Ultra-Reliable Cooperative Short-Packet Communications with Wireless Energy Transfer

01/01/2018 ∙ by Onel L. A. López, et al. ∙ 0

We analyze a cooperative wireless communication system with finite block length and finite battery energy, under quasi-static Rayleigh fading. Source and relay nodes are powered by a wireless energy transfer (WET) process, while using the harvested energy to feed their circuits, send pilot signals to estimate channels at receivers, and for wireless information transmission (WIT). Other power consumption sources beyond data transmission power are considered. The error probability is investigated under perfect/imperfect channel state information (CSI), while reaching accurate closed-form approximations in ideal direct communication system setups. We consider ultra-reliable communication (URC) scenarios under discussion for the next fifth-generation (5G) of wireless systems. The numerical results show the existence of an optimum pilot transmit power for channel estimation, which increases with the harvested energy. We also show the importance of cooperation, even taking into account the multiplexing loss, in order to meet the error and latency constraints of the URC systems.

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I Introduction

I-a Motivation

The 5G concept goes beyond broadband connectivity by bringing new kinds of services to life, enabling mission-critical control through ultra-reliable, low-latency links and connecting a massive number of smart devices, enabling the Internet of Things (IoT) [1]. It will not only interconnect people, but also interconnect and control a massive number of machines, objects, and devices, with applications having stringent requirements on latency and reliability. In fact, a novel operation mode under discussion for 5G is Ultra-Reliable Communication over a Short Term (URC-S)[2, 3, 4], which focuses on scenarios with stringent latency, e.g., ms and error probability, e.g., , requirements [2].

Powering and keeping uninterrupted operation of such massive number of nodes is a major challenge [5], which could be addressed through the WET technique. WET constitutes an attractive solution because of the coverage advantages of radio-frequency (RF) signals, specially for IoT scenarios where replacing or recharging batteries require high cost and/or can be inconvenient or hazardous (e.g., in toxic environments), or highly undesirable (e.g., for sensors embedded in building structures or inside the human body) [6, 7, 8]. Also, RF signals can carry both energy and information, which enables energy constrained nodes to harvest energy and receive information [9, 10], allowing to prolong their lifetime almost indefinitely.

The most important characteristics of WET systems are [11]: i) power consumption of the nodes on the order of ; ii) strict requirements on the reliability of the energy supply and of the data transfer; iii) information is conveyed in short packets due to intrinsically small data payloads, low-latency requirements, and/or lack of energy resources to support longer transmissions [12]. This agrees well with several URC-S scenarios with stringent latency requirements. Even though, most of the work in the field is made under the ideal assumption of communicating with large enough blocks in order to invoke Shannon theoretic arguments to address error performance, which contradicts the third characteristic of the WET systems. In fact, although performance metrics like Shannon capacity, and its extension to non-ergodic channels, have been proven useful to design current wireless systems since the delay constraints are typically above 10 ms [13], they are not necessarily appropriate in a short-packet scenario [14]. Also, for short-packet scenarios where the sizes of the preamble, the metadata and the data of the frame structure, are of the same order of magnitude, the Shannon capacity becomes an inaccurate metric for assessing the necessary blocklength to achieve a certain reliability. Instead, an essential quantity is the maximum coding rate for which [15] (and references therein) developed nonasymptotic bounds and approximations.

Our goal in this paper is to investigate wireless-powered communication with short packets, which is of great interest for URC-S use cases in 5G systems. We analyze a cooperative setup, where source and relay nodes are powered by a WET process from the destination, and practical issues such as the finite battery energy, other power consumption sources beyond data transmission, and imperfect CSI, are taken into account.

Symbol Definition Symbol Definition
Source, Relay and Destination nodes, respectively Battery capacity of and
WET blocklength Pilot signal blocklength
Information blocklength in the DC scheme Optimum information blocklength in the DC scheme
WIT blocklength in the 1st, 2nd phase with relaying Number of data and pilot symbols in the relaying scheme
Message length in bits Normalized channel in ,
Power gain coefficients in , Pilot signal transmit power for both and
Duration of each channel use Energy harvested at
Battery charge in before transmitting Energy conversion efficiency
Distance of the link , Path loss exponent
Path loss factor in , Transmit power of
Threshold for the battery saturation in Circuit and baseband processing power consumption
Available energy for transmission at Signal transmitted by

Gaussian noise vector at

Variance of ,
Received signal at from in the DC scheme Received signal at
, Received signal at from and with relaying Instantaneous SNR in in the DC scheme
Instantaneous SNR in , with relaying Instantaneous SNR in
, Delay and minimum delay , Time sharing parameter and the optimum one
Outage probability Energy insufficiency probability at
Error probability Shannon capacity for a SNR equal to
Shannon dispersion for a SNR equal to , Estimate of and error in the estimation
, Pilot symbols transmitted/received by Instantaneous SNR at the receiver with imperfect CSI
Target error probability Maximum allowable delay
Error metric for the approximations in (16) and (17) Number of terms for summation in (16)
TABLE I: Summary of Main Symbols

I-B Related Work

The most common WET techniques are based on time-switching or power splitting [16]. Authors in [17] propose two WET protocols called Time Switching-based Relaying (TSR) and Power Splitting-based Relaying (PSR) to be implemented by an energy constrained relay node while assisting the communications of a single link. The implementation of the TSR protocol is further analyzed later in [18]. In all cases, there is not a direct link between source and destination, while only the relay is energy constrained and powered by the source. Alternatively, authors in [19] consider a system where energy-constrained sources have independent information to transmit to a common destination, which is responsible for transferring energy wirelessly to the sources. The source nodes may cooperate with each other, under either decode-and-forward (DF) or network coding-based protocols, and even though the achievable diversity order is reduced due to wireless energy transfer process, it is very close to the one achieved for a network without energy constraints. Those results are extended in [20] by considering a more realistic consumption model where the circuitry power consumption is taken into account. In [21], both relay and source are powered by a WET process and the protocols: Harvest-then-Transmit (HTT) for a direct communication scenario, and Harvest-then-Cooperate (HTC) for relaying scenarios, are evaluated. However, all these works are under the ideal assumption of communicating with infinite blocklength.

Wireless-powered communication networks at finite blocklength regime have received attention in the scientific community recently. Authors of [22] attain tight approximations for the outage probability/throughput in an amplify-and-forward relaying scenario, while in [11] the authors implement retransmission protocols, in both energy and information transmission phases, to reduce the outage probability compared to open-loop communications. The impact of the number of channel uses for WET and for WIT on the performance of a system where a node charged by a power beacon attempts to communicate with a receiver, is investigated in [12]. Also, subblock energy-constrained codes are investigated in [23], while authors provide a sufficient condition on the subblock length to avoid energy outage at the receiver. In [24] we optimize a single-hop wireless system with WET in the downlink and WIT in the uplink, under quasi-static Nakagami-m fading in URC-S scenarios. The impact of a non-cooperative dual-hop setup is further evaluated in [25]. Finally and as an extension of [24], the error probability and energy consumption at finite block length and finite battery energy are characterized in [26] for scenarios with/without energy accumulation between transmission rounds with transmit power control.

The only source of energy consumption in all above works is the transmission power, which is a very impractical simplification for energy-limited systems. Also, all these works consider perfect CSI acquisition, which is a common situation in current scientific literature. However, the analysis under perfect CSI could be misleading for a wireless-powered communication network, due to its inherent energy constraints, and even more on systems with limited delay. Only few works have given a step forward on this direction in order to characterize different scenarios. Particularly interesting is the work in [27] where authors attain a closed-form approximation for the error probability at finite blocklength and imperfect CSI while showing the convenience of adapting the training sequence length and the transmission rate to reach higher reliability. However, new scenarios require deeper understanding, and battery capacity of devices and other energy consumption sources beyond transmission must be taken into account for a more realistic analysis.

I-C Contributions

This paper aims at filling the above gaps in the state of the art. Next we list the main contributions of this work:

  • We model an URC-S system under Rayleigh quasi-static fading at finite blocklength and finite battery constraint with other power consumption sources beyond data transmission power. We show that the infinite battery assumption is permissible for the scenarios under discussion since the energy harvested is very low; but considering other power consumption sources, e.g., circuit and baseband processing power, is crucial because they constitute a non-negligible cause of outage.

  • We analyze the impact of imperfect CSI acquisition from pilot signals on the system performance. Numerical results show the existence of an optimum pilot transmit power for channel estimation, which increases with the harvested energy and decreases with the information blocklength. We show that the energy devoted to the CSI acquisition, which depends on the power and utilized time, has to be taken into account due to the inherent energy and delay constraints of the discussed scenarios.

  • For ideal direct communication, we reach accurate closed-form approximations for the error probability for finite and infinite battery devices.

  • Also, cooperation appears as a viable solution to meet the reliability and delay constraints of URC-S scenarios, and it is advisable transmitting with shorter blocklengths during the broadcast phase.

The rest of this paper is organized as follows. Section II presents the system model. Sections III and IV discuss the delay and error probability metrics for scenarios with/without perfect CSI acquisition, respectively. Section V presents some numerical results, while Section VI concludes the paper.

Notation: Let denote the probability of event , while

is a normalized exponentially distributed random variable with Probability Density Function (PDF)

and Cumulative Distribution Function (CDF)

. Then, is a zero-mean circularly symmetric complex random variable with unit variance. Let denote expectation and is the absolute value operator. The Gaussian Q-function is denoted as [28, §8.250-4], [28, §8.21] is the exponential integral and is the modified Bessel function of second kind and order [28, §8.407]. Finally, represents the minimum value between and . The main symbols along the paper and their meaning are summarized in TABLE I.

Ii System Model

We consider a dual-hop cooperative network where a DF relay node, , is available for assisting the transmissions from the source, , to the destination, . All the nodes are single antenna, half-duplex devices, where is assumed to be externally powered and acts as an interrogator, e.g., an access point or a base station, requesting information from , which along with , may be seen as sensor nodes with very limited energy supply and finite battery of capacity . The communication scheme is illustrated in Fig. 1. First, powers and during channel uses through a WET process. Right after, broadcasts a known pilot signal of channel uses before sending its data during a WIT phase over the next channel uses. The pilot is used by the receivers, and , to acquire CSI. Then, tries to decode the information from and if succeeds, transmits a pilot signal of channel uses to for CSI acquisition and then forwards information received during the last WIT phase over the next channel uses. Finally, combines the received information from and to decode the message. At each WIT phase, information bits constitute the transmitted message. We assume , and notice that an equivalent direct communication (DC) scheme can be easily obtained from Fig. 1 without the assistance of , with only one pilot signal transmission ( channel uses) and one WIT ( channel uses) phase.

Fig. 1: Relaying communication scheme: powers and using channel uses, then broadcasts the pilots in channel uses and the information into channel uses, while repeats the processes using, respectively, and channel uses.

We assume Rayleigh quasi-static fading channels, where the fading process is constant over a transmission round, which spans over all the phases of the communication scheme shown in Fig. 1 ( channel uses), and it is independent and identically distributed from round to round. The normalized channel gains from node to node , where and , are denoted by , while is the power gain coefficient. Similarly, distance between nodes is denoted by . In addition, no energy accumulation is allowed between consecutive transmission rounds, thus all the harvested energy at and is used for powering their circuits and for pilot and data transmission. The pilot signal transmit power is assumed fixed and equal to for both and . Also, the duration of each channel use is denoted by .

Iii System Performance with Perfect CSI

First, as a benchmark, we assume that regardless the resources used for CSI acquisition, and , receivers get a perfect channel estimation (pCSI). We note that scenarios with pCSI are quite frequent in the literature, e.g., [17, 18, 21, 22, 11, 12, 23, 24, 25]. Moreover, we start analyzing the outage probability for the DC scheme and then we extend the results for the cooperative scheme.

Iii-a DC scheme

The energy harvested at in the WET phase, which lasts for , is [21]

(1)

and the charging state of the battery before transmitting is

(2)

since no energy is accumulated between transmission rounds and because of the finite battery at , it is not allowed to store more energy than the allowable limit . is the transmit power of when wirelessly powering (and ), is the energy conversion efficiency and accounts for the path loss in the link, where is the path loss exponent, and represents a combination of other losses as due to carrier frequency and the antenna gains [29]. Also, is the channel power gain threshold for the saturation of the battery in , which is given by

(3)

In addition, we assume that is sufficiently large such that the energy harvested from noise is negligible. Notice that (2) is a valid expression if and only if all the energy harvested by at each round is completely used during the pilot signal transmission and WIT phases of the same round. We acknowledge that energy accumulation between transmission rounds with adequate power allocation strategies could improve the system performance, however that scenario is out of the scope of this work.

Right after the WET phase, transmits a pilot signal of channel uses and power to be used by for CSI acquisition. Also, we here take into account the circuit and baseband processing power consumption, namely , which is assumed to be constant without loss of generality. When the harvested energy is insufficient for pilot transmission, WIT and other consumption sources, there is an outage and transmission fails, otherwise all the remaining energy is used for those processes. Notice that an outage event happens when the transmitter has not the sufficient power to feed its circuits and send the estimating pilot or when there is data transmission but the receiver is unable to successfully decode the received message. Thus, the available energy for transmission at and its transmit power are stated, respectively, as

(4)
(5)

where in (5) comes from substituting (2) into (4). In addition, the received signal at is

(6)

where is the codebook transmitted by , which is assumed Gaussian with zero-mean and unit-variance, . Notice that there is no guarantee that Gaussian codebooks will provide the best reliability performance at finite blocklength, and here they are merely used to gain in mathematical tractability (authors in [15] derive accurate performance approximations). is the Gaussian noise vector at with variance . Thus, by using (6) and (5), we can write the instantaneous SNR at as

(7)

where

(8)
(9)

Let be the delay in delivering a message of bits, while is the minimum delay that satisfies a given reliability constraint. Moreover, is the time sharing parameter representing the fraction of devoted to WET only. Therefore,

(10)
(11)

Notice that is measured in channel uses, while would be the delay in seconds. Finally, we define the optimum WIT blocklength, in the sense of minimizing , as . Both and are numerically investigated in Section V.

On the other hand, the reliability analysis comes from evaluating the outage probability. An outage event may be due to a low harvested energy, precluding the whole WIT process, or if after the WIT process there is a decoding error at . Hence, the outage probability is given as follows

(12)

where is the probability that the harvested energy at is insufficient simultaneously for channel estimation and for satisfying other consumption requirements at , and it is obtained through

(13)

Notice that comes from using the definitions of and given in (8) and (9), respectively, while comes from the fact that is not a random variable and it should be greater than for every practical system since otherwise the system is in outage all the time. Equality in is obtained by using the CDF expression of . Also, is the error probability when transmitting a message of information bits over channel uses and being received with SNR equal to at the destination, thus is the average probability. Both terms are accurately approximated by (14) [15], and (15) for quasi-static fading channel [30], when ,111See [15, Figs. 12 and 13] and related analyses for more insight on the accuracy of (14). Authors show that the approximate achievable rate matches almost perfectly its true value for . Many other works, such as [11, 12, 13, 22, 25, 24, 26, 27, 31, 32, 33], use this accurate approximation and/or (15) to gain in tractability. as shown next

(14)
(15)

where is the source fixed transmission rate, is the Shannon capacity, is the channel dispersion, which measures the stochastic variability of the channel relative to a deterministic channel with the same capacity [15].

Lemma 1.

The average error probability is given in (16) and (17) for finite and infinite battery, respectively.

(16)
(17)

where

(18)
Proof.

Substituting (7) into (15) and using the expression for and , we attain (16) and (17). Notice that for the infinite battery case, (7) becomes . ∎

We consider as ideal system, one where all the energy harvested by is used only during the WIT phase () while, even so, has perfect knowledge of the channel.

Theorem 1.

The outage probability given in (12) can be approximated as in (20) and (21) (on the top of the next page) for an ideal system as described in Section III-A, where , , , , for finite and infinite battery devices, respectively. is a parameter to limit the number of terms in the summation when evaluating (20).

Proof.

See Appendix A.

Corollary 1.

(21) reduces to [24, Eq.(10)] for Rayleigh fading.

To measure the accuracy of (20) and (21), we evaluate the following error metric

(19)

where is the exact error probability according to (16) and (17), for finite and infinite battery, respectively, while is the approximated value according to (20) and (21). Notice that is a parameter that must be carefully selected when evaluating the finite battery approximation given in (20) because it establishes the quantity of elements taken into account for the summation. As , expression (20) becomes more accurate but harder to evaluate and computationally heavier. Besides, a relatively small value conduces to an inaccurate approximation.

Fig. 2: as a function of with channel uses.
(20)
(21)

 

Fig. 2 shows numerically the impact of on , when and channel uses, and J. The remaining system parameters were the ones chosen in Section V. The larger the battery capacity, the larger the required for a good approximation using (16). Note that for all cases provides a good accuracy with . The accuracy of (16) and (17) was also measured for many other different setups, and in all the cases we reached similar results, e.g. for any pair with and bits. The accuracy is good because we used the linearization (first order approximation) of (14) to attain (20) and (21), which is symmetrical with respect to , and lies beneath and above of (14) for and , respectively. Thus, when integrating over all the channel realizations, the error tends to vanish.

Iii-B Relaying scheme

Cooperative technique has rekindled enormous interests from the wireless communication community over the past decade. As shown in [31], the attained spatial diversity can improve communication reliability, thus it can also reduce the system delay for a target error constraint. For an energy-constrained URC system, cooperation seems advisable to reduce the instantaneous consumption power of devices while meeting the reliability/delay constraints. Consequently, herein we consider the presence of node , which is willing to assist the communication all the time, while operating under the DF protocol. Under the finite blocklength regime, decoding errors may occur. We assume that reliably detects the errors, and consequently it does not forward the message to when an error occurs, as in [32]. In that condition, an outage event is declared for the link, and is inactive during the time reserved for its transmission, e.g., the last channel uses according to Fig. 1. The energy that could be saved in those cases is out of the scope of our analysis222Notice that in URC-S scenarios the probability of such events should be kept low, specially if relaying is crucial in achieving the ultra-reliability, and the impact of that remaining energy on the system performance can be negligible, as shown in [25] for a system without a direct link.. This behavior reduces the system complexity compared to other works, e.g. [18] for infinite blocklength and preset power relay. Nodes and are assumed to have the same characteristics, e.g., battery capacity, pilot power, power consumption profile.

Similar than in (1) and (2), the energy harvested at and the battery charge before transmitting are, respectively,

(22)
(23)

where is the channel power gain threshold for battery saturation given by

(24)

The expressions for the energy harvested at , the charge of its battery, the available energy for transmission and its transmit power are the same as in the analysis of the DC scheme, respectively (1), (2), (4) and (5), but with . Now, for , they are given by

(25)
(26)

where (note that consumes circuit power for receiving the pilots and the data from , besides the power consumed in the transmission of pilots and data to ). During the first WIT phase, broadcasts its data to and . The expression of the signal received at in this phase is , equal to (6) but with , and the signal received at is

(27)

where is given in (5) and is the Gaussian noise vector at with variance . When successfully decodes the message during the first WIT phase, it re-encodes it in channel uses, and after the pilot signal transmission, it transmits the message to during the second WIT phase. The received signal at at this phase is thus given by

(28)

where is given in (26) and is the zero-mean, unit-variance Gaussian codebook transmitted by . uses the same codebook, which is defined a priori, and when .

Let and be the instantaneous SNRs at and , respectively, for the signal transmitted by during the first WIT phase. Also, let be the instantaneous SNR at for the signal transmitted by during the second WIT phase, as long as achieved a successful decoding of the message transmitted by . Expression (7) is still useful for but with ( and according to (8) and (9), respectively) since now the transmission time depends on . Using (27) and (28) we attain the expressions for and , which are

(29)
(30)

where

(31)
(32)
(33)
(34)

The delay for the cooperative scheme is

(35)

while the time sharing parameter still obeys (11), but using (35) for the delay. An outage event for the cooperative scheme can be due to: i) a low harvested energy at , precluding the whole transmission process; ii) communication error in the link at the same time that is unable to perform a transmission due to its low harvested energy or when it fails in decoding the information from ; iii) both and were capable of completing their transmissions ( succeeded in decoding the message from ) but after combining both signals at there is still a decoding error. Thus, the outage probability can be mathematically written as333Note that in (III-B) and (12) when a complementary event, e.g. or , does not appear explicitly in the equation is because its effect is implicit when evaluating the communication errors by properly setting the integration limits.

(36)

where can be calculated as in (III-A) but with , and is the probability that the harvested energy at is insufficient simultaneously for channel estimation and for satisfying other consumption requirements at , and can be easily obtained similarly to as

(37)

can be obtained through (16) (or (17) for the infinite battery case) with , . Also, is the error probability when tries to decode the information from the combination of signals received from both WIT phases and depends on the used combination technique: Selection Combining () or Maximal Ratio Combining (). Notice that

since , and are correlated through the variable . In the first case, the expectation can be evaluated through (38) and (39) for finite and infinite battery, respectively, as shown next

(38)
(39)

where

(40)

The classical technique establishes that the signal with the highest SNR is selected since its error probability is the lowest [29]. Nonetheless, when the blocklength is small and different for each link, , this assumption does not hold always as is induced from (14), which motivates us to state the outage probability of the SC combining scheme as

(41)

where and can be obtained through (14) for each channel realization. Thus, has to select the signal to be decoded based on