RMS value (Root Mean Square)

This parameter allows us to characterize the signal according to its energy content. The energy content was related to the amplitude of the signal in a certain interval of time. For a discrete signal, which consists in N samples equally spaced, the estimate is given by the following equation:

$\begin{array}{cc}RMS=\sqrt{\frac{1}{N}{\displaystyle \sum _{k=1}^N{\left[x(k)\right]}^2}}& \text{k}=1,2,\dots ,\text{N}\end{array}\left(1\right)$

Where, N, is the number of samples, x(k) is K-th sample of the signal, and RMS is the estimate of the energy.

Spectral Estimation – AR modeling

The AR model can be viewed as an all-pole, or infinite-impulse-response (IIR) filter whose current output, s_n, is a function of both the p most recent outputs, s_n-1, s_n-2,..., s_n-p, and the current input, e_pn:

$s_n={\tilde{s}}_n+e_{pn}=-{\displaystyle \sum _{i=1}^p{a}_{pi}\cdot s_{n-i}}+e_{pn}\left(2\right)$

This AR filter can be specified in the frequency domain by taking the z-transform in equation (a). If E(z) and S(z) are the z-transforms of e_p1, e_p2,...e_pN and s₁, s₂,...s_N respectively, then:

E(z) = A(z)·S(z), where $A(z)=1+{\displaystyle \sum _{i=1}^p{a}_{pi}\cdot z^{-i}}$

$A^{-1}(z)=\frac{S(z)}{E(z)}=\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$\left(1+{\displaystyle \sum _{i=1}^p{a}_{pi}\cdot z^{-i}}\right)$}\right.\left(3\right)$

A^-1(z) is the AR model's transfer function, usually denoted by H(z). Its frequency response, H(ω), is determined by evaluating H(z) along the unit circle in the z-plane, where z = e ^jωT for a sampling period, T. Furthermore, if E(z) is a white noise input sequence then its spectrum, E(ω), will be flat and the spectrum of the output sequence, S(ω) = H(ω)·E(ω), will be equal to H(ω) scaled by constant, E(ω) = E _p ·T. In practice, however, E(z) only approximates a white noise sequence and so S(ω) can only be an estimate. This estimate $\tilde{S}$(ω), is given by:

$\tilde{S}(\omega )=\raisebox{1ex}{$E_{p}T$}\!\left/ \!\raisebox{-1ex}{${\left|1+{\displaystyle \sum _{i=1}^p{a}_{pi}\cdot e^{-ij\omega T}}\right|}^2$}\right.\left(4\right)$

The assumption on which the AR modeling technique is based can now be rephrased in the frequency domain, where it is assumed that the flat spectrum of the white noise input sequence is "coloured" by the AR model to produce an output spectrum of the desire shape.