Optimal Channel Estimation for Hybrid Energy Beamforming under Phase Shifter Impairments

02/22/2019 ∙ by Deepak Mishra, et al. ∙ Linköping University 0

Smart multiantenna wireless power transmission can enable perpetual operation of energy harvesting (EH) nodes in the internet-of-things. Moreover, to overcome the increased hardware cost and space constraints associated with having large antenna arrays at the radio frequency (RF) energy source, the hybrid energy beamforming (EBF) architecture with single RF chain can be adopted. Using the recently proposed hybrid EBF architecture modeling the practical analog phase shifter impairments (API), we derive the optimal least-squares estimator for the energy source to EH user channel. Next, the average harvested power at the user is derived while considering the nonlinear RF EH model and a tight analytical approximation for it is also presented by exploring the practical limits on the API. Using these developments, the jointly global optimal transmit power and time allocation for channel estimation (CE) and EBF phases, that maximizes the average energy stored at the EH user is derived in closed form. Numerical results validate the proposed analysis and present nontrivial design insights on the impact of API and CE errors on the achievable EBF performance. It is shown that the optimized hybrid EBF protocol with joint resource allocation yields an average performance improvement of 37% over benchmark fixed allocation scheme.

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I Introduction

Using large antenna array at the radio frequency (RF) source can enable perpetual operation of energy harvesting (EH) devices in internet of things (IoT) [1] by compensating propagation losses through energy beamforming (EBF), or enhancing information capacity via multi-stream transmission [2]. Despite these potential merits, there are two practical fundamental bottlenecks: (1) larger physical size of the antenna arrays at usable RF frequencies [3], and (2) increased signal processing complexity because the digital precoding has to be applied over hundreds of antenna elements [4]. Since, the multiantenna digital precoding is carried out at the baseband, each antenna element requires its own RF chain for the analog-to-digital conversion, and subsequent baseband-to-RF up conversion, or vice-versa. This usage of one RF chain per antenna is very inefficient, both from hardware monetary cost and energy consumption perspectives [2, 3, 4]. This has led to a growing research interest [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13] in the hybrid beamforming architectures, where all or part of the processing is based on analog beamforming which enables a substantially reduced number of RF chains in comparison to the antenna count.

I-a State-of-the-Art

We recall that accurate channel state information (CSI) is needed at the multiantenna energy source to maximize the array gains for meeting sustainable operation demand of EH IoT devices [14]. Different channel estimation (CE) schemes based on minimizing the least-squares (LS) error or linear minimum-mean square-error (MMSE) have been investigated in the literature for exploiting the fully digital energy beamforming gains [15, 16, 14]. Keeping in mind the constraints of RF EH users, various limited feedback based CE protocols [17] and resource optimization techniques [18] have also been recently studied. Further, the efficacy of received signal strength indicator (RSSI) feedback values based CE protocols has been lately investigated in [19, 20]. However, these fully-digital EBF works [14, 15, 16, 17, 19, 20, 18] adopted an overly simplified linear rectification model for their investigation, which has been recently [21, 22, 23, 24, 25, 26] shown to perform poorly for the practical RF EH circuits. The detailed investigation on RF EH performance using the statistical CSI as conducted in [25, 26] suggested that for an accurate characterization, a nonlinear EH model should be adopted during investigations.

In contrast to the multi-stream information transfer (IT) using multiantenna source, efficient RF energy transfer (RFET) involves the dynamic adjustment of the beams from different antenna elements to focus most of the radiated RF power in the direction of an intended EH user. Also, it has been proved mathematically in [7, 8] that by using two digitally controlled phase shifters (DCPS) for each antenna element the corresponding analog EBF with single RF chain can achieve exactly the same array gains as that of a fully digital system with each antenna element having its own RF chain. Different from the digital beamforming works, a highly accurate CE process for implementing hybrid EBF is more challenging to realize because here the effective channel is the product of the random fading gain and analog beam selected [3, 4, 5, 6, 7, 8, 9, 10, 13, 12, 11]. An adaptive compressed sensing (CS) based CE algorithm was proposed in [9] for a hybrid analog-digital multiple-input-multiple-output (MIMO) system. Considering the multi-user hybrid beamforming system, a minimum mean-square error (MMSE) approach was developed in [5] to estimate the effective channel. Joint least-squares (LS) based CE and analog beam selection algorithm was proposed in [10] for an uplink (UL) multiuser hybrid beamforming system. In contrast to these narrow band systems facing flat fading, CE algorithms for a single user multi-carrier hybrid MIMO system was investigated in [6] using both LS and CS approaches. More recently, a new CE approach for hybrid architecture-based wideband millimeter wave systems was proposed in [11] using the sparse nature of frequency-selective channels. However, as obtaining full-dimensional instantaneous CSI is difficult due to much lesser RF chains than the antenna elements, a low-complexity hybrid precoding approach was investigated in [12] that involves beam searching in the downlink (DL) and the analog precoder codeword index feedback in the UL. To alleviate the high hardware cost in complicated signaling procedure, a single-stage feedback scheme exploiting the second-order channel statistics for designing the digital precoder and using the feedback only for the analog beamforming was proposed in [13]. Here, it is worth noting that these works [3, 4, 5, 6, 7, 8, 9, 10, 13, 12, 11] focusing on multi-stream IT for efficient spatial multiplexing using limited RF chains at the source or user, did not investigate the joint optimal CE protocol and resource allocation to maximize the EBF gains under hardware impairments.

I-B Motivation, Novelty, and Scope

Analog EBF can address the hardware cost and space constraints in practically realizing efficient RFET from a large antenna array [4]. However, due to the usage of low-cost hardware and low-quality RF components for the ubiquitous deployment of EH devices in IoT and for making large antenna array systems economically viable, the performance of these energy sustainable systems is more prone to the RF imperfections caused by practical phase shifters (PS) and lossy combiners [27, 28, 29, 30]. This may result in a significant EH performance degradation due to the underlying practical analog phase-shifter impairments (API). Recently, an API model was introduced in [31] for investigating the efficacy of MMSE-based CE for hybrid EBF over Rayleigh channels, while assuming a linear EH model. However, this linear rectification model is only suitable when the received signal power levels are very low [23, 25]. In contrast, here we aim at investigating the degradation in the hybrid EBF gains as compared to a fully digital architecture [15, 16] for the practical multiple-input-single-output (MISO) RFET [14] over Rician fading channels [32, Ch 2.2], while adopting a more refined EH model. Rician fading is important as it incorporates the strong line-of-sight (LoS) components over RFET links [26]. Though the hybrid architecture can help in realizing significant monetary cost and energy consumption reduction due to the usage of a single RF chain, it is prone to hardware imperfections, such as phase offset errors between different DCPS pairs along with differences in their amplitude gains. However, these performance losses due to practical API, whose affect is characterized in this work, can be overcome by considering the proposed jointly optimal time and power allocation for the CE and RFET sub-phases. Further, noting the energy constraints of an EH user, we present a green transmission protocol involving optimal LS-based CE, which does not require any prior knowledge on channel statistics.

To our best knowledge, the joint impact API, CE errors, and nonlinear rectification efficiency on the optimized average stored energy at single antenna EH user due to hybrid EBF during MISO RFET over Rician channels has not been investigated yet. Moreover, the existing works [14, 15, 16, 17, 19, 20, 18] on resource allocation for optimizing the digital EBF performance under CE errors, considered an overly simplified linear RF EH model and presented either suboptimal or numerical solutions. In contrast, we focus on obtaining analytical insights on the joint design to optimally allocate resources between CE and RFET phases. The major challenge is to obtain closed-form solution for the nonconvex stored energy maximization problem.

The scope of this work involves the characterization of practical efficacy of hybrid EBF having a common single RF chain for a large array of antenna elements. The nontrivial outcomes and observations of this work for a single user DL wireless RFET scenario can be extended to multiuser simultaneous wireless information and power transfer (SWIPT) applications [1, 26]. Also, the adopted API model and optimal LS-based CE protocol for EBF with a single RF chain at multiantenna source can be extended to address the demands of hybrid EBF architectures with multiple RF chains. Further, the closed-form expressions for the joint design shed key insights on an efficient utilization of the available resources for maximizing the achievable array gains. Lastly, with the latest developments of the low-power circuits capable of harvesting power from millimeter wave energy signals [33], the proposed optimal CE and hybrid EBF designs can be used for sustainable high frequency indoor applications with much less bulkier power beacon.

I-C Key Contributions and Notations

The key contribution of this work is five fold.

  • We present a novel joint CE and energy optimization framework for maximizing the hybrid EBF efficiency during the RFET over Rician channels while incorporating API at the multiantenna source and nonlinear rectification operation of RF EH unit at the user. The considered system model and the hybrid EBF architecture are presented in Section II.

  • Global optimal LS estimator (LSE) for the effective channel, involving the product of channel vector and analog EBF design, is obtained in Section 

    III while considering the impact of API. The key statistics for the LSE, involving analog and digital channel estimators, are derived along with their respective practically-motivated tight analytical approximations.

  • Using this API-affected LSE, the average received RF power analysis is carried out in Section IV while considering the nonlinear rectification operation in the practical RF EH circuits. Tight closed-form approximations for the average harvested power and the stored energy at the EH user after replenishing the consumption in CE are also derived in Section V for the MISO RFET over the Rician fading channels, both with and without CE errors.

  • Green transmission protocol involving the joint optimal time and power allocation (PA) for CE and RFET sub-phases to maximize the stored energy at the EH user is investigated in Section VI. Apart from proving the global-optimality of the joint design, tight closed-form approximations are also derived for them to gain additional optimal system-design insights.

  • Numerical results presented in Section VII validate the proposed analysis and provide key insights on the optimal hybrid EBF protocol. The optimized performance variation with critical system parameters is conducted to quantify the achievable gains with respect to perfect CSI based and isotropic transmissions. The relative performance gain achieved by both joint and individual PA and time allocation (TA) schemes is also characterized.

Summarizing the notations used in this work, the vectors and matrices are denoted by boldface lowercase and boldface capital letters, respectively. , , , and respectively denote the Hermitian transpose, transpose, conjugate, and inverse of matrix . , , and respectively represent the zero vector, vector with all entries as one, and identity matrices. With being the trace of matrix and denoting its th element, and respectively denote the th diagonal entry of the diagonal matrix and th entry of the vector . and

respectively represent the Euclidean norm of a complex matrix and the absolute value of a complex scalar. The expectation, covariance, and variance operators have been respectively denoted using

, , and . With real and imaginary components of complex quantity defined using and , denotes the angle of . Lastly, with and denoting complex number set,

represents circularly symmetric complex Gaussian distribution with mean vector

and covariance matrix .

Fig. 1: Adopted API model for the DL hybrid EBF from using the proposed UL CE via pilot signal transmission by .

Ii System Description

We first present the system model details along with the adopted channel model and hybrid EBF architecture. Later, we also discuss the nonlinear RF EH model used in this paper.

Ii-a Nodes Architecture and MISO Channel Model

We consider a MISO wireless RFET from an antenna array based RF energy source to a single antenna RF EH user . The detailed system model diagram is presented in Fig. 1. The dedicated energy source consists of a single RF chain which is shared among the antenna elements. On other end, EH user can be a low-power sensor or IoT device programmed for performing an application-specific operation using its own micro-controller (). We assume is solely powered by the energy stored in the EH unit, being replenished via RFET from .

With , we assume flat quasi-static Rician block fading [32, Ch 2.2] where the channel impulse response for each communication link remains invariant during a coherence interval of seconds (s) and varies independently across different coherence blocks. The -to- channel is represented by an complex vector , where is a deterministic complex vector containing the LoS and specular components of the Rician channel vector , models the large-scale fading between and which includes both the distance-dependent path loss and shadowing, is the Rician factor denoting the power ratio between the deterministic and scattered components of the -to- channel. On the other hand, is a complex Gaussian random vector, with independent and identically distributed zero-mean unit-variance entries, representing the scattered components of the -to- channel. So, , where and . Here, and respectively represent the gain of th antenna at and its phase shift with respect to the reference antenna, while is the angle of arrival/departure of the specular component at from . With representing the inter-antenna separation at , .

Ii-B Practical Hybrid EBF Architecture under API

The key idea behind hybrid EBF implementation stems from the result in [7, Theorem 1], where any complex number can be alternately represented using a DCPS pair as

(1)

As shown in Fig. 1, each antenna element has two DCPSs and one combiner, which in practice suffer from amplitude and phase errors [28, 29, 30], that adversely effect the performance of the hybrid EBF. Actually, the latter involves usage of adaptive arrays comprising multiple antenna elements whose respective beam pattern is shaped by controlling the amplitudes and phases of the RF signals transmitted or received by them. A precise control over both amplitudes and phases is essential to achieve the desired performance. However, several practical constraints like finite resolution PSs, noise, mismatch in PS circuit elements, and channel uncertainty limit the practically achievable precision [28, 29, 30]. These errors cause an imbalance in the DCPS pair for each antenna element. Some of these error sources are random, and some are fixed which depend on the manufacturing errors and long-term aging effects. These API, which are unpredictable and time varying [28, 29, 30]

, can be modeled using random variables with the manufacturing or aging dependent error deciding their means and the random noise based error controlling the variance. Adopting a recently introduced API model 

[31] for characterizing amplitude and phase errors in practical DCPSs and combiners implementation, the ideal signal in (1), gets altered to

(2)

where and respectively represent the amplitude errors due to the API in the first and second DCPS in a pair. Likewise, and respectively represent the corresponding phase errors. For the ideal case with no API, and , which reduces in (2) to in (1

) . We assume that these random errors in amplitude and phase for each DCPS pair and combiner are independently and uniformly distributed across the different antennas.

Hence, with positive constants and representing the errors due to the fixed sources, the amplitude and phase errors, representing the API in practical hybrid architectures, can be respectively modeled as and defined below

(3)

where, and

, representing the errors due to random sources, follow the uniform distribution with the respective probability density functions being

and . Here, the phase errors are expressed in radians. Uniform distribution is employed because it is commonly adopted for modeling RF imperfections [34, 28] like PS and oscillator impairments leading to amplitude losses and phase errors due to the usage of low-cost hardware attributing to limited accuracy, and getting influenced by temperature variation and aging effects. Further, under the assumption , which can be easily implemented via the digital beamforming design [7], in (2) can be alternatively written as

(4)

We will be using this definition for modeling API in the analog estimator and precoder designs.

Ii-C Adopted Nonlinear RF Energy Harvesting Model

For the practical RF EH circuits, the harvested direct current (DC) power is a nonlinear function of the received RF power  [21, 23, 25, 24, 22] at the input of the RF EH unit performing the RF-to-DC rectification operation. Specifically, depends on the rectification efficiency, which itself is a function of . Recently [23, 26], a piecewise linear approximation (PWLA) was proposed for establishing the relationship between and using the function . Mathematically, considering linear pieces, is defined as

(5)

Here, W are thresholds on defining the boundaries for the linear pieces with slope and intercept W, and constant is the saturated harvested power for In practice for some harvesters, like the Powercast P1110 evaluation board [21], there is a limit on the maximum permissible received RF power (dBm) at the input of EH unit to avoid any damage to the underlying circuit components. Hence, for is sometimes not defined [23, eq. (6)].

Fig. 2: Verifying the quality of PWLA based nonlinear model [23] for the RF-EH circuit designed in [22] for efficient far-field RFET. Other two commonly adopted EH models (sigmoidal [24] and linear [25]) are also plotted.

Using (5), the PWLA for harvested versus received power (HRP) characteristic of the far-field RF EH circuit designed for efficient low-power and long-range RFET can be obtained with W as six received threshold powers dividing the HRP characteristic of RF EH circuit designed in [22] into linear pieces having slope and intercept W. As the HRP characteristics are defined only for the input power levels between dBm to dBm [22], we set mW to generate harvested power results for dBm mW.

In this work we have used this RF EH circuit for investigating the optimized hybrid EBF performance under joint API and CE errors. The goodness of the proposed PWLA model for this practical RF EH circuit [22] as shown via the log-log plot in Fig. 2, is verified by very low norm of residuals of and root mean square error (RMSE) of . In Fig. 2 we have also plotted the recently proposed sigmoidal (logistic) approximation [24] for as a function of along with the widely adopted linear fit111Generally, there are three types of linear EH models [25]: (a) linear, (b) constant-linear (CL), and (c) constant-linear-constant (CLC). In Fig. 2, we plotted the linear model which is most commonly adopted due to its analytical simplicity [14, 15, 16, 17, 19, 20, 18]. Whereas, we have considered a more generic PWLA model [23] in place of CL (having ) and CLC (i.e., ) models.. Results show that PWLA provides a much simpler and tighter fit. Therefore, we use this PWLA in (5) for analyzing the RF EH operation.

Iii Least-Squares Hybrid Channel Estimation

In this work, we refer the -to- channel as the DL and the -to- link as UL. In contrast to frequency-division duplex (FDD) systems where an estimate of CSI is obtained using feedback schemes, we consider the time-division duplex (TDD) mode of communication in MISO systems [15, 16, 14], where the channel reciprocity can be exploited. Hence adopting the TDD mode of communication, where the UL pilot and DL energy signal transmissions using the same frequency resource are separated in time, the DL channel coefficients can be obtained by estimating them from the UL pilot transmission from . We consider that each coherence interval of seconds (s) is divided into two sub-phases: (a) UL CE phase of s, and (b) DL RFET phase of s. During the CE phase, transmits a continuous-time pilot signal , having frequency with its baseband representation , satisfying . Thus, received baseband signal at is given by

(6)

where, is the energy spent during CE in Joule (J) with denoting the transmit power of and is the received complex additive white Gaussian noise (AWGN).

Iii-a Analog Channel Estimator

Adopting the antenna-switching based analog CE approach as proposed in [31, Fig. 2], where the parallel estimation of entries of vector over a duration of s reduces to a sequential estimation of each entry , each over s duration. Thus, with having only its th antenna active during the th CE sub-phase interval , the corresponding analog channel estimator is set as defined in (III-A). Under the API at as given by (II-B) in Section II-B, the practical entries of the analog channel estimator for , which remain the same for each duration are [31, eq. (7)]

(7)

Therefore, the entries of analog channel estimator matrix as set over sub-phases, with respective intervals , are defined below [31]

(8)

using the identities, and in (2) for and , respectively. For ideal (no API) scenario, with and . Hence, with this analog channel estimator, the corresponding signal received as an input to digital channel estimator block, and obtained after using (8) in (6) is given by [31, eq. (9)]

(9)

where is the effective channel to be estimated and .

Iii-B Least-Squares Based Digital Channel Estimator

For obtaining the optimal LSE, the received signal at the antennas of , as defined in (6), first undergoes the analog CE process as described by (cf. (8)) in Section III-A. Thereafter, the match filtering operation is performed to the resulting signal by setting the digital channel estimator as . Hence, LSE for the effective channel as obtained using (III-A) can be derived as shown below

(10)

where is the effective AWGN vector influenced by API having zero mean entries. Here, we would like to mention that the proposed LSE for the effective channel , as obtained using analog and digital channel estimators, and , yields the global minimum value of the objective function in the conventional LS problem [35]. Hence, is the global optimal LSE for the effective or the actual channel under API. Further, these underlying LS estimation errors due to the API and CE uncertainty are independent because their respective sources, i.e., hardware impairments and AWGN, are not related.

Iii-C Optimal Precoder Design

The optimal precoder design for the hybrid EBF should be such that it maximizes the received RF power at by focusing most of transmit power of in the direction of . Thus, the maximum ratio transmission (MRT) based precoder design should be selected at to maximize the EBF gains. Hence, for implementing the transmit hybrid EBF at to maximize the harvested DC power, the digital precoder is set as and the analog precoder is set as . Here, is the transmit power of during DL RFET and is the LSE for the channel as defined in (10). However, under API, the analog precoder gets practically altered to

(11)
Fig. 3: Depicting the alteration to the analog precoder under practical API in the hybrid EBF implementation.

The entries of this practical analog precoder design are also depicted in Fig. 3 by showing the equivalent complex baseband signal model for the analog and digital precoder designs. Here, its alternative form is shown, which is reproduced below, was obtained using (2) and (II-B).

(12)

Iv Practically Motivated Tight Approximations

Here, first in Section IV-A, we present a practically motivated approximation for API-dependent parameters. Then using it, we obtain the statistics of the key parameters based on the LSE . These statistics derived in Section IV-B and  IV-C, will be used later in Sections V and VI.

Iv-a Tight Approximation for Analog Channel Estimator Matrix

In practice, which has very low value, i.e., . In fact, as also noted in [29, 30], for the practical PSs design, the phase errors are generally much less than . This condition actually on simplification yields . It may be recalled that this practical range on amplitude and phase errors is also commonly used while investigating the performance under in-phase-and-quadrature-phase-imbalance (IQI) in practical multiantenna systems [34, 28, 36]. In fact, the in-phase (I) and quadrature-phase (Q) branches used for generating the desired complex signal , play a very similar role as to a DCPS pair and combiner in the hybrid EBF architecture [7] as depicted via (1). Furthermore, the random amplitude and phase errors due to IQI are modeled using uniform distribution [34, and references therein], which corroborates our assumption of modeling random API via uniform distribution in Section II-B. Moreover, since our proposed optimal hybrid CE and EBF protocol holds good for any generic random distribution characterizing API, later in Section VII we have also considered the Gaussian distribution for modeling randomness in API and conducted a performance comparison against the uniform one to gain insights on the impact of different API distributions.

Following the above discussion, we present a tight approximation for the ACE matrix by using the practical limits on API. As in practice, for decent quality DCPSs, . This implies that since , it results in similar practical ranges for the constants modeling API. In other words, . Finally, using it in (3) yields the following approximations which hold good for any distribution of and

(13)

Applying the above approximation (13) in API to (8) gives: , and zero for the other entries. From this result along (13), it is noted that in practice the diagonal entries of are very close to each other. Whereas, the non-diagonal entries of are very close to zero. Applying this practically motivated approximation for the API model with , the following approximation results can be obtained for matrices involving products of

(14)

where . Thus, all the three API dependent matrices and

can be practically approximated as the same scaled identity matrix. Next, we use this approximation to derive the distribution for the key API-dependent parameters.

Iv-B Statistics for LSE-based Key Parameters

Iv-B1 Effective AWGN

As discussed above, for practical API with , the entries of effective AWGN vector are independently and identically distributed (IID) with zero mean entries. Further, the covariance of can be approximated by . Here, represents the noise power spectral density in Joule (J) and entries of diagonal matrix are

(15)

So, with this approximation for practical API, we can rewrite the LSE as defined by (10) as

(16)

where is the LS estimation error [35], which is a linear function of the effective AWGN vector and independent of the effective channel vector .

Iv-B2 Norm of LSE

The real and imaginary entries of the LSE follow real and nonzero mean Gaussian distribution, i.e., and . Like , for practical API, the entries of , and hence , are also IID, So, .

Thus, the LSE , where its covariance , which can be practically approximated as . So, it includes both the unknown channel state and API information. Using this mentioned distribution for practical API,

follows the non-central chi-square distribution with

degrees of freedom and non-centrality parameter . Here, the variance is defined below

(17)

Further, the expectation of is given by

(18)

which in turn can be approximated as below after applying (14) in (18) for practical API-limits

(19)

We have validated this distribution of

by verifying the underlying probability density function (PDF) and cumulative distribution function (CDF) via simulations in Section 

VII.

Iv-B3 Conditional distribution of the channel for a given LSE

Here, we obtain the statistics (mean and variance) for the actual channel for the given LSE for the effective channel . As from (10), , the conditional expectation can be obtained as [35]

(20)

where , and . On using these statistics in (20) and simplifying, the desired expectation is obtained as

(21)

Similarly, the covariance is given by

(22)

On applying the practical API-limit based approximations as defined in (14) to the above result in (IV-B3), the following approximation for the covariance of can be obtained

(23)

Above, we had used (14) for approximating two API-dependent parameters mentioned below

(24a)
(24b)

Iv-C Received energy signal at during DL RFET phase

As discussed in Section III-C, the analog precoder design set to . Though this precoder actually gets altered to defined in (11) due to the underlying API, which are not known and thus are not compensated, we have used for the theoretical investigation in Section V and optimization in Section VI. Consequently, the corresponding random RF energy signal, as received at during the DL RFET phase, is given by . This random variable will be used for investigating the optimal resource allocation to maximize the harvested energy performance in Section VI. Below, we derive the distribution of for a given LSE .

Lemma 1

Given the LSE , follows non-central chi-square distribution with two degrees of freedom and the non-centrality parameter . Here, the statistics and denote the mean and variance of the conditional random variable , respectively.

Proof:

For the given LSE , , follows the same distribution as , i.e., nonzero complex Gaussian. Below, we obtain the required statistics to obtain the distribution, i.e., mean and variance of .

(25a)
(25b)

where (25) is obtained using (IV-B3), and (25) after using approximation (23) in (IV-B3). Hence, for the given LSE , . Hence, the normalized random variable follows the mentioned non-central chi-square distribution. Further, on using approximations (24a) and (24b) defined for practical API settings:
.

V Average Harvested Energy due to Hybrid EBF

In this section we derive the expression for the average harvested energy at the EH due to MRT from during the DL RFET using the LSE . In this regard we first revisit some basics of received power analysis over Rician channels. Thereafter, we discourse the optimal transmit hybrid EBF at based on the proposed LSE as presented in Section III. Lastly, since in general the harvested energy cannot be expressed in closed form, we present a practically-motivated tight analytical approximation for it by using the developments in Sections IV-B and IV-C.

V-a Exact Average Harvested DC Power Analysis

Following the result outlined in Lemma 3, we note that the received RF power at due to DL RFET from follows non-central chi-square distribution with two degrees of freedom, Rice factor and mean . Thus, the PDF of the received power for is given by [32]

(26)

where is the modified Bessel function of first kind with order . Further, CDF of is:

(27)

where is the first order Marcum Q-function [37].

Using the relationship from (5) along with PDF and CDF of received power defined in (26) and (27), PDF of harvested power for is given by

(28)

Thus, using (28), the mean harvested DC power is given by Although, it is difficult to obtain a closed-form expression for , an alternate representation in the form of an infinite series was derived in [26, eq. (8)]. However, for analytical tractability, we use a simpler representation in the form of a tight approximation based on the Jensen’s inequality [38], i.e., , which is defined as below

(29)

V-B Transmit Hybrid RF Energy Beamforming

As mentioned earlier in Section III-C, for implementing the transmit hybrid EBF at to maximize the harvested DC power at , the digital precoder is set as and analog precoder as . From Section IV-C, we recall that since API estimation and compensation is not investigated in this work, we have used , instead of , as the analog precoder for the underlying investigation based on the LSE for the effective channel . Further, since API are slowly varying processes because the factors influencing them like aging, hardware temperature variation, and manufacturing impairments change slowly with time, API mitigation can be incorporated via calibration methods relying on mutual coupling between antenna elements [39]. However, the detailed API compensation is out of the scope of this work.

So, ignoring API compensation, the MRT-based precoding enables that the signals emitted from different antennas add coherently at the EH user . transmits a continuous energy signal