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Waveform Design for Wireless Power Transfer with Power Amplifier and Energy Harvester Non-Linearities

Waveform optimization has shown its great potential to boost the performance of far-field wireless power transfer (WPT). Current research has optimized transmit waveform, adaptive to channel state information (CSI), to maximize the harvested power in WPT while accounting for energy harvester (EH)'s non-linearity. However, the existing transmit waveform design disregards the non-linear high power amplifiers (HPA) at the transmitter. Driven by this, this paper optimizes the multi-carrier waveform at the input of HPA to maximize the harvested DC power considering both HPA's and EH's non-linearities. Two optimization models are formulated based on whether the frequencies of the multi-carrier waveform are concentrated within the transmit pass band or not. Analysis and simulations show that, while EH's non-linearity boosts the power harvesting performance, HPA's non-linearity degrades the harvested power. Hence, the optimal waveform shifts from multi-carrier that exploits EH's non-linearity to single-carrier that reduces HPA's detrimental non-linear distortion as the operational regime of WPT becomes more sensitive to HPA's non-linearity and less sensitive to EH's non-linearity (and inversely). Simultaneously, operating towards HPA's non-linear regime by increasing the input signal power benefits the harvested power since HPA's DC power supply is better exploited, whereas the end-to-end power transfer efficiency (PTE) might decrease because of the increasing non-linear degradation. Throughout the simulations, the proposed waveforms show significant gain over those not accounting for HPA's non-linearity, especially in frequency-flat channels. We also compare the two proposed waveforms and show that the severity of HPA's non-linearity dictates which of the two proposed waveforms is more beneficial.


I. INTRODUCTION
Future networks are expected to cope with the emergence of trillions of low power sensors for applications in the Internet of Things (IoT). However, the explosion in the number of these lowpower devices is challenging traditional battery power supply or wired power supply which either needs periodic replacement or incurs deployment limits. Nevertheless, the dramatical reduction in the power consumption of these remote devices, attributed to higher efficient electronics and computation architectures, gives rise to developing far-field WPT as an alternative source of power [1], [2].
Far-field WPT generates and transmits suitable RF signals that propagate over the air before being captured and rectified into DC current via rectenna circuits at the receivers. The flexible availability of RF signals, as the power source, releases WPT from batteries or wires, which makes it more sustainable, flexible, and controllable. Since the discovery of WPT's promising potential to address the powering bottlenecks in IoT, scientists have been making every The authors are with the Department of Electrical and Electronic Engineering, Imperial College London, London SW7 2AZ, U.K. (e-mail: b.clerckx@imperial.ac.uk, yumeng.zhang19@imperial.ac.uk) effort to boost the PTE of WPT by circuit-level design [3], [4], RF signal design [5], [6], and adopting new mechanisms from novel physical effects and materials [5], [7].
Signals and waveform play a crucial role in the PTE of WPT. The RF-to-DC PTE, defined as the ratio of the generated DC power at the rectenna against the RF signal power at the transmitter, has been comprehensively experimented given different types of signals and rectenna structures [8], [9]. Experimental research verifies the advantages of the signals featuring a high peak-to-averagepower ratio (PAPR) regarding the RF-to-DC power conversion efficiency, but without consideration of CSI [9], [10]. A noteworthy contribution later is in [11] where the multi-sine waves are theoretically designed, adaptive to CSI and the rectenna's non-linear characteristics, to maximize the RF-to-DC PTE. Specifically, [11] characterized the non-linear rectenna via Taylor expansion and then expressed the generated DC power as a non-linear function of the transmitted waveform. Waveform optimization based on this model has shown a significant gain in the power harvesting performance over those assuming linear rectennas [6], [12], [13].
The model in [11] gives rise to a further flourishment in the field of signal design in far-field WPT. A corresponding low-complexity algorithm was developed in [14], and the efficiency of these waveforms were validated through prototyping and experimentation in [15]. Moreover, the model has been extended to general scenarios such as limited-feedback [16], large-scale [12], multi-input-multi-output [17], multi-user [12], [13] and opportunistic/fair-scheduling [18], [19]. For a further improvement on PTE, hybrid beamforming and intelligent reflective surfaces (IRS) have also been exploited [20]- [22]. Besides the performance advancement, WPT has also found applications in integrated systems such as wireless information and power transfer (WIPT) [23] and wireless powered backscatter communications [24]. However, the above waveform was performed without much consideration of the transmitter's architecture, especially the most non-linear and power-consuming component, HPA. Indeed, recent research has experimentally verified and characterized the performance degradation caused by non-linear HPAs in WPT/WIPT systems [25]- [29]. Effort has also been made to explore the beneficial signals that compromise both HPA's and EH's non-linearity in WPT systems [30]. [31] claimed the advantage of QPSK signals over uniformly power-allocated multi-carrier signals in the presence of HPA's and EH's non-linearity, but was based on a simplified platform compared with [15] and ignored the performance gain achievable from the CSI knowledge as in [11]. An interesting observation is in [5] which shows how the optimal modulation in WIPT changes as a function of HPA's and EH's non-linearity using machine learning techniques, i.e., to allocate more power-driven high-amplitude symbols when considering HPA's non-linearity. Hence, HPA's non-linearity not only degrades performance but also changes system preference. This drives our interest in exploring how to address HPA's non-linearity in WPT systems and how the optimal waveform changes when both HPA's and EH's non-linearity are considered for maximal harvested power.
To combat HPA's non-linear effect at the waveform-design level, mainly two lines of methods have been proposed, namely designing signals less susceptible to HPA's non-linearity and by means of digital pre-distortion (DPD). The former method decreases the input signals' exposure to HPA's non-linear region, such as PAPR reduction [32], distortion power minimization [32] and out-of-band leakage power reduction [33]. Indeed, PAPR reduction has been introduced as a transmit waveform constraint in WPT in [11]. However, this class of methods might be less efficient in the system-level design since they either neglect channels and receiver architectures [32], [33] or dismiss HPA's transfer characteristics [11]. In contrast, DPD pre-distorts the desired input signal according to HPA's transfer characteristics to linearize the transfer function of the joint pre-distorter-and-HPA structure or, if not achievable, to minimize the non-linear effect of the HPA on system performance [34]. In the latter case, the signal is designed to compensate for HPA's non-linearity and also to adapt to system characteristics.
Following the above analysis, this paper explores designing the input waveform at HPA to boost the power harvesting performance in a practical WPT system model accounting for both HPA's and rectenna's non-linearity. The contributions of the paper are summarized as follows: (1) First, we model a practical WPT system accounting for both the transmitter's and receiver's non-linearities as in Fig. 1. The transmitter is composed of a non-linear HPA and a BPF without loss of generality. On this basis, we develop the relationship between the input waveform of HPA x in (t) (the input waveform), the waveform after HPA x HPA (t), the transmit waveform at the antenna x tr (t) (the transmit waveform), the receive waveform before and after rectification. The harvested power is characterized using the non-linear rectenna model of [11]. Based on the transmitter's and rectenna's model, we formulate an end-to-end waveform optimization problem that maximizes the generated DC power given power constraints both at the HPA's input (input power constraints) and at the transmit antennas (transmit power constraints). This contrasts with all prior works on waveform optimization [5], [11], [12], [14] that consider non-linearity at the rectenna side only.
(2) Second, we formulate and solve a first optimization model where the input waveform shares the same frequencies as those of the transmit waveform (within the transmit pass band), abbreviated as Model I. In the problem formulation, we express the harvested power (objective function) and the transmit power constraints as a function of the transmit waveform and express the input power constraint as a function of the input waveform, with the relationship between the input waveform and the transmit waveform being clarified through multiple non-linear equalities. By making both the input waveform and the transmit waveform optimization variables, we separate the high-complexity of directly handling the objective function and the transmit power constraint as a function of the input waveform into two parts for successive management, i.e., to handle the objective and transmit power constraint with simpler forms first (in function of the transmit waveform) and then to deal with the non-linear equality constraints (between the input waveform and the transmit waveform). Specifically, we adopt successive convex programming (SCP) to approximate the objective function first and then use sequential quadratic programming (SQP) to handle the non-linear constraints, abbreviated as SCP-SQP.
(3) Third, we formulate and solve a second optimization model where the out-of-band frequencies of the transmit frequencies are utilized additionally when designing the input waveform of HPA, abbreviated as Model II. The aim is to concentrate the frequencies of the waveform after HPA within the pass band of the BPF in Fig. 1 (or the transmit pass band) to avoid power loss at BPF. This assumption allows the input power constraint to be re-formulated as a function of the transmit waveform, making the whole optimization program function of the transmit waveform only (instead of both input and transmit waveforms as in Model I). Consequently, this halves the number of variables in Model I and avoids numerous non-linear equality constraints, making the program more stable in large-scale problems. The resulting problem is solved by SCP and the interior-point (IP) method, abbreviated as SCP-IP.
(4) Fourth, we provide simulation results to verify the performance gain of waveforms that consider both HPA's and EH's non-linearity over those that ignore HPA's non-linearity. We also demonstrate that HPA's non-linearity, in sharp contrast with EH's non-linearity, degrades the harvested power in WPT significantly, especially in frequency-flat channels. To compensate for the power loss from HPA's non-linearity, the proposed waveforms show a tendency to allocate power to fewer sub-carriers as the HPA experiences more non-linearity. Simulations also show that operating towards the non-linear regime of HPA exploits its DC power supply more and gives larger harvested power and higher end-to-end PTE. However, if further increasing the input signal power to where HPA saturates, the harvested power saturates while the end-to-end PTE decreases correspondingly.
Organization: Section II introduces the system model and section III models the non-linear HPA and rectenna. Section IV addresses the waveform optimization given two different assumptions on the frequencies of the input signal at HPA. Section V evaluates the performance and section VI concludes the work.
Notations: Matrices and vectors are respectively denoted in form of bold upper case and bold lower case. Complex symbols are characterized by a tilde superscript, i.e., x. R{ x} and I{ x} denote the real and imaginary part of the complex number x respectively, and are referred to as x and x. x represents the amplitude of the complex number x, and x / X represents Frobenius norm of vector x or matrix X. I Nt denotes N t ×N t an identity matrix, (·) H represents the conjugate operation, (·) T represents the transpose operation, and ! denotes the factorial operation. N means the circular convolution of length N. {x n } represents the set that collects all the related x n , {x n } (n1) means to take the n 1 th entry from the set {x n }, and X i,j means to take the (i,j) th entry from X i,j . For the power notations, the AC current/voltage is denoted by lower case while the DC current/voltage is denoted by the upper case. For Fourier Transform, F {x(t)}| f=f0 means to do FT on signal x(t) and take the spectrum at frequency f 0 . Correspondingly, for discrete Fourier transform (DFT), define the index n across the frequency domain and the index k across the time domain. Then, , taking the n th entry is given by,

II. WPT SYSTEM MODEL
In this section, we introduce a practical WPT architecture with a non-linear HPA at the transmitter and a non-linear EH at the receiver. At the transmitter, the generated multi-carrier signal is first amplified and then filtered before being transmitted. The RF signal then propagates through Rayleigh fading channels and is picked up by the EH for the purpose of power harvesting. The functioning of different building blocks will be clarified throughout the paper.
We consider a single-user single-input-single-output (SISO) multi-carrier WPT system as depicted in Fig. 1. The non-linear analysis of the transmitter starts from the input of the non-linear HPA at the RF chain. For simplicity, we denote the complex signal at the input/output of the HPA as x in/HPA (t). Denote the complex transmit signal as x tr (t). We consider a multi-carrier scenario where N evenly frequency-spaced sub-carriers are finally transmitted. The frequency of the n th (n = 0,...,N − 1) transmitted sub-carrier of x tr (t) is f n = f 0 +(n−1)∆ f with f 0 being the lowest sub-carrier frequency and ∆ f being the frequency spacing.
At the transmitter, the complex signal at the input of the amplifier is: where x in (t)=| x in (t)| is the amplitude of the complex signal x in (t) and x in (t) is the phase of complex signal x in (t). w in n =w in n +j w in n denotes the complex weight of the n th sub-carrier, with w in n and w in n being the corresponding real and imaginary part respectively. κ in is the integer set which indicates the frequency range that x in (t) involves 1,2 .
Similarly, the complex signal at the output of the HPA is: 1 If κ in = [0,...,N −1], x in (t) will have the same activated frequencies with the transmit signal x tr (t). In this case, the signal x HPA (t) will have activated frequencies at the multiples of ∆ f because of the intermodulation interference from HPA's non-linearity. This corresponds to the first model in the paper 2 Otherwise, if we utilize the frequencies out of the transmit pass band at HPA's input to concentrate the frequencies of x HPA (t) within the in-band frequencies and fn = f 0 +n∆ f in Eq. (2). This corresponds to the second model in the paper where w HPA n =w HPA n +j w HPA n denotes the complex weight of the n th sub-carrier, with w HPA n and w HPA n being the corresponding real and imaginary part respectively. κ HPA represents the frequency range that signals x HPA (t) involves.
The BPF then removes frequencies out of the transmit passband [f 0 ,f 0 +(N −1)∆ f ] , which gives the real transmit signal as: where x tr (t) is the complex form of the transmit signal. w tr n =w tr n + j w tr n denotes the complex weight of the n th sub-carrier, with w tr n and w tr n being the corresponding real and imaginary part respectively. The transmitted signal x tr (t) then propagates through a multipath channel with L paths. Assume that the signal received from the l th path is: where α l denotes the amplitude suppression degree at the l th path and τ l denotes the path delay, assuming a narrowband balanced array here. ζ l is the phase shift propagating through the l th path. Then, the total complex signal at the receiver antenna is the sum over Eq. (5): where h n = L l=1 α l e j2π(−fnτ l +ζ n,l ) denotes the equivalent complex channel that the n th sub-carrier experiences during transmission.

III. HPA AND RECTENNA ANALYTICAL MODELS
This section models the HPA's and rectenna's transfer characteristics in WPT. We develop the relationship between the signals at each stage, and finally characterize the harvested power explicitly.
A. HPA Model 1) HPA model characterization: The input signal x in (t) is amplified and filtered before being transmitted. Generally, a practical HPA will exert non-linear amplitude distortion and phase shift on its input signal. The HPA's distorted effect on both amplitude and phase are normally characterized by functions with respect to the amplitude of the input signal. Mathematically, the transfer characteristic of a practical HPA can be interpreted by: where x=xe j x with x being the amplitude of x and x being the phase of x. Function A(x) represents the amplitude distortion of HPA on the input signal, and function φ(x) characterizes the phase shift after the HPA. In this paper, we focus on a classic solid-state-power amplifier (SSPA) modelled in [35]: where G is the small-signal amplifier gain of SSPA, A s is the saturation voltage of SSPA, and β (positive) is the smoothing parameter of SSPA. Eq. (9) indicates that the SSPA does not have phase shift.
Features of SSPA are clarified in Fig. 2, where the small signal input is offered an amplifier gain of G=2 approximately while the large signal input is shown to be suppressed below the saturation voltage of SSPA (A s ). The larger the smoothing parameter β, the higher the linearity SSPA exhibits below its saturation voltage. Combing Eq. (2) and Eq. (7), the complex signal at the output of the HPA (SSPA) becomes: 2) HPA non-linearity and efficiency measurement: Fig. 2 reveals that HPA's non-linear characteristics depend highly on its operation region. Hence, to show how HPA's non-linearity affects WPT's waveform preference, it would be beneficial to evaluate HPA's non-linearity numerically first. An effective measurement is the output backoff (OBO) at HPA's operating point, which is defined as the ratio of HPA's saturation power to the output RF signal's power: where x HPA (t)=| x HPA (t)| is the amplitude of the signal after HPA. Smaller OBO indicates more severe HPA's non-linearity.
Besides the severity of non-linearity, PE is also an important metric to evaluate HPA's performance, defined as the ratio of the output RF signals' power to the DC power that HPA supplies. The PE of an amplifier depends on the circuits of the amplifier. Here, we use a generalized PE approximation for all Class B HPAs (features relatively high PE) in [36]: From Eq. (12), the PE of HPA describes the efficiency of using HPA's DC power supply. Usually, higher input signal power drives more efficient use of HPA's DC power supply and produces higher output signal power, until HPA's saturation voltage is reached where the produced output power saturates.
One limitation of the PE metric in Eq. (12) is that it does not account for the power source from the input signal. When the amplifier gain is not sufficiently high or when HPA approaches its saturation region, the output power is comparable to or even less than the input power of HPA. In this context, a more effective metric of HPA efficiency is the added PE (APE), where the power of the input signal is also taken into consideration. APE is defined as the ratio of the produced signal power (difference between the input and output RF power) and the DC power supplied to amplifier: APE gives us an insight of HPA's PE considering all its power source, including the input signal power and the DC power supply. In contrast with the PE metric in Eq. (12) which prefers high input power, the APE in Eq. (13) might decrease or even become negative when the input signal power increases to HPA's highly non-linear operation regime, where the produced output power saturates regardless of the input signal power. In this condition, increasing the input signal power is not beneficial in terms of the whole system's PTE, and APE would then become an important measurement metric.
3) Analytical insights into HPA and BPF outputs: In this section, we assume that the input waveform of HPA has the same activated frequencies as those of the transmit waveform, from f 0 to f N−1 with frequency spacing ∆ f . In this case, due to HPA's non-linearity, x HPA (t) incurs distorted weights across the in-band frequency components and intermodulation leakage in the out-of-band frequencies at the multiples of ∆ f . For the convenience of the following optimization formulation, we need to express the waveforms after HPA explicitly.
Towards this, our main idea is to sample x HPA (t) in Eq. (10) and then conduct DFT. First, we need to decide the number of samples according to the Sampling Theorem. Starting from Eq. (10), we have: where x in B (t) = x in (t)e −2jπf0t is the baseband signal of x in (t). Eq. (14) indicates that the spectrum of x HPA (t) is the convolution between the spectrum of the baseband function A(x in B (t)) and the spectrum of x in (t). The latter part is easily represented by w in n from Eq. (2).
Remark 1: We avoid sampling directly on x in (t) since a large number of samples would be required to achieve our desired frequency resolution ∆ f given its carrier frequency f 0 . This will result in higher complexity or numerous equality constraints in the following optimization problem. On the other hand, sampling baseband signals A(x in B (t)) would be more practical. To sample A(x in B (t)), we approximate that the major components κ is the non-linear extending coefficient larger than 1. Then, based on the Nyquist Sampling Theorem, we set the sampling frequency at 2κ N∆ f . Further, to guarantee the frequency resolution of ∆ f , we take 2κ N samples, corresponding to a sampling interval of where with t k =kt s . From Eq. (15), we get the real part of w HPA n as: Similarly, for the imaginary part, we have: Assume that x HPA (t) is then transmitted into an ideal BPF with its passband [f 0 ,f 0 +(N −1)∆ f ], and becomes the transmitted RF signal x tr (t). We have:

B. Rectenna Model
At the receiver, the wireless signal y(t) is picked up and is converted into DC via a rectenna. We model the non-linear rectenna based on [11] whose equivalent circuit is depicted in Fig. 3. In this circuit, the receiving antenna, after picking up the RF signals, functions as a voltage source with average power ε y(t) 2 and an inner impedance R ant . Denoting the source voltage as v s (t), Assume that the following rectenna circuit has an input impedance R in matched with R ant . Hence, the input voltage for the rectenna is v in (t)=v s (t)/2=y(t) √ R ant . As shown on the right hand side of Fig. 3, the rectenna is composed of a diode and a low pass filter (LPF) 3 . The transfer characteristic of the diode is written as where k i is the corresponding Taylor coefficient related with the output voltage. Then, the high-frequency components in i d (t) are filtered out by the LPF and gives the current across the load R as To maximize the generated power at load R is equivalent to maximize the current at the load I out which, after circuit analysis in [11], is in proportional to the scaling term: where k i =i s /(i!(η 0 V 0 ) i ) with i s being the reverse bias saturation current, η 0 being the ideality factor, V 0 being the thermal voltage of the diode and R ant being the characteristic impedance of the receiving antenna. Eq. (20) only reserve the second 4 and the fourth order terms of the Taylor approximation, which have been able to capture the non-linear properties of the rectenna. Based on the scaling term z DC , the scaling term of the end-to-end PTE of the system is defined as: where P DC,HPA denotes DC power supply at the HPA. P max in + P DC,HPA refers to all the DC power supply at the transmitter 5 .

IV. OPTIMIZATION SOLUTIONS
In this section, we first formulate a generalized optimization problem with respect to the input waveform of HPA and subject to the input 6 and transmit power constraints 7 to maximize the harvested power in WPT. Then, for tractability, we specify and solve the generalized optimization model based on two different assumptions on the frequencies of the input waveform at HPA: 1) Model I: the input waveform at HPA has multi-carrier frequencies within the transmit pass band, the same as the transmit waveform.
2) Model II: the input waveform at HPA uses the frequencies out of the transmit pass band to adaptively design the input waveform that can pass the BPF after HPA in a lossless way.

A. Problem Formulation
We first formulate a general form of the optimization problem to maximize the harvested power in WPT: where P max in and P max tr are the input power constraint and the transmit power constraint respectively .
The main difficulty of solving problem (23) is that, the scaling term z DC , as a non-linear function of { w tr n } in Eq. (21), involves extremely high-complexity if writing in function of w in n by adopting the waveform relationship modelled in section III-A2. The same issue occurs for the transmit power constraint in Eq. (23c). Hence, two possible solutions we propose in the following are: 1) similarly to slack variables, we introduce { w tr n } as auxiliary variables, to divide the complexity of the objective function (also the transmit power constraint) into two parts (one part for the scaling term as a nonlinear function of { w tr n } and one part for the non-linear relationship between { w tr n } and w in n ) and to solve them separately (Model I); 2) we represent the input power constraint in Eq. (23b) as a function of the transmit waveform w in n , and make problem (23) a function of { w tr n } instead (Model II). To achieve this, we assume that the frequencies of the waveform after HPA are all concentrated in the pass band of BPF, which makes BPF lossless and avoids characterizing the inverse function of the BPF. Consequently, we can write the input power constraint with respect to { w tr n } explicitly. Correspondingly, to concentrate the frequencies of x HPA within BPF's passband after the non-linear HPA, we will need to design the out-of band frequencies of the input waveform at HPA. Details are given as follows.

B. Model I: Input Waveform with the Same Frequencies as the Transmit Waveform
As it has been mentioned, Model I solves the optimization problem where the input waveform at HPA assumes to have the same frequencies as the transmit waveform. In this context, for tractability, we collect both the input waveform of HPA w in n 6 The input power constraint at HPA avoids poor amplifier PE which occurs when the power of HPA's input signal gets close to or exceeds HPA's saturation power. 7 The transmit power constraint at antennas limits the RF exposure to human beings over propagation. and the transmit waveform { w tr n } into optimized variables so that the objective function and the transmit power constraint are in a much simpler form. The relationship between the transmit waveform and the input waveform is derived in Eq. (17) and Eq. (18). Intuitively, the auxiliary variables w in n enable us to deal with the original high complexity in Eq. (23) and Eq. (23c) separately. The corresponding optimization problem is written as: where the constraints in Eq.(24d) and Eq.(24e) feature high non-linearity. Problem (24) maximizes a convex objective, which can be efficiently solved by SCP. In SCP, the objective term is linearly approximated by its first-order Taylor expansion at a fixed operating point, forming a new tractable optimization problem whose optimal solution is used as a new operating point of the next iteration. The procedure is repeated until two successive solutions are close enough and can be viewed as the solution of problem (24). Assume where ( α (l) n , α (l) n ) are the first-order Taylor coefficients of ({w tr n },{ w tr n }) respectively at the l th iteration, shown in Eq. (26) and (27).
Correspondingly, the optimization problem (24) can be transformed into: where g I 1/2 ( w in/tr n , w in/tr n ) refers to the inequality constraint in Eq. (24b) and Eq. (24c) respectively. g E i ( w in/tr n , w in/tr n ) refers to all the equality constraints in Eq. (24d) and Eq. (24e).
Problem (28) can be solved by SQP in a highly efficient way. SQP is extended from solving the KKT conditions of an arbitrary nonlinear program by using the Newton-Raphson (NR) method (named by KKT-NR) [37]. To summarize, in KKT-NR, the NR method is used to solve the KKT functions iteratively with a closed-form solution over each iteration, and shows a fast-convergence property. In this context, SQP constructs a linearly constrained quadratic problem (LCQP) which has the same solution and which approximates the original program at the initialized point at each iteration. In this way, SQP achieves the convergence property of the KKT-NR's method. Hence in SQP, the original non-linear programming is solved by iterative approximations, where in each iteration, only a low-complex LCQP problem needs to be solved. Specifically, in each iteration, we get the direction of the optimized variables from the sub-program: where u k i (i=[1,...,2N]) and v k i (i=[1,...,2]) are the initializations of the Lagrangian multipliers at each iteration, which are obtained from the Lagrangian multipliers of the LCQP sub-program in the previous iteration.
Problem (29) is an easily handled LCQP problem. When w converges to 0, the sub-program in (29) converges to the original program as is easily seen [38].
Since SQP derives from the NR method to solve KKT conditions, it also preserves the features of the KKT-NR method, i.e., it is not a feasible-point (points that satisfy the constraints) method which means that the results from its sub-programs might not satisfy all the constraints in the original program. This is beneficial in the sense that the initialized point can be picked more randomly. However, it also gives probability that one of the sub-program gives a solution far from being feasible and requires more iterations to come back (In this condition, terminating the iteration in advance might results in worse performance). This drawback makes the results susceptible to the initializations and the functions' derivative features, especially in large-scale problems or in problems with numerous constraints. Hence, when solving problem (24), we introduce auxiliary variables to handle the complexity of the non-linear objective function and the non-linear equality constraints separately. In this way, we construct simpler sub-programs to be solved by SQP and get more sophisticated initialization points in each iteration, i.e., we use the SCP to approximate the objective function in the outer loop and use the SQP to handle the non-linear inequalities in the inner loop. The structure is named by SCP-SQP as in Algorithm 1.  DC (w (k) ), g I i (w (k) ), g I i (w (k) ), g E i (w (k) ), and g E i (w (k) ); 2:Compute w, {u k+1 i } and {v k+1 i } by solving the LCQP in (29) 3:w (k+1) ←w (k) + w; 4:Quit if w < S ,w (l) ←w (k+1) ; 5:k ←k+1; at different stages is easy to be extended to other architectures. Second, the proposed SCP-SQP algorithm is also robust to solving various non-linear programs, and can be extended to multi-input multi-output or multi-user scenarios.

C. Model II: Input Waveform Utilizing Out-of-Band Frequencies
We now take advantage of the out-of-band frequencies at HPA's input so that after HPA, the signal x HPA (t) can pass the BPF in a lossless way 8 . Another benefit of this model is that under a lossless BPF scenario, the input power can be written as a function of the transmit waveform, the same as the objective function and the transmit power constraint. Hence, compared with Model I, Model II formulates the optimization problem with respect to the transmit waveform only, which releases the numerous non-linear equality constraints between the input waveform and the transmit waveform. Specifically, we form the optimization problem as: where the constraint in Eq. (32b) is shown to be the input power constraint in Appendix A. Eq. (32b) can be proved convex in Appendix B. Problem (32) maximizes a convex objective function, which can be solved by SCP. Similarly to the SCP-SQP method, at the l th iteration, problem (32) is approximated as: Problem (33) is a convex problem with a linear objective function and two non-linear convex inequality constraints. However, problem (33) involves a highly non-linear inequality constraint in Eq. (32c), which can be efficiently solved by the IP method. The IP method uses the Barrier's method to bring the inequality constraints into the objective function equivalently and then solves the resultant program by using Gradient Descend (GD) method.
Specifically, the non-linear constraints in Eq. (33b), according to the Barrier's method, are omitted by reformulating problem (33) into: where Further, to make problem (34) differentiable, I − (x) is approximated as: where t is a parameter that sets the accuracy of the approximation. The larger the t, the closer the Consequently, for a specific t, the optimization problem (34) becomes: log(−f c,i ( w tr n , w tr n )), (38) which can be solved by GD methods such as Newton's Method.
The whole algorithm, named as SCP-IP, is summarized in Algorithm 3. Remark 3: From the mechanisms of SCP-SQP and SCP-IP, we can deduce the pros and cons of these two algorithms. Both algorithms have been proved to converge fast over iterations. For SCP-SQP, one limitation comes from the rapidly increasing number of variables and non-linear constraints given large N. Due to its origination from the KKT-NR method, this algorithm might be sensitive to bad performance, especially with large-scale problems (numerous constraints are involved), and will rely on the initialisation point heavily. In contrast, the SCP-IP strategy is only constrained by two inequalities and formulates a convex    [11] is to use the waveform from [11] directly to HPA's input. PAPR is to use the waveform from [11] but with PAPR constraints. sub-program in each iteration, which provides more robustness. However, its integral non-linear constraint makes the sub-program time-consuming on the other hand.

V. SIMULATIONS
The power harvesting and PTE performance of the proposed waveforms are evaluated under a Wi-Fi-like scenario with f 0 =5.18 GHz and bandwidth B = 10 MHz unless specified. We set a 38 dBm transmit power, a 2 dBi receive antenna gain and a 58 dB pass loss. For the simulation baseline of SSPA, set the smoothing parameter to β = 4, the saturation power to A s = 40 dBm and the small-signal gain to G = 1 without loss of generality; For the rectenna, set i s =5µA, η 0 =1.05, V 0 =25.86 mV, and R ant =50Ω.

A. Frequency-flat channels
We start from frequency-flat channels to evaluate the power harvesting and PTE performance with HPA's non-linearity and to uncover the waveform tendency considering both HPA's and EH's nonlinearity. The frequency-flat scenario is defined as h n =1 for ∀n. Fig. 4 plots z DC (top) and the end-to-end PTE (bottom) in a frequency-flat WPT system as a function of the input/transmit power constraint (P max in = P max tr ) using different input waveforms at the HPA, where z DC of an ideal amplifier is used (the black line) as a benchmark. Fig. 4 (top) demonstrates that HPA's non-linearity significantly reduces the harvested power in WPT systems, especially when the transmit power increases to a level comparable with HPA's saturation power. On the other hand, the harvested power (Fig. 4, top) verifies the performance gain of the two end-to-end optimized waveforms (Model I, red, and Model II, black) over the optimal transmit waveform in [11] (the blue line) as the input of the non-linear HPA. Nevertheless, the transmit waveform with PAPR constraints (green) can compensate for HPA's non-linearity to a certain degree.
To gain a deeper insight into HPA's non-linear effect on the power harvesting performance and the waveform tendency, Fig. 4 (bottom) plots the end-to-end PTE in Eq. (22). Fig. 5 evaluates HPA's operating points (severity of non-linearity and PE) by characterizing its OBO, PE, and APE that correspond to the simulation points in Fig. 4. Fig. 6 then plots three examples of the optimal input waveform of HPA from Model I and Model II. Details are as follows.
In the small-signal region (P max tr < −3 dBW, OBO > 10 dB), the HPA works in the linear regime and can be regarded as ideal, i.e. negligible power loss or out-of-band frequency leakage. In this case, the waveforms obtained from Model I and Model II in Fig. 6 (top) overlap, both giving the maximal harvested power the same as that from the optimal transmit waveform with ideal HPA. However, in this region, the PE of HPA is low (< 40%) as shown in Fig. 5 (middle). Increasing the input power exploits the HPA's DC power supply better, and hence gives larger PE (Fig. 5, middle), APE (Fig.  5, bottom) and also the end-to-end PTE in WPT (Fig. 4 bottom).
However, when the transmit power increases to over −3 dBW ( OBO <10 dB), the HPA's non-linear effect becomes non-negligible. In this non-linear region (−3 dBW < P max tr < 7 dBW), while the harvested power still increases with power constraints, the performance gap between different waveforms enlarges (Fig. 4). The waveforms from Model I and Model II give the best performance, followed by using the optimal transmit waveform with and without PAPR constraints. The reason for the performance gap is indicated in Fig. 5 (top), where the two end-to-end optimization waveforms provide significantly smaller OBO, or equivalently speaking, larger output power after HPA than the No-HPA waveform in [11]. This is achieved by allocating power to less sub-carriers as shown in Fig.  6 (middle). Such an allocation exploits HPA's DC power supply better and is proved effective to compensate for HPA's non-linear degradation in Fig. 5 (bottom), where the HPA's APE of the proposed input waveforms of HPA (Model I and Model II) maintains the same level as the small signal regime. Consequently, we still observe an increase in the end-to-end PTE in Fig. 4 (bottom) for the proposed waveforms, whereas the end-to-end PTE of the No-HPA waveform starts decreasing because its high-PAPR waveform comes across severe power loss from HPA's non-linear degradation effect.
If further increasing the power constraint to P max Tr > 7 dBW, the HPA's saturation effect will dominate, in which condition the harvested power tends to saturate rather than increase (Fig. 4). The resultant optimal input waveforms of HPA (Fig. 6, bottom) show a further reduction in the PAPR. Despite this reduction in PAPR, Fig. 5 (bottom) illustrates that waveform design can no longer compensate for HPA's non-linear degradation by exploiting the DC power supply in this region, giving APE lower than 0. Understandably, in this region, although the harvested power (Fig. 4) and the HPA's PE (Fig. 5, middle) keeps non-decreasing, the end-to-end PTE of all the waveforms drop with increased input power (Fig. 4 bottom).
An extension of Fig. 4 is shown in Fig. 7, where different levels of input power are compared (P max in = P max in + {−6,−4,−2,0,2} dBW) in terms of the power harvesting performance and the end-toend PTE. An indication here is that increasing the input power can compensate for HPA's non-linear degradation to some extent (Fig.  7). This is reasonable since larger input power gives larger transmit power at the antenna and therefore, larger harvested power at the receiver. However, this does not necessarily give higher end-to-end (DC-DC) power conversion efficiency since a larger input signal might suffer more severe power loss due to HPA's non-linearity.

B. Frequency-selective channels
We then evaluate the power harvesting performance of different waveforms in frequency-selective scenarios. We assume a large open space environment with an NLOS channel from model B  in [39] with 18 delay taps whose power is normalized to 1. For each realization, we assume that the complex channel gain for each tap is modeled as an independent circularly symmetric complex Gaussian random variable, with zero mean and the variance being their normalized tap power. As a comparison to the frequency-flat scenario, Fig. 8a and Fig  .8b evaluate the waveform's performance based on one realisation of a frequency-selective channel. Fig. 8a (top) illustrates the magnitude response of the channel realisation 9 . Fig. 8a (middle and bottom) displays the magnitude of waveform optimized from Model I, Model II and the No-HPA waveform in [11] in linear and non-linear region (P max in =−5/5 dBW) respectively. Fig. 8b plots the corresponding harvested power (top) and the end-to-end PTE (bottom). Fig. 8a (middle and bottom) shows that the No-HPA waveform also features relatively high PAPR in this frequency-selective channel realization. As a result, a smaller performance gap between the proposed waveforms and the No-HPA waveform is revealed in Fig. 8b, compared with that in Fig. 4. Also interestingly, with HPA's non-linearity, Fig. 8b (top) displays larger harvested power than that from the frequency-flat scenario in Fig. 4. This is reasonable since, with significant HPA's non-linearity, the proposed waveforms from Model I and Model II concentrate power to sub-carriers with the strongest channels for both frequency-flat and frequency-selective scenarios. On the other hand, the normalized frequency-selective channels offer more powerful strongest sub-carriers than the normalized frequency-flat channels. Consequently, the end-to-end PTE (Fig.  8b, bottom) with frequency-selective channels is also significantly larger than that of the frequency-flat channels in Fig. 4 (bottom). Fig. 9 further identifies the effect of the bandwidth on the average harvested power z DC over hundreds of channel realizations. It demonstrates that all these three waveforms benefit from larger bandwidth, attributing to their common preference for strong channel transmission. Larger bandwidth provides more diversity between channels and thus has a larger probability of owning a more The end-to-end PTE

Model I Model II
No-HPA in [11] (b) Fig. 8. The optimal waveforms and their power harvesting performance given a frequency-selective channel realisation, G=1,As=10dBV, β =4,N =8,B =10MHz. Fig. 8a, from top to bottom, plots the magnitude response of this channel realisation, the waveform from the three models with P max tr = −5dBW, and the waveform from the three models with P max tr =5dBW. Fig. 8b  powerful strongest sub-carrier. This is highly beneficial especially when HPA's non-linearity urges single-carrier transmission. Even for the No-HPA waveform in [11] that does not consider HPA's non-linearity, transmission with larger bandwidth gives rise to lower PAPR solutions of the No-HPA waveform and mitigates its power loss through the non-linear HPA. Fig. 10 compares the average harvested power z DC as a function of the number of sub-carriers with different saturation voltages of HPA. In [11], without HPA's non-linearity, the ideal harvested power increases proportionally to the number of sub-carriers. In contrast, in Fig. 10, the average harvested power shows a tendency to saturate when increasing the number of sub-carriers. This comes from a fine balance between HPA's non-linearity (favors low-PAPR transmission) and EH's non-linearity (favors relative high-PAPR transmission), which consequently favours only limited number of sub-carriers for transmission. The more severe the HPA's non-linearity, the more dominant HPA's non-linear effect exerts on the input waveform design. Correspondingly, in Fig. 10, smaller saturation power indicates a higher non-linear operating point of HPA (smaller OBO) and thus shows a more significant saturation tendency as well as less harvested power. So far, no significant difference has been observed between the two end-to-end optimized waveforms from Model I and Model II. The performance difference between Model I and Model II can be revealed in Fig. 11. Fig. 11 explores the effect of the smoothing parameter of different HPAs, and firstly shows that HPA with a smaller smoothing parameter produces less harvested power. This is reasonable since, from Fig. 2 parameter shows higher non-linearity over the operation regime and thus suffers more severe power loss for the same input signals.
Also, Fig. 11 indicates that for HPA with small smoothing parameters (β = 1, HPA suffers significant non-linearity even below the saturation voltage), the waveform from Model I has better power harvesting performance than Model II, whereas for HPA with large smoothing parameters (β = 15, HPA is highly linear below its saturation voltage), the waveform from Model II outperforms. This is due to their different waveform models. Model I transmits an input waveform of HPA with N sub-carriers, and generates out-of-band emission after the non-linear HPA. The out-of-band emission is filtered out by the BPF after HPA as in Fig. 1. Hence, waveform from Model I suffers power loss from both the HPA's non-linear degradation and the BPF. In contrast, in Model II, the input waveform is assumed to utilize the out-of-band frequencies at the input of HPA to concentrate the frequencies of the waveform after HPA within the BPF's pass band. Hence, the waveform from Model II is free from the power loss caused by BPF, but suffers more from HPA's non-linearity since it usually has higher PAPR on the other hand. On this basis, Model I outperforms when the non-linear degradation of HPA is more severe than the power loss from BPF, which happens for HPA with small smoothing parameter. Similarly, Model II outperforms when the power loss from BPF is more dominant, which corresponds to HPA with large smoothing parameter where the non-linearity almost only occurs in the saturation regime.

VI. CONCLUSIONS
This paper investigates the optimal input waveform of HPA that maximizes the power harvesting performance in WPT, accounting for both HPA's and rectenna's non-linearity and adaptive to the CSI. Two models are proposed and solved based on whether the input waveform of HPA has the flexibility to transmit over the frequencies out of the transmit pass band or not. Simulations verify the advantages of these two optimized input waveforms at HPA, especially with highly non-linear HPA and in frequency-flat channels. The proposed waveforms show that, as the HPA approaches more non-linearity, the optimal waveforms transform from high-PAPR to low-PAPR or even to single-carrier transmission. This reveals the trade-off between HPA's and rectenna's nonlinearity, i.e., the former causes power loss and can be compensated by low-PAPR transmission, while the latter boosts the harvested power and is better exploited with high-PAPR transmission in frequency-flat channels. We also show that while increasing the input signal power makes use of HPA's DC power supply better and generates higher harvested power, the end-to-end PTE of the WPT system decreases when operating in HPA's highly non-linear (saturation) regime, since HPA's non-linear degradation becomes more and more severe. We also compare the waveforms from the two models and show that they have their own pros and cons, in terms of the algorithms' robustness as well as the time-consumption, and that the two proposed waveforms outperform each other when implementing HPA with different smoothing characteristics.

VII. APPENDIX
A. Proof of the input power constraint in Eq. (32b) Firstly, we derive the inverse transfer characteristics of HPA (SSPA in Eq. (10)). Given the amplitude distortion function A(·) in Eq.(8) and the phase shift Φ(·) in Eq. (9), the inverse transfer function of HPA is characterized by f −1 (·) HPA , composed of the inverse amplitude function A −1 (·) and the inverse phase shift function Φ −1 (·) such that: f HPA (f −1 HPA ( x tr ))=A(A −1 (x tr ))e j( x tr +Φ(x tr )+Φ −1 (x tr )) = x tr (39) where x tr is short for the transmit signal, and x tr (t)=x tr e j x tr Hence, we obtain the inverse transfer characteristics of HPA: Then, from the Parseval's theorem (the signal power in frequency domain equals the signal power in time domain), combing Eq. (42), we have: where x in is short for the input signal at the HPA. Eq.(43) corresponds to the constraint in Eq. (32b).
B. Convexity proof of constraint in Eq. (32b) We derive the second-order derivative to prove the convexity of: f(w tr n , w tr where x tr is abbreviation for x tr (w tr n , w tr n ,t). Denote by f(w tr n , w tr n )= h(g(w tr n , w tr n )), we have h( s ] 1 β dt and g(w tr n , w tr n )=x tr 2 (w tr n , w tr n ,t)=| N n (w tr n +j w tr n )e jwnt | 2 .