Stationarity
Characterization of a Stationary Random Signal :
Consider a random signal X defined by its distribution or probability function f(x) and assumed to be stationary. We can characterise this signal with the following statistical parameters:
The probability density (ddp) :
\({f}_{x}\left({x},{t}_{i}\right)={P}\left({X}{x}\right)=\int_{{a}}^{{b}}{{f}({x}){dx}}\) (9)
This implies that the integral of f(x) over the entire real set ℜ is 1.
The statistical average is :
\({m}_{X}({t})={E}\left({X}\left({t}\right)\right)=\int_{-\infty}^{+\infty}{{X}{f}_{X}}\left({x}\right){dx}\) (10)
The mean square value, or moment of order 2, is :
\({E}({X}^\mathbf{2}({t}))=\int_{-\infty}^{+\infty}{{X}^\mathbf{2}{f}_{X}}\left({x}\right){dx}\) (11)
The variance :
\({\sigma}_{x}^\mathbf{2}\left({t}\right)=\int_{-\infty}^{+\infty}{\left[{X}({t}))-{E}({X}({t})\right]^\mathbf{2}{f}_{X}}\left({x}\right){dx}={E}\left[({X}\left({t}\right)-{{m}_{X}({t}))}^\mathbf{2}\right]\) (12)
Standard deviation :
\({\sigma}_{x}=\sqrt{{E}\left[({X}\left({t}\right)-{{m}_{X}({t}))}^\mathbf{2}\right]}\) (13)
The statistical autocorrelation function: this function, denoted as \({R}_{x}\left({t}_\mathbf{1},{t}_\mathbf{2}\right)\) Rx (t1 , t2 ), provides an indication of the statistical relationship between the values of the random signal x measured at two different times, t1 and t2 :
\({R}_{x}\left({t}_\mathbf{1},{t}_\mathbf{2}\right)={E}\left[{X}\left({t}_\mathbf{1}\right){X}\left({t}_\mathbf{2}\right)\right]=\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}{{xy}{f}_{{X}_{\left({t}_\mathbf{1}\right)}{X}_{\left({t}_\mathbf{2}\right)}}}\left({x},{y}\right){dxdy}\) (14)