Random Processes - 04 - Mean and Autocorrelation Function Example

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The previous videos provided definitions of the mean and autocorrelation function of a random process.
In this video we work with the random process X(t) = Asin(wc*t + theta) where both A and theta are random variables. We compute the mean function and autocorrelation function of this random process.
We show that the mean function is zero, and the autocorrelation function is just a function of the time difference t1-t2. Thus, this random process is a wide-sense stationary (WSS) random process (which we'll formally define late).
If you enjoyed my videos please "Like", "Subscribe", and visit adampanagos.org to setup your member account to get access to downloadable slides, Matlab code, an exam archive with solutions, and exclusive members-only videos. Thanks for watching!

[@jimmysantadeo2031]

Anyone that can explain a concept in a clear way definitely understands it, and YES YOU DO UNDERSTAND! LEGEND!

[@AdamPanagos]

Thanks!

[@usmanskp78]

Couldn't get more simpler and easy to digest! Bravo!

[@AdamPanagos]

Thanks for the kind words. Make sure to check out my website adampanagos.org where I have a lot of other videos and resources available that you might find helpful. Thanks, Adam.

[@danieljamescarter565]

You made something complicated very simple to understand, a true engineer, what a boss!

[@AdamPanagos]

Daniel James Carter Thank you! Glad you liked the video.

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[@AdamPanagos]

Thanks for the kind words, I’m glad you enjoyed the video! Make sure to check out my website adampanagos.org for additional content (600+ videos) you might find helpful. Thanks much, Adam

[@karennakye]

You are the best you simplified this for me...I went through tonnes of videos before I found this.

[@AdamPanagos]

Glad I could help, thanks for watching. Make sure to check out my website adampanagos.org for additional content you might find helpful. Thanks, Adam

[@drewphillips8817]

Adam this is such a fantastic explanation. You have no idea how much this helped on my grad school final exam!

[@AdamPanagos]

Glad I could help, thanks for watching. Make sure to check out my website adampanagos.org for additional content (600+ videos) you might find helpful. Thanks, Adam

[@davidelias8569]

Thank you so much for these videos! Cannot begin to tell you how much these have helped me!

[@AdamPanagos]

Thanks for the kind words, I’m glad you enjoyed the video! Make sure to check out my website adampanagos.org for additional content (600+ videos) you might find helpful. Thanks much, Adam

[@GiI11]

Great job man, this is incredibly clear. Now I see how autocorrelation is related to expectation values ;)

[@AdamPanagos]

+Sebastian Gil Glad it helped, thanks for the nice feedback. Adam

[@user-wt7ut4xj5r]

this was very helpful! thanks for your help

[@manjitsingh1124]

Thank you so much for the lesson Adam 👍

[@AdamPanagos]

You're welcome, thanks for watching. Adam

[@captainsquirtle4961]

Thank you so much! Searched for a good example of such calculations for quite a long time, could have saved me some hours if i found it earlier :D

[@AdamPanagos]

You're very welcome, thanks for watching. Make sure to check out my website adampanagos.org for additional content (600+ videos) you might find helpful. Thanks, Adam.

[@sibusisomtiyane6913]

Well explained thank you.

[@neelabhtiwari9914]

Thanks a lot for this video. Can you also give a similar example on finding the auto-covariance function? Both of a general stochastic process and of a stationary increment schotastic process. Thanks.

[@Tschaegger79]

THX! Good explanation, saw the same problem elsewhere and it was calculated directly with the integral for the expactation. My question is, would it be a single or a double integral if you would write the integral for the expecation? Becaue X(t1) and X(t1) or two random variables.

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Amazing! So crisp!

[@AdamPanagos]

Thanks!

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[@AdamPanagos]

+Idroj17 Glad to hear that, thanks for watching!

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You are a genius and We need more of your examples

[@AdamPanagos]

Thank you for the kind words, I appreciate you watching. Make sure to check out my website adampanagos.org for additional content (600+ videos) you might find helpful. Thanks, Adam

[@atogh]

@@AdamPanagos Done!!

[@tarunpower2516]

good job.this was very helpful.

[@AdamPanagos]

Glad to help, thanks!

[@liverpoolshivank]

Thanks a lot, this explains it really well.

[@AdamPanagos]

+Shivank Sharma Great, glad to have helped!

[@MLDawn]

In 5:31, isn't that the formula for Covariance? I beliebe it needs to be divided by variance, if we want to get the correlation. Am I missing something?

[@ramatbobby1815]

what if the value of x(t)=k where k is random variable uniformly distributed in range (-1,1) ??? is it wide-sense stationary? or it is ergodic? if it is wide-sense stationary, what it is power?? appreciate for your help sir!!!

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Hi Adam. Thanks for a very helpful video. I was wondering what tablet and software did you use? For a long time I've been wanting to buy a thinkpad tablet to write lecture notes on for easy navigation instead of writing on a paper and then later losing them. The device and program that you are using for hand writing seems very neat.

[@AdamPanagos]

Gulf Shores I use an iPad app called Doceri (www.doceri.com) for most of my videos. This app lets you record all you handwriting ahead of time and use "breakpoints" to pause as needed. Once all the writing is down you can "play" the handwriting back while recording audio over it. I find this works much better than trying to write and talk at the same time. I'd definitely recommend checking out the app, I've found it very useful. Hope that helps!

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Hi! Do you know a good book of exercises/problems about stochastic processes with solutions? I would like to practice before my exam Thanks in advance

[@3xpo]

Thanks for all your videos

[@AdamPanagos]

You're welcome, I'm glad you like them. Thanks.

[@charisrevo7869]

Thanks sir, but what if I put a constant A, Is it still independent?

[@N1NO5]

Very Helpful !

[@AdamPanagos]

Glad I could help. Thanks for watching!

[@trioxyify]

The trig-identity used above has a little mistake. sin(X)sin(Y) = (1/2) [ cos (X - Y) - cos (X + Y) ]

[@AdamPanagos]

Excellent catch! It ended up "not mattering" because that term eventually went to zero, but you're correct, it should have been a minus sign instead of a plus sign for the second term. I'll add an annotation to the video to correct it. Thanks for noting this!

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Yes, that's right

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Yess

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[@AdamPanagos]

Glad to hear, thanks for the nice comment.

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[@AdamPanagos]

Thanks for the kind words and thanks for watching!

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You are the best!!!

[@AdamPanagos]

Thanks for the kind words, I’m glad you enjoyed the video! Make sure to check out my website adampanagos.org for additional content (600+ videos) you might find helpful. Thanks much, Adam

[@muhammadnoman2635]

Thanks, dear It is really helpful

[@AdamPanagos]

Glad I could help, thanks for watching! Adam

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Superman without a cape!

[@AdamPanagos]

Thanks for the kind words, I’m glad you enjoyed the video! Make sure to check out my website adampanagos.org for additional content (600+ videos) you might find helpful. Thanks much, Adam

[@marianasoder7828]

Salvou minha prova! Obrigada

[@AdamPanagos]

Glad I could help, thanks for watching. Make sure to check out my website adampanagos.org for additional content (600+ videos) you might find helpful. Thanks, Adam

[@user-zv4qq7fv9f]

clean and clear! it very helped me thank u go a head.

[@AdamPanagos]

Thanks for the kind words, I’m glad you enjoyed the video! Make sure to check out my website adampanagos.org for additional content (600+ videos) you might find helpful. Thanks much, Adam

[@mohamedimamahmed344]

this is sooo helpfull thanks

[@AdamPanagos]

Glad I could help. Thanks for watching!

[@aasthasinghmall]

thankyou .. it was helpful :)

[@AdamPanagos]

+AASTHA SINGH Great, glad it helped!

[@Jarrod_C]

Hey I am taking a Random Processes graduate course, and was wondering where I should begin on your channel and other channels. Thanks! Further, I have no signals background, so do I need to start from the very beginning or can I jump right into the concept of random processes?

[@AdamPanagos]

I have videos for a complete graduate-level random processes course on my website here: www.adampanagos.org/rp The site also has a large amount of signals-and-systems material which would probably be best to start with as well. Hope that helps, Adam

[@rushalipandit7712]

Thank you sir 🙏

[@AdamPanagos]

You're very welcome, thanks for watching. Make sure to check out my website adampanagos.org for additional content (600+ videos) you might find helpful. Thanks, Adam.

[@doesdu7728]

good explanation sir....good work

[@AdamPanagos]

Thank you, glad you like it.

[@syedalaiyba5603]

Are you making currently more vedioz on stochastic processes

[@AdamPanagos]

Yes, I know have a full course of these videos on my website: www.adampanagos.org/rp

[@43SunSon]

good job, clean and clear.

[@AdamPanagos]

Thanks!

[@43SunSon]

Adam, do you have some videos about application of Autocorrelation? I mean, the example for real world. Thanks.

[@namonarayanmeena8361]

Thanks a lot

[@AdamPanagos]

You’re welcome, thanks for watching. If you found the video useful make sure to check out my website adampanagos.org where I have a ton of other resources available. Thanks, Adam.

[@aneeshathomas8049]

Thank you....

[@AdamPanagos]

You're welcome, thanks for watching!

[@danielrussell7877]

is it ergodic?

[@mebrateshewa8221]

thanks

[@AsmaaSamir]

Thanks alot 😀

[@AdamPanagos]

You're welcome, glad I could help.

[@thesoftwareguy2183]

how did you take the PDF of theta as 1/2pi ??

[@AdamPanagos]

Theta is a uniform random variable on [-pi, pi], so it is a constant value of 1/2pi on that interval. That's just the definition of a uniform random variable. Hope that helps, Adam

[@thesoftwareguy2183]

@@AdamPanagos Oo Yeah , Got it . THANKS!!

[@booked_society]

How did 1/2 pie come

[@prasadkulkarni5442]

good

[@jchen2932]

dose a autocorrelation function of gamma process exist？

[@amyshai5177]

life saver

[@AdamPanagos]

Glad I could help, thanks for watching. Make sure to check out my website adampanagos.org for additional content (600+ videos) you might find helpful. Thanks, Adam

[@desmondachunji4146]

good video, but if A is a random variable like you said y is it's expected value not computed

[@AdamPanagos]

It was computed. The mean of A was provided in the problem statement as muA.

[@vaishnav4035]

Why we use a different mathematical equation to find correlation when computing autocorrelation, why not normalised covarience?

[@AdamPanagos]

I'm not sure if I understand your comment regarding the notation. The notation used here for the autocorrelation function is typical of all random process texts that I'm familiar with.

[@durgar9755]

Why are we analysing random variables using these parameters

[@AdamPanagos]

All random variables have some statistical characteristics associated with them (i.e. some mean, some variance, etc.). The specific values chosen for this video weren't chosen for any particular reason, just to provide a specific example. Thanks for watching. Adam

[@siddragon5]

hey! autocorrelation and correlation mean the same thing right?

[@AdamPanagos]

Essentially, yes. I like the term autocorrelation because I think it's a little more precise. It means correlating with oneself, hence the "auto" prefix. Crosscorrelation means correlating with something else.

[@siddragon5]

+Adam Panagos ohh! Understood! Thanx alot sir! :)

[@siddragon5]

+Adam Panagos ohh! Understood! Thanx alot sir! :)

[@lupgavgavu]

Hello dear, First of all thanks for these grat jobs. In 3.30, integration outcome (cos(wct+pi)-cos(wct-pi)) is determined -2. But should it be 0 ? İf i wrong, sorry. Thank you again.

[@AdamPanagos]

No, it's definitely zero. The integration ends up equaling a term times sin(pi), and sin(pi) is zero. Hope that helps, Adam

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[@AdamPanagos]

Unfortunately, I've not studied that field in much detail myself so I don't think I'd be much help. Good luck though!