<jats:title>Abstract</jats:title><jats:p>To identify the decoherence origin, frequency spectra using multiple π-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 <jats:italic>T</jats:italic><jats:sub>2</jats:sub> in the noise spectra. Here, we show that experiments based on two theories solve these issues. As proved in the previous theory, the spectral intensity is given as the decay in the long-time limit. Unlike the initial process of decays, this definition is not only theoretically proven but also validated experimentally, since long-time behaviors are generally free from experimental artifacts. The other is the fluctuation–dissipation theory, with which the Hahn-echo <jats:italic>T</jats:italic><jats:sub>2</jats:sub> is utilized as the zero-frequency limit of the noise spectrum and as an answer to the divergent issue on the 1/<jats:italic>f</jats:italic><jats:sup><jats:italic>n</jats:italic></jats:sup> noises. As a result, arsenic nuclear spins are found to exhibit 1/<jats:italic>f</jats:italic><jats:sup>2</jats:sup> 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/<jats:italic>f</jats:italic><jats:sup>2</jats:sup> dependence indicates that the noise is dominated by a single source with characteristic frequency <jats:italic>f</jats:italic><jats:sub>c</jats:sub><jats:sup>un</jats:sup> = 170 ± 10 Hz, <jats:italic>f</jats:italic><jats:sub>c</jats:sub><jats:sup>Cr</jats:sup> = 210 ± 10 Hz and <jats:italic>f</jats:italic><jats:sub>c</jats:sub><jats:sup>Si</jats:sup> = 460 ± 30 Hz. These <jats:italic>f</jats:italic><jats:sub>c</jats:sub> values are explained by a model that the decoherence is caused by the fluctuations of next-nearest-neighboring nuclear spins.</jats:p>