# A Comprehensive Analysis of the Gravitational Wave Events with the Hilbert-
Huang Transform: From Compact Binary Coalescence to Supernova

Chin-Ping Hu Department of Physics, National Changhua University of Education,
Changhua, 50007, Taiwan  Lupin Chun-Che Lin Department of Physics, National
Cheng-Kung University, Taiwan  Kuo-Chuan Pan Institute of Astronomy, National
Tsing Hua University, Hsinchu 30013, Taiwan  Kwan-Lok Li Department of
Physics, National Cheng Kung University, 70101 Tainan, Taiwan  Chien-Chang Yen
Department of Mathematics, Fu Jen Catholic University, 24205 New Taipei City,
Taiwan  Albert K. H. Kong Institute of Astronomy, National Tsing Hua
University, Hsinchu 30013, Taiwan  C. Y. Hui Department of Astronomy and Space
Science, Chungnam National University, Daejeon, Korea (ROK) C.-P. Hu
<EMAIL_ADDRESS>
(Accepted Jul. 14, 2022)

###### Abstract

We analyze the gravitational wave signals with a model-independent time-
frequency analysis, which is improved from the Hilbert-Huang transform (HHT)
and optimized for characterizing the frequency variability on the time-
frequency map. Except for the regular HHT algorithm, i.e., obtaining intrinsic
mode functions with ensemble empirical mode decomposition and yielding the
instantaneous frequencies, we propose an alternative algorithm that operates
the ensemble mean on the time-frequency map. We systematically analyze the
known gravitational wave events of the compact binary coalescence observed in
LIGO O1 and O2, and in the simulated gravitational wave signals from core-
collapse supernovae (CCSNe) with our method. The time-frequency maps of the
binary black hole coalescence cases show much better details compared to those
wavelet spectra. Moreover, the oscillation in the instantaneous frequency
caused by mode-mixing could be reduced with our algorithm. For the CCSNe data,
the oscillation from the proto-neutron star and the radiation from the
standing accretion shock instability can be precisely determined with the HHT
in great detail. More importantly, the initial stage of different modes of
oscillations can be clearly separated. These results provide new hints for
further establishment of the detecting algorithm, and new probes to
investigate the underlying physical mechanisms.

Gravitational wave astronomy (675); Core-collapse supernovae (304); Black
holes (162); Neutron stars (1108)

††software: Matlab (MATLAB, 2018), Python3 (Van Rossum & Drake, 2009), GWpy
(Macleod et al., 2020), PyCBC (Usman et al., 2016)

##  1 Introduction

The successful detection of the gravitational wave signal from the Laser
Interferometer Gravitational-Wave Observatory (LIGO) opens a new window in
both astronomy and physics (Abbott et al., 2016a). The advanced Virgo (aVirgo)
interferometer also joined the engineer run with two aLIGO detectors since May
of 2017. At the end of 2018 for the first and second observation runs (O1 and
O2), aLIGO and the aVirgo has confirmed 11 gravitational wave (GW) transients,
including 10 merging events from two massive stellar-mass black holes (BHs)
and one from the coalescence of two neutron stars (NSs) (Abbott et al., 2019).
Until the end of 3rd observation run in March of 2020, the aVirgo and aLIGO
network have confirmed 90 GW events (The LIGO Scientific Collaboration et al.,
2021), and 50 of which have been reported during the first half of the third
observing run (O3) (Abbott et al., 2021). These GW events can be detected by
multi-resolution techniques including wavelet Q-transform (Chatterji et al.,
2004).

It has been suggested that the novel time-series analysis method, the Hilbert-
Huang transform (HHT), could precisely describe the time-frequency properties
of the GW signal owing to the high resolution of the HHT (Camp et al., 2007).
The GW signal can be extracted from the simulated data with a BH merging
signal embedded in a noisy background (Stroeer et al., 2009). Similar work has
been extended to investigate the simulated GW signal from the coalescence of
binary NSs (Kaneyama et al., 2016).

The first and strongest event GW 150914 has been analyzed with the HHT (Sakai
et al., 2017). The signal is not able to be decomposed into a single component
through empirical mode decomposition (EMD). The ensemble EMD (EEMD) has been
proposed to overcome the so-called ‘mode-mixing” or “mode-splitting” problem
(Wu & Huang, 2004, 2009) but it is only valid for a signal with moderate
frequency variability because EEMD is equivalent to a dyadic filter. This
prevents getting a precise instantaneous frequency although the signal from
different components with highest Hilbert amplitude could marginally connected
with each other. The entire time-frequency map could be consistent with the
theoretical prediction although some frequency modulations can be clearly
seen. A systematic investigation of known GW events using the HHT is carried
out by Akhshi et al. (2021). A physical time delay could be derived from the
reconstructed waveform.

In this paper, we comprehensively apply the Morlet wavelet analysis and the
HHT on the LIGO data of the known GW events to test if the HHT is eligible to
detect the trigger. Moreover, we propose an alternative method to obtain the
Hilbert spectrum by stacking the time-frequency map instead of taking the
ensemble mean of components in the data sets with artificial noises. Besides
the coalescence events in the LIGO data sets, we also tested the algorithm on
the simulated GW signal from core-collapse supernovae (CCSNe). Section 2
describes the reduction and processing of the LIGO data, and the mathematical
algorithm used in this research. We present the analysis result of two cases,
GW 150914 and GW170817, in Section 3. The time-frequency maps of other events
are shown in Appendix A. We describe the models and parameters used in
generating the CCSN signal, and show the corresponding analytical result of a
typical CCSN case in Section 4. Cases with other rotational parameters and
viewing geometries are presented in Appendix B.

##  2 Data Reduction and Time-Frequency Analysis Algorithm

###  2.1 LIGO Data Reduction

In order to examine the capability of our numerical method, we check the
performance of our algorithm on the known GW events in GWTC-1 (A
gravitational-wave transient catalog of compact binary mergers observed by
LIGO and Virgo during the first and second observing runs) within 2015–2017.
We use the 32 s data of 4KHz released from the LIGO Open Science
Center111https://www.gw-openscience.org/catalog/GWTC-1-confident/html/
(Vallisneri et al., 2015) obtained from Hanford and Livingston sites to
perform our analysis. We notice that a strong glitch and a GW candidate event
(i.e., GW 170817) occurred almost simultaneously at the Livingston site, and
therefore we downloaded the ‘clean’ data after the noise subtracting following
the glitch model described in Abbott et al. (2017a).

Table 1 lists data sets of known GW events considered in this article, and all
the signal processing was proceeded with  the python based GW analysis library
named as GWPY v2.1.0222https://gwpy.github.io/docs/stable/ (Macleod et al.,
2020). To deal with the GW data, we followed the standard whitening process to
suppress the extra colored noise at low frequencies and spectral lines led by
instrumental or background effects (see, e.g., Usman et al., 2016; Abbott et
al., 2019). This operation might affect the nonlinearily of the GW signal.  We
note that whitening process is parameter dependent and the method is not
unique (see, e.g., Tsukada et al., 2018; Huang et al., 2021). Further
developments of the de-noising techniques would be important though this is
beyond the scope of this research. We therefore whitened the time series to
enhance the higher-frequency content with a FFT integration length of 4 s and
2 s-overlap between FFTs for the given window of Hann smoothing. We also apply
a bandpass filter to reserve the data within 30–450 Hz and then use notch
filters to attenuate line noises because we want to suppress the effect from
thermal noises of mirrors, quantum noise of laser or any line noise arisen
from detector resonances to dominate the spectral behavior.

Table 1: The data files of LIGO/Virgo events investigated in this article. GW150914 | Hanford | H-H1_GWOSC_4KHZ_R1-1126259447-32.hdf5
---|---|---
Livingston | L-L1_GWOSC_4KHZ_R1-1126259447-32.hdf5
GW151012 | Hanford | H-H1_GWOSC_4KHZ_R1-1128678885-32.hdf5
Livingston | L-L1_GWOSC_4KHZ_R1-1128678885-32.hdf5
GW151226 | Hanford | H-H1_GWOSC_4KHZ_R1-1135136335-32.hdf5
Livingston | L-L1_GWOSC_4KHZ_R1-1135136335-32.hdf5
GW170104 | Hanford | H-H1_GWOSC_4KHZ_R1-1167559921-32.hdf5
Livingston | L-L1_GWOSC_4KHZ_R1-1167559921-32.hdf5
GW170608 | Hanford | H-H1_GWOSC_4KHZ_R1-1180922479-32.hdf5
Livingston | L-L1_GWOSC_4KHZ_R1-1180922479-32.hdf5
GW170729 | Hanford | H-H1_GWOSC_4KHZ_R1-1185389792-32.hdf5
Livingston | L-L1_GWOSC_4KHZ_R1-1185389792-32.hdf5
GW170809 | Hanford | H-H1_GWOSC_4KHZ_R1-1186302504-32.hdf5
Livingston | L-L1_GWOSC_4KHZ_R1-1186302504-32.hdf5
GW170814 | Hanford | H-H1_GWOSC_4KHZ_R1-1186741846-32.hdf5
Livingston | L-L1_GWOSC_4KHZ_R1-1186741846-32.hdf5
GW170817 | Hanford | H-H1_GWOSC_4KHZ_R1-1187008867-32.hdf5
Livingston | L-L1_LOSC_CLN_4_V1-1187007040-2048.hdf5
GW170818 | Hanford | H-H1_GWOSC_4KHZ_R1-1187058312-32.hdf5
Livingston | L-L1_GWOSC_4KHZ_R1-1187058312-32.hdf5
GW170823 | Hanford | H-H1_GWOSC_4KHZ_R1-1187529241-32.hdf5
Livingston | L-L1_GWOSC_4KHZ_R1-1187529241-32.hdf5

To confirm the practicability of our methods, we also check the instantaneous
frequency obtained from the simulated gravitational waveform. We generate the
waveform using the effective-one-body (EOB; Taracchini et al., 2014; Pürrer,
2016; Bohé et al., 2017) numerical-relativity (NR) model. The simulated GW
time series can be obtained by employing apps (LAL 6.19.1; LIGO Scientific
Collaboration, 2018) provided by the LIGO Algorithm
Library333https://lscsoft.docs.ligo.org/lalsuite/. The basic physical
parameters of GW150914 applied in the simulated waveform refer to those
determined in GWTC-1 (Abbott et al., 2019).

###  2.2 Hilbert-Huang Transform

The Hilbert-Huang transform (HHT) is a novel time-frequency analysis algorithm
that can trace frequency variability in a great detail (Huang et al., 1998).
This adaptive method needs no _priori_ basis, and has been widely used in many
fields of science and engineering (see, e.g., Huang et al., 1999; Huang &
Attoh-Okine, 2005; Huang & Shen, 2014). The core idea of the HHT is the
Hilbert transform of a time series $x(t)$:

$y(t)=\frac{1}{\pi}\mathcal{P}\int_{-\infty}^{\infty}\frac{x(t^{\prime})}{t-t^{\prime}}dt^{\prime},$
(1)

where $\mathcal{P}$ indicates the Cauchy principal value and $y(t)$ is the
Hilbert transform of $x(t)$. With this definition, we can have an analytical
signal

$z(t)=x(t)+iy(t)=a(t)e^{i\theta(t)}$ (2)

where $a(t)=\sqrt{x^{2}(t)+y^{2}(t)}$ is the Hilbert amplitude and
$\theta(t)=\tan^{-1}\left[y(t)/x(t)\right]$ is the instantaneous phase
function. The time derivative of $\theta(t)$

$\omega(t)=\frac{d\theta(t)}{dt}$ (3)

can be defined as the instantaneous frequency. As the development of the HHT,
a few more methods were also proposed to obtain the frequency, such as
generalized zero-crossing, and direct quadrature (Huang et al., 2009). In our
study, we mainly utilize the normalized Hilbert transform, which applying the
Hilbert transform after normalizing the intrinsic mode functions (IMFs) to the
range between $-1$ and $1$. This is designed to overcome the limitation
imposed by the Bedrosian theorem (Bedrosian, 1962; Xu & Yan, 2006). In some
cases the intra-wave modulation, i.e., frequency modulation within one cycle
that caused by possible non-linear effect of an oscillation system, is not
physically important, we use the generalized zero-crossing, which is a robust
method to calculate the inter-wave modulation (i.e., the variation of a cycle
length) frequency (Sekhar & Sreenivas, 2004). Both methods yield a fully
consistent result.

Figure 1: (a) The procedure of EEMD in this paper. We applied the post-
processing EMD from the first IMF. (b) Stacked Hilbert analysis algorithm that
is optimized for visualization of the time-frequency map.

However, the instantaneous frequency is meaningful only if the input time
series is an IMF, which must be symmetry with respect to zero, and the number
of extrema and zero-crossing must be the same or at least differs by one. This
criterion is difficult to be achieved in real data. A pre-processing is
needed. Therefore, the HHT consists of two steps: (1) decompose the time
series data into finite IMFs and (2) obtain instantaneous frequencies for each
IMFs (Wu & Huang, 2009).

The first method proposed for the decomposition is the empirical mode
decomposition (EMD). EMD is an iterative process called “sifting” that
extracts IMFs by removing the local mean values obtained from taking the
average of the upper and lower envelope of a time series (Huang et al., 1998).
It can separate time series into IMFs of different time scales, and make IMFs
symmetric with respect to zero.

In principle, there are infinite way to decompose the signal. For example,
different sifting parameters, e.g., stoppage criteria, spline method, etc. A
confidence limit for Hilbert spectrum can be obtained from an ensemble of
valid IMFs resulting from different sifting parameters. The ideal case is that
the raw time series $x(t)$ can be decomposed as

$x(t)=\sum_{j=1}^{n}c_{j}(t)+r(t)=\sum_{j=1}^{n}a_{j}(t)e^{i\int\omega_{j}(t)dt}+r(t)$
(4)

where $c_{j}(t)$ is the $j$-th IMF, and $r(t)$ is the non-oscillating
residual.

A signal with a narrow modulation timescale is expected to be decomposed in a
single IMF. However, it could be decomposed into different IMFs using EMD if
there are noises or intermittent signals. This “mode-mixing” problem can be
solved by introducing white noise with finite amplitude into the data and
taking the ensemble mean of corresponding obtained from all the simulated
sets. This is the Ensemble EMD (EEMD) (Wu & Huang, 2009). Based on the study
of white noise, the EEMD is a dyadic filter that collates signals of similar
scales together. The remaining problem is that the summation of IMFs is no
longer an IMF. Therefore, a post-processing EMD is needed (Wu & Huang, 2009).
One can apply the post-processing EMD from arbitrary components of interest.
Here we use it since the first IMF, the highest frequency mode of the signal.
The entire procedure is shown as Figure 1(a).

When applying EEMD, two more parameters are needed to be considered: the
amplitude of the input noise $\sigma_{n}$ and the number of ensemble trials
$N$.  Unlike EMD, the completeness of the signal is not guaranteed in EEMD due
to the input noise. Therefore, a large N is needed to minimize the
incompleteness of the output signal, which costs high computing resources. To
reduce the computing time, the EEMD has been improved by adding a pair of
positive and negative noises.  This operation is also named as complementary
EEMD (CEEMD Yeh et al., 2010). With this operation, a good completeness can be
achieved compared to the original version of EEMD (see, e.g., Figure 6 in Yeh
et al., 2010).  With our parameter setting, the deviation between the
summation of IMFs and the original time series is $\sim 10^{-15}$ times of the
deviation of the original time series. This algorithm has been included in the
Matlab fast EEMD package as the default EEMD algorithm, which is developed by
the Research Center for Adaptive Data Analysis at National Central University
(Wang et al., 2014).  We have successfully explored the time-frequency
properties of X-ray binaries and active galactic nuclei using this package
(see, e.g., Hu et al., 2014; Su et al., 2015; Hu et al., 2019). A pywrapper of
this package444https://github.com/HHTpy/HHTpywrapper is also available (Su,
2017). We also tried to perform the following analysis using the complete
ensemble empirical mode decomposition with adaptive
noise555https://github.com/macolominas/CEEMDAN and found that the result is
fully consistent with that obtained from the fast EEMD algorithm (Torres et
al., 2011).

To decide the best value of $\sigma_{n}$, we tried different level of and
choose the best $\sigma_{n}$ to minimize the orthogonality index OI, which is
defined as

$\textrm{OI}\equiv\frac{\sum_{t}\sum_{i\neq
j}c_{i}(t)c_{j}(t)}{\sum_{t}x^{2}(t)}$ (5)

The OI can be defined specifically for any IMF of interest or between all
IMFs. Because the GW signal usually spread in a wide frequency range, we
minimize the overall OI presented in equation 5.

However, the GW signal of the coalescence of two compact objects usually spans
a wide frequency range, which is difficult to be decomposed into a single IMF
with EEMD. This makes it difficult to trace the evolution of the signal in
detail, especially near the boundary when the signal migrates from one IMF to
another. For visualization purpose, we establish a stacked Hilbert spectrum
algorithm to trace the time-frequency property of GW signals.

The idea of our algorithm is demonstrated in Figure 1(b). Similar to EEMD, we
first add finite noises into the signal and create $N$ simulated time series.
Then, we apply EMD or EEMD on the noise-inserted time series to obtain IMFs.
After this, we did not take the ensemble mean of IMFs. Instead, we calculate
the instantaneous frequency and amplitudes of individual IMFs in each
simulated data set. Finally, we stack all the Hilbert spectra on the time-
frequency map. The basic idea is that the variability of the instantaneous
frequency introduced by the mode-splitting effect would be different in each
set of added white noise. As a result, the true frequency evolution would be
enhanced on the stacked map, and the spurious variability would be smeared
out. The detailed mathematical proof and statistical study of this method will
be presented in another paper (Yen et al. in prep).

##  3 Analysis of the LIGO O1/O2 Data

###  3.1 GW150914

We obtain the LIGO data and whiten them using the data reduction pipeline
described in Section 2.1.  We first input white noises with $\sigma=1.0$ times
the deviation of the data and perform $10^{4}$ times of such simulations (step
1 in Figure 1). This is helpful to obtain confidence intervals because the
whitened data can be treated as the Gaussian white noise except for a short
time interval that contains GW signal. Then, we try both EMD and EEMD on the
simulated time series (step 2 in Figure 1). Further input of noise in EEMD is
needed, and we observe the variability of the OI with different noise levels
between 0 and 5 times the deviation of the data (see Figure 2). We found that
the OI could be minimized with $\sigma\approx 0.3$, and therefore we use it
for the following analysis. We then stack the Hilbert spectra on the
2-dimensional map to smear the boundary effect when the GW signal was split
into two nearby IMFs.

Figure 2:  The orthogonality index versus EEMD’s noise level in the Hanford
data of GW150914 data. Blue filled area denotes the 1-sigma interval obtained
from the simulated data sets.

Figure 3: The analysis results of GW150914 observed with LIGO Hanford (left)
and Livingstone (right). (a) The whitened strain data (blue) overlaied with
the EEMD band-pass filtered signal (red). (b)–(d) The EEMD decomposed
IMF3–IMF5 with different input noise levels. The green shaded area denotes the
1-$\sigma$ interval among the 10000 simulations (e) The summation of the
Hilbert amplitude of IMF3–IMF5. The vertical orange lines denote the time of
significant change in the Hilbert amplitude (see main text). The horizontal
dashed line denotes the zero level of the Hilbert amplitude. (f) WWA spectrum
(g) WWZ spectrum (h) Stacked Hilbert spectrum with $10^{4}$ EEMD decomposed
IMFs with randomly distributed noise level (i) Stacked Hilbert spectrum with
$10^{4}$ EMD decomposed IMFs.

Figure 3 shows the decomposition result and the Hilbert spectra of GW 150914
observed with LIGO Hanford (left) and Livingstone (right). Panel (a) shows the
whitened data and the summation of IMF3–5. Because the GW signal spans from
$\sim 50$ Hz to $\sim 300$ Hz, it matches the characteristic time scales of
IMFs 3 ($\sim 340$ Hz)–IMF 5 ($\sim 80$ Hz) based on the fact that the EEMD is
a dyadic filter and the sampling rate is 4096 Hz. In this case, the summation
of IMFs 3–5 can greatly reproduce the waveform. This operation is equivalent
to applying EEMD band-pass filter to the data. We plotted the decomposed IMFs
3–5 in Figure 3 b–d, respectively. It clearly shows that the  GW signal
firstly appears in IMF5, then migrates toward the high-frequency components,
IMF4 and IMF3.

We performed the normalized Hilbert transform to calculate the Hilbert energy,
which is defined as the square of the amplitude, and instantaneous frequency
of each IMFs, and stacked those obtained from IMFs 3–5 to minimize the
contribution from other frequency ranges. This operation is done for all
$10^{4}$ simulated data sets (see Figure 1) and thus we obtained the mean
value and deviation of the band-pass filtered Hilbert energy (Figure 3e). This
instantaneous Hilbert energy can be used to detect the GW signal. We use the
Bayesian block technique to detect occurrence times of significant change in
the Hilbert energy (Scargle et al., 2013). GW150914 is the strongest event in
the LIGO O1/O2 catalog where the boundaries can be easily detected if we set
the null hypothesis probability $p_{0}$ to be $10^{-80}$.

The stacked Hilbert spectra with EEMD and EMD algorithms with $10^{4}$ times
of simulation are presented in Figure 3h and 3i, respectively. We also
performed the weighted wavelet analysis of the amplitude spectrum (WWA, Figure
3f) and the Z-transform spectrum (WWZ, Figure 3g) for comparison.  The Morlet
wavelet,

$f(z)=e^{-c\omega^{2}(t-\tau)^{2}}\left[e^{i\omega(t-\tau)-e^{-1/4c}}\right]$
(6)

is used in this analysis, where $\omega$ is the scale factor and $\tau$ is the
time shift. The parameter $c$ determines how rapidly th wavelet decays, or the
size of the Gaussian envelope. We set it as $c=0.0125$ (Foster, 1996). These
four algorithms yielded consistent results in the frequency evolution.
However, the wavelet analysis has a trade-off between the frequency and the
temporal resolutions by setting the width of the Gaussian window. The time-
frequency property would be smeared out. In comparison, the Hilbert spectrum
shows a strikingly good resolution on the time-frequency map and indicates the
precise time of the incoming  GW signal. The mode-mixing may introduce a few
nonphysical modulations when the signal crosses two IMFs. Except for this, HHT
would be an ideal tool to study the detailed time-frequency structure of the
GW signals induced by two BHs coalescence.

Figure 4: Comparison between the stacked HHT spectrum (color map), WWZ
spectrum (contour) and the instantaneous frequency of the template (blue line)
of GW 150914 observed with LIGO Hanford detector. Three slices are labeled by
the dashed line for low-frequency (a), intermediate frequency (b), and high-
frequency (c) regimes. The amplitude versus frequency plot (or spectral
density) are plotted in the lower panels. Density maps of the instantaneous
amplitude and frequency of the specific time of all simulation are shown as
black dots, where the red profile is the averaged profile for the
instantaneous frequency. The green curve denotes the result from the WWZ,
where the blue dashed line indicates the instantaneous frequency obtained from
the template.  The HHT and WWZ profiles in panels (a)–(c) are scaled for
display purpose.

To compare the analysis result between different algorithms and the model
template, we  simulated the GW waveform using the python-based open software
to explore astrophysical sources of GWs,
PyCBCv2.0.1666https://github.com/gwastro/pycbc (Usman et al., 2016), with the
approximant given by the effective-one-body (EOB; Taracchini et al., 2014;
Pürrer, 2016; Bohé et al., 2017) NR model. The instantaneous frequency of the
template can be directly calculated via Hilbert transform without decomposing
it into IMFs because the template fits the IMF criterion during the merging
phase. The resulting instantaneous frequency of the template, the WWZ
spectrum, and the stacked Hilbert spectrum are plotted in Figure 4. The time-
frequency map of the HHT can clearly describe the instantaneous frequency in
better detail compared to that of the wavelet spectrum. To demonstrate this,
we choose three slices in (a) low-frequency, (b) intermediate-frequency, and
(c) high-frequency regimes. The  slices of the HHT and WWZ spectra, as well as
the instantaneous frequency obtained from the template, are shown in panel
(a)–(c) of Figure 4. The gray color map denotes the density of the Hilbert
energy-frequency pairs of each trial. The HHT result at epoch (b) is most
ambiguous because the  GW signal switches from one IMF to another at this
time. However, the performance of the HHT remains comparable to that of the
WWZ.

GW150914 is one of the strongest signals recorded in GWTC-1. We analyzed all
the BH-BH coalescence events in GWTC-1 and present the results in the appendix
(See Figure A1 – A9). These signals have been analyzed in a combination with
the HHT and de-noise algorithm using a network array (Akhshi et al., 2021).
The instantaneous frequency can be detected but large variability in the
instantaneous frequency is occasionally seen (see, e.g., Figure 4 in Akhshi et
al., 2021). These signals could be minimized by our stacking method.

Figure 5: The analysis results of GW170817 observed with LIGO Hanford (left)
and Livingstone (right). Panel definitions are the same as those in Figure 3.

###  3.2 GW170817

GW170817 is the first confirmed NS-NS coalescence event (Abbott et al.,
2017a). A $\gamma$-ray burst and multi-wavelength afterglows is a benchmark of
the multi-messenger astronomy era (Abbott et al., 2017b). Compared to other
BH-BH coalescence obtained in GWTC-1 (Abbott et al., 2019), the signal of
GW170817 has a much longer coherence time but the amplitude is much weaker. We
tested both the WWZ and the HHT algorithms on this event. The signal can be
seen on the wavelet map with $c=0.00125$, ten times smaller than the $c$ value
for BH-BH coalescence events. Compared to the signals of BH-BH events, the
amplitude of the NS-NS signal is much lower, and the signal keeps its
coherence for a long while. The nearly coherent signal before the merging
makes it possible to be detected with this setting, but the time-frequency
property near the merging phase is smeared in time due to the large Gaussian
window. In contrast, the signal is difficult to be resolved in the HHT time-
frequency map. Because the HHT is sensitive to both the instantaneous
variability of the frequency and the noise, the signal-to-noise ratio of this
event is insufficient for the HHT to identify its intrinsic time-frequency
property. Therefore, the HHT could be powerful for future NS-NS coalescence
events with improved  GW observatories in the next generation that could have
a high signal-to-noise ratio.

##  4 Core-Collapse Supernovae

Detection of GW signals from a nearby CCSN will be the next milestone of
gravitational wave astrophysics (Abbott et al., 2016b, 2020). It would be
crucial for multi-messenger astronomy because the possible co-detection of
gravitational waves and neutrinos would allow astronomers to probe the
activity of the center of a massive star during the final evolutionary stage.
Thanks to the growth of computing power, our knowledge of the explosion engine
of a CCSN improved dramatically. Owing to the asymmetric and complex fluid
motion and the oscillation/evolution of the proto-neutron star (PNS), complex
gravitational signals with multiple components are expected to be investigated
from three-dimensional high-fidelity self-consistent CCSN simulations (Janka
et al., 2016; Burrows & Vartanyan, 2021). CCSN waveforms depend on several
physical quantities, such as the core angular momentum and the composition of
the collapsar, the nuclear equation of state, and the dynamics of fluid
instabilities. The future multi-messenger observations will place meaningful
constraints on the supernova engines and the microphysics.

Nevertheless, recent theoretical works have reached consensuses on a few main
structures of the GW signal on the time-frequency domain. These provide a
strong basis to improve searching algorithms. On the other hand, the HHT is
likely the most precise tool to investigate the detailed time-frequency
properties of the CCSNe. Here we adopted the gravitational waveforms from the
self-consistent three-dimensional CCSN simulations with neutrino transport in
Pan et al. (2021).

###  4.1 Waveforms and Simulations

The gravitational waveforms of CCSNe simulations used in this project are
adopted from the three-dimensional self-consistent CCSN simulations with the
Isotropic Diffusion Source Approximation (Liebendörfer et al., 2009) for the
neutrino transport, using the FLASH code (Fryxell et al., 2000) and are
presented in Pan et al. (2021). The data are derived from a $40M_{\odot}$
progenitor at zero-age main sequence with solar metallicity from Woosley &
Heger (2007) with three different initial rotational speeds
($\Omega_{0}=0,0.5$ and 1 rad s-1) at the onset of core-collapse. A post-
Newtonian treatment with the effective general relativity potential (Marek et
al., 2006) and the Lattimer$-$Swesty (LS220) equation of state are considered
(Lattimer & Swesty, 1991).

The GW strains are evaluated by the first time derivative of the mass
quadrupole moment with an assumption of a fixed distance $d=10$ kpc to an
observer. The second time derivative of the mass quadrupole moment is
calculated by the finite difference in post-processing, using the formula
described in Scheidegger et al. (2008) and Murphy et al. (2009). We simulated
two viewing angles, where the observers are in the infinity in the direction
of the equatorial plane or the rotational axis.

###  4.2 Analysis of Simulated CCSNe Waveforms

Here we take the gravitational waveform of the non-rotating model with an
observer viewed from the equatorial direction as an example for our HHT
analysis. The gravitational waveforms of other models with different
parameters are shown in the appendix (Figure A11 – A20). Figure 6 (a) shows
the simulated strains, which followed by the wavelet spectra (b) – (c) and the
stacked HHT spectra (d) – (e). The waveform starts from the core collapse and
reaches the core bounce at $t=0$ s. Since this model is non-rotating and the
proto-neutron star convection happens after the core bounce, the gravitational
strain shows a loud bounce signal at time zero, and then it is followed by a
low-frequency peak PNS oscillation, which could be caused by the g-mode
oscillation although the nature remains controversial (Müller et al., 2013;
Pan et al., 2021). Multiple low-frequency oscillation signals dominate the
first $\sim 0.15$ s and they can be well resolved with the HHT. In comparison,
the limitation of the wavelet spectrum prevents us to identify them. As the
frequency of the oscillation increases with time, another high-frequency mode
oscillation signal appears with similar strength as the peak PNS mode. The
low-frequency PNS oscillation vanishes at $t\sim 0.5$ s. Then, interestingly,
it followed by the quadruple radiation of the standing accretion shock
instability (SASI Blondin et al., 2003) signal,  which was suggested to be
seen in the GW data Kuroda et al. (2016); Pan et al. (2021).  The detailed
waveform of the SASI signal is model dependent, but its time-frequency
property has been shown in a few analysis either with the spectrogram or the
HHT (see, e.g., Andresen et al., 2017; Mezzacappa et al., 2020; Takeda et al.,
2021). In our analysis, the SASI signal can be seen on both the wavelet and
the HHT spectra. The wavelet and the HHT results for other simulated CCSN
events are shown in the appendix B. The Hilbert spectra of all the data sets
show much better resolution than the wavelet spectra.

Figure 6: (a) The cross-component of the gravitational waveform of a CCSN seen
from the equatorial plane with an initial stellar mass of $40$ $M_{\odot}$ and
without rotation. (b)–(c) The WWA and WWZ map of the gravitational signal.
(d)–(e) Stacked Hilbert spectra based on EEMD (d) and EMD (e).

##  5 Discussion

A  Morlet wavelet consists of a complex exponential carrier multiplied by a
Gaussian window. It provides a window size for both the low- and high-
frequency regime that contains the same number of waves. The time-frequency
map is distorted, and the signal is smeared if the frequency of a signal
changes drastically within a few cycles (see. e.g., Figure 3 (f) – (g) and
Figure 6 (b) – (c)). A narrower width of the Gaussian envelope can result in a
better time resolution, although the frequency resolution would be sacrificed.

On the other hand, the Hilbert analysis calculates the instantaneous frequency
of a signal. This helps trace the signal’s detailed evolution of the time-
frequency property. Moreover, the complex multi-component, low-frequency
signals of CCSNe can be well resolved with the HHT. For example, the SASI
signal in Figure 6 and a few multiple low-frequency signals in Figures A13 –
A20 can only be clearly resolved in the Hilbert spectra.

Nevertheless, the HHT can decompose a fast-changing signal into different
IMFs. Fake frequency modulation in the instantaneous frequency is seen when
the signal crosses two components. This can be reduced with the stacking
algorithm suggested in this study, although they cannot be entirely removed.
Moreover, the signal cannot be seen for low signal-to-noise ratio events. For
example, the NS-NS coalescence event GW 170817 can be clearly detected with
the wavelet analysis with a large Gaussian width. This is because the signal
keeps its coherence for a few more cycles compared to the BH-BH coalescence
events, but the signal-to-noise ratio within a cycle is much lower.

Before stacking the time-frequency map, the evolution of Hilbert energy has
gave hints of the GW signal. A detailed time delay among instruments can be
further characterized by cross-correlating the EMD filtered profiles (Akhshi
et al., 2021). In addition, the special pattern of the GW signal across IMFs
(see, e.g., Figure 3b – d) would form a remarkable feature in the two-
dimensional series where one dimension is time and another is IMF index. This
would be helpful for detecting the GW signal using convolution neural network.

The GW signal from a CCSN is believed to contain multiple components, where
they could across a large range in the both the frequency and the energy. The
high resolutions of the HHT in both the time and frequency domains are helpful
in distinguishing them from each other, especially the low-frequency transient
signals within $\sim 0.4$ s after the core bounce.

##  6 Summary

The HHT is a novel technique for time-frequency analysis although the mode-
splitting problem is severe when working with signals with huge frequency
variability like GW signals. We propose a method that could minimize the
unreal frequency modulation by stacking the time-frequency map instead of
taking the ensemble mean of the IMFs. This helps visualize the evolution of
the instantaneous frequency. We carried out an analysis for LIGO O1 and O2
events and found that most of their frequency evolution can be well tracked
with our algorithm. Furthermore, the EEMD filtered data is used to determine
the time delay between Livingstone and Hanford sites. Except for the known GW
sources, we also tested our algorithm on the simulated CCSN events. We found
that multiple GW components can be obtained with the HHT. Compared to the
wavelet analysis, the unprecedented resolutions in both the time and frequency
domains of the HHT are an ideal tool for further investigating the time-
frequency properties of a signal. This provides evidence that the HHT can be a
powerful tool in detecting CCSNe events for future multi-messenger campaigns.

We thank the anonymous reviewer for valuable comments that improved this
paper. This research has made use of data, software and/or web tools obtained
from the Gravitational Wave Open Science Center (https://www.gw-
openscience.org), a service of LIGO Laboratory, the LIGO Scientific
Collaboration and the Virgo Collaboration. LIGO is funded by the U.S. National
Science Foundation. Virgo is funded by the French Centre National de Recherche
Scientifique (CNRS), the Italian Istituto Nazionale della Fisica Nucleare
(INFN) and the Dutch Nikhef, with contributions by Polish and Hungarian
institutes. C.-P.H. acknowledges support from the Ministry of Science and
Technology in Taiwan through grant MOST 109-2112-M-018-009-MY3. L.C.C.L.
acknowledges support from the Ministry of Science and Technology in Taiwan
through grant Nos. MOST 110-2811-M-006-515,  MOST 110-2811-M-006-012 and MOST
110-2112-M-006-006-MY3. K.C.P. is supported by the Ministry of Science and
Technology of Taiwan through grant MOST 110-2112-M-007-019. K.L.L. is
supported by the Ministry of Science and Technology of the Republic of China
(Taiwan) through grant 110-2636-M-006-013, and he is a Yushan Young Fellow of
the Ministry of Education of the Republic of China (Taiwan). C.-C.Y. is
supported by the Ministry of Science and Technology of Taiwan through grant
MOST 109-2115-M-030 -004 -MY2. A.K.H.K. is supported by the Ministry of
Science and Technology of Taiwan under the grants 109-2628-M-007-005-RSP and
110-2628-M-007-005, and the Inter-University Research Program of the Institute
for Cosmic Ray Research (ICRR), the University of Tokyo. CYH is supported by
the National Research Foundation of Korea through grants 2016R1A5A1013277.

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##  Appendix A Resulting Spectra of LIGO Events

All the remaining LIGO O1/O2 events and corresponding time-frequency maps of
BH-BH mergers are shown in this section.

Figure A1: The analysis results of GW151012 observed with LIGO Hanford (left)
and Livingstone (right). Panels (a) – (i) have the same definition as those in
Figure 3.

Figure A2: The analysis results of GW151226 observed with LIGO Hanford (left)
and Livingstone (right). Panels (a) – (i) have the same definition as those in
Figure 3.

Figure A3: The analysis results of GW170104 observed with LIGO Hanford (left)
and Livingstone (right). Panels (a) – (i) have the same definition as those in
Figure 3.

Figure A4: The analysis results of GW170608 observed with LIGO Hanford (left)
and Livingstone (right). Panels (a) – (i) have the same definition as those in
Figure 3.

Figure A5: The analysis results of GW170729 observed with LIGO Hanford (left)
and Livingstone (right). Panels (a) – (i) have the same definition as those in
Figure 3.

Figure A6: The analysis results of GW170809 observed with LIGO Hanford (left)
and Livingstone (right). Panels (a) – (i) have the same definition as those in
Figure 3.

Figure A7: The analysis results of GW170814 observed with LIGO Hanford (left)
and Livingstone (right). Panels (a) – (i) have the same definition as those in
Figure 3.

Figure A8: The analysis results of GW170818 observed with LIGO Hanford (left)
and Livingstone (right). Panels (a) – (i) have the same definition as those in
Figure 3.

Figure A9: The analysis results of GW170823 observed with LIGO Hanford (left)
and Livingstone (right). Panels (a) – (i) have the same definition as those in
Figure 3.

##  Appendix B Resulting Spectra of CCSNe Events

All the remaining simulated CCSN GW data, corresponding wavelet and Hilbert
spectra are shown in this section.

Figure A10: (a) The cross mode of the Gravitational waveform of a CCSN with an
initial stellar mass of $40$ $M_{\odot}$ and without rotation viewed from the
polar direction. Panels (b) – (e) shows wavelet and Hilbert maps with the same
definition as those in Figure 6.

Figure A11: (a) The plus mode of the Gravitational waveform of a CCSN with an
initial stellar mass of $40$ $M_{\odot}$ and without rotation viewed from the
equatorial direction. Panels (b) – (e) shows wavelet and Hilbert maps with the
same definition as those in Figure 6.

Figure A12: (a) The plus mode of the Gravitational waveform of a CCSN with an
initial stellar mass of $40$ $M_{\odot}$ and without rotation viewed from the
polar direction. Panels (b) – (e) shows wavelet and Hilbert maps with the same
definition as those in Figure 6.

Figure A13: (a) The cross mode of the Gravitational waveform of a CCSN with an
initial stellar mass of $40$ $M_{\odot}$ and slow rotation (0.5 rad s-1)
viewed from the equatorial direction. Panels (b) – (e) shows wavelet and
Hilbert maps with the same definition as those in Figure 6.

Figure A14: (a) The cross mode of the Gravitational waveform of a CCSN with an
initial stellar mass of $40$ $M_{\odot}$ and and slow rotation (0.5 rad s-1)
viewed from the polar direction. Panels (b) – (e) shows wavelet and Hilbert
maps with the same definition as those in Figure 6.

Figure A15: (a) The plus mode of the Gravitational waveform of a CCSN with an
initial stellar mass of $40$ $M_{\odot}$ and slow rotation (0.5 rad s-1)
viewed from the equatorial direction. Panels (b) – (e) shows wavelet and
Hilbert maps with the same definition as those in Figure 6.

Figure A16: (a) The plus mode of the Gravitational waveform of a CCSN with an
initial stellar mass of $40$ $M_{\odot}$ and and slow rotation (0.5 rad s-1)
viewed from the polar direction. Panels (b) – (e) shows wavelet and Hilbert
maps with the same definition as those in Figure 6.

Figure A17: (a) The cross mode of the Gravitational waveform of a CCSN with an
initial stellar mass of $40$ $M_{\odot}$ and fast rotation (1 rad s-1) viewed
from the equatorial direction. Panels (b) – (e) shows wavelet and Hilbert maps
with the same definition as those in Figure 6.

Figure A18: (a) The cross mode of the Gravitational waveform of a CCSN with an
initial stellar mass of $40$ $M_{\odot}$ and and fast rotation (1 rad s-1)
viewed from the polar direction. Panels (b) – (e) shows wavelet and Hilbert
maps with the same definition as those in Figure 6.

Figure A19: (a) The plus mode of the Gravitational waveform of a CCSN with an
initial stellar mass of $40$ $M_{\odot}$ and fast rotation (1 rad s-1) viewed
from the equatorial direction. Panels (b) – (e) shows wavelet and Hilbert maps
with the same definition as those in Figure 6.

Figure A20: (a) The plus mode of the Gravitational waveform of a CCSN with an
initial stellar mass of $40$ $M_{\odot}$ and and fast rotation (1 rad s-1)
viewed from the polar direction. Panels (b) – (e) shows wavelet and Hilbert
maps with the same definition as those in Figure 6.