Performance Tradeoff Analysis of Hybrid Signaling SWIPT Systems with Nonlinear Power Amplifiers

Abstract

Simultaneous wireless information and power transfer (SWIPT) is a promising technology to achieve wide-area energy transfer by sharing the same radio frequency (RF) signal and infrastructure of legacy wireless communication systems. To enlarge the effective range of energy transfer in practice, it is desirable to have a hybrid signaling SWIPT scheme, which combines a high-power multitone energy signal with a low-power broadband information signal. This paper presents a systematic study on the performance of hybrid signaling SWIPT systems with memoryless nonlinear transmitter power amplifiers (PAs). Using PA efficiency and signal-to-noise-and-distortion ratio (SNDR) as the metrics to measure the efficiency of energy transfer and information transmission, respectively, we derive the tradeoff between these two metrics for two PA classes, two nonlinear PA models, and two SNDR definitions. Our results reveal insights into the fundamental performance tradeoff inherent in SWIPT systems using hybrid signaling schemes.

1. Introduction

Along with the rapid commercialization of the 5th generation (5G) mobile communication technology, research and development of the 6th generation (6G) network is gaining momentum [1]. Enabled by 5G and 6G technologies, Internet of things (IoT) is quickly becoming ubiquitous in people’s daily lives. A key challenge hindering the proliferation of IoT is to provide power-deficient sensor devices with a reliable and sustainable energy supply [2]. Conventional energy supply technologies, such as wired charging [3], near-field wireless charging [4], and renewable energy harvesting (EH) [5], are limited by either the short distance or the availability of energy sources; hence, their applicability is constrained to specific scenarios. Exploiting the fact that radio signals are widely used for modern wireless communication systems, radio frequency energy transfer (RFET) technology was proposed as a ubiquitous energy supply technology that can continuously harvest the weak energy of far-field radio signals from the environment [6]. Simultaneous wireless information and power transfer (SWIPT) systems [7] take a step further to propose that concurrent information transmission and energy transfer can be integrated into a single system and jointly optimized at both the signaling and device levels.

In a typical SWIPT scenario, information and energy transmitters share the same transmitting device, whereas receivers for energy harvesting (EH) and information decoding (ID) are not necessarily co-located. Based on whether the EH and ID receivers are co-located, SWIPT systems have two operational models: standalone and co-located models. The former uses separated antennas on different devices for EH and ID, respectively, as shown in Figure 1a. The latter has one device for signal reception and needs to separate the analog signal received by the antenna (or antenna array) into two signals by means of time division [8,9,10], power division [11,12,13,14], space division [15,16,17], or frequency-splitting [18]. The separated signals are then fed into the ID receiver and EH receiver, respectively, as shown in Figure 1b. The potential benefits of SWIPT systems include improved spectrum utilization, reduced overall energy consumption, and reduced cost with shared infrastructure. Nevertheless, significant challenges remain in SWIPT systems to balance the conflicting goals of information transmission and energy transfer and ensure the coexistence of the two systems [19,20,21].

As a potentially disruptive technology, SWIPT has attracted significant research attention in recent years. Papers [22,23,24] analyzed the trade-off between the spectrum efficiency and energy efficiency of information transmission from the perspective of information theory. The authors in [25,26,27] studied the optimization of coding, modulation, and resource allocation. The problem of optimal power allocation in wireless propagation channels under transmit power constraints was investigated in [28,29]. Paper [30] studied the issue of receiver operation and dynamic power allocation. Paper [31] exploited the non-orthogonal-multiple-access (NOMA) technique and proposed a transmit power allocation scheme to improve the spectrum efficiency of a SWIPT system. A comprehensive survey of recent advances in SWIPT systems was given in [32]. The nonlinear model of energy harvesting is also an important research topic for SWIPT systems [20]. Paper [33,34,35] studied the intelligent reflecting surface (IRS) system and investigated how to use IRS to improve the energy collection efficiency of a SWIPT system. In order to highlight the tradeoff between SNDR and PA efficiency, the above discussion focused on the transmitter side with the basic single-input-single-output (SISO) hybrid signal. When extended to the whole SWIPT system, there remain many important and need research areas, such as energy harvesting circuit, signal modulation, multiple-input-multiple-output (MIMO), energy waveform design, and multipath channel estimation [19,20,21]. A research area that requires immediate attention is how to enhance the power efficiency of the total SWIPT system, which is a combination of the PA DC-to-RF conversion efficiency, RF-to-RF transmission efficiency, and the energy harvesting circuit RF-to-DC conversion efficiency. Especially when the signal power is low, the efficiencies are coupled with each other due to the nonlinearity of the energy harvester circuit [19]. Another promising area is the MIMO-OFDM modulation scheme; the use of multi-antenna beamforming can further increase the DC power of the energy harvesting circuit. However, it is more sensitive to the nonlinear distortion of PA, especially the phase distortion, and more efforts are needed to estimate the dynamic nature of a MIMO channel [21]. For the whole SWIPT system, more performance metrics at the receiver side are needed to evaluate the efficiency of information transmission, such as signal-to-interference-plus-noise ratio (SINR) and bit-error-rate (BER). This also leads to interesting new challenges in terms of the information rate and delivered energy trade-off for the whole system. These works, along with the majority of theoretical SWIPT studies in the literature, imply an idealized assumption that energy receivers can effectively harvest radio signals of arbitrary power strength. However, in practice, due to the physical characteristics of electronic devices, RF energy harvesting is only feasible when the received signal strength is above a certain threshold. Compared with ID receivers that typically work at a signal strength of around $-90$ dBm [36], EH receivers require a much higher signal strength of around $-40$ dBm to function properly [37]. Such a drastic difference should be taken into account during the design of SWIPT signaling schemes.

Two types of SWIPT signaling schemes have been proposed in the literature: identical signaling schemes and hybrid signaling schemes. The former uses a common transmit signal for EH and ID, as illustrated in Figure 2a. The advantage of this scheme is that it encourages the maximum reuse of existing information signals transmitted by legacy wireless communication systems. However, as wireless communications systems are designed for high spectrum utilization, tight transmit power constraints are imposed on information-bearing wideband RF signals. As a result, the effective range of energy transfer in a SWIPT system with identical signaling would be very short; hence, this system has limited practical value.

To overcome the drawback of identical signaling, a hybrid signaling scheme has been proposed [38], as shown in Figure 2b. It consists of a low-power broadband information signal and a high-power, small-bandwidth energy signal [39]. The energy signal is typically an unmodulated multitone sinusoid signal, whose signal waveform can be optimized to improve the energy harvesting efficiency by increasing the peak-to-average power ratio (PAPR) [40,41,42,43]. To date, reported research on hybrid signaling SWIPT has covered multiple topics, including the topics of beamforming and optimal power allocation [44,45], the tradeoff between information transmission and energy transfer [46], and the impacts of baseband modulation on the efficiency of energy and data transmission [41].

In the existing SWIPT research literature, most studies have focused on the receiver side, while assuming that the transmitter performs ideal linear amplification on the signal. Receiver-side nonlinearity has also been studied in energy harvesting schemes for multiple applications such as IoT [47]. On the contrary, transmitter-side nonlinearity is still under-investigated. In practice, the high-power energy signal transmission may drive the transmit PA to work on the nonlinear region and cause serious interference with the information signal. Although there exists a wealth of conventional studies that investigate nonlinear transmit PA in communication systems, the signaling schemes in these studies are restricted to pure information signals. In the HSS-SWIPT system considered in this paper, the transmitted signal is a mixture of a high-power energy signal and low-power information signal. Nonlinear amplification will result in spectrum spreading of the information signal. Furthermore, the two signals will produce intermodulation distortion, which will further degrade the information signal quality. On one hand, for high-power energy signals, we prefer the PA to work in the nonlinear region to improve the energy conversion efficiency; on the other hand, for low-power information signals, nonlinear distortions of the signal should be avoided as far as possible [48,49]. As a result, achieving high efficiency of energy conversion and low distortion of the information signal become conflicting objectives. As nonlinear PA is practically unavoidable, it is of great importance to investigate the tradeoff between energy transfer and information transmission in a hybrid signaling SWIPT [50].

This paper is among the first efforts to analyze the performance of hybrid signaling SWIPT systems with nonlinear PA. Because nonlinear PA is a transmitter-side phenomenon, we restrict our study to performance metrics at the transmitter side. In other words, receiver-side particularities such as modulation and circuit designs are deliberately made irrelevant and neglected. This helps us to decouple the problem and obtain deeper insights. More specifically, we use the transmitter-side signal-to-noise-and-distortion ratio (SNDR) [51,52] to measure the efficiency of information transmission, and the transmitter PA efficiency to measure the efficiency of energy transfer. This paper makes the following contributions: first, the SNDR metrics are redefined for the SWIPT system with hybrid signaling, which consists of a high-power energy signal and a small-power information signal; second, the SNDR and the PA efficiency are analytically evaluated for SWIPT systems with nonlinear PAs; finally, using the input back-off (IBO) as the intermediate parameter, we quantify the tradeoff between the SNDR and the PA efficiency.

The remainder of this paper is organized as follows. Section 2 introduces the system model and presents definitions of the performance metrics. Theoretical evaluations of PA energy conversion efficiency and information signal SNDR are presented in Section 3 and Section 4, respectively. Section 5 analyzes the tradeoff relationship between PA efficiency and SNDR in hybrid signaling SWIPT systems. Finally, conclusions are drawn in Section 6.

2. System Model

Figure 3 shows the system model of the hybrid signaling SWIPT system studied in this paper. We assumed that the information signal $s\left(t\right)$ uses an orthogonal frequency division multiplexing (OFDM) signal and the energy signal $e\left(t\right)$ uses a multitone sine signal [41]. Mathematically, these two signals are defined as

$$s\left(t\right)=\sum _{k=0}^{K-1}{B}_k\left(t\right)exp\left(j\left(2\pi f_{k}t+{\varphi }_k\right)\right),$$

(1)

$$e\left(t\right)=\sum _{n=-N}^N{D}_{n}exp\left(j\left(2\pi \left(f_n+n{\mathsf{\Delta }}_f\right)t+{\varphi }_n\right)\right)$$

(2)

where $B_k\left(t\right)$ is the baseband waveform signal, $D_n$ is the amplitude of the sinusoid signal at each frequency point, $f_k$ and ${\varphi }_k$ denote the frequency and initial phase of the OFDM subcarrier, respectively, $f_n$ and ${\varphi }_n$ denote the center frequency and initial phase of the sinusoid signal, respectively, and ${\mathsf{\Delta }}_f$ is the frequency offset of the multitone sine signal.

The two signals are superimposed and fed into a PA before transmission into a wireless channel. The receiver uses power splitting to separate the received signal into the information signal and energy signal, which are subsequently used for ID and EH, respectively. As mentioned above, the focus of our paper is on the nonlinear distortion at the transmitter side caused by the PA. Common RF PAs can be classified into two categories: memoryless PA and PA with memory. In a memoryless PA, the output is an instantaneous function of the input. Therefore, any change in the input signal results in instantaneous reactions at the output. In contrast, in a PA with memory, the output is also a function of previous input values. Our paper focuses on the case of memoryless PA, leaving the more complicated case of PA with memory as future work. The signal distortion and power efficiency of a PA depend on its nonlinear distortion model and input power back-off (IBO). The PA models and the concept of IBO will be subsequently introduced as preliminaries.

2.1. Memoryless Nonlinear PA Models

As illustrated in the system model in Figure 3, the complex baseband information signal can be written as $s\left(t\right)=r_s\left(t\right)exp\left(j{\varphi }_s\left(t\right)\right)$. Similarly, the complex baseband energy signal can be written as $e\left(t\right)=r_e\left(t\right)exp\left(j{\varphi }_e\left(t\right)\right)$. It follows that the total complex baseband input signal, which is obtained by the superposition of the information signal and energy signal, is given by $x\left(t\right)=s\left(t\right)+e\left(t\right)=r\left(t\right)exp\left(j\varphi \left(t\right)\right)$. The output signal $y\left(t\right)=r_o\left(t\right)exp\left(j{\varphi }_o\left(t\right)\right)$ is the complex baseband output signal of the nonlinear PA. For the memoryless PA, the output signal $y\left(t\right)$ can be regarded as a function of the envelope of the input signal $r\left(t\right)$, i.e.,

$$y\left(t\right)=g\left(r\left(t\right)\right)exp\left(j\left(\varphi \left(t\right)+f\left(r\left(t\right)\right)\right)\right)$$

(3)

where $g\left(r\left(t\right)\right)$ and $f\left(r\left(t\right)\right)$ are the magnitude response and phase response of $r\left(t\right)$, respectively. The magnitude response and phase response are commonly referred to as AM-AM and AM-PM characteristics. Because the time variable t is not important in a memoryless PA model, it will be neglected in the remainder of this paper for clarity.

2.1.1. Ideal Linear PA Model

For comparison purposes, we first introduce an ideal linear PA model. In such a model, the amplifier has no phase distortion, i.e., we have $f\left(r\right)=0$ for any r. The AM-AM characteristic of the linear PA model can be expressed as [46]

$$g\left(r\right)=\left\{\begin{array}{cc}{r}_{o,\max }\left({\displaystyle \frac{r}{r_{\max }}}\right),\hfill & r<r_{\max },\hfill \\ r_{o,\max },\hfill & r\ge r_{\max },\hfill \end{array}\right.$$

(4)

where $r_{o,\max }$ is the maximum output signal envelope and $r_{\max }$ is the maximum input signal envelope for linear amplification input. These two variables are related by

$$r_{o,\max }=A{r}_{\max }$$

(5)

where A represents the constant amplitude (envelope) gain of the idealized PA. The characteristics of this ideal model can be used as a reference benchmark for other nonlinear PA models [53].

2.1.2. Saleh Model

The Saleh model was first obtained by statistical analysis of the input and output data of travelling-wave tube amplifiers (TWTA) [54]. It has a simple structure, but can describe the nonlinear characteristics of TWTA accurately. The model also has high fitting performance for the measured data of other memoryless PAs. Therefore, it is widely used as a memoryless PA model in modern communication systems [55].

Our paper adopts a normalized Saleh PA model, whose AM-AM and AM-PM characteristics are given by

$$g\left(r\right)={\displaystyle \frac{a_1{\displaystyle \frac{r}{r_{\max }}}}{1+b_1{\left({\displaystyle \frac{r}{r_{\max }}}\right)}^2}},$$

(6)

$$f\left(r\right)={\displaystyle \frac{a_2{\left({\displaystyle \frac{r}{r_{\max }}}\right)}^2}{1+b_2{\left({\displaystyle \frac{r}{r_{\max }}}\right)}^2}}$$

(7)

respectively. Without loss of generality, the values of model parameters are taken from numerical values reported in [52]. Specifically, we have $a_1=r_{o,\max }$, $b_1=1/4$, $a_2=\pi /12$, and $b_2=1/4$.

2.1.3. Memoryless Polynomial Model

The memoryless model based on polynomial equations is essentially the complex baseband form of Taylor series used in the analysis of memoryless nonlinear systems. Considering that the band-pass signal after the PA only retains the signal near the central frequency, other frequency components will be filtered out. Thus, only odd-order nonlinear terms should be retained. The mathematical expression of the memoryless polynomial model is given by [56]

$$y\left(t\right)=\sum _{k=0}^K{a}_{2k+1}{\left|x\left(t\right)\right|}^{2k}x\left(t\right).$$

(8)

The polynomial model has wider applicability than the Saleh model and can be used to fit most memoryless RF PAs. When the model coefficients $a_{2k+1}$ are complex-valued, the system is called a quasi-memoryless system, in which the AM-PM characteristic will change with $\left|x\left(t\right)\right|$. When the coefficients are real numbers, the system is called a strictly memoryless system, such that the AM-PM characteristic is constant at 0, i.e., there is no phase distortion. To facilitate theoretical analysis, this paper adapts a strict memoryless PA model with a small nonlinear order ($K=2$). After normalizing the polynomial model, we obtain

$$\begin{array}{c}y=r_{o,\max }\left({\displaystyle \frac{x}{r_{\max }}}\right)+c_1{r}_{o,\max }{\left|{\displaystyle \frac{x}{r_{\max }}}\right|}^2\left({\displaystyle \frac{x}{r_{\max }}}\right)+c_2{r}_{o,\max }{\left|{\displaystyle \frac{x}{r_{\max }}}\right|}^4\left({\displaystyle \frac{x}{r_{\max }}}\right).\hfill \end{array}$$

(9)

According to the normalized limit criterion of the ideal linear model, the model coefficients can be calculated to obtain $c_1=-3/16$ and $c_2=1/64$. The detailed derivation process is given in Appendix A. The AM-AM characteristics of the polynomial model can then be written as

$$\begin{array}{c}g\left(r\right)=r_{o,\max }{\displaystyle \frac{r}{r_{\max }}}+c_1{r}_{o,\max }{\left({\displaystyle \frac{r}{r_{\max }}}\right)}^3+c_2{r}_{o,\max }{\left({\displaystyle \frac{r}{r_{\max }}}\right)}^5.\hfill \end{array}$$

(10)

Because the coefficients of the model are all real numbers, there is no AM-PM distortion, i.e., $f\left(r\right)=0$.

2.2. Input Power Back-Off

Figure 4 shows the AM-AM characteristics of the three PA models described above. It can be seen that in both the Saleh and polynomial models, when the input signal amplitude is less than $r_{\max }$, a smaller value of $r/r_{\max }$ leads to a better linear amplification property. The IBO is defined as the ratio of $r_{\max }^2$ to the average power [53], i.e.,

$$\xi ={\displaystyle \frac{r_{\max }^2}{P_{\mathrm{in}}}}$$

(11)

where $P_{\mathrm{in}}=E\left[r^2\right]=E\left[{\left|x\right|}^2\right]$ is the average power of the input signal and $E\left[·\right]$ denotes the expectation function.

In actual applications, the IBO is usually defined with respect to the 1 dB compression point instead of the maximum limit. This is called effective IBO, which is defined as [53]

$${\xi }_{\mathrm{eff}}={\displaystyle \frac{r_{1\mathrm{dB}}^2}{P_{\mathrm{in}}}}=k^2\xi$$

(12)

where k is the relative 1 dB compression point, and $r_{1\mathrm{dB}}=k{r}_{\max }$. The 1 dB compression point is defined by the following equation:

$$G_{\mathrm{dB}}\left(r_{1\mathrm{dB}}\right)=10lg{\left|{\displaystyle \frac{g\left(r\right)}{Ar}}\right|}^2=-1.$$

(13)

Based on the above equation, the value of $r_{1\mathrm{dB}}$ can be calculated from the PA’s AM-AM characteristic. It follows that the 1 dB compression coefficient k can be obtained as $k=r_{1\mathrm{dB}}/r_{\max }$. For the Saleh model and polynomial model in Figure 4, k is given by 0.6986 [53] and 0.7817, respectively. In the rest of this paper, the term IBO refers to effective IBO.

3. Calculation of PA Efficiency

The average efficiency of PA is defined as the ratio of the average power of the generated RF signal for actual transmission, which is denoted as $P_{\mathrm{RF}}$, and the direct current (DC) input power supplied by the DC source to the transistor PA circuit, which is denoted as $P_{\mathrm{DC}}$. Both $P_{\mathrm{RF}}$ and $P_{\mathrm{DC}}$ are functions of the output envelope level $r_o$. It follows that the average PA efficiency is given by [41,56]:

$${\eta }_{\mathrm{av}}={\displaystyle \frac{P_{o,\mathrm{av}}}{P_{i,\mathrm{av}}}}={\displaystyle \frac{E\left[P_{\mathrm{RF}}\left(r_o\right)\right]}{E\left[P_{\mathrm{DC}}\left(r_o\right)\right]}}.$$

(14)

The RF PAs commonly used in communication systems include Class A, Class B, and Class AB types. In recent years, some new PA circuit structures have emerged, such as envelope tracking PAs and Doherty PAs. These new types of PAs can effectively improve the efficiency at the cost of higher nonlinear distortion. In communication systems, the nonlinear distortion in PAs can be corrected by linearization techniques [57,58,59,60]. In this paper, we restrict our discussions on classic Class A and Class B PAs, noting that the efficiency of Class AB PAs is between Class A and Class B. The empirical efficiency models for Class A and Class B PAs are presented in [61]. The maximum average efficiency achieved by these two types of PAs can be written as [53]

$${\eta }_{\mathrm{av},A}={\displaystyle \frac{1}{2}}{\displaystyle \frac{E\left[r_o^2\right]}{r_{o,\max }^2}},$$

$${\eta }_{\mathrm{av},B}={\displaystyle \frac{\pi }{4}}{\displaystyle \frac{E\left[r_o^2\right]}{r_{o,\max }E\left[r_o\right]}}.$$

(15)

This shows that the efficiency function is the first and second non-central moments of the output signal envelope [53]. Moreover, the standardized nth non-central moment is defined as [52]

$$M_n\triangleq {\displaystyle \frac{E\left[r_o^n\right]}{r_{o,\max }^n}}={\displaystyle \frac{1}{{\left(A{r}_{\max }\right)}^n}}{\int }_0^{\infty }{g}^n\left(r\right)f_r\left(r\right)dr$$

(16)

where $f_r\left(r\right)$ is the probability density function (PDF) of signal envelope r. It follows that we can rewrite (15) as

$${\eta }_{\mathrm{av},A}={\displaystyle \frac{1}{2}}{M}_2\left(r_{\max }\right),{\eta }_{\mathrm{av},B}={\displaystyle \frac{\pi }{4}}{\displaystyle \frac{M_2\left(r_{\max }\right)}{M_1\left(r_{\max }\right)}}.$$

(17)

In this paper, we assume that the information signal and energy signal are a multi-carrier OFDM signal and multitone sine signal, respectively. When the subcarrier numbers are large enough, both signals can be approximated as complex Gaussian signals in the baseband. Thus, the composite signal can also be treated as a complex Gaussian signal. The PDF of its envelope is given by $(2r/P_{\mathrm{in}})/exp\left(-r^2/P_{\mathrm{in}}\right)$. Given such a complex Gaussian signal approximation, the first and second non-central moments with the Saleh model are given by [53]

$$M_1\left(\xi \right)=4\sqrt{\pi \xi }-8\pi \xi exp\left(4\xi \right)\mathrm{erfc}\left(2\sqrt{\xi }\right),$$

(18)

$$M_2\left(\xi \right)=16\xi \left[\left(1+4\xi \right)exp\left(4\xi \right)E_1\left(4\xi \right)-1\right]$$

(19)

where $\mathrm{erfc}\left(x\right)=1-\mathrm{erf}\left(x\right)$ is the complementary error function and $E_n\left(x\right)={\int }_1^{\infty }{\displaystyle \frac{exp\left(-xt\right)}{t^n}}dt$ is the exponential integral.

For the polynomial model, substituting Equation (10) into Equation (16), the moments can be calculated as

$$\begin{array}{c}{M}_1\left(\xi \right)=\left({\displaystyle \frac{\sqrt{\pi }}{2\sqrt{\xi }}}+{\displaystyle \frac{3c_1\sqrt{\pi }}{4\xi \sqrt{\xi }}}+{\displaystyle \frac{15c_2\sqrt{\pi \xi }}{8{\xi }^2}}\right)\mathrm{erf}\left(\sqrt{\xi }\right)-{\displaystyle \frac{6c_1+10c_2\xi +15c_2}{4\xi exp\left(\xi \right)}},\hfill \end{array}$$

(20)

$$\begin{array}{c}{M}_2\left(\xi \right)={\displaystyle \frac{\left(1-exp\left(-\xi \right)\right)}{\xi }}-{\displaystyle \frac{4c_1\left(1-\left(\xi +1\right)exp\left(-\xi \right)\right)}{{\xi }^2}}\hfill \\ -{\displaystyle \frac{\left(12c_2+6c_1^2+2c_2\right)\left(1-\left({\xi }^2+2\xi +2\right)exp\left(-\xi \right)\right)}{2{\xi }^3}}\hfill \\ -{\displaystyle \frac{16c_1{c}_2}{{\xi }^4}}\left(3-{\displaystyle \frac{\left({\xi }^3+3{\xi }^2+6\xi +6\right)exp\left(-\xi \right)}{2}}\right)\hfill \\ -{\displaystyle \frac{5c_2^2\left(1-\left({\xi }^4+4{\xi }^3+12{\xi }^2+24\xi +24\right)exp\left(-\xi \right)\right)}{{\xi }^5}}\hfill \end{array}$$

(21)

respectively.

We note that Equation (21) is associated with the specific polynomial model introduced in Equation (10). The coefficients in Equation (21) may change subject to different polynomial models.

4. SNDR Analysis

In the hybrid signaling SWIPT scenario, we have a mixture of a high-power energy signal and a low-power information signal. When the hybrid signal passes through nonlinear PAs [62], both signals will generate undesirable distortion signals that act as interference with the information signal. As a commonly used metric of signal distortion in communication systems, SNDR is broadly defined as the ratio of the useful signal to the total distortion power and noise power. The exact definitions of SNDR can take several forms, as proposed by C. Zhao [63], P. Zillmann [64], and R. Raich [65]. This paper adopts the definition in [65]. Considering an information transmission system, the output signal y of the PA can be written as the sum of two parts:

$$y=\alpha x+d.$$

(22)

The first item on the right-hand side is a linear amplification of the useful information signal, while the second item is the distortion signal, which is independent of x. The linear amplification factor $\alpha$ is defined as

$$\alpha ={\displaystyle \frac{E\left[x^{*}y\right]}{E\left[{\left|x\right|}^2\right]}}.$$

(23)

The power ${\epsilon }_d$ of the distortion term is defined as

$${\epsilon }_d=E\left[{\left|d\right|}^2\right]=E\left[{\left|y\right|}^2\right]-{\left|\alpha \right|}^{2}E\left[{\left|x\right|}^2\right].$$

(24)

Given the above definitions, the SNDR can be defined as

$$\mathrm{SNDR}={\displaystyle \frac{{\left|\alpha \right|}^{2}E\left[{\left|x\right|}^2\right]}{{\epsilon }_d+{\sigma }_v^2}}$$

(25)

where ${\sigma }_v^2$ is the noise power. Because the SWIPT signal consists of a large energy signal and a small information signal, the PA could have different linear amplification effects with respect to these two signals. When the effects of nonlinearity are not severe, the amplification coefficients of the two components can be approximately treated as the same. Otherwise, different amplification coefficients should be applied. For a comprehensive discussion, we further distinguish the following two cases of SNDR definitions.

4.1. Identical Amplification Coefficient

In this case, $y=\alpha \left(s+e\right)+d=\alpha x+d$. Because the energy signal and the information signal have an identical amplification factor, the definition of $\alpha$ is the same as that of the traditional SNDR defined in (23). Unlike the traditional SNDR definition, the power of the useful signal in SNDR definition should be expressed as the power of the information signal. Therefore, according to Equation (25), the SNDR with respect to the information signal s can be obtained as

$$\mathrm{SNDR}={\displaystyle \frac{{\left|\alpha \right|}^{2}E\left[{\left|s\right|}^2\right]}{{\epsilon }_d+{\sigma }_v^2}}.$$

(26)

Here, the linear coefficient $\alpha$ is a function of the signal envelope r, i.e.,

$$\alpha ={\displaystyle \frac{{\int }_0^{\infty }rg\left(r\right)exp\left(jf\left(r\right)\right)f_r\left(r\right)dr}{{\int }_0^{\infty }{r}^2{f}_r\left(r\right)dr}}.$$

(27)

where $f_r\left(r\right)$ denotes the PDF of signal envelope r. The power ${\epsilon }_d$ of the distortion term can be evaluated as

$${\epsilon }_d={\int }_0^{\infty }{\left[g\left(r\right)\right]}^2{f}_r\left(r\right)dr-{\left|\alpha \right|}^2{\int }_0^{\infty }{r}^2{f}_r\left(r\right)dr.$$

(28)

4.2. Different Amplification Coefficients

In the case of different amplification coefficients, we have $y=\alpha s+\beta e+d$. The exact definitions of the two amplification coefficients $\alpha$ and $\beta$ are given by

$$\alpha ={\displaystyle \frac{E\left[s^{*}\left(y-\beta e\right)\right]}{E\left[{\left|s\right|}^2\right]}},$$

(29)

$$\beta ={\displaystyle \frac{E\left[e^{*}\left(y-\alpha s\right)\right]}{E\left[{\left|e\right|}^2\right]}},$$

(30)

respectively. We can see that according to the above definitions, the two coefficients are mutually coupled and very difficult to calculate. Fortunately, in the SWIPT scenario, we can simplify the calculation with proper approximations. Exploiting the fact that the information signal s is small compared with the energy signal e, $y-\alpha s$ can be approximately treated as equal to y when we calculate $\beta$. It follows that we have $y\approx \beta e+d$. The amplification coefficient of the energy signal is then given by

$$\beta ={\displaystyle \frac{E\left[e^{*}y\right]}{E\left[{\left|e\right|}^2\right]}}$$

(31)

where

$$\begin{array}{c}{\displaystyle E\left[e^{*}y\right]={\int }_0^{\infty }{\int }_0^{\infty }\left|e\right|g\left(r\right)exp\left(j\left({\varphi }_x-{\varphi }_e+f\left(r\right)\right)\right)·f_{ey}\left(e,y\right)dedy}\hfill \end{array}$$

(32)

and $f_{ey}\left(e,y\right)$ is the joint PDF of energy signal e and output signal y. Although function $f_{ey}\left(e,y\right)$ cannot be expressed in closed form, it can be numerically evaluated.

The power of the distortion term can be written as

$${\epsilon }_d=E\left[{\left|d\right|}^2\right]=E\left[{\left|y\right|}^2\right]-{\left|\alpha \right|}^{2}E\left[{\left|s\right|}^2\right]-{\left|\beta \right|}^{2}E\left[{\left|e\right|}^2\right].$$

(33)

Finally, the SNDR can be expressed as

$$\begin{array}{c}\mathrm{SNDR}={\displaystyle \frac{{\left|\alpha \right|}^{2}E\left[{\left|s\right|}^2\right]}{{\epsilon }_d+{\sigma }_v^2}}={\displaystyle \frac{{\left|\alpha \right|}^{2}E\left[{\left|s\right|}^2\right]}{E\left[{\left|y\right|}^2\right]-{\left|\alpha \right|}^{2}E\left[{\left|s\right|}^2\right]-{\left|\beta \right|}^{2}E\left[{\left|e\right|}^2\right]+{\sigma }_v^2}}.\hfill \end{array}$$

(34)

5. Numerical Results and Discussion

This section presents numerical results based on the above analysis. The default values of parameters used in this section are set as follows. The maximum amplitude of linear amplification $r_{\max }$ is set to be 3, the ideal amplification gain coefficient of PA A is 10, the effective IBO ranges from −5 dB to 15 dB. The power of the information signal is set to be constant, the power ratio of the energy signal to the information signal $P_{\mathrm{E}}/P_{\mathrm{I}}$ ranges from 4 to 256, and the noise power ${\sigma }_v^2$ is set to −100 dB. Complex Gaussian signals are used to approximate the multi-carrier information signal and multitone energy signal. In some figures, theoretical results and simulation results are presented for comparison purposes. The theoretical results are obtained from numerical computation of closed-form or semi-closed-form equations, while the simulation results are obtained from Monte Carlo simulations and sample-based numerical evaluation. The presented results are statistically averaged over 50 trials.

Figure 5 compares the PA efficiency of Class A and Class B PAs as a function of the effective IBO. Results for both the Saleh model and polynomial model are shown. The theoretical and simulation results are calculated based on Equations (17) and (15), respectively, which are shown to agree well. As expected, the efficiency of PA generally decreases with the increase in IBO. The efficiency of Class B exceeds that of Class A, with a gain ranging from 10 dB to 40 dB. The efficiency differences are more significant at smaller values of IBO. Moreover, the efficiency given by the polynomial model is slightly better than that of the Saleh model. This is because the polynomial model exhibits a slightly higher AM-AM curve, as shown in Figure 4. Such difference diminishes with increasing IBO. Figure 5 successfully establishes the relationship between PA efficiency and IBO. The next step is to link IBO with SNDR, such that we can use IBO as an intermediate parameter to characterize the tradeoff between PA efficiency and SNDR.

Figure 6 shows the relationship between effective IBO and SNDR under different energy-to-information signal power ratios. Results are shown for the two different cases discussed in Section 4. Case I refers to the case where we have the same amplification coefficients, while Case II denotes the case with different coefficients. For Case I, we can have both theoretical and simulation results based on the equations presented in Section 4.1. For Case II, we can only have simulation results based on Equation (34). For notation simplicity, the SNDR values obtained by Case I theory, Case I simulation, and Case II simulation are denoted as I(1), I(2), and II(2) in the figure legend, respectively.

In Figure 6, SNDR values are shown to increase almost linearly with increasing IBO (both in dB values). The SNDR difference between Case I and Case II is not very significant, only reaching a maximum value of 2 dB when the IBO is 15 dB. Moreover, the SNDR given by the polynomial model is slightly better than the Saleh model. This is because the memoryless polynomial model in (9) assumes no AM-PM distortion. Another finding presented in Figure 6 is that the SNDR decreases with increasing $P_E/P_I$ power ratio. This is expected because a higher-power energy signal will generate more interference with the information signal. Figure 6 establishes the relationship between SNDR and IBO.

Building on the results shown in Figure 5, Figure 6 and Figure 7 show the tradeoff between SNDR and PA efficiency with different PA classes and different PA models. The power ratio is fixed at 64. We can see that, given a targeted SNDR value, the two PA classes contribute to the most significant efficiency difference at around 20 dB. The two different PA models also lead to a small difference of less than 5 dB. The two different cases of SNDR definition give almost identical curves. A slightly concave-shaped tradeoff relationship is observed for SNDR and efficiency.

Similar to Figure 7, Figure 8 shows the relationship between SNDR and PA efficiency with different power ratios. We can see that increasing the power ratio in an exponential fashion has a similar effect as shifting the tradeoff curve horizontally towards the left-hand side along the X-axis. This implies a negative linear relationship between the power ratio (in dB) and the SNDR. Again, the differences caused by using the Saleh and polynomial models are limited to 5 dB.

To pinpoint the results more accurately,

Table 1. Efficiency of different PAs with different values of SNDR and power ratios.

Power Ratio		Class A				Class B
		Saleh PA		Polynomial PA		Saleh PA			Polynomial PA
		SNDR (dB)
		10	20	10	20	10	20	30	10	20	30
4	Efficiency (%)	14.4	4.5	20.7	8.1	45.9	26.1	14.2	54.3	34.8	20.1
16		7.8	2.4	12.3	4.7	35.1	19.4	10.9	42.9	26.6	15.1
64		4.2	1.2	6.8	2.3	24.9	13.6	-	32.5	18.9	-
256		1.9	-	3.6	1.0	17.1	-	-	23.7	13.8	-

shows the efficiency of different PAs with different values of SNDR and power ratios. We can see that as the energy-to-information signal power ratio increases, the corresponding SNDR under the same efficiency is significantly reduced.

In the literature of nonlinear PA analysis, it is a common practice to measure the nonlinear distortion effect by observing the power spectral density (PSD) of signals after a nonlinear PA. In the frequency domain, nonlinear distortion is manifested as power leakage into the adjacent bands. The adjacent channel power ratio (ACPR) is defined as the ratio of the signal power at the central frequency versus the power at an adjacent frequency (i.e., frequency offset). Similar to SNDR, ACPR is also a metric for nonlinear distortion. Figure 9 aims to show the correlation between these two commonly used metrics. A simulation is performed to measure the ACPR and SNDR. The information signal is taken as an OFDM signal with a bandwidth of 5 MHz, while the energy signal is a multitone sine signal. In Figure 9, when the SNDR is measured to be 10 dB, 20 dB, and 30 dB, the corresponding ACPR values at the 5 MHz offset are roughly −25 dB, −34 dB, and −41 dB, respectively. This means that the variations in the theoretically calculated SNDR are mirrored by ACPR in the simulations. This further validates the practical value of the SNDR metrics.

From the above discussion, we can see that maximizing SNDR and PA efficiency represents two conflicting objectives in a SWIPT system. The performance of a hybrid signaling SWIPT system is fundamentally characterized by the tradeoff relationship between SNDR and PA efficiency. In practice, an appropriate SNDR efficiency operational point should be carefully chosen for a targeted application.

6. Conclusions

This paper presents a systematic analysis of the performance of hybrid signaling SWIPT systems with nonlinear PAs. Using PA efficiency and SNDR as performance metrics, we have presented a quantitative analysis of the conflicting design objectives in SWIPT. Our study has taken into account the variations in PA types, PA models, and SNDR metrics. Theoretical and simulation results have revealed a concave-shaped tradeoff relationship between the SNDR and PA efficiency. Furthermore, a nearly linear relationship has been revealed between the energy-to-information signal power ratio (in dB) and the SNDR. It has been shown that, for the SNDR value, the PA efficiency differences between Class A and Class B PAs are around 20 dB. Our study reveals a fundamental tradeoff in SWIPT systems and provides useful guidelines for the design of practical SWIPT systems with hybrid signaling schemes.