About stationarity Before talking about non-stationarity, which is a 'non-property', we must define what we call stationarity. A deterministic signal is said to be stationary if it can be written as a discrete sum of sinusoids : $$\begin{eqnarray*} x(t)&=&\sum_{k \in {\cal N}} A_k \cos{[2\pi \nu_k t + \Phi_k]}... ...k \exp{[j(2\pi \nu_k t + \Phi_k)]} \mbox{for a complex signal} \end{eqnarray*}$$ i.e. as a sum of elements which have constant instantaneous amplitude and instantaneous frequency. In the random case, a signal $x(t)$ is said to be wide-sense stationary (or stationary up to the second order) if its expectation is independent of time and its autocorrelation function $E[x(t_1)x^*(t_2)]$ depends only on the time difference $t_2-t_1$. We can then show that the associated analytic signal has constant instantaneous amplitude and frequency expectations, which can be connected to the deterministic case. So a signal is said to be non-stationary if one of these fundamental assumptions is no longer valid. For example, a finite duration signal, and in particular a transient signal (for which the length is short compared to the observation duration), is non-stationary.
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