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To identify the decoherence origin, frequency spectra using multiple{\pi}-pulses have been extensively studied. However, little has been discussedon how to define the spectral intensities from multiple-echo decays and how toincorporate the Hahn-echo T_2 in the noise spectra. Here, we show thatexperiments based on two theories solve these issues. With the previous theoryclarifying that the spectral intensity should be given as the decay in thelong-time limit, the intensity can be deduced without experimental artifactsusually entailed in the initial process. The other is thefluctuation-dissipation theory, with which the Hahn-echo T_2 is utilized as thezero-frequency limit of the noise spectrum and as an answer to the divergentissue on the 1/f^n noises. As a result, arsenic nuclear spins are found toexhibit 1/f2 dependences over two orders of magnitude in all the substrates ofun-doped, Cr-doped semi-insulating and Si-doped metallic GaAs at 297 K. The1/f^2 dependence indicates single noise source that is characterized by thecharacteristic 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 thefluctuations of next-nearest-neighboring nuclear spins.

1/f^2 spectra of decoherence noise on ^75^As nuclear spins in bulk GaAs