If I understand correctly, a signal is stationary if:

time domain: it's generated from the same distribution at each instant time.

frequency domain: its frequency content does not change in time.

My question is, are these both definition related? I mean, are they equal? If yes, how are they related? if you have some articles explaining it, it will be helpful. Cheers

  • $\begingroup$ Where do you find these definitions ? I just don't understand how a signal can change in time while its frequency content does not change. $\endgroup$ – AlexTP Jun 2 '17 at 13:51
  • $\begingroup$ You do not understand correctly and so it is difficult to find written material refuting your specific misunderstandings of the situation. $\endgroup$ – Dilip Sarwate Jun 2 '17 at 13:52
  • 2
    $\begingroup$ Usually when you talk about a stationary signal or process we're working in the time domain. Whatever results when going from the time domain to the frequency domain is simply from the nature of the signal. Depending on what frequency domain information you're looking at, going backwards from the frequency domain to the time domain is not a good idea. For example different time signals can have the same power spectral densities. Trying to classify the "stationariness" of one particular signal in both time and frequency domain individually doesn't really help you. $\endgroup$ – Envidia Jun 2 '17 at 14:48
  • $\begingroup$ @AlexTP Just like sinusoidal signal with a specific frequency. It changes in time, but the frequency content is constant. $\endgroup$ – abkoesdw Jun 2 '17 at 14:48
  • $\begingroup$ @AlexTP please check page 5, last paragraph: link $\endgroup$ – abkoesdw Jun 2 '17 at 14:52

Your definitions are not correct.

For a Strict Sense Stationary process (signal) the joint distribution of your process' value for all instants of time must be independent of time, in other words if x(t) were your process, P(x(t1),x(t2),x(t3),...) must be independent of time's origin. For a Wide Sense Stationary process the joint probability of process value up to 2 instant of time must be independent of time, in other word P(x(t1),x(t2)) must be independent of time's origin.

Now if you have a process which at least is wide sense stationary process then your process power spectrum doesn't change by time.

To estimate a process' power spectrum from a sample signal in addition to stationarity your process also have to be ergodic, then square magnitude of Fourier transform of your signal is an estimation of power spectrum which also doesn't change with time for long enough signal.

The classic reference is stochastic processes by A. Papoulis.


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