dekkyrkex4

2022-10-08

Let x be a block vector with two vector elements x=(a,b), where a and b are vectors of size n and m, respectively.

How do I express $rms(x)$ in terms of $rms(a)$, $rms(b)$, $m$ and $n$?

How do I express $rms(x)$ in terms of $rms(a)$, $rms(b)$, $m$ and $n$?

Cooper Gillespie

Beginner2022-10-09Added 4 answers

From the conventional definition of the RMS value of a finite dimensional vector,

$\begin{array}{r}{x}_{\text{RMS}}=\sqrt{\frac{1}{n+m}\sum _{i=1}^{n+m}{x}_{i}^{2}}=\sqrt{\frac{1}{n+m}(\sum _{i=1}^{n}{a}_{i}^{2}+\sum _{j=1}^{m}{b}_{j}^{2})}=\sqrt{\frac{1}{n+m}(n{a}_{\text{RMS}}^{2}+m{b}_{\text{RMS}}^{2})}.\end{array}$

$\begin{array}{r}{x}_{\text{RMS}}=\sqrt{\frac{1}{n+m}\sum _{i=1}^{n+m}{x}_{i}^{2}}=\sqrt{\frac{1}{n+m}(\sum _{i=1}^{n}{a}_{i}^{2}+\sum _{j=1}^{m}{b}_{j}^{2})}=\sqrt{\frac{1}{n+m}(n{a}_{\text{RMS}}^{2}+m{b}_{\text{RMS}}^{2})}.\end{array}$

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