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There are nice technical definitions in textbooks and wikipedia, but I'm having a hard time understanding what differentiates stationary and non-stationary signals in practice?

Which of the following discrete signals are stationary? why?:

  1. white noise - YES (according to every possible information found)
  2. colored noise - YES (according to Colored noises: Stationary or non-stationary? )
  3. chirp (sinus with changing frequency) - ?
  4. sinus - ?
  5. sum of multiple sinuses with different periods and amplitudes - ?
  6. ECG, EEG, PPT and similar - ?
  7. Chaotic system output (mackey-glass, logistic map) - ?
  8. Record of outdoors temperature - ?
  9. Record of forex market currency pair development - ?

Thank you.

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  • $\begingroup$ Is this a homework question? $\endgroup$ – A_A Apr 12 '16 at 12:23
  • $\begingroup$ @A_A No. I am preparing the presentation of my results and I want to be prepared for tricky questions. So this question come out of my thoughts. $\endgroup$ – matousc Apr 12 '16 at 12:37
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    $\begingroup$ Good question, by the way! :-) $\endgroup$ – Peter K. Apr 12 '16 at 16:26
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There is no stationary signal. Stationary and non-stationary are characterisations of the process that generated the signal.

A signal is an observation. A recording of something that has happened. A recording of a series of events as a result of some process. If the properties of the process that generates the events DOES NOT change in time, then the process is stationary.

We know what a signal $x(n)$ is, it is a collection of events (measurements) at different time instances ($n$). But how can we describe the process that generated it?

One way of capturing the properties of a process is to obtain the probability distribution of the events it describes. Practically, this could look like a histogram but that's not entirely useful here because it only provides information on each event as if it was unrelated to its neighbour events. Another type of "histogram" is one where we could fix an event and ask what is the probability that the other events happen GIVEN another event has already taken place. So, if we were to capture this "monster histogram" that describes the probability of transition from any possible event to any other possible event, we would be able to describe any process.

Furthermore, if we were to obtain this at two different time instances and the event-to-event probabilities did not seem to change then that process would be called a stationary process. (Absolute knowledge of the characteristics of a process in nature is rarely assumed of course).

Having said this, let's look at the examples:

  1. White Noise:

    • White noise is stationary because any signal value (event) is equally probable to happen given any other signal value (another event) at any two time instances no matter how far apart they are.
  2. Coloured Noise:

    • What is coloured noise? It is essentially white-noise with some additional constraints. The constraints mean that the event-to-event probabilities are now not equal BUT this doesn't mean that they are allowed to change with time. So, Pink noise is filtered white noise whose frequency spectrum decreases following a specific relationship. This means that pink noise has more low frequencies which in turn means that any two neighbouring events would have higher probabilities of occurring but that would not hold for any two events (as it was in the case of white noise). Fine, but if we were to obtain these event-to-event probabilities at two different time instances and they did not seem to change, then the process that generated the signals would be stationary.
  3. Chirp:

    • Non stationary, because the event-to-event probabilities change with time. Here is a relatively easy way to visualise this: Consider a sampled version of the lowest frequency sinusoid at some sampling frequency. This has some event-to-event probabilities. For example, you can't really go from -1 to 1, if you are at -1 then the next probable value is much more likely to be closer to -0.9 depending of course on the sampling frequency. But, actually, to generate the higher frequencies you can resample this low frequency sinusoid. All you have to do for the low frequency to change pitch is to "play it faster". AHA! THEREFORE, YES! You can actually move from -1 to 1 in one sample, provided that the sinusoid is resampled really really fast. THEREFORE!!! The event-to-event probabilities CHANGE WITH TIME!, we have by passed so many different values and went from -1 to 1 in this extreme case....So, this is a non-stationary process.
  4. Sinus(oid)

    • Stationary...Self-explanatory, given #3
  5. Sum of multiple sinuses with different periods and amplitudes

    • Self explanatory given #1, #2,#3 and #4. If the periods and amplitudes of the components do not change in time, then the constraints between the samples do not change in time, therefore the process will end up stationary.
  6. ECG, EEG, PPT and similar

    • I am not really sure what PPT is but ECG and EEG are prime examples of non-stationary signals. Why? The ECG represents the electrical activity of the heart. The heart has its own oscillator which is modulated by signals from the brain AT EVERY HEARTBEAT! Therefore, since the process changes with time (i.e. the way that the heart beats changes at each heart beat) then it is considered non-stationary. The same applies for the EEG. The EEG represents a sum of localised electrical activity of neurons in the brain. The brain cannot be considered stationary in time since a human being performs different activities. Conversely, if we were to fix the observation window we could claim some form of stationarity. For example, in neuroscience, you can say that 30 subjects were instructed to stay at rest with their eyes closed while EEG recordings were obtained for 30 seconds and then say that FOR THOSE SPECIFIC 30 SEC AND CONDITION (rest, eyes closed) THE BRAIN (as a process) IS ASSUMED TO BE STATIONARY.
  7. Chaotic system output.

    • Similar to #6, chaotic systems could be considered stationary over brief periods of time but that's not general.
  8. Temperature recordings:

    • Similar to #6 and #7. Weather is a prime example of a chaotic process, it cannot be considered stationary for too long.
  9. Financial indicators:

    • Similar to #6,#7,#8,#9. In general cannot be considered stationary.

A useful concept to keep in mind when talking about practical situations is ergodicity. Also, there is something that eventually creeps up here and that is the scale of observation. Look too close and it's not stationary, look from very far away and everything is stationary. The scale of observation is context dependent. For more information and a large number of illustrating examples as far as the chaotic systems are concenred, I would recommend this book and specifically chapters 1,6,7,10,12 and 13 which are really central on stationarity and periodicity.

Hope this helps.

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  • $\begingroup$ Great answer, thanks. But Still I have one question. You said: "Therefore, since the process changes with time (i.e. the way that the heart beats changes at each heart beat) then it is considered stationary" about ECG. Why is it stationary when it is changes in time? $\endgroup$ – matousc Apr 13 '16 at 14:17
  • $\begingroup$ Thank you, that was a typographic mistake which I have corrected. Since we are on this anyway, can you please tell me what PPT stands for? $\endgroup$ – A_A Apr 13 '16 at 14:46
  • $\begingroup$ It is Plethysmograph. The PPT shortcut is maybe not really common. Next time I will use the full name. $\endgroup$ – matousc Apr 13 '16 at 15:23
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    $\begingroup$ Most of the details in this answer are incorrect. White noise is independent regardless of how close the samples are, not how far apart. For stationary colored noise, the "event to event" probabilities must be the same (whatever that means); it is the dependence between $X(t_1)$ and $X(t_2)$ that must be a function only of $t_1-t_2$ etc. $\endgroup$ – Dilip Sarwate Apr 13 '16 at 15:42
  • $\begingroup$ @DilipSarwate: Question itself is quite tricky. By saying "Most of the details in this answer are incorrect", and providing only one example, you are somehow making the whole answer sound incorrect. I don't fully agree with that. Would you mind writing a separate answer, which in your opinion, is correct? After that I shall remove your comment. It's up to the OP to decide which answer should be accepted. $\endgroup$ – jojek Apr 13 '16 at 19:20
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@A_A's good answer misses one point: stationarity or nonstationarity are generally only applied to stochastic signals, not deterministic signals.

In general, when statistical tests are applied for stationarity or nonstationarity, the deterministic component must be removed first.

Hence, in my view, numbers 3, 4, and 5 are non-sensical questions because they contain no stochastic component and, therefore, cannot be considered either stationary or nonstationary.

Item #3, if the sinusoid has stationary noise added to it, could be considered a cyclostationary process, as the mean of the process changes (though generally with cyclostationary processes it is assumed the variance also changes with time).

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