# Statistical Analysis of Multiple Antenna Strategies for Wireless Energy Transfer

Wireless Energy Transfer is emerging as a potential solution for powering small energy-efficient devices. We propose strategies that use multiple antennas at a power station, which wirelessly charges a large set of single-antenna devices. Proposed strategies operate without Channel State Information (CSI), we attain the distribution and main statistics of the harvested energy under Rician fading channels with sensitivity and saturation energy harvesting (EH) impairments. A switching antenna strategy, where single antenna with full power is transmitting at the time, guarantees the lowest variance in the harvested energy, thus providing the most predictable energy source, and it is particularly suitable for powering sensor nodes with highly sensitive EH hardware operating under non-LOS (NLOS) conditions; while other WET schemes perform alike or better in terms of the average harvested energy. Under NLOS switching antennas is better, when LOS increases transmitting simultaneously with equal power in all antennas is best. Moreover, spatial correlation is not beneficial unless the power station transmits simultaneously through all antennas, raising a trade-off between average and variance of the harvested energy since both metrics increase with the spatial correlation. Moreover, the performance gap between CSI-free and CSI-based strategies decreases quickly as the number of devices increases.

## Authors

• 9 publications
• 38 publications
• 10 publications
• 1 publication
• ### On CSI-free Multi-Antenna Schemes for Massive Wireless Energy Transfer

Wireless Energy Transfer (WET) is emerging as a potential green enabler ...
02/06/2020 ∙ by Onel L. A. López, et al. ∙ 0

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

Wireless Energy Transfer (WET) is emerging as a potential green enabler ...
02/06/2020 ∙ by Onel L. A. López, et al. ∙ 0

• ### Throughput Optimization in FDD MU-MISO Wireless Powered Communication Networks

In this paper, we consider a frequency-division duplexing (FDD) multiple...
09/28/2018 ∙ by Arman Ahmadian, et al. ∙ 0

• ### Massive Wireless Energy Transfer: Enabling Sustainable IoT Towards 6G Era

Recent advances on wireless energy transfer led to a promising solution ...
12/11/2019 ∙ by Onel L. Alcaraz Lopez, et al. ∙ 0

• ### CSI-free vs CSI-based multi-antenna WET schemes for massive low-power Internet of Things

Wireless Energy Transfer (WET) is a promising solution for powering mass...
05/29/2020 ∙ by Onel L. A. López, et al. ∙ 0

• ### MIMO with Energy Recycling

Multiple input single output (MISO) point-to-point communication system ...
02/05/2018 ∙ by Y. Ozan Basciftci, et al. ∙ 0

• ### FarSense: Pushing the Range Limit of WiFi-based Respiration Sensing with CSI Ratio of Two Antennas

The past few years have witnessed the great potential of exploiting chan...
07/09/2019 ∙ by Youwei Zeng, et al. ∙ 0

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## I Introduction

With the advent of the Internet of Things (IoT) era, there is an increasing interest in energy efficient technologies in order to prolong the battery life time of the devices. The recent trends in energy harvesting (EH) techniques provide a fundamental efficient method that avoids the replacing or recharging batteries procedures, which may be costly, inconvenient or hazardous, e.g., in toxic environments, for sensors embedded in building structures or inside the human body [1]. Many types of EH schemes, according to the energy source, have been considered, based on solar, piezoelectric, wind, hydroelectric, and wireless radio frequency (RF) signals [2]. While harvesting energy from environmental sources is dependent on the presence of the corresponding energy source, RF-EH provides key benefits in terms of being wireless, readily available in the form of transmitted energy (TV/radio broadcasters, mobile base stations and handheld radios), low cost, and having small form factor implementation.

Three main transmit scenarios can be distinguished in RF-EH networks, namely Wireless Energy Transfer (WET) [3], Wireless Powered Communication Network (WPCN) [4] and Simultaneous Wireless Information and Power Transfer (SWIPT) [5]. In the first scenario a power transmitter transfers energy to EH receivers to charge their batteries, without any information exchange, while WPCN refers to those cases where the EH receiver uses the energy harvested in a first phase to transmit its information in a second phase. Finally, in the third scenario a hybrid transmitter is transferring wireless energy and information signals using the same waveform to multiple receivers. More details on each of these scenarios can be found in [6], along with a survey on energy beamforming (EB) techniques. In this work we focus on WET scenarios, that also could be seen as an element of WPCN systems111This is because WPCN setups consist of WET, which is followed by a wireless information transfer phase., while readers can refer to [7] for a review and discussion on recent progress on SWIPT technologies.

### I-a Related Work

Many recent works have considered specifically WET and WPCN setups in different contexts and scenarios. An overview of the key networking structures and performance enhancing techniques to build an efficient WPCN is provided in [8], while authors also point out new and challenging research directions. A power beacon that constantly broadcasts wireless energy in a cellular network for RF-EH was proposed in [9]. These power beacons are deployed in conjunction with base stations to provide power coverage and signal coverage in the network, while the deployment of this hybrid network under an outage constraint on data links was designed using stochastic-geometry tools. In [10], a hybrid access point (AP) was proposed where the AP broadcasts wireless power in the downlink followed by data transmission using the harvested energy in the uplink in a time-division duplex (TDD) manner. Also in TDD setups, works in [11, 12, 13, 14, 15] consider the transmission of separately short energy and information packets (stringent delay constraints) in ultra-reliable WPCN scenarios under different channel conditions, e.g., Rayleigh or Nakagami-m fading. Authors either analyze the performance of the information transmission phase [11], or optimize it by using power [12] and rate [13] control, or cooperative schemes under perfect [14] and imperfect [15] Channel State Information (CSI). 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].

Yet, WET requires shifts in the system architecture and in its resource allocation strategies for optimizing the energy supplying, thus, avoiding energy outages. In that regard, authors in [18]

study the probability density function (PDF), the cumulative distribution function (CDF), and the average of the energy harvested from signals transmitted by multiple sources. Interestingly, such information allows to determine the best strategies when operating under different channel conditions. Additionally, multi-antenna EB, where the energy-bearing signals are weighted at the multiple transmit antennas before transmission, has been proposed very recently

[19, 20]. The average throughput performance of EB in a WPCN, consisting in one hybrid AP with multiple antennas and a single-antenna user, is investigated in [19]. The impact of various parameters, such as the AP transmit power, the energy harvesting time, and the number of antennas on the system throughput is analyzed. In [20], authors propose an EB scheme that maximizes the weighted sum of the harvested energy and the information rate in multiple-input single-output (MISO) WPCN. They show that their proposed scheme achieves the highest performance compared to existing work. In practice, the benefits of EB in WET crucially depend on the available CSI at the transmitter. An efficient channel acquisition method for a point-to-point multiple-input multiple-output (MIMO) WET system is designed in [21] by exploiting the channel reciprocity. Authors provide useful insights on when channel training should be employed to improve the transferred energy. Meanwhile, the training design problem is studied in [22] for MISO WET systems in frequency-selective channels.

### I-B Contributions and Organization of the Paper

The problem of CSI acquisition in WET systems is critical and limits the practical significance of previous works. This is because WET systems are inherently energy-limited, and part of the harvested energy would need to be used for CSI acquiring purpose [23]

. In fact, the required energy resources to that end cannot be neglected when there is a large number of antennas and/or if the estimation takes place at the EH side since it requires complex baseband signal processing. Even when previous problems could be addressed in some particular scenarios, there is still the problem of CSI acquisition in multi-user setups, specially in IoT use cases where the broadcast nature of wireless transmissions could be exploited for powering a massive number of devices simultaneously. In such cases, effective CSI-free strategies are of vital importance.

This paper addresses CSI-free WET with multiple transmit antennas, while assuming practical characteristics of EH hardware. The main contributions of this work can be listed as follows:

• We present and analyze several strategies for the use of multiple antennas at a dedicated power station that powers a set of RF-EH devices without any CSI. The performance analysis considers the harvested energy at the receiver, and comparisons with ideal CSI-based schemes are carried out;

• We attain the distribution and main statistics, e.g., mean and variance, of the harvested energy in correlated Rician fading channels under the operation of each of the WET schemes and ideal EH operation. These results are extended to more practical scenarios where sensitivity and saturation EH impairments come to play. The Rician fading assumption is general enough to include a class of channels, ranging from a fully random Rayleigh fading channel without line of sight (LOS) to a fully deterministic LOS channel, by varying the Rician factor ;

• We found that switching antennas such that only one antenna with full power is transmitting at a time, guarantees the lowest variance in the harvested energy, thus providing the most predictable energy source, and it is particularly suitable for powering sensor nodes with highly sensitive EH hardwares and operating under non LOS (NLOS) conditions; while the other schemes perform better (or at least equal) in terms of the average harvested energy.

• While under NLOS it is better switching antennas, under some LOS it is better transmitting simultaneously with equal power by all antennas. Additionally, an increase in the spatial correlation is generally prejudicial, except when transmitting simultaneously with equal power by all antennas, for which there is a trade-off between average and variance of the harvested energy since both metrics increase with the spatial correlation;

• Numerical results validate our analytical findings and demonstrate the suitability of the CSI-free over the CSI-based strategies as the number of devices increases.

Next, Section II presents the system model, while Section III introduces the WET strategies under study. Their performance under Rician fading is investigated in Section IV, while Section V presents numerical results. Finally, Section VI concludes the paper.

Notation:

Boldface lowercase letters denote 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 . By

we denote the identity matrix, and by

we denote a vector of ones. The superscript denotes the transpose, denotes the determinant, and by we denote the diagonal matrix with elements . The norm of vector is [24, Eq.(3.2.13)]. denotes the set of complex numbers and is the imaginary unit. Meanwhile, is the conjugate value of , and is the absolute operation, or cardinality of the set according to the case. and denote expectation and variance, respectively, while is the probability of event . is a Gaussian random vector with and covariance ,

is a Rician random variable (RV) with factor

[25, Ch.2], while is a gamma random variable with PDF and CDF given by

 fV(v)=(m/a)mΓ(m)vm−1e−mv/a,FV(v)=1−Γ(m,mv/a)Γ(m),v≥0, (1)

where and is the complete and incomplete gamma function, respectively. Additionally, is the non-central chi-squared RV with degrees of freedom and parameter , thus, its PDF and CDF are given by [25, Eqs.(2-1-118) and (2-1-121)]

 fZ(z)=12e−(z+ψ)/2(zψ)φ/4−1/2Iφ/2−1(√ψz), FZ(z)=1−Qφ/2(√ψ,√z), z≥0, (2)

where is the -th order modified Bessel function of the first kind [25, Eq.(2-1-120)] and is the Marcum Q-function [25, Eq.(2-1-122)]. According to [25, Eq.(2-1-125)] we have that

 E[Z]=φ+ψ,VAR[Z]=2(φ+2ψ). (3)

Table I summarizes the main symbols used throughout this paper.

## Ii System Model

Consider the scenario in Fig. 1, in which a dedicated power station equipped with antennas, powers wirelessly a set of single-antenna sensor nodes located nearby. 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. The fading channel coefficient between the -th antenna of and the th sensor node is denoted as , while is a vector with the channel coefficients from the power station antennas to .

In general, during WET may transmit with up to energy beams to broadcast energy to all sensors in . Then, the received signal at is given by

 y(t)j=√ϱjhTjl∑k=1wkx(t)k+n(t)j, (4)

where is the block index, denotes the precoding vector for generating the th energy beam, and is its energy-carrying signal. It is assumed that ’s are i.i.d RVs with zero mean and unit variance. Without loss of generality we set , while accounts for the path loss of the link times the overall transmit power of . Finally, is the Additive White Gaussian Noise (AWGN) at . Then, by considering negligible the noise energy, the incident RF power at the th EH receiver is given by

 ξrfj=ϱjl∑k=1∣∣hTjwk∣∣2. (5)

Now, the harvested energy222We use the terms energy and power indistinctly, which can be interpreted as if block duration is normalized., , can be written as a function of as where is a non-decreasing function of its argument. In general is nonlinear and analytical analyses are cumbersome, but starting from the linear model the accuracy can be significantly improved by considering tree main factors that limit strongly the performance of a WET receiver [26, 27, 28, 12]: (i) its sensitivity , which is the minimum RF input power required for energy harvesting; (ii) its saturation level , which is the RF input power for which the diode starts working in the breakdown region, and from that point onwards the output DC power keeps practically constant; and (iii) the energy efficiency in the interval , which we assume as constant. Therefore, we can write as

 ξj=g(ξrfj)=⎧⎪ ⎪ ⎪⎨⎪ ⎪ ⎪⎩0,ξrfj<ϖ1ηξrfj,ϖ1≤ξrfj<ϖ2ηϖ2,ξrfj≥ϖ2. (6)

Notice that the linear model assumed in most of the literature does not take into account the sensitivity and saturation phenomena, which is equivalent to operate with and . Thus, taking these impairments into consideration in the scenarios under discussion is an important contribution from a practical perspective.

## Iii WET Strategies

First, in Subsection III-A we characterize the performance of WET for three different strategies at without any CSI, while two alternative strategies that require full CSI are presented as benchmarks in Subsection III-B.

### Iii-a WET Strategies without CSI

Since no CSI is available and cannot depend on the channel coefficients, does not form energy beams to reach efficiently each of the . Therefore, for these kind of strategies it is only necessary focusing on the performance of an arbitrary user , while also setting .

#### Iii-A1 One Antenna (OA)

Is the simplest strategy because only one out of antennas is used for powering the devices, thus, transmitting with full power.Then, using (5) we obtain

 ξjOA=g(ϱj|hi,j|2), (7)

where . Notice that in this case is a vector of zeros with entry in the th element. There is no difference whether is equipped with only one or several antennas when operating with the strategy.

#### Iii-A2 All Antennas at Once (AA)

The strategy does not exploit multiple antennas, thus, it does not take advantage of that degree of freedom. One obvious and simple alternative is transmitting with all antennas but with reduced power at each, , thus,333Notice that any other allocation of transmit powers is not advisable since does not know how the channels are behaving.

 ξjAA=g(ϱjM∣∣M∑i=1hi,j∣∣2). (8)

The and schemes are the extreme cases of a more general strategy where out of antennas are selected to power the sensors. As a consequence, the -out-’s performance is limited by that of the and strategies.

#### Iii-A3 Switching Antennas (SA)

Instead of transmitting with all antennas at once, may transmit with full power by one antenna at a time such that all antennas are used during a block. 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 that of the sub-blocks. That is

 ξjSA=1MM∑i=1g(ϱj|hi,j|2). (9)

Note that in each sub-block is given as in the strategy, but the chosen is different in each sub-block.

### Iii-B Benchmark WET Strategies

For the sake of describing some benchmark strategies, herein we consider that full CSI is available at . Sensors could use a “small” amount of energy444Coming from a short phase where are first powered by one of the previous CSI-free strategies, or even from some residual energy after previous rounds. to send some pilot symbols to in order to acquire the CSI. This requires reciprocal channels, thus, should be listening at the time of the transmission. Otherwise, has to send the pilots and wait for a feedback from the sensor(s) informing the CSI. Whatever the case, it seems unsuitable in a setup where multiple energy constrained sensors require the powering service from , specially if we also consider the multiple access problem. Therefore, we present the following CSI-based strategies only as benchmarks for those presented in the previous subsection. We assume that knows also the EH hardware characteristics, e.g., of the EH sensors and the goal is to maximize the overall energy harvested by .

#### Iii-B1 Best Antenna (OA−CSI)

This strategy is the counterpart for the previous scheme when full CSI is available. In this case, the antenna that provides the greatest amount of overall harvested energy is selected out of the overall set. Therefore, and

 |S|∑j=1ξjOA−CSI=maxi=1,...,M|S|∑j=1g(ϱj|hi,j|2), (10)

since this time is a vector of zeros with entry in the selected antenna index. Of course, in a multi-user system where the nodes benefit from the WET phase simultaneously, the best antenna is usually not the same for all users. Different from (10), another possible implementation may be that in which some users are optimized first and then the others, but in all these scenarios the complexity scales quickly with the number of users while reducing the overall performance.

#### Iii-B2 Best Transmit Beamforming (AA−CSI)

This strategy is the counterpart for the previous scheme when full CSI is available. In this case, instead of transmitting with the same power over each antenna, precompensates through for the channel and EH hardware effects before transmission such that the overall harvested energy at is maximized. Hence,

 |S|∑j=1ξjAA−CSI=max{wk}|S|∑j=1g(ϱjM∑k=1∣∣hTjwk∣∣2), (11)

where is set to since that is the maximum possible number of energy beams. In case the system performance maximizes with an smaller , which may be the case when , then, some of the optimum beamformers are all-zeros vectors. Notice that the scheme is less sensitive to CSI imperfections than the . This is because the former only relies on the power gain of the channel, while the latter requires the full characterization, envelope and phase, of the channel coefficients.

### Iii-C Comparison of the WET Strategies for |S|=1 and under Ideal EH Linear Model

Some useful insights come from setting and using the ideal EH linear model such that and . In this case, we denote as the harvested energy at the unique sensor node while we avoid using the subindex , then, (7), (8), (9), (10) and (11) can be rewritten as

 ξ0OA =ηϱ|hi|2, (12) ξ0AA =ηϱM∣∣M∑i=1hi∣∣2, (13) ξ0SA =ηϱMM∑i=1|hi|2, (14) ξ0OA−CSI =ηϱmaxi=1,⋯,M|hi|2, (15) ξ0AA−CSI =ηϱmaxw1|hTw1|2(a)=ηϱM∑i=1|hi|2, (16)

where comes from setting which is the optimum precoding vector for Maximum Ratio Transmission (MRT) [29] in a MISO system.

###### Theorem 1.

The following relations are satisfied:

 ξ0OA ≤ξ0OA−CSI≤ξ0AA−CSI=Mξ0SA, (17) ξ0AA ≤ξ0AA−CSI=Mξ0SA. (18)
###### Proof.

According to (12), (15), (16) and (14) we have that

 ξ0OA =ηϱ|hi|2≤ηϱmaxi=1,...,M|hi|2=ξ0OA−CSI(a)≤ηϱM∑i=1∣∣hi∣∣2=ξ0AA−CSI=Mξ0SA, (19)

where the equality in is only when . Thus, (17) is satisfied. Now, for the second part of the proof we proceed from (13), (16) and (14) as follows.

 ξ0AAηϱ (20)

where comes from applying the triangular inequality and generalizing as shown in [30, Section 1.1.7], follows from using each time the norm notation, while from the inequality between the arithmetic and quadratic mean. ∎

## Iv Analysis under Rician Fading

Herein we assume that channels undergo Rician fading, which is a very general assumption that allows modeling a wide variety of channels by tuning the Rician factor , e.g., when the channel envelope is Rayleigh distributed, while when there is a fully deterministic LOS channel. Since this work deals mainly with CSI-free WET schemes, and for such scenarios the characterization of one sensor’s performance is similar to that of the others, we focus our attention to the performance of a generic node, thus we avoid using subindex . Additionally, the performance gap between the CSI-free and CSI-based WET schemes is maximum for and analyzing such scenario when CSI is available allows getting analytical expressions along with some useful insights. Previous assumptions imply that the envelope distribution of is Rician distributed with factor , e.g., . Specifically, the channels are Gaussian with independent real and imaginary parts, with , where and are respectively the covariance matrix and mean vector of [31].

For performance evaluation it is enough considering , e.g., equal mean over all the fading paths, and uniformly spatial correlated fading, such that the antenna elements are correlated555Spatial correlation occurs due to insufficient spacing between antenna elements, small angle spread, existence of few dominant scatterers, and the antenna geometry. The general concept of spatial correlation is usually linked only to the specific positive spatial correlation, even when the negative correlation is also physically possible, mainly due to the use of decoupling networks and antenna geometry effects. On the other hand, notice that correlation between the coefficients is highly probable in the kind of systems we are investigating here because of the short range transmissions [32]. Even though, by setting we are also able of modeling completely independent fading realizations over all the antennas. between each other with coefficient . Thus,

 R=σ2⎡⎢⎣1ρ⋯ρρ1⋯ρ⋮⋮⋱⋮ρρ⋯1⎤⎥⎦M×M, (21)

where is the variance of each . In order to guarantee that is positive definite and consequently a viable covariance matrix, is lower bounded by [33], thus . Additionally, factor is connected to and as

 κ =μ22σ2, (22)

and normalizing the channel power gain as [31], e.g., , we attain

 σ2 =12(1+κ),μ2=κ1+κ. (23)

### Iv-a Distribution of the Harvested Energy under Ideal EH Linear Model

Now, we proceed to characterize the distribution of the harvested energy when using each of the WET strategies analyzed in the previous section under the ideal EH linear model.

#### Iv-A1 OA

Obviously, the spatial correlation has no impact on the performance of the scheme since under that scheme selects only one antenna without using any information related with the other antennas. From (12) we proceed as follows

 ξ0OA =ηϱ|hi|2=ηϱ(α2+β2)(a)=ηϱσ2(^α2+^β2)(b)∼ηϱσ2χ2(2,μ2σ2)(c)∼ηϱ2(1+κ)χ2(2,2κ), (24)

where comes from normalizing the variance of and such that , comes from the direct definition of a non-central chi-squared RV [25, Ch.2], and follows after using (22) and (23).

#### Iv-A2 Aa

Now we focus on the performance of the scheme. Based on (13) we have that

 ξ0AA (d)∼1+(M−1)ρ2(1+κ)ηϱχ2(2,2Mκ1+(M−1)ρ), (25)

where comes from using and and notice that are still Gaussian RVs [34] with mean and variance according to (21), thus, follows after variance normalization such that . Finally, comes from using the definition of a non-central chi-squared RV [25, Ch.2], while from using (22) and (23).

#### Iv-A3 Sa

Let us proceed with the analysis of the scheme. From (14), we write as

 ξ0SA =ηϱMM∑i=1(α2i+β2i)=ηϱM(αTα+βTβ). (26)

Since and are i.i.d between each other we focus on the product and the results are also valid for . Let us define which is distributed as , then

 α =R1/2v+1√2μ αTα (27)

where last step comes from simple algebraic transformations. Notice that

 R=σ2BΛBT, (28)

which is the spectral decomposition of [35, Ch.21]. In (28), is a diagonal matrix containing the eigenvalues of and is a matrix whose column vectors are the orthogonalized eigenvectors of . In order to find the eigenvalues, s, of , we require solving for , for which we proceed as follows

 det(σ−2R−λI) (a)=(1−λ−ρ)M⎛⎜ ⎜ ⎜ ⎜⎝1+[ρ,ρ,⋯,ρ]⎡⎢ ⎢ ⎢ ⎢⎣11−λ−ρ11−λ−ρ⋮11−λ−ρ⎤⎥ ⎥ ⎥ ⎥⎦⎞⎟ ⎟ ⎟ ⎟⎠(b)=(1−λ−ρ)M−1(1−λ+ρ(M−1)), (29)

where comes from using the Matrix determinant lemma [35], while follows after some algebraic manipulations. Now, two different eigenvalues are easily obtained by matching (29) with . These are with multiplicity and with multiplicity , thus

 Λ=Diag[1−ρ,⋯,1−ρ,1+(M−1)ρ]. (30)

Meanwhile, the corresponding eigenvectors, , satisfy , thus,

• for we have

 ⎡⎢⎣1ρ⋯ρρ1⋯ρ⋮⋮⋱ρρρ⋯1⎤⎥⎦[e1e2⋮eM] =(1−ρ)[e1e2⋮eM] ⟶ ⎡⎢ ⎢⎣e1+ρ∑Mi=2eiρe1+e2+ρ∑Mi=3ei⋮ρ∑M−1i=1ei+eM⎤⎥ ⎥⎦ =⎡⎢⎣(1−ρ)e1(1−ρ)e2⋮(1−ρ)eM⎤⎥⎦ ⎡⎢ ⎢⎣ρ∑Mi=1eiρ∑Mi=1ei⋮ρ∑Mi=1ei⎤⎥ ⎥⎦ =[00⋮0] ⟶ M∑i=1ei=0, (31)
• and for we have

 ⎡⎢⎣1ρ⋯ρρ1⋯ρ⋮⋮⋱ρρρ⋯1⎤⎥⎦[e1e2⋮eM] =(1+ρ(M−1))[e1e2⋮eM] ⟶ ⎡⎢ ⎢⎣e1+ρ∑Mi=2eiρe1+e2+ρ∑Mi=3ei⋮ρ∑M−1i=1ei+eM⎤⎥ ⎥⎦=⎡⎢ ⎢ ⎢ ⎢⎣(1+ρ(M−1))e1(1+ρ(M−1))e2⋮(1+ρ(M−1))eM⎤⎥ ⎥ ⎥ ⎥⎦ ⎡⎢ ⎢⎣ρ∑Mi=1eiρ∑Mi=1ei⋮ρ∑Mi=1ei⎤⎥ ⎥⎦ =⎡⎣ρMe1ρMe2⋮ρMeM⎤⎦ ⟶ e1=e2=⋯=eM. (32)

Thus, the eigenvectors associated to satisfy (31), while the eigenvector associated to satisfies (32

). After orthogonalization by using the Gram-Schmidt process

[36], and normalization, the resulting vectors still satisfy either (31) or (32) according to the case. For the latter, the resulting vector is , therefore

 (33)

where for .

Now, substituting (28) into (27) yields

 αTα =(v+(σ2BΛBT)−121√2μ)Tσ2BΛBT(v+(σ2BΛBT)−121√2μ) (a)=(BTv+σ−1Λ−12BT1√2μ)Tσ2Λ(BTv+σ−1Λ−12BT1√2μ) (34)

where comes after some algebraic transformations, follows from taking and , for which using (33) and (30) yields

 d=[0,0,⋯,0,M1+(M−1)ρ]T, (35)

and finally comes from setting . Notice that is the -th eigenvalue of . Using (34) into (26) yields

 ξ0SA (a)=ηϱσ2MM∑i=1λii[(ςi+di1√2μσ)2+(ωi+di1√2μσ)2] (b)=ηϱσ2M[(1−ρ)2(M−1)∑i=1ς2i+(1+(M−1)ρ)2∑i=1(ωi+√M2(1+(M−1)ρ)μσ)2], (36)

where comes from defining to use when evaluating the term in (26), which has the same form given in (34), while follows from using (35). In we also regrouped similar terms, which allows writing after using the direct definition of a non-central chi-square distribution [25, Ch.2] along with (22) and (23).

###### Remark 1.

Therefore, under uniform spatial correlation, is distributed as a linear combination of a chi-square RV and a non-central chi-square RV, with and degrees of freedom, respectively. Unfortunately, it seems intractable finding a closed-form expression for the distribution of , except for

• , for which , which can be easily verified by using the direct definition of a non-central chi-squared RV;

• , for which . The latter term results from the fact that when the PDF of is 1 at , and otherwise.

• , for which .

###### Remark 2.

For full positive correlation, e.g., , the performance of the scheme matches that of the . This is an expected result since even by switching antennas the energy harvested at keeps the same.

#### Iv-A4 OA−CSI

According to (15) finding the distribution of is equivalent to the problem of finding the distribution of the Signal-to-Noise Ratio in a correlated Rician single-input multiple-output (SIMO) channel scenario, where the receiver with antennas uses Selection Combining (SC). Thus, we can directly use [37, Eq.(21)] to state

 Fξ0OA−CSI(ηϱx) =e−κρ∞∫0[1−Q(√2ρt1−ρ,√2(1+κ)x1−ρ)]Me−tI0(2√κtρ)dt, (37) fξ0OA−CSI(x) =ddxFξ0OA−CSI(x)=M(1+κ)ηϱ(1−ρ)e−κρ−1+κηϱ(1−ρ)x∞∫0e−t1−ρI0(2√κtρ)× (38)

For two specific correlation setups it is even possible to get simplified expressions as follows

• For the channel coefficients are i.i.d, thus,

 Fξ0OA−CSI(ηϱx) =P[ξ0OA−CSI<ηϱx]=P[maxi=1,...,M|hi|2

where in we take advantage of the i.i.d property of the channel realizations when and comes from making where . Now,

 fξ0OA−CSI(x) =ddxFξ0OA−CSI(x)=ddxFZ(2(1+κ)x/(ηϱ))M (a)=2(1+κ)MηϱFZ(2(1+κ)ηϱx)M−1fZ(2(1+κ)ηϱx), (40)

where

comes from using the chain rule. Notice that both,

and are given in (2) with and .

• For the performance under the scheme matches that of the strategy since the fading behaves instantaneously equal over all the antennas.

Finally, notice that when is the minimum, , the performance gain of this scheme over that of the strategy should be the maximum.

#### Iv-A5 AA−CSI

For the scheme the characterization is much easier since according to (20), thus, by using (36) we attain

 ξ0AA−CSI∼ ηϱ2(1+κ)[(1−ρ)χ2(2(M−1),0)+(1+(M−1)ρ)χ2(2,2κM1+(M−1)ρ)]. (41)

Notice that the analysis in Remark 1 can be extended to this case straightforwardly.

#### Iv-A6 Comparisons and Remarks

Since the distribution of the harvested energy is related in all the cases with the non-central chi-squared distribution except for the scheme, we are able to find the mean and variance statistics according to (3). When we can also provide an upper bound for the mean of by using [38, The. 2.1] such that

 E[ξ0OA−CSI]ηϱ ≤1MM∑i=1E[|hi|2]+√M−1M ⎷M∑i=1VAR[|hi|2]+M∑i=1E[|hi|