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I have started reading the following book: The scientist and Engineer guide to digital processing. At the beginning of chapter two, the following is said when talking about mean deviation v.s. standard deviation:

In most cases, the important parameter is not the deviation from the mean, but the power represented by the deviation from the mean. For example, when random noise signals combine in an electronic circuit, the resultant noise is equal to the combined power of the individual signals, not their combined amplitude.

I think this is something crucial but I don't quite understand it. Why is amplitude not important? My impression was that noise is something that becomes problematic when it can cause bit detection errors at the receiver. (i.e going above the minimum high logic level when a low level is sent or going below the maximum low logic level when a high level is sent.)

I am sure I am missing something important, can someone expand on or rephrase the citation?

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  • \$\begingroup\$ I think the concept here is noise on a channel and how it affects the channel throughput (by degrading signal to noise ratio (SNR)). \$\endgroup\$
    – user57037
    Commented Oct 12, 2022 at 0:24
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    \$\begingroup\$ Note that the average amplitude of the noise is ZERO ! You might consider the average of the absolute value of the amplitude, but a) this is difficult to handle mathematically, and b) for uncorrelated signals that add, it turns out that the average of the signal squared is what combines (that in fact can be the definition of uncorrelated) \$\endgroup\$
    – jp314
    Commented Oct 12, 2022 at 1:02

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When adding uncorrelated signals, it is the average power rather than the average amplitude that adds up as a statistic measure, even though the momentary value at any point of time is added non-squared to yield the resulting signal.

As a result, the "standard deviation" might have a more tangible unit, but for the actual statistical prediction, the variance (its square) is much more relevant.

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Why is amplitude not important?

It is important but, when incoherent noise sources come together (i.e. add in the time domain), the only thing that can be said is that their individual powers add together.

So, if noise-1 has an amplitude of (say) 30 volts RMS and, noise-2 has an amplitude of 20 volts amplitude, then the resultant power into 1 Ω is \$V_1^2 + V_2^2\$ or 900 watts + 400 watts. This is 1300 watts into 1 Ω.

The resultant voltage is the square root of the power i.e. \$\sqrt{V_1^2 + V_2^2}\$ or 36.05 volts RMS.

Of course you may say that the power is not defined by the voltage squared because the resistance in which they are applied to isn't defined but, the very act of adding the square of the RMS voltages then taking the square root of that addition, cancels out the need to know the resistance.

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    \$\begingroup\$ Well, 900 volt-squared + 400 volt-squared. It's only watts if it's working into a 1 ohm sink. \$\endgroup\$
    – TimWescott
    Commented Oct 12, 2022 at 0:00
  • \$\begingroup\$ @TimWescott I did mention that in my last paragraph but, just for you, I'll make it clearer. \$\endgroup\$
    – Andy aka
    Commented Oct 12, 2022 at 9:55
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Its because average value of noise amplitude may be zero or very low compared to average noise power which may be positive and significant. So, noise power becomes more relevant. Signal to noise ratio is a term used to indicate quality of a signal.

Signal to noise ratio is measured as:

$$ \frac{S}{N} = \frac{P_{signal}}{P_{noise}} = \frac {V_s^2}{V_n^2} $$

Reference:

Page 120, Electronic communications,4th edition,Roddy and Coolen.

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  • \$\begingroup\$ The average amplitude of noise is non-zero. \$\endgroup\$
    – tobalt
    Commented Oct 12, 2022 at 10:55
  • \$\begingroup\$ @tobalt: Only if it has a DC component. You can certainly have noise in a highpass or bandpass channel, and its average amplitude will be exactly zero. \$\endgroup\$
    – Ben Voigt
    Commented Oct 12, 2022 at 18:36
  • \$\begingroup\$ The signals whose noise the author discussed can be electromagnetic waves. I think the noise here is in a antenna receiver which is amplified by a intermediate amplifier and combiner and processed in a typical cellular system. EM waves which are sinusoidal waveforms can have zero average when they super impose on each other while creating noise \$\endgroup\$
    – Amit M
    Commented Oct 12, 2022 at 18:41
  • \$\begingroup\$ @BenVoigt the average amplitude of noise is by definition nonzero. You are describing the mean value. \$\endgroup\$
    – tobalt
    Commented Oct 12, 2022 at 19:03
  • \$\begingroup\$ tobalt: The squared value of noise amplitude may be calculated for each radio channel. \$\endgroup\$
    – Amit M
    Commented Oct 12, 2022 at 19:10

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