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From numerical simulation and regression analysis I discovered that the root-mean-square amplitude of white noise with bandwidth $\Delta\!f$ is proportional to $\sqrt{\!\Delta\!f}$. How can this be derived mathematically ?

Petoetje59
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  • There are various conventions. Please state your definition of "white noise" and "bandwidth". – Lee David Chung Lin May 27 '19 at 14:26
  • This is a sequel to another question that got answered : https://math.stackexchange.com/questions/3235359/puzzling-random-number-property

    When I generate this and save it as a 16 bit mono audio file sampled at 5512 Hz I hear the white noise. The full bandwidth is (nearly) half the sampling speed (Nyquist criterium), thus 2756 Hz. I'm interested in determining the RMS only from a stretch of frequencies in its spectrogram, like 1375$\pm$ 150 Hz. The bandwidth of this stretch is thus 150 Hz.

     – Petoetje59 May 27 '19 at 18:31

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I think I found and grasped the explanation... In white noise the energy (power) is the same for each frequency - by definition. So for a bunch of successive frequencies (bandwidth) the average RMS power will become proportionally larger. Electric power is proportional to voltage squared. Thus the voltage (RMS amplitude) of white noise will be proportional to the square root of its bandwidth.

Correct me if my reasoning is wrong.

Petoetje59
  • 165