RMS Value of Vectors

Question

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 $n$?

Thank you!

Answer 1

From the conventional definition of the RMS value of a finite dimensional vector, \begin{align*} x_\text{RMS} = \sqrt{\frac{1}{n+m}\sum_{i=1}^{n+m}x_i^2} = \sqrt{\frac{1}{n+m}\left(\sum_{i=1}^n a_i^2 + \sum_{j=1}^m b_j^2\right)} = \sqrt{\frac{1}{n+m}(na_\text{RMS}^2 + mb_\text{RMS}^2)}. \end{align*}