Abstract
In recent years, sub-synchronous oscillation accidents caused by wind power integration have received extensive attention. The recorded constant-amplitude waveforms can be induced by either linear or nonlinear oscillation mechanisms. Hence, the nonlinear behavior needs to be distinguished prior to choosing the analysis method. Since the 1960s, the higher-order statistics (HOS) theory has become a powerful tool for the detection of nonlinear behavior (DNB) in production quality control wherein it has mainly been applied to mechanical condition monitoring and fault diagnosis. This study focuses on the hard limiters of the voltage source converter (VSC) control systems in the wind farms and attempts to detect the nonlinear behavior caused by bi- or uni-lateral saturation hard limiting using the HOS analysis. First, the conventional describing function is extended to obtain the detailed frequency domain information on the bi- and uni-lateral saturation hard limiting. Furthermore, the bi- and tri-spectra are introduced as the HOS, which are extended into bi- and tri-coherence spectra to eliminate the effects of the linear parts on the harmonic characteristics of hard limiting in the VSC control system, respectively. The effectiveness of the HOS in the DNB and the classification of the hard-limiting types is proven, and its detailed derivation and estimation procedure is presented. Finally, the quadratic and cubic phase coupling in the signals is illustrated, and the performance of the proposed method is evaluated and discussed.
WITH the implementation of the energy development strategy in China, the installation capacity of wind power has increased [
Current research on nonlinear oscillation analysis can be broadly classified into two categories.
1) Under small perturbations, constant-amplitude oscillations induced by the negative damping of the voltage source converter (VSC) generally begin with divergent oscillations near the equilibrium point of the system. Owing to the influence of the nonlinear factors in the VSC, such as hard limiting and pulse-width modulation (PWM) saturation, the oscillations will not continue to diverge, thereby resulting in constant-amplitude self-sustained oscillations [
2) Under large disturbances such as faults, self-sustained oscillations can be caused by the effects of nonlinear parts such as hard limiting. Meanwhile, if the corresponding hard-limiting parts are removed, the system can return to its original equilibrium point after the large disturbance. Current research on the mechanism of oscillations under large disturbances is still in the preliminary stage. Reference [
The mechanism and corresponding analysis method of the nonlinear oscillations are different from those of linear oscillations. The direct analysis of the amplitude and frequency of the SSO using a linearized method and adoption of the corresponding measures to suppress the oscillation are unacceptable. Hence, before choosing the analysis method, it is important to determine the type of oscillation (linear or nonlinear) from the waveform records.
Since its emergence in the early 1960s, the higher-order statistics (HOS) theory has become a powerful tool for the condition monitoring and fault diagnosis of the mechanical equipment. Its applications include harmonic retrieval [
Inspired by the aforementioned ideas in 1995, the researchers of power systems applied the bi-spectral analysis to the fault diagnosis and condition monitoring of the three-phase induction motors to analyze and identify motor asymmetric faults and stator winding failure [
In wind farms, [
However, the current applications of the bi-spectral analysis focus on the mechanical defects in the power systems, and few studies have analyzed the nonlinearity of the control modules in these systems, particularly in wind farms.
In this study, we focus on the hard limiters in the VSC, which is a crucial component of wind turbines and SVGs. To apply the HOS to the hard-limiting DNB in the control system of VSCs in wind farms, four aspects must be considered: ① suitability of the HOS for characterizing the nonlinearity of the hard limiters in the VSC control system and selection of an appropriate HOS; ② approach to determine the characteristics using only the waveforms collected at the terminal of the VSC; ③ method to detect the nonlinear behavior caused by the bi- or uni-lateral saturation hard limiting, which is considered as the source of the nonlinearity in this study; and ④ approach to improve the effectiveness and quality of the spectrum.
By fully addressing the aforementioned aspects, this study seeks to analytically prove the effectiveness of the HOS applied to the DNB induced by hard limiting in the VSC control system.
The rest of this paper is organized as follows. Section II extends the conventional DF and analyzes the two types of hard limiting. In Section III, the HOS is introduced. In Section IV, the VSC control system is modeled while the HOS-based hard-limiting DNB is analyzed and confirmed. The detailed procedures for the hard-limiting DNB in the VSC control system are described in Section V. Its effectiveness is demonstrated through case studies in Section VI. Finally, Section VII draws the conclusions.
The DF [
As shown in
(1) |

Fig. 1 Typical hard limiter with a sinusoidal input.
where and are the amplitude and frequency of the sinusoidal input signal, respectively.
The hard-limit output is a periodic nonsinusoidal signal, which can be expanded into a Fourier series as expressed below:
(2) |
where is the magnitude of the DC component; and and are the coefficients of the cosine and sine parts of the
In the conventional DF-based analysis method, the nonlinear part is considered oddly symmetrical and the linear part is considered low-pass. Consequently, , and can be approximated as:
(3) |
where and , and we have
(4) |
The ratio of the first-order Fourier series of to the magnitude of the input signal is defined as the DF of the nonlinear part, i.e., .
(5) |
To extend the DF method, the higher-order harmonic coefficients and in (2) are calculated as:
(6) |
As shown in
(7) |

Fig. 2 Time-domain characteristics of bi- and uni-lateral saturation hard limiting. (a) Bi-lateral saturation. (b) Uni-lateral saturation.
where is the upper hard limit.
Here, the bi-lateral saturation hard limiting is oddly symmetrical, and the output periodic signal is an odd function. Hence, the coefficients of the DC and cosine components in the Fourier series are 0, that is, . According to (4), the fundamental Fourier coefficient of the output signal can be obtained as:
(8) |
The
(9) |
The detailed calculation results are presented in Appendix A Table AI.
As shown in
(10) |
where is the offset of the sinusoidal input signal, and is the upper hard limit.
The
(11) |
The detailed calculation results are presented in Appendix A Table AII.
The definition of the HOS is introduced to analyze the harmonic characteristics of the nonlinear parts. The eigenfunction method is an important tool in statistical analysis since it can easily define the higher-order moments and cumulants.
The first joint eigenfunction of continuous random variables is defined as:
(12) |
where is the probability density function; and is the expectation function.
The
(13) |
(14) |
For a stationary continuous random signal , we set , which are later substituted into (14). Consequently, the
(15) |
The
(16) |
Generally, a higher-order cumulant spectrum is referred to as a higher-order spectrum. Particularly, the third-order spectrum is referred to as the bi-spectrum because it is a two-frequency energy spectrum, which is represented by Similarly, the fourth-order spectrum is referred to as the tri-spectrum, expressed as .
A typical direct-drive wind farm has 30-60 generators. Several direct-drive permanent magnet synchronous generators (PMSGs) are connected to form a string structure. The strings are further connected to a point of common coupling (PCC) and finally to the grid with a voltage level of 35 kV through a series of boosting transformers and transmission lines (equivalent to a set of impedances) [

Fig. 3 Model of wind farm connected to grid.

Fig. 4 Structure of a typical direct-drive wind generator.
Moreover, static reactive power compensation equipment, such as SVGs, is generally installed in wind farms. In this study, the SVG model adopts a double closed-loop control scheme. The d-axis control loop stabilizes the DC bus voltage, and the q-axis control loop varies based on the control mode. When the SVG operates in the constant-voltage control mode, the control target of the loop is the terminal voltage, and when it operates in the constant reactive power control mode, the control target of the loop is the output reactive power. Owing to its ability to produce high-quality output voltage waveforms with reduced harmonic distortion and low switching frequencies, many existing SVGs in wind farms are based on cascaded H-bridge topology [
Thus, we obtain a unified VSC control system for the PMSGs and SVGs in the wind farms. The only difference is the choice of control targets in the d- and q-axis control loops, as shown in

Fig. 5 Structure of VSC control system.
In
While the nonlinear parts take effect inside the control system, the accident waveform record collected from the phasor measurement unit provides only voltage and current information at the terminal of the VSC. Hence, the relationship between the HOS of the hard-limit output and that of the terminal electrical quantities must be derived.
When the hard limiting in the d-axis inner control loop of the current takes effect, is produced by the nonlinear part after proportional-integral (PI) control, as shown in
(17) |
where represents the small-signal perturbation; and is the fundamental angular frequency.
Without considering the dynamics of the PWM, we assume that the VSC output tracks the reference signal:
(18) |
(19) |
From
(20) |
where is the hard-limit output in the d-axis inner control loop of the current.
Then, we obtain the relationship between and by combining (17) and (20), as expressed below:
(21) |
Similarly, when the hard limiting in the q-axis inner control loop of the current takes effect, the relationship between and is expressed as:
(22) |
where is the hard-limit output in the q-axis inner control loop of the current.
When the hard limit in the d-axis outer control loop of the voltage takes effect, the output is produced by the nonlinear part after PI control. Thus, (20) can be rewritten as:
(23) |
where is the output of the hard limit in the d-axis outer control loop of the voltage; and is the transfer function of the inner control loop of the current.
Then, we obtain the relationship between and by combining (17) and (23), as expressed below:
(24) |
Similarly, when the hard limit in the q-axis outer control loop of the voltage takes effect, the relationship between and is expressed as:
(25) |
where is the output of the hard limit in the q-axis outer control loop of the voltage.
Hence, the output of the nonlinear part in the control system can always be obtained from the current measured at the terminal after passing through the linear part by combining (21), (22), (24), and (25). The system shown in

Fig. 6 Typical linear system.
The relationship between the HOS of and that of can be obtained based on the definition and properties of the HOS [
(26) |
where and are the
(27) |
(28) |
Define as the bi-coherence [
(29) |
By combining (27), (28), and (29), we can obtain:
(30) |
Further, we define the tri-coherence [
(31) |
Similarly, it can be proven that .
Consequently, the linear part does not change the bi- or tri-coherence of the system. Obtaining the waveforms of and at the terminal of the VSC and performing the HOS analysis enable the detection of the nonlinearity of the VSC system.
Based on the results in Section II-C, when a self-sustained oscillation occurs, the output of the uni-lateral saturation hard limiter contains the second harmonic of the oscillation frequency with the same phase as that of the fundamental frequency. Without loss of generality, let its initial phase be zero, i.e.,
(32) |
where denotes the oscillation frequency; and and are the coefficients of the fundamental and second Fourier harmonics, respectively.
A Fourier transform is performed twice on the second-order autocorrelation function of . Consequently, the bi-spectrum of can be expressed as:
(33) |
The bi-spectrum has 12 symmetrical regions [

Fig. 7 Related figures of bi-spectrum and bi-coherence spectrum. (a) Symmetric region of bi-spectrum. (b) Peak of bi-coherence spectrum.
Therefore, to completely describe the entire bi-spectrum, only a symmetrical region in the resulting is required for analysis. Considering can be calculated as:
(34) |
where is the Dirac delta function [
(35) |
where is a finite maximum at when the input signal is discretized. In (34), is a finite maximum if and only if ; otherwise, it is zero. Therefore, a peak can be observed at the x-y coordinates in the three-dimensional (3D) graph of . Furthermore, because
is on the symmetry axis , when the area is extended to (areas 1 and 2 in
The power spectrum of is derived as:
(36) |
Based on (29), (34), and (36), the bi-coherence spectrum can be calculated as:
(37) |
Since , it can be proven that the bi-coherence spectrum of reaches its peak if and only if , as shown in
(38) |
In (35), Therefore, the range of the corresponding bi-coherence value of each x-y coordinate in the bicoherence spectrum is . The larger the bi-coherence value, the stronger the nonlinear phase coupling between the two frequencies corresponding to the coordinate (the stronger the nonlinearity).
In (32), when extends to , it can be proven that in the 3D graph of , there are peaks at the x-y coordinates , and their corresponding bi-coherence values are equal to one. Furthermore, it can be proven that the conclusion remains unchanged when , which accurately represents the output of the uni-lateral saturation hard limiter according to (11).
Based on the results in Section II-B, when , which accurately represents the output of the bi-lateral saturation hard limiter according to (9), it can be proven that in the 4D graph of , peaks exist at the x-y-z coordinates , with the peak values of 1. The range of the corresponding tri-coherence value of each x-y-z coordinate in the tri-coherence spectrum is . Furthermore, the larger the tri-coherence value, the stronger the nonlinear phase coupling among the three frequencies corresponding to the coordinate (the stronger the nonlinearity).
The detailed derivation process is presented in Appendix B.
Based on Section IV-B, the nonlinearity induced by hard limiting in the VSC control system can be detected by transforming the current waveform into and at the terminal of the VSC and performing the HOS (bi-/tri-coherence) analysis. The nonlinearity of represents the nonlinearity in the d-axis control loop of the VSC control system, whereas the nonlinearity of represents the nonlinearity in the q-axis.
Type | Phase coupling | Uni-lateral saturation | Bi-lateral saturation |
---|---|---|---|
Fourier series | |||
Bi-coherence | Peaks | No peaks | |
Tri-coherence | Peaks | Peaks |

Fig. 8 Flowchart of nonlinearity detection and classification.
In this study, when detection is performed at the VSC terminal, only hard limiting related nonlinearity exists in the control system (average-value model), and all the other function blocks in the control system can be represented by the transfer functions (linear blocks). Therefore, we assume that hard limiting is the only source of nonlinearity in
The specific procedure for applying the hard-limiting DNB to the VSC control system is based on the theoretical analysis described in Section IV. The corresponding steps are summarized in

Fig. 9 Key points of procedure for hard-limiting DNB.
To obtain the time series for the calculation, we first collect the accident waveform records of the current at the terminal of the VSC. The studied signals , and are sampled from the three-phase currents , and , respectively. We begin the sampling from the initial time and set the sampling interval to . The sampling point is denoted by , and the sample length is , i.e.,
(39) |
Step 1: perform a dq transformation on the three-phase sampling signals , , and . The initial phase can be obtained by applying a PLL algorithm as expressed below:
(40) |
where ; and , and are the transformation results in the dq0 coordinate system.
Step 2: set the transformation result of the previous step as the subsequent signal processing object, i.e.,
(41) |
Step 3: divide into segments, wherein each segment length is (). Each segment is recorded as .
Step 4: select an appropriate window function, such as a Hanning window, which is expressed as:
(42) |
Multiply each segment of the signal by the window function, and use the obtained results for further calculations to reduce leakage errors, as expressed below:
(43) |
Step 5: for each segment , subtract its mean value as:
(44) |
Step 6: perform the fast Fourier transform (FFT) on each segment :
(45) |
Step 7: when dealing with the FFT results, consider a small parameter (for instance, ). Sweep , and for any , if , then let . This step can further increase the difference in the order of magnitudes between the white noise and peak value in the spectrum. Thus, the analysis and conclusion of the peak value is not affected by the appearance of the values close to in the bi- and tri-coherence spectrum.
Step 8: the estimated values of the power spectrum, bi-spectrum, and tri-spectrum of are expressed as:
(46) |
(47) |
(48) |
Step 9: calculate the bi-coherence spectrum as:
(49) |
Step 10: the obtained bi-coherence spectrum is a 3D graph. Its x-y coordinates are the frequencies and its coordinate is the corresponding bi-coherence value, whose theoretical value range is .
Step 11: calculate the tri-coherence spectrum as:
(50) |
Step 12: determination of DNB thresholds.
Peak value threshold: define as the nonlinear threshold (preferably 0.3). A peak in the bi- or tri-coherence spectrum whose value is greater than is considered to characterize the existence of the quadratic or cubic phase coupling, and the coordinates of the peak represent the corresponding frequencies. The conclusion and classification of the nonlinearity can be completed using the process provided in
Flatness threshold: in addition to the index for checking the peak values, a nonlinear index can be defined based on the graph flatness [
(51) |
where is the average of the estimated squared bi-coherence; and is the standard deviation of . Based on the power-quality standard [
For a single VSC with slight harmonic pollution, the peak value threshold is enough for the DNB. Contrarily, for the multi-VSCs or complex systems, the flatness threshold can be considered. However, only the existence of nonlinear behavior in the control system can be verified, and misjudgment may occur when using the flatness threshold to locate the nonlinear oscillatory source, which is illustrated and discussed in Section VI-C.
Step 13: the above steps implement the DNB on the d-axis control loop of the VSC control system. To study the q-axis control loop, return to Step 2: let and repeat Steps 3-12.
Consequently, the nonlinear behavior caused by the bi- or uni-lateral saturation hard limiting in the d- or q-axis control loop can be detected. Some of these steps increase the resolution and effectiveness of the HOS. Step 4 reduces the spectrum leakage, Steps 6 and 8 eliminate the effect of the random phases, and Step 7 adds credibility to the presence of peaks.
Three cases are presented and discussed to evaluate the effectiveness of the proposed method for the hard-limiting DNB in the VSC control system.
Case 1 considers an artificially constructed signal that is abstracted from the harmonic characteristics of the uni-lateral saturation hard limiter in Section II-C to further demonstrate whether is a necessary condition of phase coupling in HOS.
Case 2 sets up a grid-connected PMSG model to demonstrate the effectiveness of the proposed process in detecting the nonlinearity owing to the uni-lateral saturation hard limiting by collecting accident waveform records at the terminals of the VSCs. In this case, the nonlinearity owing to bi-lateral saturation hard limiting is also detected later using the tri-coherence spectrum.
Case 3 sets up an IEEE 9-bus system with three SVGs and one static VAR compensator (SVC), wherein the self-sustained oscillation is induced by two SVGs. This case shows that the HOS can only detect the presence of nonlinearity but not locate the source of the nonlinear oscillations.
Here, the following signal is considered in MATLAB.
(52) |
where is the sampling frequency; and is a -20 dB Gaussian white noise.
In (52), let , and record as . Further, let , , and record as .

Fig. 10 Detection of quadratic phase coupling. (a) Frequency spectrum of . (b) Bi-coherence spectrum of . (c) Frequency spectrum of . (d) Bi-coherence spectrum of .
Previous studies [
In this study, according to (11), the output of the uni-lateral saturation hard limiter can be expressed as:
(53) |
Therefore, the peaks with values greater than will appear in its bi-coherence spectrum similarly to a “chessboard” because any two integer multiples of the fundamental frequency have the property of the quadratic phase coupling. However, if the bi-coherence could detect only those satisfying and simultaneously, no peaks in the bi-coherence spectrum will exist, which does not match the actual situation.
A detailed grid-connected PMSG model is set up in PSCAD/EMTDC with the structure of the VSC control system shown in

Fig. 11 PMSG model for DNB.
First, the parameters are adjusted to make the hard limit of PI in the d-axis outer control loop of the voltage take effect.
Symbol | Description | Value |
---|---|---|
Vg | Grid voltage | 0.69 kV |
f0 | Fundamental frequency | 50 Hz |
PN | Rated capacity of PMSG | 1.5 MW |
PLL | ||
P | Active power | 0.34 MW |
C | DC capacitor | 200 mF |
R | Connection resistance | 0.001 Ω |
L | Connection inductance | 0.35 mH |
Rg | Grid-side resistance | 0.005 Ω |
Lg | Grid-side inductance | 0.4 mH |
DC-voltage controller | ||
Reactive power controller | ||
Inner-loop current controller |
The simulation is implemented as follows.
: use a voltage source to charge the DC capacitor. In the initial state, the PMSG is off-grid, and the active power and reactive power references are both zero.
: the DC capacitor side is switched to the power source, and the PMSG is connected to the grid.
: the active power is set to be 0.34 MW.
: is set to be .
: is set to be .

Fig. 12 d-axis current reference.
Stage I: after and before any hard limit takes effect, the system can be analyzed with small-signal linearization. Based on the parameters presented in
Stage II: after the oscillation causes to reach the hard limit of PI in the d-axis outer control loop of the voltage, the hard limit becomes uni-laterally saturated and the system cannot be analyzed by the linearization method. At the PCC, a constant-amplitude self-sustained oscillation of 33.8 Hz is observed, as shown in

Fig. 13 Voltage and current at PCC in d-axis.

Fig. 14 Bi-coherence and power spectra of and . (a) Bi-coherence spectrum of . (b) Power spectrum of . (c) Bi-coherence spectrum of . (d) Power spectrum of .
Next, we adjust the parameters to examine the effectiveness of the tri-coherence spectrum. We set the upper and lower limits of the d-axis current reference to and set those of the d-axis voltage reference to . Then, the hard limit of takes effect and induces a bi-lateral saturation self-sustained oscillation.

Fig. 15 Bi- and tri-coherence spectra of . (a) Bi-coherence spectrum of . (b) Tri-coherence spectrum of .

Fig. 16 Topology of case study system.
Parameter | Value |
---|---|
KPυd, KIυd, KPυq, KIυq (SVG voltage control loop) | 2.5 p.u., 1000 p.u., 2 p.u., 20 p.u. |
Vref1, Vref2, Vref3 (reference of terminal voltage control) |
1.005 p.u., 1.005 p.u., 1.005 p.u. ( s) 1.005 p.u., 1.000 p.u., 1.005 p.u. ( s) |
KPi, KIi (current control loop) | 40 p.u., 6250 p.u. |
Xl1, Xl2, Xl3 (connection impedance) | 0.0051 p.u., 0.0038 p.u., 0.0256 p.u. |
R6-10, R8-11 (line resistance) | 0.0017 p.u., 0.0054 p.u. |
X6-10, X8-11 (line impedance) | 0.0092 p.u., 0.0178 p.u. |
XT1, XT2, XT3 (transformer impedance) | 0.0586 p.u., 0.0586 p.u., 0.0576 p.u. |
The case study system is simulated using PSCAD/EMTDC. In the case, a self-sustained oscillation is induced by mismatching the reference terminal voltages of SVG1 and SVG2, which are denoted by and , respectively.

Fig. 17 Voltage and current waveforms. (a) Voltage amplitude. (b) Three-phase. (c) Three-phase currents of SVC.
Calculated as (51),
Part | μ (p.u.) | Threshold μp (p.u.) |
---|---|---|
SVG1 | 0.3727 | 0.1000 |
SVG2 | 0.1598 | |
SVG3 | 0.1043 | |
SVC | 0.0079 |
This study proposes a method based on the HOS analysis for the hard-limiting DNB in the VSC control system in wind farms, wherein the PMSGs and VSGs are modeled using a unified VSC control model. The contributions are summarized as follows.
1) The effectiveness of the bi- and tri-spectra is proven when characterizing the nonlinear behavior induced by hard limiting in the VSC control system.
2) The effects of the linear parts in the VSC control system are eliminated using the bi- and tri-coherence, which facilitate the hard-limiting DNB with only the waveforms collecte at the terminal of the VSC. Additionally, the phase equation is proven to be a sufficient and unnecessary condition of the phase coupling phenomenon, which is not examined thoroughly in previous studies.
3) The detailed procedure is proposed to detect and classify the nonlinear behavior caused by the bi- or uni-lateral saturation hard limiting.
4) The data processing problems are solved in the DNB procedures based on the HOS, such as the “0/0” phenomenon and spectrum leakage, to improve the resolution and quality of the spectra.
Future studies include two aspects.
First, the HOS analysis is a measurement-based method. However, the extension of the proposed method to any equipment requires further examination. This study confirms the applicability of the HOS-based DNB to VSCs, which exist in energy storage equipment and flexible DC transmission systems, in addition to wind power systems as discussed in this study. Here, we do not investigate eliminating the effects of different parts of other devices (in the path from the hard limiter to the terminal).
Second, as discussed in Section VI-C, the DNB is not sufficient to constitute an effective control measure and generator tripping strategy when a self-sustained oscillation accident occurs. Therefore, a nonlinear oscillatory source localization method needs to be introduced.
Appendix
0 | ||
1 | ||
2 | ||
3 | ||
4 | ||
5 | ||
6 | ||
7 |
0 | ||
1 | ||
2 | ||
3 | ||
4 | ||
5 | ||
6 | ||
7 |
Based on the results in Section II-B, when self-sustained oscillation occurs, the output of the bi-lateral saturation hard limiter contains the
(B1) |
where , , and are the Fourier coefficients of the
The Fourier transform is performed three times on the third-order autocorrelation function of . Thus, the tri-spectrum of can be derived as:
(B2) |
Additionally, the tri-spectrum has 96 symmetrical regions [
(B3) |
is a finite maximum if and only if or ; otherwise, it is zero. Therefore, two peaks can be observed at the x-y-z coordinates and in the four-dimensional (4D) graph of . For an easy intuitive visualization of the graphs, when the area is extended to , four peaks at , , and occur.
The power spectrum of is expressed as:
(B4) |
According to (B3) and (B4), the tri-coherence spectrum can be expressed as:
(B5) |
In (B5), the tri-coherence spectrum of reaches its peak if and only if , , , or .
When , (B5) is calculated as (B6). When , , or , (B5) is calculated as (B7).
(B6) |
(B7) |
In (B5), . Therefore, the range of the corresponding tri-coherence value of each x-y-z coordinate in the tri-coherence spectrum is . The larger the tri-coherence value, the stronger the nonlinear phase coupling among the three frequencies corresponding to the coordinate (the stronger the nonlinearity).
In (B1), when extends to , according to (9), which accurately represents the output of the bi-lateral saturation hard limiter, it can be proved that in the 4D graph of , peaks exist at the x-y-z coordinates .
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