A transient TRansport Analysis method for Modulation (t-TRAM) has been developed. This method consists of two methods. One is an instantaneous modulation analysis method using the Hilbert transform (HT) with the analytical mode decomposition (AMD). In the modulation experiments, plasma transport can be changed periodically by modulated external input. Profiles of the modulation amplitude and phase change in time due to the change in the plasma transport. The HT with AMD method can evaluate an instantaneous analytic signal in the presence of changes in the modulation amplitude and phase in time. The other is a transport analysis method to evaluate the diffusion coefficient and the convection velocity from the analytic signal. The t-TRAM has enabled us to evaluate a momentum diffusion coefficient and momentum convection velocity in the presence of a momentum source. The t-TRAM is applied to momentum modulation experiments in JT-60U. The evaluation reveals that a normalized momentum flux has a linear dependence on a normalized momentum gradient during a half cycle of the modulation.

## I. INTRODUCTION

Modulation experiments are commonly used to evaluate the diffusion coefficient and the convection velocity of magnetically confined plasmas experimentally.^{1–8} In the experiments, the plasma transport changes periodically by the modulated external input, e.g., gas puff injection, electron cyclotron resonant heating, and neutral beam injection (NBI). The diffusion coefficient and the convection velocity are evaluated from the modulation amplitude and phase, which are commonly evaluated by the fast Fourier Transform (FFT). In order to apply the FFT to the modulation experiments, the FFT requires one or more modulation periods in which the plasma condition is kept constant. These plasma operations are generally difficult and time-consuming, and they limit the experimental condition. If non-linearity^{9} and the hysteresis relation^{10,11} between the flux and the gradient exist, the diffusion coefficient and the convection velocity will change during one cycle of the modulation. The FFT cannot evaluate these changes of the plasma transport in time.

The data obtained with the modulation experiment have several frequency components: not only the components of the modulation frequency (a few Hz) and its higher harmonics but also its nearby components due to the change in the plasma transport, the frequency components of the linear trend, and the noise components. The candidates for time-frequency domain transformation methods are the short-time Fourier transform (STFT), the wavelet transform, the Hilbert transform (HT), and so on. However, the STFT and the wavelet transform cannot evaluate the change in the amplitude and the phase during one cycle of the modulation due to the Gabor limit.^{12} The HT is one of the promising methods because it can evaluate the instantaneous amplitude and phase from an analytic signal.^{13} Here, the analytic signal is the complex function of time without any negative frequency component. Since the HT cannot separate the frequency components, the modulation component and its nearby components should be extracted from the data in advance. The candidates for the methods extracting these components are a band-pass filter using the FFT and the empirical mode decomposition (EMD).^{14} In the case of the band-pass filter using the FFT, the decomposed signal distorts due to the Gibbs phenomenon at the edge of the data, and its effect is significant when the number of cycles is small. The EMD is often used with the HT, which is called Hilbert–Huang transform (HHT).^{14} The EMD separates the data into a finite number of intrinsic mode functions, which are well-behaved for the HT. However, the EMD faces several difficulties due to the empirical nature of its decomposition process when the signal contains several frequency components in a narrow frequency range. Thus, when the modulation frequency is low of a few Hz, it is difficult to extract only the modulation component and its nearby components from the data because the signal contains the frequency components of the higher harmonics in a narrow frequency range.

The analytical mode decomposition (AMD) is one of the promising methods to extract the frequency components in the narrow frequency range, which was developed to overcome the challenges with the EMD.^{15} Thus, the AMD is also well-behaved for the HT. The AMD can decompose a time series into two components whose Fourier spectra do not vanish over two mutually exclusive frequency ranges separated by a constant bisecting frequency. The AMD can decompose the non-stationary and nonlinear signal with the amplitude-decaying and the frequency-changing components. The feature is suitable for the modulation analysis.

This paper presents the transient transport analysis method for modulation (t-TRAM), using the HT with AMD method, and the diffusion coefficient and the convection velocity calculated from the analytic signal. Note that throughout this paper, the time function is treated as a discrete value in time. In Sec. II, the HT with AMD method and the transport analysis method from the analytic signal are explained. In Sec. III, analysis results of the momentum modulation experiment in JT-60U using the t-TRAM are shown. In Sec. IV, the effect of the source term on the transport analysis is investigated. In Sec. V, the linearity of momentum transport is discussed. Finally, we will summarize them in Sec. VI.

## II. ANALYSIS METHOD

In this section, the t-TRAM is explained, which consists of the HT with AMD method and the transport analysis method from the analytic signal. These validations will be shown in Appendix B.

### A. HT with AMD method

The analytic signal is evaluated by the HT. Let *S*(*t*) be a time series signal, and the analytic signal *Z*(*t*) is determined as

Here, j is an imaginary unit, *ω* is an angular frequency, $F$ indicates the Fourier transform, $F\u22121$ indicates the inverse Fourier transform, and *u*(*ω*) is the Heaviside step function, which is

From the analytic signal, the instantaneous amplitude and phase of *S*(*t*) are evaluated by |*Z*(*t*)| and arg[*Z*(*t*)], respectively.

In the modulation experiment, *S*(*t*) contains not only the components of the modulation frequency and its higher harmonics but also the frequency components due to the change in the plasma transport, the frequency components of the linear trend, and the noise components. The frequency components due to the change in the plasma transport are assumed to surround the modulation frequency, and their frequency are less than the second-harmonic frequency of the modulation. To extract the modulation frequency and its nearby components from *S*(*t*), the AMD is applied,^{15} the theorem of which is described in Appendix A. Here, the application for the band-pass filter using the AMD is shown. The signal *S*(*t*) can be decomposed by the following equations:

where *ω*_{i} (*ω*_{i} > *ω*_{i−1}) is a bisecting frequency. The signal *x*_{i}(*t*) contains the frequency components lower than *ω*_{i} because Eq. (1b) works as the low-pass filter. Thus, the signal *s*_{i}(*t*) contains the frequency components from *ω*_{i−1} to *ω*_{i} of *S*(*t*). In this paper, the two bisecting frequencies *ω*_{i−1}/2*π* and *ω*_{i}/2*π* are set to 0.5 and 3.0 Hz, respectively, for the purpose of its application to the momentum modulation experiments in JT-60U, in which the modulation frequency is 2 Hz.

In the case of the finite length data, the HT makes an error at the edges of the data. As we described in the Introduction, since it is difficult to keep the plasma parameters constant, the data length of the modulation experiments is limited. Thus, the time-reversed data are connected to both the edges of the original data in order to obtain the long distance between the edges of the data and the region of interest.^{16} This procedure is called “flip” in this paper. The data *S*_{Flip=n}(*t*), which are produced by performing flip procedure n times on *S*(*t*), are described as follows:

When the data have a discontinuous point, the large instantaneous frequency occurs there. Thus, the HT in Eq. (1b) makes an error due to the Nyquist frequency. In order to bring *S*_{Flip=n}(*t*) a continuously differentiable function, *t*_{0} and *t*_{1} are determined as the time when the time derivative of *S*(*t*) is close to zero. Therefore, the length of the analysis time window is smaller than that of the original data. In this paper, the number of flips was determined as 3 from the discussion in Appendix B 1.

### B. Transport analysis method from analytic signal

Using the diffusion coefficient *D*_{X} and the convection velocity *V*_{X} of fluid moment *X*, such as particle, heat, and momentum, Fick’s law with the convection term and the continuity equation for the modulation are described as follows:

Here, the cylindrical plasma is assumed, and *r*, Γ_{X}, and *S*_{X} are the radial position, the flux, and the source term, respectively. The positive *V*_{X} indicates the outward convection. In the conventional transport analysis method, Eqs. (2a) and (2b) are deformed as $\Gamma \u0303X=\u2212DX\u2207X\u0303+VXX\u0303$ and $\u2212j\omega X\u0303=\u2212\u2207\u22c5\Gamma \u0303X+S\u0303X$, respectively, assuming that the modulation frequency of *X* is constant in time and equal to the modulation frequency of *S*_{X}. Here, $\Gamma \u0303X$, $S\u0303X$, and $X\u0303$ are the phasor of Γ_{X}, *S*_{X}, and *X*, respectively. However, when the modulation frequency changes in time, *D*_{X} and *V*_{X} can be described as follows:

Here, *α*, *β*, *S*_{Re}, and *S*_{Im} are the real and the imaginary parts of the complex modulation component of *X* and the real and the imaginary parts of the complex modulation component of *S*_{X}, respectively. These complex modulation components are calculated as the analytic signal by the HT with AMD method.

## III. APPLICATION TO MODULATION EXPERIMENT

Using the t-TRAM, the temporal evolution of the diffusion coefficient and the convection velocity is evaluated. As an example of the modulation experiments, the JT-60U momentum modulation experiment (No. 49555) is treated. The target plasma is H-mode plasma with the plasma current of 1.0 MA, the toroidal magnetic field of 3.8 T, the major radius *R*_{maj} of 3.4 m, the volume-averaged minor radius at the plasma edge *a* of 1.03 m, the triangularity of 0.33, and the elongation of 1.4. Figure 1 shows the temporal evolution of the neutral beam injection (NBI) power, the electron density *n*_{e}, and the electron temperature *T*_{e} obtained with the Thomson scattering measurement^{17} and the ion temperature *T*_{i} and the toroidal rotation velocity *V*_{ϕ} obtained with charge exchange spectroscopy (CXRS).^{18} The chords of NBI are shown in Fig. 2. The two tangential NBs (Nos. 9 and 10) and the three perpendicular NBs (Nos. 4, 6, and 14) are injected stationary. The power of the two perpendicular NBs (Nos. 3 and 13) is modulated. The values of *n*_{e}, *T*_{e}, and *T*_{i} are almost kept constant even though the modulated NBs are injected. The value of *V*_{ϕ} decreases while the modulated NBs are injecting and increases while the modulated NBs are off. In the experiment, the tangential NB of No. 10, which delivers the torque density one order of magnitude larger than the perpendicular NB, accidentally stopped at *t* ∼ 8.1 s. Thus, *V*_{ϕ} suddenly drops at *t* = 8.1 s. As shown in Fig. 11, the transitional change produces a relatively large error, especially in the phase. For accuracy, the time window of *t* < 8.2 s is eliminated from the analysis for the HT with AMD method.

Figures 3(a) and 3(b) show the amplitude *A*_{V} (=|*α*_{V} + j*β*_{V}|) and the phase *θ*_{V} (=arg[*α*_{V} + j*β*_{V}]) of the modulated *V*_{ϕ} at *t* = 9.2 s evaluated by the HT with AMD method, respectively. Here, *α*_{V} and *β*_{V} are the real part and the imaginary part of the analytic signal of the modulated *V*_{ϕ}, respectively. In Figs. 3(a) and 3(b), *A*_{V} and *θ*_{V} calculated by the conventional modulation analysis method, in which a sinusoidal fitting was applied,^{6} are also shown as a reference. Furthermore, the ion density *n*_{i} profile, the evaluation method of which is shown in the next paragraph, is shown in Fig. 3(a). The fitted curves of *α*_{V} and *β*_{V} are the eighth order even polynomial function. Both the results are quite similar in 0.4 < *ro*/*a* < 0.8. Here, *ro* indicates the volume-averaged minor radius. Figures 3(c) and 3(d) show the profiles of the momentum diffusion coefficient *χ*_{ϕ} and the momentum convection velocity *V*_{conv} at *t* = 8.8, 9.2, and 9.6 s evaluated by the t-TRAM. The detailed calculation method is shown in the next paragraph. These results indicate that the profiles of *χ*_{ϕ} and *V*_{conv} can be evaluated by the t-TRAM, even in the situation where momentum transport varies in time. In these figures, the profiles of *χ*_{ϕ} and *V*_{conv} evaluated by the conventional transport analysis method are also shown. The profiles of *χ*_{ϕ} and *V*_{conv} evaluated by the t-TRAM are not significantly different from those evaluated by the conventional methods. Note that with the measurement error of *V*_{ϕ} being considered, the t-TRAM was performed 10 000 times. In Fig. 3, the errors in the results of the t-TRAM are Monte Carlo errors. On the other hand, the errors in the results of the conventional method are fitting errors.

Figures 4(a) and 4(b) show the spatio-temporal evolution of *α*_{V} and *β*_{V} evaluated by the HT with AMD method, respectively. Here, *α*_{V} and *β*_{V} are fitted to the eighth order even polynomial function in the radial direction. The signals can be evaluated correctly from 8.3 to 9.75 s, which is the intersection of the analysis time windows determined when the flip technique is applied to the signal at the each position. Figure 4(c) shows the *n*_{i} profile that is evaluated from the *n*_{e} profile. Here, impurities are assumed to be only carbon, and an effective charge is assumed to be *Z*_{eff} = 3.0 and constant in the radial direction. The modulated momentum flux $M\u0303$ is evaluated by

Here, *n*_{i} is assumed not to be modulated by NB. The values of *χ*_{ϕ} and *V*_{conv} without the source term are evaluated, substituting *α*_{M}, *β*_{M}, *χ*_{ϕ}, and *V*_{conv} into *α*, *β*, *D*_{X}, and *V*_{X} in Eq. (3), respectively. Figures 4(d) and 4(e) show *χ*_{ϕ} and *V*_{conv}, respectively, in 0.2 < *ro*/*a* < 0.8 since there is no line of sight of CXRS for *ro*/*a* < 0.2 and *ro*/*a* > 0.9. As shown by red plots in Fig. 3(b), the gradient of the phase becomes close to zero outside *ro*/*a* ∼ 0.8, where *χ*_{ϕ} and *V*_{conv} are unstable. The profiles of *χ*_{ϕ} and *V*_{conv} in 0.8 < *ro*/*a* < 0.9 are also eliminated from Figs. 4(d) and 4(e). Inside of *ro*/*a* = 0.4, *χ*_{ϕ} increases, and *V*_{conv} decreases with increasing time as shown in Figs. 4(d) and 4(e). Before *t* ∼ 8.8 s, the propagation speeds of *α*_{V} and *β*_{V} are slow and then increase gradually inside of *ro*/*a* = 0.4 as shown in Figs. 4(a) and 4(b). This behavior of *α*_{V} and *β*_{V} suggests that inward momentum transport increases after *t* ∼ 8.8 s. Thus, the changes in *χ*_{ϕ} and *V*_{conv} correspond to the changes in *α*_{V} and *β*_{V} in this time-space domain. These changes cannot be evaluated by the conventional modulation analysis method as it averages *α*_{V} and *β*_{V} in the analysis time window, and the t-TRAM clarifies that the momentum transport changes in time.

## IV. EFFECT OF SOURCE TERM ON MOMENTUM TRANSPORT EVALUATION

The effect of the source term on the momentum transport evaluation is now investigated. The torque density from the modulated NBs is calculated by the Orbit Following Monte Carlo (OFMC) code.^{19,20} In order to evaluate the temporal evolution of the torque density profile, the OFMC code is performed every 0.1 s from 8.2 to 10.0 s. Figure 5(a) shows the torque density profile obtained with the OFMC code. Here, the *j* × *B* and collisional torque densities to the main ion (deuterium) and the impurity (carbon) are considered. The cubic spline interpolation is applied to the *j* × *B* torque density profile in order to match the spatial resolution of the *j* × *B* torque density profile to those of *α*_{V} and *β*_{V}. Because the collisional torque density contains the Monte Carlo error, a 26th order even polynomial fitting function, whose order is determined by Akaike information criteria,^{21} is applied to the collisional torque density in the radial direction. Then, the sum of the *j* × *B* and the collisional torque densities is linearly interpolated at 2.5 ms intervals. The analytic signal of the modulated torque density is evaluated by the HT with AMD method as shown in Figs. 5(b) and 5(c). The modulation phase of the torque density is not uniform in the radial direction. The complex momentum source *S*_{M} is described as *S*_{M} = *T*_{ρ}/*m*_{D}*R*_{maj}, where *T*_{ρ} and *m*_{D} are the analytic signal of the torque density and deuterium mass, respectively. The temporal evolution of *χ*_{ϕ} without the source term $\chi \varphi w/o$, *χ*_{ϕ} with the source term $\chi \varphi w/$, *V*_{conv} without the source term $Vconvw/o$, and *V*_{conv} with the source term $Vconvw/$ at *ro*/*a* = 0.3, 0.5, and 0.7 is shown in Fig. 6. Figure 7 shows Δ*χ*_{ϕ} and Δ*V*_{conv} that are defined as $\chi \varphi w/o\u2212\chi \varphi w/$ and $Vconvw/o\u2212Vconvw/$, respectively. Since the source term in the edge region is large compared with that in the core region, |Δ*χ*_{ϕ}| and |Δ*V*_{conv}| at *ro*/*a* = 0.7 are larger than those at *ro*/*a* = 0.3 and 0.5. However, since |*χ*_{ϕ}| and |*V*_{conv}| in the core region are smaller than those in the edge region, the values of $|\Delta \chi \varphi |/|\chi \varphi w/o|$ and $|\Delta Vconv|/|Vconvw/o|$ in *ro*/*a* < 0.5 are comparable to those at *ro*/*a* ∼ 0.7. Thus, the source term should be included in the momentum transport analysis due to the correct evaluation even though the source term is small inside the position.

## V. DISCUSSION

The relation between the momentum gradient and momentum transport is investigated to clarify whether the non-linearity and/or the hysteresis relation between momentum transport and the momentum gradient exist in this discharge. Figures 8(a) and 8(b) show the relation between the normalized momentum flux *M*_{mod}/*n*_{i}*V*_{ϕ} and the normalized momentum gradient ∇(*n*_{i}*V*_{ϕ})/*n*_{i}*V*_{ϕ} at *ro*/*a* = 0.3 and 0.5, respectively. These results show that the linear dependence of *M*_{mod}/*n*_{i}*V*_{ϕ} on ∇(*n*_{i}*V*_{ϕ})/*n*_{i}*V*_{ϕ} exists at both the positions during a half cycle of the modulation: *t* = 8.3–8.55, 8.55–8.8, 8.8–9.05, 9.05–9.3, 9.3–9.55, and 9.55–9.8 s.

At *ro*/*a* = 0.3, *M*_{mod}/*n*_{i}*V*_{ϕ} for the same ∇(*n*_{i}*V*_{ϕ})/*n*_{i}*V*_{ϕ} increases at *t* ∼ 8.8 s, and the trend of *M*_{mod}/*n*_{i}*V*_{ϕ} on ∇(*n*_{i}*V*_{ϕ})/*n*_{i}*V*_{ϕ} also increases. This increase in the momentum flux corresponds to the increase in the penetration speed of *α*_{V} and *β*_{V} in the core region, as shown in Figs. 4(a) and 4(b). At *ro*/*a* = 0.5, *M*_{mod}/*n*_{i}*V*_{ϕ} for the same ∇(*n*_{i}*V*_{ϕ})/*n*_{i}*V*_{ϕ} decreases at *t* ∼ 8.6 s, and it does not change during *t* = 8.6–9.5 s and then it increases at *t* ∼ 9.6 s again. Therefore, the t-TRAM clarifies the variations of momentum transport and its dependence on the momentum gradient in time. Note that investigating the physics mechanism of this variation is left for future work.

## VI. SUMMARY

This paper describes the transient transport analysis for the modulation (t-TRAM) using the Hilbert transform with the analytical mode decomposition (HT with AMD method) and the transport analysis method from the analytic signal. The t-TRAM is applied to the momentum modulation experiments in JT-60U. The t-TRAM can evaluate the temporal evolution of *χ*_{ϕ} and *V*_{conv}. These values are not significantly different from the averaged values of *χ*_{ϕ} and *V*_{conv} evaluated by the conventional methods.

The effects of the source term on the evaluation of *χ*_{ϕ} and *V*_{conv} are investigated. The momentum source at *ro*/*a* ∼ 0.7 is larger than that at *ro*/*a* < 0.5. The differences of *χ*_{ϕ} and *V*_{conv} between with and without the source term normalized by *χ*_{ϕ} and *V*_{conv} without the source term are evaluated, respectively. These values in *ro*/*a* < 0.5, however, are comparable to those at *ro*/*a* ∼ 0.7 where the large source term exists. This result suggests that the source term affects the analysis results of *χ*_{ϕ} and *V*_{conv} even in the core region with the small source term. Therefore, the source term should be included in the modulation analysis.

Finally, the linearity and the hysteresis relation between the momentum flux normalized by the momentum and the momentum gradient normalized by the momentum are investigated. The result clarifies that the momentum flux has the linear dependence on the momentum gradient in the half cycle of the modulation in the JT-60U H-mode plasma, and the dependence of the momentum gradient on the momentum flux changes in time.

## ACKNOWLEDGMENTS

The authors are grateful to JT-60 experimental team members for the deep discussion and the valuable comments on this study. We would like to thank Dr. Terakado for providing the sophisticated CXRS data.

## DATA AVAILABILITY

Raw data were generated at the JT-60U large scale facility. Derived data supporting the findings of this study are available from the corresponding author upon reasonable request.

### APPENDIX A: AMD THEOREM

In this appendix, we describe the AMD theorem, according to Ref. 22. Let $x(t)\u2208L2(R)$ be a time series with *n* significant frequency components (*ω*_{i}: *i* = 1, 2, 3, ⋅⋅⋅ *n*, *ω*_{i} > 0 and invariant). Here, $R$ denotes the set of real numbers. The signal *x*(*t*) can be decomposed into *n* signals $xi(d)(t)$ whose Fourier spectra are equal to $F[x(t)](\omega )$ over *n* mutually exclusive frequency ranges (|*ω*| < *ω*_{b1}), ⋅⋅⋅, (*ω*_{b(i−1)} < |*ω*| < *ω*_{bi}), ⋅⋅⋅, (*ω*_{b(n−1)} < |*ω*|), that is,

Here, *ω*_{bi} ∈ (*ω*_{i}, *ω*_{i+1}) (*i* = 1, 2, ⋅⋅⋅, *n* − 1) represent bisecting frequencies. Each signal has a narrow bandwidth in the frequency domain and can be determined by

First, it is confirmed that *x*(*t*) is split into two signals whose Fourier spectra are non-vanishing over two mutually exclusive frequency ranges about a bisecting frequency. The signal *x*(*t*) can be expressed as a summation of two signals, that is,

Here, the Fourier spectra $F[s1(t)](\omega )$ and $F[s\u03041(t)](\omega )$ vanish for |*ω*| > *ω*_{b} and |*ω*| < *ω*_{b}, respectively. The bisecting frequency *ω*_{b} is an arbitrary positive value. Since $F[s1(t)](\omega )$ and $F[s\u03041(t)](\omega )$ are equal to $F[x(t)](\omega )$ over two mutually exclusive frequency ranges of open intervals (−*∞*, *ω*_{b}) and (*ω*_{b}, + *∞*), respectively, the following equations hold from the Parseval’s theorem in Fourier transforms:

Therefore, both *s*_{1}(*t*) and $s\u03041(t)$ are also in $L2(R)$.

Next, the bisecting process is applied to derive the decomposed signal. Let *s*_{c}(*t*) = cos(*ω*_{b}*t*) and *s*_{s}(*t*) = sin(*ω*_{b}*t*). From Eq. (A1), the Hilbert transform of *s*_{k}(*t*)*x*(*t*), (k = c, s), can be described as

Because the Fourier components of $F[ss(t)](\omega )$ and $F[sc(t)](\omega )$ are zero except for |*ω*| = *ω*_{b}, they are non-vanishing over mutually exclusive frequencies with $F[s1(t)](\omega )$ and $F[s\u03041(t)](\omega )$. The first term on the right-hand side of Eq. (A2) has a low pass function of *s*_{1}(*t*) and a high pass function of *s*_{k}(*t*). The second term on the right-hand side of Eq. (A2) has a high pass function of $s\u03041(t)$ and a low pass function of *s*_{k}(*t*). Thus, according to the Bedrosian theorem,^{23} Eq. (A2) can be transformed as follows:

Solving Eq. (A3) for *s*_{1}(*t*) and $H[s\u03041(t)](t)$, we obtain

Since $H[sc(t)](t)$ and $H[ss(t)](t)$ are equal to sin(*ω*_{b}*t*) and − cos(*ω*_{b}*t*), respectively, Eqs. (A4a) and (A4b) are simplified as

From Eq. (A1), $s\u03041(t)$ and the Hilbert transform of *s*_{1}(*t*) can be derived as

respectively. By selecting *ω*_{b} = *ω*_{b1}, *ω*_{b2}, ⋅⋅⋅, *ω*_{b(n−1)}, a time series can be bisected into two signals at the various frequencies as follows:

Each decomposed signal $xi(d)$ (*i* = 1, 2, ⋅⋅⋅, *n* − 1) can be expressed as

### APPENDIX B: VALIDATION

#### 1. Effect of flip technique

In this subsection, the effects of the flip technique and the length of the analysis time window on the accuracy of the HT with AMD are evaluated. From these results, how many times the flip procedure is executed for the certain length of the analysis time window is determined.

Figures 9(a1)–9(a5) show the comparison of the HT with AMD analysis results for synthetic data in the cases of Flip = 0 and 3. The synthetic signal are defined as *S*(*t*) = sin(2*F*_{C}*πt*), where *F*_{C} indicates the frequency of the carrier wave, and the analysis time window is from 0.125 to 1.875 s. Here, the carrier wave carries information, such as amplitude, frequency, and phase modulation, which correspond to the frequency components due to the plasma transport in the modulation experiment. In these figures, *F*_{C} and the sampling interval are set to 2 Hz and 2.5 ms, respectively, in order to clarify that the HT with AMD method is applicable to the data of the modulation experiment. The symbols of *A*, *A*_{given}, *θ*, and *θ*_{given} are the analysis result of the amplitude, the given amplitude, the analysis result of the phase, and the given phase, respectively. It should be noted that *θ* is defined as the phase of the signal from which 2*F*_{C}*π*(*t* − *t*_{0}) is subtracted. In order to evaluate the analysis error, the amplitude error ratio and phase difference are defined as Δ*A*/*A* = (*A* − *A*_{given})/*A*_{given} and Δ*θ* = *θ* − *θ*_{given}, respectively. A significant error exists near the edges of the data in the case of Flip = 0. However, in the case of Flip = 3, the analysis errors near the edges of the data diminish. The absolute values of Δ*A*/*A* and Δ*θ* are almost zero in the case of Flip = 3. In order to determine the number of flips, the dependence of the errors of the amplitude and the phase on the number of flips is evaluated as shown in Figs. 9(b) and 9(c). In these figures, the length of the analysis time window is scanned from 0.25 to 1.75 s. These errors are determined as the root mean square (RMS) values of |Δ*A*/*A*| and |Δ*θ*| in the analysis time window. Here, |*x*| indicates the absolute value of *x*. At Flip = 1, both the RMS values of |Δ*A*/*A*| and |Δ*θ*| are drastically suppressed to 0.1 and 0.3 rad, respectively, in all cases. The effectiveness of the flip is approximately the same in Flip = 3–5 cases in which the RMS values of |Δ*A*/*A*| and |Δ*θ*| are suppressed to 0.04 and 0.1 rad, respectively. Thus, the number of flips was determined as three in this paper. From this result, it is further clarified that the analysis can be evaluated accurately even for signals with only one half of the modulation period.

#### 2. Validation of HT with AMD method

In this subsection, we show the validation of the instantaneous amplitude and phase evaluation by the HT with AMD method using synthetic data. In order to clarify that the HT with AMD method is applicable to the data of the modulation experiment, the sampling interval (2.5 ms) and the carrier wave frequency (*F*_{C} = 2 Hz) are imitated to the experimental data.

Figure 10 shows the dependence of the analysis error on the amplitude modulation (AM) and the frequency modulation (FM). The synthetic signal *S*(*t*) is defined as

where *A*_{AM}, *A*_{FM}, and *N*(*t*) are the amplitude of the AM, the amplitude of the FM, and the Gaussian noise whose standard deviation is 0.1, respectively. The values of *A*_{AM} and *A*_{FM} are constant, and *t*_{m} is set to 1 s. The analysis time window is determined individually for each signal, which is approximately *t* = 0.16–1.9 s. Figures 10(a1)–10(a5) show the example of the HT with AMD analysis for *S*(*t*) in the case of *A*_{AM} = 0.5 and *A*_{FM} = 1.5 rad/s^{2}. This method can reproduce *S*(*t*), the amplitude modulation, and the phase variation, as shown in Figs. 10(a1)–10(a3). As shown in Figs. 10(a4) and 10(a5), |Δ*A*/*A*| and |Δ*θ*| are suppressed to 0.04 and 0.06 rad, respectively. Figures 10(b) and 10(c) show the dependence of the maximum values of |Δ*A*/*A*| and |Δ*θ*| on *A*_{AM} and *A*_{FM}. In the case of *A*_{AM} < 0.6 and *A*_{FM} < 3 rad/s^{2}, the maximum values of |Δ*A*/*A*| and |Δ*θ*| are suppressed to 0.1 and 0.1 rad, respectively. These values of *A*_{AM} and *A*_{FM} are considered to be much larger than the change in the modulation amplitude and phase due to the change in the plasma transport. Therefore, when the amplitude and the phase change slowly and smoothly in time, the HT with AMD method can evaluate *A* and *θ* with sufficient accuracy.

It is investigated whether the HT with AMD method can evaluate the signals with the transitional changes in the amplitude and the phase. The synthetic signal *S*(*t*) with the transitional changes is defined as follows:

where *A*_{PM}, *t*_{T}, and *w*_{d} indicate the amplitude of the phase modulation (PM), the transition timing (*t*_{T} = 1 s), and the time width in which 50% of the maximum change occurs around *t*_{T}, respectively. The analysis time window is approximately *t* = 0.12–1.84 s. Figure 11 shows the effect of the transitional changes in the analysis error in the case of *w*_{d} = 0.1. Figures 11(a1)–11(a5) show the example of the HT with AMD analysis result for *S*(*t*) in the case of *A*_{AM} = 0.2 and *A*_{PM} = 0.2 rad/s. The decomposed signal shown in Fig. 11(a1) appears to be consistent with *S*(*t*). However, the amplitude error exists near the transition timing as shown in Figs. 11(a2) and 11(a4), and the phase error also exists in the whole analysis time window as shown in Figs. 11(a3) and 11(a5). The effects of *A*_{AM} and *A*_{PM} on the maximum values of |Δ*A*/*A*| and |Δ*θ*| are summarized in Figs. 11(b) and 11(c). The error, especially for the phase, is more affected by *A*_{PM} compared with *A*_{AM}. When the transitional phase change occurs, the instantaneous frequency, which is a time derivative of the phase, increases/decreases drastically. Therefore, the band-pass filter using the AMD eliminates the frequency components outside the frequency band of the filter. Since the modulation frequency is low, the narrow band-pass filter is required to exclude the second harmonic components. Therefore, the significant transition timing, such as L–H transition and temporal stop of NBI, should be removed from the analysis time window.

Here, it should be noted as follows: In the case of Figs. 10(b) and 10(c), the largest |Δ*A*/*A*| and |Δ*θ*| are found at the edge of the data because the errors occur in determining the edges. On the other hand, in the case of Figs. 11(b) and 11(c), the largest |Δ*A*/*A*| and |Δ*θ*| are found near the transition timing when the instantaneous frequency changes drastically. Thus, the difference in the properties of the two cases leads to the difference between Figs. 10 and 11.

#### 3. Validation of transport analysis method from analytic signal

In order to validate Eq. (3), they are applied to the JT-60U experimental data (No. 49555). The details of the experiment are described in Sec. III. Figures 12(a) and 12(b) show *A*_{V}, *n*_{i}, and *θ*_{V}. Assuming $X=niAVexp{j(\omega t+\theta V)}$ and *S*_{X} = 0, *χ*_{ϕ} and *V*_{conv} are evaluated by the conventional transport analysis method and the developed transport analysis method shown in Subsection II B, as shown in Figs. 12(c) and 12(d). Here, *χ*_{ϕ} and *V*_{conv} are substituted for *D*_{X} and *V*_{X} in Eq. (3), respectively. The modulation frequency *ω*/2*π* is fixed to 2 Hz for both analyses. The time interval *δt* is set to 2.5 ms, which is the same as the time resolution of CXRS. The values of *χ*_{ϕ} and *V*_{conv} are almost identical between the conventional and the developed transport analysis methods. As shown in Figs. 12(e) and 12(f), the differences of *χ*_{ϕ} and *V*_{conv} between the conventional and the developed transport analysis methods increase with increasing *δt*. These differences stem from the discretization error. Because the difference increases approximately quadratically with the increase of *δt*, *δt* should be set less than 10 ms for the error suppression.