Fundamentals of Wireless Information and Power Transfer: From RF Energy Harvester Models to Signal and System Designs

Fundamentals of Wireless Information and Power Transfer: From RF Energy Harvester Models to Signal and System Designs

Bruno Clerckx Senior Member, IEEE Rui Zhang Fellow, IEEE Robert Schober Fellow, IEEE Derrick Wing Kwan Ng Senior Member, IEEE Dong In Kim Senior Member, IEEE and H. Vincent Poor Fellow, IEEE B. Clerckx is with the Electrical and Electronic Engineering Department at Imperial College London, London SW7 2AZ, UK (email: b.clerckx@imperial.ac.uk). R. Zhang is the Department of Electrical and Computer Engineering, National University of Singapore, Singapore 117583 (e-mail: elezhang@nus.edu.sg). R. Schober is with the Institute of Digital Communications, Friedrich-Alexander-University Erlangen-Nurnberg (FAU), Germany (email: robert.schober@fau.de). D. W. K. Ng is with the School of Electrical Engineering and Telecommunications, University of New South Wales, Sydney, NSW 2052, Australia (email: w.k.ng@unsw.edu.au). D. I. Kim is with the School of Information and Communication Engineering, Sungkyunkwan University (SKKU), Korea (email: dikim@skku.ac.kr). H. V. Poor is with the Department of Electrical Engineering, Princeton University, Princeton, NJ 08544 USA (e-mail: poor@princeton.edu). This work has been partially supported by the EPSRC of UK, under grant EP/P003885/1.
Abstract

Radio waves carry both energy and information simultaneously. Nevertheless, Radio-Frequency (RF) transmission of these quantities have traditionally been treated separately. Currently, we are experiencing a paradigm shift in wireless network design, namely unifying wireless transmission of information and power so as to make the best use of the RF spectrum and radiations as well as the network infrastructure for the dual purpose of communicating and energizing. In this paper, we review and discuss recent progress on laying the foundations of the envisioned dual purpose networks by establishing a signal theory and design for Wireless Information and Power Transmission (WIPT) and identifying the fundamental tradeoff between conveying information and power wirelessly. We start with an overview of WIPT challenges and technologies, namely Simultaneous Wireless Information and Power Transfer (SWIPT), Wirelessly Powered Communication Network (WPCN), and Wirelessly Powered Backscatter Communication (WPBC). We then characterize energy harvesters and show how WIPT signal and system designs crucially revolve around the underlying energy harvester model. To that end, we highlight three different energy harvester models, namely one linear model and two nonlinear models, and show how WIPT designs differ for each of them in single-user and multi-user deployments. Topics discussed include rate-energy region characterization, transmitter and receiver architecture, waveform design, modulation, beamforming and input distribution optimizations, resource allocation, and RF spectrum use. We discuss and check the validity of the different energy harvester models and the resulting signal theory and design based on circuit simulations, prototyping and experimentation. We also point out numerous directions that are promising for future research.

Wireless information and power transfer, wireless power transfer, wireless powered communications, wireless energy harvesting communications, rate-energy region, linear and nonlinear energy harvester modeling, signal and system design, prototyping, experimentation.

I Introduction

Wireless communications via Radio-Frequency (RF) radiation has been around for more than a century and has significantly shaped our society in the past 40 years. Wireless is however not limited to communications. Wireless powering of devices using near-field Inductive Power Transfer has become a reality with several commercially available products and standards. However, its range is severely limited (less than one meter). On the other hand, far-field Wireless Power Transfer (WPT) via RF (as in wireless communication) could be used over longer ranges. It has long been regarded as a possibility for energising low-power devices but it is only recently that it has become recognized as feasible due to reductions in the power requirements of electronics and smart devices [1, 2]. Indeed, in 20 years from now, according to Koomey’s law [3], the amount of energy needed for a given computing task will fall by a factor of 10000 compared to what it is now, thus further continuing the trend towards low-power devices. Moreover, the world will see the emergence of trillions of Internet-of-Things (IoT) devices. This explosion of low-power devices calls for a re-thinking of wireless network design.

Recent research advocates that the future of wireless networking goes beyond conventional communication-centric transmission. In the same way as wireless (via RF) has disrupted mobile communications for the last 40 years, wireless (via RF) will disrupt the delivery of mobile power. However, current wireless networks have been designed for communication purposes only. While mobile communication has become a relatively mature technology, currently evolving towards its fifth generation, the development of mobile power is in its infancy and has not even reached its first generation. Today, not a single standard on far-field WPT exists. Wireless power will bring numerous new opportunities: no wires, no contacts, no batteries, genuine mobility and a perpetual, predictable, dedicated, on-demand, and reliable energy supply as opposed to intermittent ambient energy-harvesting technologies (e.g. solar, thermal, vibration). This is highly relevant in future networks with ubiquitous and autonomous low-power and energy limited devices, device-to-device communications, and the IoT with massive connections.

Interestingly, although radio waves carry both energy and information simultaneously, RF transmission of these quantities have traditionally been treated separately. Imagine instead a wireless network, e.g. WiFi, in which information and energy flow together through the wireless medium. Wireless communication, or Wireless Information Transfer (WIT), and WPT would then refer to two extreme strategies, respectively, targeting communication-only and power-only. A unified design of Wireless Information and Power Transmission (WIPT) would on the other hand have the ability to softly evolve and compromise in between those two extremes to make the best use of the RF spectrum/radiation and the network infrastructure to communicate and energize. This will enable trillions of low-power devices to be connected and powered anywhere, anytime, and on the move.

The integration of wireless power and wireless communications brings new challenges and opportunities, and calls for a paradigm shift in wireless network design. As a result, numerous new research problems need to be addressed that cover a wide range of disciplines including communication theory, information theory, circuit theory, RF design, signal processing, protocol design, optimization, prototyping, and experimentation.

I-a Overview of WIPT Challenges and Technologies

WIT and WPT are fundamental building blocks of WIPT and the design of efficient WIPT networks fundamentally relies on the ability to design efficient WIT and WPT. In the last 40 years, WIT has seen significant advances in RF theory and signal theory. Traditional research on WPT in the last few decades has focused extensively on RF theories and techniques concerning the energy receiver with the design of efficient RF, circuit, antenna, rectifier, and power management unit solutions [4, 5, 6], but recently a new and complementary line of research on communications and signal design for WPT has emerged in the communication literature [7]. Moreover, there has been a growing interest in bridging RF, signal, and system designs in order to bring those two communities closer together and to get a better understanding of the fundamental building blocks of an efficient WPT network architecture [8].

The engineering requirements and design challenges of the envisioned network are numerous: 1) Range: Delivery of wireless power at distances of 5-100 meters (m) for indoor/outdoor charging of low-power devices; 2) Efficiency: Boosting the end-to-end power transfer efficiency (up to a fraction of a percent/a few percent), or equivalently the DC power level at the output of the rectenna(s) for a given transmit power; 3) Non-line of sight (NLoS): Support of Line of sight (LoS) and NLoS to widen the practical applications of WIPT networks; 4) Mobility support: Support of mobile receivers, at least for those at pedestrian speed; 5) Ubiquitous accessibility: Support of ubiquitous power accessibility within the network coverage area; 6) Safety and health: Resolving the safety and health issues of RF systems and compliance with the regulations; 7) Energy consumption: Limitation of the energy consumption of energy-constrained RF powered devices; 8) Seamless integration of wireless communication and wireless power: Interoperability between wireless communication and wireless power via a unified WIPT.

Solutions to tackle challenges 1)-7) are being researched and have been discussed extensively in [9, 10, 6, 7, 8]. They cover a wide range of areas spanning sensors, devices, RF, communication, signal and system designs for WPT. This survey article targets challenge 8) by reviewing the fundamentals of WIPT signal and system designs. In WPT and WIT, the emphasis of the system design is to exclusively deliver energy and information, respectively. On the contrary, in WIPT, both energy and information are to be delivered. A WIPT system should therefore be designed such that the RF radiation and the RF spectrum are exploited in the most efficient manner to deliver both information and energy. Such a system design requires the characterization of the fundamental tradeoff between how much information and how much energy can be delivered in a wireless network and how signals should be designed to achieve the best possible tradeoff between them.

As illustrated in Fig. 1, WIPT can be categorized into three different types:

  • Simultaneous Wireless Information and Power Transfer (SWIPT): Energy and information are simultaneously transferred in the downlink from one or multiple access points to one or multiple receivers. The Energy Receiver(s) (ER) and Information Receiver(s) (IR) can be co-located or separated. In SWIPT with separated receivers, ER and IR are different devices, the former being a low-power device being charged, the latter being a device receiving data. In SWIPT with co-located receivers, each receiver is a single low-power device that is simultaneously being charged and receiving data.

  • Wirelessly Powered Communication Network (WPCN): Energy is transferred in the downlink and information is transferred in the uplink. The receiver is a low-power device that harvests energy in the downlink and uses it to send data in the uplink.

  • Wirelessly Powered Backscatter Communication (WPBC): Energy is transferred in the downlink and information is transferred in the uplink but backscatter modulation at a tag is used to reflect and modulate the incoming RF signal for communication with a reader. Since tags do not require oscillators to generate carrier signals, backscatter communications benefit from orders-of-magnitude lower power consumption than conventional radio communications.

Moreover, a network could also include a mixture of the above three types of transmissions with multiple co-located and/or separated Energy Transmitter(s) (ET) and Information Transmitter(s) (IT).

Fig. 1: Different WIPT architectures.

I-B Objectives and Organization

This paper reviews and summarizes recent advances and contributions in the area of WIPT. The main objective of this article is to give a systematic treatment of signal theory and design for WIPT and use it to characterize the fundamental tradeoff between conveying information and energy in a wireless network. This tradeoff is commonly referred to as rate-energy (R-E) tradeoff. Various review papers on WIPT have appeared in past years [11, 12, 13, 14, 19, 15, 16, 17, 18, 20, 21]. Emphasis was put at that time on characterizing the R-E tradeoff under the assumption of a very simple linear model of the energy harvester. Interestingly, the importance of the energy harvester model for WIPT design was never raised and the validity of this linear model never questioned in that WIPT literature. In recent years, there has been an increasing interest in the WIPT literature to depart from the linear model. However what we know about WIPT design from those review papers is fundamentally rooted in the underlying linear model. It turns out that WIPT design radically changes once we change the energy harvester model and adopt more realistic nonlinear models of the energy harvester.

Hence, in contrast to those existing tutorial and review papers, we here aim at showing how crucial the energy harvester model is to WIPT signal and system designs and how WIPT signal and system designs revolve around the underlying energy harvester model. To that end, we highlight three different energy harvester models, namely one linear model and two nonlinear models, and show how WIPT designs differ for each of them. In particular, we show how the modeling of the energy harvester can have tremendous influence on the design of the Physical (PHY) and Medium Access Control (MAC) layers of WIPT networks. We rigorously review how the different models can favor different waveforms, modulations, input distributions, beamforming, transceiver architectures, and resource allocation strategies as well as a different use of the RF spectrum. We first consider single-user (point-to-point) WIPT and then extend to multi-user scenarios. We discuss the validity of the different energy harvester models and the resulting signal and system designs through experimentation and prototyping. Finally, we point out directions that are promising for future research.

The rest of this article is organized as follows. In the next subsection, we first give some insights into the crucial role of energy harvester modeling and its impact onto signal designs. We then jump into the core parts of the paper. Section II introduces three models for the energy harvester (rectenna), namely the diode linear model, the diode nonlinear model, and the saturation nonlinear model. Section III is dedicated to the study of the fundamental tradeoff between rate and energy in single-user (point-to-point) WIPT for each of the three rectenna models. Special emphasis is given to how deeply the rectenna model influences the R-E tradeoff and WIPT signal and system design. Section IV extends the discussion to multi-user WIPT. Section V discusses recent prototyping and experimentation efforts to validate the signal theory and designs. Section VI concludes the paper.

Throughout the paper, a special emphasis is put on SWIPT as it can be seen as the most involved and disruptive scenario, where wireless communications and wireless power are closely intertwined. Nevertheless, the analysis and ideas reviewed in the paper can also find applications in WPCN and WPBC, as pointed out throughout the manuscript.

I-C The Crucial Role of Energy Harvester Modeling

Fig. 2: The block diagram of a generic WPT system [7].

In order to motivate the importance of the energy harvester modeling, recall first the block diagram of a generic WPT system illustrated in Fig. 2. The end-to-end power transfer efficiency can be decomposed as

(1)

where , , and denote the DC-to-RF, RF-to-RF, and RF-to-DC power conversion/transmission efficiency, respectively.

A natural approach to come up with an efficient WPT architecture would be to concatenate techniques designed specifically to maximize , , and . One could therefore use an efficient Power Amplifier (PA), smart channel-adaptive signals, and an efficient rectenna to maximize , , and , respectively. Doing so, the RF and signal designs are completely decoupled. WPT/WIPT RF designers would deal with efficient PA and rectenna designs and WPT/WIPT signal designers focus on maximizing assuming and constant, i.e., assuming and are not a function of the transmit/received signals but only a function of the PA and rectenna designs, respectively. Though not explicitly stated this way, this is the design philosophy adopted in the early works on SWIPT, WPCN and WPBC, see e.g. [22, 23, 24, 26, 25, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 11, 12, 13, 14, 15, 16, 17, 18].

SWIPT was first considered in [22]. The tradeoff between information rate and delivered energy, the so-called R-E region, was characterized for point-to-point discrete channels, and a Gaussian channel subject to an amplitude constraint on the input. SWIPT was then studied in a frequency-selective Gaussian channel under an average power constraint in [23]. In [24], the term SWIPT was first coined and SWIPT was investigated for multi-user MIMO systems, where practical receivers to realize both RF energy harvesting and information decoding were proposed. Since then, SWIPT has attracted significant interests in the communication literature with works covering a wide range of topics, among others MIMO broadcasting [26, 25], architecture [27, 28], interference channel [29, 30, 31], broadband system [32, 33, 34], relaying [35, 36]. In parallel, much attention has been drawn to WPCN [37, 38] and WPBC [39, 40].

Interestingly, while the above literature addresses complicated scenarios with multiple transmitters and receivers and complicated R-E tradeoff characterizations, results are based on the assumptions that and are constant. Indeed, the DC-to-RF conversion efficiency has been assumed equal to unity and the energy harvester has been abstracted using a linear relationship stating that the output DC power of the energy harvester is equal to its input RF power multiplied by a constant RF-to-DC conversion efficiency [7]. Such a linear model for the energy harvester has the benefit of being analytically easily tractable.

Another approach to designing efficient WPT and WIPT architectures has emerged more recently and relies on observations made in the RF literature that the RF-to-DC conversion efficiency is not a constant but a nonlinear function of the input signal (power and shape) [41, 42, 43, 44, 45, 8]. Assuming constant is indeed over-simplified and is not validated by circuit simulations and measurements. This observation has as consequence that the maximization of is not achieved by maximizing , and independently from each other, and therefore, simply concatenating an efficient PA, an -maximizing signal, and an efficient rectenna [8]. Efficiencies , and are indeed coupled with each other due to the energy harvester nonlinearity [46, 47, 7]. The RF-to-DC conversion efficiency is not only a function of the rectenna design but also of its input signal shape and power and therefore a function of the transmit signal (beamformer, waveform, modulation, power allocation) and the wireless channel state. Similarly, depends on the transmit signal and the channel state and so does , since it is a function of the transmit signal Peak-to-Average Power Ratio (PAPR). Hence, signal design not only influences but also and in general settings. Being able to predict the influence of the signal design on and requires the development of nonlinear models for the PA and the energy harvester, respectively. Of particular interest in this paper is the modeling of the energy harvester and the influence of the signal design on and .

Notations: In this paper, scalars are denoted by italic letters. Boldface lower- and upper-case letters denote vectors and matrices, respectively. denotes the space of complex matrices. denotes the imaginary unit, i.e., . denotes statistical expectation and represents the real part of a complex number. denotes an identity matrix and denotes an all-zero vector/matrix. For an arbitrary-size matrix , its complex conjugate, transpose, Hermitian transpose, and Frobenius norm are respectively denoted as , , , and . denotes the th element of matrix . For a square Hermitian matrix , denotes its trace, while and denote its largest eigenvalue and the corresponding eigenvector, respectively. In the context of random variables, i.i.d. stands for independent and identically distributed. The distribution of a Circularly Symmetric Complex Gaussian (CSCG) random variable with zero-mean and variance is denoted by ; hence with the real/imaginary part distributed as . stands for “distributed as”. We use the notation .

Ii Analytical Models for the Rectenna

The energy receiver in Fig. 2 consists of an energy harvester comprising a rectenna (antenna and rectifier) and a power management unit (PMU). Since the quasi-totality of electronics requires a DC power source, a rectifier is required to convert RF to DC. The recovered DC power then either powers a low-power device directly, or is stored in a battery or a super-capacitor for higher power low duty-cycle operations. It can also be managed by a DC-to-DC converter as part of the PMU before being stored. In the sequel, we will not discuss the PMU but only the rectenna models. We first start by giving a short overview of rectennas before jumping into the rectenna models.

Ii-a Rectenna Behavior

Fig. 3: Examples of single series, voltage doubler, and diode bridge rectifiers, designed for an average RF input power of -20dBm at 5.18GHz. is the voltage source of the antenna [48]. R1 models the antenna impedance. C1 and L1 form the matching network. D1, D2, D3, and D4 refer to the Schottky diodes. C2 and R2 form the low-pass filter with R2 being the output load.

A rectenna harvests electromagnetic energy, then rectifies and filters it using a low pass filter. Various rectifier technologies (including the popular Schottky diodes, CMOS, active rectification, spindiode, backward tunnel diodes) and topologies (with single and multiple diode rectifier) exist [4, 5, 6]. Examples of single series, voltage doubler and diode bridge rectifiers consisting of 1, 2 and 4 Schottky diodes respectively are illustrated in Fig. 3 [48]. In its simplest form, the single series rectifier is made of a matching network (to match the antenna impedance to the rectifier input impedance) followed by a single diode and a low-pass filter, as illustrated by the circuit at the top in Fig. 3.

Assuming =1 Watt (W), 5-dBi Tx/Rx antenna gain, a continuous wave (CW) at 915MHz, of state-of-the-art rectifiers is 50% at 1m, 25% at 10m and about 5% at 30m [2]. Hence, (and as well) decreases as the range increases. Viewed differently, this implies that decreases as the input power to the rectifier decreases. Indeed, of state-of-the-art rectifiers drops from 80% at 10 mW to 40% at 100 W, 20% at 10 W and 2% at 1 W [5, 2]. This is due to the rectifier sensitivity with the diode not being easily turned on at low input power. For typical input powers between 1 W and 1 mW, low barrier Schottky diodes remain the most competitive and popular technology [5, 6]. A single diode is commonly preferred at low power (1-500 W) because the amount of input power required to switch on the rectifier is minimized. Multiple diodes (voltage doubler/diode bridge/charge pump) are on the other hand favoured at higher input power, typically above 500W [6, 4]. Topologies using multiple rectifying devices each one optimized for a different range of input power levels also exist and can enlarge the operating range versus input power variations [49]. This can be achieved using e.g. a single-diode rectifier at low input power and multiple diodes rectifier at higher power.

Fig. 4: Harvested DC power vs average input power and RF-to-DC conversion efficiency with a single-series rectifier obtained from circuit simulations [52]. The input signal is a continuous wave at 5.18 GHz and rectifier designed for -20dBm input power.

Fig. 4 illustrates the dependency of to the average signal power at the input of the rectifier. Using circuit simulations and a single-series rectifier similar to the one at the top of Fig. 3, we plot the DC power harvested at the load as a function of the input power to the rectifier when a CW (i.e. a single sinewave) signal is used for excitation [52]. We also display the RF-to-DC conversion efficiency . This circuit was designed for 10W input power but as we can see it can operate typically between 1W and 1mW. Clearly, the RF-to-DC conversion efficiency is not a constant, but depends on the input power level. It is about 2% at 1W, 15% at 10W and 35% at 100W, which is inline with the values reported from the literature in the previous paragraph. Beyond 1mW input power, the output DC power saturates and suddenly significantly drops, i.e., the rectifier enters the diode breakdown region. Indeed, the diode SMS-7630 becomes reverse biased at 2 Volts (V), corresponding to an input power of about 1mW. To operate beyond 1mW, a rectifier with multiple diodes (similarly to the ones in Fig. 3) would be preferred so as to avoid the saturation problem [4, 6, 49].

The above discussion illustrates the dependency of on the rectifier design and the average received signal power level . Actually is also a function of the rectifier’s input signal shape and not only power. This was first highlighted in [41, 42], wherein the authors proposed the use of a multisine waveform instead of a continuous wave (single sinewave) to provide a higher charge pump efficiency and thus to increase the range of RFID readers. A multisine is characterized by a high PAPR, and the envelope of the transmitted RF signal is designed so that there are large peaks, while the average power is kept the same as in the continuous wave case. Consider indeed multiple in-phase sinewaves (with equal magnitudes) at frequencies , , as the voltage source of the rectenna. As the number of tones increases, the time domain waveform appears as a sequence of pulses with a period equal to illustrated by the red curve in Fig. 5. The signal power is therefore concentrated into a series of high energy pulses, each of which triggers the diode that then conducts and helps charging the output capacitor. Once a pulse has passed, the diode stops conducting and the capacitor is discharging, as illustrated by the blue curve in Fig. 5. The larger the number of tones , the larger is the magnitude of the pulses and therefore the larger is the output voltage at the time of discharge. Since peaks of high power drive the rectenna with a much higher efficiency than the average low level input, they contribute more to the output DC voltage, and the rectifier sensitivity, range and RF-to-DC conversion efficiency increase. A more systematic way to design and optimize multisine waveforms for WPT was proposed in [46]. Though limited to deterministic multisine signals, the discussion illustrates a key starting point of the paper, namely the fact that the RF-to-DC conversion efficiency is influenced by the input signal shape and power to the rectifier.

Fig. 5: Example of the multisine waveform (red) used as a voltage source at the input of the rectenna and the rectified output voltage (blue). See also Fig. 6 for an illustration of a rectenna and the corresponding and .

Modeling the dependency of on the input signal shape and power is very challenging. This is so because RF-based energy harvesting circuits consist of various components such as resistors, capacitors, and diodes that introduce various nonlinearities [5, 6, 50, 51]. This ultimately makes rectenna modeling and analysis an important and challenging research area [6, 50, 51]. Moreover the practical implementation of rectenna is hard and subject to several losses due to threshold and reverse-breakdown voltages, devices parasitics, impedance matching, and harmonic generation [5]. In the sequel, we introduce various models for the rectenna. The first two models, the so-called diode linear model and diode nonlinear model, are driven by the physics of the diode and relate the output DC current/power to the input signal through the diode current-voltage (I-V) characteristics [46]. The diode linear model is a particular case of the diode nonlinear model and is obtained by ignoring the diode nonlinearity [27]. The third model, the so-called saturation nonlinear model, models the saturation of the output DC power at large RF input power due to the diode breakdown. In contrast to the first two models, the third model is circuit-specific and obtained via curve fitting based on measured data [47].

It is important to note that more complicated models can be found in the RF literature, where one could for instance derive mathematical (differential) equations to describe the exact input-output characteristic of an RF-based energy harvesting circuit based on its schematic such as in Fig. 3. However, RF-based energy harvesting circuits may consist of various multistage rectifying circuits. This leads to complicated analytical expressions which are intractable for signal and resource allocation algorithm design. More importantly, such an approach may rely on specific implementation details of energy harvesting circuits and the corresponding mathematical expressions may differ significantly across different types of energy harvesting circuits. In contrast, the three models described in the sequel are driven by a tradeoff between accuracy and tractability. They may appear oversimplified from an RF perspective but the goal here is to extract the key elements of the energy harvester that influences signal and resource allocation design and enables insights for signal and system designs.

Ii-B The Antenna Model

Fig. 6: Antenna equivalent circuit (left) and a single diode rectifier (right) [46]. The rectifier comprises a non-linear device (diode) and a low-pass filter (consisting of a capacitor C and a load RL).

A lossless antenna is modeled as a voltage source followed by a series resistance111Assumed real for simplicity. A more general model can be found in [48]. (Fig. 6 left hand side). Let denote the input impedance of the rectifier and the matching network. Let also denote the RF signal impinging on the receive antenna. Assuming perfect matching (, ), the available RF power is transferred to the rectifier and absorbed by , so that , , and . We also assume that the antenna noise is too small to be harvested.

Ii-C The Diode Linear and Nonlinear Models

Let us now abstract the rectifiers in Fig. 3 into the simplified representation in Fig. 6 (right hand side). We consider for simplicity a rectifier composed of a single series diode followed by a low-pass filter with a load. We consider this setup as it is the simplest rectifier configuration. Nevertheless the model presented in this subsection is not limited to a single series diode but also holds for more general rectifiers with many diodes as shown in [48].

Denote 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 function

(2)

with the reverse bias saturation current , the thermal voltage , the ideality factor assumed to be equal to , around a quiescent operating point . We have

(3)

where and , . Choosing222We here assume a steady-state response and an ideal rectification. Namely the low pass filter is ideal such that is at constant DC level (we drop the dependency on ). Similarly the output current is also at constant DC level . , we can write .

The DC current delivered to the load and the harvested DC power are then given by

(4)

respectively. Note that the operator has the effect of taking the DC component of the diode current but also averaging over the potential randomness carried by the input signal . Indeed, in WIPT applications, commonly carries information and is therefore changing at every symbol period due to the randomness of the input symbols it carries. This randomness due to modulation impacts the diode current and the amount of harvested energy, which is captured in the model by taking an expectation over the distribution of the input symbols [52].

In order to make the signal design tractable and get further insights, we truncate the Taylor expansion at the order. This leads to

(5)

where is an even integer with . The diode nonlinear model truncates the Taylor expansion at the order but retains the fundamental nonlinear behavior of the diode while the diode linear model truncates at the second order term (). Note that the rectifier characteristics are a function of and therefore a function of , which makes it difficult to express explicitly as a function of based on (5). Fortunately, it is shown in [46] that from a transmit signal optimization perspective, maximizing in (5) (subject to an RF transmit power constraint), and therefore in (4), is equivalent to maximizing the quantity

(6)

where . Parameters and are now independent of the quiescent operating point . Readers are referred to [46, 48, 52] for more details on this model.

The diode linear model is obtained by truncating at order 2 such that . Under the linear model, since is a constant independent of the input signal, the best transmit strategy for maximizing , subject to a transmit RF power constraint, is equivalent to the one that maximizes the average input power to the rectenna [46]. In other words, the diode linear model assumes that the RF-to-DC conversion efficiency of the rectifier is a constant independent of [7]. The diode linear model can therefore equivalently be written as with a constant independent of the rectifier’s input signal power and shape.

This is the energy harvester model first introduced in [53] and adopted in the early works on WIPT [27]. It has since then been used extensively throughout the WIPT literature, with among others [22, 23, 24, 26, 25, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 11, 12, 13, 14, 15, 16, 17, 18]. Such a model holds whenever the higher order terms are found negligible. This occurs in the very low input power, , regime or equivalently whenever the voltage drop across is the diode is small as illustrated by region R1 in Fig. 7. Such a regime is commonly denoted as the square-law regime of the diode in the RF literature [4]. According to [54], such a regime occurs for below -20dBm with a continuous wave (CW) input signal. When the input signal is a multisine, the higher order terms become increasingly important as the number of sinewaves increases. This has as a consequence that the square-law regime (where the diode linear model is valid) is shifted towards a lower range of average input power, namely below -30dBm [46, 7, 55]. Recall nevertheless that power levels below -30dBm are very low for operating state-of-the-art rectifiers since the Schottky diode is not easily turned on.

The diode nonlinear model is obtained by truncating to a higher order term with [57, 46]. Choosing for simplicity, and the nonlinearity is characterized through the presence of the fourth-order term . Such a model holds whenever the higher order terms are found non-negligible. This occurs in region R2 in Fig. 7. Region R2 is often called transition region in the RF literature [4]. The transition region ranges from to 0 dBm average input power, when a CW input signal is considered. When using a multisine input signal, the transition region shifts to a lower range of average input powers, e.g. dBm, as given in [55]. Generally speaking, the diode behavior is known in the RF literature to be highly nonlinear in the low power regime of -30dBm to 0dBm, as discussed in [8] and references therein.

For the diode nonlinear model, finding the best transmit strategy so as to maximize , subject to an RF transmit power constraint, does not lead to the same solution as the one that maximizes . This model accounts for the dependence of the RF-to-DC conversion efficiency of the rectifier on the input signal (waveform shape, power, and modulation format) [46, 7]. The diode nonlinear model is a simple form of a memoryless polynomial model that has been widely adopted and validated in the RF literature [56, 43, 4]. It has since then been used in various signal design literature for WPT [58, 59, 48, 60], SWIPT [61, 52, 63, 62, 64, 65, 66, 67, 68] and WPBC [69, 70].

Fig. 7: Diode characteristic showing the three regions of diode operation [4]. R1 and R2 correspond to the diode operation of the diode linear model and the diode nonlinear model, respectively. R3 corresponds to the region where the diode acts as a resistor.
Remark 1

As noted in [46], the Taylor series expansion around a quiescent point is a small-signal model that is valid only for the nonlinear operating region of the diode. If the input voltage amplitude becomes large, the diode will be driven into the large-signal operating region where the diode behavior is dominated by the diode series resistance and the I-V relationship is linear as illustrated by region R3 in Fig. 7 [4].

Ii-D The Saturation Nonlinear Model

The saturation nonlinear model characterizes another source of nonlinearity in the rectenna that originates from the saturation of the output DC power beyond a certain input RF power due to the diode breakdown333Though the term “diode” is not highlighted in “saturation nonlinear model” in contrast to the previous two models, we need to keep in mind that saturation also originates from the diode behavior.. As illustrated in Fig. 4, sharply decreases once the rectifier operates in the diode breakdown region444Operating diodes in the breakdown region is not the purpose of a rectifier and should be avoided as much as possible. 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.. The diode breakdown occurs when the diode is reversed biased with a voltage across the diode being larger than the diode breakdown voltage VBR, as illustrated in Fig. 7. At such a voltage, the breakdown is characterized by a sudden increase of the current flowing in the opposite direction (hence the negative sign of the current in Fig. 7 around the breakdown voltage). This can occur typically when the input power to the rectifier is too large for the power regime it has been designed for.

The saturation nonlinear model is a tractable parametric model proposed in [47], and is applicable to SWIPT systems for a given pre-defined signal waveform and only based on the average received RF power . Unlike the diode nonlinear model discussed in the previous subsection that is based on the physics of the diode, the nonlinear parametric saturation model is fit to measurement results obtained from practical RF-based energy harvesting circuits (excited using the pre-defined signal waveform) via curve fitting. Specifically, the total harvested power at an energy harvesting receiver, , is modeled as:

(7)
(8)

is a sigmoid (logistic) function which has the received RF power, , as input. Constant denotes the maximal harvested power at the energy harvesting receiver when the energy harvesting circuit is driven to saturation due to an exceedingly large input RF power. Constants and capture the joint effects of resistance, capacitance, and circuit sensitivity. In particular, reflects the nonlinear charging rate (e.g. the steepness of the curve) with respect to the input power and determines the minimum turn-on voltage of the energy harvesting circuit.

(a) Linear scale.
(b) Logarithmic scale.
Fig. 8: A comparison between the harvested power for the proposed model in (7) and measurement data obtained for two different practical energy harvesting circuits with a continuous wave (CW) as input signal. Fig (a) and (b) contain the same information but using two different scales, namely linear scales for (a) and logarithmic scales for (b). The measurement data of the upper and lower subfigure (for both (a) and (b)) have been taken from [50] and [51], respectively, showing the different dynamic ranges in harvested energy of practical energy harvesting circuits. The parameters , , and in (7) are calculated with a standard curve fitting tool. The RF-to-DC conversion efficiency of the energy harvesting receiver for the linear energy harvesting model is set to and in the conventional linear model for the upper and lower subfigure, respectively.

This model isolates the resource allocation algorithm for practical SWIPT systems from the specific implementation details of the energy harvesting circuit and signal waveform distribution. In practice, for a given waveform of the adopted RF signal, parameters , , and of the model in (7) can be obtained by applying a standard curve fitting algorithm to measurement results of a given energy harvesting hardware circuit. In Fig. 8, we show two examples for the curve fitting for the saturation nonlinear energy harvesting model. For the upper and lower subfigure in Fig. 8 (a) and (b), the parameters are mW, , and mW, , , for input powers in the mW and W range, respectively. As can be observed, in the high power regime ( dBm mW), the parametric nonlinear model closely matches the experimental results provided in [50] and [51] for the wireless power harvested by a practical energy harvesting circuit. Fig. 8 also illustrates the inability of the conventional (diode) linear model to capture the nonlinear characteristics of practical energy harvesting circuits in the high received RF power regime. In the low power regime, both the conventional (diode) linear model and the saturation nonlinear model experience some discrepancies. The saturation nonlinear model has been widely adopted in the literature for resource allocation algorithm design, e.g. [71][77].

Ii-E Comparisons of The Rectenna Models

Table I provides a comparison of the three models. Further comparisons between the diode linear and nonlinear models can be found in [46, 52, 7]. In particular, it was observed from circuit simulations that the diode nonlinear model more accurately characterizes the rectenna behavior in the practical low power regime. For more discussions on the similarities and differences between the diode nonlienar model and the saturation nonlinear models, the readers are referred to Remark 5 in [52].

Diode Linear Model Diode Nonlinear Model Saturation Nonlinear Model
Operation Regime Characterizes the diode behavior at very low power (below -30dBm) Characterizes the diode behavior at low power (-30dBm to 0dBm) Characterizes the diode/rectenna behavior at high power in/around the diode breakdown region (above 0dBm)
Constant Function of the rectifier input signal power and shape Function of the rectifier input signal power and shape
Philosophy Driven by simplicity Driven by the physics of the rectenna Curve fitting based on measured data
Beamforming Suitable for beamforming design Suitable for beamforming design Suitable for beamforming design
Modulation and Waveform Does not reflect dependence on input signal power and shape. Cannot be used for modulation and waveform design. Does reflect dependence on input signal power and shape. Can be used for modulation and waveform design. Fitted to a given pre-defined signal. Cannot be used for modulation and waveform design design.
Resource Allocation (RA) Suitable for RA optimization Suitable for RA optimization Suitable for RA optimization
Impact Neutral Diode nonlinearity is beneficial Saturation is detrimental. Avoidable by proper (adaptive) rectifier design.
Rectenna Valid for rectifiers with single diode and multiple diodes Parameters are circuit-specific
Applications For system-level performance evaluations For PHY layer signal design and performance evaluations For system-level performance evaluations
TABLE I: Comparisons of the rectenna models.

Ii-F Extension and Future Work

In the following, we review some interesting future research directions. The challenge is finding accurate but tractable models for the energy harvesters that can be used for signal and system design. Software-based models of the energy harvester exist but are insufficiently fast and not insightful to derive new signal design and optimization. Nonetheless, they are very handy when it comes to validating analytical models. On the other hand, simple models such as the linear model can be over-simplified and do not reflect the rectenna behavior accurately enough. The nonlinear models described above try to keep some level of tractability while also improving upon the accuracy compared to the linear model. Nevertheless, much remains to be done in designing rectenna model that are suited to signal and system designs. We here mention a few interesting research avenues.

First, we may think of developing a combined diode and saturation nonlinear model so as to tackle both sources of nonlinearity at once and cope with a wider range of input power.

Second, we may want to provide alternative or enhanced models for the diode and saturation nonlinearities or for the general energy harvester. Some alternative models have emerged in [68, 78, 79, 80, 81]. In view of Fig. 8(b), more works are also needed to better capture the harvester behavior in the low-power regime. Moreover, those models are always assuming CW input signals. It would also be beneficial to design new signals using the diode nonlinear model, validate it through circuit simulations, and then fit data using some curve fitting tool mechanism. The resulting model could then be used for system level evaluations and would capture the dependence on input signal shape and power. The sensitivity is another important characteristic of the energy harvester in the low-power regime that needs to be further investigated [82, 83].

Third, we may need to consider other sources of nonlinearity in the energy harvester, such as the impedance mismatch and the rectifier output harmonics. Modeling accurately the impedance mismatch due to variation in the input signal power (accounting for fading) and shape is a challenge. Unfortunately, due to the dynamic nature of the wireless channel, the input power and signal change dynamically, implying that impedance matching cannot always be guaranteed.

Fourth, nonlinearities were considered at the receiver side but also exist at the transmitter side. Modeling PA nonlinearities jointly with the EH nonlinearity would result in more efficient WPT and WIPT signal designs. One way forward studied in [46] consists in designing transmit signal to maximize the harvested DC power subject to an average power constraint and transmit PAPR constraints. Such a design leads to a new tradeoff since low PAPR signals are preferred at the transmitter but high PAPR signals at the input of the energy harvester.

Fifth, the design and modeling of energy harvester for other frequency bands, e.g. millimeter-wave band, is also of high interests. At those frequencies, the diode linear model was also shown not to accurately model the rectification behavior of the diode [84].

Iii Single-User WIPT

In this section, we first introduce the signal model used throughout the manuscript. We then discuss various receiver architectures and formulate the R-E region maximization problem. The core part of the section is dedicated to characterizing the R-E region (and the corresponding signal design strategies) for the three energy harvester models.

Iii-a Signal and System Model

We consider a single-user point-to-point MIMO SWIPT system in a general multipath environment. This setup is referred to as “SWIPT with co-located receivers” in Fig. 1. The transmitter is equipped with antennas that transmit information and power simultaneously to a receiver equipped with receive antennas. We consider the general setup of a multi-subband transmission (with a single subband being a special case) employing orthogonal subbands where the subband has carrier frequency and all subbands employ equal bandwidth , . The carrier frequencies are evenly spaced such that with the inter-carrier frequency spacing (with ).

The SWIPT signal transmitted on antenna , , is a multi-carrier modulated waveform with frequencies , , carrying independent information symbols on subband . The transmit SWIPT signal at time on antenna is given by

(9)

with the baseband equivalent signal given by

(10)

where denotes the complex-valued information and power carrying symbol at time index , modeled as a random variable generated in an i.i.d. fashion. has bandwidth .

The transmit SWIPT 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 receive antenna for subband . The signal received at antenna () from transmit antenna can be expressed as

(11)

We have assumed so that, for each subband, are narrowband signals, thus , . Variable is the baseband channel frequency response between transmit antenna and receive antenna at frequency .

The total signal and noise received at antenna is the superposition of the signals received from all transmit antennas, i.e.,

(12)

where is the antenna noise, denotes the channel vector from the transmit antennas to receive antenna and denotes the signals transmitted by the antennas in subband . Next, the processing depends on the exact SWIPT receiver architecture. Nevertheless, a commonality exists among all considered types of receivers. Namely, from an energy perspective, (or a fraction of it) is conveyed to an ER, where energy is harvested directly from the RF-domain signal. From an information perspective, (or a fraction of it) is conveyed to an IR, where it is first downconverted and filtered to produce the baseband signal for subband

(13)

where is the downconverted received filtered noise, accounting for both the antenna and the RF-to-baseband processing noise. Sampling with a sampling frequency to produce the sampled outputs at time instants (multiples of the sampling period), we can write the baseband system model as follows

(14)

with . Due to the assumption of i.i.d. channel inputs and the discrete memoryless channel, we can drop the time index and simply write

(15)

We model as an i.i.d. and CSCG random variable with variance , i.e., , where is the total Additive White Gaussian Noise (AWGN) power originating from the antenna () and the RF-to-baseband processing ().

After stacking the observations from all receive antennas, we obtain

(16)

where , , and denotes the MIMO channel matrix from the transmit antennas to the receive antennas at subband .

Ignoring the noise power, the total RF power received by all antennas of the receiver can be expressed as

(17)

where the positive semidefinite input covariance matrix at subband is defined as . The total average transmit power is expressed as

(18)

with . For convenience, we also define as the transmit power in subband . Throughout the manuscript, we will assume that the total average transmit power is subject to the constraint .

Finally, we assume perfect Channel State Information at the Transmitter (CSIT) and perfect Channel State Information at the Receiver (CSIR).

Iii-B Receiver Architectures

Various architectures for the integrated information and energy receivers in Fig. 1 have been proposed.

(a) Ideal Receiver
(b) TS Receiver
(c) PS Receiver
Fig. 9: Three receiver architectures for SWIPT: (a) Ideal receiver (using the same signal for both the ID and EH receivers); (b) TS receiver (switching the signal to either ID or EH receiver); and (c) PS receiver (splitting a portion of the signal to ID receiver and the rest to EH receiver).

An Ideal Receiver (Fig. 9(a)) is assumed to be able to decode information and harvest energy from the same signal [22, 23]; however, this cannot so far be realized by practical circuits. With such an architecture, is conveyed to the energy harvester (EH) and also simultaneously RF-to-baseband downconverted and conveyed to the information decoder (ID). Different R-E tradeoffs could be realized by varying the design of the transmit signals to favor rate or energy.

A Time Switching (TS) Receiver (Fig. 9(b)) consists of co-located ID and EH receivers, where the ID receiver is a conventional baseband information decoder; the EH receiver’s structure follows that in e.g. Fig. 3 [24, 27, 29]. In this case, the transmitter divides the transmission block into two orthogonal time slots, one for transferring power and the other for transmitting data. At each time slot, the transmitter could optimize its transmit waveforms for either energy transfer or information transmission. Accordingly, the receiver switches its operation periodically between harvesting energy and decoding information in the two time slots. Then, different R-E tradeoffs could be realized by varying the length of the energy transfer time slot, jointly with the transmit signals.

In a Power Splitting (PS) Receiver (Fig. 9(c)), the EH and ID receiver components are the same as those of a TS receiver. The transmitter optimizes the transmitted signals jointly for information and energy transfer and the PS receiver splits the received signal into two streams, where one stream with PS ratio is used for EH, and the other with power ratio is used for ID [24, 27, 28]. Hence, assuming perfect matching (as in Section II-B), the input voltage signals and are respectively conveyed to the EH and the ID. Different R-E tradeoffs are realized by adjusting the value of jointly with the transmit signals.

Iii-C Rate-Energy Region and Problem Formulation

The focus of this paper is the characterization of the Rate-Energy (R-E) tradeoff and the corresponding signaling strategies for the various receiver architectures for the linear and nonlinear EH models. We define the R-E region as the set of all pairs of rate and energy such that simultaneously the receiver can communicate at rate and harvested energy . The R-E region in general is obtained through a collection of input distributions that satisfies the average transmit power constraint . Mathematically, we can write

(19)

where refers to the mutual information between the channel input and the channel output on subband and , function of , refers to (4) and (7) for the (linear and nonlinear) diode model and the saturation nonlinear model, respectively. For the diode models, since directly relates to the current and therefore (defined in (6)), it is more convenient to define the R-E region in terms of , such that inequality in (19) is replaced by .

In order to characterize the R-E region, one solution is to obtain the capacity (supremization of the mutual information over all possible distributions of the input) of a complex AWGN channel subject to an average power constraint and a receiver delivered/harvested energy constraint , for different values of . Namely,

(20)
(21)
(22)

where is interpreted as the minimum required or target harvested energy. Here again, for the diode models, it is more convenient to formulate Problem (20)-(22) in terms of metric such that constraint simply writes as .

In the rest of this paper, we focus on the case when the power of the processing noise is much larger than that of the antenna noise, i.e., , such that . As explained in [24], the above setting results in the worst-case R-E region for the practical PS receiver. This can be inferred by considering the other extreme case of and hence . In this case, it can be easily shown that the achievable rate for the ID receiver is independent of the PS ratio, and thus the optimal strategy for PS is to use an infinitesimally small split power of the received signal for ID and the remaining for EH, which achieves the same box-like R-E region (see Fig. 10) as the ideal receiver [24]. As a result, we mainly consider the R-E region for the worst case of , which serves as a performance lower bound for practical PS receivers.

Iii-D Rate-Energy Tradeoff with The Diode Linear Model

In this subsection, we study the R-E tradeoff for the diode linear model starting with the simplest case of a SISO single-subband transmission. We then extend the results to multi-subband transmission and multi-antenna transmission, before drawing some general conclusions about SWIPT signal and architecture design for the diode linear model.

Iii-D1 Single-Subband Transmission

Let us first assume a SISO () single-subband () transmission and the ideal receiver. The system model in (15) simplifies to and the delivered power can be expressed as , where we assumed that the noise is negligible for energy harvesting. Problem (20)-(22) can then be written as

(23)
(24)
(25)

Following [22, 85], the optimal input distribution555We here consider an average power constraint only. Under average power and amplitude constraints, the optimal capacity achieving distribution is discrete with a finite number of mass points for the amplitude and continuous uniform over for the phase [86, 87, 22]. is CSCG with average transmit power , namely , and there is no tradeoff between rate and energy, as noticed in [23]. In other words, the R-E region is a rectangle characterized by (26) illustrated in Fig. 10.

For the TS and PS receivers, CSCG input is again optimal for the diode linear model. TS leads to a triangular R-E region characterized by (27) where is the fraction of time used for energy harvesting. PS leads to a concave-shape R-E region characterized by (28) where is the PS ratio. Hence, in the single-subband case with the diode linear model, the tradeoff between rate and energy is actually induced by the receiver architecture, not by the transmit signal.

(26)
(27)
(28)

Comparing the three considered regions, we observe that . Hence, a TS receiver is outperformed by a PS receiver, and they are both outperformed by the ideal receiver. This is further illustrated in Fig. 10.

Fig. 10: Comparison of R-E tradeoffs of different SWIPT receivers with the diode linear model. The parameters are set as follows: , the channel power is , the transmit power is , the noise power is .

Iii-D2 Multi-Subband Transmission

Let us now consider the SISO multi-subband transmission such that (15) becomes in subband . This was first investigated in [23] for the ideal receiver. Following [23], the use of independent CSCG inputs in each subband, i.e., , is optimal and the R-E tradeoff results from the power allocation across subbands. Indeed, while the maximization of energy subject to an average sum power constraint favors allocating all power to a single subband, namely the one corresponding to the strongest channel , the maximization of rate subject to an average sum power constraint in general allocates power to multiple subchannels following the standard water-filling (WF) solution [85]. Hence, there exists a non-trivial tradeoff between rate and energy in the multi-subband case and the best power allocation can be formulated as the solution of the optimization problem

(29)
(30)
(31)

which yields a modified WF solution [23]. Specifically, let and denote the optimal dual variables corresponding to the transmit sum-power constraint (30) and the total harvested power constraint (31). Then, the optimal transmit power allocation is given by [23]

(32)

. It can be observed that if the energy harvesting constraint (31) is not active, i.e., , (32) reduces to the conventional WF power allocation with a constant water-level for all subbands. However, when the energy harvesting constraint is tight, i.e., , the water-level is higher on subbands with stronger channel power. This indicates that due to the energy harvesting constraint, the power allocation among subbands is more greedy (i.e., more power is assigned to stronger subbands) than the conventional WF power allocation.

The TS architecture relies on time-sharing between conventional WF (for rate maximization) and transmission on the strongest subband (for energy maximization). In the PS architecture, the PS ratio (same for all the subbands) and the power allocations across subbands can be jointly optimized [33]. Similarly to the single-subband case, also holds for the multi-subband case. In fact, this result can be obtained from [24], which considers the general MIMO system model (see next subsection for more details). As shown in [24], for arbitrary MIMO channel matrix , under the so-called uniform power splitting (UPS) scheme, in which the PS ratios in each dimension of are identical, the corresponding R-E region is always no smaller than that achieved by applying TS in each dimension of . As a result, in a multi-subband SISO system, follows directly by restricting in [24] to an -by- diagonal channel.

Hence, in the multi-subband case for the diode linear model, a tradeoff between rate and energy is induced by the power allocation strategy at the transmitter additionally to the tradeoff already induced by the receiver architecture (as in the single-subband case).

Iii-D3 Multi-Antenna Transmission

Let us now consider a MIMO transmission and assume a single subband for simplicity such that (16) becomes . Similarly to the SISO case, following [88], the maximization of the mutual information subject to average transmit power and received power constraints is achieved by CSCG inputs. Problem (20)-(22) becomes

(33)
(34)
(35)

In the above problem formulation, we assume that each receive antenna is equipped with an energy harvester and the constraint reflects that the total harvested energy across all rectennas should be larger than . The choice of the input covariance leads to a non-trivial R-E tradeoff [24]. Let us write the eigenvalue decomposition . The harvested energy is maximized by choosing the covariance matrix as where denotes the eigenvector corresponding to the dominant eigenvalue of . Rate maximization on the other hand is obtained through multiple eigenmode transmission (spatial multiplexing) along the eigenvectors of and with a power allocation across eigenmodes based on the conventional MIMO WF solution [88], i.e., leading to a covariance matrix of the form with the diagonal matrix obtained from the standard WF power allocation solution. The optimal solution of the R-E region maximization problem (33)-(35) can also be expressed in form of a multiple eigenmode transmission with , where the diagonal matrix is obtained from a modified WF power allocation [24]. As explained in Section III-D2, the above optimal precoder design with the modified WF power allocation is more general than the optimal power allocation for the multi-subband SISO system considered in [23], since the channel model in [23] is a special case of that considered in [24] with being diagonal.

The TS architecture relies on time-sharing between the conventional eigenmode transmission (for rate maximization) and aligning one energy beam towards the eigenvector corresponding to the strongest eigenvalue of (for energy maximization) [24]. In contrast, with the PS architecture, the transmit precoder and PS ratios of receive antennas can be jointly optimized to obtain various points on the boundary of the achievable R-E region. Moreover, as mentioned in Section III-D2, a low-complexity UPS scheme is considered in [24] under which the PS ratios are identical for all receive antennas. Let denote the corresponding R-E region. Then, it follows from [24] that .

Note that in the MISO setup (), , with and the optimal covariance matrix for energy and rate maximization coincide. The transmitter simply performs conventional Maximum Ratio Transmission (MRT), with , which maximizes both energy and rate. Hence, there is no R-E tradeoff and the R-E region is a rectangle characterized by (26) with replaced by , and therefore enlarged compared to the SISO case thanks to the beamforming gain. Similarly, for the TS and PS receivers, the R-E regions are given by (27) and (28), respectively, with replaced by .

Remark 2

Note that while CSCG is optimal for the ideal, TS, and PS receivers in single-subband, multi-subband and multi-antenna transmissions for maximizing the R-E region under the diode linear model, from an energy maximization-only perspective, any input distribution with an average power would be optimal. In particular a continuous wave (CW) would do as well as a CSCG input while modulated and unmodulated waveforms are equally suitable from an energy maximization perspective under the diode linear model. Hence, in TS, the R-E region can also be achieved by time sharing with CSCG inputs during the information transmission phase and with CW during the power transmission phase.

Observation 1

The use of the diode linear model leads to three important observations. First, the strategy that maximizes maximizes . Second, CSCG inputs are sufficient and optimal to achieve the R-E region boundaries. Third, .

Iii-E Rate-Energy Tradeoff with The Diode Nonlinear Model

The first systematic signal designs for WPT accounting for the diode nonlinearity appeared in [57, 46]. Uniquely, this nonlinearity was shown to be beneficial for system performance and be exploitable (along with a beamforming gain and a channel frequency diversity gain) through suitable signal designs. It was observed that signals designed accounting for the diode nonlinearity are more efficient than those designed based on the diode linear model. Interestingly, while the diode linear model favours narrowband transmission with all the power allocated to a single subband (as in Section III-D2), the diode nonlinear model favours a power allocation over multiple subbands and those with stronger frequency-domain channel gains are allocated more power. The optimum power allocation strategy results from a compromise between exploiting the diode nonlinearity and the channel frequency diversity.

The works [57, 46] assumed deterministic multisine waveforms. Designing SWIPT requires the transmit signals to carry information and therefore to be subject to some randomness. This raises an interesting question: How do modulated signals perform in comparison to deterministic signals for energy transfer? Recall from Remark 2 that modulated and unmodulated inputs are equally suitable for energy maximization according to the diode linear model. Interestingly, it was shown in [52] that modulation using CSCG inputs leads to a higher harvested energy at the output of the rectifier compared to an unmodulated input, despite exhibiting the same average power at the input to the rectenna. This gain comes from the large fourth order moment offered by CSCG inputs, which is exploited by the rectifier nonlinearity and modeled by the fourth order term in . Indeed with CSCG inputs , , which is twice as large as what is achieved with unmodulated CW inputs with the same average power [52].

This highlights that the signal theory and design for SWIPT, such as modulation, waveform, and input distribution, are actually influenced by the diode nonlinearity. This motivates the design of SWIPT signals that intelligently exploit the diode nonlinearity.

Iii-E1 Single-Subband Transmission

Assuming a SISO transmission and the ideal receiver, Problem (20)-(22) becomes

(36)
(37)
(38)

As shown in [64], a baseband equivalent for (left hand side of (38)) is not only a function of as with the diode linear model but also a function of , , with .

Interestingly, the presence of the higher moments of the input distribution has a significant impact on the choice of the input distribution . In [64], assuming general non-zero mean Gaussian inputs and with , it is found that the supremum in Problem (36)-(38) is achieved with and by zero mean asymmetric Gaussian inputs. CSCG input obtained by equally distributing power between the real and the imaginary dimensions, i.e., and is optimal for rate maximization. However, as the harvested power constraint increases, the input distribution becomes asymmetric with the power allocated to the real part increasing and the one to the imaginary part decreasing (or inversely) up to a point where the rate is minimized and the energy is maximized by allocating the transmit power to only one dimension, e.g. . This is because allocating power to one dimension leads to a higher fourth moment. Indeed, for in contrast to with . The R-E region obtained with asymmetric Gaussian inputs is illustrated in Fig. 11.

Hence, in contrast with the diode linear model, a R-E tradeoff exists in SISO single-subband transmission with an ideal receiver for the diode nonlinear model. The tradeoff is induced by the presence of the fourth moment of the received signal in . Moreover, the R-E region achieved by the diode nonlinear model-motivated input distribution leads to an enlarged R-E region compared to that achieved by the diode linear model-motivated input distribution. In other words, the diode nonlinearity is beneficial to SWIPT system performance if properly exploited.

Fig. 11: R-E region with asymmetric Gaussian inputs in single-subband transmission (