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What is a Stationary Random Process?

Iain Explains Signals, Systems, and Digital Comms

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Explains the concept of stationarity in random processes, using an example and diagrams.
* Note that I unfortunately forgot to mention that Stationarity also requires that the joint distribution of X(t1) and X(t2) is the same as the joint distribution of X(t1+Δ) and X(t2+Δ), for any t1, any t2, and for all Δ. In summary, this means that the way in which X(t1) and X(t2) are related, is the same as the way in which X(t1+Δ) and X(t2+Δ) are related. In other words, the level of dependency between two values that are spaced Δ apart, doesn't change over time. Or from yet another perspective, how independent they are from each other, doesn't change over time.
* If you would like to support me to make these videos, you can join the Channel Membership, by hitting the "Join" button below the video, and making a contribution to support the cost of a coffee a month. It would be very much appreciated.
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Опубликовано:

19 мар 2023

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Комментарии : 27

@payman_azari Год назад

thanks Professor, even if I have some prior knowledge, your videos help me to rewire my brain.

@iain_explains Год назад

That's great. I'm so glad they help.

@palisthashrestha8829 Год назад

Really appreciate how you show things visually!

@iain_explains Год назад

I'm glad you like the approach.

@andrus3125 Год назад

Thank you Professor. Waiting for more videos about statistical analysis of random processes

@iain_explains Год назад

I'm glad you like the videos. Are there any specific topics you'd like me to cover on Random Processes?

@mq6605 Год назад

@@iain_explains Please make a video about a system which is stationary in time average and a system which is stationary in ensemble average , and ergodicity.

@iain_explains Год назад

I've got a video coming up on this topic, in the next couple of weeks.

@mq6605 Год назад

@@iain_explains Thank you Professor.

@aniketpimparkar640 11 месяцев назад

Under rated gem 💎

@iain_explains 11 месяцев назад

I'm glad you like the channel.

@khalifi2100 Год назад

More generally, for a stationary process, the joint distribution of X(t1) and X(t2) is the same as the joint distribution of X(t1+Δ) and X(t2+Δ). In particular, if a process is stationary, then its analysis is usually simpler as the probabilistic properties do not change by time.

@iain_explains Год назад

Thanks for this. Yes, I forgot to mention the joint distributions. That's a pain. I was mostly thinking about the relationship between stationarity and ergodicity (which someone had asked me about). I'll add a note to the description below the video. Thanks again!

@seslocrit9365 Год назад

Could you do a video on how to construct the PDF? It makes intuitibe sense but I am having a hard time actually making it.

@iain_explains Год назад

I'm not sure what you mean by the phrase "how to construct". Have you seen my video: "What is a Probability Density Function (pdf)?" ru-vid.com/video/%D0%B2%D0%B8%D0%B4%D0%B5%D0%BE-jUFbY5u-DMs.html

@pltcmod4425 7 месяцев назад

Hello Professor, thanks for the video. I have a question. Is it sufficient to say that a random variable is stationary if it looks like white noise when plotted against time? Thanks in advance!

@iain_explains 6 месяцев назад

No, it's not sufficient to just look at the waveforms. Also, it might be that there is time-correlation between the samples, but the RP can still have the same probability distribution at all times (and hence it is stationary, but doesn't look like "white noise"). More details are in these videos: "Are Stationary Random Processes Always Ergodic?" ru-vid.com/video/%D0%B2%D0%B8%D0%B4%D0%B5%D0%BE-onxzu2xUQ4E.html and "What is Autocorrelation?" ru-vid.com/video/%D0%B2%D0%B8%D0%B4%D0%B5%D0%BE-hOvE8puBZK4.html

@user-ks8zu2nf8d 9 месяцев назад

Question: By saying that the PDFs of the two RV are the same that does that mean the variance and the mean are the same? Or does it mean that the PDFs are either both Rayleigh or both Gaussian?

@iain_explains 9 месяцев назад

It means they are the same. ... Since they are the same, then yes, they will have the same mean, the same variance, the same distribution, the same everything ... because they are the same.

@user-ks8zu2nf8d 9 месяцев назад

Thanks a lot@@iain_explains

@ImranMoezKhan Год назад

If I understood correctly, this means the process PDF does not depend on time? Perhaps "static" random process would have been be a more appropriate when the term was being coined :-).

@iain_explains Год назад

I'm not so sure about which term is better, ... and I wasn't even alive when the term was being coined ... 🤔😁

@tuongnguyen9391 11 месяцев назад

Does the fact that stationarity happen for short recording time play an important role in DSP ?

@iain_explains 11 месяцев назад

Sorry, I'm not sure what you're asking.

@tuongnguyen9391 11 месяцев назад

@@iain_explains I mean " do a lot of DSP algorithm can only work if stationary hold ?"

@iain_explains 11 месяцев назад

Most DSP algorithms assume stationarity of noise processes. But it depends on which DSP algorithms we're talking about, whether they assume stationarity of the `signal' component too.

@tuongnguyen9391 11 месяцев назад

@@iain_explains oh thank you there is so much subtlety