac{\sum_{k=1}^{j-1} y_k}{M} \le \delta < \frac{\sum_{k=1}^{j} y_k}{M}, j \in \{2, 3, \dots, M'\}. \end{cases} \qquad (48) $$

where, for all $i \in \{1, 2, \dots, M\}$, energy $e_i$ is in (28); the positive integer $M'$ is in (31); and, for all $j \in \{1, 2, \dots, M'\}$, $\bar{e}_j$ is in (31) and $y_j$ is in (32).

*Proof.* From (42), it follows that,

$$ \sum_{i=1}^{M} 1_{\{e_i < B\}} \le M\delta. \qquad (49) $$

This implies that, in order to achieve an EOP less than or equal to $\delta$, the number of codewords that have energy less than $B$ (given by $\sum_{i=1}^{M} 1_{\{e_i < B\}}$) can be at most equal to $\lfloor M\delta \rfloor$. This allows calculating the upper bound on the energy transmission rate $B$ as follows.

From (32) and the definition of the EOP in (40), it can be seen that the EOP $\delta$ can only take values from the following set:

$$ \left\{ 0, \frac{y_1}{M}, \frac{y_1+y_2}{M}, \dots, \frac{\sum_{j=1}^{M'-1} y_j}{M}, 1 \right\}. \qquad (50) $$

For $0 \le \delta < \frac{y_1}{M}$, at most $\lfloor M\delta \rfloor = y_1$ codewords can have energy less than $B$ which is possible only if

$$ B \le \bar{e}_1. \qquad (51) $$

For $\frac{y_1}{M} \le \delta < \frac{y_1+y_2}{M}$, at most $\lfloor M\delta \rfloor = y_1 + y_2$ codewords can have energy less than $B$ which is possible only if

$$ B \le \bar{e}_2. \qquad (52) $$

The upper bound on $B$ can similarly be calculated for all possible values of $\delta$ in (50). Therefore, the upper bound for $B$ is given by the following:

$$ B \le \begin{cases} \bar{e}_1 & \text{if } 0 \le \delta < \frac{y_1}{M}, \\ \bar{e}_j & \text{if } \frac{\sum_{k=1}^{j-1} y_k}{M} \le \delta < \frac{\sum_{k=1}^{j} y_k}{M}, j \in \{2, 3, \dots, M'\}. \end{cases} \qquad (53) $$

This completes the proof.

<figure><img src="image_3.png" /></figure>
**Theorem 4.1.** A code $\mathcal{C}$ of the form in (12) from the family $C(C, \mathbf{A}, \mathbf{L}, \boldsymbol{\alpha}, \mathbf{p}, \mathbf{r})$ in (62) is an $(n, M, \mathcal{X}, \epsilon, B, \delta)$-code for the random transformation in (9), if the following inequalities are simultaneously satisfied:

$$ \epsilon \ge 1 - \frac{1}{M} \sum_{i=1}^{M} \prod_{c=1}^{C} \left(1 - \exp\left(-\frac{r_c^2}{\sigma^2}\right)\right)^{n \sum_{\ell=1}^{L_c} P_{u(i)}(x_c^{(\ell)})} \quad (63a) $$

$$ R(\mathcal{C}) \le \log_2 \left( \sum_{c=1}^{C} \left| \frac{\pi}{2 \arcsin \frac{r_c}{2A_c}} \right| \right) \quad (63b) $$

$$ \delta \ge \frac{1}{M} \sum_{i=1}^{M} 1 \left\{ \sum_{c=1}^{C} \sum_{\ell=1}^{L_c} P_{u(i)}(x_c^{(\ell)}) (k_1 A_c^2 + k_2 A_c^4) < \frac{B}{n} \right\} \quad (63c) $$

$$ B \le \bar{e}_{j+} \quad (63d) $$

where, the information rate $R(\mathcal{C})$ is defined in (36b); the energy level $\bar{e}_{j+}$ is defined in (42); and the constants $k_1$ and $k_2$ are the same as in (21); and the type $P_{u(i)}$ is defined in (31).

*Proof.* The proofs are provided in Appendix D in [44]. □

Theorem 4.1 characterizes the conditions on the parameters $C, \mathbf{A}, \mathbf{L}, \boldsymbol{\alpha}, \mathbf{p}$ and $\mathbf{r}$ such that all codes within the family $C(C, \mathbf{A}, \mathbf{L}, \boldsymbol{\alpha}, \mathbf{p}, \mathbf{r})$ in (62) are $(n, M, \mathcal{X}, \epsilon, B, \delta)$-codes. Hence, given some target information transmission rate $R$; energy requirement $B$; DEP $\epsilon$; and EOP $\delta$, Theorem 4.1 provides the conditions on the parameters $(C, \mathbf{A}, \mathbf{L}, \boldsymbol{\alpha}, \mathbf{p}, \mathbf{r})$ for the existence of an $(n, M, \mathcal{X}, \epsilon, B, \delta)$-code. Such a code can be constructed by designing the codewords and the decoding regions as suggested in Section 4-B and Section 4-C, respectively.

**Remark 4.1.** It is essential to note the following about the choice of decoding sets in (57) and the derived bound on the achievable DEP in (63a). Firstly, the circular construction of decoding sets in (57) is a choice made in order to obtain closed form tractable expressions for the lower bound on the DEP in (63a). This is enabled by the inherent circular symmetry of the complex AWGN noise. The bound in (63a) reveals the dependence of the DEP on the various parameters of the code such as the type. This is instructive for studying the trade-offs between the parameters of the constructed family of codes. Secondly, this choice of decoding sets provides guidelines for constructing the set of channel input symbols $\mathcal{X}$ while ensuring that the DEP does not exceed a required value which is controlled using the choice of the radii of the decoding sets in (61).

## 5. PERFORMANCE ANALYSIS

The performance of the constructed family of codes can be evaluated by comparing the achievability bounds identified for the constructed family of codes $C(C, \mathbf{A}, \mathbf{L}, \boldsymbol{\alpha}, \mathbf{p}, \mathbf{r})$ in Theorem 4.1 with the necessary conditions in Theorem 3.1 that any $(n, M, \mathcal{X}, \epsilon, B, \delta)$-code must satisfy. This comparison is presented in the form of information-energy regions in Fig. 1 where the information transmission rates are plotted as a function of the energy transmission rates of the codes. For this comparison, consider constant composition $(n, M, \mathcal{X}, \epsilon, B, \delta)$-codes in the family $C(C, \mathbf{A}, \mathbf{L}, \boldsymbol{\alpha}, \mathbf{p}, \mathbf{r})$ in (62) that employ

the set of channel input symbols $\mathcal{X}$ of the form in (46) with number of layers $C = 3$. The duration of the transmission in channel uses is $n = 80$. The radii of the decoding regions $r_c$ are assumed to be the same for all the layers i.e., for all $c \in \{1, 2, \dots, C\}$, the radius $r_c = r$ in (57). The amplitude of the first layer is $A_1 = 30$. Amplitudes of the second and third layers $A_2$ and $A_3$ are determined by $r$ to satisfy (60). The points on the curves in Fig. 1 are obtained by varying $\epsilon$ and the probability vector $\mathbf{p}$ in (55).

The first important observation from Fig. 1 is that the achievability results for the codes constructed in this work match the necessary conditions identified in Theorem 3.1 except for the DEP $\epsilon$. This is illustrated by the overlap of the necessary conditions and the achievable information-energy curves. However, for the same information rate and energy pair, the DEP for the achievable curves does not match the DEP bound identified by the necessary conditions. The sub-optimality in DEP is a result of the choice of circular decoding regions in (57) (See Remark 4.1).

Fig. 1 also shows a clear trade-off between the information and energy transmission rates that can be simultaneously supported by a given code. The maximum achievable information transmission rate is $R = 4.93$ bits/channel use. This $R$ is achieved by a code in which all the symbols in the set of channel inputs $\mathcal{X}$ are used with the same frequency. The maximum energy that can be delivered at $R = 4.93$ bits/channel use is $B = 3.2 \times 10^5$ energy units. This corresponds to the point $D_1$ in Fig. 1. The maximum achievable $B$ is $3.8 \times 10^5$ energy units. This is achieved by a code that exclusively uses the symbols in the first layer i.e., the probability vector $\mathbf{p}$ in (55) is $\mathbf{p} = (1, 0, 0)^T$. The maximum $R$ that can be achieved at $B = 3.8 \times 10^5$ energy units is $R = 3.8$ bits/channel use. This corresponds to the point $D_2$ in Fig. 1. The curves between the points $D_1$ and $D_2$ in Fig. 1 illustrate the trade-off between the information and energy transmission. As $B$ is increased from $3.2 \times 10^5$ energy units at point $D_1$, $R$ begins to decrease. Similarly, as $R$ is increased from $3.8$ bits/channel use at point $D_2$, $B$ begins to decrease.

## 6. DISCUSSION

1) Trade-offs: The results presented in this work reveal several interesting insights into the trade-offs between the information transmission rate $R$, the energy requirement $B$, the DEP $\epsilon$ and the EOP $\delta$. For instance, as shown by (36c) and (63c), $\delta$ increases as $B$ increases and vice versa. The consequence of this relationship is that a smaller $\delta$ can be achieved at the cost of lower $B$. Similarly, for increasing $B$ in (36d) and (63d), a larger value of $\delta$ has to be tolerated. These relationships also reveal that both $B$ and $\delta$ can be improved by a code that has higher values of $e_i$ in (21). This is achieved by using the symbols with greater energy more frequently in the code.

From the necessary condition in (36a), it follows that $\epsilon$ decreases as the distance between the symbols in the set of channel inputs $\mathcal{X}$ in (3) increases. This implies that, a lower DEP can be achieved by increasing the distances between
Consider the special case in which all the M codewords carry a different amount of energy. That is, for all $i \in \{1, 2, \dots, M\}$, $e_1 \neq e_2 \neq \dots \neq e_M$. Furthermore, assume without loss of generality that

$$e_1 < e_2 < \dots < e_M. \tag{54}$$

For this particular case, the upper bound on B is given by the following:

$$B \le e_i, \quad \text{if} \quad \frac{(i-1)}{M} \le \delta < \frac{i}{M}, \quad i \in \{1, 2, \dots, M\}. \tag{55}$$

Note that (55) is a special case of (53) for $M' = M$.

## B. Average Decoding Error Probability

The average DEP of a given code $\mathcal{C}$ of the form in (16) depends on the choice of the decoding sets $\mathcal{D}_1, \mathcal{D}_2, \dots, \mathcal{D}_M$. For all $i \in \{1, 2, \dots, M\}$, consider the smallest sets $\mathcal{D}_{i,1}, \mathcal{D}_{i,2}, \dots, \mathcal{D}_{i,n}$ such that the decoding sets $\mathcal{D}_i$ are of the following form:

$$\mathcal{D}_i \subseteq \mathcal{D}_{i,1} \times \mathcal{D}_{i,2} \times \dots \times \mathcal{D}_{i,n}. \tag{56}$$

Note that these sets exist by construction as follows. For all $\mathbf{y} = (y_1, y_2, \dots, y_n)^\mathrm{T} \in \mathcal{D}_i$, let for all $m \in \{1, 2, \dots, n\}$, $y_m \in \mathcal{D}_{i,m}$. Hence, the inclusion in (56) holds.

The following lemma proves that it is sub-optimal to have the decoding set of a symbol $u_m(i)$ be dependent on the codeword $i \in \{1, 2, \dots, M\}$ and/or the position of the symbol $m \in \{1, 2, \dots, n\}$ in the codeword. In other words, for all $(i, i') \in \{1, 2, \dots, M\}^2$, all $(m, m') \in \{1, 2, \dots, n\}^2$, and all $\ell \in \{1, 2, \dots, L\}$, if $u_m(i) = u_{m'}(i') = x^{(\ell)}$, with $x^{(\ell)} \in \mathcal{X}$, it is sub-optimal to have $\mathcal{D}_{i,m} \neq \mathcal{D}_{i',m'}$.

**Lemma 4.4.** Consider an $(n, M, \epsilon, B, \delta)$-code $\mathcal{C}$ for the random transformation in (9) of the form in (16) with the set of channel input symbols $\mathcal{X}$ in (3) and decoding sets of the form in (56). For all $i \in \{1, 2, \dots, M\}$, all $m \in \{1, 2, \dots, n\}$, and all $\ell \in \{1, 2, \dots, L\}$, if $u_m(i) = x^{(\ell)}$, with $x^{(\ell)} \in \mathcal{X}$, the sets $\mathcal{D}_{i,m}$ in (56) are given by the following:

$$\mathcal{D}_{i,m} = \bar{\mathcal{E}}_{\ell}, \tag{57}$$

where, for all $\ell \in \{1, 2, \dots, L\}$,

$$\bar{\mathcal{E}}_{\ell} \triangleq \mathcal{D}_{j, \omega(j, \ell)}, \tag{58}$$

with,

$$j \in \arg \max_{k \in \{1, 2, \dots, M\}} \int_{\mathcal{D}_{k, \omega(k, \ell)}} f_{Y|X}(y|x^{(\ell)}) dy, \tag{59}$$

where,

$$\omega(k, \ell) \in \arg \max_{m \in \{1, 2, \dots, n\}} \int_{\mathcal{D}_{k,m}} f_{Y|X}(y|u_m(k) = x^{(\ell)}) dy. \tag{60}$$

Given any other $(n, M, \epsilon, B, \delta)$-code $\mathcal{C}'$ for the random transformation in (9) of the form in (16), with the same set of channel input symbols $\mathcal{X}$ in (3) and decoding sets $\mathcal{D}_i$ of the form in (56), the average DEP $\gamma$ in (22) of $\mathcal{C}$ and $\mathcal{C}'$ satisfy the following:

$$\gamma(\mathcal{C}') \ge \gamma(\mathcal{C}). \tag{61}$$

**Proof.** From (21), for all $i \in \{1, 2, \dots, M\}$, the DEP for codeword $i$ of code $\mathcal{C}'$ is given by

$$\gamma_i(\mathcal{C}') = 1 - P_{Y|X=\mathbf{u}(i)}(\mathcal{D}_i) \tag{62}$$

$$\ge 1 - P_{Y|X=\mathbf{u}(i)}(\mathcal{D}_{i,1} \times \mathcal{D}_{i,2} \times \dots \times \mathcal{D}_{i,n}) \tag{63}$$

$$= 1 - \int_{\mathcal{D}_{i,1} \times \mathcal{D}_{i,2} \times \dots \times \mathcal{D}_{i,n}} \prod_{m=1}^{n} f_{Y|X}(y|u_m(i)) dy \tag{64}$$

$$= 1 - \prod_{m=1}^{n} \int_{\mathcal{D}_{i,m}} f_{Y|X}(y|u_m(i)) dy, \tag{65}$$

where $P_{Y|X}$ is the probability measure induced by the pdf $f_{Y|X}$ in (10); for all $i \in \{1, 2, \dots, M\}$ and all $m \in \{1, 2, \dots, n\}$, $\mathbf{u}(i)$ and $u_m(i)$ are in (15a); the inequality in (63) follows from (56); equality in (64) follows from (10); and (65) follows from (64) due to Fubini's theorem. Let $\omega: \{1, 2, \dots, M\} \times \{1, 2, \dots, L\} \to \{1, 2, \dots, n\}$ be defined as follows

$$\omega(i, \ell) \in \arg \max_{m \in \{1, 2, \dots, n\}} \int_{\mathcal{D}_{i,m}} f_{Y|X}(y|u_m(i) = x^{(\ell)}) dy, \tag{66}$$
<figure><img src="image_1.png" /><figcaption>Fig. 1: A comparison of the information-energy region defined by the necessary conditions (Theorem 3.1) versus the achievable information-energy region (Theorem 4.1) for constant composition codes in the family $C(C, A, L, \alpha, p, r)$.</figcaption></figure>

<figure><img src="image_2.png" /><figcaption>Fig. 2: Bounds on the information transmission rate $R$ in (36b) as a function of the harvested energy $e$ in (65).</figcaption></figure>

the channel input symbols. However, with the peak-amplitude constraint in place, increasing the distance between symbols implies that the number of symbols $L$ decreases. This in turn, decreases the upper bound on the information rate $R$ in (36b). In (63c), $\epsilon$ decreases as a function of the radii $r_c$ of the decoding regions in (57). On the other hand, from (63b) it follows that the information rate $R$ increases as the radii $r_c$ decrease. These trade-offs between $R$, $B$, $\epsilon$, and $\delta$ are more comprehensively illustrated in the following example.

Consider a family of constant composition $(n, M, \mathcal{X}, \epsilon, B, \delta)$-codes $\mathcal{C}$ in the family $C(C, A, L, \alpha, p, r)$ in (62). The set of channel input symbols $\mathcal{X}$ in (46) is composed of two layers with 5 symbols in each layer, i.e., $C = 2$ and $L_1 = L_2 = 5$. The radius of the first layer is $A_1 = 20$ millivolts and the radius of the second layer $A_2$ is varied to illustrate the

trade-offs between the various parameters. The frequency with which symbols from the first layer appear in the code is $p_1 = p = 1 - p_2$. That is, the vector $p$ in (55) is given by

$$ \boldsymbol{p} = (p, (1-p))^{\top}. \qquad (64) $$

The duration of the transmission in channel uses is $n = 100$. Since $\mathcal{C}$ is a constant composition code, from (34), it holds that, for all $i \in \{1, 2, \dots, M\}$,

$$ e_i = e, \qquad (65) $$

where $e \in [0, \infty)$ is calculated as in (35).

Fig. 2 shows the trade-offs between the information transmission rate $R$ in (36b) and the harvested energy $e$ in microwatts ($\mu W$) in (65) as a function of $p$ in (64). Each curve in the figure is generated for some value of $A_2 < A_1$ by varying the value of $p \in [0, 1]$. The following trade-offs can be observed from this figure. The harvested energy $e$ increases as $p$ increases. This is because higher $p$ corresponds to the symbols from the first layer $c = 1$ which have higher energy (since $A_1 > A_2$) being used more frequently in $\mathcal{C}$. For a fixed value of $A_2$ in Fig. 2, the information rate $R$ first increases and then decreases as a function of $e$ in (65). For each of these curves, the maximum $R = 2.13$ bits/channel use corresponds to the uniform type, i.e., $p = 0.5$. For $p$ lesser or greater than 0.5, the bound on $R$ decreases. Furthermore, the bounds on $R$ are independent of the values of $A_1$ and $A_2$. This is due to the fact that the information rate $R$ in (36b) is only a function of the type. The harvested energy $e$ also increases as $A_2$ increases. This is because, higher values of $A_2$ imply higher energy contained in the symbols in the second layer which in turn increases $e$.

<figure><img src="image_3.png" /><figcaption>Fig. 3: A comparison of the bounds on the information rate from the necessary conditions in (36b) (Necessary 1) and (43) (Necessary 2) for finite block-length SIET with the finite block-length converse in [46] as a function of the block-length $n$.</figcaption></figure>

2) Comparison with state of the art: Fig. 3 compares the bound on the information rate in (36b) (Necessary 1) and (43) (Necessary 2) with the finite block-length converse bound
with $x^{(\ell)} \in \mathcal{X}$, and $\mathcal{X}$ in (3). For all $i \in \{1, 2, \dots, M\}$, let the sets $\mu_{i1}, \mu_{i2}, \dots, \mu_{iL}$ be a partition of $\{1, 2, \dots, n\}$ such that, for all $\ell \in \{1, 2, \dots, L\}$, the set $\mu_{i\ell}$ is given by

$$ \mu_{i\ell} = \{m \in \{1, 2, \dots, n\} : u_m(i) = x^{(\ell)}\}. \qquad (67) $$

Hence, for all $i \in \{1, 2, \dots, M\}$, from (65), it follows that,

$$ \gamma_i(\mathcal{C}') \ge 1 - \prod_{\ell=1}^{L} \left( \prod_{m \in \mu_{i\ell}} \int_{\mathcal{D}_{i,m}} f_{Y|X}(y|x^{(\ell)}) dy \right) \qquad (68) $$

$$ \ge 1 - \prod_{\ell=1}^{L} \left( \prod_{m \in \mu_{i\ell}} \int_{\mathcal{D}_{i,\omega(i,\ell)}} f_{Y|X}(y|x^{(\ell)}) dy \right) \qquad (69) $$

$$ = 1 - \prod_{\ell=1}^{L} \left( \int_{\mathcal{D}_{i,\omega(i,\ell)}} f_{Y|X}(y|x^{(\ell)}) dy \right)^{n P_{u(i)}(x^{(\ell)})}, \qquad (70) $$

where, the inequality in (69) follows from (66). For all $\ell \in \{1, 2, \dots, L\}$, let the sets $\bar{\mathcal{E}}_1, \bar{\mathcal{E}}_2, \dots, \bar{\mathcal{E}}_L$ be such that

$$ \bar{\mathcal{E}}_{\ell} \triangleq \mathcal{D}_{j,\omega(j,\ell)}, \qquad (71) $$

$$ \text{with } j \in \arg \max_{k \in \{1, 2, \dots, M\}} \int_{\mathcal{D}_{k, \omega(k, \ell)}} f_{Y|X}(y|x^{(\ell)}) dy. \qquad (72) $$

Hence, from (70) and (71), for all $i \in \{1, 2, \dots, M\}$, it follows that

$$ \gamma_i(\mathcal{C}') \ge 1 - \prod_{\ell=1}^{L} \left( \int_{\bar{\mathcal{E}}_\ell} f_{Y|X}(y|x^{(\ell)}) dy \right)^{n P_{u(i)}(x^{(\ell)})}. \qquad (73) $$

From (22) and (73), it follows that the average DEP $\gamma(\mathcal{C}')$ is given by

$$ \gamma(\mathcal{C}') \ge \frac{1}{M} \sum_{i=1}^{M} \left( 1 - \prod_{\ell=1}^{L} \left( \int_{\bar{\mathcal{E}}_\ell} f_{Y|X}(y|x^{(\ell)}) dy \right)^{n P_{u(i)}(x^{(\ell)})} \right) \qquad (74) $$

$$ = 1 - \frac{1}{M} \sum_{i=1}^{M} \prod_{\ell=1}^{L} \left( \int_{\bar{\mathcal{E}}_\ell} f_{Y|X}(y|x^{(\ell)}) dy \right)^{n P_{u(i)}(x^{(\ell)})}, \qquad (75) $$

$$ = \gamma(\mathcal{C}). \qquad (76) $$

This completes the proof. $\square$

The following lemma introduces a lower bound on the DEP $\epsilon$ in (23) for an $(n, M, \epsilon, B, \delta)$-code $\mathcal{C}$ of the form in (16).

**Lemma 4.5.** Given an $(n, M, \epsilon, B, \delta)$-code $\mathcal{C}$ for the random transformation in (9) of the form in (16) and decoding sets of the form in (56), the average DEP $\epsilon$ in (23) satisfies the following:

$$ \epsilon \ge 1 - \frac{1}{M} \sum_{i=1}^{M} \exp \left( -nH(P_{u(i)}) - nD(P_{u(i)}||F) + n \log L \right), \qquad (77) $$

where, $P_{u(i)}$ is the type defined in (17); $L$ is in (4); the function $F$ is a pmf on $\mathcal{X}$ in (3) such that for all $\ell \in \{1, 2, \dots, L\}$,

$$ F(x^{(\ell)}) = \frac{\int_{\bar{\mathcal{E}}_{\ell}} f_{Y|X}(y|x^{(\ell)}) dy}{\sum_{j=1}^{L} \int_{\bar{\mathcal{E}}_{j}} f_{Y|X}(y|x^{(j)}) dy}; \qquad (78) $$

and for all $\ell \in \{1, 2, \dots, L\}$, the set $\bar{\mathcal{E}}_\ell$ is in (71).

**Proof.** From (75), the average DEP of the $(n, M, \epsilon, B, \delta)$-code $\mathcal{C}$ is given by

$$ \gamma(\mathcal{C}) \ge 1 - \frac{1}{M} \sum_{i=1}^{M} \prod_{\ell=1}^{L} \left( \int_{\bar{\mathcal{E}}_\ell} f_{Y|X}(y|x^{(\ell)}) dy \right)^{n P_{u(i)}(x^{(\ell)})}. \qquad (79) $$
on the information rate in [46, Theorem 54] for a QPSK (i.e., L = 4) constellation and signal to noise ratio (SNR) equal to 18 dB. It should be noted that the converse bounds in [46] are concerned with information transmission alone. The following observations can be made from these plots. Firstly, the proposed bounds are very close to the finite block-length converse for small block-lengths. As the block-length increases, the difference between the two bounds increases (Converse and Necessary 1/Necessary 2 curves in Fig. 3). Secondly, the bounds in (36b) and (43) are very close. This shows that the approximation in (43) is tight, especially as the block-length increases. Finally, it should be noted that the bounds in (36b) and (43) are for the case of constant composition codes whereas the bound from [46] has no such restriction. In fact, the information rate R can be at most $log_2 L$ in this case. As can be observed from Fig. 3, the proposed bounds actually come very close to the maximum value of $log_2 L$ as the block-length n increases.

<figure><img src="image_4.png" /><figcaption>Fig. 4: Scaling of the lower bound on the DEP in (36a) as a function of the block-length n.</figcaption></figure>

3) Additional results: Fig. 4 illustrates how the lower bound on the DEP in (36a) scales as the block-length n increases. As expected, the DEP decays exponentially as the block-length n increases.

<figure><img src="image_5.png" /><figcaption>Fig. 5: Variation of the harvested energy for the constructed codes as a function of the PAPR of the set of channel input symbols.</figcaption></figure>

Peak-to-average power ratio (PAPR) is an important property

of constellations with regard to their energy transmission capability. The large number of variables involved in our analysis make it intractable to directly express these bounds in terms of the PAPR. However, PAPR is a function of the amplitudes $A_c$ which appear in the bound on the EOP $\delta$ in (63c). In particular,

$$PAPR = \frac{A_1^2}{\sum_{c=1}^{C} p_c A_c^2} \qquad (66)$$

Therefore, even though PAPR does not explicitly appear in our results, its impact is implicitly present in the form of various parameters of the proposed construction. To illustrate the impact of PAPR on the energy transmission capability of the constructed codes, the harvested energy B is plotted as a function of the PAPR in Fig. 5. The plot is generated for codes constructed with a constellation with three layers, i.e., C = 3. The number of symbols per layer $L_c$ are varied to create constellations with different PAPR and energy values. We observe that, as the PAPR of the underlying constellation increases, so does the energy harvested from the code.

## REFERENCES

[1] S. u. Zuhra, S. M. Perlaza, and E. Altman, "Simultaneous Information and Energy Transmission with Finite Constellations," in Proceedings of the IEEE Information Theory Workshop (ITW), Kanazawa, Japan, Oct. 2021, pp. 1-6.

[2] S. u. Zuhra, S. M. Perlaza, H. V. Poor, and M. Skoglund, "Information-Energy Trade-offs with EH Non-linearities in the Finite Block-Length Regime with Finite Constellations," in Proceedings of the IEEE Information Theory Workshop (ITW), Mumbai, India, Nov. 2022.

[3] S. u. Zuhra, S. M. Perlaza, H. V. Poor, and E. Altman, "Achievable Information-Energy Region in the Finite Block-Length Regime with Finite Constellations," in Proceedings of the IEEE International Symposium on Information Theory (ISIT), Espoo, Finland, Jun. 2022, pp. 2106-2111.

[4] N. Tesla, Apparatus for Transmitting Electrical Energy. New York, NY: United States Patent Office, Dec. 1914, vol. US1119732 A.

[5] P. Grover and A. Sahai, "Shannon Meets Tesla: Wireless Information and Power Transfer," in Proceedings of the IEEE International Symposium on Information Theory, 2010, pp. 2363-2367.

[6] A. Iqbal, A. Smida, M. Al-Hasan, I. B. Mabrouk, and T. A. Denidni, "Highly Isolated Wireless Power Transfer and Information Co-Delivery Using a Pacemaker Duplex Antenna," IEEE Transactions on Microwave Theory and Techniques, p. 1-13, 2024.

[7] L. R. Varshney, "Transporting Information and Energy Simultaneously," in Proceedings of the IEEE International Symposium on Information Theory (ISIT), Toronto, ON, Canada, Jul. 2008, pp. 1612-1616.

[8] S. B. Amor and S. M. Perlaza, "Fundamental Limits of Simultaneous Energy and Information Transmission," in Proceedings of the International Conference on Telecommunications (ICT), Thessaloniki, Greece, May 2016, pp. 1-5.

[9] N. Khalfet and I. Krikidis, "The Capacity of SWIPT Systems over Rayleigh-Fading Channels with HPA," in Proceedings of the IEEE Information Theory Workshop (ITW), Kanazawa, Japan, 2021, pp. 1-6.

[10] S. B. Amor, S. M. Perlaza, I. Krikidis, and H. V. Poor, "Feedback Enhances Simultaneous Energy and Information Transmission in Multiple Access Channels," in Proceedings of the IEEE International Symposium on Information Theory (ISIT), Barcelona, Spain, Jul. 2016, pp. 1974-1978.
Evaluating the product of integrals in (79), the following holds:

$$ \prod_{\ell=1}^{L} \left( \int_{\bar{\mathcal{E}}_\ell} f_{Y|X}(y|x^{(\ell)}) dy \right)^{n P_{\mathbf{u}(i)}(x^{(\ell)})} = \exp \left( \log \prod_{\ell=1}^{L} \left( \int_{\bar{\mathcal{E}}_\ell} f_{Y|X}(y|x^{(\ell)}) dy \right)^{n P_{\mathbf{u}(i)}(x^{(\ell)})} \right), \quad (80) $$

$$ = \exp \left( \sum_{\ell=1}^{L} n P_{\mathbf{u}(i)}(x^{(\ell)}) \log \int_{\bar{\mathcal{E}}_\ell} f_{Y|X}(y|x^{(\ell)}) dy \right), \quad (81) $$

$$ = \exp \left( \sum_{\ell=1}^{L} n P_{\mathbf{u}(i)}(x^{(\ell)}) \left( \log \frac{\int_{\bar{\mathcal{E}}_\ell} f_{Y|X}(y|x^{(\ell)}) dy}{\sum_{j=1}^{L} \left( \int_{\bar{\mathcal{E}}_j} f_{Y|X}(y|x^{(j)}) dy \right)} + \log \sum_{j=1}^{L} \left( \int_{\bar{\mathcal{E}}_j} f_{Y|X}(y|x^{(j)}) dy \right) + \log P_{\mathbf{u}(i)}(x^{(\ell)}) - \log P_{\mathbf{u}(i)}(x^{(\ell)}) \right) \right). \quad (82) $$

Plugging (78) in (82) yields,

$$ \prod_{\ell=1}^{L} \left( \int_{\bar{\mathcal{E}}_\ell} f_{Y|X}(y|x^{(\ell)}) dy \right)^{n P_{\mathbf{u}(i)}(x^{(\ell)})} = \exp \left( \sum_{\ell=1}^{L} \left( n P_{\mathbf{u}(i)}(x^{(\ell)}) \log \frac{F(x^{(\ell)})}{P_{\mathbf{u}(i)}(x^{(\ell)})} + n P_{\mathbf{u}(i)}(x^{(\ell)}) \log \left( \sum_{j=1}^{L} \int_{\bar{\mathcal{E}}_j} f_{Y|X}(y|x^{(j)}) dy \right) + n P_{\mathbf{u}(i)}(x^{(\ell)}) \log P_{\mathbf{u}(i)}(x^{(\ell)}) \right) \right) \quad (83) $$

$$ = \exp \left( -nH(P_{\mathbf{u}(i)}) - nD(P_{\mathbf{u}(i)}||F) + n \log \sum_{j=1}^{L} \int_{\bar{\mathcal{E}}_j} f_{Y|X}(y|x^{(j)}) dy \right). \quad (84) $$

Using (84) in (74) yields,

$$ \gamma(\mathscr{C}) \ge 1 - \frac{1}{M} \sum_{i=1}^{M} \exp \left( -nH(P_{\mathbf{u}(i)}) - nD(P_{\mathbf{u}(i)}||F) + n \log \sum_{j=1}^{L} \int_{\bar{\mathcal{E}}_j} f_{Y|X}(y|x^{(j)}) dy \right) \quad (85) $$

$$ \ge 1 - \frac{1}{M} \sum_{i=1}^{M} \exp \left( -nH(P_{\mathbf{u}(i)}) - nD(P_{\mathbf{u}(i)}||F) + n \log L \right). \quad (86) $$

Finally, from (23) and (86), it follows that

$$ \epsilon \ge 1 - \frac{1}{M} \sum_{i=1}^{M} \exp \left( -nH(P_{\mathbf{u}(i)}) - nD(P_{\mathbf{u}(i)}||F) + n \log L \right), \quad (87) $$

which completes the proof. □

The following lemma provides another lower bound on the average DEP $\epsilon$ that recovers the optimal decoding sets for the channel input symbols as defined by the maximum a posteriori (MAP) decision rule [48, Chapter 21]. For all $\ell \in \{1, 2, \dots, L\}$, define the sets $\mathcal{E}_\ell$ as follows:

$$ \mathcal{E}_{\ell} \triangleq \left\{ \begin{array}{ll} \{y \in \mathbb{C} : \forall \ell' \in \{1, 2, \dots, L\}, P_{\mathscr{C}}(x^{(\ell)}) f_{Y|X}(y|x^{(\ell)}) \ge P_{\mathscr{C}}(x^{(\ell')}) f_{Y|X}(y|x^{(\ell')})\} & \text{if } \ell = 1, \\ \{y \in \mathbb{C} : \forall \ell' \in \{1, 2, \dots, L\}, P_{\mathscr{C}}(x^{(\ell)}) f_{Y|X}(y|x^{(\ell)}) \ge P_{\mathscr{C}}(x^{(\ell')}) f_{Y|X}(y|x^{(\ell')})\} & \text{otherwise.} \end{array} \right\} \cup \bigcup_{k=1}^{\ell-1} \mathcal{E}_k \quad (88) $$

Note that the definition of the sets in (88) is not unique because the set of channel input symbols $\mathcal{X}$ in (3) is not an ordered set. Therefore, one could go through the $L$ symbols in $\mathcal{X}$ in any order ($L!$ possibilities) to define the sets in (88) without impacting the resulting DEP.
[11] N. Khalfet and S. M. Perlaza, "Simultaneous Information and Energy Transmission in the Two-User Gaussian Interference Channel," *IEEE Journal on Selected Areas in Communications*, vol. 37, no. 1, pp. 156-170, Jan. 2019.

[12] B. Clerckx, "Wireless Information and Power Transfer: Nonlinearity, Waveform Design, and Rate-Energy Tradeoff," *IEEE Transactions on Signal Processing*, vol. 66, no. 4, pp. 847-862, 2018.

[13] S. B. Amor, S. M. Perlaza, I. Krikidis, and H. V. Poor, "Feedback Enhances Simultaneous Wireless Information and Energy Transmission in Multiple Access Channels," *IEEE Transactions on Information Theory*, vol. 63, no. 8, pp. 5244-5265, 2017.

[14] S. M. Perlaza, A. Tajer, and H. V. Poor, "Simultaneous Information and Energy Transmission: A Finite Block-length Analysis," in *Proceedings of the IEEE International Workshop on Signal Processing Advances in Wireless Communications (SPAWC)*, Kalamata, Greece, Jun. 2018, pp. 1-5.

[15] N. Khalfet, S. M. Perlaza, A. Tajer, and H. V. Poor, "On Ultra-reliable and Low Latency Simultaneous Information and Energy Transmission Systems," in *Proceedings of the IEEE International Workshop on Signal Processing Advances in Wireless Communications (SPAWC)*, Cannes, France, Jul. 2019, pp. 1-5.

[16] N. Khalfet, C. Psomas, S. Chatzinotas, and I. Krikidis, "Semantic Communications for Simultaneous Wireless Information and Power Transfer," *IEEE Transactions on Communications*, pp. 1-1, 2024.

[17] M. Varasteh, B. Rassouli, and B. Clerckx, "Wireless Information and Power Transfer over an AWGN Channel: Nonlinearity and Asymmetric Gaussian Signaling," in *Proceedings of the IEEE Information Theory Workshop (ITW)*, Kaohsiung, Taiwan, Nov. 2017, pp. 181-185.

[18] B. Clerckx and E. Bayguzina, "Waveform Design for Wireless Power Transfer," *IEEE Transactions on Signal Processing*, vol. 64, no. 23, pp. 6313-6328, 2016.

[19] N. Shanin, L. Cottatellucci, and R. Schober, "Markov Decision Process Based Design of SWIPT Systems: Non-Linear EH Circuits, Memory, and Impedance Mismatch," *IEEE Transactions on Communications*, vol. 69, no. 2, pp. 1259-1274, 2021.

[20] S. Abeywickrama, R. Zhang, and C. Yuen, "Refined Nonlinear Rectenna Modeling and Optimal Waveform Design for Multi-User Multi-Antenna Wireless Power Transfer," *IEEE Journal of Selected Topics in Signal Processing*, vol. 15, no. 5, pp. 1198-1210, 2021.

[21] S. Shen and B. Clerckx, "Joint Waveform and Beamforming Optimization for MIMO Wireless Power Transfer," *IEEE Transactions on Communications*, vol. 69, no. 8, pp. 5441-5455, 2021.

[22] O. L. A. López, F. A. Monteiro, H. Alves, R. Zhang, and M. Latva-Aho, "A Low-Complexity Beamforming Design for Multiuser Wireless Energy Transfer," *IEEE Wireless Communications Letters*, vol. 10, no. 1, pp. 58-62, 2021.

[23] Y. Zeng, B. Clerckx, and R. Zhang, "Communications and Signals Design for Wireless Power Transmission," *IEEE Transactions on Communications*, vol. 65, no. 5, pp. 2264-2290, 2017.

[24] P. Mukherjee, C. Psomas, and I. Krikidis, "Differential Chaos Shift Keying-Based Wireless Power Transfer with Nonlinearities," *IEEE Journal of Selected Topics in Signal Processing*, vol. 15, no. 5, pp. 1185-1197, 2021.

[25] J. Kim, B. Clerckx, and P. D. Mitcheson, "Signal and System Design for Wireless Power Transfer: Prototype, Experiment and Validation," *IEEE Transactions on Wireless Communications*, vol. 19, no. 11, pp. 7453-7469, 2020.

[26] A. Khalili, S. Zargari, Q. Wu, D. W. K. Ng, and R. Zhang, "Multi-Objective Resource Allocation for IRS-Aided SWIPT," *IEEE Wireless Communications Letters*, vol. 10, no. 6, pp. 1324-1328, 2021.

[27] G. M. Kraidy, C. Psomas, and I. Krikidis, "Fundamentals of Circular QAM for Wireless Information and Power Transfer," in *Proceedings of the IEEE International Workshop on Signal Processing Advances in Wireless Communications (SPAWC)*, 2021, pp. 616-620.

[28] T. D. P. Perera, D. N. K. Jayakody, S. K. Sharma, S. Chatzinotas, and J. Li, "Simultaneous Wireless Information and Power Transfer (SWIPT): Recent Advances and Future Challenges," *IEEE Communications Surveys & Tutorials*, vol. 20, no. 1, pp. 264-302, Dec. 2018.

[29] J. Huang, C.-C. Xing, and C. Wang, "Simultaneous Wireless Information and Power Transfer: Technologies, Applications, and Research Challenges," *IEEE Communications Magazine*, vol. 55, no. 11, pp. 26-32, Nov. 2017.

[30] B. Clerckx, R. Zhang, R. Schober, D. W. K. Ng, D. I. Kim, and H. V. Poor, "Fundamentals of Wireless Information and Power Transfer: From RF Energy Harvester Models to Signal and System Designs," *IEEE Journal on Selected Areas in Communications*, vol. 37, no. 1, pp. 4-33, 2019.

[31] B. Clerckx, J. Kim, K. W. Choi, and D. I. Kim, "Foundations of Wireless Information and Power Transfer: Theory, Prototypes, and Experiments," *Proceedings of the IEEE*, vol. 110, no. 1, pp. 8-30, Jan. 2022.

[32] L. Liu, R. Zhang, and K.-C. Chua, "Wireless Information Transfer with Opportunistic Energy Harvesting," *IEEE Transactions on Wireless Communications*, vol. 12, no. 1, pp. 288-300, 2013.

[33] N. Shanin, L. Cottatellucci, and R. Schober, "Rate-Power Region of SWIPT Systems Employing Nonlinear Energy Harvester Circuits with Memory," in *Proceedings of the IEEE International Conference on Communications (ICC)*, Dublin, Ireland, Jun. 2020, pp. 1-7.

[34] E. Goudeli, C. Psomas, and I. Krikidis, "Sequential Decoding for Simultaneous Wireless Information and Power Transfer," in *Proceedings of the International Conference on Telecommunications (ICT)*, 2017, pp. 1-5.

[35] G. Lin, Y. Zhou, W. Jiang, X. He, X. Zhou, G. He, and P. Yang, "LF-SWIPT: Outage Analysis for SWIPT Relaying Networks Using Lossy Forwarding With QoS Guaranteed," *IEEE Internet of Things Journal*, pp. 1-1, 2022.

[36] H. T. Thien, P.-V. Tuan, and I. Koo, "A Secure-Transmission Maximization Scheme for SWIPT Systems Assisted by an Intelligent Reflecting Surface and Deep Learning," *IEEE Access*, vol. 10, pp. 31851-31867, 2022.

[37] J. Liu, C.-H. R. Lin, Y.-C. Hu, and P. K. Donta, "Joint Beamforming, Power Allocation, and Splitting Control for SWIPT-Enabled IoT Networks with Deep Reinforcement Learning and Game Theory," *Sensors*, vol. 22, no. 6, p. 2328, 2022.

[38] B. Clerckx, K. Huang, L. R. Varshney, S. Ulukus, and M.-S. Alouini, "Wireless Power Transfer for Future Networks: Signal Processing, Machine Learning, Computing, and Sensing," *IEEE Journal of Selected Topics in Signal Processing*, vol. 15, no. 5, pp. 1060-1094, 2021.

[39] R. Zhang and C. K. Ho, "MIMO Broadcasting for Simultaneous Wireless Information and Power Transfer," *IEEE Transactions on Wireless Communications*, vol. 12, no. 5, pp. 1989-2001, 2013.

[40] C. Song, C. Ling, J. Park, and B. Clerckx, "MIMO Broadcasting for Simultaneous Wireless Information and Power Transfer: Weighted MMSE Approaches," in *Proceedings of the IEEE Globecom Workshops*, 2014, pp. 1151-1156.

[41] M. Varasteh, J. Hoydis, and B. Clerckx, "Learning to Communicate and Energize: Modulation, Coding, and Multiple Access Designs for Wireless Information-Power Transmission," *IEEE Transactions on Communications*, vol. 68, no. 11, pp. 6822-6839, 2020.

[42] D. Tse and P. Viswanath, *Fundamentals of Wireless Communication*. Cambridge University Press, 2005.

[43] S. u. Zuhra, S. M. Perlaza, H. V. Poor, and E. Altman, "Simultaneous Information and Energy Transmission with Finite Constellations," Inria Sophia Antipolis - Méditerranée, Research Report RR-9409, Jan. 2022.

[44] S. ul Zuhra, S. M. Perlaza, H. V. Poor, and M. Skoglund, "Simultaneous Information and Energy Transmission with Short Packets and Finite Constellations," 2024. [Online]. Available: https://arxiv.org/abs/2408.07807

[45] A. Lapidoth, *A Foundation in Digital Communication*, 2nd ed. Cambridge University Press, 2017.
**Lemma 4.6.** Given an $(n, M, \epsilon, B, \delta)$-code $\mathscr{C}$ for the random transformation in (9) of the form in (16) such that (56) holds, the average DEP $\epsilon$ satisfies the following:

$$ \epsilon \geq 1 - \sum_{\ell=1}^{L} \int_{\mathcal{E}_{\ell}} P_{\mathscr{C}}(x^{(\ell)}) f_{Y|X}(y|x^{(\ell)}) dy, \qquad (89) $$

where, $P_{\mathscr{C}}$ is the type defined in (18); $L$ is in (4); $x^{(\ell)} \in \mathcal{X}$ in (3), and for all $\ell \in \{1, 2, \dots, L\}$, the set $\mathcal{E}_{\ell}$ is defined in (88).

*Proof.* From (75), the average DEP of the $(n, M, \epsilon, B, \delta)$-code $\mathscr{C}$ is given by

$$ \gamma(\mathscr{C}) \geq 1 - \frac{1}{M} \sum_{i=1}^{M} \prod_{\ell=1}^{L} \left( \int_{\bar{\mathcal{E}}_{\ell}} f_{Y|X}(y|x^{(\ell)}) dy \right)^{n P_{\mathbf{u}(i)}(x^{(\ell)})} . \qquad (90) $$

The second term on the right hand side of (90) can be written as follows.

$$ \frac{1}{M} \sum_{i=1}^{M} \prod_{\ell=1}^{L} \left( \int_{\bar{\mathcal{E}}_{\ell}} f_{Y|X}(y|x^{(\ell)}) dy \right)^{n P_{\mathbf{u}(i)}(x^{(\ell)})} = \frac{1}{M} \sum_{i=1}^{M} \exp \left( \log \left( \prod_{\ell=1}^{L} \left( \int_{\bar{\mathcal{E}}_{\ell}} f_{Y|X}(y|x^{(\ell)}) dy \right)^{n P_{\mathbf{u}(i)}(x^{(\ell)})} \right) \right) \qquad (91) $$

$$ = \frac{1}{M} \sum_{i=1}^{M} \exp \left( \sum_{\ell=1}^{L} n P_{\mathbf{u}(i)}(x^{(\ell)}) \log \left( \int_{\bar{\mathcal{E}}_{\ell}} f_{Y|X}(y|x^{(\ell)}) dy \right) \right) \qquad (92) $$

$$ \leq \frac{1}{M} \sum_{i=1}^{M} \exp \left( n \log \left( \sum_{\ell=1}^{L} P_{\mathbf{u}(i)}(x^{(\ell)}) \int_{\bar{\mathcal{E}}_{\ell}} f_{Y|X}(y|x^{(\ell)}) dy \right) \right) \qquad (93) $$

$$ = \frac{1}{M} \sum_{i=1}^{M} \exp \left( \log \left( \sum_{\ell=1}^{L} P_{\mathbf{u}(i)}(x^{(\ell)}) \int_{\bar{\mathcal{E}}_{\ell}} f_{Y|X}(y|x^{(\ell)}) dy \right)^n \right) \qquad (94) $$

$$ = \frac{1}{M} \sum_{i=1}^{M} \left( \sum_{\ell=1}^{L} P_{\mathbf{u}(i)}(x^{(\ell)}) \int_{\bar{\mathcal{E}}_{\ell}} f_{Y|X}(y|x^{(\ell)}) dy \right)^n \qquad (95) $$

$$ \leq \frac{1}{M} \sum_{i=1}^{M} \left( \sum_{\ell=1}^{L} P_{\mathbf{u}(i)}(x^{(\ell)}) \int_{\bar{\mathcal{E}}_{\ell}} f_{Y|X}(y|x^{(\ell)}) dy \right) \qquad (96) $$

$$ = \sum_{\ell=1}^{L} \left( \frac{1}{M} \sum_{i=1}^{M} P_{\mathbf{u}(i)}(x^{(\ell)}) \right) \int_{\bar{\mathcal{E}}_{\ell}} f_{Y|X}(y|x^{(\ell)}) dy \qquad (97) $$

$$ = \sum_{\ell=1}^{L} P_{\mathscr{C}}(x^{(\ell)}) \int_{\bar{\mathcal{E}}_{\ell}} f_{Y|X}(y|x^{(\ell)}) dy \qquad (98) $$

$$ = \sum_{\ell=1}^{L} \int_{\bar{\mathcal{E}}_{\ell}} P_{\mathscr{C}}(x^{(\ell)}) f_{Y|X}(y|x^{(\ell)}) dy, \qquad (99) $$

where, the inequality in (93) follows from the concavity of the log function and Jensen's inequality; the inequality in (96) follows because $\sum_{\ell=1}^{L} P_{\mathbf{u}(i)}(x^{(\ell)}) \int_{\bar{\mathcal{E}}_{\ell}} f_{Y|X}(y|x^{(\ell)}) dy \leq 1$; and the equality in (98) follows from (18). From (99) it follows that,

$$ \frac{1}{M} \sum_{i=1}^{M} \left( \prod_{\ell=1}^{L} \left( \int_{\bar{\mathcal{E}}_{\ell}} f_{Y|X}(y|x^{(\ell)}) dy \right)^{n P_{\mathbf{u}(i)}(x^{(\ell)})} \right) \leq \sum_{\ell=1}^{L} \int_{\bar{\mathcal{E}}_{\ell}} P_{\mathscr{C}}(x^{(\ell)}) f_{Y|X}(y|x^{(\ell)}) dy \qquad (100) $$

$$ \leq \sum_{\ell=1}^{L} \int_{\mathcal{E}_{\ell}} P_{\mathscr{C}}(x^{(\ell)}) f_{Y|X}(y|x^{(\ell)}) dy, \qquad (101) $$

where, for all $\ell \in \{1, 2, \dots, L\}$, the set $\mathcal{E}_{\ell}$ is in (88). From (90) and (101), it follows that

$$ \gamma(\mathscr{C}) \geq 1 - \sum_{\ell=1}^{L} \int_{\mathcal{E}_{\ell}} P_{\mathscr{C}}(x^{(\ell)}) f_{Y|X}(y|x^{(\ell)}) dy. \qquad (102) $$

Finally, from (23) and (102), it follows that

$$ \epsilon \geq 1 - \sum_{\ell=1}^{L} \int_{\mathcal{E}_{\ell}} P_{\mathscr{C}}(x^{(\ell)}) f_{Y|X}(y|x^{(\ell)}) dy. \qquad (103) $$
[46] Y. Polyanskiy, H. V. Poor, and S. Verdú, "Channel Coding Rate in the Finite Blocklength Regime," *IEEE Transactions on Information Theory*, vol. 56, no. 5, pp. 2307-2359, 2010.

<figure><img src="image_6.png" /></figure>

**Sadaf ul Zuhra** is a researcher at Ericsson Research, Sweden since 2025. She was a postdoctoral research associate at Princeton University, USA from 2023 to 2024. From 2021 to 2022, she was a postdoctoral researcher at INRIA, Sophia Antipolis, France. She received the Ph.D. degree in Electrical engineering from Indian Institute of Technology (IIT) Bombay, India in 2020. Her research interests are in the areas of information theory and machine learning as applied to wireless networks.

<figure><img src="image_7.png" /></figure>

**Mikael Skoglund** (S'93-M'97-SM'04-F'19) received the Ph.D. degree in 1997 from Chalmers University of Technology, Sweden. In 1997, he joined the Royal Institute of Technology (KTH), Stockholm, Sweden, where he was appointed to the Chair in Communication Theory in 2003. At KTH, he heads the Division of Information Science and Engineering, as well as the Department of Intelligent Systems. Dr. Skoglund has worked on problems in source-channel coding, coding and transmission for wireless communications, Shannon theory, information-theoretic security, information theory for statistics and learning, information and control, and signal processing. He has authored and co-authored more than 200 journal and 450 conference papers. Dr. Skoglund is a Fellow of the IEEE. During 2003-08 he was an associate editor for the IEEE Transactions on Communications. In the interval 2008-12 he was on the editorial board for the IEEE Transactions on Information Theory and starting in the Fall of 2021 he joined it once again. He has served on numerous technical program committees for IEEE sponsored conferences, he was general co-chair for IEEE ITW 2019 and TPC co-chair for IEEE ISIT 2022. He is an elected member of the IEEE Information Theory Society Board of Governors.

<figure><img src="image_8.png" /></figure>

**Samir M. Perlaza** is a permanent member of the scientific staff at INRIA, the French Institute for Research in Computer Science and Applied Mathematics; a visiting research collaborator in the Department of Electrical and Computer Engineering at Princeton University; and an associate researcher in the Mathematics Laboratory GAATI at the University of French Polynesia. He received the M.Sc. and Ph.D. degrees from École Nationale Supérieure des Télécommunications (Telecom ParisTech) in 2008 and 2011, respectively. From 2008 to 2011, he was also a research engineer at France Télécom - Orange Labs (Paris, France). He was a postdoctoral researcher in the Alcatel-Lucent Chair in Flexible Radio at Supélec in 2011 and at Princeton University in 2012 - 2013. Dr. Perlaza's research interests are in the areas of information theory, game theory, data sciences, and their applications in wireless networks, power systems, and artificial intelligence. Among his publications in these areas is the book "Advanced Data Analytics for Power Systems" (Cambridge University Press, 2021). Dr. Perlaza has served as an Editor of the IEEE Transactions in Communications and the IET Smart Grid Journal. Recognition of his work includes the Alban Fellowship and the Marie Skłodowska-Curie Fellowship, both from the European Commission.

<figure><img src="image_9.png" /></figure>

**H. Vincent Poor** received the Ph.D. degree in EECS from Princeton University in 1977. From 1977 until 1990, he was on the faculty of the University of Illinois at Urbana-Champaign. Since 1990 he has been on the faculty at Princeton, where he is currently the Michael Henry Strater University Professor. During 2006 to 2016, he served as the dean of Princeton's School of Engineering and Applied Science. He has also held visiting appointments at several other universities, including most recently at Berkeley and Cambridge. His research interests are in the areas of information theory, machine learning and network science, and their applications in wireless networks, energy systems and related fields. Among his publications in these areas is the recent book Machine Learning and Wireless Communications. (Cambridge University Press, 2022). Dr. Poor is a member of the National Academy of Engineering and the National Academy of Sciences and is a foreign member of the Chinese Academy of Sciences, the Royal Society, and other national and international academies. He received the IEEE Alexander Graham Bell Medal in 2017.
This completes the proof.

<figure><img src="image_4.png" /></figure>

The following result simplifies the lower bound on the DEP in Lemma 4.6 for the case of homogeneous codes (Definition 3.3) as follows.

**Corollary 4.7.** Given a homogeneous $(n, M, \epsilon, B, \delta)$-code $\mathcal{C}$ for the random transformation in (9) of the form in (16) such that (56) holds, the DEP $\epsilon$ satisfies that

$$ \epsilon \geq 1 - \exp \left( -nH(P_{\mathcal{C}}) - nD(P_{\mathcal{C}}||F) + n \log L \right), \qquad (104) $$

where, $P_{\mathcal{C}}$ is the type defined in (18) and the pmf $F$ is defined in (78).

The lower bound on the DEP in (104) provides interesting insights into the trade-offs between the DEP and the energy transmission rate $B$ for an $(n, M, \epsilon, B, \delta)$-code $\mathcal{C}$. As the divergence between the type $P_{\mathcal{C}}$ in (18) and the pmf $F$ in (78) increases, the lower bound on the DEP increases. Furthermore, if the type $P_{\mathcal{C}}$ of a symbol $x^{(\ell)}$ is increased/decreased to increase the harvested energy $B$, the bound in (104) dictates that the corresponding decoding set $\mathcal{E}_{\ell}$ should be proportionately made bigger/smaller to respect the same DEP $\epsilon$.

Though the previous lemmas provide interesting insights into the behavior of the lower bound on the DEP of an $(n, M, \epsilon, B, \delta)$-code, these are often difficult to calculate for large sets of channel inputs. The following lemma introduces a more tractable lower bound on the DEP $\epsilon$ that can be easily calculated for any form of the set of channel input symbols.

**Lemma 4.8.** Consider an $(n, M, \epsilon, B, \delta)$-code $\mathcal{C}$ for the random transformation in (9) of the form in (16). For all $\ell \in \{1, 2, \dots, L\}$, the complex $\bar{x}^{(\ell)} \in \mathcal{X}$ in (3), denotes the nearest neighbor of the symbol $x^{(\ell)} \in \mathcal{X}$, i.e.,

$$ \bar{x}^{(\ell)} \in \arg \min_{x \in \mathcal{X} \setminus \{x^{(\ell)}\}} |x^{(\ell)} - x|. \qquad (105) $$

Then, the DEP $\epsilon$ satisfies,

$$ \epsilon \ge Q \left( \frac{|x^{(\ell^*)} - \bar{x}^{(\ell^*)}|}{\sqrt{2\sigma^2}} - \frac{\sigma}{\sqrt{2}|x^{(\ell^*)} - \bar{x}^{(\ell^*)}|} \log \left( \frac{P_{\mathcal{C}}(\bar{x}^{(\ell^*)})}{P_{\mathcal{C}}(x^{(\ell^*)})} \right) \right), \qquad (106) $$

where, the function $Q$ is defined in (2), the real $\sigma^2$ is defined in (12) and $\ell^* \in \{1, 2, \dots, L\}$ is such that

$$ \ell^* \in \arg \max_{\ell \in \{1, 2, \dots, L\}} \left( 1 - Q \left( \frac{|x^{(\ell)} - \bar{x}^{(\ell)}|}{\sqrt{2\sigma^2}} - \frac{\sigma}{\sqrt{2}|x^{(\ell)} - \bar{x}^{(\ell)}|} \log \left( \frac{P_{\mathcal{C}}(\bar{x}^{(\ell)})}{P_{\mathcal{C}}(x^{(\ell)})} \right) \right) \right). \qquad (107) $$

**Proof.** From Lemma 4.6, the average DEP $\epsilon$ for code $\mathcal{C}$ satisfies the following:

$$ \epsilon \ge 1 - \sum_{\ell=1}^{L} P_{\mathcal{C}}(x^{(\ell)}) \int_{\mathcal{E}_{\ell}} f_{Y|X}(y|x^{(\ell)}) dy, \qquad (108) $$

with the set $\mathcal{E}_{\ell}$ in (88). For all $\ell \in \{1, 2, \dots, L\}$ denote by $\bar{x}^{(\ell)}$, a nearest neighbor of $x^{(\ell)}$ as in (105). Denote by $\hat{\mathcal{E}}_{\ell}$, the following region

$$ \hat{\mathcal{E}}_{\ell} = \{ y \in \mathbb{C} : P_{\mathcal{C}}(x^{(\ell)}) f_{Y|X}(y|x^{(\ell)}) \ge P_{\mathcal{C}}(\bar{x}^{(\ell)}) f_{Y|X}(y|\bar{x}^{(\ell)}) \}. \qquad (109) $$

From the definition of $\mathcal{E}_{\ell}$ in (88) and the definition of $\hat{\mathcal{E}}_{\ell}$ in (109), it follows that,

$$ \mathcal{E}_{\ell} \subseteq \hat{\mathcal{E}}_{\ell} \qquad (110) $$

Therefore, from (110) and (108) it follows that,

$$ \epsilon \ge 1 - \sum_{\ell=1}^{L} P_{\mathcal{C}}(x^{(\ell)}) \int_{\hat{\mathcal{E}}_{\ell}} f_{Y|X}(y|x^{(\ell)}) dy \qquad (111) $$

$$ = 1 - \sum_{\ell=1}^{L} P_{\mathcal{C}}(x^{(\ell)}) \left( 1 - Q \left( \frac{|x^{(\ell)} - \bar{x}^{(\ell)}|}{\sqrt{2\sigma^2}} - \frac{\sigma}{\sqrt{2}|x^{(\ell)} - \bar{x}^{(\ell)}|} \log \left( \frac{P_{\mathcal{C}}(\bar{x}^{(\ell)})}{P_{\mathcal{C}}(x^{(\ell)})} \right) \right) \right), \qquad (112) $$

where, the equality in (112) follows from (111) due to [48, Lemma 20.14.1]. Let $\ell^* \in \{1, 2, \dots, L\}$ such that

$$ \ell^* \in \arg \max_{\ell \in \{1, 2, \dots, L\}} \left( 1 - Q \left( \frac{|x^{(\ell)} - \bar{x}^{(\ell)}|}{\sqrt{2\sigma^2}} - \frac{\sigma}{\sqrt{2}|x^{(\ell)} - \bar{x}^{(\ell)}|} \log \left( \frac{P_{\mathcal{C}}(\bar{x}^{(\ell)})}{P_{\mathcal{C}}(x^{(\ell)})} \right) \right) \right). \qquad (113) $$
From (112) and (113) it follows that,

$$ \epsilon \ge 1 - \sum_{\ell=1}^{L} P_{\mathcal{C}}(x^{(\ell)}) \left( 1 - Q \left( \frac{|x^{(\ell)} - \bar{x}^{(\ell)}|}{\sqrt{2}\sigma^2} - \frac{\sigma}{\sqrt{2}|x^{(\ell)} - \bar{x}^{(\ell)}|} \log \left( \frac{P_{\mathcal{C}}(\bar{x}^{(\ell)})}{P_{\mathcal{C}}(x^{(\ell)})} \right) \right) \right) \quad (114) $$

$$ = 1 - \left( 1 - Q \left( \frac{|x^{(\ell)} - \bar{x}^{(\ell)}|}{\sqrt{2}\sigma^2} - \frac{\sigma}{\sqrt{2}|x^{(\ell)} - \bar{x}^{(\ell)}|} \log \left( \frac{P_{\mathcal{C}}(\bar{x}^{(\ell)})}{P_{\mathcal{C}}(x^{(\ell)})} \right) \right) \right) \quad (115) $$

$$ = Q \left( \frac{|x^{(\ell)} - \bar{x}^{(\ell)}|}{\sqrt{2}\sigma^2} - \frac{\sigma}{\sqrt{2}|x^{(\ell)} - \bar{x}^{(\ell)}|} \log \left( \frac{P_{\mathcal{C}}(\bar{x}^{(\ell)})}{P_{\mathcal{C}}(x^{(\ell)})} \right) \right). \quad (116) $$

This completes the proof. □

### C. Information Transmission Rate

A first upper bound on the information rate is obtained by upper bounding the number of codewords that a code might possess given the particular types $P_{u(1)}$, $P_{u(2)}$, ..., $P_{u(n)}$ in (17); or the average type $P_{\mathcal{C}}$ in (18). The following lemma introduces such an upper bound for the case of a homogeneous code.

**Lemma 4.9.** Given a homogeneous $(n, M, \epsilon, B, \delta)$-code $\mathcal{C}$ for the random transformation in (9) of the form in (16), the information transmission rate $R(\mathcal{C})$ in (19) is such that

$$ R(\mathcal{C}) = \frac{1}{n} \log \left( \frac{n!}{\prod_{\ell=1}^{L} (nP_{\mathcal{C}}(x^{(\ell)}))!} \right) \le \log L, \quad (117) $$

where, $P_{\mathcal{C}}$ is the type defined in (18) and $L$ is the number of channel input symbols in (4).

*Proof.* The largest number of codewords of length $n$ that can be formed using $L$ channel input symbols is $L^n$. In this case, the codewords do not exhibit the type $P_{\mathcal{C}}$. Hence, from (19), in the absence of a constraint on the type $P_{\mathcal{C}}$, the largest information transmission rate is $\log L$ bits per channel use. This justifies the inequality on the right-hand side of (117).

Alternatively, given a code type $P_{\mathcal{C}}$ that satisfies (24), the number of codewords that can be constructed is given by

$$ M = \binom{n}{nP_{\mathcal{C}}(x^{(1)})} \binom{n - nP_{\mathcal{C}}(x^{(1)})}{nP_{\mathcal{C}}(x^{(2)})} \cdots \binom{n - \sum_{\ell=1}^{L-1} n P_{\mathcal{C}}(x^{(\ell)})}{nP_{\mathcal{C}}(x^{(L)})} = \frac{n!}{\prod_{\ell=1}^{L} (nP_{\mathcal{C}}(x^{(\ell)}))!} \quad (118) $$

Therefore, the information rate $R(\mathcal{C})$ in (19) satisfies

$$ R(\mathcal{C}) = \frac{1}{n} \log \left( \frac{n!}{\prod_{\ell=1}^{L} (nP_{\mathcal{C}}(x^{(\ell)}))!} \right), \quad (119) $$

which completes the proof. □

Equality in (117) holds when the type induced by the codewords of the code $\mathcal{C}$ is uniform, i.e., for all $x \in \mathcal{X}$ it holds that

$$ P_{\mathcal{C}}(x) = \frac{1}{L}. \quad (120) $$

Lemma 4.9 can be written in terms of the entropy of the type $P_{\mathcal{C}}$ as shown in the following result.

**Lemma 4.10.** Given a homogeneous $(n, M, \epsilon, B, \delta)$-code $\mathcal{C}$ for the random transformation in (9) of the form in (16), the information transmission rate $R(\mathcal{C})$ in (19) is such that

$$ R(\mathcal{C}) \le H(P_{\mathcal{C}}) + \frac{1}{n^2} \left( \frac{1}{12} - \sum_{\ell=1}^{L} \frac{1}{12 P_{\mathcal{C}}(x^{(\ell)}) + 1} \right) + \frac{1}{n} \left( \log(\sqrt{2\pi}) - \sum_{\ell=1}^{L} \log \sqrt{2\pi P_{\mathcal{C}}(x^{(\ell)})} \right) - \frac{\log n}{n} \left( \frac{L-1}{2} \right), \quad (121) $$

where, $P_{\mathcal{C}}$ is the type defined in (18) and $L$ is the number of channel input symbols in (4).

*Proof.* From (117), the following holds

$$ R(\mathcal{C}) \le \frac{1}{n} \log(n!) - \frac{1}{n} \sum_{\ell=1}^{L} \log \left( (nP_{\mathcal{C}}(x^{(\ell)}))! \right), \quad (122) $$
and using the Stirling's approximation [49] on the factorial terms yields

$$ (nP_{\mathscr{C}}(x^{(\ell)}))! \ge \sqrt{2\pi} (nP_{\mathscr{C}}(x^{(\ell)}))^{nP_{\mathscr{C}}(x^{(\ell)})+\frac{1}{2}} \exp \left(-nP_{\mathscr{C}}(x^{(\ell)}) + \frac{1}{12nP_{\mathscr{C}}(x^{(\ell)})+1}\right), \quad \text{and} \quad (123) $$

$$ n! \le \sqrt{2\pi n} n^{n+\frac{1}{2}} \exp \left(-n + \frac{1}{12n}\right). \quad (124) $$

From (123) and (124), it follows that,

$$ \log((nP_{\mathscr{C}}(x^{(\ell)}))!) \ge \log(\sqrt{2\pi}) + \left(nP_{\mathscr{C}}(x^{(\ell)}) + \frac{1}{2}\right) \log(nP_{\mathscr{C}}(x^{(\ell)})) - nP_{\mathscr{C}}(x^{(\ell)}) + \frac{1}{12nP_{\mathscr{C}}(x^{(\ell)})+1} \quad (125) $$

$$ = \log(\sqrt{2\pi}) + nP_{\mathscr{C}}(x^{(\ell)}) \log(P_{\mathscr{C}}(x^{(\ell)})) + \frac{1}{2} \log(P_{\mathscr{C}}(x^{(\ell)})) \\ + \left(nP_{\mathscr{C}}(x^{(\ell)}) + \frac{1}{2}\right) \log(n) - nP_{\mathscr{C}}(x^{(\ell)}) + \frac{1}{12nP_{\mathscr{C}}(x^{(\ell)})+1} \quad \text{and,} \quad (126) $$

$$ \log(n!) \le \log(\sqrt{2\pi}) + \left(n + \frac{1}{2}\right) \log(n) - n + \frac{1}{12n} \\ = n \log(n) - n + \frac{1}{12n} + \frac{1}{2} \log(2\pi n). \quad (127) $$

The sum in (122) satisfies,

$$ \sum_{\ell=1}^{L} \log \left( (nP_{\mathscr{C}}(x^{(\ell)}))! \right) \ge L \log(\sqrt{2\pi}) - nH(P_{\mathscr{C}}) + \frac{1}{2} \sum_{\ell=1}^{L} \log(P_{\mathscr{C}}(x^{(\ell)})) + n \log(n) \\ + \frac{L}{2} \log(n) - n + \sum_{\ell=1}^{L} \frac{1}{12nP_{\mathscr{C}}(x^{(\ell)})+1}. \quad (128) $$

Using (127) and (128) in (122) yields,

$$ R(\mathscr{C}) \le \log(n) - 1 + \frac{1}{12n^2} + \frac{1}{2n} \log(2\pi n) - \frac{L}{n} \log(\sqrt{2\pi}) + H(P_{\mathscr{C}}) - \frac{1}{2n} \sum_{\ell=1}^{L} \log(P_{\mathscr{C}}(x^{(\ell)})) - \log(n) \\ - \frac{L}{2n} \log(n) + 1 - \frac{1}{n} \sum_{\ell=1}^{L} \frac{1}{12nP_{\mathscr{C}}(x^{(\ell)})+1} \quad (129) $$

$$ \le H(P_{\mathscr{C}}) + \frac{1}{n^2} \left( \frac{1}{12} - \sum_{\ell=1}^{L} \frac{1}{12P_{\mathscr{C}}(x^{(\ell)})+1} \right) + \frac{1}{2n} \left( \log(2\pi n) - \sum_{\ell=1}^{L} \log(2\pi n P_{\mathscr{C}}(x^{(\ell)})) \right) \quad (130) $$

$$ = H(P_{\mathscr{C}}) + \frac{1}{n^2} \left( \frac{1}{12} - \sum_{\ell=1}^{L} \frac{1}{12P_{\mathscr{C}}(x^{(\ell)})+1} \right) + \frac{1}{n} \left( \log(\sqrt{2\pi}) - \sum_{\ell=1}^{L} \log(\sqrt{2\pi} P_{\mathscr{C}}(x^{(\ell)})) \right) - \frac{\log n}{n} \left( \frac{L-1}{2} \right), \quad (131) $$

which completes the proof. $\square$

Note that all terms in (131), except the entropy $H(P_{\mathscr{C}})$, vanish with the block-length $n$. This implies that the information rate is essentially constrained by the entropy of the channel input symbols. In particular, note that $H(P_{\mathscr{C}}) \le \log L$.

## D. Information and Energy Trade-Off

The bounds provided in subsections 4-A, 4-B and 4-C together define the converse information-energy region for $(n, M, \epsilon, B, \delta)$-codes for the random transformation in (9) of the form in (16) for a given set of channel inputs. These bounds prove to be a function of the type $P_{\mathscr{C}}$ and quantify the trade-offs between the information rate $R$, the energy rate $B$, the DEP $\epsilon$ and the EOP $\delta$ for these codes. The following theorem illustrates the dependence between $R$, $B$, $\epsilon$, and $\delta$ for the class of homogeneous codes.
**Theorem 4.11.** Given a homogeneous $(n, M, \epsilon, B, \delta)$-code $\mathcal{C}$ for the random transformation in (9) of the form in (16), the following hold:

$$ \delta = \mathbb{1}_{\{e_{\mathcal{C}} < B\}}; \qquad (132) $$

$$ R(\mathcal{C}) = \frac{1}{n} \log \left( \frac{n!}{\prod_{\ell=1}^{L} (n P_{\mathcal{C}}(x^{(\ell)}))!} \right); \quad \text{and} \qquad (133) $$

$$ \epsilon \ge Q \left( \frac{|x^{(\ell)} - \bar{x}^{(\ell)}|}{\sqrt{2\sigma^2}} - \frac{\sigma}{\sqrt{2}|x^{(\ell)} - \bar{x}^{(\ell)}|} \log \left( \frac{P_{\mathcal{C}}(\bar{x}^{(\ell)})}{P_{\mathcal{C}}(x^{(\ell)})} \right) \right), \qquad (134) $$

where, for all $\ell \in \{1, 2, \dots, L\}$, the complex $\bar{x}^{(\ell)} \in \mathcal{X}$ is in (105) with $\mathcal{X}$ in (3); the function $Q$ is defined in (2); the real $\sigma^2$ is in (12); $\ell^* \in \{1, 2, \dots, L\}$ is defined in (107); the real $e_{\mathcal{C}} \in [0, \infty)$ is in (30); and $P_{\mathcal{C}}$ is the type defined in (18).

*Proof.* The result follows from Lemma 4.2, Lemma 4.9, and Lemma 4.8. $\square$

<figure><img src="image_5.png" /></figure>

Theorem 4.11 illustrates the trade-offs between all the parameters of a homogeneous $(n, M, \epsilon, B, \delta)$-code $\mathcal{C}$ for the random transformation in (9). The upper bound on the information rate in (133) and the lower bound on the DEP in (134) depend on the type $P_{\mathcal{C}}$. Though it is not made obvious from (132), the lower bound on the EOP also depends on the type $P_{\mathcal{C}}$ via (30). The information rate is essentially upper bounded by the entropy of $P_{\mathcal{C}}$. Hence, codes whose codewords are such that every channel input symbol is used the same number of times are less constrained in terms of the information rate. This is the case in which $P_{\mathcal{C}}$ is a uniform distribution. Alternatively, using a uniform type $P_{\mathcal{C}}$ might dramatically constrain the energy transmission rate. For instance, assume that the set of channel input symbols is such that for at least one pair $(x^{(1)}, x^{(2)}) \in \mathcal{X}^2$, with $\mathcal{X}$ in (3), it holds that $|x^{(1)}| < |x^{(2)}|$. Then, from (30), using the symbol $x^{(1)}$ equally often as $x^{(2)}$ certainly constraints the energy $e_{\mathcal{C}}$ and hence, the energy rate $B$. Codes that might potentially exhibit the largest energy rates are those in which the symbols that have the largest magnitude are used more often. This clearly deviates from the uniform distribution and thus, constraints the information rate $R$.

# 5. INFORMATION-ENERGY ACHIEVABLE REGION

This section characterizes an achievable information-energy region for SIET. An achievable region defines tuples of information rate, energy rate, DEP and EOP that can be achieved by some code. As the name suggests, for a tuple to be achievable, a code needs to exist that can achieve the tuple. Therefore, the first step in the characterization of an achievable region is the construction of a code which is accomplished in Section 5-A. This is followed by characterizing the bounds on the various parameters that can be achieved by the constructed code in Section 5-B.

## A. Code Construction

The process of characterizing an achievable information-energy capacity region for SIET in the finite block-length regime with finite set of channel input symbols begins with the construction of an $(n, M)$-code $\mathcal{C}$ of the form in (16). The construction of the code begins with the construction of the channel input symbols. The set of channel input symbols is a modulation constellation represented by a finite subset of $\mathbb{C}$. Consider a set of channel input symbols formed by $C$ layers, with $C \in \mathbb{N}$. A layer is a subset of symbols in $\mathbb{C}$ that have the same magnitude. For all $c \in \{1, 2, \dots, C\}$, denote by $L_c \in \mathbb{N}$ the number of symbols in the $c^{th}$ layer and let $A_c \in \mathbb{R}^+$ and $\alpha_c \in [0, 2\pi]$ be the amplitude and phase shift of the symbols in layer $c$. Denote such a layer by $\mathcal{U}(A_c, L_c, \alpha_c)$. That is,

$$ \mathcal{U}(A_c, L_c, \alpha_c) \triangleq \left\{ x_c^{(\ell)} = A_c \exp \left( i \left( \frac{2\pi}{L_c} \ell + \alpha_c \right) \right) \subseteq \mathbb{C} : \ell \in \{0, 1, 2, \dots, (L_c - 1)\} \right\}, \qquad (135a) $$

where $i$ is the complex unit. Using this notation, the set of channel input symbols can be described by the following set

$$ \mathcal{X} = \bigcup_{c=1}^{C} \mathcal{U}(A_c, L_c, \alpha_c). \qquad (135b) $$

The vector of the amplitudes in (135b) is denoted by

$$ \mathbf{A} = (A_1, A_2, \dots, A_C)^{\mathrm{T}}; \qquad (135c) $$

the vector of the number of symbols in each layer in (135b) is denoted by

$$ \mathbf{L} = (L_1, L_2, \dots, L_C)^{\mathrm{T}}; \qquad (135d) $$

and the vector of the phase shifts of the symbols in each layer in (135b) is denoted by

$$ \alpha = (\alpha_1, \alpha_2, \dots, \alpha_C)^{\mathrm{T}}. \qquad (135e) $$
The total number of symbols $L$ in (4) for $\mathcal{X}$ in (135b) is

$$L = \sum_{c=1}^{C} L_c. \qquad (135f)$$

Without any loss of generality, assume that

$$A_1 > A_2 > \dots > A_C. \qquad (135g)$$

For all $c \in \{1, 2, \dots, C\}$, the symbols in $\mathcal{U}(A_c, L_c, \alpha_c)$ in (135a), are equally spaced along a circle of radius $A_c$ (see Fig. 1). The set of channel input symbols $\mathcal{X}$ in (135b) is thus made up of points uniformly distributed along $C$ concentric circles.

The construction of the $(n, M)$-code $\mathcal{C}$ is accomplished as follows. For all $c \in \{1, 2, \dots, C\}$, let $p_c$ be the frequency with which symbols of the $c^{th}$ layer appear in $\mathcal{C}$. The resulting probability vector is denoted by

$$\mathbf{p} = (p_1, p_2, \dots, p_C)^{\mathrm{T}}. \qquad (135h)$$

For transmitting message index $i \in \{1, 2, \dots, M\}$, the transmitter uses the codeword

$$\mathbf{u}(i) = (u_1(i), u_2(i), \dots, u_n(i)) \in \mathcal{X}^n, \qquad (135i)$$

with $\mathcal{X}$ in (135b). Using the probability vector $\mathbf{p}$ in (135h) which determines the type $P_{\mathcal{C}}$ in (18), and the set of symbols $\mathcal{X}$ in (135b), the set of codewords $\{\mathbf{u}(1), \mathbf{u}(2), \dots, \mathbf{u}(M)\}$ is completely defined.

For all $c \in \{1, 2, \dots, C\}$, it holds that,

$$p_c = \frac{1}{Mn} \sum_{\ell=1}^{L_c} \sum_{i=1}^{M} \sum_{m=1}^{n} \mathbb{1}_{\{u_m(i)=x_c^{(\ell)}\}} \qquad (135j)$$

$$= \frac{1}{M} \sum_{\ell=1}^{L_c} \sum_{i=1}^{M} P_{\mathbf{u}(i)}(x_c^{(\ell)}), \qquad (135k)$$

where, for all $i \in \{1, 2, \dots, M\}$ and all $m \in \{1, 2, \dots, n\}$, the complex $u_m(i)$ in (135i) is the $m^{th}$ channel input symbol of the codeword $\mathbf{u}(i)$ and $x_c^{(\ell)}$ is in (135a). The equality in (135k) follows from (17). The symbols within a layer are used with the same frequency in $\mathcal{C}$. Hence, for all $c \in \{1, 2, \dots, C\}$ and for all $\ell \in \{1, 2, \dots, L_c\}$, the frequency with which the symbol $x_c^{(\ell)}$ appears in $\mathcal{C}$ is given by

$$P_{\mathcal{C}}(x_c^{(\ell)}) \triangleq \frac{1}{Mn} \sum_{i=1}^{M} \sum_{m=1}^{n} \mathbb{1}_{\{u_m(i)=x_c^{(\ell)}\}}, \qquad (135l)$$

$$= \frac{p_c}{L_c}. \qquad (135m)$$

The decoding region for codeword $\mathbf{u}(i)$ is denoted by $\mathcal{D}_i$ and that for the symbol $u_m(i)$ is denoted by $\mathcal{D}_{i,m}$ such that the following constraint on $\mathcal{D}_i$ holds:

$$\mathcal{D}_i = \mathcal{D}_{i,1} \times \mathcal{D}_{i,2} \times \dots \times \mathcal{D}_{i,n}, \qquad (135n)$$

where, for all $i \in \{1, 2, \dots, M\}$, $c \in \{1, 2, \dots, C\}$, $t \in \{1, 2, \dots, n\}$, and $\ell \in \{1, 2, \dots, L_c\}$, when $u_m(i) = x_c^{(\ell)}$, then, $\mathcal{D}_{i,m} = \mathcal{G}_c^{(\ell)}$.

The decoding set $\mathcal{G}_c^{(\ell)} \subseteq \mathbb{C}$ associated with symbol $x_c^{(\ell)}$ is a circle of radius $r_c \in \mathbb{R}^+$ centered at $x_c^{(\ell)}$. That is,

$$\mathcal{G}_c^{(\ell)} = \{y \in \mathbb{C} : |y - x_c^{(\ell)}|^2 \leq r_c^2\}, \qquad (135o)$$

$$= \{y \in \mathbb{C} : (\Re(y) - \Re(x_c^{(\ell)}))^2 + (\Im(y) - \Im(x_c^{(\ell)}))^2 \leq r_c^2\}. \qquad (135p)$$

The radii $r_1, r_2, \dots, r_C$ are chosen such that the decoding sets $\mathcal{G}_c^{(\ell)}$ are mutually disjoint. To ensure that the decoding regions are mutually disjoint, for all $c \in \{1, 2, \dots, C\}$, the amplitudes $A_c$ in (135a) are chosen to satisfy the following:

$$A_{c+1} - A_c \geq r_{c+1} + r_c. \qquad (135q)$$

The vector of these radii is denoted by

$$\mathbf{r} = (r_1, r_2, \dots, r_C)^{\mathrm{T}}. \qquad (135r)$$
This defines a family of $(n, M)$-codes denoted by

$$ C(C, \mathbf{A}, \mathbf{L}, \alpha, \mathbf{p}, \mathbf{r}), \qquad (135s) $$

with the number of layers $C$ in (135b), $\mathbf{A}$ in (135c), $\mathbf{L}$ in (135d), $\alpha$ in (135e) $\mathbf{p}$ in (135h) and $\mathbf{r}$ in (135r). The following section characterizes the achievable region for codes in the family $C(C, \mathbf{A}, \mathbf{L}, \alpha, \mathbf{p}, \mathbf{r})$.

## B. Achievability Bounds

The following lemma provides a lower bound on the DEP $\epsilon$ of codes from the family $C(C, \mathbf{A}, \mathbf{L}, \alpha, \mathbf{p}, \mathbf{r})$ in (135s).

**Lemma 5.1.** Consider an $(n, M)$-code $\mathcal{C}$ of the form in (135) from the family $C(C, \mathbf{A}, \mathbf{L}, \alpha, \mathbf{p}, \mathbf{r})$ in (135s). The code $\mathcal{C}$ is an $(n, M, \epsilon)$-code if the parameters $r_1, r_2, \dots, r_C$ in (135p) satisfy the following:

$$ \epsilon \ge 1 - \frac{1}{M} \sum_{i=1}^{M} \prod_{c=1}^{C} \left(1 - e^{-\frac{r_c^2}{\sigma^2}}\right)^{n \sum_{\ell=1}^{L_c} P_{u(i)}(x_c^{(\ell)})}, \qquad (136) $$

where, the type $P_{u(i)}$ is defined in (17), the real $\sigma^2$ is defined in (12), and $x_c^{(\ell)} \in \mathcal{U}(A_c, L_c, \alpha_c)$, with $\mathcal{U}(A_c, L_c, \alpha_c)$ in (135a).

*Proof.* From (21) and (22), the average DEP of code $\mathcal{C}$ is given by

$$ \gamma(\mathcal{C}) = 1 - \frac{1}{M} \sum_{i=1}^{M} \int_{D_i} f_{Y|X}(y|u(i)) dy \qquad (137) $$

$$ = 1 - \frac{1}{M} \sum_{i=1}^{M} \int_{D_{i,1} \times D_{i,2} \times \dots \times D_{i,n}} \prod_{m=1}^{n} f_{Y|X}(y|u_m(i)) dy \qquad (138) $$

$$ = 1 - \frac{1}{M} \sum_{i=1}^{M} \prod_{m=1}^{n} \int_{D_{i,m}} f_{Y|X}(y|u_m(i)) dy \qquad (139) $$

$$ = 1 - \frac{1}{M} \sum_{i=1}^{M} \prod_{c=1}^{C} \prod_{\ell=1}^{L_c} \left( \int_{G_c^{(\ell)}} f_{Y|X}(y|x_c^{(\ell)}) dy \right)^{n P_{u(i)}(x_c^{(\ell)})}, \qquad (140) $$

where, the equality in (138) follows due to (10) and (135n), and (139) follows due to Fubini's theorem. Using (12) in (140) yields,

$$ \gamma(\mathcal{C}) = 1 - \frac{1}{M} \sum_{i=1}^{M} \prod_{c=1}^{C} \prod_{\ell=1}^{L_c} \left( \int_{G_c^{(\ell)}} \frac{1}{\pi \sigma^2} \exp \left( - \frac{(\Re(y) - \Re(x_c^{(\ell)}))^2 + (\Im(y) - \Im(x_c^{(\ell)}))^2}{\sigma^2} \right) dy \right)^{n P_{u(i)}(x_c^{(\ell)})} \qquad (141) $$

Evaluating the integral term in (141) for all $\ell \in \{1, 2, \dots, L_c\}$ yields,

$$ \int_{G_c^{(\ell)}} \frac{1}{\pi \sigma^2} \exp \left( - \frac{(\Re(y) - \Re(x_c^{(\ell)}))^2 + (\Im(y) - \Im(x_c^{(\ell)}))^2}{\sigma^2} \right) dy \\ = \int_{\Im(x_c^{(\ell)}) - r_c}^{\Im(x_c^{(\ell)}) + r_c} \int_{\Re(x_c^{(\ell)}) - \sqrt{r_c^2 - (v - \Im(x_c^{(\ell)}))^2}}^{\Re(x_c^{(\ell)}) + \sqrt{r_c^2 - (v - \Im(x_c^{(\ell)}))^2}} \frac{1}{\pi \sigma^2} e^{-\frac{(u - \Re(x_c^{(\ell)}))^2 + (v - \Im(x_c^{(\ell)}))^2}{\sigma^2}} du dv, \qquad (142) $$

$$ = \int_{-r_c}^{r_c} \int_{-\sqrt{r_c^2 - v^2}}^{\sqrt{r_c^2 - v^2}} \frac{1}{\pi \sigma^2} e^{-\frac{u^2 + v^2}{\sigma^2}} du dv, \qquad (143) $$

$$ = \int_{0}^{\pi} \int_{0}^{\frac{r_c}{\sigma}} \frac{1}{2\pi} e^{-\zeta^2} \zeta d\zeta d\eta, \qquad (144) $$

$$ = (1 - e^{-\frac{r_c^2}{\sigma^2}}). \qquad (145) $$

The equality in (144) is obtained from the change of variables $u = \sigma\zeta \cos\eta, v = \sigma\zeta \sin\eta$. Plugging (145) in (141) yields,

$$ \gamma(\mathcal{C}) = 1 - \frac{1}{M} \sum_{i=1}^{M} \prod_{c=1}^{C} \prod_{\ell=1}^{L_c} \left(1 - e^{-\frac{r_c^2}{\sigma^2}}\right)^{n P_{u(i)}(x_c^{(\ell)})}, \qquad (146) $$

$$ = 1 - \frac{1}{M} \sum_{i=1}^{M} \prod_{c=1}^{C} \left(1 - e^{-\frac{r_c^2}{\sigma^2}}\right)^{n \sum_{\ell=1}^{L_c} P_{u(i)}(x_c^{(\ell)})}. \qquad (147) $$
From (23), for $\mathcal{C}$ to be an $(n, M, \epsilon)$-code, the following must hold:

$$ \gamma(\mathcal{C}) \leq \epsilon. \qquad (148) $$

This implies that,

$$ \epsilon \geq 1 - \frac{1}{M} \sum_{i=1}^{M} \prod_{c=1}^{C} \left(1 - e^{-\frac{r_c^2}{\sigma^2}}\right)^{n \sum_{\ell=1}^{L_c} P_{u(i)}(x_c^{(\ell)})}, \qquad (149) $$

which is the desired result.

<figure><img src="image_6.png" /></figure>

From Definition 3.3 and (135m), it follows that, for a homogeneous code $\mathcal{C}$ of the form in (135) from the family $\mathcal{C}(C, A, L, \alpha, p, r)$ in (135s), it holds that,

$$ P_{u(i)}(x_c^{(\ell)}) = \frac{p_c}{L_c}. \qquad (150) $$

The following result for homogeneous codes follows from (150) and Lemma 5.1.

**Corollary 5.2.** Consider a homogeneous $(n, M)$-code $\mathcal{C}$ of the form in (135) from the family $\mathcal{C}(C, A, L, \alpha, p, r)$ in (135s). The code $\mathcal{C}$ is an $(n, M, \epsilon)$-code if the parameters $r_1, r_2, \dots, r_C$ in (135p) satisfy the following:

$$ \epsilon \geq 1 - \prod_{c=1}^{C} \left(1 - e^{-\frac{r_c^2}{\sigma^2}}\right)^{np_c}, \qquad (151) $$

where, the real $\sigma^2$ is defined in (12), and $x_c^{(\ell)} \in \mathcal{U}(A_c, L_c, \alpha_c)$, with $\mathcal{U}(A_c, L_c, \alpha_c)$ in (135a).

For the special case where for all $c \in \{1, 2, \dots, C\}$, $r_c = r$ in (135r), the following lemma provides a lower bound on the radius $r$.

**Lemma 5.3.** Consider an $(n, M)$-code $\mathcal{C}$ of the form in (135) from the family $\mathcal{C}(C, A, L, \alpha, p, r)$ in (135s) such that, for all $c \in \{1, 2, \dots, C\}$, $r_c = r$ in (135p). The code $\mathcal{C}$ is an $(n, M, \epsilon)$-code if the parameter $r$ satisfies:

$$ r \geq \sqrt{\sigma^2 \log \left( \frac{1}{1 - (1 - \epsilon)^{\frac{1}{n}}} \right)}, \qquad (152) $$

where, the real $\sigma^2$ is defined in (12).

*Proof.* If the parameters $r_c$ in (135p) are such that, for all $c \in \{1, 2, \dots, C\}$, $r_c = r$, the average DEP in (147) is given by:

$$ \gamma(\mathcal{C}) = 1 - \frac{1}{M} \sum_{i=1}^{M} \prod_{c=1}^{C} \left(1 - e^{-\frac{r_c^2}{\sigma^2}}\right)^{n \sum_{\ell=1}^{L_c} P_{u(i)}(x_c^{(\ell)})}, \qquad (153) $$

$$ = 1 - \frac{1}{M} \sum_{i=1}^{M} \left(1 - e^{-\frac{r_c^2}{\sigma^2}}\right)^{n \sum_{c=1}^{C} \sum_{\ell=1}^{L_c} P_{u(i)}(x_c^{(\ell)})}, \qquad (154) $$

$$ = 1 - \frac{1}{M} \sum_{i=1}^{M} \left(1 - e^{-\frac{r_c^2}{\sigma^2}}\right)^n, \qquad (155) $$

$$ = 1 - \left(1 - e^{-\frac{r_c^2}{\sigma^2}}\right)^n. \qquad (156) $$

From (23), for $\mathcal{C}$ to be an $(n, M, \epsilon)$-code, the following must hold:

$$ \epsilon \geq 1 - \left(1 - e^{-\frac{r_c^2}{\sigma^2}}\right)^n. \qquad (157) $$

This implies that

$$ r \geq \sqrt{\sigma^2 \log \left( \frac{1}{1 - (1 - \epsilon)^{\frac{1}{n}}} \right)}. \qquad (158) $$

This completes the proof.

<figure><img src="image_7.png" /></figure>

The information rate achievable by a code is a function of the number of channel input symbols $L$ in (135f) which in turn is a function of the number of symbols in each layer of the constructed set of channel inputs $\mathcal{X}$ in (135b). The following lemma provides an upper bound on the number of symbols in a layer $c \in \{1, 2, \dots, C\}$ denoted by $L_c$.
<figure><img src="image_8.png" /><figcaption>Fig. 1: Graphical representation of the symbols in layer c defined in (135a)</figcaption></figure>

**Lemma 5.4.** Consider an $(n, M)$-code $\mathcal{C}$ of the form in (135) from the family $\mathbf{C}(C, \mathbf{A}, \mathbf{L}, \alpha, \mathbf{p}, \mathbf{r})$ in (135s). Then, for all $c \in \{1, 2, \dots, C\}$, the number of symbols in layer c of $\mathcal{X}$ is given by

$$L_c \leq \left\lfloor \frac{\pi}{2 \arcsin \frac{r_c}{2A_c}} \right\rfloor, \qquad (159)$$

where, $r_c$ is the radius of the decoding regions $\mathcal{G}_c^{(1)}, \mathcal{G}_c^{(2)}, \dots, \mathcal{G}_c^{(L_c)}$ in (135p) and $A_c$ is the amplitude in (135a).

*Proof.* For an $(n, M)$-code $\mathcal{C}$ in the family $\mathbf{C}(C, \mathbf{A}, \mathbf{L}, \alpha, \mathbf{p}, \mathbf{r})$, for all $c \in \{1, 2, \dots, C\}$, the radius $r_c$ of the decoding regions in (135r) and the amplitude $A_c$ in (135c) determine the number of symbols $L_c$ that can be accommodated in the layer c.

Layer c of the form in (135a) is illustrated in Fig. 1. Symbols in layer c are distributed uniformly along the circle of radius $A_c$ centered at the origin $O$. The maximum number of symbols that can be accommodated in layer c is equal to the number of non-overlapping circles of radius $r_c$ corresponding to the decoding regions defined in (135p) that can be placed along the circumference of the circle of radius $A_c$. From Fig. 1, a circle of radius $r_c$ centered at a symbol in layer c subtends angle $\angle SOU = \beta$ at $O$. Therefore, the maximum number of symbols $L_c$ that can be accommodated along the circle of radius $A_c$ is given by

$$L_c \leq \left\lfloor \frac{2\pi}{\beta} \right\rfloor. \qquad (160)$$

The following lemma determines the value of the angle $\beta$ in Fig. 1.

**Lemma 5.5.** Consider a circle of radius $A_c$ centered at $O$ and a circle of radius $r_c$ centered on the circumference of the first circle as shown in Fig. 1. The two circles intersect at points $S$ and $U$. Then, the angle $\angle SOU$ is given by the following:

$$\angle SOU = \beta = 4 \arcsin \frac{r_c}{2A_c}. \qquad (161)$$

*Proof.* In Fig. 1, consider the circle of radius $r_c$ centered at $T$ and the larger circle of radius $A_c$ centered at the origin $O$. The circles intersect at points $S$ and $U$. The angle subtended by the major arc $SU$ at $O$ is the reflex angle $2\pi - \beta$. Since the angle subtended by an arc of a circle at its centre is two times the angle that it subtends anywhere on the circumference, it holds that

$$2\pi - \beta = 2\angle STU, \qquad (162)$$

which implies that

$$\angle STU = \frac{2\pi - \beta}{2}. \qquad (163)$$
The line segment *TO* bisects angles ∠STU and ∠SOU. Therefore, the following hold:

$$ \angle STO = \frac{2\pi - \beta}{4}, \qquad (164) $$

$$ \angle SOT = \frac{\beta}{2}. \qquad (165) $$

From the triangle △SOT, it holds that:

$$ \frac{\sin(\angle SOT)}{ST} = \frac{\sin(\angle STO)}{SO}. \qquad (166) $$

This implies that,

$$ \frac{\sin\left(\frac{\beta}{2}\right)}{r_c} = \frac{\sin\left(\frac{2\pi-\beta}{4}\right)}{A_c}, \qquad (167) $$

$$ = \frac{1}{A_c} \sin\left(\frac{\pi}{2} - \frac{\beta}{4}\right), \qquad (168) $$

$$ = \frac{1}{A_c} \cos\left(\frac{\beta}{4}\right). \qquad (169) $$

From (169), it follows that,

$$ \frac{2}{r_c} \sin\left(\frac{\beta}{4}\right) \cos\left(\frac{\beta}{4}\right) = \frac{1}{A_c} \cos\left(\frac{\beta}{4}\right), \qquad (170) $$

which implies that,

$$ \sin\left(\frac{\beta}{4}\right) = \frac{r_c}{2A_c}, \quad \text{and} \qquad (171) $$

$$ \beta = 4 \arcsin \frac{r_c}{2A_c}. \qquad (172) $$

This completes the proof.

<figure><img src="image_9.png" /></figure>

Substituting the value of $\beta$ from (161) in (160), the number of symbols in layer c of the set $\mathcal{X}$ is given by

$$ L_c \le \left\lfloor \frac{\pi}{2 \arcsin \frac{r_c}{2A_c}} \right\rfloor. \qquad (173) $$

This completes the proof.

<figure><img src="image_10.png" /></figure>

The information transmission rate $R(\mathcal{C})$ is a function of the number of symbols $L$ in (135f). Therefore, using Lemma 5.4, the achievable bound on the information transmission rate $R(\mathcal{C})$ for code $\mathcal{C}$ from the family $\mathcal{C}(C, \mathbf{A}, \mathbf{L}, \boldsymbol{\alpha}, \mathbf{p}, \mathbf{r})$ in (135s) can now be calculated. The following lemma provides this result.

**Lemma 5.6.** For a homogeneous $(n, M)$-code $\mathcal{C}$ of the form in (135) from the family $\mathcal{C}(C, \mathbf{A}, \mathbf{L}, \boldsymbol{\alpha}, \mathbf{p}, \mathbf{r})$ in (135s), the information transmission rate $R(\mathcal{C})$ satisfies the following:

$$ R(\mathcal{C}) = \frac{1}{n} \log \left( \frac{n!}{\prod_{c=1}^{C} \left( \left( n \frac{p_c}{L_c} \right)! \right)^{L_c}} \right) \le \log \sum_{c=1}^{C} \left\lfloor \frac{\pi}{2 \arcsin \frac{r_c}{2A_c}} \right\rfloor, \qquad (174) $$

where, for all $c \in \{1, 2, \dots, C\}$, the radius $r_c$ is in (135p), and the amplitude $A_c$ is in (135a).

*Proof.* The largest number of codewords of length $n$ that can be formed with $L$ different channel input symbols is $L^n$. Hence, from (19), it follows that

$$ R(\mathcal{C}) = \frac{\log M}{n} \qquad (175) $$

$$ \le \frac{\log L^n}{n} \qquad (176) $$

$$ = \log L \qquad (177) $$

$$ \le \log \sum_{c=1}^{C} \left\lfloor \frac{\pi}{2 \arcsin \frac{r_c}{2A_c}} \right\rfloor, \qquad (178) $$

where, the inequality in (178) follows from Lemma 5.4 and (135f).
The bound in (178) can only be achieved by a code for which the type $P_{\mathscr{C}}$ is uniform, i.e., for all $x \in \mathcal{X}$ with $\mathcal{X}$ in (135b), it holds that,

$$ P_{\mathscr{C}}(x) = \frac{1}{L}. \qquad (179) $$

Given the type $P_{\mathscr{C}}$ that satisfies (24), the number of codewords that can be constructed is given by

$$ M = \binom{n}{nP_{\mathscr{C}}(x_1^{(1)})} \binom{n-nP_{\mathscr{C}}(x_1^{(1)})}{nP_{\mathscr{C}}(x_1^{(2)})} \cdots \binom{n-\sum_{\ell=1}^{L_1-1} nP_{\mathscr{C}}(x_1^{(\ell)})}{nP_{\mathscr{C}}(x_1^{(L_1)})} \times \binom{n-\sum_{\ell=1}^{L_1} nP_{\mathscr{C}}(x_1^{(\ell)})}{nP_{\mathscr{C}}(x_2^{(1)})} \\ \times \binom{n-\sum_{\ell=1}^{L_1} nP_{\mathscr{C}}(x_1^{(\ell)}) - nP_{\mathscr{C}}(x_2^{(1)})}{nP_{\mathscr{C}}(x_2^{(2)})} \cdots \binom{n-\sum_{\ell=1}^{L_2-1} nP_{\mathscr{C}}(x_2^{(\ell)})}{nP_{\mathscr{C}}(x_2^{(L_2)})} \times \cdots \times \binom{n-\sum_{c=1}^{C-1} \sum_{\ell=1}^{L_c} nP_{\mathscr{C}}(x_c^{(\ell)})}{nP_{\mathscr{C}}(x_C^{(1)})} \\ \times \binom{n-\sum_{c=1}^{C-1} \sum_{\ell=1}^{L_c} nP_{\mathscr{C}}(x_c^{(\ell)}) - nP_{\mathscr{C}}(x_C^{(1)})}{nP_{\mathscr{C}}(x_C^{(2)})} \cdots \binom{n-\sum_{c=1}^{C-1} \sum_{\ell=1}^{L_c} nP_{\mathscr{C}}(x_c^{(\ell)}) - \sum_{\ell=1}^{L_c-1} nP_{\mathscr{C}}(x_C^{(\ell)})}{nP_{\mathscr{C}}(x_C^{(L_c)})} \qquad (180) $$

$$ = \binom{n}{nP_{1}^{\frac{p_1}{L_1}}} \binom{n-n\frac{p_1}{L_1}}{nP_{1}^{\frac{p_1}{L_1}}} \cdots \binom{n-\sum_{\ell=1}^{L_1-1} n\frac{p_1}{L_1}}{nP_{1}^{\frac{p_1}{L_1}}} \times \binom{n-np_1}{nP_{2}^{\frac{p_2}{L_2}}} \binom{n-np_1-n\frac{p_2}{L_2}}{nP_{2}^{\frac{p_2}{L_2}}} \cdots \binom{n-np_1-\sum_{\ell=1}^{L_2-1} n\frac{p_2}{L_2}}{nP_{2}^{\frac{p_2}{L_2}}} \times \cdots \times $$

$$ \left( n - \sum_{c=1}^{C-1} np_c \right) \left( n - \sum_{c=1}^{C-1} np_c - n \frac{p_c}{L_c} \right) \cdots \left( n - \sum_{c=1}^{C-1} np_c - \sum_{\ell=1}^{L_c-1} n \frac{p_c}{L_c} \right) \qquad (181) $$

$$ = \frac{n!}{\prod_{c=1}^{C} \left( \left( n \frac{p_c}{L_c} \right)! \right)^{L_c}} \qquad (182) $$

Therefore, the information transmission rate $R(\mathscr{C})$ is given by

$$ R(\mathscr{C}) = \frac{\log M}{n} = \frac{1}{n} \log \left( \frac{n!}{\prod_{c=1}^{C} \left( \left( n \frac{p_c}{L_c} \right)! \right)^{L_c}} \right). \qquad (183) $$

The result follows from (178) and (183). $\square$

Given the number of layers $C$ in (135b), the vector of amplitudes $\mathbf{A}$ in (135c), the vector of the number of symbols in each layer $\mathbf{L}$ in (135d), the vector of phase shifts $\boldsymbol{\alpha}$ in (135e), the probability vector $\mathbf{p}$ in (135h) and the radius vector $\mathbf{r}$ in (135r), infinitely many sets of channel input symbols $\mathcal{X}$ in (135b) can be generated by rotating $\mathcal{X}$. Rotating $\mathcal{X}$ implies changing the phase shift of all the symbols in $\mathcal{X}$ by the same angle. It is easy to see that the information transmission rate $R(\mathscr{C})$ of the code $\mathscr{C}$ is independent of such rotations since it only depends on the number of symbols $\mathbf{L}$ in (135f) and the probability vector $\mathbf{p}$. However, the impact of such rotations on the DEP $\gamma(\mathscr{C})$ and the EOP $\theta(\mathscr{C}, \mathbf{B})$ of $\mathscr{C}$ is not immediately obvious. The following lemma proves that the DEP and the EOP also remain unchanged for any rotation of the set $\mathcal{X}$.

**Lemma 5.7.** Consider two $(n, M)$-codes $\mathscr{C}$ and $\mathscr{C}'$, both of the form in (135) from the family $\mathbf{C}(\mathbf{C}, \mathbf{A}, \mathbf{L}, \boldsymbol{\alpha}, \mathbf{p}, \mathbf{r})$ in (135s). For all $c \in \{1, 2, \dots, C\}$ and all $\ell \in \{1, 2, \dots, L_c\}$, denote the symbols in layer $c$ in (135a) of the code $\mathscr{C}$ by $x_c^{(\ell)}$ and those of $\mathscr{C}'$ by $\hat{x}_c^{(\ell)}$. For all $c \in \{1, 2, \dots, C\}$, all $\ell \in \{1, 2, \dots, L_c\}$, and $\omega \in [0, 2\pi]$, it holds that,

$$ \hat{x}_c^{(\ell)} = e^{i\omega} x_c^{(\ell)}. \qquad (184) $$

All other parameters of the codes $\mathscr{C}$ and $\mathscr{C}'$ are identical. Then, the average DEP $\gamma$ in (22) and the average EOP $\theta$ in (37) of $\mathscr{C}$ and $\mathscr{C}'$ satisfy the following:

$$ \gamma(\mathscr{C}) = \gamma(\mathscr{C}'), \text{ and} \qquad (185) $$

$$ \theta(\mathscr{C}, \mathbf{B}) = \theta(\mathscr{C}', \mathbf{B}). \qquad (186) $$

*Proof.* From (141), the average DEP for $\mathscr{C}$ is given by the following:

$$ \gamma(\mathscr{C}) = 1 - \frac{1}{M} \sum_{i=1}^{M} \prod_{c=1}^{C} \prod_{\ell=1}^{L_c} \left( \int_{G_c^{(\ell)}} \frac{1}{\pi \sigma^2} \exp \left( - \frac{(\Re(y) - \Re(x_c^{(\ell)}))^2 + (\Im(y) - \Im(x_c^{(\ell)}))^2}{\sigma^2} \right) dy \right)^{n P_{u(i)}(x_c^{(\ell)})}, \qquad (187) $$

$$ = 1 - \frac{1}{M} \sum_{i=1}^{M} \prod_{c=1}^{C} \prod_{\ell=1}^{L_c} \left( \int_{y:|y-x_c^{(\ell)}| \le r_c^2} \frac{1}{\pi \sigma^2} \exp \left( - \frac{|y-x_c^{(\ell)}|^2}{\sigma^2} \right) dy \right)^{n P_{u(i)}(x_c^{(\ell)})}, \qquad (188) $$
where, the expression in (188) follows from (187) due to (11) and (135o). The change of variable $y = z + x_c^{(\ell)} - \hat{x}_c^{(\ell)}$ in the integral in (188) yields

$$ \int_{y:|y-x_c^{(\ell)}|^2 \le r_c^2} \frac{1}{\pi \sigma^2} \exp\left(-\frac{|y-x_c^{(\ell)}|^2}{\sigma^2}\right) dy = \int_{z:|z-\hat{x}_c^{(\ell)}|^2 \le r_c^2} \frac{1}{\pi \sigma^2} \exp\left(-\frac{|z-\hat{x}_c^{(\ell)}|^2}{\sigma^2}\right) dz. \quad (189) $$

Using (189) in (188) yields:

$$ \gamma(\mathcal{C}) = 1 - \frac{1}{M} \sum_{i=1}^{M} \prod_{c=1}^{C} \prod_{\ell=1}^{L_c} \left( \int_{z:|z-\hat{x}_c^{(\ell)}|^2 \le r_c^2} \frac{1}{\pi \sigma^2} \exp\left(-\frac{|z-\hat{x}_c^{(\ell)}|^2}{\sigma^2}\right) dz \right)^{n P_{u(i)}(x_c^{(\ell)})} . \quad (190) $$

The empirical probability of usage of symbols is equal for the symbols $x_c^{(\ell)}$ and $\hat{x}_c^{(\ell)}$. That is, for all $I \in \{1, 2, \dots, M\}$, all $c \in \{1, 2, \dots, C\}$ and all $\ell \in \{1, 2, \dots, L_c\}$, it holds that

$$ P_{u(i)}(x_c^{(\ell)}) = P_{u(i)}(\hat{x}_c^{(\ell)}). \quad (191) $$

From (190) and (191), it follows that

$$ \gamma(\mathcal{C}) = 1 - \frac{1}{M} \sum_{i=1}^{M} \prod_{c=1}^{C} \prod_{\ell=1}^{L_c} \left( \int_{z:|z-\hat{x}_c^{(\ell)}|^2 \le r_c^2} \frac{1}{\pi \sigma^2} \exp\left(-\frac{|z-\hat{x}_c^{(\ell)}|^2}{\sigma^2}\right) dz \right)^{n P_{u(i)}(\hat{x}_c^{(\ell)})} \quad (192) \\ = \gamma(\mathcal{C}'), \quad (193) $$

which is the desired result in (185).

From (184), for all $c \in \{1, 2, \dots, C\}$, all $\ell \in \{1, 2, \dots, L_c\}$, and $\omega \in [0, 2\pi]$, it holds that,

$$ \hat{x}_c^{(\ell)} = e^{i\omega} x_c^{(\ell)}. \quad (194) $$

This implies that

$$ |\hat{x}_c^{(\ell)}| = |x_c^{(\ell)}|. \quad (195) $$

Using (40) and (29), the EOP for $\mathcal{C}$ is given by

$$ \theta(\mathcal{C}, B) = \frac{1}{M} \sum_{i=1}^{M} \mathbb{1}\{k_1 \sum_{c=1}^{C} \sum_{\ell=1}^{L_c} n P_{u(i)}(x_c^{(\ell)}) |x_c^{(\ell)}|^2 + k_2 \sum_{c=1}^{C} \sum_{\ell=1}^{L_c} n P_{u(i)}(x_c^{(\ell)}) |x_c^{(\ell)}|^4 < B\} \quad (196) $$

$$ = \frac{1}{M} \sum_{i=1}^{M} \mathbb{1}\{k_1 \sum_{c=1}^{C} \sum_{\ell=1}^{L_c} n P_{u(i)}(\hat{x}_c^{(\ell)}) |\hat{x}_c^{(\ell)}|^2 + k_2 \sum_{c=1}^{C} \sum_{\ell=1}^{L_c} n P_{u(i)}(\hat{x}_c^{(\ell)}) |\hat{x}_c^{(\ell)}|^4 < B\} \quad (197) $$

$$ = \theta(\mathcal{C}', B), \quad (198) $$

where, the equality in (197) follows from (191) and (195). This completes the proof. $\square$

Lemma 5.7 proves that the DEP and the EOP of a code $\mathcal{C}$ are independent of the rotations of the underlying set of channel input symbols $\mathcal{X}$. In fact, for the given construction of codes in this section, the DEP and the EOP are actually independent of the vector of phase shifts $\alpha$ in (135e) altogether. The following lemma proves this result.

**Lemma 5.8.** Consider $(n, M)$-codes $\mathcal{C}$ and $\mathcal{C}'$ of the form in (135) from the family $\mathcal{C}(C, A, L, \alpha, p, r)$ in (135s). The codes $\mathcal{C}$ and $\mathcal{C}'$ are identical except for the phase shift $\alpha_c$ in (135a). More specifically, the set of channel input symbols of $\mathcal{C}$ and $\mathcal{C}'$ are $\mathcal{X}$ and $\mathcal{X}'$, respectively such that

$$ \mathcal{X} = \bigcup_{c=1}^{C} \mathcal{U}(A_c, L_c, \alpha_c), \quad \text{and} \quad (199) $$

$$ \mathcal{X}' = \bigcup_{c=1}^{C} \mathcal{U}(A_c, L_c, \alpha_c'). \quad (200) $$

Then, the average DEP $\gamma$ in (22) and the average EOP $\theta$ in (37) of $\mathcal{C}$ and $\mathcal{C}'$ satisfy the following:

$$ \theta(\mathcal{C}, B) = \theta(\mathcal{C}', B), \quad \text{and} \quad (201) $$

$$ \gamma(\mathcal{C}) = \gamma(\mathcal{C}'). \quad (202) $$
Proof. Denote the $l^{th}$ symbol in layer $c$ of $\mathscr{C}$ by $x_c^{(l)}$ and that of $\mathscr{C}'$ by $\bar{x}_c^{(l)}$. From (135a), it follows that

$$x_c^{(l)} = A_c \exp \left( i \left( \frac{2\pi}{L_c} l + \alpha_c \right) \right) \qquad (203)$$

$$= A_c \exp \left( i \left( \frac{2\pi}{L_c} l + \alpha_c + \alpha'_c - \alpha'_c \right) \right) \qquad (204)$$

$$= A_c \exp \left( i \left( \frac{2\pi}{L_c} l + \alpha'_c \right) \right) \exp \left( i (\alpha_c - \alpha'_c) \right) \qquad (205)$$

$$= \exp \left( i (\alpha_c - \alpha'_c) \right) \bar{x}_c^{(l)} \qquad (206)$$

$$= \exp(i\omega_c) \bar{x}_c^{(l)}, \qquad (207)$$

where, $\omega_c = \alpha_c - \alpha'_c \in [0, 2\pi]$. This implies that for all $c \in \{1, 2, \dots, C\}$ and all $l \in \{1, 2, \dots, L_c\}$, it holds that

$$|x_c^{(l)}| = |\bar{x}_c^{(l)}|. \qquad (208)$$

The result in (201) then follows from (208) and Lemma 5.7.

From (188), the average DEP for $\mathscr{C}$ is given by

$$\gamma(\mathscr{C}) = 1 - \frac{1}{M} \sum_{i=1}^{M} \prod_{c=1}^{C} \prod_{\ell=1}^{L_c} \left( \int_{y:|y-x_c^{(\ell)}|^2 \le r_c^2} \frac{1}{\pi\sigma^2} \exp \left( -\frac{|y-x_c^{(\ell)}|^2}{\sigma^2} \right) dy \right)^{n P_{u(i)}(x_c^{(\ell)})} \qquad (209)$$

$$= 1 - \frac{1}{M} \sum_{i=1}^{M} \prod_{c=1}^{C} \prod_{\ell=1}^{L_c} \left( \int_{y:|y-x_c^{(\ell)}|^2 \le r_c^2} \frac{1}{\pi\sigma^2} \exp \left( -\frac{|y-x_c^{(\ell)}|^2}{\sigma^2} \right) dy \right)^{n P_{u(i)}(\bar{x}_c^{(\ell)})}, \qquad (210)$$

where, the equality in (210) follows from (209) because $P_{u(i)}(x_c^{(\ell)}) = P_{u(i)}(\bar{x}_c^{(\ell)})$. Using the change of variable $y = z + x_c^{(\ell)} - \bar{x}_c^{(\ell)}$ in (210) yields

$$\gamma(\mathscr{C}) = 1 - \frac{1}{M} \sum_{i=1}^{M} \prod_{c=1}^{C} \prod_{\ell=1}^{L_c} \left( \int_{z:|z-\bar{x}_c^{(\ell)}|^2 \le r_c^2} \frac{1}{\pi\sigma^2} \exp \left( -\frac{|z-\bar{x}_c^{(\ell)}|^2}{\sigma^2} \right) dz \right)^{n P_{u(i)}(\bar{x}_c^{(\ell)})} \qquad (211)$$

$$= \gamma(\mathscr{C}'), \qquad (212)$$

which completes the proof of (202). $\square$

The following lemma provides the achievable bound on the EOP $\delta$ in (41) for codes from the family $\mathcal{C}(C, A, L, \alpha, p, r)$ in (135s).

**Lemma 5.9.** Consider an $(n, M, \epsilon)$-code $\mathscr{C}$ of the form in (135) from the family $\mathcal{C}(C, A, L, \alpha, p, r)$ in (135s). The code $\mathscr{C}$ is an $(n, M, \epsilon, B, \delta)$-code if the following holds:

$$\delta \ge \frac{1}{M} \sum_{i=1}^{M} \mathbb{1}\{ (k_1 \sum_{c=1}^{C} \sum_{\ell=1}^{L_c} n P_{u(i)}(x_c^{(\ell)}) A_c^2 + k_2 \sum_{c=1}^{C} \sum_{\ell=1}^{L_c} n P_{u(i)}(x_c^{(\ell)}) A_c^4) < B \}, \qquad (213)$$

where, $k_1$ and $k_2$ are positive real constants defined in (29).

Proof. From (135a), for all $c \in \{1, 2, \dots, C\}$ and all $l \in \{1, 2, \dots, L_c\}$, the symbols $x_c^{(l)} \in \mathcal{U}(A_c, L_c, \alpha_c)$ in (135a) are given by

$$x_c^{(l)} = A_c \exp \left( i \left( \frac{2\pi}{L_c} l + \alpha_c \right) \right). \qquad (214)$$

This implies that

$$|x_c^{(l)}| = A_c. \qquad (215)$$
From (40) and (29), the EOP for the code $\mathcal{C}$ is given by:

$$ \theta(\mathcal{C}, B) = \frac{1}{M} \sum_{i=1}^{M} \mathbb{1} \left\{ \left( k_1 \sum_{c=1}^{C} \sum_{\ell=1}^{L_c} n P_{u(i)}(x_c^{(\ell)}) |x_c^{(\ell)}|^2 + k_2 \sum_{c=1}^{C} \sum_{\ell=1}^{L_c} n P_{u(i)}(x_c^{(\ell)}) |x_c^{(\ell)}|^4 \right) < B \right\} \quad (216) $$

$$ = \frac{1}{M} \sum_{i=1}^{M} \mathbb{1} \left\{ \left( k_1 \sum_{c=1}^{C} \sum_{\ell=1}^{L_c} n P_{u(i)}(x_c^{(\ell)}) A_c^2 + k_2 \sum_{c=1}^{C} \sum_{\ell=1}^{L_c} n P_{u(i)}(x_c^{(\ell)}) A_c^4 \right) < B \right\}. \quad (217) $$

From (41) and (217), the code $\mathcal{C}$ is an $(n, M, \epsilon, B, \delta)$-code if the following holds:

$$ \frac{1}{M} \sum_{i=1}^{M} \mathbb{1} \left\{ \left( k_1 \sum_{c=1}^{C} \sum_{\ell=1}^{L_c} n P_{u(i)}(x_c^{(\ell)}) A_c^2 + k_2 \sum_{c=1}^{C} \sum_{\ell=1}^{L_c} n P_{u(i)}(x_c^{(\ell)}) A_c^4 \right) < B \right\} < \delta. \quad (218) $$

This completes the proof. $\square$

The achievable bound on the EOP $\delta$ in (41) for a homogeneous code $\mathcal{C}$ from the family $C(C, A, L, \alpha, p, r)$ in (135s) is given by the following lemma:

**Lemma 5.10.** Consider a homogeneous $(n, M, \epsilon)$-code $\mathcal{C}$ of the form in (135) from the family $C(C, A, L, \alpha, p, r)$ in (135s). The code $\mathcal{C}$ is an $(n, M, \epsilon, B, \delta)$-code if, for all $c \in \{1, 2, \dots, C\}$, the parameters $p_c$ in (135j) satisfy the following:

$$ \delta = \mathbb{1} \left\{ \left( k_1 \sum_{c=1}^{C} n p_c A_c^2 + k_2 \sum_{c=1}^{C} n p_c A_c^4 \right) < B \right\}. \quad (219) $$

where, $k_1$ and $k_2$ are positive real constants defined in (29).

*Proof.* From (40) and (30), the EOP for the homogeneous code $\mathcal{C}$ is given by:

$$ \theta(\mathcal{C}, B) = \frac{1}{M} \sum_{i=1}^{M} \mathbb{1} \left\{ \left( k_1 \sum_{c=1}^{C} \sum_{\ell=1}^{L_c} n P_{\epsilon}(x_c^{(\ell)}) |x_c^{(\ell)}|^2 + k_2 \sum_{c=1}^{C} \sum_{\ell=1}^{L_c} n P_{\epsilon}(x_c^{(\ell)}) |x_c^{(\ell)}|^4 \right) < B \right\} \quad (220) $$

$$ = \mathbb{1} \left\{ \left( k_1 \sum_{c=1}^{C} \sum_{\ell=1}^{L_c} n P_{\epsilon}(x_c^{(\ell)}) A_c^2 + k_2 \sum_{c=1}^{C} \sum_{\ell=1}^{L_c} n P_{\epsilon}(x_c^{(\ell)}) A_c^4 \right) < B \right\} \quad (221) $$

$$ = \mathbb{1} \left\{ \left( k_1 \sum_{c=1}^{C} \sum_{\ell=1}^{L_c} n \frac{p_c}{L_c} A_c^2 + k_2 \sum_{c=1}^{C} \sum_{\ell=1}^{L_c} n \frac{p_c}{L_c} A_c^4 \right) < B \right\} \quad (222) $$

$$ = \mathbb{1} \left\{ \left( k_1 \sum_{c=1}^{C} n p_c A_c^2 + k_2 \sum_{c=1}^{C} n p_c A_c^4 \right) < B \right\}. \quad (223) $$

The equality in (221) follows from (215) and (222) follows from (135m). From (41) and (223), the code $\mathcal{C}$ is an $(n, M, \epsilon, B, \delta)$-code if the following holds:

$$ \delta = \mathbb{1} \left\{ \left( k_1 \sum_{c=1}^{C} n p_c A_c^2 + k_2 \sum_{c=1}^{C} n p_c A_c^4 \right) < B \right\}. \quad (224) $$

This completes the proof. $\square$

The following lemma provides an upper bound on the energy transmission rate $B$ for $(n, M, \epsilon)$-code $\mathcal{C}$ of the form in (135) from the family $C(C, A, L, \alpha, p, r)$.

**Lemma 5.11.** Consider an $(n, M, \epsilon)$-code $\mathcal{C}$ of the form in (135) from the family $C(C, A, L, \alpha, p, r)$ in (135s). The code $\mathcal{C}$ is an $(n, M, \epsilon, B, \delta)$-code if the following holds:

$$ B \le \begin{cases} \bar{e}_1 & \text{if } 0 \le \delta < \frac{y_1}{M}, \\ \bar{e}_j & \text{if } \frac{\sum_{k=1}^{j-1} y_k}{M} < \delta \le \frac{\sum_{k=1}^{j} y_k}{M}, j \in \{2, 3, \dots, M'\}. \end{cases} \quad (225) $$

where, for all $i \in \{1, 2, \dots, M\}$, energy $e_i$ is in (28); the positive integer $M'$ is in (31); and for all $j \in \{1, 2, \dots, M'\}$ $\bar{e}_j$ is in (31) and $y_j$ is in (32).

*Proof.* The proof follows on the same lines as that for Lemma 4.3 where, for all $i \in \{1, 2, \dots, M\}$, the energy $e_i$ in (28) is given by

$$ e_i = \sum_{c=1}^{C} \sum_{\ell=1}^{L_c} n P_{u(i)} (x_c^{(\ell)}) A_c^2 + k_2 \sum_{c=1}^{C} \sum_{\ell=1}^{L_c} n P_{u(i)} (x_c^{(\ell)}) A_c^4. \quad (226) $$

$\square$
The information transmission rate $R(\mathcal{C})$ of a code $\mathcal{C}$ is dictated by the number of channel input symbols $L$ in (135f) and the probability vector $\mathbf{p}$ in (135h) that defines the empirical frequency with which each symbol appears in $\mathcal{C}$. The following lemma provides the value of $\mathbf{p}$ that achieves the upper bound on $R(\mathcal{C})$ for a given set of channel input symbols.

**Lemma 5.12.** Consider an $(n, M)$-code $\mathcal{C}'$ of the form in (135) from the family $\mathbf{C}(C, \mathbf{A}, \mathbf{L}, \boldsymbol{\alpha}, \mathbf{p}, \mathbf{r})$ in (135s) with $\mathbf{p} = (p_1, p_2, \dots, p_C)^\mathrm{T}$ in (135h) such that, for all $c \in \{1, 2, \dots, C\}$,

$$p_c = \frac{L_c}{L}, \qquad (227)$$

where, $L_c$ and $L$ are as defined in (135a) and (135f) respectively. Then, given any other $(n, M)$-code $\mathcal{C}$ in $\mathbf{C}(C, \mathbf{A}, \mathbf{L}, \boldsymbol{\alpha}, \mathbf{p}, \mathbf{r})$ that is identical to $\mathcal{C}'$ except for the probability distribution $\mathbf{p}$, it holds that,

$$R(\mathcal{C}') \geq R(\mathcal{C}), \qquad (228)$$

where, $R(\mathcal{C}')$ and $R(\mathcal{C})$ are the information transmission rates for $\mathcal{C}'$ and $\mathcal{C}$, respectively.

*Proof.* For the $(n, M)$-code $\mathcal{C}'$ in the family $\mathbf{C}(C, \mathbf{A}, \mathbf{L}, \boldsymbol{\alpha}, \mathbf{p}, \mathbf{r})$ with $p_c$ of the form in (227), from (135m), for all $c \in \{1, 2, \dots, C\}$ and all $\ell \in \{1, 2, \dots, L_c\}$, the type $P_{\mathcal{C}'}$ is given by the following:

$$P_{\mathcal{C}'}(x_c^{(\ell)}) = \frac{p_c}{L_c} = \frac{1}{L}. \qquad (229)$$

For the $(n, M)$-code $\mathcal{C}'$ with the type $P_{\mathcal{C}'}$ of the form in (229), the number of codewords that can be represented using $L$ symbols is given by $M = L^n$. From (19), the information transmission rate for code $\mathcal{C}'$ is given by

$$R(\mathcal{C}') = \frac{\log M}{n} = \log L. \qquad (230)$$

For the $(n, M)$-code $\mathcal{C}$, from (19) and Lemma 4.9, it follows that,

$$R(\mathcal{C}) \leq \log L = R(\mathcal{C}'). \qquad (231)$$

This completes the proof. $\square$

The following lemma provides the value of $\mathbf{p}$ that achieves the lower bound on the EOP $\theta(\mathcal{C}, B)$ for code $\mathcal{C}$ from the family $\mathbf{C}(C, \mathbf{A}, \mathbf{L}, \boldsymbol{\alpha}, \mathbf{p}, \mathbf{r})$ in (135s).

**Lemma 5.13.** Consider an $(n, M)$-code $\mathcal{C}$ of the form in (135) from the family $\mathbf{C}(C, \mathbf{A}, \mathbf{L}, \boldsymbol{\alpha}, \mathbf{p}, \mathbf{r})$ in (135s) with $\mathbf{p} = (p_1, p_2, \dots, p_C)^\mathrm{T}$ in (135h) such that,

$$p_c = \begin{cases} 1 & \text{if } c = 1, \\ 0 & \text{otherwise.} \end{cases} \qquad (232)$$

Then, given any other $(n, M)$-code $\mathcal{C}'$ in the family $\mathbf{C}(C, \mathbf{A}, \mathbf{L}, \boldsymbol{\alpha}, \mathbf{p}, \mathbf{r})$ that is identical to $\mathcal{C}$ except for the probability distribution $\mathbf{p}$, it holds that

$$\theta(\mathcal{C}', B) \geq \theta(\mathcal{C}, B), \qquad (233)$$

where $\theta$ is the average EOP in (37).

*Proof.* Since $p_c = 0$ for all $c \in \{2, 3, \dots, C\}$, from (135k) it follows that, for all $i \in \{1, 2, \dots, M\}$, all $c \in \{2, 3, \dots, C\}$ and all $\ell \in \{1, 2, \dots, L_c\}$, the type

$$P_{u(i)}(x_c^{(\ell)}) = 0. \qquad (234)$$

Since $P_{u(i)}$ is a pmf as defined in (17), it holds that, for all $i \in \{1, 2, \dots, M\}$,

$$\sum_{c=1}^{C} \sum_{\ell=1}^{L_c} P_{u(i)}(x_c^{(\ell)}) = 1. \qquad (235)$$

From (234) and (235), it follows that,

$$\sum_{\ell=1}^{L_1} P_{u(i)}(x_1^{(\ell)}) = 1. \qquad (236)$$
From (40) and (29), the EOP for the code $\mathcal{C}$ is given by

$$ \theta(\mathcal{C}, B) = \frac{1}{M} \sum_{i=1}^{M} \mathbb{1}\{k_1 \sum_{c=1}^{C} \sum_{\ell=1}^{L_c} n P_{u(i)}(x_c^{(\ell)}) |x_c^{(\ell)}|^2 + k_2 \sum_{c=1}^{C} \sum_{\ell=1}^{L_c} n P_{u(i)}(x_c^{(\ell)}) |x_c^{(\ell)}|^4 < B\} \quad (237) $$

$$ = \frac{1}{M} \sum_{i=1}^{M} \mathbb{1}\{k_1 \sum_{\ell=1}^{L_1} n P_{u(i)}(x_1^{(\ell)}) A_1^2 + k_2 \sum_{\ell=1}^{L_1} n P_{u(i)}(x_1^{(\ell)}) A_1^4 < B\} \quad (238) $$

$$ = \frac{1}{M} \sum_{i=1}^{M} \mathbb{1}\{k_1 n A_1^2 + k_2 n A_1^4 < B\}, \quad (239) $$

where, the equality in (238) follows from (234) and (239) follows from (236). Similarly, the EOP for the code $\mathcal{C}'$ is given by

$$ \theta(\mathcal{C}', B) = \frac{1}{M} \sum_{i=1}^{M} \mathbb{1}\{k_1 \sum_{c=1}^{C} \sum_{\ell=1}^{L_c} n P_{u(i)}(x_c^{(\ell)}) |x_c^{(\ell)}|^2 + k_2 \sum_{c=1}^{C} \sum_{\ell=1}^{L_c} n P_{u(i)}(x_c^{(\ell)}) |x_c^{(\ell)}|^4 < B\} \quad (240) $$

$$ = \frac{1}{M} \sum_{i=1}^{M} \mathbb{1}\{k_1 \sum_{c=1}^{C} \sum_{\ell=1}^{L_c} n P_{u(i)}(x_c^{(\ell)}) A_c^2 + k_2 \sum_{c=1}^{C} \sum_{\ell=1}^{L_c} n P_{u(i)}(x_c^{(\ell)}) A_c^4 < B\} \quad (241) $$

$$ \geq \frac{1}{M} \sum_{i=1}^{M} \mathbb{1}\{k_1 \sum_{c=1}^{C} \sum_{\ell=1}^{L_c} n P_{u(i)}(x_c^{(\ell)}) A_1^2 + k_2 \sum_{c=1}^{C} \sum_{\ell=1}^{L_c} n P_{u(i)}(x_c^{(\ell)}) A_1^4 < B\} \quad (242) $$

$$ = \frac{1}{M} \sum_{i=1}^{M} \mathbb{1}\{k_1 n A_1^2 + k_2 n A_1^4 < B\} \quad (243) $$

$$ = \theta(\mathcal{C}, B), \quad (244) $$

where, the equality in (241) follows from (215); the inequality in (242) follows from (135g); the equality in (243) follows from (235); and (244) follows from (239). This completes the proof. □

### C. An Information-Energy Achievable Region

The achievable upper bound on the information rate $R$ in (19), lower bound on the average DEP $\epsilon$ in (23) and lower bound on the average EOP $\delta$ in (41) determined in Section 5-B together define the achievable region of the codes of the form in (135) from the family $\mathcal{C}(C, A, L, \alpha, p, r)$ in (135s). The following theorem provides the achievable region thus defined, for the class of homogeneous codes in the family $\mathcal{C}(C, A, L, \alpha, p, r)$.

**Theorem 5.14.** A homogeneous $(n, M)$-code $\mathcal{C}$ of the form in (135) from the family $\mathcal{C}(C, A, L, \alpha, p, r)$ in (135s) is an $(n, M, \epsilon, B, \delta)$-code if the number of layers $C$, and for all $c \in \{1, 2, \dots, C\}$, the parameters $A_c, L_c, r_c$, and $p_c$ satisfy the following:

$$ \epsilon \geq 1 - \prod_{c=1}^{C} \left(1 - e^{-\frac{r_c^2}{\sigma^2}}\right)^{np_c}, \quad (245a) $$

$$ L_c \leq \left\lfloor \frac{\pi}{2 \arcsin \frac{r_c}{2A_c}} \right\rfloor, \quad (245b) $$

$$ M \leq \frac{n!}{\prod_{c=1}^{C} \left(\left(n \frac{p_c}{L_c}\right)\right)^{L_c}}, \quad (245c) $$

$$ \delta = \mathbb{1}\{k_1 \sum_{c=1}^{C} n p_c A_c^2 + k_2 \sum_{c=1}^{C} n p_c A_c^4 < B\}, \quad (245d) $$

where, the real $\sigma^2$ is defined in (12).

*Proof.* The result follows from Corollary 5.2; and Lemmas 5.4, 5.6, 5.10. □

## 6. DISCUSSION

The converse and achievable bounds presented in this work reveal several interesting insights into the trade-offs between the information transmission rate $R$, the energy transmission rate $B$, the DEP $\epsilon$ and the EOP $\delta$. For instance, the converse as well as the achievable lower bounds on $\delta$ in (42) and (213), respectively, increase as $B$ increases and vice versa. The consequence of this relationship is that a lower $\delta$ can be achieved at the cost of lower $B$. Similarly, for increasing the converse or achievable upper bounds on $B$ in (48) and (225), respectively, a higher value of $\delta$ has to be tolerated. These bounds also reveal that both $B$ and $\delta$ can be improved by a code that has higher values of $e_i$ in 28. This is achieved by using the symbols with greater
<figure><img src="image_11.png" /><figcaption>Fig. 2: Converse bounds on the information transmission rate $R$ in (117) as a function of the harvested energy $e$ in (247).</figcaption></figure>

energy more frequently in the code. In fact, for the constructed codes in the family $C(C, A, L, \alpha, p, r)$ in (135s), the largest $B$ and the least $\delta$ are achieved by a code with $p_1 = 1$ in (135h).

The converse lower bound on the DEP $\epsilon$ in (134) decreases as the distance between the symbols in the set of channel inputs $\mathcal{X}$ in (135b) increases. This implies that, a lower DEP can be achieved by increasing the distances between the channel input symbols. However, within the peak-amplitude constraint, increasing the distance between symbols implies that the number of symbols $L$ decreases. This in turn, decreases the converse upper bound on the information rate $R$ in (117). The achievable lower bound on $\epsilon$ in (245a) decreases as a function of the radii $r_c$ of the decoding regions in (135p). On the other hand, the upper bound on the achievable information rate $R$ in (174) increases as the radii $r_c$ decrease. The trade-offs between $R$, $B$, $\epsilon$, and $\delta$ are further illustrated using the following example.

Consider a homogeneous $(n, M, \epsilon, B, \delta)$-code $\mathcal{C}$ in the family $C(C, A, L, \alpha, p, r)$ in (135s) with the peak-amplitude constraint $P = 20$ millivolts in (15d). The set of channel input symbols $\mathcal{X}$ in (135b) is composed of two layers with 5 symbols in each layer, i.e., $C = 2$ and $L_1 = L_2 = 5$. The radius of the first layer is $A_1 = P$ and the radius of the second layer $A_2$ is varied to illustrate the trade-offs between various parameters of $\mathcal{C}$. The frequency with which symbols from the first layer appear in the code is $p_1 = p = 1 - p_2$. That is, the vector $p$ in (135h) is given by

$$p = (p, (1-p))^{\top}. \qquad (246)$$

The duration of the transmission in channel uses is $n = 100$. Since $\mathcal{C}$ is a homogeneous code, from (30), it holds that, for all $i \in \{1, 2, \dots, M\}$,

$$e_i = e, \qquad (247)$$

where $e \in [0, \infty)$ is calculated as in (30).

Fig. 2 shows the trade-offs between the converse bound on the information transmission rate $R$ (117) and the harvested energy $e$ in microwatts ($\mu W$) in (247) as a function of $p$ in (246). Each curve in the figure is generated for some value of $A_2 < A_1$ by varying the value of $p \in [0, 1]$. The following trade-offs can be observed from this figure. The harvested energy $e$ increases as $p$ increases. This is because higher $p$ corresponds to the symbols from the first layer $c = 1$ which have higher energy (since $A_1 > A_2$) being used more frequently in $\mathcal{C}$. For a fixed value of $A_2$ in Fig. 2, the bound on the information rate $R$ first increases and then decreases as a function of $e$ in (247). For each of these curves, the maximum $R = 2.13$ bits/channel use corresponds to uniform type, i.e., $p = 0.5$. For $p$ lesser or greater than 0.5, the bound on $R$ decreases. Furthermore, the bounds on $R$ are independent of the values of $A_1$ and $A_2$. This is due to the fact that the information rate $R$ is only a function of the number of codewords $M$ in (118).
<figure><img src="image_12.png" /><figcaption>Fig. 3: The bounds on the harvested energy $e$ in (247) as a function of $p$ in (246).</figcaption></figure>

<figure><img src="image_13.png" /><figcaption>Fig. 4: The bounds on the DEP $\epsilon$ in (134) as a function of $p$ in (246).</figcaption></figure>

Fig. 3 shows the variation of the harvested energy $e$ in (247) as a function of $p$ in (246). The value of $e$ increases as $p$ increases. This is because increasing $p$ means that the symbols from the first layer which have greater energy are used more frequently in $\mathcal{C}$ which increases the harvested energy. Furthermore, the harvested energy $e$ also increases as $A_2$ increases. This is because, higher values of $A_2$ imply higher energy contained in the symbols in the second layer which in turn increases $e$. Fig. 4 shows the variation of the converse bound on the DEP $\epsilon$ in (134) as a function of the probability $p$ in (246). It is observed that, as $A_2$ increases, the bound on $\epsilon$ increases. This is because, increasing $A_2$ results in the distances between the symbols in the two layers of $\mathcal{X}$ in (135b) to decrease which results in an increase in $\epsilon$ according to (134).
7. EXAMPLES

In this section, the characterized converse and achievable information-energy regions for SIET are illustrated using the following numerical example. Consider homogeneous ($n, M, \epsilon, B, \delta$)-codes in $C(C, \mathcal{A}, \mathcal{L}, \boldsymbol{\alpha}, \mathbf{p}, \mathbf{r})$ in (135s) that employ the set of channel input symbols $\mathcal{X}$ of the form in (135b) with number of layers $C = 3$. The duration of the transmission in channel uses is $n = 80$.

In Fig. 5, the achievable bounds on the energy transmission rate $B$ in (225) are plotted for homogeneous ($n, M, \epsilon, B, \delta$)-codes in the family $C(C, \mathcal{A}, \mathcal{L}, \boldsymbol{\alpha}, \mathbf{p}, \mathbf{r})$ as a function of the achievable bounds on the DEP $\epsilon$ in (245a). The figure also shows the converse bounds on $B$ in (48) as a function of the converse bounds on $\epsilon$ in (134). The amplitude of the first layer of the set of channel inputs $\mathcal{X}$ is fixed at $A_1 = 50$. Amplitudes of the second and third layers $A_2$ and $A_3$ are determined by the radii of the decoding regions according to (135q). For all $c \in \{1, 2, \dots, C\}$, the number of symbols in layer $c$ i.e., $L_c$, is determined by the radii $r_c$ and the amplitudes $A_c$ according to (245b). The probability vector in (135h) is $\mathbf{p} = (0.5, 0.3, 0.2)^T$. The points on the curves are generated by varying $r_c$ between 2 and 10.

<figure><img src="image_14.png" /><figcaption>Fig. 5: Converse (48) and achievable (225) bounds on the energy transmission rate $B$ for homogeneous codes in the family $C(C, \mathcal{A}, \mathcal{L}, \boldsymbol{\alpha}, \mathbf{p}, \mathbf{r})$ as a function of the converse and achievable bounds on the DEP $\epsilon$ in (134) and (245a), respectively.</figcaption></figure>

Fig. 5 shows several trade-offs between the energy transmission rate $B$, the DEP $\epsilon$, and the information transmission rate $R$. Since the codes under consideration are homogeneous, the EOP $\delta \in \{0, 1\}$. Here, the $\delta$ is fixed to be 0 and the upper bounds on $B$ in (225) and (48) are calculated accordingly. It is observed that the converse and achievable upper bounds on $B$ increase as $\epsilon$ increases. This is due to the fact that increasing $\epsilon$ allows decreasing the radii of the decoding regions $r_c$ in (135r) according to (245a). At the same time, decrease in $r_c$ allows increasing the amplitudes $A_2$ and $A_3$ according to (135q). Increased amplitudes $A_2$ and $A_3$ result in higher values of the energy $e_{\mathcal{C}}$ in (30) which increases $B$. The information rate $R$ also increases as a function of the DEP $\epsilon$ in Fig. 5. This is because, increasing $\epsilon$ allows decreasing the radii $r_c$ according to (245a). At the same time, decrease in $r_c$ increases the number of symbols in a layer $L_c$ according to (245b) which increases the converse and achievable bounds on $R$ in (117) and (174), respectively.

Fig. 6 illustrates the converse and achievable information-energy regions of the constructed homogeneous ($n, M, \epsilon, B, \delta$)-codes in the family $C(C, \mathcal{A}, \mathcal{L}, \boldsymbol{\alpha}, \mathbf{p}, \mathbf{r})$ in (135s). In this case, the radii of the decoding regions $r_c$ are assumed to be the same for all the layers i.e., for all $c \in \{1, 2, \dots, C\}$, the radius $r_c = r$ in (135p). The value of $r$ is obtained according to (245a). The amplitude of the first layer is $A_1 = 30$. Amplitudes of the second and third layers $A_2$ and $A_3$ are determined by $r$ according to (135q). The points on the curves in Fig. 6 are obtained by varying $\epsilon$ and the probability vector $\mathbf{p}$ in (135h).

Fig. 6 shows the following trade-offs between the information and energy transmission rates in the converse and achievable curves. Firstly, the maximum achievable information transmission rate is $R = 4$ bits/channel use. This $R$ is achieved by a code in which all the symbols in the set of channel inputs $\mathcal{X}$ are used with the same frequency. The maximum energy transmission
<figure><img src="image_15.png" /><figcaption>Fig. 6: Converse and achievable information-energy regions for homogeneous codes in the family $C(C, A, L, \alpha, p, r)$.</figcaption></figure>

rate that can be achieved at $R = 4$ bits/channel use is $B = 8.3 \times 10^4$ energy units. This corresponds to the point $D_1$ in Fig. 6. Secondly, the maximum achievable $B$ is $3.8 \times 10^5$ energy units. This is achieved by a code that exclusively uses the symbols in the first layer i.e., the probability vector $p$ in (135h) is $p = (1, 0, 0)^T$. The maximum $R$ that can be achieved at $B = 3.8 \times 10^5$ energy units is $R = 2.46$ bits/channel use. This corresponds to the point $D_2$ in Fig. 6. Thirdly, the curves between the points $D_1$ and $D_2$ in Fig. 6 show the trade-off between the information and energy transmission rates. As $B$ is increased from $8 \times 10^4$ energy units at point $D_1$, $R$ begins to decrease. Similarly, as $R$ is increased from 2.46 bits/channel use at point $D_2$, $B$ begins to decrease.

The codes constructed in this work are optimal in the sense of the converse results in Section 4 except for the DEP $\epsilon$. Fig.5 shows that for the same values of the converse and achievable bounds on the energy transmission rate $B$, the lower bound on the DEP $\epsilon$ for the achievability curve is much higher than the converse. Similarly, for the same value of the information rate $R$ in the converse and achievability curves, the corresponding $\epsilon$ for the achievability curve is higher than that of the converse. Fig. 6 shows that the converse and achievable information-energy rate curves for $C$ overlap. However, for the same information and energy rate pair, the DEP for the achievable curves is higher than the converse. The sub-optimality in DEP arises due to the choice of circular decoding regions in (135p).

## REFERENCES

[1] L. R. Varshney, "Transporting Information and Energy Simultaneously," in *Proc. IEEE International Symposium on Information Theory (ISIT)*, Toronto, ON, Canada, Jul. 2008, pp. 1612–1616.

[2] S. B. Amor and S. M. Perlaza, "Fundamental Limits of Simultaneous Energy and Information Transmission," in *Proc. International Conference on Telecommunications (ICT)*, Thessaloniki, Greece, May 2016, pp. 1–5.

[3] N. Khalfet and I. Krikidis, "The Capacity of SWIPT Systems over Rayleigh-Fading Channels with HPA," in *Proc. IEEE Information Theory Workshop (ITW)*, Kanazawa, Japan, 2021, pp. 1–6.

[4] P. Grover and A. Sahai, "Shannon Meets Tesla: Wireless Information and Power Transfer," in *Proc. IEEE International Symposium on Information Theory*, 2010, pp. 2363–2367.

[5] S. B. Amor, S. M. Perlaza, I. Krikidis, and H. V. Poor, "Feedback Enhances Simultaneous Energy and Information Transmission in Multiple Access Channels," in *Proc. IEEE International Symposium on Information Theory (ISIT)*, Barcelona, Spain, Jul. 2016, pp. 1974–1978.

[6] N. Khalfet and S. M. Perlaza, "Simultaneous Information and Energy Transmission in the Two-User Gaussian Interference Channel," *IEEE Journal on Selected Areas in Communications*, vol. 37, no. 1, pp. 156–170, Jan. 2019.

[7] S. M. Perlaza, A. Tajer, and H. V. Poor, "Simultaneous Information and Energy Transmission: A Finite Block-length Analysis," in *Proc. IEEE International Workshop on Signal Processing Advances in Wireless Communications (SPAWC)*, Kalamata, Greece, Jun. 2018, pp. 1–5.

[8] N. Khalfet, S. M. Perlaza, A. Tajer, and H. V. Poor, "On Ultra-reliable and Low Latency Simultaneous Information and Energy Transmission Systems," in *Proc. IEEE International Workshop on Signal Processing Advances in Wireless Communications (SPAWC)*, Cannes, France, Jul. 2019, pp. 1–5.
[9] S. u. Zuhra, S. M. Perlaza, and E. Altman, "Simultaneous Information and Energy Transmission with Finite Constellations," in Proc. IEEE Information Theory Workshop (ITW), Kanazawa, Japan, Oct. 2021, pp. 1-6.

[10] S. u. Zuhra, S. M. Perlaza, H. V. Poor, and E. Altman, "Achievable Information-Energy Region in the Finite Block-Length Regime with Finite Constellations," in Proc. IEEE International Symposium on Information Theory (ISIT), Espoo, Finland, Jun. 2022, pp. 2106-2111.

[11] I. Csiszár, "The Method of Types," IEEE Transactions on Information Theory, vol. 44, no. 5, pp. 2505-2523, Oct. 1998.

[12] S. u. Zuhra, S. M. Perlaza, H. V. Poor, and M. Skoglund, "Information-Energy Trade-offs with EH Non-linearities in the Finite Block-Length Regime with Finite Constellations," in Proc. IEEE Information Theory Workshop (ITW), Mumbai, India, Nov. 2022.

[13] M. Varasteh, B. Rassouli, and B. Clerckx, "Wireless Information and Power Transfer over an AWGN Channel: Nonlinearity and Asymmetric Gaussian Signaling," in Proc. IEEE Information Theory Workshop (ITW), Kaohsiung, Taiwan, Nov. 2017, pp. 181-185.

[14] B. Clerckx, "Wireless Information and Power Transfer: Nonlinearity, Waveform Design, and Rate-Energy Tradeoff," IEEE Transactions on Signal Processing, vol. 66, no. 4, pp. 847-862, 2018.

[15] B. Clerckx and E. Bayguzina, "Waveform Design for Wireless Power Transfer," IEEE Transactions on Signal Processing, vol. 64, no. 23, pp. 6313-6328, 2016.

[16] S. Abeywickrama, R. Zhang, and C. Yuen, "Refined Nonlinear Rectenna Modeling and Optimal Waveform Design for Multi-User Multi-Antenna Wireless Power Transfer," IEEE Journal of Selected Topics in Signal Processing, vol. 15, no. 5, pp. 1198-1210, 2021.

[17] S. Shen and B. Clerckx, "Joint Waveform and Beamforming Optimization for MIMO Wireless Power Transfer," IEEE Transactions on Communications, vol. 69, no. 8, pp. 5441-5455, 2021.

[18] O. L. A. López, F. A. Monteiro, H. Alves, R. Zhang, and M. Latva-Aho, "A Low-Complexity Beamforming Design for Multiuser Wireless Energy Transfer," IEEE Wireless Communications Letters, vol. 10, no. 1, pp. 58-62, 2021.

[19] Y. Zeng, B. Clerckx, and R. Zhang, "Communications and Signals Design for Wireless Power Transmission," IEEE Transactions on Communications, vol. 65, no. 5, pp. 2264-2290, 2017.

[20] P. Mukherjee, C. Psomas, and I. Krikidis, "Differential Chaos Shift Keying-Based Wireless Power Transfer with Nonlinearities," IEEE Journal of Selected Topics in Signal Processing, vol. 15, no. 5, pp. 1185-1197, 2021.

[21] J. Kim, B. Clerckx, and P. D. Mitcheson, "Signal and System Design for Wireless Power Transfer: Prototype, Experiment and Validation," IEEE Transactions on Wireless Communications, vol. 19, no. 11, pp. 7453-7469, 2020.

[22] A. Khalili, S. Zargari, Q. Wu, D. W. K. Ng, and R. Zhang, "Multi-Objective Resource Allocation for IRS-Aided SWIPT," IEEE Wireless Communications Letters, vol. 10, no. 6, pp. 1324-1328, 2021.

[23] N. Shanin, L. Cottatellucci, and R. Schober, "Markov Decision Process Based Design of SWIPT Systems: Non-Linear EH Circuits, Memory, and Impedance Mismatch," IEEE Transactions on Communications, vol. 69, no. 2, pp. 1259-1274, 2021.

[24] G. M. Kraidy, C. Psomas, and I. Krikidis, "Fundamentals of Circular QAM for Wireless Information and Power Transfer," in Proc. IEEE International Workshop on Signal Processing Advances in Wireless Communications (SPAWC), 2021, pp. 616-620.

[25] T. D. P. Perera, D. N. K. Jayakody, S. K. Sharma, S. Chatzinotas, and J. Li, "Simultaneous Wireless Information and Power Transfer (SWIPT): Recent Advances and Future Challenges," IEEE Communications Surveys & Tutorials, vol. 20, no. 1, pp. 264-302, Dec. 2018.

[26] J. Huang, C.-C. Xing, and C. Wang, "Simultaneous Wireless Information and Power Transfer: Technologies, Applications, and Research Challenges," IEEE Communications Magazine, vol. 55, no. 11, pp. 26-32, Nov. 2017.

[27] B. Clerckx, R. Zhang, R. Schober, D. W. K. Ng, D. I. Kim, and H. V. Poor, "Fundamentals of Wireless Information and Power Transfer: From RF Energy Harvester Models to Signal and System Designs," IEEE Journal on Selected Areas in Communications, vol. 37, no. 1, pp. 4-33, 2019.

[28] B. Clerckx, J. Kim, K. W. Choi, and D. I. Kim, "Foundations of Wireless Information and Power Transfer: Theory, Prototypes, and Experiments," Proceedings of the IEEE, vol. 110, no. 1, pp. 8-30, Jan. 2022.

[29] L. Liu, R. Zhang, and K.-C. Chua, "Wireless Information Transfer with Opportunistic Energy Harvesting," IEEE Transactions on Wireless Communications, vol. 12, no. 1, pp. 288-300, 2013.

[30] N. Shanin, L. Cottatellucci, and R. Schober, "Rate-Power Region of SWIPT Systems Employing Nonlinear Energy Harvester Circuits with Memory," in Proc. IEEE International Conference on Communications (ICC), Dublin, Ireland, Jun. 2020, pp. 1-7.

[31] E. Goudeli, C. Psomas, and I. Krikidis, "Sequential Decoding for Simultaneous Wireless Information and Power Transfer," in Proc. International Conference on Telecommunications (ICT), 2017, pp. 1-5.

[32] G. Lin, Y. Zhou, W. Jiang, X. He, X. Zhou, G. He, and P. Yang, "LF-SWIPT: Outage Analysis for SWIPT Relaying Networks Using Lossy Forwarding With QoS Guaranteed," IEEE Internet of Things Journal, pp. 1-1, 2022.

[33] H. T. Thien, P.-V. Tuan, and I. Koo, "A Secure-Transmission Maximization Scheme for SWIPT Systems Assisted by an Intelligent Reflecting Surface and Deep Learning," IEEE Access, vol. 10, pp. 31851-31867, 2022.

[34] J. Liu, C.-H. R. Lin, Y.-C. Hu, and P. K. Donta, "Joint Beamforming, Power Allocation, and Splitting Control for SWIPT-Enabled IoT Networks with Deep Reinforcement Learning and Game Theory," Sensors, vol. 22, no. 6, pp. 2328, 2022.

[35] B. Clerckx, K. Huang, L. R. Varshney, S. Ulukus, and M.-S. Alouini, "Wireless Power Transfer for Future Networks: Signal Processing, Machine Learning, Computing, and Sensing," IEEE Journal of Selected Topics in Signal Processing, vol. 15, no. 5, pp. 1060-1094, 2021.

[36] R. Zhang and C. K. Ho, "MIMO Broadcasting for Simultaneous Wireless Information and Power Transfer," IEEE Transactions on Wireless Communications, vol. 12, no. 5, pp. 1989-2001, 2013.

[37] C. Song, C. Ling, J. Park, and B. Clerckx, "MIMO broadcasting for simultaneous wireless information and power transfer: Weighted MMSE approaches," in 2014 IEEE Globecom Workshops (GC Wkshps), 2014, pp. 1151-1156.
[38] M. Varasteh, J. Hoydis, and B. Clerckx, "Learning to Communicate and Energize: Modulation, Coding, and Multiple Access Designs for Wireless Information-Power Transmission," *IEEE Transactions on Communications*, vol. 68, no. 11, pp. 6822-6839, 2020.

[39] M. B. Goktas and Z. Ding, "A Wireless Power Transfer Assisted NOMA Transmission Scheme for 5G and Beyond mMTC," *IEEE Wireless Communications Letters*, pp. 1-1, 2022.

[40] C. E. García, M. R. Camana, and I. Koo, "Low-Complexity PSO-Based Resource Allocation Scheme for Cooperative Non-Linear SWIPT-Enabled NOMA," *IEEE Access*, vol. 10, pp. 34207-34220, 2022.

[41] D. Mishra and G. C. Alexandropoulos, "Jointly Optimal Spatial Channel Assignment and Power Allocation for MIMO SWIPT Systems," *IEEE Wireless Communications Letters*, vol. 7, no. 2, pp. 214-217, 2018.

[42] Q. Wu and R. Zhang, "Joint Active and Passive Beamforming Optimization for Intelligent Reflecting Surface Assisted SWIPT Under QoS Constraints," *IEEE Journal on Selected Areas in Communications*, vol. 38, no. 8, pp. 1735-1748, 2020.

[43] J. Park and B. Clerckx, "Joint Wireless Information and Energy Transfer in a Two-User MIMO Interference Channel," *IEEE Transactions on Wireless Communications*, vol. 12, no. 8, pp. 4210-4221, 2013.

[44] A. Dytso, M. Egan, S. M. Perlaza, H. V. Poor, and S. S. Shitz, "Optimal Inputs for Some Classes of Degraded Wiretap Channels," in *Proc. IEEE Information Theory Workshop (ITW)*, 2018, pp. 1-5.

[45] A. Dytso, S. Yagli, H. V. Poor, and S. Shamai Shitz, "The Capacity Achieving Distribution for the Amplitude Constrained Additive Gaussian Channel: An Upper Bound on the Number of Mass Points," *IEEE Transactions on Information Theory*, vol. 66, no. 4, pp. 2006-2022, 2020.

[46] S. Yagli, A. Dytso, H. V. Poor, and S. S. Shitz, "An Upper Bound on the Number of Mass Points in the Capacity Achieving Distribution for the Amplitude Constrained Additive Gaussian Channel," in *Proc. IEEE International Symposium on Information Theory (ISIT)*, 2019, pp. 1907-1911.

[47] D. Tse and P. Viswanath, *Fundamentals of Wireless Communication*. Cambridge University Press, 2005.

[48] A. Lapidoth, *A Foundation in Digital Communication*, 2nd ed. Cambridge University Press, 2017.

[49] H. Robbins, "A Remark on Stirling's Formula," *The American mathematical monthly*, vol. 62, no. 1, pp. 26-29, 1955.

[50] J. G. Proakis and M. Salehi, *Digital Communications*, 5th ed. McGraw-Hill Higher Education, 2008.