How are the Fourier Series, Fourier Transform, DTFT, DFT, FFT, LT and ZT Related?

Explains how the Fourier Series (FS), Fourier Transform (FT), Discrete Time Fourier Transform (DTFT), Discrete Fourier Transform (DFT), Fast Fourier Transform (FFT), Laplace Transform (LT) and Z-Transform (ZT) are all Related?
(* Note that for the DFT, unfortunately I forgot to say the last part of the explanation. I made the point that the DFT effectively assumes that the finite-length sampled signal is actually one period of an infinitely long periodic signal. I then drew the Fourier transform for the infinitely long discrete time signal. What I forgot to do, was to then say that the output of the DFT is only the finite part of that Fourier transform between omega=0 and omega=2pi. And that it is a discrete sequence with the same number of elements as the number of samples in the time domain sampled signal (and that the delta functions are finite). I fixed it on the summary sheet on my website: drive.google.com/file/d/1fh7T... . For more details see: "How does the Discrete Fourier Transform DFT relate to Real Frequencies?" • How does the Discrete ... )
* 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.
Check out my search for signals in everyday life, by following my social media feeds:
Facebook: profile.php?...
Instagram: / iainexplains
Related videos: (see www.iaincollings.com)
• Fourier Series and Eigen Functions of LTI Systems • Fourier Series and Eig...
• What is the Fourier Transform? • What is the Fourier Tr...
• What is the Fourier Transform used for? • What is the Fourier Tr...
• Fourier Transform Equation Explained • Fourier Transform Equa...
• Typical Exam Question on Fourier Transform Properties • Typical Exam Question ...
• Discrete Time Basis Functions • Discrete Time Basis Fu...
• How does the Discrete Fourier Transform DFT relate to Real Frequencies? • How does the Discrete ...
• Laplace Transform Equation Explained • Laplace Transform Equa...
• Laplace Transform Region of Convergence Explained • Laplace Transform Regi...
• What is the Z Transform? • What is the Z Transform?
• Z Transform Region of Convergence Explained • Z Transform Region of ...
• Is Phase important in the Fourier Transform? • Is Phase important in ...
For a full list of Videos and accompanying Summary Sheets, see the associated website: www.iaincollings.com
.

[@damiensaga6941]

Thank you so much for this series of videos. This is the first time i've seen such a comprehensive explanation of the purpose behind Z and Laplace transform and the region of convergences !

[@iain_explains]

Thanks for your comment. I'm so glad you found the videos helpful!

[@TheRockStar04261999]

I have my signals 2 final on dtfs, dtft dft and z transform tomorrow. Thank you so much for these videos, they're really helpful

[@iain_explains]

I'm glad they have been helpful. Good luck tomorrow!!

[@ramon1930]

@@iain_explains Profesor Iain. I have a curiosity, can i use the series expansion obtained from Dft in fuzzy logic? Generally i use Z transform to do it.

[@adeddy8138]

Is there video playlist on Fourier series?

[@sofia671]

what a great video, your explanations have helped me in 3 subjects so far, thank you so much Iain! hugs from Argentina

[@iain_explains]

I'm glad they've been helpful. That's so great to hear!

[@GS-qe3pt]

jajajajaja, a mi también

[@giulio2797]

Thank you so much for the time you put in. I found this video extremely helpful

[@iain_explains]

I'm so glad!

[@darkcnotion]

Excellent

[@wwayaka8508]

Thank you so much for the videos! It was great and really helpful, truely appreciate your work :)!

[@iain_explains]

Glad it was helpful!

[@sinan4495]

Thank you for this amazing summary

[@iain_explains]

Glad it was helpful!

[@BentHestad]

A brilliant overview, Sir! Thanks! Greetings from Trondheim, Norway!

[@iain_explains]

I'm so glad you liked it! It's great to know that it's connecting all around the globe. Unfortunately I've never been to Norway, although one of my good friends during my PhD was a Norwegian student who spent a year here in Australia. I don't think I'll ever forget the "rotten fish delicacy" his mother used to send out to remind him of home! 😁

[@techankhamun838]

Very extremely useful! Liked & Subscribed! Thanks a lot!

[@iain_explains]

That's great to hear. I'm glad it was helpful.

[@rodas4yt137]

Am two minutes in and I'm already sure this is what I was looking for. Love when that happens, thanks!

[@iain_explains]

I'm glad you liked the video. If you'd like to see more like this, check out iaincollings.com where you'll find a categorised listing of all the videos on the channel, as well as summary sheets.

[@user-zr1vd7vt7o]

Thanks !

[@RibeDaniel]

Buen trabajo 👍.

[@iain_explains]

Thanks. I'm glad you liked the video.

[@kaursingh637]

MY LORD = MOST ENLIGHTENING --VERY GOOD -EXCELLENT - AMARJIT- INDIA

[@user-fl1lu7lv9k]

that's soo helpful thank you so much

[@iain_explains]

Glad it was helpful!

[@srijandwivedi294]

Thanks a lot Sir😊

[@iain_explains]

I'm glad it helped.

[@mertpurtas8913]

Thank you . This video explained everything ı had trouble with it .

[@iain_explains]

That's great to hear. I'm glad it was helpful.

[@sudiptamandal1518]

Sir Can You Please now make videos on digital filters?By the way your playlist really helped me to grow interest in Digital Signal processing. Thank You so much.😊

[@iain_explains]

Thanks for the suggestion. Yes, it's on my list (but it's a long list, sorry). Hopefully soon.

[@harshitsharma2262]

thankyou sir

[@rachitjoshi23]

LOVE YOUR BEAUTIFUL VIDEOS. Even a layman can become an expert after watching them

[@iain_explains]

Thanks for your nice comment. It's great to hear that they're helping.

[@soumyaneogy9522]

excellent explanation

[@iain_explains]

Glad it helped

[@emirhanbilgic2475]

amazing video, thanks alot and greetings from Turkey

[@iain_explains]

Thanks for watching!

[@Tommy-js3to]

Awesome video, thanks!

[@iain_explains]

Glad you found it helpful.

[@mnada72]

So enlightening, Thanks 😊

[@iain_explains]

Glad you enjoyed it!

[@yazeedAlkhrijah]

Thank you fro the video!

[@iain_explains]

My pleasure. I'm glad you found it helpful.

[@fardadansari1885]

You are always great

[@iain_explains]

Thanks so much. I'm really glad you like the videos.

[@hjen4861]

FT is complicated to me. This is timely. Thank you, Sir

[@iain_explains]

Glad it helped.

[@steve_killemall6182]

Brilliantly explained professor. Thank you very much! If there is any way I can express my gratitude, please let me know. Greetings from Greece :)

[@iain_explains]

Thanks for your nice comment. I'm glad you have found the videos helpful. I'm planning to set up a Patreon page, so people can support what I'm doing if they wish, and also to potentially run interactive sessions, but I don't have anything set up yet. For now, it's just great to know that you have found the channel useful. Thanks.

[@mqasimkhan6920]

the best lecture ever.thanks a lot

[@iain_explains]

I'm so glad you liked it.

[@anushasanpoudel3034]

tomorrow is my DSP exam , what a great timing. Thanks Iain !

[@iain_explains]

Best of luck! I'm glad this video has helped.

[@oggamer2244]

Me too hahaha, what a world.

[@bec_Divyansh]

Great video sir!

[@iain_explains]

Glad you liked it!

[@nebiyoubelay4600]

Dear Professor I have seen your CP-OFDM (Cyclic Prefix ) explanation in another video and really enjoyed it , If you have some spare time I appreciate you also to explain about W-OFDM (Wide band), F-OFDM (Filtered) and other types? Thank you a lot.

[@iain_explains]

Thanks for the suggestion. I've added those topics to my "to do" list.

[@mytech3833]

Very nice explanation, sir!. Thank you! It could be nice if you could formulize them too!

[@iain_explains]

Thanks. Have you seen my webpage? I've got videos on each transform, where I explain the formulas. iaincollings.com

[@sivasaikrishnamarthy178]

Excellent explanation sir

[@iain_explains]

Thanks. I'm glad you liked it.

[@susantpanigrahi1149]

Awesome video. But I think as in DTFT it is periodic in frequency domain, 10:00 the period should be -pi to pi.

[@iain_explains]

The discrete time basis functions repeat every 2pi. So that means 0 frequency is the same as 2pi, 4pi, ... and also the same as -2pi, -4pi, ... See this video for more explanation: "Discrete Time Basis Functions" kzbin.info/www/bejne/hmjUY4yDhs6Uqpo

[@tusarmondal8767]

thank u sir..

[@iain_explains]

You're welcome

[@user-saint]

Amazing

[@iain_explains]

Thanks

[@PE-gw5gu]

Thanks sir, it was so useful and helpful . 🙏🏻🙏🏻🌹🌹🌹🌹

[@iain_explains]

That's great to hear.

[@eitanas85]

Dear professor, Do you have a video explaining the Hilbert transform when it used to extract the instantaneous amplitude and phase, and calculate the phase locking value? Thank you for uploading great content. Very much appreciated. Eitan

[@iain_explains]

Thanks for the suggestion. The Hilbert transform is on my "to do" list. It's not very intuitive, so I'm giving some thought to how best to explain it.

[@T0NYD1CK]

Caveat: My memory is not what it was! However, if we start from the Fourier Transform, we end up with positive and negative frequencies. If you think of the spectrum of a cosine wave it has two impulses: one at the positive frequency and one at its negative counterpart. If you now think in 3D, you can imagine those impulses rotating around the frequency axis where the positive frequency rotates towards you "out of the paper" and the negative frequency also rotates but "into the paper." If you now make a phasor sum of both those components you get your cosine wave back if you plot the resultant amplitude against time. (If you turn the picture round so you are looking straight down the frequency axis you would see two phasors rotating in opposite directions which you can then sum to give a purely real wave.) There is another way of doing this by not having negative frequencies. You just keep the positive frequency, double its amplitude and rotate that about the frequency axis. The result, when plotted against time, is the same as the Fourier approach. What we now have is a rotating phasor that is drawing out a helix in 3D space. (Mathematically, that is what exp(jωt) looks like.) When looking at the projection on the real plane we see a cosine wave but what does it look like on the imaginary plane? That is what the Hilbert Transform tells us. In this case, the imaginary projection would be a sine wave. I hope that helps.

[@kushalnanote4800]

Useful explaination sirr

[@iain_explains]

I'm glad you found it helpful.

[@vedatbegec6927]

Sir very good explanation thanks a lot

[@iain_explains]

Glad you liked it.

[@fnegnilr]

What a boss!

[@muhannadaydi2235]

First of all, thank you for uploading great content. Secondly, I have a question, what is the difference between Fast Fourier Transform (FFT) and Fractional Fourier Transform (FrFT) ? and what is its applications? I searched on KZbin on it but I don't find videos explain it.

[@iain_explains]

Thanks for the suggested topic. I'll add it to my "to do" list. I'm not familiar with the FrFT, so I'll need to look into it.

[@amandhakal8939]

DA CA DP CP Discrete - Aperiodic (CTFS) Continuous - Aperiodic (CTFT) Discrete - Periodic (DTFS) Continuous - Periodic ( DTFT)

[@sathyanarayanankulasekaran5928]

this is the best cheat sheet video of signal processing

[@iain_explains]

I'm glad it's helpful.

[@bigmak845]

Hi Iain! Thanks for the great video. I notice that the magnitude of the DTFT example has some negative regions. Is that actually just a plot of the real part of the DTFT rather than the magnitude which should always be positive?

[@iain_explains]

Yes, that's right. Let's call it a 'minor typo'. Actually that particular function is real valued (there is no complex component), so it's a plot of the actual function.

[@bigmak845]

Thanks for the quick response!

[@zyctc000]

For DTFT, isn’t the frequency domain’s period determined by the sample frequency?

[@iain_explains]

In discrete time, all the discrete values are spaced apart by 1 sample time. More explanation can be found here: kzbin.info/www/bejne/bV6Xpnh-pL5ko5Y

[@Archturian8880]

Thank you for a good lecture. This lecture makes me understand region of convergence. However, I am a bit confused with transforming from time domain to frequency domain and frequency domain to image domain. I don't get this relationship. some lectures talk about only one part time domain to freq. domain or only image to freq domain. In reality, it seems the process includes all of this steps, time domain >> frequency domain (k space)>> image (object) domain.

[@iain_explains]

I'm not sure what you mean, sorry. This video does not talk about images. What do you mean by "image domain" in your question?

[@adeddy8138]

Is there a full playlist of dft ,dtft ,ctft and CFT and fft?

[@iain_explains]

My playlist on the Fourier transform can be found here: kzbin.info/aero/PLx7-Q20A1VYJlVLBCkuOBoBnaUdd5Qyms

[@adeddy8138]

Is there no video on Fourier series and dft , fft ,dtft sir. I am unable to find such videos in your channel

[@iain_explains]

I don't tend to pay too much attention to the Fourier Series, because there aren't really any periodic signals in the real world that go for an infinite amount of time. I prefer to think in terms of the Fourier Transform. However I do have one video on the FS, and I also have some on the DFT/FFT. Have you checked out my webpage? iaincollings.com Here's the link to the video on FS: "Fourier Series and Eigen Functions of LTI Systems" kzbin.info/www/bejne/nYPUZH5qj7Z-n5o

[@adeddy8138]

@@iain_explains yeah I have checked your website after I finish the entire playlist of signals and systems I will replay your videos again and make notes or downloads your summery sheets depending on the time I have. thank you very much

[@anwerarif894]

Thanks .. Why in communications we use Fourier and in control system in stability ues Laplace??

[@iain_explains]

Great question. The Laplace transform is a generalisation of the Fourier transform that allows for functions (eg. signals, system responses, ...) that have infinite energy (eg. the impulse response of an unstable system). In Communications, we're mostly dealing with communication channels that are inherently stable (if the input has finite energy, then the output will have finite energy) and we're interested in frequency domain aspects (eg. inter-channel interference, bandwidth efficiency, ...), so the Fourier transform is appropriate. In Control Systems, there's feedback (for controlling in the "plant") and this can lead to instabilities if not designed appropriately (eg. positive feedback in guitar amplifiers), so the Laplace transform is needed, in order to investigate aspects of stability in cases where a function potentially has infinite energy.

[@anwerarif894]

Thank you, Mr . I understand from your comment that Fourier does not work in unstable systems. Why is Laplace not widely used in communications?

[@iain_explains]

Yes, that's right. As I said, in Communications, we're mostly dealing with communication channels that are inherently stable (if the input has finite energy, then the output will have finite energy) and we're interested in frequency domain aspects (eg. inter-channel interference, bandwidth efficiency, ...), so the Fourier transform is appropriate.

[@anwerarif894]

@@iain_explains Thank you sir .

[@frederikvanaverbeke8840]

Hi, little remark, just to dot the i's and bar the t's... when you say that the spectrum in DT (the basisfunctions) repeats around 2pi, on the omega-axis, do you actually mean that they repeat around 2pi*fs, fs being the sample rate? I was just wondering because w is in radians times Hz. So, any point on it should be too, no? So, basisfunction repeats around 2pi*fs and -2pi*fs etc...? Is that correct?

[@iain_explains]

No, w is _not_ radians times Hz. It is just radians. In discrete-time, the "time" samples are just numbers stored in a vector. They are just indexed by integers.

[@lutzvonwangenheim9682]

do the samples taken by the DFT is following the Nyquist sampling rate criterion ?

[@iain_explains]

It depends how fast you take the samples.

[@lutzvonwangenheim9682]

hello lain, as you have mentioned that a sinusoidal which is having infinite energy is defined by the delta function in the Fourier domain so my question is how this can be done as impulse function is itself an unpractical signal which exists only theoretically is there any point I am missing means can you give an insight into this? means i want to say that is there any boundation or condition apllied in defining sineij terms of the delta function Thanks !

[@iain_explains]

Well, when you think about it, the sinusoidal signal sin(wt) is also an impractical signal which only exists theoretically (because it starts at negative infinite time, and goes until positive infinite time.) If you think about multiplying sin(wt) by a "window function" (eg. rect(t) ) to limit its duration to a finite range of time, then in the frequency domain you would be convolving the delta function with the Fourier transform of the rect function, which is a sinc function. These videos might help: "How to Understand the Delta Impulse Function" kzbin.info/www/bejne/rqmqlHxvjLuiebs and "Fourier Transform Duality Rect and Sinc Functions" kzbin.info/www/bejne/qIbKc5t7pcqkrtE

[@te9781]

I'm confused isn't the discrete time Signal is a group of impulse delta functions ? And the fourier transform of delta function is 1 meaning it's got continuous frequency components? How we are getting discrete frequency components in for example DTFS instead of 1

[@iain_explains]

Good question. Hopefully these points will help to explain it: 1) In general, the DTFT is a continuous valued function. See this video for more explanation: "Fourier Transform of Discrete Time Signals are not Discrete" kzbin.info/www/bejne/d4C0cp-oqayIsNE 2) The FT of a delta function has a magnitude of 1 (as you point out), but it also has a phase which is a function of frequency (depending on its time offset in the time-domain). This phase is most often not plotted, and is sometimes overlooked. The phases from all the different time-domain delta functions add up to give an overall function (in the frequency domain) that is not a constant magnitude. 3)The plots that show discrete frequency components for the DTFS and DFT (the 3rd and 4th plots on the far right hand side) both correspond to sinusoids in the time-domain. For time-domain signals that are periodic, the Fourier transform will consist of discrete impulses. This video explains this more: "Why do Periodic Signals have Discrete Frequency Spectra?" kzbin.info/www/bejne/rXKWh4uvoZ6rjMk 4) and finally, note that there is a slight error in the plot for the DFT. I explain this in the notes below the video, and I've fixed it in the Summary Sheet on my website: drive.google.com/file/d/1fh7TzeT4HCeoRECnHiQYmjFRSeWLYnDI/view

[@te9781]

@@iain_explains Thank you so much 🙏

[@gist_plenty]

Thank you. I'm not sure I get the explanation on why a discrete signal is both aperiodic and periodic. I thought that a discrete signal is a sampled signal already. I was of the thought that discrete signals were aperiodic.

[@iain_explains]

I think you're talking about the DFT, right? If you've sampled a signal for a finite period of time, you will have a vector of a certain length (depending on the sampling rate you used). The Fourier transform is defined as an integral over all time - not just over the time period that you sampled over. So the question is, what to do? One approach would be to assume that the signal is in fact zero outside the period of time that you sampled for. Another approach is to assume that the signal keeps repeating itself outside the period of time that you sampled for. The DFT takes the second approach.

[@gist_plenty]

@@iain_explains Thanks. Yes DFT/FFT since I'm considering seismic signals. And I believe you mentioned the FT is for processing finite signals? I read a material that finite signals were aperiodic so is the seismic signals aperiodic or assumed to be periodic in the DFT. Your videos are great by the way. I work with transform software processes but I'm still trying to understand HOW f(t) transforms to F(w). Like how 2π/T actually works. Your videos are helping me though.

[@iain_explains]

Have you seen my video: "Fourier Transform Equation Explained" kzbin.info/www/bejne/boeZeZxjoLVse6c

[@olayomateoreynaud9956]

The only thing I don´t understand is that the DFT does not give impulses, as you said in the video. It gives a vector of finite values.

[@iain_explains]

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

[@olayomateoreynaud9956]

@@iain_explains Thanks for replying :). I meant that If you have a vector of values, lets say x = [1, 2, 3], and you perform the DFT (for example, with MATLAB: fft(x) ), the result is a vector of 3 finite values. If I undertood well, the result shown in 15:00 is made up of impulses, with infinite values.

[@iain_explains]

Ah yes, I remember now. Unfortunately I wasn't as accurate as I should have been in that diagram. I drew "continuous" impulses (delta functions), when I should have drawn "finite/discrete" impulses only over a finite range of frequencies. I fixed it on the summary sheet on my website: drive.google.com/file/d/1fh7TzeT4HCeoRECnHiQYmjFRSeWLYnDI/view

[@olayoreynaud3312]

@@iain_explains I see now. Thank you! :)

[@faheemtassadaq]

I just wish I had a teacher like him in my engineering college..

[@iain_explains]

Oh well, at least you've got me on KZbin! 😁

[@faheemtassadaq]

@@iain_explains you are a rockstar Sir!! Hats off to you!!

[@anjurs7777]

Sir, Can you make a video on FFT alogorithm .

[@iain_explains]

Thanks for the suggestion. I'll add it to my "to do" list.

[@anjurs7777]

@@iain_explains thank you sir

[@kaursingh637]

my lord= i do not under stand difference between different types of fourier transform= thank u sir = amarjit= india

[@iain_explains]

Glad you found it useful.

[@neuroscience2012]

Thank you a lot for your videos! They are very helpful. However, I'm a bit confused about one point. In your other video ( kzbin.info/www/bejne/on3UZHdjq5mehrc ) you said that the CTFT is periodically replicated/has repeats on the sampling frequency, but here it is not. When is it and when is it not?

[@iain_explains]

When a continuous-time signal is sampled with a sequence of (ideal continuous-time) delta functions, the resultant "continuous-time sampled signal" has a (continuous time) Fourier transform that repeats at the sample rate. For all other (non-sampled) continuous-time signals, there is no frequency repetition. In other words, the repetition is because of the sampling.

[@neuroscience2012]

@@iain_explains Thank you very much!

[@johndunn5272]

Instead of a book to explain this in signals and systems theory the universities should have a special oscilloscope that allows periodic and non periodic signals to be represented and explained in terms of all these various transforms and relationships...a specific machine to hone in on these concepts for education.