Study of the Windowing and Overlap-Add Operation for a Super-Gaussian Random Vibration Test

Random vibration environmental testing employs the specified statistical properties of the real world vibration to reproduce the desired excitations on the shaker table for fatigue test purposes. Smooth and safe operation is the essential requirement for a longduration test. Traditionally, the windowing and overlap-add (WOA) method is applied to the acceleration signals of the shaker table, and previous studies have indicated that this operation reduces the kurtoses of the processed signals. To protect the test equipment from abrupt changes in the input voltage, the WOA method is proposed to operate on the input voltage signals in a frame-by-frame form for super-Gaussian environmental testing. To figure out the impacts of the proposed operation on the response kurtoses of a shaker table, we express the system transfer function in the time domain, and theWOAmethod is analysed considering the transfer function of a dynamic system. Based on the analysis, a further study is made to explain the mechanism of the kurtosis decrease due to theWOAmethod.)rough the study, we find that the kurtosis reduction conclusion is not applicable to all types of super-Gaussian signals, and the kurtoses can be invariable and even increased by allocating the positions of the highexcursion peaks of super-Gaussian signals when the WOA method is applied. A window function is recommended for zeromemory nonlinear (ZMNL) transformation to move the positions of the high-excursion peaks of a super-Gaussian signal, providing a novel way of adjusting kurtosis whenWOAmethod is applied.)e proposedWOAmethod and window function are first verified in a single-input-single-output (SISO) numerical simulation to test their effectiveness under different reference kurtoses. )en, they are evaluated in a two-input-two-output shaker table test. )e test results demonstrate that the proposed window function can prevent the kurtosis decrease with the application of the WOA method.


Introduction
e windowing operation is widely used in the signal processing domain. For example, in spectral analysis, computers are not capable of handling signals with infinite length. A common practice is to split the signals into segments on which Fourier transformation is applied. However, the truncation may result in a distortion of the frequency spectra of the original signals [1]. To mitigate this issue, the intercepted signals are multiplied by window functions. Another operation, known as the overlap-add (OLA) method, reassembles the different pieces of processed signals into infinite-length signals. Usually, the OLA method is often applied after the windowing operation. e windowing and overlap-add (WOA) operation is also an important tool in random vibration environmental testing. In the test, products are exposed to the vibrational environments they endure during their service lifetime. e vibration environments we apply in the laboratory should possess the same statistical properties as those in the real world. ese properties include the power spectra density (PSD), probability density function (PDF), kurtosis and skewness for the non-Gaussian random test, and stationarity for the stationary random test. ese objects frequently deviate from their desired values due to disturbances and some indispensable algorithms. e WOA method is the dispensable method in this test. e vibration test serves for testing the reliability or fatigue assessment, which will take a considerably long time. e shaker table excitations with long-enough duration are required. However, the memory of most test instruments is unable to save and process the entire signals with the desired length. As a result, the excitations are generated segment by segment. Each segment should possess the desired statistical properties. WOA method is used to connect two segments in case discontinuity between the two segments of signals occurs. In this area, the major concern lies in the impact of the WOA method on the above statistical properties. Many research works have been conducted to mitigate the side effects. In 1993, time-domain randomization was recommended for the Gaussian stationary random test by Smallwood and Paez [2]. e method can produce infinite-length input signals without changing the PSD of the output signals. In addition, several window functions and overlap factors were studied for the realization of the minimum stationarity errors. In 1997, Smallwood [3] put forward a procedure for super-Gaussian tests. In that article, he indicated that the generation of continuous and infinite inputs by the means of time-domain randomization presents some difficulties. If the WOA operation, the core component of time-domain randomization, is implemented in a super-Gaussian test, the output kurtoses are reduced, and the distribution of the output moves towards a Gaussian distribution. In 2005, Steinwolf et al. [4] investigated the kurtosis reduction arising from the WOA method for fatigue estimation and put forward a equation to describe the numerical relationship of the reduction. ey located the specific numerical relationship between the kurtosis of the signal before WOA operation and the signal's kurtosis after the operation. In 2019, González and López-Valcarce [5] put forward a window design method to improve the stationarity for both the Gaussian and non-Gaussian random tests. ey reported that the OLA method shifted the random process from super-Gaussian to Gaussian. In general, the current viewpoint is that the WOA operation reduces the kurtosis of the super-Gaussian signal to be processed. Currently, there are only two methods to solve the problem. One is to simply employ the kurtosis control algorithm [6]. e other is to use Steinwolf's equation [4]. e equation is not always precise, and additional kurtosis control efforts are still needed after the correction. How to eliminate the kurtosis loss effectively and accurately is the concern of this article.
After 2010, fatigue studies for structures under super-Gaussian random excitations are widely conducted. A series of articles has been published in the area [7][8][9][10]. In fatigue estimation, the acceleration signals of the shaker table are seen as the ground excitations of structures. To maintain the continuity of the long-duration excitations, the WOA method is applied to the acceleration signals. is type of operation can be found in many literature reports [11][12][13][14]. e foremost input of the whole test system is the input voltage of the shakers. We can control the shape of the PSD, kurtosis, and other properties of any location in the system (points on shaker table or on under-test structures) by  modifying the input voltage. e continuity of the input voltage is the basic requirement for a smooth and safe experiment. Usually, the loss of WOA operation leads to considerably large peaks in the voltage level at the connecting area of adjacent frames. e consequence of the large voltage difference is the abrupt amplitude change of the shaker table, which may damage the shaker coil and disrupt an ongoing fatigue test. Articles about the WOA operation on input voltage signals have not been reported in the literature.
Motivated by the above considerations, we apply the WOA method to input voltage signals of a shaker table test system. First, the WOA operation and its impact on output kurtoses are studied. Here, the output denotes the acceleration response of a shaker table, which usually acts as excitations of the structures fixed on the table in a fatigue or reliability test. en, the reason for the kurtosis reduction caused by the WOA operation is researched. Finally, a window function is introduced in the process of generating super-Gaussian signals using a zero-memory nonlinear (ZMNL) transformation. With this windowing operation, the kurtosis reduction can be removed. e article is organized as follows. We introduce the definition of Gaussian and non-Gaussian random processes in Section 2, including their differences in kurtoses, the amplitude of time-domain signals, and PDFs. en, the state-space method of formulating transfer function matrices of a dynamic system is introduced in Section 3. In Section 4, we introduce the WOA method.
en, the impact of the WOA method on the kurtoses of the shaker table's acceleration signals is deduced. A numerical simulation is provided here to verify this deduction. In Section 5, a windowed ZMNL transformation is proposed. A numerical example is given to verify the benefits of the proposed method. A super-Gaussian field test is given in Section 6 to validate the proposed method in a two-inputtwo-output case. Finally, a discussion is presented and conclusion summarized in Section 7.

Gaussian and Super-Gaussian
Random Processes e kurtosis of any time-domain signal x(t) is defined as the fourth central moment divided by the square of its second central moment. In particular, when the mean of x(t) is zero, the kurtosis can be expressed as [15] K � E x 4 (t) where E denotes the expected value. e kurtosis of a Gaussian signal is approximately 3. A non-Gaussian signal with a kurtosis greater than 3 is called a leptokurtic or super-Gaussian signal; it is platykurtic or sub-Gaussian when this value is less than 3. Intuitively, leptokurtic time histories have much more high-excursion peaks than Gaussian and platykurtic signals, as seen in Figure 1. e time signals pictured in Figure 1 share the same amplitude spectra but have different kurtoses. With a rise in kurtosis, high peaks 2 Shock and Vibration increase in quantity and height. is conclusion can also be reached by observing the PDFs of the three signals shown in Figure 2. Actually, the kurtosis characterizes the sharpness of the PDF head and the width of the PDF tails. A super-Gaussian signal has a sharper PDF head and wider PDF tails than a Gaussian or sub-Gaussian signal does.

State-Space Method
e time-domain procedure for the super-Gaussian test used in the submission was presented by Cui et al. [16]. e formulation of this procedure starts with expressing the system transfer function using state-space matrices. e equation of motion of an N degree-of-freedom system can be written as where € x, _ x, and x are the system's response components representing the acceleration, velocity, and displacement, respectively; M, C, and K are the mass, damping, and stiffness matrices, respectively. f is the vector of external excitation forces that act on the structures, and L f is the mapping matrix for the input forces. e equation of motion shown in (2) can also be expressed in a continuous statespace form as Subscript c denotes matrices for a continuous system. Vector y is the output of the structure and can be assembled from the measurements with where R a contains the output influence matrices for the measured accelerations. Equation (5) can be rewritten as where In practice, the measurements are recorded over discrete time intervals, and the continuous state-space equation should be transformed into discrete form. erefore, (3) and (6) can be converted into the following equations: where A d , B d , C d , and D d are the discrete version of the system state-space matrices. Several sample and hold algorithms [17,18] are useful for solving discrete state-space matrices. e regularly adopted algorithm is the zero-order hold, and the corresponding A d , B d , C d , and D d are expressed as where dt is the discrete time interval.

Shock and Vibration
For a known input sequence f and initial state z(0), the state vectors at any time step can be written as us, the system output can be derived based on (8) and (10): For zero initial conditions, i.e., z(0) � 0, (11) can be represented as a moving average model: where the values of H i are called the system Markov parameters, which are often used for the time-domain force identification. Equation (12) can also be rewritten in the form of the matrix convolution equation as where Here, H ∈ R (N m ·n)×(N f ·n) , y 0 ∈ R (N m ·n)×1 , and f 0 ∈ R (N f ·n)×1 , where N m is the number of sensors; n is the number of sampling points of every block, and N f is the number of excitations. According to (11), the system Markov parameters can be expressed as Acquiring system Markov parameters, as shown in (15), the desired time-domain response signals for environmental testing are required so that we can compute the input forces for the under-test dynamical systems. e traditional environmental test procedure is defined in the frequency domain, and there are no such time-domain signals.
Nevertheless, we can construct this kind of signal using reference spectra defined in the frequency-domain test procedure.
e Cholesky decomposition of the reference spectra R is where the superscript H denotes the operation of the conjugate transpose. Multiplying the Cholesky factor L with the phase matrix P as in (17), the frequency spectra of the desired time-domain signals are obtained. en, the desired timedomain signals y r are obtained by the inverse Fourier transformation on Y. ese time-domain signals are referred to as reference signals because they are transformed from the reference spectra To obtain the drive loads for a test system, (13) should be generalized for both zero and non-zero initial condition cases. us, (13) is expressed in the form where y k � y r (k · n) y r (k · n + 1) ⋮ y r (k · n +(n − 1)) Any n-length driving forces can be represented as where z can be obtained using (10) and the observability matrix O is computed after the realization of the system. It is worth mentioning that the system transfer matrix H should be handled by truncated singular value decomposition (TSVD). is truncation brings noise filtering for environmental testing. Details can be found in [19] and are not discussed in this article.
To obtain the inputs by (20) in a super-Gaussian test, we should transform the Gaussian reference signal y r into a super-Gaussian reference signal y r with the specified kurtosis. e ZMNL functions we adopt in this article are where A is used to rescale the PSD as changed by the ZMNL transformation. Note that the ZMNL transformation may cause magnitude distortion of the PSD if the dynamic range is large, according to Kihm et al. and Steinwolf [6,9]. To find suitable parameter a for the ZMNL transformation using (21), the MATLAB function fminsearch is recommended for a given kurtosis to locate the unknown parameters so that the error function is minimized. In (22), K y,i denotes the kurtosis of the i th block of the response signals and K r,i denotes the desired kurtosis of the i th block of the reference signals.
To summarize, the flow chart of the time-domain procedure for a super-Gaussian test is shown in Figure 3. First, reference spectra R are decomposed by the Cholesky decomposition into L. e frequency spectra Y of the reference signals for a Gaussian test are obtained using (17). Afterward, the Gaussian reference signal y r is gained by inverse Fourier transformation on Y. e super-Gaussian reference signal y r can be obtained using (21). To acquire the input signals f for a super-Gaussian test, it is divided by the system transfer matrix H as in (20). e input signals f are then sent to stimulate the under-test articles, and the response signals y of the under-test articles are gathered. In this step, the power spectra and kurtoses of y are obtained and compared with their reference values. is loop continues until the power spectra S yy and the kurtoses K y converge to their references R and K r , respectively.

WOA Operation for the Gaussian Test.
e time-domain randomization proposed by Smallwood and Paez [2] is divided into 2 parts. e first part is to convert one frame pseudorandom signal into a vast number of similar but different signals. e second part is to window and overlapadd these signals for the generation of infinite nonperiodic true random signals. To express our method for a super-Gaussian test, the original WOA operation for a Gaussian test is necessary to be introduced. e windowing operation is used to mitigate leakage when we cut off a segment of signals for analysis. Frequently used window functions include the half-sine, Potter, and Hanning windows. In environmental testing, we typically use the half-sine window. Figure 4 demonstrates the schematic diagram of a windowing operation employing the half-sine window. After being windowed, the blocks of data are then overlapped and added to each other as outlined in Figure 5. In Figure 5, n is the length of a frame of data. e 2 nd frame of data is delayed with respect to the 1 st frame by n/2. erefore, the overlap factor is 2. e overlap factor p is defined as the ratio of the block length n to the number of samples over which a block of data is delayed with respect to the previous block in the overlap-add operation. In this paper, we set this factor of all overlapping operations to be 2.

WOA Operation for a Super-Gaussian Test.
If the signals being processed by the WOA method follow a Gaussian distribution, the kurtoses of the signals are not changed. is conclusion has been verified by Steinwolf [4]. In fact, he also quantified how the kurtoses were changed by a half-sine window and overlap-add operation with the expression [4] where K 1 is the averaged kurtoses of all blocks of signals before the WOA operation, K 2 is the averaged kurtoses of all blocks of signals after the operation, and p is the overlap factor. e military standard [20] stipulates we should gather at least 50 blocks of signals to acquire K 1 if we employ the arithmetical average. As mentioned previously, the overlap factor is 2 in this article, so (23) can be simplified to [4] Steinwolf's equation (24) is deduced for super-Gaussian signals being processed but is also suitable for sub-Gaussian situations. However, it does not reveal how the kurtoses of the response signals are changed if we window and overlapadd the input voltage signals. Here, we explain the relationship based on (18). e k th block of input signals f k and the k th block of super-Gaussian reference signals y k are related to the expression shown in (25) if the initial condition z(kn) � 0.  Figure 3: Flow chart of a super-Gaussian test procedure.
Shock and Vibration 5 If we window the input signals f k , the windowed inputs are where w n represents the window function. Substituting (26) into (25), we have From (27), we know that the window operation on the input signals is equivalent to the same window operation on the response signals.
en, we study the overlap-add operation between the k th block of the signals and the (k + 1) th block of the signals. First, we delay the (k + 1) th block of the input signal f k+1 by n/ 2 and add the corresponding segment of f delayed k+1 to f k . e overlap-added input signal is Half-sine window 6 Shock and Vibration

Substituting (28) into (25), the equation is
Because the first n/2 rows of f delayed k+1 are zeros, the term Hf delayed k+1 can be simplified as Shock and Vibration e (k + 1) th block of the input signals f k+1 and the (k + 1) th block of the super-Gaussian reference signals y k+1 are related to the expression in (29) if the initial condition z((k + 1)n) � 0: As seen from (31), the non-zero rows of the right term in (30) correspond to the first n/2 rows of the right term of (31). As a result, (30) can be expressed as and with (32), (29) becomes Equation (33) indicates that the overlap-add operation on input signals is equivalent to the same operation on the response signals.
To summarize, as shown in (30) and (33), we find that if we apply the WOA operation on the voltage signals of a test system, the responses of the shaker table are transformed in the same way.
is conclusion is reached under the assumption that the initial condition of every block signal is zero. Numerical simulation shows that omitting the initial conditions in (20) does not change the response spectra or response kurtoses. erefore, when we study the response kurtoses, the effects of the initial conditions can be neglected. Since the system Markov parameters and the frequency response functions all represent the same dynamic system, the above conclusion is expected to apply to environmental testing using both the time-domain and frequency-domain procedures. is conclusion will be verified in the numerical simulation.
Leaving out other factors that may change the response kurtoses, under ideal conditions, the response kurtoses should be equal to the reference kurtoses if WOA operation is not applied. Once the WOA is applied on the input voltage signals in a super-Gaussian test, according to (30) and (33), the response kurtoses can be anticipated based on Steinwolf's equation [4]. When the window function is a half-sine window and the overlap factor is 2, the kurtoses of the response signals are where K y denotes the averaged kurtoses of the response signals and K r represents the desired reference kurtoses.

Numerical Example.
In this finite element simulation, the WOA operation is introduced into both the time-domain and frequency-domain super-Gaussian test procedures. e timedomain procedure is outlined in Figure 3. e frequencydomain procedure utilizes a frequency response function to compute the input voltage signals, which can be found in [3]. As seen in Figure 6, a model of a cantilever beam consists of 5 plane beam elements whose nodes have one translational degree of freedom and one rotational degree of freedom. e reference spectrum of the simulation is described in Table 1.
e parameters of the Bernoulli-Euler beam are listed in Table 2. Above all, the damping of this structure is defined in terms of (35), where ζ k is the k th modal damping ratio; M k and K k are the k th modal mass and stiffness, respectively. To eliminate the impact of damping on the response kurtoses, the modal damping ratio of every mode is set to 0.05 in this simulation. e reference spectra are still those described in Table 1.

Shock and Vibration
First, we compute the discrete input using (20) and apply the WOA operation to the input. e continuous input and discrete input are both shown in Figure 7. From the figure, we can see that there are more high-excursion peaks in the discrete input voltages than in the continuous input voltages. e sampling period is 0.4 s, and the positions of these peaks are located at the frame boundaries of the discrete input. It is obvious that the WOA operation can remove discontinuity at the adjacent area of different frames. Figure 8 shows the autopower spectral density (ASD) of the control point generated by the continuous input, and Figure 9 demonstrates the corresponding kurtoses of all 50 frames of responses. e reference kurtosis of the simulation is 10, and the averaged kurtosis of the response is 8.05, which fits (34). To further validate the relationship between the reference kurtosis and the averaged response kurtosis, the averaged response kurtoses are shown in Figure 10 with the change of the reference kurtoses. e solid curve in this figure represents (34), the plus symbols denote the results from the time-domain procedure, and the stars show the results from the frequency-domain procedure. e three lines match well when the reference kurtoses are below 20, which confirms      Shock and Vibration that (34) is credible to some degree. When the reference kurtoses are above 20, the averaged response kurtoses from both procedures deviate from the theoretical curve. Nevertheless, the overall trend of the averaged response kurtoses fits that of (34), and most of the deviations are acceptable. Although the influences of the WOA operation on response kurtoses are deduced using the state-space model, the conclusion is also valid using the frequency-domain approach of computing the inputs.

The Improved Method for a Super-Gaussian Test with a Continuous Input
Knowing the relationship between averaged response kurtosis and the reference kurtosis with the WOA operation applied, we seem to be able to simply employ (34) to reset the reference kurtosis K r ′ so that the averaged response kurtosis K y can be equal to the true reference kurtosis K r with the expression where K r ′ is used in (22) to generate the reference signals. However, (34) is not valid to all kinds of super-Gaussian signals. e kurtosis of the signal operated by WOA method is highly related with the position of the high-excursion peaks in the signal. e support of the conclusion will be introduced in this section. Together, an improved method is recommended in this section to eliminate the kurtosis reduction caused by the WOA operation. e method starts by studying the reason for the kurtosis reduction.

Kurtosis Reduction Induced by the WOA Operation.
According to the conclusion we draw in Section 4.2, the WOA operation on the input is equivalent to the same operation on the response. Under ideal conditions, the response should be equal to the reference signals we build at the beginning. To determine the reason for the kurtosis reduction induced by the WOA operation, we can apply the WOA operation on the reference signals. Figure 11 demonstrates the process of the windowing operation on one frame of the reference signals using a half-sine function. Intuitively, super-Gaussian signals are characterized by high-excursion peaks. e number and heights of these peaks are highly related to the kurtoses of the signals. In Figures 11 and 12, the X coordinates denote time, and the Y coordinates represent the amplitude of the signal. e sampling period of one frame of the signal is 0.4 s, which indicates that the point in Figure 11 with X � 0.2117 is located in the middle area of the signal. After windowing, the amplitudes of the high peaks in the middle area of the signal decrease much less than those at either ends. Figure 12 outlines the OLA operation between the first windowed frame and the second windowed frame. Only the n-length signal after an OLA operation is provided in the figure. Because the second frame of the windowed signal is delayed by n/2 with respect to the first frame of the windowed signal, the middle area of the first frame of the windowed signal corresponds to the left end of the second frame. Both ends of the windowed signals have small amplitudes, which causes the high peaks in the middle area of the first frame to barely change by the adding operation. In contrast, the peaks in the right end of the first frame correspond to the peaks with large amplitudes of the second frame, and the same applies in the middle area, resulting in large changes to the peak amplitudes at the end. As a result, if the high-excursion peaks are distributed in the marginal area of every frame of the signals, the highexcursion peaks are reduced in number by the WOA operation, and kurtoses of the windowed and overlapadded signals may be substantially decreased. ere were several methods that generate the super-Gaussian signal with specified locations of high-excursion peaks. For example, an on-band phase manipulation method [21] creates the super-Gaussian signal with only one high peak per data block at specified locations. A method recommended by Zhang et al. [22] also have the same feature. To prevent the high-excursion peaks and kurtoses from being decreased by the WOA operation, we should move the high-excursion peaks' location.

Windowed ZMNL Transformation.
e ZMNL transformation employs specific functions to transform a Gaussian signal into a super-Gaussian signal. e theory of this transformation is to maintain most of the data of the Gaussian signal and to amplify the data with relatively large amplitudes. e ZMNL function has the following property:  where f(x) denotes the ZMNL function and x represents the data of a Gaussian signal. When x approaches zero, the data are incorporated into the new super-Gaussian signal. Otherwise, x values with large amplitudes are increased, and high-excursion peaks appear in a super-Gaussian transformation. Based on this property, we can shrink the amplitude of a Gaussian signal in the area near both ends with the window function where we use c to control the area that the high peaks generate; b is less than 1 and is used to decrease the amplitudes of the data outside the central area. e original Gaussian signal is multiplied by (38) to reshape the amplitude of the signal. en, we process this windowed signal with the ZMNL transformation. e ZMNL function we use in this article is the hyperbolic sine function. e expression of the transformation is To make the kurtosis of y r equal to the reference kurtosis, we need to find a suitable a and A so that Er in (22) is minimized. Equation (38) is able to decrease the amplitude of y r in the marginal area of one frame signal. Because the amplitudes of the windowed signal in the middle positions are relatively large, it is more likely that the high-excursion peaks are created in the central positions. A schematic diagram of this process is given in Figure 13.
Because the mechanism of creating non-Gaussian features is similar for all types of ZMNL transformation, the method proposed in this section is suitable for any ZMNL function. It is worth mentioning that the method recommended by Zhang et al. [22] is also capable of adjusting the positions and the numbers of high-excursion peaks.

Kurtosis Increase Induced by the WOA Operation.
Steinwolf's equation is deduced under the assumption that the high-excursion peaks are produced with random positions. If we can move these peaks to the middle area of every block of signals, as shown in Figure 13, the OLA method might increase the kurtoses. If the overlap factor is 2, the OLA method delays the next frame of the signal by n/2 and adds these signals together in a frame-by-frame form. Since the high peaks are moved to the middle, the interval between two adjacent high peaks is decreased to approximately n/2, less than the signal before the WOA operation. In addition, because the peaks are in the middle, the windowing operation barely changes the amplitude of these peaks. In general, the density of these high peaks is increased, and their amplitudes are nearly unchanged. e kurtoses of every block of the processed signals are increased.
To explain the issue intuitively, a numerical example is given. A signal is processed with the windowed ZMNL method first, and the processed signal is then windowed with the half-sine function and overlap-added. e signals before and after the WOA operation are presented in Figure 14.
From the upper figure, we can see that the time interval of high peaks is 0.4 s, which is also the sampling time of one frame signal. e time interval indicates that the windowed ZMNL transformation can move high-excursion peaks to the middle of every frame of signals. e figure below shows the signal after the WOA operation. e time interval is approximately 0.2 s, which is half of the sampling time of one frame signal. is result suggests that the WOA method can increase the density of high-excursion peaks. e kurtosis of the signal before the WOA operation is 8, as shown in Figure 15. From the figure below, we can see that the kurtoses of the signal after the WOA operation are increased substantially.

Discussion.
According to the conclusions reached in the previous sections, we can see that the kurtosis of the signal after the WOA operation changes with respect to the positions of the high peaks. If these peaks are distributed randomly, the resulting kurtosis can be obtained using (34). e kurtosis gained by (34) is not the minimum kurtosis that the WOA method can create. By moving all the high peaks to both ends of every frame of signals, the minimum kurtosis can be reduced to 3. Since the window function proposed in (38) amplifies the signal in the middle instead of at both ends, the windowed ZMNL transformation in Section 5.2 is not capable of reducing the kurtosis to 3. By adjusting the parameters b and c of (38), the distribution of high peaks can be adjusted, and any value of output kurtosis in its interval can be achieved. e minimum value of the interval can be obtained using (34). When b and c are 0.1 and 3 in (38), the maximum averaged kurtoses obtained from a simulation are listed in Table 3.
In the experiment, we aim to make the response kurtoses equal to their preset references, which can also be achieved by adjusting b and c. A simulation is given for this purpose.
In this simulation, we still use the same parameter settings described in Section 4.3. c and b in (38) are set to 50 and 0.65, respectively. First, we set the reference kurtosis to be 10. e proposed ZMNL transformation is used to produce super-Gaussian signals with high peaks located in the middle, and the WOA operation is applied on the inputs in a frame-byframe form. e super-Gaussian response, its ASD, and the kurtoses of all 50 blocks of signals are demonstrated in Figures 16-18, respectively. e auto PSD of the response provided in Figure 18 indicates that the improved ZMNL transformation does not disturb the power spectrum. e mean value of the kurtoses shown in Figure 17 is 9.93, almost equal to its reference kurtosis. For further validation, a certain range of reference kurtoses and the corresponding averaged response kurtoses are documented in Figure 19, as in Figure 11. e averaged response kurtoses of the simulation are nearly identical to the reference kurtoses, and the figure indicates that the improved ZMNL transformation can eliminate the kurtosis reduction brought by the WOA operation regardless of the type of procedure employed. e kurtosis increase caused by the WOA method can also be used to compensate for the kurtosis loss induced by other factors, for example, low damping. To examine the low damping effect, the damping coefficient is set to 0.2%. When the windowed ZMNL transformation is not applied and the WOA method is applied to the input signal, the response kurtoses are as outlined in Figure 20. e reference kurtosis is 12, and the averaged kurtosis in Figure 20 is only 6.4, less than 9.75 (the kurtosis value obtained by (34)). When the windowed ZMNL transformation is applied with b � 0.4 and c � 20, the response kurtoses are as shown in Figure 21. e averaged kurtosis in Figure 21 is 11.5, nearly identical to the reference kurtosis.

Experimental Validation
e windowed ZMNL transformation has been validated in the numerical simulation; in addition, an experiment is conducted on a vibration table with two inputs and two outputs. In the experiment, two sensors are mounted on a vibration table. Each of them is used to collect the acceleration of different directions. We employ NI USB 6363 to send input voltage signals to the vibration table, and the voltage signals are acquired at the same time also by NI USB 6363. e data acquisition software is coded with LabVIEW and implemented on a computer. e reference kurtoses of the two points are where sensors are located at 5 and 9,        respectively.
e parameters b and c are 300 and 0.6, respectively. Figure 22 shows the input voltage signals of the initial 10 s. e input signals are obviously super-Gaussian. Super-Gaussian inputs have much more high-excursion peaks than Gaussian or sub-Gaussian inputs, which makes the input voltage tend to exceed the voltage tolerances of the vibration table. erefore, the WOA operation on the input voltage signals is absolutely necessary. e tolerated input-output voltage of the NI instrument is only 5; this limitation restricts us from conducting the super-Gaussian test with high kurtoses. To avoid this problem, we adopt relatively low reference spectra in this test, as seen in Figure 23. e horizontal portion of the reference spectra is only 10 −7 g 2 / Hz. e processed input voltage signals, response signals, and the corresponding kurtoses are shown in Figures 22-25, respectively. e mean values of the kurtoses in Figure 25 are 4.98 and 9.01, almost equal to the preset reference kurtoses. Figure 23 demonstrates the power spectral densities of the response signals. e cross-power spectral densities, defined with the phase and coherence, are not disturbed by the windowed ZMNL transformation.
Providing a validation with kurtoses ranging from 4 to 40, such as in Figures 10 or 19, presents some difficulty. As  mentioned above, the hardware cannot accommodate inputs with large amplitudes when the reference kurtoses are too large. Here, a similar result with a shorter kurtosis range is given in Figure 26. In the interval from 4 to 12, the response kurtoses generally fit the K r � K y curve.

Conclusion
In this paper, the state-space method is employed to deduce how response kurtoses are influenced by the WOA operation on the input voltage signals. e derivation in Section 4 and the numerical simulation in Section 4.3 verify that the WOA operation on the input signals is equivalent to the same operation on the corresponding response signals. Based on this finding, the reason why the WOA operation causes a kurtosis loss is studied. en, the window operation is proposed to be added into the ZMNL transformation. With this windowed ZMNL method, high-excursion peaks of super-Gaussian signals can be centred in the middle position of every frame, and kurtoses can be invariable and even increased. A discussion about the parameter settings for the windowed ZMNL method is also presented. A singleinput single-output numerical simulation and a two-input two-output shaker table test validate the effectiveness of the proposed method.

Data Availability
e data used to support the findings of this study are available from the corresponding author upon request.

Conflicts of Interest
e authors declare no conflicts of interest.