# Why is power level area under the autocorrelation function of the white-noise signal?

A paper I am reading (Linear and Nonlinear Encoding Properties of an Identified Mechanoreceptor on the Fly Wing Measured with Mechanical Noise Stimuli) defines power level for a white-noise signal as the area under the autocorrelation function of the signal.

What exactly is power level? I couldn't find a proper definition for it. Is it average or RMS power?

Should I take the autocorrelation in frequency domain or time domain?

autocorrStim = conv(stim,-stim);
powerLevel = sum(autocorrStim);


stim is the white-noise signal. Is this implementation correct?

Excerpt from a textbook on white-noise analysis (Marmarelis and Marmarelis, 1978)

• hey, they probably (? we don't know what Dickinson 1992a really is… please provide an actual citation!) just mean the power when they say power level. Power is, well, the physical entity power; seeing that they're defining this through the ACF, that is the expected power over an infinitely long observation interval of a weak-sense stationary process. So, it's expected, not "average" nor "RMS"; these two terms are something you apply to a deterministic signal or an observation that's already done, whereas expectation is a quality of a random variable. Sep 15 at 13:48
• That seems wrong. The power is typically $<x^2[n]> = r_{xx}[0]$. I.e. it's the zero tap of the autocorrelation, not the sum over the whole thing. Sep 15 at 14:21
• Sorry for not sharing the correct citation earlier. I have added the image of the relevant lines from the paper. Thanks for the explanation about power in deterministic signal. The signal I have is a finite length white noise that I have recorded (feedback of mechanically delivered white noise stimulus). So, can I treat it as deterministic signal? Sep 15 at 14:34
• @Hilmar I have added a screenshot from the textbook. According to the textbook, the total power delivered is given by the integral of the power spectrum. Am I reading it correctly?. Sep 15 at 14:54
• @user3581147 so, yes, you're right, integral over the PSD, and Hilmar is right, ACF at 0; the text you cited/highlighted in yellow is in contradiction to the formula (and is thus wrong) Sep 15 at 16:42

## 1 Answer

It looks like the first book is misquoting the second book. The second book says nothing about integrating the autocorrelation. It cites (correctly) that the power is the autocorrelation at tap 0 (i.e. $$r_{xx}[0]$$) and the integral of the PSD which is the Fourier Transform of the autocorrelation.

• That's true. Thanks! I will use integral over PSD. Sep 18 at 7:16