  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.
