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Fast Fact: Root Mean Squared Unraveled

      RMS or Root Mean Squared is defined as the Root of the Mean of the Square of each sampled value of a signal (for discrete values) or the Root of the Definite Integral of the Square of each bounded function (for continuous values). But what exactly does it mean? It is actually like a means of central tendency (I use the word "like" because it doesn't exactly function the same way as an average) only that negative integers are treated as positive values by squaring them in the first place. After squaring, you take the "mean" then take the square root again to reverse the effects of squaring. Thus, the entire signal is sufficiently represented by one constant value. This constant value can be taken as DC (direct current) since DC is a constant unchanging voltage. Hence, when we take an RMS measurement of an AC signal, we are actually getting an equivalent DC signal that can replace that AC signal in terms of power delivery.


      But why not use Cube-Root Mean Cube or higher exponentials? Well, I've tried it in Matlab, and when you observe the change in the final value when you tweak the signal, you'll find that the sensitivity to change is highest when squared and gets worse when you "CMC" it or treat it with higher exponentials. For example, take the vector [10 20 12 15 7 9]. Its RMS, CMC, etc. is 14.6002, 15.2561, and 15.8017 respectively. When you try and tweak one component of the vector, say increase the first element by 3, yielding [13 20 12 15 7 9], the resulting RMS, CMC, etc. is 14.9889, 15.5366, and 15.9942. Now let us get the sensitivity of each:


RMS


14.9889-14.6002=0.3887


CMC (Cube-root Mean Cube)


15.5366-15.2561=0.2805


Etc. (actually raised each to the 4th , got the mean, and took the 4th root)


15.9942-15.8017=0.1925


     It is directly observable that RMS yields optimum sensitivity to change. Also, CMC is disqualified from the beginning since it doesn't remove negative components at all.

(Note: I only made up the CMC term. I'm not sure whether such a term was really coined. Hahaha...)

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