On CSI-free Multi-Antenna Schemes for Massive Wireless Energy Transfer

02/06/2020 ∙ by Onel L. A. López, et al. ∙ UTEM UFSC 0

Wireless Energy Transfer (WET) is emerging as a potential green enabler for massive Internet of Things (IoT). Herein, we analyze Channel State Information (CSI)-free multi-antenna strategies for powering wirelessly a large set of single-antenna IoT devices. The CSI-free schemes are AA-SS (AA-IS), where all antennas transmit the same (independent) signal(s), and SA, where just one antenna transmits at a time such that all antennas are utilized during the coherence block. We characterize the distribution of the provided energy under correlated Rician fading for each scheme and find out that while AA-IS and SA cannot take advantage of the multiple antennas to improve the average provided energy, its dispersion can be significantly reduced. Meanwhile, AA-SS provides the greatest average energy, but also the greatest energy dispersion, and the gains depend critically on the mean phase shifts between the antenna elements. We find that consecutive antennas must be π phase-shifted for optimum average energy performance under AA-SS. Our numerical results evidenced that correlation is beneficial under AA-SS, while a greater line of sight (LOS) and/or number of antennas is not always beneficial under such scheme. Meanwhile, both AA-IS and SA schemes benefit from small correlation, large LOS and/or large number of antennas.

READ FULL TEXT VIEW PDF
POST COMMENT

Comments

There are no comments yet.

Authors

page 1

page 2

page 3

page 4

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

Internet of Things (IoT), where every thing is practically transformed into an information source, represents a major technology trend that is revolutionizing the way we interact with our surrounding environment so that we can make the most of it. There are two general categories of IoT use cases [1]: i) critical IoT, with stringent requirements on reliability, availability, and low latency, e.g. in remote health care, traffic safety and control, industrial applications and control, remote manufacturing, surgery; and ii) massive IoT, where sensors typically report to the cloud, and the requirement is for low-cost devices with low energy consumption and good coverage. In such massive deployments, these IoT nodes are not generally supposed to be transmitting continuously, but they do require to operate for long periods of time without batteries replacement, specially since many of them could be placed in hazardous environments, building structures or the human body. One important enabler under consideration is Wireless Energy Transfer (WET), so IoT nodes would be equipped with an energy harvesting (EH) circuitry that allows them to harvest energy from incoming radio frequency (RF) signals [2].

WET holds vast potential for replacing batteries or increasing their lifespans. In fact, RF-EH devices can become self-sustaining with respect to the energy required for operation, thereby obtaining an unlimited operating lifespan while demanding negligible maintenance [3]. This is crucial for the future society since the processing of battery wastes is already a critical problem. The most effective approach for reducing battery wastes is to avoid using them, for which WET is an attractive clean solution. Notice that with the continuous advances on circuitry technology aiming at reducing further the power consumption of low-cost devices111Currently there is available a wide-range of IoT devices that seem suitable for relying on a RF-EH module as the main power supply due to their extremely low power consumption profiles. For instance, accelerometer ADXL362 (W in active) and Light ISL29033 (W in active) [4]. In fact, the first wireless battery-free bio-signal processing system on chip was introduced by [5] and it is able to monitor various bio-signals via electrocardiogram, electromyogram, and electroencephalogram. The total size of the chip is 8.25 and consumes 19W to measure heart rate., e.g. printed sensors technology [6], it is expected an exponential growth on WET-enabled IoT applications.

The IoT paradigm intrinsically includes wireless information transfer (WIT), thus WET appears naturally combined with WIT. In such case two main architectures can be distinguished in literature: i) Wireless Powered Communication Network (WPCN), where WET occurs in the downlink in a first phase and WIT takes place in the second phase; and ii) Simultaneous Wireless Information and Power Transfer (SWIPT), where WET and WIT occur simultaneously. Readers can refer to [7] to review the recent progress on both architectures, while herein the discussions will focus merely on WPCN and pure WET setups. Notice that in most of practical applications WET duration would be significantly larger than WIT in order to harvest usable amounts of energy [2]. Actually, some use cases require operating under WET almost permanently while WIT happens sporadically, e.g. due to event-driven traffic. Therefore, enabling efficient WET is mandatory for realizing the IoT paradigm and constitutes the scope of this work.

I-a Related Work

Recent works have specifically considered WET and WPCN setups in different contexts and scenarios. Key networking structures and performance enhancing techniques to build an efficient WPCN are discussed in [8], where authors also point out challenging research directions. Departing from the simple Harvest-then-Transmit (HTT) scheme [9, 10, 11] several other protocols have been proposed over the past few years to boost the WPCN performance such as the Harvest-then-Cooperate (HTC) system studied in [12, 13, 14] and the power control scheme relying on energy accumulation between transmission rounds discussed in [15]. Authors either analyze the performance of the information transmission phase, or optimize it by using power control or cooperative schemes. Some scheduling strategies that allow a direct optimization of the energy efficiency of the network are also proposed in [16]. Additionally, an energy cooperation scheme that enables energy cooperation in battery-free wireless networks with WET is presented in [17]. Meanwhile, the deployment of single-antenna power beacons (PBs) for powering the mobiles in an uplink cellular network is proposed in [18]. Therein, authors investigate the network performance under an outage constraint on data links using stochastic-geometry tools, while they corroborate the effectiveness of relying on directed WET instead of using isotropic antennas.

Yet, shifts in the system architecture and in the resource allocation strategies for optimizing the energy supply to massive IoT deployments are still required. In [2] we discuss several techniques that seem suitable for enabling WET as an efficient solution for powering the future IoT networks. They are

  • Energy beamforming (EB), which allows the energy signals at different antennas to be carefully weighted to achieve constructive superposition at intended receivers. The larger the number of antennas installed at the PB, the sharper the energy beams can be generated in some particular spatial directions. The EB benefits for WPCNs have been investigated for instance in [19] in terms of average throughput performance, while in [20] authors propose an EB scheme that maximizes the weighted sum of the harvested energy and the information rate in a multiple-input single-output (MISO) system. However, the benefits of EB in practice depend on the available CSI at the transmitter, and although there has been some works proposing adequate channel acquisition methods, e.g. [21, 22], this still constitutes a serious limitation. This is due to the harsh requirements in terms of energy and scheduling policies, which become even more critical as the number of EH devices increases;

  • Distributed Antenna Systems (DAS), which are capable of eliminating blind spots while homogenizing the energy provided to a given area and supporting ubiquitous energy accessibility. The placement optimization of single-antenna energy and information access points in WPCNs is investigated in [23], where authors focus on minimizing the network deployment cost subject to energy harvesting and communication performance constraints. On the other hand, authors in [24]

    study the probability density function (PDF), the cumulative distribution function (CDF), and the average of the energy harvested in DAS, while they determine appropriate strategies when operating under different channel conditions by using such information. Although works in this regard have avoided the use of multiple transmit antennas, we would like to highlight the fact that multiple separate PBs, each equipped with multiple transmit antennas, could alleviate the issue of CSI acquisition when forming efficient energy beams in multiple-users setups, since each PB may be responsible for the CSI acquisition procedure of a smaller set of EH devices;

  • CSI-limited/CSI-free schemes. Even without accounting for the considerable energy resources demanded by CSI acquisition, the performance of CSI-based systems decays quickly as the number of served devices increases. Therefore, in massive deployment scenarios the broadcast nature of wireless transmissions should be intelligently exploited for powering simultaneously a massive number of IoT devices with minimum or non CSI. Very recently, and for the first time, we proposed in [25] several CSI-free multi-antenna schemes that a PB could utilize to power efficiently a large set of nearby EH devices, while we analyzed their performance in Rician correlated fading channels. We found out that i) the switching antenna () strategy, where a single antenna with full power transmits at a time, provides the most predictable energy source, and it is particularly suitable for powering sensor nodes with highly sensitive EH hardware operating under non line of sight (NLOS) conditions; and ii) transmitting simultaneously the same signal with equal power in all antennas (, but herein referred as ) is the most beneficial scheme when LOS increases and it is the only scheme that benefits from spatial correlation. However, the performance analysis was carried out under the idealistic assumption of channels sharing the same mean phase.

I-B Contributions and Organization of the Paper

This paper builds on CSI-free WET with multiple transmit antennas to power efficiently a large set of IoT devices. Different from our early work in [25], herein we do consider the mean phase shifts between antenna elements, which is a practical and unavoidable phenomenon. Based on such modeling we arrive to conclusions that are similar in some cases but different in others to those in [25]. Specifically, the main contributions of this work can be listed as follows:

  • We analyze the CSI-free multi-antenna strategies proposed in [25], e.g. and , in addition to a new scheme named All Antennas transmitting Independent Signals (

    ), but under shifted mean phase channels. We do not consider any other information related to devices such as topological deployment, battery charge; although, such information could be crucial in some setups. Our derivations are specifically relevant for scenarios where it is difficult and/or not worth obtaining such information, e.g. when powering a massive number of low-power EH devices uniformly distributed in a given area and possibly with null/limited feedback to the PB;

  • By considering the non-linearity of the EH receiver we demonstrated that those devices far from the PB and more likely to operate near their sensitivity level, benefit more from the scheme than from . However, those closer to the PB and more likely to operate near saturation, benefit more from ;

  • We attain the distribution and some main statistics of the RF energy at the EH receiver in correlated Rician fading channels under each WET scheme. Notice that the Rician fading assumption is general enough to include a class of channels, ranging from Rayleigh fading channel without LOS to a fully deterministic LOS channel, by varying the Rician factor ;

  • While and cannot take advantage of the multiple antennas to improve the average statistics of the incident RF power, the energy dispersion can be significantly reduced, hence, reducing the chances of energy outage. Meanwhile, the gains attained by in terms of average RF energy delivery depend critically on the mean phase shifts between the antenna elements. In that regard, we show the considerable performance gaps between the idealistic proposed in [25] and such scheme when considering channels with different mean phases. Even under such performance degradation still provides the greatest average harvested energy when compared to and , although its associated energy outage probability is generally the worst;

  • We attained the optimum preventive phase shifting for maximizing the average energy delivery or minimizing its dispersion for each of the schemes. We found that when transmitting the same signal simultaneously over all the antennas (, or equivalently in [25]), consecutive antennas must be phase-shifted for optimum average energy performance. Meanwhile, under other optimization criterion and/or different schemes, there is no need of carrying out any preventive phase shifting;

  • Our numerical results corroborate that correlation is beneficial under , specially under poor LOS where channels are more random. A very counter-intuitive result is that a greater LOS and/or number of antennas is not always beneficial when transmitting the same signal simultaneously through all antennas. Meanwhile, since correlation (LOS and number of antennas) is well-known to decrease (increase) the diversity, and schemes are affected by (benefited from) an increasing correlation (LOS factor and number of antennas).

Next, Section II presents the system model, while Section III presents and discusses the CSI-free WET strategies. Their performance under Rician fading is investigated in Sections IV and V, while SectionVI presents numerical results. Finally, Section VII concludes the paper.
Notation:

Boldface lowercase letters denote column vectors, while boldface uppercase letters denote matrices. For instance,

, where is the -th element of vector ; while , where is the -th row -th column element of matrix . By

we denote the identity matrix, and by

we denote a vector of ones. Superscripts and denote the transpose and conjugate transpose operations, while is the determinant, and by we denote the diagonal matrix with elements . and are the set of complex and real numbers, respectively; while is the imaginary unit. Additionally, and are the absolute and modulo operations, respectively, while denotes the euclidean norm of . and

denote expectation and variance, respectively, while

is the probability of event . and are a Gaussian real random vector and a circularly-symmetric Gaussian complex random vector, respectively, with mean vector and covariance matrix . Additionally,

denotes the PDF of random variable (RV)

, while is a non-central chi-squared RV with degrees of freedom and parameter . Then, according to [26, Eq.(2-1-125)]

the first two central moments are given by

, and . Finally, denotes the Bessel function of first kind and order [27, §10.2].

Ii System model

Consider the scenario in which a PB equipped with antennas powers wirelessly a large set of single-antenna sensor nodes located nearby. Since this work deals only with CSI-free WET schemes, and for such scenarios the characterization of one sensor is representative of the overall performance, we focus our attention to the case of a generic node . The fading channel coefficient between the th PB’s antenna and is denoted as , while is a vector with all the antennas’ channel coefficients.

Ii-a Channel model

Quasi-static channels are assumed, where the fading process is considered to be constant over the transmission of a block and independent and identically distributed (i.i.d) from block to block. Without loss of generality we set the duration of a block to 1 so the terms energy and power can be indistinctly used. Specifically, we consider channels undergoing Rician fading, which is a very general assumption that allows modeling a wide variety of channels by tuning the Rician factor [26, Ch.2], e.g. when the channel envelope is Rayleigh distributed, while when there is a fully deterministic LOS channel. Therefore,

(1)

is the normalized channel vector [28, Ch.5], where is the deterministic LOS propagation component such that is the mean phase shift of the th array element with respect to the first antenna. Additionally, accounts for an initial phase shift, while represents the scattering (Rayleigh) component. We assume a real covariance matrix for gaining in analytical tractability, which means that real and imaginary parts of are i.i.d and also with covariance [29]. Assume half-wavelength equally-spaced antenna elements, e.g. as in a uniform linear array (ULA), yielding

(2)

where is the azimuth angle relative to the boresight of the transmitting antenna array. Such angle depends on both transmit and receive local conditions, e.g. antenna orientation, node’s location, and consequently it is different for each sensor . Additionally, let use denote by the average RF power available at if the PB transmits with full power over a single antenna. Under such single-antenna setup, the available RF energy at the input of the EH circuitry is given by , where is the index of the active antenna. Notice that includes the effect of both path loss and transmit power.

Ii-B Preventive adjustment of mean phases

As mentioned earlier, and consequently , are different for each sensor, which prevents us from making any preventive phase adjustment based on an specific . However, maybe we could still use the topological information embedded in (2), which tell us that increases with for a given , to improve the statistics of the harvested energy. To explore this, let us consider that the PB applies a preventive adjustment of the signal phase at the th array element given by , while without loss of generality we set . Then, the equivalent normalized channel vector seen at certain sensor becomes , where

(3)

Now, departing from (1) we have that

(4)

where last line comes from simple algebraic operations and using the fact that since is diagonal with unit absolute values’ entries. Without loss of generality, by conveniently setting [30] so imposes the effect of LOS (constant) component on real and imaginary parts of the scattering (Rayleigh) component , we rewrite (II-B) as , where and are independently distributed as

(5)

where .

Ii-C EH transfer function

Finally, and after going through the channel, the RF energy is harvested at the receiver end. The EH circuitry is characterized by a non-decreasing function modeling the relation between the incident and harvested RF power at . In most of the works is assumed to be linear for analytical tractability, e.g. [9, 10, 11, 12, 13, 14, 16, 17, 18, 19, 20, 21, 22, 23], which implies that where is a RF-to-DC conversion efficiency constant, thus, independent of the input power. However, in practice the conversion efficiency actually depends on the input power and consequently, the relationship between the input power and the output power is nonlinear [7, 31, 24, 25]. In this work, we consider the following EH transfer function [31]

(6)

which is known to describe accurately the non-linearity of EH circuits by properly fitting parameters , while is the harvested power at saturation.

Iii CSI-free multi-antenna WET strategies

Herein we overview CSI-free multiple-antenna WET strategies for an efficient wireless powering, while discussing some related practicalities.

Iii-a All Antennas transmitting the Same Signal (AA-SS)

Under this scheme the PB transmits the same signal simultaneously with all antennas but with equal, hence proportionally reduced, power at each. We have previously proposed such scheme in [25], where it is referred as , however, herein we refer to it as to highlight explicitly its difference with respect to the scheme (see next subsection). Under this scheme, the RF signal at the receiver side and ignoring the noise is given by , where is normalized such that . Then, the energy harvested by is given by

(7)
(8)

is the available RF energy. Notice that the terms energy and power can be used indistinctly since the block duration is normalized and the PB does not change its strategy over time.

Iii-B All Antennas transmitting Independent Signals (AA-IS)

Instead of transmitting the same signal over all antennas, the PB may transmit power signals independently generated across the antennas. This is for alleviating the issue of destructive signal combination at . We refer to this scheme as AA-IS, for which the RF signal at the receiver side and ignoring the noise is given by , where each is normalized such that . Then, the harvested energy is given by

(9)
(10)

For both, and , the signal power over each antenna is of the total available transmit power. Additionally, notice that RF chains are required since all the antennas are simultaneously active. This is different from the CSI-free scheme discussed next.

Iii-C Switching Antennas (SA)

Instead of transmitting with all antennas at once, the PB may transmit the (same or different) signal with full power by one antenna at a time such that all antennas are used during a block. This is the SA scheme we proposed in [25]. In this case just one RF chain is required, hence, reducing circuit power consumption, hardware complexity and consequently the economic cost.

Assuming equal-time allocation for each antenna, the system is equivalent to that in which each sub-block duration is of the total block duration, and the total harvested energy accounts for the sum of the sub-blocks. The RF signal at the receiver side during the th sub-block, and ignoring the noise, is given by , where is normalized such that , then

(11)
(12)

is the incident RF power during the th sub-block.

Notice that for the simple, but commonly adopted in literature, linear EH model, both (9) and (11) match. However, in practice is non-linear, and consequently (9) may differ significantly from (11). We depart from (6) to write the second derivative of as

(13)

which allows us to conclude that is convex (concave) for (). Then, for certain channel vector realization and using Jensen’s inequality, we have that

(14)
Remark 1.

Above result implies that devices far from the PB and more likely to operate near their sensitivity level, benefit more from the scheme than from . However, those closer to the PB and more likely to operate near saturation, benefit more from .

In the following we analyze the statistics of the RF energy available at for harvesting under the and schemes. For such schemes, the harvested energy comes from mapping the RF energy through the EH transfer function , as shown in (7) and (9). For the mapping is much more convoluted, specially because the available RF energy varies (although possibly correlationally) within the same coherence block as illustrated in (11). However, our analysis in the previous paragraphs suggests that the statistics of and may be approximated, which is an issue we discuss numerically in detail in Section VI.

Iv RF available energy under AA-SS

Next, we characterize the distribution of the RF power at under the scheme.

Theorem 1.

Conditioned on the mean phase shits of the powering signals, the distribution of the RF power at the input of the energy harvester under the operation is given by

(15)

where ,

(16)
(17)

and is given in (2) as a function of .

Proof.

See Appendix A.

Notice that (15) matches [25, Eq.(25)] just in the specific case of un-shifted mean phases, e.g. . In such case becomes .

Iv-a On the impact of different phase means

The impact of different phase means on the system performance is strictly determined by in (15), and can be better understood by checking the main statistics, e.g. mean and variance, of the incident RF power, which can be easily obtained from (15) by using as

(18)
(19)
Fig. 1: vs (top), and vs for according to (23) (bottom). We set .

Therefore, both mean and variances increases with . Meanwhile, it is easy to check that is maximized for , for which , thus, the entire analysis carried out in [25] on this AA-SS scheme provides upper-bounds for both the mean and variance of the harvested energy. However, different phase means cause in practice a degradation on the diversity order of . Let us assume no preventive adjustment of mean phases is carried out, e.g. , to illustrate in Fig. 1a the impact of such different channel phase means. Specifically, we show for different values of . Notice that the performance diverges fast from the one claimed in [25] as moves away from .

Remark 2.

The number of minimums of matches , thus, as increases the chances of operating close to a minimum increase as well, which deteriorates significantly the system performance in terms of average incident RF power.

Iv-B Preventive adjustment of mean phases

Herein, we discuss on how to set the vector for optimizing the system performance. For clarity, and using (17) followed by some algebraic transformations, we rewrite (16) as follows

(20)

Assume uniformly distributed in 222This fits scenarios where the corresponding to each sensor is unknown, or alternatively, scenarios where there is a very large number of sensors homogeneously distributed in space such that . Although our analysis here holds specifically for such uniform angle distribution, our procedures and ideas can be extended to other scenarios., e.g. , then the problem translates to optimize over . Substituting (2) into (20) and integrating over we attain

(21)

which comes from using the integral representation of [27, Eq.(10.9.1)]. Obviously, (18) and (19) still hold but using instead of .

Iv-B1 Average energy maximization

Solving the problem of maximizing the average incident RF power is equivalent to solve . Now, since we have that

(22)

Using the fact that is positive (negative) if

is even (odd), we can easily observe that the upper bound in (

22) can be attained by setting () for even (odd) in (21), e.g.

(23)
Remark 3.

Above results means that consecutive antennas must be phase-shifted for optimum average energy performance under the scheme.

Despite optimum as in (22) cannot be simplified, an accurate approximation is

(24)

which comes from standard curve-fitting.

In Fig. 1b we show the impact of above preventive phase shifting on for different angles . By comparing Fig. 1a (no preventive phase shifting) and Fig. 1b (preventive phase shifting given in (23)), notice that the number of minimums keeps the same, the best performance occurs now for , while the worst situation happens when ; and there are considerable improvements in terms of area under the curves, which are expected to conduce to considerable improvements when averaging over .

Iv-B2 Energy dispersion minimization

As metric of dispersion we consider the variance. Therefore, herein we aim to solve ; although notice that this in turns minimize the average available RF energy as well333It is easily verifiable that the phase shifting that satisfies , minimizes the coefficient of variation parameter, which is given by , and constitutes probably a more suitable dispersion metric..

Different from the maximization problem, the problem of minimizing is not such easy to handle. We resorted to MatLab numerical solvers and realized that optimal solutions diverge significantly for different values of . However, we found that approximates extraordinarily to the optimum function value. Therefore, minimizes the variance of the incident RF power, and not preventive phase shifting is required in this case. This means that

(25)

where comes from standard curve-fitting.

Remark 4.

Results in (24) and (25) evidence that both and share a polynomial dependence on . In case of such relation is linear, while is roughly times greater than .

Iv-B3 Validation

In Fig. 2 we show as a function of for i) the phase shifting in (23), which maximizes the average incident RF power, ii) , e.g. no preventive phase shifting, which minimizes both the variance and average statistics, and iii) the scenario discussed in [25] which is constrained to . We utilized MatLab numerical optimization solvers for minimizing , but such approach was efficient just for . For greater , MatLab solvers do not always converge and are extremely time-consuming. Meanwhile, notice that indeed approaches extremely to . Approximations in (24) and (25) are accurate.

Iv-B4 On the optimization gains

By using (18), (24) and (25), the dB gain in average incident RF energy with respect to the non preventive shifting scheme, which in turns minimizes the energy dispersion according to our discussion in Subsection IV-B2, can be obtained as follows

(26)

where last line comes from the fact that the argument of the logarithm is a decreasing function of since for , and .

Fig. 2: vs .

Now, we evaluate the costs in terms of the variance increase of such average energy maximization shifting. By using (19), (24) and (25) we have that

(27)

where comes from using the lower-bound of , e.g. . Above results imply for instance that the gain in the average incident RF energy is above 3.47 dB when and , while the variance can increase dB maximum as well.

V RF available energy under AA-IS

Next, we characterize the distribution of the RF power at the receiver end under the scheme.

Theorem 2.

Conditioned on the mean phase shits of the powering signals, the approximated distribution of the RF power available as input to the energy harvester under the is

(28)

where

(29)
Proof.

See Appendix B.

For , (2) matches [25, Eq.(39)]444Notice that [25, Eq.(39)] describes the distribution of the RF incident power under un-shifted mean phases and the SA operation considering the whole block time. Therefore, both (2) and [25, Eq.(39)] must match when according to our discussions in Subsection III-C.. However, when mean phase shifts between antenna elements increase, both expressions diverge.

V-a On the impact of different mean phases

Fig. 3: vs for (top), and vs for and (bottom).

It is not completely clear from (2) whether phase shifts are advantageous or not under this scheme. Let us start by checking the average statistics of as follows

(30)

where . Notice that the larger , the greater . However, independently of the value of as shown in Fig. 3a for the case where no preventive adjustment of mean phases is carried out, e.g. .

Remark 5.

Therefore, channel mean phase shifts do not strictly bias the average harvested energy, which is intuitively expected since transmitted signals are independent to each other. Meanwhile, such result is very different from what happened under the scheme for which mean phase shifts always influenced (negatively) on such metric.

Additionally, notice that when is maximum (perfect correlation), could provide up to times more energy on average than , whose average statistics are not affected in any way by . Previous statement holds as long as there is not a strong LOS component; however, as takes greater values, which is typical of WET systems due to the short range characteristics, and become more relevant, which favors the scheme.

Let us investigate now the impact on the variance as follows

(31)

where last line comes from taking advantage of to write just as a function of . Since , it is obvious that just when , e.g. negative correlation of some antennas, the system performance benefits from having different mean phases.

V-B Preventive adjustment of mean phases

Herein, we discuss on how to set the vector for optimizing the system performance. Again, is taken randomly and uniformly from as in Subsection IV-B.

V-B1 Average energy maximization

As shown in Fig. 3a, channel mean phase shifts do not impact significantly on the average incident RF power. Then, it is intuitively expected that none preventive phase shifting would improve such average statistics. To corroborate this we require computing

for which

(32)

where is given in (21) and last line comes from integrating over by using the integral representation of [27, Eq.(10.9.1)]. Due to the extreme non-linearity of (32) we resort to exhaustive search optimization solvers of MatLab to find and compare its performance with the non preventive phase shifting scheme for which . Specifically, we show in Fig. 3b their associated performance in terms of normalized by since the average incident RF power depends strictly on such ratio. Results evidence that although the performance gap is large in relative terms, it is not in absolute values. That is, not even the optimum preventive phase shifting allows increasing significantly, e.g. .

Remark 6.

Then, and based on (V-A), the average incident RF power is approximately . Therefore, this scheme cannot take advantage of the multiple antennas to improve the average statistics of the incident RF power in any way.

V-B2 Energy dispersion minimization

As in Subsection IV-B2 we consider the variance of the incident RF power as the dispersion measure. Then, we have that

(33)

where last line comes from using directly our previous results in Subsection IV-B.

Remark 7.

Since the average incident RF energy is not affected by any phase shifting, we can conclude that (33) provides the optimum preventive phase shifting. This means that phase shifting is not required when , as in most of the practical systems.

On the other hand, observe from (31) that the variance decreases with , thus, although the is not more advantageous than single antenna transmissions in terms of the provided average RF energy, it benefits significantly from the multiple antennas to reduce the energy dispersion.

Finally, based on Subsection III-C it is expected that the phase shifting given in (33) approaches the optimum for the SA scheme as well, hence we adopt it in such scenario.

Schemes
”scheme” () [25] (unrealistic) ()
TABLE I: CSI-free schemes and corresponding preventive phase shifting

Vi Numerical Results

Herein we present simulation results on the performance of the discussed CSI-free multiple-antenna schemes under the non-linear EH model given in (6) with W ( dBm) and . We evaluate the average harvested energy, and energy outage probability, which refers to the probability that the RF energy falls below an energy threshold dBm, thus, disabling the EH process555 must be obviously not smaller than the EH sensitivity.. Notice the energy outage probability is highly related with both mean and variance statistics. The EH hardware parameters were already utilized in [25] and agree with the EH circuitry experimental data in [32].

Fig. 4: PDF of the harvested energy under and for dBm, and under for dBm.

We take uniformly and randomly from , while comparing the corresponding results to those with equal mean phases, , which are not attainable in practice and are just presented as benchmark. For clarity we summarize the schemes under consideration in Table I. Notice that in case of and we just consider the preventive phase shifting given in the first line of (33) since we assume positive spatial correlation, e.g. . Specifically, we assume exponential spatial correlation with coefficient such that , hence,

(34)

where last step comes from using a geometric series compact representation. Such model is physically reasonable since correlation decreases with increasing distance between antennas. Unless stated otherwise we set and to account for certain LOS and correlation, while we assume the PB is equipped with a moderate-to-small number of antennas .

Vi-a On the distribution of the harvested energy

In Fig. 4 we illustrate the PDF of the energy harvested under each of the schemes. Specifically, Fig. 4a shows the PDF for two different path-loss profiles, while considering and schemes. We observe that under the harvested energy under large path loss presents slightly better statistics than under , while is superior when operating under better average channel statistics. Therefore, we corroborate our statements in Remark 1: devices far from the PB indeed benefit more from than from , while those closer to the PB benefit more from the latter. We also show that the statistics improve by considering the channel mean phases, therefore, the performance of these schemes is better in a practical setup than the foreseen by [25]. Something different occurs under as shown in Fig. 4b. As we discussed in Section IV, the un-shifted mean phase assumption is the most optimistic under the operation of , and notice that the performance gains with respect to what can be attained in practice, e.g. under the and discussed in this work, are extremely notorious. Regarding vs , we can observe the performance gains in terms of average and variance of the harvested energy, respectively, of one with respect to the other. We must say that although provides less disperse harvested energy values, they could be extremely small compared to the ones achievable under