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I am reading about noise from this lecture (page 9/31). However, I don't understand this point. Could anyone explain why mean squared value is zero?

The mean squared value of a random noise signal at a single precise frequency is zero.

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  • \$\begingroup\$ If the mean squared value wasn't zero, it would be a systematic error, not a random one. \$\endgroup\$ Oct 28 '16 at 20:11
  • \$\begingroup\$ I think the key word here is "random" noise. Meaning over time you should have the same amount of random positive magnitude noise as negative magnitude noise. Law of large numbers means that this cancels out eventually. Since it is also randomly spread across frequencies, it will balance everywhere (on average). \$\endgroup\$
    – jbord39
    Oct 28 '16 at 20:11
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Noise is continuous spectrum in this case and measured here in \$ \mu V/Hz^{0.5} \$

Thus for a given ratio, the voltage also goes to zero as Hz =>0. In practice we cannot make a band pass filter with zero bandwidth, so it is academic.

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The power spectral density is finite and the 'single precise frequency' is arbitrarily narrow so the value of the power in the noise (mean squared value) in a given bandwidth must go to zero as the bandwidth goes to zero.

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