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to know the amount of the noise and it's spectral density You may use the FFT Analyzer ( also called Dynamic Signal Analyzer or Low Frequency Spectrum Analyzer).
To know how much the noise is "white" and gaussian You have to sample, store and then perform the Allan Variance
Depending on the spectral bandwidth of interest, a fast system multimeter could be used with FFT processing, e. g. in spreadsheet program or any available measurement tool (Labview or whatever).
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