Understanding the Z-Transform

Рет қаралды 94,704

Жыл бұрын

This intuitive introduction shows the mathematics behind the Z-transform and compares it to its similar cousin, the discrete-time Fourier transform. Mathematically, the Z-transform is straightforward-it’s just a bunch of multiplications and additions, and you’ll learn how to solve an equation for a few different signals. However, understanding how to solve a Z-transform equation isn’t as important as understanding why the math is the way it is. Therefore, the majority of this tech talk discusses what you are actually doing when you take the Z-transform of a signal.
Want to see all of the references in a nice, organized list? Check out this journey on Resourcium: bit.ly/3DeS8DX
Check out these links to learn more about Z-transform:
- DTFT visualization MATLAB App: bit.ly/DTFT_MATLAB_App
- MATLAB Documentation: Discrete Fourier Transform: bit.ly/3NfFtGV
- MATLAB Function, ztrans: bit.ly/40MSnjy
- Creating Discrete-Time Models: bit.ly/438xsZZ
--------------------------------------------------------------------------------------------------------
Get a free product trial: goo.gl/ZHFb5u
Learn more about MATLAB: goo.gl/8QV7ZZ
Learn more about Simulink: goo.gl/nqnbLe
See what's new in MATLAB and Simulink: goo.gl/pgGtod
© 2023 The MathWorks, Inc. MATLAB and Simulink are registered trademarks of The MathWorks, Inc.
See www.mathworks.com/trademarks for a list of additional trademarks. Other product or brand names may be trademarks or registered trademarks of their respective holders.

Пікірлер: 116

@BrianBDouglas 11 ай бұрын

If you want to play around with the DTFT MATLAB App from this video, you can find it here: github.com/aerojunkie/control-tools/tree/master/DTFT%20MATLAB%20App

@fahimrahman5196 9 ай бұрын

Thanks Brian

@ShahadetHosenPolash 3 ай бұрын

How you design those matlab app? How can i learn this? any tutorial video?

@nurahmedomar Жыл бұрын

In university, we were taught how to do the DFT and Z-transform, but never taught why. And this video explains them in crystal clear. Great Job!

@HansScharler 10 ай бұрын

Thanks for the feedback!

@harrytsai0420 Жыл бұрын

OMG! I've been waiting for the series of discrete-time control for nearly 5 years!

@BrianBDouglas Жыл бұрын

Hopefully, it's worth the wait!

@xptransformation3564 Жыл бұрын

@@BrianBDouglashow did you get such a perfect understanding for all this?

@erickappel4120 Жыл бұрын

I was blown away by the clearness of the explanation! Looking forward to more!

@diegoperezromero6610 Жыл бұрын

This is by far the best video about the dtft and z transform i've ever seen. Very good and very detailed explanation.

@erickappel4120 11 ай бұрын

Excelllent explanation! It is the first time someone made the DFT's relation to finite length input clear to me! Thankyou!!!

@MarkNewmanEducation Жыл бұрын

Amazing explanation. Thanks so much for explaining things in a more intuitive way.

@Michael-ze6oz 8 ай бұрын

Your explanations and teachings are awesome. This opens my eyes to so many things, and in just 1 video!. Brain, you have been blessed with an amazing gift. Thank you for sharing this with us.

@jmanius1 Жыл бұрын

Brian is basically my mentor at this point

@BrianBDouglas Жыл бұрын

🤓 aww thanks!

@jmanius1 Жыл бұрын

@@BrianBDouglas You responding has literally made my day. Thank you for your awesome videos!

@martinsanchez-hw4fi Жыл бұрын

Is there a list of the Matlab videos made by Brian?

@MATLAB 5 ай бұрын

Hi @martinsanchez-hw4fi, You can find all of our Tech Talks on this page: bit.ly/MATLAB-Tech-Talks

@wellid2087 8 ай бұрын

One of the best introductions I have ever watched

@user-tj8gy5ew8c Ай бұрын

bless my soul that i landed on this video. the use of complex domain in dft was a "magic" for me. today i stand near to the oz who does the "magic"

@DRAMBgo Жыл бұрын

Crazy explanation Brian, thank you!

@terrycrews2281 Жыл бұрын

been watching ur vids since 2014 ty so much for this ,brian

@teebee3881 Жыл бұрын

This helps so much, please make more of these

@Adhithya2003 10 ай бұрын

The most intutive and lucid explanation of DTFT and Z transform I have ever seen 💖

@BrianBDouglas 10 ай бұрын

I'm glad you liked it!

@grxoxl 5 ай бұрын

Honestly, the best explanation of the DFT I've ever seen)

@user-pn6qx1vu3q Жыл бұрын

please make more of this ! it really really help !!!!

@PrestonRogersJPL 7 ай бұрын

This was the best explanation on DTFT I have ever seen! Bravo!

@BrianBDouglas 7 ай бұрын

I appreciate that!

@emiliomartineziii2980 6 ай бұрын

This was truly amazing! This guy has MIT level brainpower. Like woah, his ability to explain and efficiently articulate this stuff is unreal. This channel will be my saving grace in dsp for sure! Already subscribed too. He's a legend for sure!

@BrianBDouglas 5 ай бұрын

🤓 haha thanks so much!

@yinkaleoogundiran1171 5 ай бұрын

Or MIT has Brian Douglas’ level brainpower? 😊

@hamedmajidian4451 Жыл бұрын

You are the best looking forward to watching more and more!

@MrRaghunar 10 ай бұрын

Excellent explanation, great job!

@222_Official 5 ай бұрын

Amazing and very intuitive video, thanks!

@vincentjupiterayuste2358 Жыл бұрын

Thank you Sir Brian

@Pedritox0953 Жыл бұрын

Great video!

@mixguru9669 Жыл бұрын

My control systems professor should watch this video.

@sukursukur3617 Жыл бұрын

I am very sure he didnt learn the background of the concept. That is why he cant convey its knowledge properly.

@Amine-gz7gq Жыл бұрын

@@sukursukur3617 KZfaq FTW

@giantbee9763 3 ай бұрын

Brian is amazing! This video is extremely clear :) One note on time delay [n-1] implying z^-1, it wasn't clear to me at first how the first example with a pulse function translated to all general cases, but it later occured to me that y^0, which in a delayed sequence matched z^-1, and every term afterward matches the original function, so hence it is the case for the general case. Such that time delay [n-2] does not imply z^-2, and so on. If I have understood correctly.

@clarklau1292 4 ай бұрын

I learned about DFT and z-transform many years ago, but it wasn't until I watched this video that I truly understood what the 'e' and 'z' in these two transformations actually do.

@ernstuzhansky Жыл бұрын

Many thanks Brian!

@Volticymo 8 ай бұрын

Can’t wait for z domain video

@j.fkamaldeen Жыл бұрын

Thanks alot Brian 🎉

@franciskimonge1851 11 ай бұрын

my guy just made fourier transforms, laplace transorm and z transform all make sense , thank you

@BrianBDouglas 11 ай бұрын

Good to hear! Thanks!

@BingtheLizard 7 ай бұрын

I've been able to solve a Fourier Transform in analytical form back at uni, without understanding what it was for. I've been able to implement a Discrete Fourier Transform algorithm in code to do simple frequency analysis of a signal, without understanding why the algorithm was the way it is. With your explanation, I think I finally get the "why" of the Discrete Fourier Transform, the way it uses correlation. Thanks for this explanation; I feel like one more thing is clicking into place.

@BrianBDouglas 7 ай бұрын

Glad it helped!

@em2129 3 ай бұрын

You are a wizard. Thank you!

@philipprottweiler2924 Жыл бұрын

Awesome explanation! When will the video on the Z-Domain be published? Really looking forward to it

@BrianBDouglas 11 ай бұрын

I hope before the end of the year. I have 4-5 more videos in my queue before it though :(

@Eta_Carinae__ Жыл бұрын

We know that we can use FT on account of e^{i /omega n} forming an orthogonal basis for all n. I'm wondering, since z^{-n} spans all complex exponentials (assuming no limit like n>0), if there are any non-zero inner products between differently-based exponential functions. I mean, just by definition you're sweeping over scalar multiples of a basis, making all elements (r e^{i /omega})^{-n} LD on (e^{i /omega})^{-n}, right?

@user-iu7qg9lo7y Жыл бұрын

this is teaching at its finest

@BalajiSankar Жыл бұрын

Thank you.

@Neuromante73 Жыл бұрын

Hi Brian, since I am studying how to develop MATLAB apps, would it be possible for you to share the app that you have used during the discussion on the DTFT? Thanks

@Amine-gz7gq 2 ай бұрын

he put the github link in the description

@braedenlarson9122 5 ай бұрын

So cool!

@joez9162 Жыл бұрын

@Brian I’d love to see you do an explanation of ADRC controllers!

@heitorabreu3045 4 ай бұрын

love you Brian

@user-pn6qx1vu3q Жыл бұрын

I never understand why we use complex number to do the transform, until now !! TO CONSIDER THE PHASE !!

@user-yy5vg3mf6m 2 ай бұрын

Common man I had math exam yesterday, where were you 😢

@franciscorainero8673 Жыл бұрын

Is the shown Matlab application available on any link? (The one that calculates the FFT in min 7:50)

@BrianBDouglas 11 ай бұрын

I just put it on GitHub. Check the pinned comment above!

@blessingshenjere484 2 ай бұрын

Like how he explains. Like a documentary

@user-xq6dq7xe5e 9 ай бұрын

is there any open source on the internet about detail on z domain

@martinsanchez-hw4fi Жыл бұрын

Thanks for te awesome content. Where can I find the list with Brian videos for this Channel?

@BrianBDouglas 11 ай бұрын

I organize all of my MATLAB videos here: engineeringmedia.com/videos

@gradientenfeld Жыл бұрын

Is this tool you showed for the DTFT somewhere available?

@BrianBDouglas 11 ай бұрын

I just put it on GitHub. Check the pinned comment above!

@5vart5ol 9 ай бұрын

I love this, Im only 9 min in and it is all so clear to me now! Thank you. (Unless the next 10 min gives me dementia).

@BrianBDouglas 9 ай бұрын

Did it stick? :)

@5vart5ol 9 ай бұрын

It has indeed. Thank you.

@LokeKS Ай бұрын

brilliant

@khanhtruong3254 10 ай бұрын

Hi Brian, thanks for great explanation. At your DTFT demo 10:10, your input data signal clearly has magnitude of 1, so you choose the probing signals also have magnitude of 1 so it fits perfectly in your "dot product". If the input data signal is a combination of many frequencies with different magnitudes, it seems that we still keep the magnitude of probing signals (sine and cosine) as 1. I wonder why that still works? Shouldn't we adjust magnitude of probing signals according to the input signal?

@BrianBDouglas 10 ай бұрын

I think I understand your question, but I might be off a bit. Let me give it a try. If a particular frequency in the input signal had an amplitude that wasn't 1 there would still be correlation between it and the probing signal with an amplitude of 1. Since we're multiplying the two signals, having a larger amplitude in one is just like multiplying the product of the two signals by a gain. So that frequency would have a larger value in the lower plot. Which is exactly what you want because you want to see "how much" of each frequency there is in the original signal. For example, if the signal is 5*sin(t) and the probing signal is sin(t), then the product of the two is 5*sin^2(t) and so the summation would be 5 times larger. Was that your question and did my answer make sense?

@binhnguyenquoc3249 10 ай бұрын

This is how I see it, lets assume the input signal is a sum of many sinusoidal signals like x1.sin(w1.t) + x2.sin(w2.t) + x3.sin(w3.t) + .... When you have infinite data point, you can prove mathematically that with each probing signal, ONLY the exact probing signal that have the same frequency as one of the input's frequency components would have a NON-ZERO dot product (you can intuitively understand the reason why this happen by taking the integration of the product of two sinusoidal signals with different frequencies, the value would be 0), so the value of x1, x2, x3 or any amplitude of the input's frequency component is not that important (except when you need to know the amplitude of the component frequency) since the output plot would only have pulses at those exact frequencies (when you have infinite data point).

@joelevi9823 2 ай бұрын

I can't find the link to the video about the Z domain.. will be happy to get it

@manfredbogner9799 7 ай бұрын

Very good

@BrianBDouglas 7 ай бұрын

Thanks!

@borisbvt5473 10 ай бұрын

So, is z-transform kind of discreet Laplace Transform?

@BrianBDouglas 9 ай бұрын

Yep!

@rob9756 Жыл бұрын

Is z transform discrete Laplace transform?

@matwiz20xx Жыл бұрын

Yes, that's right. This is why you have to see what is the application for which you are going to choose certain transform. For instance, z-transform is used in digital filter design, whereas the Laplace transform is mainly used in applications such as AC circuit analysis, and so on.

@martinsanchez-hw4fi Жыл бұрын

Why an impulse would be integrated to a constant value of 1?

@PankajSingh-dc2qp 3 ай бұрын

The input signal at @ 1:31 is delta[n] and not u[n].

@tim110-handle 7 ай бұрын

when will the z domain video come out?

@BrianBDouglas 6 ай бұрын

I was hoping by now! But the video got pushed for a bit to get other work done. It will be here one day though. Sorry!

@briancoon308 5 ай бұрын

Did that z domain video ever get published?

@BrianBDouglas 5 ай бұрын

I'm almost done with it! I had to change what I'm going to cover in it because I found a video by Youngmoo Kim that covers pretty much what I wanted to and it's brilliantly done. I'm going to call it out in my video and cover a few different things now. I'd put the link but my comment will be removed but search KZfaq for "Applied DSP No. 9: The z-Domain and Parametric Filter Design"

@pijnappel03 6 ай бұрын

DTFT: 5:28

@kenakackrmn3898 2 ай бұрын

5:25 sampai 12:09

@PankajSingh-dc2qp 3 ай бұрын

There is no integrator in discrete time. Discrete time has "accumulator".

@eduardojreis 6 ай бұрын

16:50 - Where is the Z-Domain video?

@BrianBDouglas 6 ай бұрын

I'm just starting it now ☺ It took a while to get to it. Any particular questions you'd like answered in it while you're thinking about it? Thanks!

@eduardojreis 6 ай бұрын

@@BrianBDouglas Thank you so much for the feedback. Very much appreciated you asking! Yes! I do have questions that, if possible, I'd love to see them answers. 1) Can you give a couple more examples of how to go from the zero+poles representation to the filter equation? 2) Can you explain butter filters design and their relation to the Z-transform? What does the butter function does? 3) Can you explain `sos` for filter design? Why `sos` is more stable than the `butter` function in some cases? These are some topics I came across when first watching this video. I would find very helpful to have them addressed, but if you consider a bit out of scope for the next video you're planning that is fine. I looking forward to it anyways, your videos are great and I've learned a lot! Thank you for all the effort and quality put into them!

@reviewchan9806 3 ай бұрын

If the components have both exponential and periodic signals, wouldn't that just the the Laplace transform?

@BrianBDouglas 3 ай бұрын

The Laplace transform is for continuous signals, the Z-transform is the discrete equivalent.

@hamedhojatian3539 8 ай бұрын

Could it BE any more simpler to explain DFT?

@user-xq6dq7xe5e 9 ай бұрын

when the video on the z domain will be made and published

@BrianBDouglas 9 ай бұрын

I don't know, it keeps getting pushed back :( But I'll get it out eventually.

@KingBhoomie Жыл бұрын

1:27 Input u[n] generally refers to unit step function. And Unit impulse is represented by δ[n]

@SRIGHT0 2 ай бұрын

Halo saa disuru bu guru njelasin

@Amine-gz7gq Жыл бұрын

Thank you very much Brian. I love you man (I'm not gay 😂). I have a master degree in CS and system control but I never really understood system control and that's why when I have free time I enjoy watching your videos. software engineering sucks, I spend my day refactoring complex, buggy programs written by idiots and unconscious people. I'd like to do something more meaningful and intellectually stimulating like systems controls. Unfortunately, I wasn't taught the latter and maths very well, and thanks to some books and youtube videos, I'm in the process of fixing that. I have a question : Why don't we use exponential functions based on 'e' to scan/probe the sampled signal ? Is it because the signal is sampled and it doesn't make sense to use a function that expresses continuous growth/decay ? after all we can pose r = e^something.

@BrianBDouglas 11 ай бұрын

The middle of page 608 here: www.dspguide.com/CH33.PDF explains why we use r instead of e. Hope that helps!

@40NoNameFound-100-years-ago Жыл бұрын

I am wondering why Mathworks took so long time to make such videos? We are in 2023 and version 2023 has been released. These control methods were introduced and had been modified since version 2006 .

@jimapost1496 6 ай бұрын

respect to matlab

@Amine-gz7gq 2 ай бұрын

For the videos, yes, but not for their software, which should be free for home users.

@tidin_tss 3 ай бұрын

респект авторам, но ничего не понятно для чего его использовать вообще

@kdre76 5 ай бұрын

L 1

@mightyparry 9 ай бұрын

in cre di ble

@BrianBDouglas 9 ай бұрын

Glad you liked it!

@Alexis-ym9ph 10 ай бұрын

This explanation is actually pretty bad in comparison to explanation of other authors on KZfaq)

@youtube-username-placeholder 10 ай бұрын

I respectfully disagree😅

@Amine-gz7gq 2 ай бұрын

This subject (like many others) is so complex that a single video is not enough to explain it in depth. You need to watch several videos by different people to improve your understanding of the subject.