Chapitre 4

Stationary signals (ergodic or non-ergodic signals) exhibit an average value that is independent of time. They are considered ergodic if the statistical average at a certain point in time is the same as taking the average of several samples or a long enough time average on a single instance of these tests.

• A stationary random signal is ergodic if,

  \({E}\left[{X}\left({t}\right)\right]=\int_{-\infty}^{+\infty}{{X}({t}){f}_{X}}\left({x},{t}\right){dx}\)

  \({\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ ={\mu}}_{x}=\lim\limits_{T \to \infty }{\frac{\mathbf{1}}{{T}}}\int_{-\frac{{T}}{\mathbf{2}}}^{+\frac{{T}}{\mathbf{2}}}{X}\left({t},{\omega}\right){dt}=\bar{{X}\left({t},{\omega}\right)}\) (15)

This ergodic hypothesis is, however, difficult to verify. It is frequently admitted that the usual random processes are ergodic.

• Time average :

  \(\bar{{X}\left({t}\right)}=\lim\limits_{T \to \infty }{\frac{\mathbf{1}}{{T}}}\int_{-\frac{{T}}{\mathbf{2}}}^{+\frac{{T}}{\mathbf{2}}}{X}\left({t}\right){dt}\equiv{m}_{x}\) (16)

• Signal strength is :

  \({P}=\bar{{X}^\mathbf{2}\left({t}\right)}=\lim\limits_{T \to \infty }{\frac{\mathbf{1}}{{T}}}\int_{-\frac{{T}}{\mathbf{2}}}^{+\frac{{T}}{\mathbf{2}}}{{X}^\mathbf{2}\left({t}\right){dt}\equiv{E}\left[{X}^\mathbf{2}\right]}\) (17)

• Power of variations from the mean :

  \({P}_{{m}_{x}}=\lim\limits_{T \to \infty }{\frac{\mathbf{1}}{{T}}}\int_{-\frac{{T}}{\mathbf{2}}}^{+\frac{{T}}{\mathbf{2}}}{\left[{X}\left({t}\right)-\bar{{X}\left({t}\right)}\right]^\mathbf{2}.{dt}}={P}-\bar{{X}^\mathbf{2}\left({t}\right)}\equiv{\sigma}_{x}^\mathbf{2}\) (18)

• The temporal autocorrelation function \({C}_{{ss}}\left({\tau}\right)\) :

  \(C_{ss}\left(\tau\right)=\lim\limits_{T \to \infty }{\frac{1}{T}}\int_{-\frac{T}{2}}^{+\frac{T}{2}}{X\left(t\right).X\left(t-\tau\right).dt}\) (19)