Wireless Information and Power Transfer: Nonlinearity, Waveform Design and Rate-Energy Tradeoff

Wireless Information and Power Transfer: Nonlinearity, Waveform Design and Rate-Energy Tradeoff

Bruno Clerckx Bruno Clerckx is with the EEE department at Imperial College London, London SW7 2AZ, UK (email: b.clerckx@imperial.ac.uk). This work has been partially supported by the EPSRC of UK, under grant EP/P003885/1. The material in this paper was presented in part at the ITG WSA 2016 [1].
Abstract

The design of Wireless Information and Power Transfer (WIPT) has so far relied on an oversimplified and inaccurate linear model of the energy harvester. In this paper, we depart from this linear model and design WIPT considering the rectifier nonlinearity. We develop a tractable model of the rectifier nonlinearity that is flexible enough to cope with general multi-carrier modulated input waveforms. Leveraging that model, we motivate and introduce a novel WIPT architecture relying on the superposition of multi-carrier unmodulated and modulated waveforms at the transmitter. The superposed WIPT waveforms are optimized as a function of the channel state information so as to characterize the rate-energy region of the whole system. Analysis and numerical results illustrate the performance of the derived waveforms and WIPT architecture and highlight that nonlinearity radically changes the design of WIPT. We make key and refreshing observations. First, analysis (confirmed by circuit simulations) shows that modulated and unmodulated waveforms are not equally suitable for wireless power delivery, namely modulation being beneficial in single-carrier transmissions but detrimental in multi-carrier transmissions. Second, a multi-carrier unmodulated waveform (superposed to a multi-carrier modulated waveform) is useful to enlarge the rate-energy region of WIPT. Third, a combination of power splitting and time sharing is in general the best strategy. Fourth, a non-zero mean Gaussian input distribution outperforms the conventional capacity-achieving zero-mean Gaussian input distribution in multi-carrier transmissions. Fifth, the rectifier nonlinearity is beneficial to system performance and is essential to efficient WIPT design.

Nonlinearity, optimization, waveform, wireless power, wireless information and power transfer

I Introduction

Wireless Information and Power Transfer/Transmission (WIPT) is an emerging research area that makes use of radiowaves for the joint purpose of wireless communications or Wireless Information Transfer (WIT) and Wireless Power Transfer (WPT). WIPT has recently attracted significant attention in academia. It was first considered in [2], where the rate-energy tradeoff was characterized for some discrete channels, and a Gaussian channel with an amplitude constraint on the input. WIPT was then studied in a frequency-selective AWGN channel under an average power constraint [3]. Since then, WIPT has attracted significant interests in the communication literature with among others MIMO broadcasting [4, 5, 6], architecture [7], interference channel [8, 9, 10], broadband system [11, 12, 13], relaying [14, 15, 16], wireless powered communication [17, 18]. Overviews of potential applications and promising future research avenues can be found in [20, 19].

Wireless Power Transfer (WPT) is a fundamental building block of WIPT and the design of an efficient WIPT architecture fundamentally relies on the ability to design efficient WPT. The major challenge with WPT, and therefore WIPT, is to find ways to increase the end-to-end power transfer efficiency, or equivalently the DC power level at the output of the rectenna for a given transmit power. To that end, the traditional line of research (and the vast majority of the research efforts) in the RF literature has been devoted to the design of efficient rectennas [21, 22] but a new line of research on communications and signal design for WPT has emerged recently in the communication literature [23].

A rectenna is made of a nonlinear device followed by a low-pass filter to extract a DC power out of an RF input signal. The amount of DC power collected is a function of the input power level and the RF-to-DC conversion efficiency. Interestingly, the RF-to-DC conversion efficiency is not only a function of the rectenna design but also of its input waveform (power and shape) [24, 25, 26, 27, 28, 29, 30]. This has for consequence that the conversion efficiency is not a constant but a nonlinear function of the input waveform (power and shape).

This observation has triggered recent interests on systematic wireless power waveform design [29]. The objective is to understand how to make the best use of a given RF spectrum in order to deliver a maximum amount of DC power at the output of a rectenna. This problem can be formulated as a link optimization where transmit waveforms (across space and frequency) are adaptively designed as a function of the channel state information (CSI) so as to maximize the DC power at the output of the rectifier. In [29], the waveform design problem for WPT has been tackled by introducing a simple and tractable analytical model of the diode nonlinearity through the second and higher order terms in the Taylor expansion of the diode characteristics. Comparisons were also made with a linear model of the rectifier, that only accounts for the second order term, which has for consequence that the harvested DC power is modeled as a conversion efficiency constant (i.e. that does not reflect the dependence w.r.t. the input waveform) multiplied by the average power of the input signal. Assuming perfect Channel State Information at the Transmitter (CSIT) can be attained, relying on both the linear and nonlinear models, an optimization problem was formulated to adaptively change on each transmit antenna a multisine waveform as a function of the CSI so as to maximize the output DC current at the energy harvester. Important conclusions of [29] are that 1) multisine waveforms designed accounting for nonlinearity are spectrally more efficient than those designed based on a linear model of the rectifier, 2) the derived waveforms optimally exploit the combined effect of a beamforming gain, the rectifier nonlinearity and the channel frequency diversity gain, 3) the linear model does not characterize correctly the rectenna behavior and leads to inefficient multisine waveform design, 4) rectifier nonlinearity is key to design efficient wireless powered systems. Following [29], various works have further investigated WPT signal and system design accounting for the diode nonlinearity, including among others waveform design complexity reduction [32, 33, 31, 34], large-scale system design with many sinewaves and transmit antennas [32, 33], multi-user setup [32, 33], imperfect/limited feedback setup [35], information transmission [36] and prototyping and experimentation [37]. Another type of nonlinearity leading to an output DC power saturation due to the rectifier operating in the diode breakdown region, and its impact on system design, has also appeared in the literature [30, 38].

Interestingly, the WIPT literature has so far entirely relied on the linear model of the rectifier, e.g. see [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 20, 19]. Given the inaccuracy and inefficiency of this model and the potential of a systematic design of wireless power waveform as in [29], it is expected that accounting for the diode nonlinearity significantly changes the design of WIPT and is key to efficient WIPT design, as confirmed by initial results in [1].

In this paper, we depart from this linear model and revisit the design of WIPT in light of the rectifier nonlinearity. We address the important problem of waveform and transceiver design for WIPT and characterize the rate-energy tradeoff, accounting for the rectifier nonlinearity. In contrast to the existing WIPT signal design literature, our methodology in this paper is based on a bottom-up approach where WIPT signal design relies on a sound science-driven design of the underlying WPT signals initiated in [29].

First, we extend the analytical model of the rectenna nonlinearity introduced in [29], originally designed for multi-carrier unmodulated (deterministic multisine) waveform, to multi-carrier modulated signals. We investigate how a multi-carrier modulated waveform (e.g. OFDM) and a multi-carrier unmodulated (deterministic multisine) waveform compare with each other in terms of harvested energy. Comparison is also made with the linear model commonly used in the WIPT literature. Scaling laws of the harvested energy with single-carrier and multi-carrier modulated and unmodulated waveforms are analytically derived as a function of the number of carriers and the propagation conditions. Those results extend the scaling laws of [29], originally derived for unmodulated waveforms, to modulated waveforms. We show that by relying on the classical linear model, an unmodulated waveform and a modulated waveform are equally suitable for WPT. This explains why the entire WIPT literature has used modulated signals. On the other hand, the nonlinear model clearly highlights that they are not equally suitable for wireless power delivery, with modulation being beneficial in single-carrier transmission but detrimental in multi-carrier transmissions. The behavior is furthermore validated through circuit simulations. This is the first paper where the performance of unmodulated and modulated waveforms are derived based on an tractable analytical model of the rectifier nonlinearity and the observations made from the analysis are validated through circuit simulations.

Second, we introduce a novel WIPT transceiver architecture relying on the superposition of multi-carrier unmodulated and modulated waveforms at the transmitter and a power-splitter receiver equipped with an energy harvester and an information decoder. The WIPT superposed waveform and the power splitter are jointly optimized so as to maximize and characterize the rate-energy region of the whole system. The design is adaptive to the channel state information and results from a posynomial maximization problem that originates from the nonlinearity of the energy harvester. This is the first paper that studies WIPT and the characterization of the rate-energy tradeoff considering the diode nonlinearity.

Third, we provide numerical results to illustrate the performance of the derived waveforms and WIPT architecture. Key observations are made. First, a multi-carrier unmodulated waveform (superposed to a multi-carrier modulated waveform) is useful to enlarge the rate-energy region of WIPT if the number of subbands is sufficiently large (typically larger than 4). Second, a combination of power splitting and time sharing is in general the best strategy. Third, a non-zero mean Gaussian input distribution outperforms the conventional capacity-achieving zero-mean Gaussian input distribution in multi-carrier transmissions. Fourth, the rectifier nonlinearity is beneficial to system performance and is essential to efficient WIPT design. This is the first paper to make those observations because they are direct consequences of the nonlinearity.

Organization: Section II introduces and models the WIPT architecture. Section III optimizes WIPT waveforms and characterizes the rate-energy region. Section IV derives the scaling laws of modulated and unmodulated waveforms. Section V evaluates the performance and section VI concludes the work.

Notations: Bold lower case and upper case letters stand for vectors and matrices respectively whereas a symbol not in bold font represents a scalar. refers to the Frobenius norm a matrix. refers to the DC component of a signal. refers to the expectation operator taken over the distribution of the random variable ( may be omitted for readability if the context is clear). refers to the conjugate of a scalar. and represent the transpose and conjugate transpose of a matrix or vector respectively. The distribution of a circularly symmetric complex Gaussian (CSCG) random vector with mean and covariance matrix is denoted by and stands for “distributed as”.

Ii A Novel WIPT Transceiver Architecture

In this section, we introduce a novel WIPT transceiver architecture and detail the functioning of the various building blocks. The motivation behind the use of such an architecture will appear clearer as we progress through the paper.

Ii-a Transmitter and Receiver

We consider a single-user point-to-point MISO WIPT system in a general multipath environment. The transmitter is equipped with antennas that transmit information and power simultaneously to a receiver equipped with a single receive antenna. We consider the general setup of a multi-carrier/band transmission (with single-carrier being a special case) consisting of orthogonal subbands where the subband has carrier frequency and equal bandwidth , . The carrier frequencies are evenly spaced such that with the inter-carrier frequency spacing (with ).

Uniquely, the WIPT signal transmitted on antenna , , consists in the superposition of one multi-carrier unmodulated (deterministic multisine) power waveform at frequencies , for WPT and one multi-carrier modulated communication waveform at the same frequencies for WIT111 can be implemented using e.g. OFDM., as per Fig 1(a). The modulated waveform carries independent information symbols on subband . Hence, the transmit WIPT signal at time on antenna writes as

(1)

where we denote the complex-valued baseband signal transmitted by antenna at subband for the unmodulated (deterministic multisine) waveform as and for the modulated waveform as . is constant across time (for a given channel state) and is therefore the weighted summation of sinewaves inter-separated by Hz, and hence occupies zero bandwidth. On the other hand, has a signal bandwidth no greater than with symbols assumed i.i.d. CSCG222following the capacity achieving input distribution in a Gaussian channel with average power constraint. random variable with zero-mean and unit variance (power), denoted as . Denoting the input symbol , we further express the magnitude and phase of as follows with and . Hence and .

The transmit WIPT signal propagates through a multipath channel, characterized by paths. Let and be the delay and amplitude gain of the path, respectively. Further, denote by the phase shift of the path between transmit antenna and the receive antenna at subband . Denoting , the signal received at the single-antenna receiver due to transmit antenna can be expressed as the sum of two contributions, namely one originating from WPT and the other from WIT , namely

(2)

where we have assumed so that and for each subband are narrowband signals, thus and , . The quantity is the channel frequency response between antenna and the receive antenna at frequency .

Stacking up all transmit signals across all antennas, we can write the transmit WPT and WIT signal vectors as

(3)
(4)

where . Similarly, we define the vector channel as . The total received signal comprises the sum of (2) over all transmit antennas, namely

(5)

The magnitudes and phases of the sinewaves can be collected into matrices and . The entry of and write as and , respectively. Similarly, we define matrices such that the entry of matrix and write as and , respectively. We define the average power of the WPT and WIT waveforms as and . Due to the superposition of the two waveforms, the total average transmit power constraint writes as .

Following Fig 1(b), using a power splitter with a power splitting ratio and assuming perfect matching (as in Section II-C1), the input voltage signals and are respectively conveyed to the energy harvester (EH) and the information decoder (ID).

Remark 1

As it will appear clearer throughout the paper, the benefit of choosing a deterministic multisine power waveform over other types of power waveform (e.g. modulated, pseudo-random) is twofold: 1) energy benefit: multisine will be shown to be superior to a modulated waveform, 2) rate benefit: multisine is deterministic and therefore does not induce any rate loss at the communication receiver.

Remark 2

It is worth noting the effect of the deterministic multisine waveform on the input distribution in (1). Recall that . Hence and the effective input distribution on a given frequency and antenna is not zero mean333If using OFDM, and are OFDM waveforms with CSCG inputs and non-zero mean Gaussian inputs, respectively.. The magnitude is Ricean distributed with a K-factor on frequency and antenna given by .

Remark 3

The superposition of information and power signals has appeared in other works, but for completely different purposes; namely for multiuser WIPT in [6], collaborative WIPT in interference channel in [43, 44], and for secrecy reasons in [45, 46]. Since those works relied on the linear model, the superposition was not motivated by the rectifier nonlinearity. Moreover, the properties of the power signals are completely different. While the power signal is a deterministic multisine waveform leading to non-zero mean Gaussian input and the twofold benefit (Remark 1) in this work, it is complex (pseudo-random) Gaussian in those works.

(a) Transmitter (b) Receiver with waveform cancellation
Fig. 1: Transceiver (Tx and Rx) architecture for WIPT with superposed communication and power waveform (WF).

Ii-B Information Decoder

Since does not contain any information, it is deterministic. This has for consequence that the differential entropy of and are identical (because translation does not change the differential entropy) and the achievable rate is always equal to

(6)

where is the variance of the AWGN from the antenna and the RF-to-baseband down-conversion on tone .

Naturally, is larger than the maximum rate achievable when , i.e. , which is obtained by performing Maximum Ratio Transmission (MRT) on each subband and water-filling power allocation across subbands.

The rate (6) is achievable irrespectively of the receiver architecture, e.g. with and without waveform cancellation. In the former case, after down-conversion from RF-to-baseband (BB) and ADC, the contribution of the power waveform is subtracted from the received signal (as illustrated in Fig 1(b))444If using OFDM, conventional OFDM processing (removing the cyclic prefix and performing FFT) is then conducted in the BB receiver.. In the latter case, the “Power WF cancellation” box of Fig 1(b) is removed and the BB receiver decodes the translated version of the codewords.

Ii-C Energy Harvester

In [29], a tractable model of the rectifier nonlinearity in the presence of multi-carrier unmodulated (deterministic multisine) excitation was derived and its validity verified through circuit simulations. In this paper, we reuse the same model and further expand it to modulated excitation. The randomness due to information symbols impacts the amount of harvested energy and needs to be captured in the model.

Fig. 2: Antenna equivalent circuit (left) and a single diode rectifier (right).

Ii-C1 Antenna and Rectifier

The signal impinging on the antenna is and has an average power . A lossless antenna is modelled as a voltage source followed by a series resistance555Assumed real for simplicity. A more general model can be found in [31]. (Fig 2 left). Let denote the input impedance of the rectifier with the matching network. Assuming perfect matching (, ), due to the power splitter, a fraction of the available RF power is transferred to the rectifier and absorbed by , so that the actual input power to the rectifier is and . Hence, can be formed as . We also assume that the antenna noise is too small to be harvested.

Let us now look at Fig 2(right) and consider a rectifier composed of a single series diode666The model holds also for more general rectifiers as shown in [31]. followed by a low-pass filter with load. Denoting the voltage drop across the diode as where is the input voltage to the diode and is the output voltage across the load resistor, a tractable behavioral diode model is obtained by Taylor series expansion of the diode characteristic equation (with the reverse bias saturation current, the thermal voltage, the ideality factor assumed equal to ) around a quiescent operating point , namely where and , . Assume a steady-state response and an ideal low pass filter such that is at constant DC level. Choosing , we can write .

Under the ideal rectifier assumption and a deterministic incoming waveform , the current delivered to the load in a steady-state response is constant and given by . In order to make the optimization tractable, we truncate the Taylor expansion to the order. A nonlinear model truncates the Taylor expansion to the order but retains the fundamental nonlinear behavior of the diode while a linear model truncates to the second order term.

Ii-C2 Linear and Nonlinear Models

After truncation, the output DC current approximates as

(7)

Let us first consider a multi-carrier unmodulated (multisine) waveform, i.e. . Following [29], we get an approximation of the DC component of the current at the output of the rectifier (and the low-pass filter) with a multisine excitation over a multipath channel as

(8)

where and are detailed in (9) and (11), respectively (at the top of next page). The linear model is a special case of the nonlinear model and is obtained by truncating the Taylor expansion to order 2 ().

(9)
(10)
(11)

Let us then consider the multi-carrier modulated waveform, i.e. . It can be viewed as a multisine waveform for a fixed set of input symbols . Hence, we can also write the DC component of the current at the output of the rectifier (and the low-pass filter) with a multi-carrier modulated excitation and fixed set of input symbols over a multipath channel as . Similar expressions as (9) and (11) can be written for and for a fixed set of input symbols . However, contrary to the multisine waveform, the input symbols of the modulated waveform change randomly at symbol rate . For a given channel impulse response, the proposed model for the DC current with a modulated waveform is obtained as

(12)

by taking the expectation over the distribution of the input symbols . For with even, the DC component is first extracted for a given set of amplitudes and phases and then expectation is taken over the randomness of the input symbols . Due to the i.i.d. CSCG distribution of the input symbols, is exponentially distributed with and is uniformly distributed. From the moments of an exponential distribution, we also have that . We can then express (13) and (14) as a function of and . Note that this factor of does not appear in (11) due to the absence of modulation, which explains why (11) and (14) enjoy a multiplicative factor of and , respectively. Here again, the linear model is obtained by truncating to .

(13)
(14)
(15)

Let us finally consider the superposed waveform, i.e. . Both and waveforms now contribute to the DC component

(16)

Taking for instance and further expanding the term using the fact that , , and , can be written as

(17)
Observation 1

The linear model highlights that there is no difference in using a multi-carrier unmodulated (multisine) waveform and a multi-carrier modulated (e.g. OFDM) waveform for WPT, since according to this model the harvested energy is a function of , as seen from (9) and (13). Hence modulated and unmodulated waveforms are equally suitable. On the other hand, the nonlinear model highlights that there is a clear difference between using a multi-carrier unmodulated over a multi-carrier modulated waveform in WPT. Indeed, from (13) and (15) of the modulated waveform, both the second and fourth order terms exhibit the same behavior and same dependencies, namely they are both exclusively function of . That suggests that for a multi-carrier modulated waveform with CSCG inputs, the linear and nonlinear models are equivalent, i.e. there is no need in modeling the fourth and higher order term. On the other hand, for the unmodulated waveform, the second and fourth order terms, namely (9) and (11), exhibit clearly different behaviors with the second order term being linear and the fourth order being nonlinear and function of terms expressed as the product of contributions from different frequencies.

Remark 4

The linear model is motivated by its simplicity rather than its accuracy and is the popular model used throughout the WIPT literature, e.g. [4]. Indeed, it is always assumed that the harvested DC power is modeled as where is the RF-to-DC conversion efficiency assumed constant. By assuming constant, those works effectively only care about maximizing the input power (function of ) to the rectifier, i.e. the second order term (or linear term) in the Taylor expansion. Unfortunately this is inaccurate as is not a constant and is itself a function of the input waveform (power and shape) to the rectifier, as recently highlighted in the communication literature [28, 29, 23, 30] but well recognized in the RF literature [21, 22]. This linear model was shown through circuit simulations in [29] to be inefficient to design multisine waveform but also inaccurate to predict the behavior of such waveforms in the practical low-power regime (-30dBm to 0dBm). On the other hand, the nonlinear model, rather than explicitly expressing the DC output power as with a function of the input signal power and shape, it directly expresses the output DC current as a function of (and therefore as a function of the transmit signal and wireless channel) and leads to a more tractable formulation. Such a nonlinear model with has been validated for the design of multisine waveform in [28, 29, 31] using circuit simulators with various rectifier topologies and input power and in [37] through prototyping and experimentation. Nevertheless, the use of a linear vs a nonlinear model for the design of WPT based on other types of waveforms and the design of WIPT has never been addressed so far.

Remark 5

The above model deals with the diode nonlinearity under ideal low pass filter and perfect impedance matching. However there exist other sources of nonlinearities in a rectifier, e.g. impedance mismatch, breakdown voltage and harmonics. Recently, another nonlinear model has emerged in [30]. This model accounts for the fact that for a given rectifier design, the RF-to-DC conversion efficiency is a function of the input power and sharply decreases once the input power has reached the diode breakdown region. This leads to a saturation nonlinearity where the output DC power saturates beyond a certain input power level. There are multiple differences between those two models.
First, our diode nonlinearity model assumes the rectifier is not operating in the diode breakdown region. Circuit evaluations in [29, 31] and in Section V-B also confirm that the rectifier never reached the diode breakdown voltage under all investigated scenarios. We therefore do not model the saturation effect. On the other hand, [30] assumes the rectifier can operate in the breakdown region and therefore models the saturation. However, it is to be reminded that operating diodes in the breakdown region is not the purpose of a rectifier and should be avoided. A rectifier is designed in such a way that current flows in only one direction, not in both directions as it would occur in the breakdown region. Hence, [30] models a saturation nonlinearity effect that occurs in an operating region where one does not wish to operate in. In other words, the rectifier is pushed in an input power range quite off from the one it has originally been designed for. This may only occur in applications where there is little guarantee to operate the designed rectifier below that breakdown edge.
Second, the diode nonlinearity is a fundamental, unavoidable and intrinsic property of any rectifier, i.e. any rectifier, irrespectively of its design, topology or implementation, is always made of a nonlinear device (most commonly Schottky diode) followed by a low pass filter with load. This has for consequence that the diode nonlinearity model is general and valid for a wide range of rectifier design and topology (with one and multiple diodes) as shown in [31]. Moreover, since it is driven by the physics of the rectenna, it analytically links the output DC metric to the input signal through the diode I-V characteristics. On the other hand, the saturation nonlinearity in [30] is circuit-specific and modeled via curve fitting based on measured data. Hence changing the diode or the rectifier topology would lead to a different behavior. More importantly, the saturation effect, and therefore the corresponding nonlinearity, is actually avoidable by properly designing the rectifier for the input power range of interest. A common strategy is to use an adaptive rectifier whose configuration changes as a function of the input power level, e.g. using a single-diode rectifier at low input power and multiple diodes rectifier at higher power, so as to generate consistent and non-vanishing over a significantly extended operating input power range [39, 40].
Third, the diode nonlinearity model accomodates a wide range of multi-carrier modulated and unmodulated input signals and is therefore a function of the input signal power, shape and modulation. The saturation nonlinearity model in [30] is restricted to a continuous wave input signal and is a function of its power. Hence it does not reflect the dependence of the output DC power to modulation and waveform designs.
Fourth, the diode nonlinearity is a beneficial feature that is to be exploited as part of the waveform design to boost the output DC power, as shown in [29]. The saturation nonlinearity is detrimental to performance and should therefore be avoided by operating in the non-breakdown region and using properly designed rectifier for the input power range of interest.
Fifth, the diode nonlinearity is more meaningful in the low-power regime (-30dBm to 0dBm with state-of-the-art rectifiers777At lower power levels, the diode may not turn on.) while the saturation nonlinearity is relevant in the high power regime (beyond 0dBm input power).

Iii WIPT Waveform Optimization and Rate-Energy Region Characterization

Leveraging the energy harvester model, we now aim at characterizing the rate-energy region of the proposed WIPT architecture. We define the achievable rate-energy region as

(18)

Assuming the CSI (in the form of frequency response ) is known to the transmitter, we aim at finding the optimal values of amplitudes, phases and power splitting ratio, denoted as ,,,, so as to enlarge as much as possible the rate-energy region. We derive a methodology that is general to cope with any truncation order 888We display terms for but the derived algorithm works for any ..

Characterizing such a region involves solving the problem

(19)
subject to (20)

Following [29], (19)-(20) can equivalently be written as

(21)
subject to (22)

with where we define . Assuming , a diode ideality factor and , we get and . For , similarly to (17), we can compute as in (23).

(23)

This enables to re-define the achievable rate-energy region in terms of rather than as follows

(24)

This definition of rate-energy region will be used in the sequel.

Iii-a WPT-only: Energy Maximization

In this section, we first look at energy maximization-only (with no consideration for rate) and therefore assume . We study and compare the design of multi-carrier unmodulated (multisine) waveform () and modulated waveforms () under the linear and nonlinear models. Since a single waveform is transmitted (either unmodulated or modulated), the problem simply boils down to the following for

(25)

where for multi-carrier unmodulated (multisine) waveform, and for the multi-carrier modulated waveform.

The problem of multisine waveform design with a linear and nonlinear rectenna model has been addressed in [29]. The linear model leads to the equivalent problem subject to whose solution is the adaptive single-sinewave (ASS) strategy

(26)

The ASS performs a matched (also called MRT) beamformer on a single sinewave, namely the one corresponding to the strongest channel . On the other hand, the nonlinear model leads to a posynomial maximization problem that can be formulated as a Reversed Geometric Program and solved iteratively. Interestingly, for multisine waveforms, the linear and nonlinear models lead to radically different strategies. The former favours transmission on a single frequency while the latter favours transmission over multiple frequencies. Design based on the linear model was shown to be inefficient and lead to significant loss over the nonlinear model-based design.

The design of multi-carrier modulated waveform is rather different. Recall that from (13) and (15), both the second and fourth order terms are exclusively function of . This shows that both the linear and nonlinear model-based designs of multi-carrier modulated waveforms for WPT lead to the ASS strategy and the optimum should be designed according to (26). This is in sharp contrast with the multisine waveform design and originates from the fact that the modulated waveform is subject to CSCG randomness due to the presence of input symbols . Note that this ASS strategy has already appeared in the WIPT literature, e.g. in [11, 41] with OFDM transmission.

Iii-B WIPT: A General Approach

We now aim at characterizing the rate-energy region of the proposed WIPT architecture. Looking at (6) and (23), it is easy to conclude that matched filtering w.r.t. the phases of the channel is optimal from both rate and harvested energy maximization perspective. This leads to the same phase decisions as for WPT in [28, 29], namely

(27)

and guarantees all arguments of the cosine functions in ((9), (11)) and in ((13), (14)) to be equal to 0. and are obtained by collecting and into a matrix, respectively.

(28)

With such phases and , can be finally written as (III-B). Similarly we can write

(29)

where .

Recall from [47] that a monomial is defined as the function