On the Secrecy Capacity of a Full-Duplex Wirelessly Powered Communication System in the Presence of a Passive Eavesdropper

In this paper, we investigate the secrecy capacity of a point-to-point, full-duplex (FD) wirelesly powered communication system in the presence of a passive eavesdropper. The considered system is comprised of an energy transmitter (ET), an energy harvesting user (EHU), and a passive eavesdropper (EVE). The ET transmits radio-frequency energy which is used for powering the EHU as well as for generating interference at EVE. The EHU uses the energy harvested from the ET to transmit confidential messages back to the ET. As a consequence of the FD mode of operation, both the EHU and the ET are affected by self-interference, which has contrasting effects at the two nodes. In particular, the self-interference impairs the decoding of the received message at the ET, whilst it serves as an additional energy source at the EHU. For this system model, we derive an upper and a lower bound on the secrecy capacity. For the lower bound, we propose a simple achievability scheme which offers rates close to the upper bound on the secrecy capacity. Our numerical results show significant improvements in terms of achievable rate when the proposed communication scheme is employed compared to its half-duplex counterparts, even for high self-interference values.

Authors

• 3 publications
• 11 publications
• 2 publications
• Artificial-Noise-Aided Secure Channel with a Full-duplex Source

This paper consider a new secure communication scene where a full-duplex...
10/19/2017 ∙ by Xinyue Hu, et al. ∙ 0

• On the Secrecy Rate of Spatial Modulation Based Indoor Visible Light Communications

In this paper, we investigate the physical-layer security for a spatial ...
06/22/2019 ∙ by Jin-Yuan Wang, et al. ∙ 0

• Secure Users Oriented Downlink MISO NOMA

This paper proposes a secure users oriented multiple-input and single-ou...
03/01/2019 ∙ by Hui-Ming Wang, et al. ∙ 0

• Improving the Secrecy of Distributed Storage Systems using Interference Alignment

Regenerating codes based on the approach of interference alignment for w...
01/05/2018 ∙ by Natasa Paunkoska, et al. ∙ 0

• Secrecy Outage Analysis of Non-Orthogonal Spectrum Sharing for Heterogeneous Cellular Networks

In this paper, we investigate physical-layer security for a heterogeneou...
01/27/2019 ∙ by Yulong Zou, et al. ∙ 0

• On the Secrecy Performance of NOMA Systems with both External and Internal Eavesdroppers

Sharing resource blocks in NOMA systems provides more opportunity to the...
06/10/2019 ∙ by Milad Abolpour, et al. ∙ 0

• Rate Balancing in Full-Duplex MIMO Two-Way Relay Networks

Maximizing the minimum rate for a full-duplex multiple-input multiple-ou...
01/21/2019 ∙ by Erfan Khordad, et al. ∙ 0

This week in AI

Get the week's most popular data science and artificial intelligence research sent straight to your inbox every Saturday.

I Introduction

The security of wireless communication is of critical societal interest. Traditionally, encryption has been the primary method which ensures that only the legitimate receiver receives the intended message. Encryption algorithms commonly require that some information, colloquially referred to as a key, is shared only among the legitimate entities in the network. However, key management makes the encryption impractical in architectures such as radio-frequency identification (RFID) networks and sensor networks, since certificate authorities or key distributers are often not available and limitations in terms of computational complexity make the use of standard data encryption difficult [1], [2]. This problem with network security will be increasingly emphasised in the foreseeable future because of paradigms such as the Internet of Things (IoT). The IoT, as a “network of networks”, will provide ubiquitous connectivity and information-gathering capabilities to a massive number of communication devices. However, the low-complexity hardware and the severe energy constraints of the IoT devices present unique security challenges. To ensure confidentiality in such networks, exploitation of the physical properties of the wireless channel has become an attractive option [2]. Essentially, the presence of fading, interference, and path diversity in the wireless channel can be leveraged in order to degrade the ability of potential intruders to gain information about the confidential messages sent through the wireless channel [2]. This approach is commonly known as physical layer security, or alternatively as information-theoretic security [3].

Shannon and Wyner have laid a solid foundation for studying secrecy of many different system models in [4],[5], including communication systems powered by energy harvesting (EH), which have attracted significant attention recently [6], [7]. EH relies on harvesting energy from ambient renewable and environmentally friendly sources such as, solar, thermal, vibration or wind, or, from dedicated energy transmitters. The latter gives rise to wirelesly powered communication networks (WPCNs) [8]. EH is often considered as a suitable supplement to IoT networks, since most IoT nodes have low power requirements on the order of microwatts to milliwatts, which can be easily met by EH. In addition, when paired with physical layer security, WPCNs can potentially offer a secure and ubiquitous operation [9]. An EH network with multiple power-constrained information sources has been studied in [10]

, where the authors derived an exact expression for the probability of a positive secrecy capacity. In

[11] and [12], the secrecy capacity of the EH Gaussian multiple-input-multiple-output (MIMO) wire-tap channel under transmitter- and receiver-side power constraints has been derived. The secrecy outage probability of a single-input-multiple-output (SIMO) and multiple-input-single-output (MISO) simultaneous wireless information and power transfer (SWIPT) systems were characterized in [13] and [14]-[15], respectively. Relaying networks with EH in the presence of a passive eavesdropper have been studied in [16]. Defence methods with EH friendly jammers, have been proposed in [17] and [18], where the secrecy capacity and the secrecy outage probability have been derived.

In addition to physical layer security, another appealing option for networks with scarce resources such as WPCNs, is the full-duplex (FD) mode of operation. Recent results in the literature, e.g., [19]-[20], have shown that it is possible for transceivers to operate in the FD mode by transmitting and receiving signals simultaneously and in the same frequency band. The FD mode of operation can lead to doubling (or even tripling, see [21]) of the spectral efficiency of the network in question.

Motivated by these advances in FD communication and the applicability of physical layer security to WPCNs, in this paper, we investigate the secrecy capacity of a FD wirelessly powered communication system. Unlike our prior work which does not have an eavesdropper and therefore does not consider secrecy constraints [22], the network in this paper is comprised of an energy transmitter (ET) and an energy harvesting user (EHU) in the presence of a passive eavesdropper (EVE). In this system, the ET sends radio-frequency (RF) energy to the EHU, whereas, the EHU harvests this energy and uses it to transmit confidential information back to the ET. The signal transmitted by the ET serves a second purpose by acting as an interference signal for EVE. Both the ET and the EHU are assumed to operate in the FD mode, hence, both nodes transmit and receive RF signals in the same frequency band and at the same time. As a result, both are subjected to self-interference. The self-interference hinders the decoding of the information signal received from the EHU at the ET. At the EHU, the self-interference increases the amount of energy that can be harvested by the EHU [23]. Meanwhile, EVE is passive and only aims to intercept the confidential message transmitted by the EHU to the ET. For the considered system model, we derive an upper and a lower bound on the secrecy capacity. Furthermore, we provide a simple achievability scheme for the lower bound on the secrecy capacity. The proposed scheme in this paper is relatively simple and therefore easily applicable in practice in wirelessly powered IoT networks which require secure information transmissions. For example, sensors which are embedded in the infrastructure, like buildings, bridges or the power grid, monitor their environment and generate measurements. The generated measurements often contain sensitive information. An Unmanned Aerial Vehicle (UAV) can fly close to the sensors in order to power them, and then receive the generated data packets from the sensors. The proposed scheme in this paper can be used in this scenario and it will guarantees that such sensitive information will never be intercepted by an illegitimate, third party.

The rest of the paper is organized as follows. Section II provides the system and channel models. Sections III and  IV present the upper and the lower bounds on the secrecy capacity, respectively. In Section V, we provide numerical results and we conclude the paper in Section VI. Proofs of theorems/lemmas are provided in the Appendices.

Ii System Model and Problem Formulation

We consider a system model comprised of an EHU, an ET, and an EVE. In order to improve the spectral efficiency of the considered system, both the EHU and the ET are assumed to operate in the FD mode, i.e., both nodes transmit and receive RF signals simultaneously and in the same frequency band. Thereby, the EHU receives energy signals from the ET and simultaneously transmits information signals to the ET. Similarly, the ET transmits energy signals to the EHU and simultaneously receives information signals from the EHU. The signal transmitted from the ET also serves as interference to the EVE, and thereby increases its noise floor. Due to the FD mode of operation, both the EHU and the ET are subjected to self-interference, which has opposite effects at the two nodes, respectively. More precisely, the self-interference signal has a negative effect at the ET since it hinders the decoding of the information signal received from the EHU. As a result, the ET should be designed with a self-interference suppression apparatus, which can suppress the self-interference at the ET and thereby improve the decoding of the desired signal received from the EHU. On the other hand, at the EHU, the self-interference signal is desired since it increases the amount of energy that can be harvested by the EHU. Hence, the EHU should be designed without a self-interference suppression apparatus in order for the energy contained in the self-interference signal to be harvested, i.e., the EHU should perform energy recycling as proposed in [23]. Meanwhile, EVE remains passive and only receives, thus it is not subjected to self-interference.

Ii-a Channel Model

Let and

denote random variables (RVs) which model the fading channel gains of the EHU-ET and ET-EHU channels in channel use

, respectively. Due to the FD mode of operation, the EHU-ET and the ET-EHU channels are identical and as a result the channel gains and are assumed to be identical, i.e., . Moreover, let and denote RVs which model the fading channel gains of the EHU-EVE and ET-EVE channels in channel use , respectively. We assume that all channel gains follow a block-fading model, i.e., they remain constant during all channel uses in one block, but change from one block to the next, where each block consists of (infinitely) many channel uses.

In the -th channel use, let the transmit symbols at the EHU and the ET be modeled as RVs, denoted by and , respectively. Moreover, in channel use , let the received symbols at the EHU, the ET, and EVE be modeled as RVs, denoted by , , and , respectively. Furthermore, in channel use , let the RVs modeling the AWGNs at the EHU, the ET, and the EVE be denoted by , , and , respectively, such that , , and , where

denotes a Gaussian distribution with mean

and variance

. Moreover, let the RVs modeling the additive self-interferences at the EHU and the ET in channel use be denoted by and , respectively.

By using the notation defined above, the input-output relations describing the considered channel in channel use can be written as

 Y1i =ViX2i+I1i+N1i, (1) Y2i =ViX1i+I2i+N2i, (2) Y3i =FiX1i+GiX2i+N3i. (3)

Ii-B Self-Interference Model

A general model for the self-interference at the EHU and the ET is given by [24]

 I1i =M∑m=1~Q1,m(i)Xm1i, (4) I2i =M∑m=1~Q2,m(i)Xm2i, (5)

where is an integer and and model the -th component of the self-interference channel between the transmitter- and the receiver-ends at the EHU and the ET in channel use , respectively. As shown in [24], the components in (4) and (5) for which

is odd carry non-negligible energy and the remaining components carry negligible energy and therefore can be ignored. Furthermore, the higher order components carry less energy than the lower order terms. As a result, we can justifiably adopt the first-order approximation of the self-interference in (

4) and (5), and model and as

 I1i =~Q1iX1i, (6) I2i =~Q2iX2i, (7)

where and are used for simplicity of notation. Thereby, the adopted self-interference model takes into account only the linear component of (4) and (5), i.e., the component for . The linear self-interference model has been widely used, e.g. in [24], [25].

By inserting (6) and (7) into (1) and (2), respectively, we obtain

 Y1i =ViX2i+~Q1iX1i+N1i, (8) Y2i =ViX1i+~Q2iX2i+N2i. (9)

To model the worst-case of linear self-interference, we note the following. Since the ET knows which symbol it has transmitted in channel use , the ET knows the outcome of the RV , denoted by . As a result of this knowledge, the noise that the ET “sees” in its received symbol given by (9), is , where is a constant. Hence, the noise that the ET “sees”,

, will represent the worst-case of noise, under a second moment constraint, if and only if

is an independent and identically distributed (i.i.d.) Gaussian RV111This is due to the fact that the Gaussian distribution has the largest entropy under a second moment constraint, see [26].. Therefore, in order to derive results for the worst-case of linear self-interference, we assume that in the rest of the paper. Meanwhile,

is distributed according to an arbitrary probability distribution with mean

and variance .

Now, since and can be written equivalently as and , where and are the means of and , respectively, and and denote the remaining zero-mean components of and , respectively, we can write and in (8) and (9), respectively, as

 Y1i =ViX2i+¯q1iX1i+Q1iX1i+N1i, (10) Y2i =ViX1i+¯q2iX2i+Q2iX2i+N2i. (11)

Since the ET always knows the outcome of ,

, and since given sufficient time it can always estimate the deterministic component of its self-interference channel,

, the ET can remove from its received symbol , given by (11), and thereby reduce its self-interference. In this way, the ET obtains a new received symbol, denoted again by , as

 Y2i =ViX1i+Q2iX2i+N2i. (12)

Note that since in (12) changes independently from one channel use to the next, the ET cannot estimate and remove from its received symbol even though the ET knows the outcome of . Thus, in (12) is the residual self-interference at the ET where the ET knows the outcome of . On the other hand, since the EHU benefits from the self-interference, it does not remove from its received symbol , given by (10), in order to have a self-interference signal with a much higher energy, which it can then harvest. Hence, the received symbol at the EHU is given by (10).

In this paper, we are interested in the secrecy capacity of the channel characterised by the input-output relationships given by (10), (12), and (3).

Ii-C Energy Harvesting Model

The energy harvested by the EHU in channel use is given by [23]

 Ein,i=η(ViX2i+¯q1X1i+Q1iX1i)2, (13)

where is the energy harvesting efficiency coefficient. For convenience, we have assumed unit time and thus we use the terms power and energy interchangeably in the sequel. The EHU stores in its battery, which is assumed to have an infinitely large storage capacity. Let denote the amount of harvested energy in the battery of the EHU at the end of the -th channel use. Moreover, let be the extracted energy from the battery in the -th channel use. Then, , can be written as

 Bi=Bi−1+Ein,i−Eout,i. (14)

Since in channel use the EHU cannot extract more energy than the amount of energy stored in its battery at the end of channel use , the extracted energy from the battery in channel use , , can be obtained as

 Eout,i=min{Bi−1,X21i+Pp}, (15)

where is the transmit energy of the desired transmit symbol in channel use , , and is the processing energy cost of the EHU [27]. The processing cost, , models the system level power consumption at the EHU, i.e., the energy spent due to the electrical components in the electrical circuit such as AC/DC convertors and RF amplifiers as well as the energy spent for processing. Note that the ET also requires energy for processing. However, the ET is assumed to be equipped with a conventional power source which is always capable of providing the processing energy without affecting the energy required for transmission.

Now, if the total number of channel uses satisfies , if the battery of the EHU has an unlimited storage capacity, and furthermore

 E{Ein,i}≥E{X21i}+Pp (16)

holds, where denotes statistical expectation, then the number of channel uses in which the extracted energy from the battery is insufficient and thereby holds is negligible compared to the number of channel uses in which the extracted energy is sufficient for both transmission and processing [28]. In other words, when the above three conditions hold, in almost all channel uses, there will be enough energy to be extracted from the EHU’s battery for both processing, , and for the transmission of the desired transmit symbol , , and thereby holds.

Iii Upper Bound on the Secrecy Capacity

For the considered channel, we propose the following theorem which establishes an upper bound on the secrecy capacity.

Theorem 1

Assuming that the average power constraint at the ET is , an upper bound on the secrecy capacity of the considered channel is given by

 maxp(x1|x2,v),p(x2|v)∑x2∈X2∑v∈VI(X1;Y2|X2=x2,V=v)p(x2|v)p(v) −∑v∈V∑g∈G∑f∈FI(X1;Y3|V=v,G=g,F=f)p(v)p(g)p(f) Subjectto C1:∑x2∈X2∑v∈Vx22p(x2|v)p(v)≤PET C2:∫x1∑x2∈X2∑v∈V(x21+Pp)p(x1|x2,v)p(x2|v)p(v)dx1≤ ∫x1∑x2∈X2∑v∈VEinp(x1|x2,v)p(x2|v)p(v)dx1 C3:∑x2∈X2p(x2|v)=1 C4:∫x1p(x1|x2,v)dx1=1, (17)

where denotes the conditional mutual information. In (17), lower-case letters , , , and represent realizations of the random variables , , , and , respectively, and their support sets are denoted by , , , and , respectively. Constraint C1 in (17) constrains the average transmit power of the ET to , and C2 is due to (16), i.e., due to the fact that EHU has to have harvested enough energy for both processing and transmission of symbol

. The maximum in the objective function is taken over all possible conditional probability distributions of

and , given by and , respectively.

Proof:

Please refer to Appendix A, where the converse is provided.

Iii-a Simplified Expression of the Upper Bound on the Secrecy Capacity

The optimal input distributions at the EHU and the ET that are the solutions of the optimization problem in (17) and the resulting simplified expressions of the upper bound on the secrecy capacity are provided by the following lemma.

Lemma 1

The optimal input distribution at the EHU, found as the solution of the optimization problem in (17), is zero-mean Gaussian with variance , i.e., , where can be found as the solution of

 v2σ22+x22α2+(1+v2PEHU(x2,v)σ22+x22α2)∑f∈Ff2f2PEHU(x2,v)+σ23p(f) = (1+v2PEHU(x2,v)σ22+x22α2)λ2(1−η(¯g12+α1)), (18)

where is chosen such that C2 in (17) holds with equality.

On the other hand, the optimal input distribution at the ET, found as the solution of the optimization problem in (17), has the following discrete form

 p(x2|v)=p(x2=0)δ(x2)+12J∑j=1p(x2=x2j)(δ(x2−x2j)+δ(x2+x2j)), (19)

where denotes the Dirac delta function. Finally, the simplified expression of the upper bound on the secrecy capacity in (17), denoted by , is given by

 Cus=12∑v∈VJ∑j=1log(1+v2PEHU(x2,v)σ22+x22jα2)p(x2=x2j)p(v) +∑v∈V∑g∈G∑f∈F⎡⎢ ⎢⎣∫∞−∞1√2πσ2y3J∑j=1p(x2=x2j)e−(y3−x2j)22σ2y3 ×ln⎛⎜ ⎜⎝1√2πσ2y3J∑j=1p(x2=x2j)e−(y3−x2j)22σ2y3⎞⎟ ⎟⎠dy3−∫∞−∞1√2πσ23J∑j=1p(x2=x2j)e−(z−x2j)22σ23 ×ln⎛⎜ ⎜⎝1√2πσ23J∑j=1p(x2=x2j)e−(z−x2j)22σ23⎞⎟ ⎟⎠dz3⎤⎥ ⎥⎦p(v)p(g)p(f). (20)

Iv Lower Bound on the Secrecy Capacity - An Achievable Secrecy Rate

From Lemma 1, we can see that the upper bound on the secrecy capacity cannot be achieved since the EHU has to know in each channel use , in order for the EHU to calculate (1). In other words, the EHU can not adapt and the data rates of its codewords accordingly. The knowledge of at the EHU is not possible since the input distribution at the ET, given by (19), is discrete with a finite number of probability mass points. However, if we set the input distribution at the ET to be binary such that , , takes values from the set , then the EHU can know in each channel use since , , and therefore this rate can be achieved. Hence, to obtain an achievable lower bound on the secrecy capacity, we propose the ET to use the following input distribution

 p(x2|v)=12(δ(x2−x0(v))+δ(x2+x0(v))). (21)

The value of will be determined in the following.

Iv-a Simplified Expression of the Lower Bound on the Secrecy Capacity

The simplified expression for the lower bound on the secrecy capacity resulting from the ET using the distribution given by (21), is provided by the following lemma.

Lemma 2

Let us define as

 I(x)=2√2πxe−x2/2∫∞0e−y2/2xcosh(y)ln(cosh(y))dy. (22)

Depending on the channel qualities, we have three cases for the achievable secrecy rate.

Case 1: If the following conditions hold

 12∑v∈Vlog(1+v2PEHU(x2,v)σ22+PETα2)p(v)+λ1PET = λ2((1−η(¯q12+α1))∑v∈VPEHU(x2,v)p(v)−ηPETΩV), (23)

and

 12∑v∈Vlog(1+v2PEHU(x2,v)σ22+PETα2)p(v)>∑v∈V∑g∈G∑f∈F[12ln(2πeσ2y3) +PETf2PEHU(x2,v)+σ23−I⎛⎜ ⎜⎝√PET√f2PEHU(x2,v)+σ23⎞⎟ ⎟⎠ −12ln(2πeσ23)−PETσ23+I(√PETσ3)]p(v)p(g)p(f), (24)

where is the root of (1) for and , then the input distribution at the ET has the following form

 p(x2|v)=12(δ(x2−√PET)+δ(x2+√PET)),∀v. (25)

On the other hand, the input distribution at the EHU is zero-mean Gaussian with variance , i.e., , where can be found as the solution of (1) for .

For Case 1, the achievable secrecy rate, denoted by , is given by

 Cls=12∑v∈Vlog(1+v2PEHU(√PET,v)σ22+PETα2)p(v) +∑v∈V∑g∈G∑f∈F⎡⎢ ⎢⎣∫∞−∞12√2πσ2y3⎛⎜⎝e−(y3−√PET)22σ2y3+e−(y3+√PET)22σ2y3⎞⎟⎠ ×ln⎛⎜ ⎜⎝12√2πσ2y3⎛⎜⎝e−(y3−√PET)22σ2y3+e−(y3+√PET)22σ2y3⎞⎟⎠⎞⎟ ⎟⎠dy3 −∫∞−∞12√2πσ23⎛⎜⎝e−(z3−√PET)22σ23+e−(z3+√PET)22σ23⎞⎟⎠ ×ln⎛⎜⎝e−(z3−√PET)22σ23+e−(z3+√PET)22σ23⎞⎟⎠dz3⎤⎥⎦p(v)p(g)p(f). (26)

Case 2: If (2) does not hold, and

 12∑v∈Vlog(1+v2PEHU(x2,v)σ22+x20(v)α2)p(v)>∑v∈V∑g∈G∑f∈F[12ln(2πeσ2y3) +x20(v)f2PEHU(x2,v)+σ23−I⎛⎜ ⎜⎝x0(v)√f2PEHU(x2,v)+σ23⎞⎟ ⎟⎠ −12ln(2πeσ23)−x20(v)σ23+I(x0(v)σ3)]p(v)p(g)p(f) (27)

holds, then the input distribution at the ET is given by

 p(x2|v)=12(δ(x2−x0(v))+δ(x2+x0(v))), (28)

whereas the input distribution at the EHU is zero-mean Gaussian with variance . In this case, and are the roots of the system of equations comprised of (1) for and the following equation

 12log(1+v2PEHU(x0(v),v)σ22+x20(v)α2)−λ1x20(v) = λ2((1−η(¯q12+α1))PEHU(x0(v),v)−ηv2x20(v)). (29)

For Case 2, the achievable secrecy rate is given by

 Cls=12∑v∈Vlog(1+v2PEHU(x0(v),v)σ22+x20(v)α2)p(v) +∑v∈V∑g∈G∑f∈F⎡⎢ ⎢⎣∫∞−∞12√2πσ2y3⎛⎜⎝e−(y3−x0(v))22σ2y3+e−(y3+x0(v))22σ2y3⎞⎟⎠ ×ln⎛⎜ ⎜⎝12√2πσ2y3⎛⎜⎝e−(y3−x0(v))22σ2y3+e−(y3+x0(v))22σ2y3⎞⎟⎠⎞⎟ ⎟⎠dy3 −∫∞−∞12√2πσ23⎛⎜⎝e−(z3−x0(v))22σ23+e−(z3+x0(v))22σ23⎞⎟⎠ ×ln⎛⎜⎝e−(z3−x0(v))22σ23+e−(z3+x0(v))22σ23⎞⎟⎠dz3⎤⎥⎦p(v)p(g)p(f). (30)

Case 3: If neither (2) nor (2) holds, then, the achievable secrecy rate is .

Proof:

In order for C1 in (17) to hold, or equivalently for C1 in (B) to hold, there are two possible cases for . In Case 1, C1 in (B) is satisfied if is set to take values from the set . If (63) for does not hold, then is set to take values from the set , where is given by (2) in order for C1 in (B) to be satisfied. Now, since , where follows a Gaussian probability distribution, and is distributed according to (25) and (28) for Case 1 and Case 2, respectively, we obtain the expressions in (2) and (2) by using (B) and (B).

Lemma 2 gives insights into the achievability scheme of the derived lower bound on the secrecy capacity. When Case 1 of Lemma 2 holds, the achievability scheme is very simple. In particular, the ET only chooses between or in every channel use. When Case 2 of Lemma 2 holds, from (2) we see that the ET adapts its transmit power to the channel fading states of the EHU-ET channel, , and increases its transmit power when is larger, and conversely, it lowers its transmit power when is not as favourable. As for the EHU, we first note that, since the EHU knows the square of the transmit symbol of the ET in a given channel use, the EHU can adapt its transmit power and its rate in the given channel use according to the expected self-interference at the ET, which depends on the value of . Secondly, the EHU also takes advantage of the better channel fading states of the EHU-ET channel, , and increases its transmit power and rate when is larger, and conversely, it lowers its transmit power and rate when is not as strong. Thirdly, since is chosen such that constraint C2 in (17) holds, the transmit power of the EHU depends on the processing cost . Thereby, when Case 2 holds, the ET also takes into account the processing cost of the EHU.

Iv-B Achievability of the Lower Bound on the Secrecy Capacity

We set the total number of channel uses (i.e., symbols) to , where denotes the total number of time slots used for the transmission and denotes the number of symbols transmitted per time slot, where , , , and .

Let denote a set comprised of the time slots during which the EHU has enough energy harvested and thereby transmits a codeword, and let denote a set comprised of the time slots during which the EHU does not have enough energy harvested and thereby it is silent. Let and , where denotes the cardinality of a set.

Transmissions at the ET: During the channel uses of a considered time slot with fading realisation , the ET’s transmit symbol is drawn from the probability distribution given in Lemma 2. Thus, in each channel use of the considered time slot, the ET transmits either or with probability if Case 1 in Lemma 2 holds, or transmits or with probability if Case 2 in Lemma 2 holds.

Reception of Energy and Transmission of Information at the EHU: The EHU first generates all binary sequences of length , where

 REHU=12∑v∈Vlog(1+v2PEHU(x2,v)σ22+x22α2)p(v), (31)

where and can be found from Lemma 1 depending on which case holds. Then the EHU uniformly assigns each generated sequence to one of groups, where is given by (2) for Case 1 of Lemma 2, or by (2) for Case 2 of Lemma 2. The confidential message drawn uniformly from the set is then assigned to a group. Next, the EHU randomly select a binary sequence from the corresponding group to which

is assigned, according to the uniform distribution. This binary sequence is then mapped to a codeword comprised of

symbols, which is to be transmitted in time slots. The symbols of the codeword are drawn according to a zero-mean, unit-variance Gaussian distribution. Next, the codeword is divided into blocks, where each block is comprised of symbols. The length of each block is assumed to coincide with a single fading realization, and thereby to a single time slot.

The EHU will transmit in a given time slot only when it has harvested enough energy both for processing and transmission in the given time slot, i.e., only when its harvested energy accumulates to a level which is higher than , where is the fading gain in the time slot considered for transmission. Otherwise, the EHU is silent and only harvests energy. When the EHU transmits, it transmits the next untransmitted block of symbols of its codeword. To this end, each symbol of this block is first multiplied by , where can be found from Lemma 2, and then the block of symbols is transmitted over the wireless channel to the ET. The EHU repeats this procedure until it transmits all blocks of its entire codeword for which it needs time slots.

Receptions at the ET: When the ET receives a transmitted block by the EHU, it checks if the power level of the received block is higher than the noise level at the ET or not. If affirmative, the ET places the received block in its data storage, without decoding. Otherwise the received block is discarded.

Now, in time slots, the ET receives the entire codeword transmitted by the EHU. In order for the ET to be able to decode the transmitted codeword, the rate of the transmitted codeword must be equal to or lower than the capacity of the EHU-ET’s channel, given by

 CEHU−ET=12∑v∈Vlog(1+v2PEHU(x2,v)σ22+x22α2)p(v). (32)

Note that the rate of the transmitted codeword is , given by (31). Now, since , the ET is able to decode the codeword transmitted by the EHU. Next, since the ET knows the binary sequences corresponding to each group, by decoding the transmitted codeword the ET determines the group to which the transmitted codeword belongs to. As a result, the ET is able to decode the secret message .

In the time slots, the achieved secrecy rate is given by . It was proven in [28] that when the EHU is equipped with a battery with an unlimited storage capacity and when C2 in (17) holds, then as . Thereby, the achieved secrecy rate in time slots is given by , which is the actual lower bound of the channel secrecy capacity given by Lemma 2.

Receptions at the EVE: EVE receives the transmitted blocks by the EHU and the ET. Similarly to the ET, EVE places the received block in its data storage, without decoding.

In time slots, the EVE also receives the entire codeword transmitted by the EHU. In addition, EVE receives the signal from the ET, comprised of randomly generated symbols (see Lemma 2), which acts as noise to EVE and impairs the ability of EVE to decode the codeword from the EHU. To show that the EVE will not be able to decode the secret message, we use properties of the multiple access channel, resulting from the EHU and the ET transmitting at the same time. The multiple-access capacity region at the EVE formed by the transmission of the EHU and the ET is given by . The EVE will be able to decode the EHU’s codeword only if one of the following two cases holds, i.e., when or when , where is the entropy of the signal generated by the ET and is given by

 RET=−[p(x2)log2p(x2)+p(x2)log2p(x2)]. (33)

In (33), , see lemma 2. As a result,

 RET=log2=1. (34)

Case 1: For the EHU’s codeword to be decodable at the EVE in this case,
and have to hold. For we have

 I(X2;Y3|V,F,G)=∑v∈V∑g