  To identify the decoherence origin, frequency spectra using multiple
{\pi}-pulses have been extensively studied. However, little has been discussed
on how to define the spectral intensities from multiple-echo decays and how to
incorporate the Hahn-echo T_2 in the noise spectra. Here, we show that
experiments based on two theories solve these issues. With the previous theory
clarifying that the spectral intensity should be given as the decay in the
long-time limit, the intensity can be deduced without experimental artifacts
usually entailed in the initial process. The other is the
fluctuation-dissipation theory, with which the Hahn-echo T_2 is utilized as the
zero-frequency limit of the noise spectrum and as an answer to the divergent
issue on the 1/f^n noises. As a result, arsenic nuclear spins are found to
exhibit 1/f2 dependences over two orders of magnitude in all the substrates of
un-doped, Cr-doped semi-insulating and Si-doped metallic GaAs at 297 K. The
1/f^2 dependence indicates single noise source that is characterized by the
characteristic frequency f_c_^un^=170 Hz, f_c_^Cr^=210 Hz and f_c_^Si^ =460 Hz.
These f_c values are explained by a model that the decoherence is caused by the
fluctuations of next-nearest-neighboring nuclear spins.
