# Confining Burst Energy Function and Spectral Fringe Pattern of FRB 20121102A
with MultiFrequency Observations

Fen Lyu Astronomical Research Center, Shanghai Science & Technology Museum,
Shanghai, 201306, China  Guangxi Key Laboratory for Relativistic Astrophysics,
School of Physical Science and Technology, Guangxi University, Nanning 530004,
China  Shanghai Astronomical Observatory, CAS, Nandan Road 80, Shanghai
200030, China  Ji-Gui Cheng Guangxi Key Laboratory for Relativistic
Astrophysics, School of Physical Science and Technology, Guangxi University,
Nanning 530004, China  En-Wei Liang* Guangxi Key Laboratory for Relativistic
Astrophysics, School of Physical Science and Technology, Guangxi University,
Nanning 530004, China  Can-Min Deng Guangxi Key Laboratory for Relativistic
Astrophysics, School of Physical Science and Technology, Guangxi University,
Nanning 530004, China  Tao An Shanghai Astronomical Observatory, CAS, Nandan
Road 80, Shanghai 200030, China  Qing Lin  Astronomical Research Center,
Shanghai Science & Technology Museum, Shanghai, 201306, China

###### Abstract

The observed spectral shapes variation and tentative bimodal burst energy
distribution (E-distribution) of fast radio burst (FRB) 20121102A with the
FAST telescope are great puzzles. Adopting the published multifrequency data
observed with the FAST and Arecibo telescopes at $L$ band and the GBT
telescope at $C$ band, we investigate these puzzles through Monte Carlo
simulations. The intrinsic energy function (E-function) is modeled as
$dp/dE\propto E^{-\alpha_{\rm E}}$, and the spectral profile is described as a
Gaussian function. A fringe pattern of its spectral peak frequency ($\nu_{\rm
p}$) in 0.5-8 GHz is inferred from the $\nu_{\rm p}$ distribution of the GBT
sample. We estimate the likelihood of $\alpha_{\rm E}$ and the standard
deviation of the spectral profile ($\sigma_{\rm s}$) by utilizing the
Kolmogorov–Smirnov (K-S) test probability for the observed and simulated
specific E-distributions. Our simulations yields $\alpha_{\rm
E}=1.82^{+0.10}_{-0.30}$ and $\sigma_{\rm s}=0.18^{+0.28}_{-0.06}$ ($3\sigma$
confidence level) with the FAST sample. These results suggest that a single
power-law function is adequate to model the E-function of FRB 20121102A. The
variations of its observed spectral indices and E-distributions with
telescopes in different frequency ranges are due to both physical and
observational reasons, i.e. narrow spectral width for a single burst and
discrete $\nu_{p}$ fringe pattern in a broad frequency range among bursts, and
the selection effects of the telescope bandpass and sensitivity. The putative
$\nu_{p}$ fringe pattern cannot be explained with the current radiation
physics models of FRBs. Some caveats of possible artificial effects that may
introduce such a feature are discussed.

Radio transient sources (2008); Radio bursts (1339): Individual FRB 20121102A

##  1 Introduction

Fast radio bursts (FRBs) are bright (typical fluence of $\sim$Jy ms and
brightness temperature of $T_{\mathrm{B}}\geq 10^{35}{\rm K}$), enigmatic
millisecond-duration radio bursts (Lorimer et al. 2007; Cordes & Chatterjee
2019; Petroff et al. 2022). More than 800 FRBs have been detected so far
(Petroff et al., 2016; CHIME/FRB Collaboration et al.,
2021)111https://www.herta-experiment.org/frbstats/catalogue. A high dispersion
measure (88-3038 pc cm-3; Bhardwaj et al. 2021; CHIME/FRB Collaboration et al.
2021) of most FRBs 222https://www.chime-frb.ca/catalog indicates their
extragalactic origin, which is confirmed with the host galaxy identification
for some FRBs (Macquart et al., 2020). Their progenitors are thought to be
flaring magnetars (Lyubarsky, 2014; Beloborodov, 2017; Metzger et al., 2019;
Beloborodov, 2020; Lu et al., 2020), the giant pulses of young pulsars (Popov
& Postnov, 2010; Lyutikov et al., 2016; Connor et al., 2016; Lyu et al.,
2021), neutron stars (Wang et al., 2016; Gu et al., 2016; Dai et al., 2016;
Ioka & Zhang, 2020), the crust of strange stars (Geng et al., 2021), or
primordial black hole mergers (Deng et al., 2018; Deng, 2021). The possible
association between FRB 200428 and the Galactic magnetar SGR 1934 + 2154
favors the magnetar origin for at least some FRBs (CHIME/FRB Collaboration et
al., 2020; Bochenek et al., 2020).

The observed FRB spectrum strongly varies. Describing the spectrum with a
single power-law function, $S_{\nu}\propto\nu^{\beta}$, Macquart et al. (2019)
showed that $\beta$ varies from -15 to 10 among 23 FRBs observed with the
Australian SKA Pathfinder (ASKAP). CHIME observations show that an FRB with a
longer duration tends to have a narrower spectral shape, which often peaks in
the instrumental bandpass due to the instrument selection effect; and a
shorter burst tends to have a power-law-like spectrum covering most or all of
the instrumental bandpass (e.g., Scholz et al. 2016; CHIME/FRB Collaboration
et al. 2019; Pleunis et al. 2021; CHIME/FRB Collaboration et al. 2021). Bright
FRBs observed by ASKAP exhibit strong similar spectral modulation with unclear
origins (Shannon et al., 2018; Macquart et al., 2019). It is uncertain whether
the observed diverse spectra are due to their intrinsic properties or affected
by the propagation effects (such as diffractive scintillation; Murase et al.
2016; Macquart et al. 2019; Yang & Zhang 2020; Wada et al. 2021; Aggarwal
2021). The radiation physics of FRBs is extensively discussed (see Zhang 2020;
Xiao et al. 2021 for recent reviews). The proposed models (Platts et al.,
2019) can be grouped into two types, i.e. synchrotron maser emission models
(Lyubarsky, 2014; Beloborodov, 2017, 2020; Metzger et al., 2019; Margalit et
al., 2020; Lyubarsky, 2020) and the curvature radiation models (Yang & Zhang,
2018; Wang et al., 2018; Lu et al., 2020).

Among the detected FRBs, most of them are one-off events, and about two dozen
exhibit repeating behaviors. It remains an open question whether or not all
FRBs are repeaters (Palaniswamy et al., 2018; Caleb et al., 2019; Ai et al.,
2021; Zhong et al., 2022). FRB 20121102A is the first discovered repeating FRB
(Spitler et al., 2016). It has a burst rate as high as $\sim 122$ hr-1 (Li et
al., 2021a). It even exhibits a possible periodic activity of $\sim$ 160-day
(Rajwade et al., 2020; Cruces et al., 2021). It locates in the star-forming
region of a dwarf host galaxy at redshift $z\simeq 0.193$ (Tendulkar et al.,
2017). The observed high rotation measure ($\mathrm{RM}\sim
10^{5}\,\mathrm{rad}\,\mathrm{m}^{-2}$) indicates that its environment is
extremely magnetized (Michilli et al., 2018). Moreover, it is one of only two
identified FRBs associated with a compact, luminous persistent radio source
(PRS; Chatterjee et al. 2017).

FRB 20121102A is one of the most extensively monitored FRB sources with
different telescopes at frequencies from 0.5 to 8 GHz with a wealth of
observational features (e.g., Spitler et al. 2014; Scholz et al. 2016; Spitler
et al. 2016; Chatterjee et al. 2017; Tendulkar et al. 2017; Scholz et al.
2017; Zhang et al. 2018; Michilli et al. 2018; Josephy et al. 2019; Gourdji et
al. 2019; Houben et al. 2019; Rajwade et al. 2020; Majid et al. 2020; Fonseca
et al. 2020; Caleb et al. 2020; Oostrum et al. 2020; Li et al. 2021a; Hewitt
et al. 2022). Observations indicate that the burst dynamic spectra of FRB
20121102A are highly variable. They are usually described with a Gaussian
function. Note that a “spectral index” is usually for a power-law function
spectrum. We prefer adopting “spectral slope” instead of “spectral index” for
avoiding any confusion about the Gaussian spectral model hereafter ( please
refer to details under bullet point 4 in Section 3.2). Both the spectral width
and peak frequency vary among bursts detected in different bandpass and
different observation sessions (Spitler et al., 2016; Scholz et al., 2016; Law
et al., 2017; Gajjar et al., 2018; Zhang et al., 2018; Hewitt et al., 2022).
For example, the spectral indices of the bursts of FRB 20121102A observed with
the Arecibo telescope at L-band ($\sim$1.4 GHz) vary from $-10.4\pm 1.1$ to
$13.6\pm 0.4$ (Spitler et al., 2016). In addition, the bursts seem to be
active in some specific frequencies. Search for simultaneous bursts in
different frequencies have been conducted by some groups (e.g. Scholz et al.
2016; Law et al. 2017; Gourdji et al. 2019; Houben et al. 2019; Majid et al.
2020; Caleb et al. 2020). Law et al. (2017) presented results from a multi-
telescope campaign of FRB 20121102A using the VLA at 3 GHz and 6 GHz, the
Arecibo telescope at 1.4 GHz, the Effelsberg telescope at 4.85 GHz, the first
station of the Long Wavelength Array (LWA1) at 70 MHz, and the Arcminute
Microkelvin Imager Large Array (AMI-LA) at 15.5 GHz. Four of the nine bursts
detected with the VLA had simultaneous observing coverage at different
frequencies. Only one was detected simultaneously at two different observing
frequencies with Arecibo (1.15-1.73 GHz) and the VLA (2.5-3.5 GHz), and none
were detected during simultaneous LWA1, Effelsberg, or AMI-LA observations,
despite the instantaneous sensitivities of these telescopes are better than or
comparable to the VLA. Gourdji et al. (2019) showed that among 41 bursts
detected with Arecibo at 1.4 GHz, no bursts were seen with the VLA during
their simultaneous observations. Karastergiou et al. (2015) conducted a
1446-hour survey for FRBs at 145 MHz, covering a total of 4193 square degrees
in the sky with the LOFAR radio telescope. No FRBs above a signal-to-noise
threshold of 10 were detected. Houben et al. (2019) performed $\sim 20$ h of
simultaneous observations with the Effelsberg 100 m radio telescope and LOFAR
to search for burst activity of FRB 20121102A at 1.4 GHz and 150 MHz. They
found nine bursts at 1.4 GHz but no low-frequency burst was simultaneously
detected with LOFAR. Note that LOFAR detected bursts from repeating FRB
20180916B down to 0.11 GHz (Pastor-Marazuela et al., 2021; Pleunis et al.,
2021). The non-detection of FRB 20121102A with LOFAR implies that FRB20121102A
is not very active at this low frequency. Majid et al. (2020) reported six
bursts from FRB 20121102A at the 2.25 GHz frequency band, but none of these
bursts is detected in the 8.36 GHz band during a campaign of 5.7 hr continuous
simultaneous observation using the 70 m Deep Space Network radio telescope.

Extensive multi-frequency observations of FRB 20121102A make it the best
candidate for revealing the radiation physics of FRBs. More interestingly, a
tentative bimodal distribution of the specific isotropic equivalent energy
($E_{\mu_{\rm c}}$) at the central frequency ($\mu_{\rm c}$) is found from
1652 bursts observed with the FAST telescope (Li et al., 2021a). Note that the
estimate of the burst energy depends on the burst spectrum and the bandwidth
of the telescope. Aggarwal (2021) argued that the bimodal feature disappears
if the burst energy is calculated over the entire FAST bandpass rather than on
the center frequency of the instrument. That motivates us to explore the
intrinsic spectrum and energy function (E-function) underlying this
phenomenon. In this paper, we employ large burst samples of FRB 20121102A
observed with the FAST and Arecibo telescopes at L-band (1-2 GHz) as well as
the GBT Telescope at C-band (4-8 GHz) to investigate the intrinsic spectra of
the bursts from FRB 20121102A via Monte Carlo simulations over a broadband
frequency range. Our paper is organized as follows. Our selected samples are
presented in Section 2. Our simulations are reported in Section 3. Discussion
and conclusions are given in Sections 4 and 5, respectively. Throughout, we
adopt a flat $\Lambda$CDM universe with the cosmological parameters
$H_{0}$=67.7$\mathrm{~{}km}$ $\mathrm{~{}s}^{-1}$ $\mathrm{Mpc}^{-1}$,
$\Omega_{m}=0.31$ (Planck Collaboration et al., 2016).

##  2 Data

As mentioned above, FRB 20121102A has been extensively observed with multi-
frequency telescopes in different fluence thresholds, which is estimated as
(Caleb et al., 2016; Li et al., 2021b),

$F_{\rm th}=\frac{T_{\text{sys }}\kappa<S/N>}{G\sqrt{B\Delta\tau N_{p}}},$ (1)

where $T_{\text{sys }}$ is the system temperature in the unit of Kelvin (K),
$\kappa$ is a digitization factor, $<S/N>$ is the signal-to-noise ratio, $G$
is the system gain in the unit of $\mbox{K Jy}^{-1}$, $B$ is the bandwidth in
the unit of Hz, $\Delta\tau$ is the integration time in seconds, and $N_{p}$
is the number of polarizations summed. The observed specific energy
($E_{\nu}$) at the peak frequency $\nu$ which is given by Zhang (2018),

$E_{\nu}^{\rm obs}\simeq\left(10^{39}{\rm
erg}\right)\frac{4\pi}{1+z}\left(\frac{D_{\rm
L}}{10^{28}{\rm~{}cm}}\right)^{2}\left(\frac{F^{\rm obs}_{\nu}}{{\rm
Jy}\cdot{\rm ms}}\right)\left(\frac{\nu}{\rm GHz}\right),$ (2)

where $F^{\rm obs}_{\nu}$ is the specific fluence at $\nu$ and $D_{L}$ is the
luminosity distance ($D_{L}$ = 972 Mpc for FRB 20121102A). Hereafter, we
denote the simulated observable with a superscript “sim” and denote the
observational data with a superscript “obs”. We employ the data of FRB
20121102A observed with FAST, Arecibo, and GBT telescopes for our analysis
(Zhang et al., 2018; Li et al., 2021a; Hewitt et al., 2022). We describe the
selected samples below.

As the largest single-dish telescope in the world, FAST has a 19-beam array
receiver in L-band ranging from 1.05 GHz to 1.45 GHz with the central
frequency of $\mu_{\rm c}=1.25$ GHz. Its frequency resolution is 0.122 MHz. Li
et al. (2021a) reported the results of a monitoring campaign for FRB 20121102A
with the FAST telescope. During an observational period of 47 days, which took
place from August 29 to October 29, 2019, the total observational time is 59.5
hours. A sample of 1652 bursts (the FAST sample) with a fluence criterion of
$F^{\rm FAST}_{\rm th}=0.015$ Jy ms in the $7\sigma$ confidence level,
assuming 1 ms burst duration. The frequency coverage ($\Delta\nu^{\rm obs}$)
of these bursts ranges from 0.1 GHz to 0.5 GHz. The peak frequency ($\nu_{\rm
p}^{\rm obs}$) of the spectrum is not available in the FAST sample. Li et al.
(2021a) calculated the specific burst energy $E^{\rm obs}_{\mu_{\rm c}}$ at
$\mu_{\rm c}$. We collect the data of $E^{\rm obs}_{\mu_{\rm c}}$,
$\Delta\nu^{\rm obs}$, and the burst duration ($T^{\rm obs}$) from Li et al.
(2021a).

The Arecibo telescope is sensitive at L-band in the frequency range of
1.15-1.73 GHz in 64 channels. Its central frequency is $\mu_{c}=1.44$ GHz. Its
time resolution of the sampled signal is $10.24~{}\mu$s. An observational
campaign for FRB 20121102A with the Arecibo telescope was conducted from
November 2015 to October 2016. The total observational time is $\sim 59$
hours. A sample of 478 bursts (the Arecibo sample) was detected (Hewitt et
al., 2022). The bursts were selected with $F^{\rm Arecibo}_{\rm th}=0.057$ Jy
ms in the $6\sigma$ confidence level assuming 1 ms burst duration. The
isotropic equivalent burst energy ($E_{\Delta\nu}^{\rm obs}$) over the upper
and lower frequency edges ([$\nu_{\text{high }}$, $\nu_{\text{low}}$]) of
these bursts are available in Hewitt et al. (2022), where $\Delta\nu^{\rm
obs}=\nu_{\text{high }}-\nu_{\text{low}}$. We collect the data of
$E_{\Delta\nu}^{\rm obs}$, $T^{\rm obs}$, $\Delta\nu^{\rm obs}$, and burst
edges ([$\nu_{\text{high }}$, $\nu_{\text{low}}$]) of these bursts from Hewitt
et al. (2022).

Green Bank Telescope (GBT) covers the C-band frequency range from 4 to 8 GHz
(Gajjar et al., 2018; Zhang et al., 2018). The temporal and the frequency
resolutions are 350 $\mu$s and 366 kHz, respectively. Estimating the GBT
observational fluence threshold with Eq. (1) by adopting the performance of
the GBT at C-band as $T_{\rm sys}=25$ K, $\beta=2$, $N_{p}=2$, $G=2$ K Jy-1,
$B=4$ GHz, and $<S/N>=6$, we obtain $F^{\rm GBT}_{\nu,\rm th}=0.0265$ Jy ms,
which is roughly consistent with the full-band fluence limit (0.30 Jy ms) in
Zhang et al. (2018). A 5-hour observation campaign for FRB 20121102A with the
GBT telescope on 2017 August 26 detected 93 bursts (the GBT sample).

Note that the bursts observed with GBT in the low-frequency end (4-4.5 GHz)
were not analyzed by Gajjar et al. (2018) and Zhang et al. (2018) owing to the
radio frequency interference (RFI). With a broad bandpass, the peak frequency
of the spectrum ($\nu_{\rm p}^{\rm obs}$) and the radiating frequency range
edges ($\nu_{\text{low}}$ and $\nu_{\text{high }}$), the specific observed
fluence ( $F^{\rm obs}$) and the burst duration $T^{\rm obs}$ values of these
bursts are available in Gajjar et al. (2018); Zhang et al. (2018). We take
these data from Gajjar et al. (2018); Zhang et al. (2018) and regard observed
burst energy as the specific energy $E^{\rm obs}_{\nu_{\rm p}}$ at $\nu^{\rm
obs}_{\rm p}$.

Figure 1 shows the distributions of the data for the FAST, Arecibo, and GBT
samples. One can observe that the $\nu_{\text{p}}^{\rm obs}$ distribution of
the GBT sample shows peaks at 4.75, 5.64, 6.28, 7.05, and 7.50 GHz. We
empirically fit the distribution with multiple normal distribution functions,
i.e. $dp/d_{\nu_{\rm p}}\propto\sum_{i}N^{\rm i}(\nu_{\rm p};\nu_{\rm
p,c,i},p_{\rm i},\sigma_{\rm i})$, where $\nu_{\rm p,c,i}$, $p_{\rm i}$,
$\sigma_{\rm i}$ are the center value, the relative probability, and the
$1\sigma$ of the component $i$th, respectively. The $\nu_{\rm p}^{\rm obs}$
distribution of the Arecibo sample can be roughly fitted with a single normal
function. Our fits are also illustrated in Figure 1.

The $E^{\rm obs}_{\nu_{\rm p}}$ distribution of the GBT sample is well
consistent with $E^{\rm obs}_{\nu_{\rm c}}$ distribution of the Arecibo
sample. The $p_{\rm KS}$ value derived from the non-parametric Kolmogorov-
Smirnov (K-S) test is 0.35. They narrowly range in $10^{38-39}$ erg. It is
possible that the $\nu_{\rm p}$ of burst in the Arecibo sample would be in the
Arecibo band (see further discussion in §3.1). Therefore, we regard
$E_{\Delta\nu}^{\rm obs}$ as $E^{\rm obs}_{\nu_{\rm p}}$ and estimate the
$\nu_{\rm p}^{\rm obs}$ of these burst as $\nu^{\rm obs}_{\rm p}\simeq\nu^{\rm
obs}_{\rm c}=(\nu_{\text{high }}+\nu_{\text{low}}$)/2 for the bursts in the
Arecibo sample. The $E^{\rm obs}_{\mu_{\rm c}}$ distribution of the FAST
sample shows a bimodal feature (Li et al., 2021a). Its high-$E^{\rm
obs}_{\mu_{\rm c}}$ also ranges in $10^{38-39}$ erg, being consistent with the
$E^{\rm obs}_{\nu_{\rm p}}$ distributions of the GBT and the Arecibo samples.

The distributions of the spectral widths of the bursts in the FAST and Arecibo
samples are $\Delta\nu^{\rm obs}=0.1\sim 0.5$ GHz with a typical value of
$\Delta\nu=0.23$ GHz and $\Delta\nu^{\rm obs}=0.1\sim 0.7$ GHz with a typical
value of 0.32 GHz, respectively. The burst durations $T^{\rm obs}$ of the FAST
and Arecibo samples are comparable, and they are much longer than that of the
bursts in the GBT sample. Fitting the distribution of $\log T^{\rm obs}$ with
a normal function, we have $\log(T^{\rm obs}/{\rm ms})=0.62\pm 0.25$, $0.60\pm
0.33$, and $0.10\pm 0.26$ for the FAST, Arecibo, and GBT samples,
respectively. This indicates that bursts in the high frequency band are
usually shorter than that in the low frequency band (e.g. Gajjar et al. 2018).

Figure 1: Histogram of $\nu_{\rm p}^{\rm obs}$, $\log E_{\nu}^{\rm obs}$
($\log E_{\mu_{\rm c}}^{\rm obs}$ for the FAST sample, $\log E_{\nu_{\rm
c}}^{obs}$ for the Arecibo sample, and $\log E_{\nu_{\rm p}}^{\rm obs}$ for
the GBT sample), radiating frequency range $\Delta\nu^{\rm obs}$, and $\log
T^{\rm obs}$. The dashed lines are our fits with a function of one or multiple
Gaussian component(s). The grey line in the left-top panel is the constructed
$\nu_{\rm p}$ distribution in 0.5-8 GHz inferred from the $\nu_{\rm p}^{\rm
obs}$ distributions of the GBT and Arecibo samples as discussed in §3. The
FAST, Arecibo and GBT bandpasses are also marked with arrows.

##  3 Simulation Analysis

###  3.1 Model Assumptions

  * •

Intrinsic Energy Function (E-function): We model the intrinsic E-function of
the bursts in the range of $[10^{37},10^{42}]$ erg with a single power-law
function,

$\frac{dp(E)}{dE}\propto E^{-\alpha_{\rm E}}.$ (3)

  * •

Intrinsic Spectral Profile: The spectra of bursts from FRB 20121102A seem to
have a Gaussian envelope (Law et al., 2017; Gajjar et al., 2018; Josephy et
al., 2019). Therefore, we model the spectral profile as a Gaussian function

$F_{\nu}=\frac{F}{\sigma_{\rm
s}\sqrt{2\pi}}\exp\left[\frac{-\left(\nu-\nu_{\rm p}\right)^{2}}{2\sigma_{\rm
s}^{2}}\right],$ (4)

where $\nu_{\rm p}$ is the peak frequency, $\sigma_{\rm s}$ is the standard
deviation, and $F$ ($=E/4\pi D_{L}^{2}$) is the fluence of the spectrum.

  * •

Intrinsic Spectral Fringe Pattern: As shown in Figure 1, the detection
probability distribution of $\nu^{\rm obs}_{\rm p}$ in the 4-8 GHz observed
with the GBT telescope is not uniform. It shows a fringe feature. Each fringe
component $i$ can be fitted with a normal distribution function $N_{\rm
i}(\nu_{\rm p};\nu_{\rm p,c,i},p_{\rm i},\sigma_{\rm i})$. Since the peaks at
5.64, 6.28, and 7.05 GHz are less suffered the frequency-cut effect, we regard
it as a basic block of the fringes and extrapolate it to the range from 0.5
GHz to 5 GHz. The constructed probability distribution of $\nu_{\rm p}$ in the
range from 0.5 GHz to 8 GHz is also shown in Figure 1. It reads as

$dp/d{\nu_{\rm p}}\propto\sum_{i}N(\nu_{\rm p};\nu_{\rm p,c,i},p_{\rm
i},\sigma_{\rm i}).$ (5)

The parameters of $\nu_{p,c}$, $p$, $\sigma$ of each component are summarized
in Table 1. Interestingly, the bandpass of the Arecibo telescope covers the
$\nu_{\rm p}$ distribution peak at 1.57 GHz333Note that the relative
probability of the $\nu_{\rm p}$ distribution between the GBT and Arecibo
samples cannot be directly compared. since the two telescopes have different
sensitivity, indicating that the $\nu_{\rm p}$ for a large fraction of bursts
in the Arecibo sample should be in the bandpass of the telescope. No burst in
the 2-4.5 GHz is available in the selected FAST, Arecibo, and GBT samples.

Table 1: The intrinsic $\nu_{\rm p}$ distribution of FRB 20121102A in the frequency coverage [0.5, 8] GHz inferred from the observed $\nu_{\rm p}^{\rm obs}$ distributions of the GBT and Arecibo samples. $i$th fringe | ${p_{\rm i}}$ | $\nu_{\rm p,c,i}$ | $\sigma_{\rm i}$
---|---|---|---
1 | 0.05 | 0.87 | 0.23
2 | 0.04 | 1.57 | 0.41
3 | 0.06 | 2.46 | 0.24
4 | 0.04 | 3.28 | 0.41
5 | 0.05 | 4.05 | 0.23
6 | 0.04 | 4.75 | 0.41
7 | 0.05 | 5.64 | 0.24
8 | 0.04 | 6.28 | 0.50
9 | 0.06 | 7.05 | 0.23

###  3.2 Simulation Procedure

Utilizing the FAST, Arecibo, and GBT samples, we constrain $\alpha_{\rm E}$
and $\sigma_{\rm s}$ of FRB 20121102A with Monte Carlo simulations. We outline
our simulation procedure as follows.

  1. 1.

We assume that the $\alpha_{\rm E}$ value uniformly distributes in the range
of [1,4], then randomly pick up a $\alpha_{\rm E}$ value.

  2. 2.

For a given $\alpha_{\rm E}$ value, we simulate a burst. Its energy $E^{\rm
sim}$ is generated from the intrinsic $E-$function of Eq. (3). We assume that
the simulated burst duration distributions for the FAST, Arecibo and GBT
samples are the same as the observed ones (as shown in Figure 1), and generate
the burst duration ($T^{\rm sim}$) from the $\log T^{\rm obs}$ probability
distributions.

  3. 3.

We simulate an intrinsic radiation spectrum, which is described as Eq. (4),
for a given burst. Faber et al. (2021) divided the entire 4 GHz bandwidth of
the GBT observation (Gajjar et al., 2018) into 8 sub-bands. Each band spans
$\sim$ 500 MHz. They show that the spectral width ranges from 120 to 650 MHz.
Therefore, we assume that $\sigma_{\rm s}$ of the Gaussian spectrum uniformly
distributes in the range of 0.1-0.8 GHz, then randomly pick up a $\sigma_{\rm
s}$ value. The $\nu_{\rm p}$ value of the spectrum is generated based on Eqs.
(5). We define the observable spectral range as $[\nu_{1},\nu_{2}]$, where
$\nu_{1}=\nu_{\rm p}-\sigma_{\rm s}$ and $\nu_{2}=\nu_{\rm p}+\sigma_{\rm s}$.
We exclude those bursts that have $\nu_{1}\leq 0$. Since the bandpasses of the
FAST and Arecibo telescopes are below 3 GHz and $\sigma_{\rm s}$ is limited as
$<0.8$ GHz, we take only the fringe components 1-3 (0.5-3 GHz) of the
$dp/d\nu_{\rm p}$ distribution for generating the $\nu_{\rm p}$ for the
simulated FAST and Arecibo samples. Similarly, we take only the fringe
components 5-8 (4-8 GHz) for generating the $\nu_{\rm p}$ in the simulated GBT
sample.

  4. 4.

We calculate the specific energy and spectral slope of the simulated burst in
the observable frequency range. We define the observable frequency range of a
simulated burst in the spectral range $[\nu_{1},\nu_{2}]$ with an instrument
in the bandpass $[\mu_{1},\mu_{2}]$ as $[\lambda_{1},\lambda_{2}]$, where
$\lambda_{1}=\max(\nu_{1},\mu_{1})$ and $\lambda_{2}=\min(\nu_{2},\mu_{2})$.
We exclude those bursts whose observable ranges are out of the instrumental
bandpass, i.e. $\nu_{1}\geq\mu_{2}$ or $\nu_{2}\leq\mu_{1}$. We do not
discriminate the $\nu_{\rm p}$-in-band and $\nu_{\rm p}$-out-band scenarios,
and calculate the specific energy at the central frequency, i.e. $\nu^{\rm
sim}_{c}=(\lambda_{1}+\lambda_{2})/2$, with Eq. (2). We measure the global
feature of an observable spectrum by calculating its spectral slope with
$\beta^{\rm sim}=\log_{10}(F^{\rm sim}_{\lambda_{2}}/F^{\rm
sim}_{\lambda_{1}})/\log_{10}(\lambda_{2}/\lambda_{1})$. It presents the
global shape of a spectrum, i.e. a rising, decaying, or flattening spectrum in
the range $[\lambda_{1},\lambda_{2}]$.

  5. 5.

We check the detectability of the simulated burst with the FAST, Arecibo, and
GBT telescopes. We calculate the equivalent flux density of the burst in the
unit of Jy ms with $S_{\nu}^{\rm sim}=F^{\rm sim}/(\omega^{\rm sim}_{\rm
eq}T^{\rm sim})$, where $F^{\rm sim}$ is the observable fluence given by
$F^{\rm sim}=\int_{\lambda_{1}}^{\lambda_{2}}F_{\nu}d\nu$, and $\omega^{\rm
sim}_{\rm eq}$ is the equivalent spectral width of the burst, which is defined
as the full-width-half-maximum (FWHM) in the range of
[$\lambda_{1}$,$\lambda_{2}$]. Note that the baseline for calculating the FWHM
is taken as the minimum flux between $F(\lambda_{1})$ and $F(\lambda_{2})$,
but not zero. A burst is detectable if its $S_{\nu}^{\rm sim}$ is larger than
the telescope thresholds, i.e. $S^{\rm FAST}_{\nu,\rm th}=0.015$ Jy ms,
$S^{\rm Arecibo}_{\nu,\rm th}=0.057$ Jy ms, and $S^{\rm GBT}_{\nu,\rm
th}=0.0265$ Jy ms.

###  3.3 Results

Figure 2 shows the contours of $\log p_{\mathrm{KS}}$ in the $\alpha_{\rm
E}$-$\sigma_{\rm s}$ plane for the three samples. It is found that the
observed E-distributions of the three samples can be reproduced by our
simulations at a confidence level of $3\sigma$. As marked with magenta stars
in Figure 2, the derived $\\{\alpha_{\rm E},\sigma_{\rm s}\\}$ set with the
maximum likelihood is $\\{1.82,0.18\\}$ from the FAST sample ($p_{\rm
KS}=0.08$), $\\{2.09,0.17\\}$ from the Arecibo sample ($p_{\rm KS}=0.31$), and
$\\{3.16,0.19\\}$ from the GBT sample ($p_{\rm KS}=0.08$). Comparisons of the
distributions of the burst specific energy (at $\mu_{\rm c}$ for the FAST
sample, $\nu_{\rm c}$ for the Arecibo sample, and $\nu_{\rm p}$ for the GBT
sample) and burst duration between the observed and simulated samples by
adopting the derived maximum likelihood parameter sets are also shown in
Figure 2. It is found that the $E_{\nu_{\rm c}}^{\rm obs}$ and $T^{\rm obs}$
distributions of the three samples are well reproduced. Note that we do not
make any dependence of the burst energy $E$ (or fluence $F$) on the burst
duration, and the $T^{\rm sim}$ values are bootstrapped from the $T^{\rm obs}$
distributions. This eventually makes the distributions of $T^{\rm sim}$ and
$T^{\rm obs}$ are roughly consistent, yielding $P_{\rm KS}=4.56\times
10^{-6}$, 0.23, and $3.40\times 10^{-6}$ for the FAST, Arecibo, and GBT
samples, respectively.

The $\log p_{\mathrm{KS}}$ contour for the FAST sample illustrates two
distinct regions, i.e. $\\{\sigma_{\rm s}<0.5,\alpha_{\rm E}<2.3\\}$ and
$\\{\sigma_{\rm s}>0.5,\alpha_{\rm E}>2.3\\}$. In the $\\{\sigma_{\rm
s}>0.5,\alpha_{\rm E}>2.3\\}$ region, the $p_{\rm KS}$ values are $<10^{-4}$.
In addition, as mentioned above, the spectral width of the bursts observed
with GBT is typically in the range of 120 $\sim$ 650 MHz (Faber et al., 2021).
Therefore, the potential parameter region of $\\{\sigma_{\rm
s}>0.5,\alpha_{\rm E}>2.3\\}$ is ruled out in our analysis. The most
preferable parameter region is at the region of $\\{\sigma_{\rm
s}<0.5,\alpha_{\rm E}<2.3\\}$, which gives $\alpha_{\rm E}=1.8^{+0.1}_{-0.3}$
and $\sigma_{\rm s}=0.18^{+0.28}_{-0.06}$ by adopting $p_{\rm KS}>10^{-4}$.
The bimodal $\log E^{\rm obs}_{\mu_{\rm c}}$ distribution of the FAST sample
can be reproduced with this optimal parameter set. A similar feature is also
seen in the Arecibo sample, but the two-parameter regions cannot be well
separated. The $\alpha_{\rm E}$ value is constrained as $\alpha_{\rm
E}=2.09^{+0.30}_{-0.40}$, but $\sigma_{\rm s}$ cannot be constrained with
$p_{\rm KS}>10^{-4}$. Since the size of the GBT sample is very small (93
bursts), it gives $\alpha_{\rm E}>2.1$ but loses any constraint on
$\sigma_{\rm s}$ with $p_{\rm KS}>10^{-4}$.

Adopting optimal parameter sets of $\\{\alpha_{\rm E},\sigma_{\rm s}\\}$,
Figure 3 displays the $\nu^{\rm sim}_{\rm p}$ and $\beta^{\rm sim}$
distributions of the simulated samples. The derived $\nu^{\rm sim}_{\rm p}$
distribution of the simulated FAST sample peaks at 0.87 GHz and 1.57 GHz,
being consistent with the pre-set $\nu_{\rm p}$ distribution of the 1st and
2nd fringe components. The bandpass of the FAST telescope spans the two peaks.
Most simulated FAST bursts ($75\%$) are of $\nu^{\rm sim}_{\rm p}$-off-band
bursts. The inferred $\beta^{\rm sim}$ is dramatically different from burst to
burst, ranging from -8 to 10. The $\nu^{\rm sim}_{\rm p}$ distribution of the
simulated Arecibo sample, which covers from 1 GHz to 1.85 GHz, is broader than
the $\nu^{\rm obs}_{\rm c}$ distribution. Most simulated Arecibo bursts
($65\%$) are $\nu^{\rm sim}_{\rm p}$-in-band burst. The $\beta^{\rm sim}$
distribution ranges from -8 to 12, but the $\beta^{\rm sim}$ values of about
one-third of simulated Arecibo bursts are $\sim 0$, indicating that $\nu^{\rm
sim}_{\rm p}$ of these bursts are very closed to the central frequency of the
Arecibo telescope or the characterized frequency range
$[\lambda_{1},\lambda_{2}]$ is completely in the bandpass. The broad frequency
range of the GBT telescope (4-8 GHz) makes the most simulated bursts $\nu^{\rm
sim}_{\rm p}$-in-band bursts, and their $\beta^{\rm sim}$ values are $\sim 0$.

Figure 2: The $\log p_{\mathrm{KS}}$ contours in the $\sigma_{\rm
s}-\alpha_{\rm E}$ plane (left panels) and comparisons of the distributions of
the burst specific energy (at $\mu_{\rm c}$ for the FAST sample, $\nu_{\rm c}$
for the Arecibo sample, or $\nu_{\rm p}$ for the GBT sample) and burst
duration between the observed and simulated samples by adopting the maximum
likelihood parameter sets of $\\{\alpha_{\rm E},\sigma_{\rm s}\\}$ (magenta
stars in the left panels), i.e. $\\{1.82,0.18~{}{\rm GHz}\\}$ for the FAST
sample ($P_{\mathrm{KS}}$=0.08), $\\{2.09,0.17~{}{\rm GHz}\\}$ for the Arecibo
sample ($P_{\mathrm{KS}}$=0.31), and $\\{3.16,0.22~{}{\rm GHz}\\}$ for the GBT
sample ($P_{\mathrm{KS}}$=0.10). The cyan, blue, and green lines in the left
panels mark the contours of $\log P_{\mathrm{KS}}$=-4, -3, and -2,
respectively.

Figure 3: Distributions of $\nu^{\rm sim}_{\rm p}$ and $\beta^{\rm sim}$ of
the simulated samples in comparison with the observed ones if available. The
simulated samples are generated by adopting the maximum likelihood parameter
sets of $\\{\alpha_{\rm E},\sigma_{\rm s}\\}$ (magenta stars in Figure 2). The
shaded region marks the bandpasses of the FAST and Arecibo telescopes.

##  4 Discussion

Figure 4: Comparisons of the $\nu_{\rm p}$ histograms between different sub-
samples in the GBT sample:left panel— 21 bursts found by Gajjar et al. (2018)
through a regular method vs. extra 72 bursts discovered by Zhang et al. (2018)
through a neural network machine learning algorithm from the same GBT data;
right panel— low- vs. high-fluence burst samples by averagely divided the
total sample with a division $F_{\rm p}^{obs}$ $\geq$46.2 Jy $\mu$s. The gray
line is the constructed $\nu_{\rm p}$ distribution for the total sample as
shown in Figure 1.

###  4.1 Putative Spectral Fringe Pattern and Narrow Spectral Width

As shown in Figure 1, we construct a modulated $\nu_{\rm p}$ distribution for
the bursts of FRB 20121102A based on the Arecibo and GBT observations in
L-band and C-band. The interval among the $\nu_{\rm p}$-probability peaks is
$\sim 0.8$ GHz. Such a discrete $\nu_{\rm p}$ distribution suggests that the
radiations of FRB 20121102A in a broad frequency range show a fringe pattern.
As mentioned in §1, a search for simultaneous bursts of FRB 20121102A in
different frequencies indicates that the bursts are active in some preferred
frequency bands, likely favoring the putative $\nu_{\rm p}$ fringe pattern
derived from broadband (4-8 GHz) observations with the GBT telescope.

The putative $\nu_{\rm p}$ fringe pattern and the narrowness of the radiating
spectrum should give insight into the radiation physics of FRBs. The proposed
radiation models are classified into two groups, i.e. synchrotron maser in the
relativistic shocks far away from the central engine (Lyubarsky, 2014;
Beloborodov, 2017; Metzger et al., 2019 for the case of magnetized shocks and
see Waxman (2017); Deng et al. (2021) for the case of weakly magnetized
shocks) and the coherent curvature radiation (CR) or coherent inverse Compton
scattering (ICS) of bunching electrons close-in the magnetosphere (Yang &
Zhang, 2018; Zhang, 2022). Although the CR spectrum of a single bunch appears
to oscillate at a typical narrow frequency and shows a discrete structure, the
collective CR spectrum of bunches is characterized as several power-law
segments in a broad frequency range (Yang & Zhang, 2018). The observed narrow
spectra of FRBs are thought to be due to the absorption of low frequency radio
emission, but the fringe pattern of $\nu_{p}$ cannot be expected. In the ICS
radiation model, the characteristic frequency of the coherent emission depends
on the Lorenz factor of the bunching electrons, as well as the frequency and
the incident angle of the seed photons, which is possible to produce a narrow
spectrum if all of these parameters take typical values in the framework the
model (Zhang, 2022). Nonetheless, this model does not predict such a fringe
pattern of $\nu_{p}$ as well. For the synchrotron maser emission models,
analysis of particle-in-cell (PIC) simulations by Plotnikov & Sironi (2019)
shows that the synchrotron maser produces a narrow radiation spectrum peaking
at $\rm{a~{}few}\times\nu_{e}$ in the shock frame, where $\nu_{e}$ is the
frequency of the pair plasma ahead of the shock. However, the observed peak
frequency requires a Lorentz transformation from the shock frame to the
observer, and the Lorentz factor of the shock depends on the burst energy, the
burst time scale, and the property of the interacting medium. Therefore, it
also does not predict the fringe pattern of $\nu_{\rm p}$ in the synchrotron
maser model. The fringe pattern of $\nu_{\rm p}$ found in this work, if it is
true, strongly challenges the two kinds of models mentioned above. New
radiation models should be considered for generating such a fringe pattern
intrinsically. They should be independent of the external environment and
propagation effects. We should note that potential analogs of the spectral
fringe pattern are also found in the high frequency interpulse of Crab and the
zebra patterns in solar radio spectra (Karlický, 2013; Eilek & Hankins, 2016).
They may give some hints for revealing the nature of the spectral fringe
pattern.

Several statistical and observational caveats to the apparent fringe pattern
should be addressed. First, it is unclear whether the fringe pattern results
from artificial effects, such as the signal-to-noise ratio (or fluence)
threshold, the detection algorithm, the frequency ranges, and the burst
bandwidths. Note that among the 93 bursts in the GBT sample for our analysis,
21 bursts were previously reported in Gajjar et al. (2018) and extra 72 bursts
were discovered in Zhang et al. (2018) through the narrow bandwidth search by
the use of a neural network machine learning algorithm. The neural network
algorithm may have biases toward certain frequency ranges and burst
bandwidths. As explained in Zhang et al. (2018), their neural network machine
learning algorithm modulates the spectra of their training data set. This
might have caused a bias if the modulations did not cover the potential
parameter space fully. We check whether the $\nu_{p}$ fringe pattern persists
in the samples of the 21 bursts and the 72 bursts. As shown in Figure 4, one
can see that the $\nu_{p}$ fringe pattern exists and is consistent in the two
sub-samples at $\nu_{p}>5.5$ GHz. The K-S test gives a probability of $p_{\rm
KS}$=0.23. The bursts with $\nu_{p}<5.5$ GHz are only found with the neural
network algorithm. Second, it is uncertain whether the fringe pattern is
suffered a bias of fluence threshold selection effect. Threshold of signal-to-
noise ratio for burst searching is taken as 6 in both Gajjar et al. (2018) and
Zhang et al. (2018). We averagely separate the global GBT sample into high-
fluence and low-fluence groups with a fluence division of $F$=46.2 Jy $\mu$s
and compare the $\nu_{p}$ fringe pattern in the two groups. As shown in Figure
4, the $\nu_{p}$ fringe pattern in the low- and high-fluence groups are also
statistically consistent ($p_{\rm KS}$=0.39), indicating that the $\nu_{p}$
fringe pattern persists in both the low and high fluence bursts. Finally, the
$\nu_{p}$ fringe pattern suffered a great risk of statistical fluctuation
effect based on a small sample of bursts identified from the GBT data.
Especially, it is quite uncertain to extrapolate the fringe pattern in the GBT
band to low frequency ranges. Notice that 9 bursts detected with VLA in the
frequency coverage of 2.5-3.5 GHz likely imply a $\nu_{\rm p}$ fringe in this
frequency range (Law et al., 2017). The spectra of four out of the 9 bursts
(bursts 57623, 57643, 57645, and 57648) peak at $\sim 2.8$ GHz, being likely
inconsistent with the inferred fringe pattern, which is a trough around 2.8
GHz as shown in the top left panel of Figure 1. The such small number of
statistics still makes concerns about the apparent fringe pattern. In
addition, the apparent fringe pattern is derived from individual bursts at
different times observed with GBT, but not from the simultaneous bursts
observed in a broad frequency band. Broadband simultaneous observations in
dense frequency coverage with the Square Kilometre Array (SKA), which has an
extreme sensitivity and broad bandpass ($50\,\mathrm{MHz}\sim
20\,\mathrm{GHz}$, see the document available at the SKA website 444
https://www.skatelescope.org/; see also Dewdney et al. 2009) should offer an
opportunity to verify this fringe pattern.

From Figure 2, the preferred $\sigma_{\rm s}$ value constrained with the FAST
sample is $\sigma_{\rm s}=0.18^{+0.28}_{-0.06}$ GHz, and $\sigma_{\rm s}\leq
500$ MHz (the best $\sigma_{\rm s}$=0.17 GHz) with the Arecibo sample. As
shown in Figure 1, the spectral widths of the bursts in the FAST and Arecibo
samples are $\Delta\nu^{\rm FAST}=0.1\sim 0.5$ GHz with a typical value of
$\Delta\nu=0.22$ GHz and $\Delta\nu^{\rm Arecibo}=0.1\sim 0.7$ GHz with a
typical value of 0.31 GHz. Taking $2\sigma_{\rm s}$ as the spectral width of
the simulated bursts, we find that it is comparable to the observed ones. Note
that the interval of the $\nu_{\rm p}$ fringes is $\sim 0.8$ GHz, which is
longer than the derived spectral width. This is reasonable since the observed
$\nu_{\rm p}$ fringes indicate that the radiating spectrum should be narrower
than the fringe interval, otherwise the fringes could be smeared out by the
broad radiating spectrum. The narrow spectrum feature makes the observed
spectral shapes among bursts dramatically different, highly depending on the
$\nu_{\rm p}$ is in or out of the bandpass. The diverse and time-varied
spectral indices among bursts observed with narrow-bandpass telescopes (such
as the FAST and Arecibo telescopes; Spitler et al., 2016) are well explained
with the narrow spectral width and the fringe pattern of $\nu_{\rm p}$.

###  4.2 Intrinsic E-function of the FRB 20121102A bursts

The observed E-distribution of FRB 20121102A is variable in different
observational epochs (Scholz et al., 2016; Spitler et al., 2016; Chatterjee et
al., 2017; Scholz et al., 2017; Michilli et al., 2018; Zhang et al., 2018;
Gourdji et al., 2019). Li et al. (2021a) reported a two-component distribution
of the specific energy distribution at $\mu_{\rm c}$ for the FAST sample, but
it does not show up in the Arecibo and GBT samples, as shown in Figure 1. Our
simulation analysis shows that the observed $E-$distribution of the three
samples can be well reproduced by modeling the intrinsic E-function with a
single power-law function. We compare the burst energy distributions of the
simulated FAST and Arecibo samples 555We do not add in the intrinsic
E-distribution of the simulated GBT sample for comparison since the
$\alpha_{\rm E}$ and $\sigma_{\rm s}$ values lose constraints with the GBT
sample. in Figure 5. With a higher sensitivity, the intrinsic burst energy of
the simulated FAST sample is lower than that of the simulated Arecibo sample,
but no bimodal feature is observed.

Note that the specific energy of a burst depends on the specific frequency.
For a narrow spectrum with a Gaussian profile, its $E_{\nu_{\rm p}}$ would be
a reasonable representation of the burst energy. However, the bandpass of the
FAST telescope covers the valley between the fringe peaks at $\nu_{\rm
p}=0.87$ and $\nu_{\rm p}=1.57$ GHz. The $\nu_{\rm p}$ for most bursts in the
simulated FAST sample is out of the FAST bandpass. Therefore, the $E_{\nu_{\rm
c}}$ value is not representative of the intrinsic burst energy. Our
simulations show that FAST can detect these $\nu_{\rm p}$-out-band bursts with
its high sensitivity (see Figure 3). This should make an excess of the low
energy bursts. Therefore, the bimodal $E_{\nu_{\rm c}}$ distribution of the
FAST sample should result from the intrinsic $\nu_{\rm p}$ fringe pattern and
the narrow radiating spectrum as well as the detection of $\nu_{\rm p}$-out-
band bursts with a high fluence sensitivity of the FAST telescope. This is
also supported by the fact that the bimodal feature disappeared by calculating
the burst energy over the bandpass (Aggarwal, 2021).

Figure 5: Distributions of the intrinsic energy $E$ of the simulated FAST and
Arecibo samples.

##  5 Conclusions

In this paper, we have investigated the intrinsic radiating spectrum and
energy distribution of FRB 20121102A through Monte Carlo simulations by
adopting multi-frequency observations with the FAST (1.05-1.45 GHz), Arecibo
(1.15-1.73 GHz), and GBT (4-8 GHz) telescopes. Our results are summarized
below.

  * •

Using the GBT sample of FRB 20121102A, we find a fringe feature of the
$\nu^{\rm obs}_{\rm p}$ distribution, which can be fitted with a series of
normal functions with peaks at 4.75, 5.58, 6.28, 7.06 GHz, indicating that the
bursts are active in these preferred frequencies. The intervals among the
peaks are $\sim 0.8$ GHz. Combing the GBT and Arecibo samples, we construct
the intrinsic $\nu_{\rm p}$ fringe pattern in 0.5-8 GHz. Current simultaneous
broadband observations in sparse frequency coverage still do not reveal
simultaneous burst activities in these preferred $\nu_{\rm p}$.

  * •

We investigate the intrinsic E-distribution and radiating spectrum profile
through Monte Carlo simulations. By modeling the intrinsic energy distribution
as a single power-law function and depicting the spectrum profile with a
Gaussian, our simulations show that the maximum likelihood parameter set
$\\{\alpha_{\rm E},\sigma_{\rm s}\\}$ derived from the FAST and Arecibo
samples are $\\{1.82,0.18\\}$ and $\\{2.09,0.17\\}$, respectively. The
$\sigma_{\rm s}$ value well agrees with the observed spectral range of the
FAST and Arecibo observations. The observed E-distributions of the three
samples are well reproduced with the parameter sets. Especially, the bimodal
E-distribution and the significant spectral slope variation of the FAST sample
result from the spectral fringe pattern and narrow Gaussian spectral profile
as well as the detection of $\nu_{\rm p}$-out-band bursts with high
sensitivity of the FAST telescope.

In conclusion, our results suggest that the $\nu_{\rm p}$ among bursts of FRB
20121102A illustrates a fringe pattern in a broad energy frequency range, and
the variations of the observed E-distribution and spectral slope are
physically due to both the intrinsic $\nu_{\rm p}$ fringe pattern and the
narrowness of the radiating spectrum, and observationally due to the bandpass
selection and sensitivity of different telescopes.

## acknowledgments

We very much appreciate thoughtful and constructive comments and suggestions
from the referee. We also thank Di Li, Bing Zhang, Xue-Feng Wu, Fa-Yin Wang,
Pei Wang, Wei-Yang Wang, and Yuan-Pei Yang for their helpful discussion. We
acknowledge the use of the public data from the FAST/FRB Key Project. F.L. is
supported by Shanghai Post-doctoral Excellence Program. E.W.L and J.G.C. are
supported by the National Natural Science Foundation of China (grant Nos.
12133003). C.M.D. is supported by the National Natural Science Foundation of
China (grant No. 12203013) and the Guangxi Science Foundation (grant Nos.
2021AC19263).

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