Nonlinear Multi-Carrier System with Signal Clipping: Measurement, Analysis, and Optimization

Power amplifiers (PAs) exhibit nonlinear characteristics when faced with high-power input signals, which is commonly referred to as PA nonlinearity[1, 2, 3, 4, 5, 6, 7, 8]. The rapid progress of multiple-input and multiple-output (MIMO) technology, coupled with the growing number of subcarriers, has resulted in a significant rise in the peak-to-average power ratio (PAPR) of orthogonal frequency division multiplexing (OFDM) devices. This increase in PAPR poses a major challenge in practical OFDM systems, as it can lead to reduced power efficiency, degraded system performance, and potential distortion in the transmitted signals. PA’s nonlinearity has become a vital concern, as it may cause significant performance degradation when faced with high PAPR [9, 10]. To ensure system linearity, designers are compelled to reduce the input power of the amplifier, albeit at the cost of energy efficiency[11]. Reducing PAPR, as a result, has emerged as a critical problem in OFDM systems and has attracted significant research attention.

Several established PAPR reduction algorithms [12, 13, 14, 15] have been extensively studied. However, these conventional algorithms, such as partial transmit sequence (PTS) and selected mapping (SLM), involve complex signal processing, making them unsuitable for resource-limited or battery-driven devices, which is common in sensors and Internet of Things networks.

Signal clipping offers a less complex PAPR reduction approach [16, 17, 18]. In OFDM systems, clipping is realized in the time-domain signals to limit their magnitude to a predetermined threshold. This approach effectively limits the PAPR of OFDM signals within an upper bound with relatively small computation requirements and is therefore very popular in resource-constrained OFDM devices [19]. However, as a nonlinear operation, signal clipping introduces additional frequency components that never appear in the original signal, bringing both out-of-band radiation and in-band distortion to the original signal. Although radiation spilling beyond the target bandwidth can be restricted by filtering, how to deal with the in-band distortion is still a challenging problem.

Prior research has made efforts to investigate and mitigate the in-band distortion caused by signal clipping. In [20], a novel Bayesian approach was implemented to recover the clipped signals of an OFDM system, aiming to minimize the impact of in-band distortion at the receiver side. Another technique, proposed in a separate publication [21], is known as the repeated clipping and filtering (RCF) method. The RCF technique focuses on reducing distortion on individual tones of the OFDM signal. Researchers have also explored optimization techniques to enhance the performance of signal clipping for PAPR reduction in OFDM systems. In a couple of studies [22, 23], the authors optimized both the clipping and filtering stages to achieve improved PAPR reduction while simultaneously mitigating in-band distortion. Alternative approaches have been investigated to tackle the computational complexity associated with clipped OFDM systems. For example, [24] applies convolutional neural networks (CNN) to reduce computational complexity while maintaining PAPR reduction performance. Additionally, compressed sensing techniques were employed in another study [25] to recover the signal clipping noise in clipped OFDM signals.

However, the joint impact of PA nonlinearity and clipping distortion has received little attention in previous studies. Intuitively, the choice of clipping level introduces a trade-off between PA nonlinearity and clipping noise: a higher clipping level reduces PA nonlinearity but increases clipping noise, while lower clipping levels decrease clipping noise but amplify nonlinear distortion. A previous research [26] tried to understand the problem via simulations. However, the complex settings in real-world engineering scenarios necessitate a theoretical analysis of the trade-off between these two distortion sources. Otherwise, with simulations only, we are unable to comprehensively understand the problem, let alone optimize the system setting to achieve the best performance.

In light of this, the primary objective of this paper is to thoroughly investigate the trade-off between clipping distortion and PA nonlinearity in practical OFDM systems and make optimizations accordingly. By addressing this trade-off, we aim to enhance the overall system performance and mitigate the adverse effects of both clipping distortion and PA nonlinearity. The contributions of this research can be summarized as follows:

• •

  To simplify the representation of PA nonlinearity, we utilize the Bessel-Fourier PA (BFPA) model. Through an analysis of the inter-modulation product (IMP) in the PA’s output, we derive the signal-to-noise ratio (SNR) for the studied system, where both nonlinear PAs and signal clipping are taken into consideration.

• •

  Our optimization problem is approached from three distinct scenarios. In the first scenario, we consider a known power level for the PA’s nonlinear distortion. By deriving the optimal clipping level, we aim to minimize the symbol error rate (SER). Moving on to the second scenario, we assume the power of clipping distortion is known. In this case, we derive the optimal operating point for the PA that minimizes the SER. Lastly, we address the scenario where both PA nonlinearity and clipping distortion are unknown variables. We prove the existence of a global minimum SER and derive the optimal signal clipping level and PA operating point. Besides, we obtain a closed-form expression for the global SER lower bound, which enhances our understanding of the system’s performance.

• •

  We thoroughly examine the impact of various system parameters on both the SER and total degradation (TD) in the presence of signal clipping noise and PA nonlinearity. Our investigation offers a comprehensive analysis of how these system parameters contribute to the overall performance degradation.

This paper is organized as follows. Section II presents the framework of the studied nonlinear clipped OFDM system. Section III-A builds the BFPA model, Section III-B analyzes the PA nonlinear distortion, and Section III-C derives the closed-form expression of SER. Then in Section IV, SER optimizations with different constraints are presented in detail. Section V gives simulation results and explains our experimental observations. Section VI concludes this paper.

This paper uses the following notation conventions. we denote matrices by bold capital letters, e.g., $\mathbf{M}$. And we denote the element of a matrix and the conjugate of the element by ${{\left[\mathbf{M}\right]}{i,j}}$ and $\left[\mathbf{M}\right]{i,j}^{*}$, respectively. We represent the Frobenius norm of a matrix by ${{\left|\mathbf{H}\right|}_{F}}$. We denote the expectation operator and the variance operator by $E(\cdot)$ and $Var(\cdot)$, respectively. Furthermore, $\mathsf{\mathcal{C}\mathcal{N}}(m,{\sigma}^{2})$ represents a complex Gaussian random variable with a mean of $m$ and a variance of ${{\sigma}^{2}}$.

We first present the general framework of the studied system in Fig. 1. The number of subcarriers is denoted as $N_{S}$. Additionally, the system we considered is comprised of $N_{R}$ receive antennas and $N_{T}$ transmit antennas. We then represent the source bit stream by vector $\mathbf{U}$ and denote the output of the $M$-order Quadrature Amplitude Modulation (M-QAM) modulator by matrix $\mathbf{M}$. Further, the modulated symbol transmitted through the $t^{th}$ transmitter antenna (where $t=1,2,...,N_{T}$) during the $s^{th}$ time slot (where $s=1,2,...,N_{S}$) is denoted as $\left[\mathbf{M}\right]_{t,s}$.

We make an assumption that the inverse fast Fourier transform (IFFT) module has a length $N_{S}$, which is equal to the number of subcarriers. Consequently, the signal obtained after the OFDM modulator can be expressed as

where $\mathbf{E}^{+}$ represents the IFFT operation within the OFDM modulation. The $(k,s)$ element within the matrix can be written as [27]

We define the system input power as $E_{U}$, which represents the system operating point. Additionally, we denote the input power of the signal clipping blocks as $E_{C}$. Furthermore, at the $t^{th}$ transmit antenna, the $s^{th}$ post-IFFT symbol transmitted is denoted by ${\left[\mathbf{C}\right]}_{t,s}$. The expression for $E_{U}$ can be written as

We then express the output signal of the clipping module as

where the phase of the transmitted symbol is denoted by $\angle(.)$, and the signal clipping level is represented by $\eta$. From (4), we see that the output power of the clipped signal is upper-bounded by $\eta^{2}E_{U}$.

Building upon Equation (4), the scaling factor $\beta$ for signal power is written as,

We then write the signal matrix after clipping as

where ${{\mathbf{D}}_{clip}}$ denotes the signal clipping distortion, and we have ${{\mathbf{D}}_{clip}}\sim G\left(0,{{D}_{k}}{{E}_{U}}\right)$.

We write the coefficient of clipping distortion power as

where the over-sampling factor is written as $J$, and the $k^{th}$ output of the $JN$-point fast Fourier transform (FFT) is denoted by $FFT\left\{\right\}_{k}$.

The power of the signal clipping module’s output can be expressed as

As in Fig. 1, we represent the input of nonlinear PAs by ${\mathbf{S}}$, which is the clipped output of prior modules. And we represent the output symbol of PA by $\mathbf{S}_{T}$, which is expressed as

where $\mathbf{D}_{non}$ represents the nonlinear distortion of amplifiers and $\alpha$ denotes the PA’s linear gain factor.

Radio-frequency signals received at the receiver antenna can be written as

where the channel noise is represented by $\mathbf{D}_{ch}$, and the uncorrelated elements of $\mathbf{D}_{ch}$ can be written as $\mathsf{\mathcal{C}\mathcal{N}}(0,\sigma_{ch}^{2})$. Further, the quasi-static Rayleigh channel applied in the system is denoted by $\mathbf{H}$.

The output signal of the FFT module (i.e., the OFDM demodulator) is expressed as

where the $s^{th}$ post-FFT symbol obtained through the $r^{th}\left(r=1,2,...,N_{R}\right)$ receiver antenna is denoted by $\left[\mathbf{M_{R}}\right]_{r,s}$, and we know

As in [28], we model the PA’s nonlinear in-band distortion $\mathbf{D}_{non}\mathbf{E}^{-}$ as a Gaussian signal. The output of the M-QAM demodulator can be written as

where the sum of system impairments is denoted by $\mathbf{W}$, and the output symbol of the source bit stream is written as $\mathbf{U}$.

We write $\mathbf{W}$ as

In subsection A, we present how we measure a PA chip and give the measurement results for building the PA model. And then in subsection B, we use the BA model to simplify the SNR analysis. Finally, we derive the SER performance in subsection C.

Section III gives the SER of the studied system in (31), while this section further optimizes the SER in different scenarios.

Before technical details, we give an outline of the following three subsections and make the following notations. Subsection A assumes a constant $\eta$ and optimizes the system operating point. The optimal SNR, the optimal, system operating point, and the optimal SER performance are denoted as $\gamma^{opt}_{E_{U}}$, $E_{U}^{opt}$, and $SER^{opt}_{E_{U}}$, respectively. Subsection B assumes a constant $E_{U}$ and optimizes the signal clipping level. We denote the optimal signal clipping level, the optimal SNR, and the optimal SER as $\eta^{opt}$, $\gamma^{opt}_{\eta}$, and $SER^{opt}_{\eta}$, respectively. Subsection C undertakes the joint optimization of ${E_{U}}$ (power of the desired signal) and $\eta$ (signal clipping level). The optimal signal clipping level and the optimal PA operating point are denoted by $\eta^{g}$ and ${E^{g}_{U}}$, respectively, where the superscript $g$ represents the global optimum. The resulting optimal values for the SNR and SER are denoted as $\gamma^{g}$ and $SER^{g}$, respectively.

At the beginning of this section, we explain why the increase in the number of subcarriers makes PA’s nonlinearity a more challenging problem. Here we focus on a linear OFDM system’s complementary cumulative distribution function (CCDF), which represents the probability of the system’s actual PAPR exceeding a targeted PAPR threshold [40]. As we can see from Fig. 4, given the same PAPR threshold (say 9dB), OFDM systems with larger $N_{S}$ have significantly higher CCDFs. In this regard, with the rapid increase of $N_{S}$, modern wireless communication systems have much higher CCDF than before, making the OFDM system more sensitive to PA’s nonlinearity. The above observation justifies the motivation of considering nonlinear amplifiers in this research.

The rest of this section validates our analytical results by simulating the studied OFDM system with signal clipping distortion and PA nonlinear distortion. We then study how the number of antennas affects the system’s SER performance and examine the total degradation (TD) performance of the system.

In the following simulations, we set $N_{S}$ as 64, which corresponds to the length of the OFDM modulator/demodulator block. The over-sampling rate of the OFDM symbols is set to 4. Furthermore, unless stated otherwise, we consider a MIMO configuration with a $2\times 2$ antenna setting and employ 4-QAM as the default modulation scheme. We assume that the quasi-static Rayleigh channel is normalized, and we refer readers to [41] for implementation details of the channel.

Fig. 5, Fig. 6, and Fig. 7 are presented to examine the influence of the PA operating point $E_{U}$ on the SER performance under various system configurations while keeping $\eta$ constant. These figures provide a comprehensive understanding of the relationship between $E_{U}$ and SER, with a comparison of simulated and analytical curves.

In Fig. 5, it is evident that each nonlinear case exhibits an SER lower bound that is dependent on $E_{U}$. This phenomenon can be explained as follows: when $E_{U}$ is not large, the PA stays in its linear region, hence its nonlinearity could be negligible. The increase of $E_{U}$ raises the PA’s input power and leads to a higher SNR and a lower SER. However, when $E_{U}$ is increased beyond a certain point, further increasing it results in very high nonlinear distortion. An SER lower bound can be observed when the detrimental effect of nonlinear distortion counteracts the amplification of the desired signal. Here the PA’s operating point that results in this SER lower bound is denoted as $E^{opt}_{U}$. Beyond $E^{opt}_{U}$, further increase $E_{U}$ deteriorates the SER performance.

From Fig. 6, we can see that the lower bound of system SER and $E^{opt}_{U}$ is positively correlated with $\sigma_{ch}^{2}$, the channel noise. This is, a higher noise level necessitates a higher $E_{U}$ to reach the optimal SER, but the resulting SER lower bound will be larger compared to scenarios with lower noise levels. Furthermore, Fig. 7 demonstrates a negative relationship between $E^{opt}_{U}$, the SER lower bound, and $\eta$, the clipping level.

Fig. 8 investigates $\eta$’s impact on the SER when $E_{U}$ is constant. Both analytical and simulated SER curves are presented, considering the presence of significant nonlinear distortion in the PA. Notably, each curve exhibits an SER lower bound that is clipping level-dependent. When the PA operating point is sufficiently high, such as $E_{U}=5$ dB or $0$ dB, increasing the clipping level initially leads to a decrease in SER. This is because signal clipping can help mitigate the nonlinear distortion of a PA by reducing the signal’s PAPR. However, further, increasing $\eta$ beyond the optimal value can result in significant degradation of the system’s SER performance because clipping noise now dominates the system’s impairment.

In Fig. 9, we treat SER as a function of both PA operating point $E_{U}$ and the signal clipping level $\eta$, with a consideration of nonlinear PAs. As anticipated in Section IV, we can observe a global optimal SER in the figure. Specifically, when the channel noise $\sigma_{ch}^{2}$ is set to $-30$ dB, the global optimal SER is determined to be $8.57\times 10^{-6}$, which aligns precisely with the simulated results.

Fig. 10 presents how SER and the optimal signal clipping level change with the system operating point $E_{U}$ under various noise conditions. It can be seen that there is a global system lower bound of SER when system operating point $E_{U}$ sweeps. Moreover, the optimal signal clipping level $\eta_{opt}$ is a convex function, where its convex point corresponds to the system operating point. Besides, the global system lower bound is positively related to the channel noise $\sigma_{ch}^{2}$, and the global system lower bound of SER is also positively related to $\sigma_{ch}^{2}$. This can be explained by (51): $\gamma^{g}$ is positively related to the channel noise.

Fig. 11 presents how the number of transmit/receive antennas affects the system SER when nonlinear amplifiers are considered. As the figure shows, using more transmit/receive antennas can significantly reduce the system’s SER, although it comes with increased system complexity.

Fig. 12 presents an investigation into the relationship between the PA output back off (OBO) and the overall system TD. The OBO refers to the reduction in output power from the PA’s maximum allowed output power, indicating the extent to which the PA operates in its nonlinear region. On the other hand, TD represents the SNR loss compared to that of an ideal linear amplifier in achieving a specific level of SER. For each target SER, the TD graphs in Fig. 12 exhibit a truncation point in the low-value OBO range, with the associated SER lower bound being equal to the target SER. Notably, as the PA operates within its highly nonlinear range, the TD graphs noticeably deviate from those obtained with a linear PA. It is understandable that as the target SER decreases, the OBO value at which the truncation point occurs increases. This trend becomes evident when examining the TD graph for 4-QAM signals, particularly when the target SER ranges from $10^{-2}$ to $10^{-4}$. Furthermore, a decrease in the parameter $\eta$ leads to an increase in the OBO value. This observation is apparent when analyzing the TD graph for 4-QAM signals with varying $\eta$ values from 100 to 2.

This paper studies the performance of OFDM systems with considerations of signal clipping distortion and PA nonlinearity distortion. By modeling the PA and analyzing its IMPs, we derive the system SNR and SER in a polynomial form. These derivations offer the advantage of low complexity while maintaining reasonable accuracy. Additionally, we conduct joint optimization of the clipping distortion and the PA nonlinearity to obtain the global minimum SER. This optimization process identifies the optimal signal clipping level and the optimal PA operating point. Furthermore, we examine the influence of system parameters on the SER performance under various configurations.