Fourier Transform Equation Explained ("Best explanation of the Fourier Transform on all of YouTube")

Signal waveforms are used to visualise and explain the equation for the Fourier Transform.
Something I should have been more clear about in the video, is that the equation I wrote is the inverse FT, not the FT. The reason I did this is that I like to start by thinking about the time domain signal x(t) (ie. the actual signal that we observe in the real world), because time is one of the main dimensions that we observe/experience, and we are familiar with signals in this domain. I wrote the equation for the time domain signal in terms of the frequency components (which is actually the Inverse Fourier Transform equation). Once you see the signal from this perspective (ie. as being made up of a sum of individual single-frequency waveforms), then you will understand that there are two ways of viewing the same signal (ie. time-domain and frequency-domain), and that they are related by the Fourier Transform and the Inverse Fourier Transform. That's the important point.
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Related videos: (see www.iaincollings.com)
• Visualising the Fourier Transform • Visualising the Fourie...
• What is the Fourier Transform? • What is the Fourier Tr...
• What is the Fourier Transform used for? • What is the Fourier Tr...
• Fourier Transform and Inverse Fourier Transform: What's the difference? • Fourier Transform and ...
• Is Phase important in the Fourier Transform? • Is Phase important in ...
• Fourier Transform of Cosine Function: • Fourier Transform of Cos
• Fourier Transform of Cosine with Phase Shift: • Fourier Transform of C...
• Fourier Transform of a Sum of Delta Functions: • Fourier Transform of a...
• Typical Exam Question on Fourier Transform Properties • Typical Exam Question ...
• 2D Fourier Transform Explained with Examples • 2D Fourier Transform E...
• What is Negative Frequency?: • What is Negative Frequ...
• Laplace Transform Equation Explained • Laplace Transform Equa...
• Laplace Transform Region of Convergence Explained • Laplace Transform Regi...
• How are the Fourier Series, Fourier Transform, DTFT, DFT, FFT, LT and ZT Related? • How are the Fourier Se...
For a full list of Videos and accompanying Summary Sheets, see the associated website: www.iaincollings.com
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[@justinpower4475]

It blows my mind that some dude cooked this up 300 years ago. Thanks for the great explanation.

[@iain_explains]

Well, he wasn't just "some dude" I guess.

[@HalfinchLonomia]

Thats what a life free of social media and other distractions gets you

[@notcrediblesolipsism3851]

​@@HalfinchLonomia unmitigated genius helps too

[@cosmologicalturtle9528]

@@HalfinchLonomiaThere was a substantial amount of people who never saw social media who _still_ never invented the Fourier transform

[@OzieCargile]

Best explanation of the Fourier Transform on all of KZbin. Clear. Concise. To the point.

[@iain_explains]

Thanks for your nice comment. I'm glad you found the video helpful.

[@S24W2]

This video is the proof that when I can't understand something like this from KZbin videos, it's actually not me, it's the way the other videos are explaining it, their teaching style obviously doesn't work. This guy's teaching style does work, so thank you 😊

[@iain_explains]

It's great to hear that you're finding my videos more useful than other ways of explaining these concepts.

[@ronneyismael605]

Wow. you explained so nicely. Love from India

[@iain_explains]

Thanks. Glad you found it helpful.

[@bman6180]

That was great. I love how you showed the waveforms. It clears everything up.

[@iain_explains]

Glad it was helpful!

[@daves465]

Just should have shown that the frequency/cycle time of the waves is f1 = w1/2*pi, T1 = 2*pi/w1, f2 = w2/2*pi etc ... @@iain_explains

[@Nereus00]

thanks, people dont understand that a good explanation video shouldn't last more than 5 mins. you just go straight to the point :)

[@iain_explains]

I'm glad you like the format.

[@hansoogi]

This is amazing. Measured, clear explanation using graphics. Nicely done.

[@iain_explains]

Glad it was helpful!

[@inquisitiverakib5844]

Lucid and crystal clear explanation ever on this👏💖

[@iain_explains]

Glad you liked it.

[@rakeshpillai9662]

Different content helps different people, and I just wanted to say thanks a bunch cause your videos help a ton. Your content is similar to what I'm learning so it clicks the best for me. My prof seems to have quite a few similarities with your teaching haha.

[@iain_explains]

Sounds like your Prof might have been watching my videos too! 😁

[@lity1124]

hhha

[@jkye_314]

Wow Sublime. I am CE student and honeslty i hate signals and systems course of my univesisity cause our professor dont give the "mind map" idea and just show this random formula and pretedend that u can understand by youself. In the end, i decided to abandone the course, but i was been fascinated by the fourier transform, cause it revolutionized the digital world. It could be said this is the base of all things started :). Thanks Professor to give the real concept of this equation.

[@iain_explains]

I'm glad you found the video useful. Based on what you've said, you might be interested to take a look at some of my other videos too, including on the Laplace Transform and on Sampling. And yes, I totally agree, these concepts underlie so much of all of engineering. Check out my webpage for a full list of videos: www.iaincollings.com

[@user-tn9xv9rj7k]

Beautifully explained !

[@iain_explains]

Glad you liked it

[@mnada72]

That was a Brilliant Explanation 👏

[@iain_explains]

Glad you liked it

[@jesusbermudez6775]

Thank you. I've never understood why transform, but now it has become clear - it can be represented in the frequency or time dimension.

[@iain_explains]

I'm glad the video was helpful.

[@hubercats]

Great explanation! Thank you.

[@iain_explains]

Glad it was helpful!

[@user-gc2uf1ud4l]

It made me understand the Fourier transform formula very clearly, thank you

[@iain_explains]

You're very welcome! I'm glad it helped.

[@gijoraj623]

Thank you very much for the insightful video on FT

[@iain_explains]

Glad it was helpful!

[@seahorsefencing]

I have a bs in computer engineering and this helped a lot for my class. Thanks!

[@iain_explains]

That's great to hear. I'm so glad it helped.

[@georgesadler7830]

Thank you for the video. This is a great explanation.

[@iain_explains]

Glad you liked it!

[@pcbworks]

OMG! What a great explanation. I wish my professors could explain the theory behind it so that we could enjoy the digital comm rather suffering from it

[@iain_explains]

I'm glad my videos have helped.

[@GammaRay25]

Nice explanation, a recording of your voice speaking was a great analogy

[@iain_explains]

Thanks. I've got plans to make a video using an actual voice recording, and demonstrating filtering ... when I get time.

[@abdohatem8152]

great explanation ,thank you!

[@iain_explains]

Glad it was helpful!

[@HarpreetSingh-ke2zk]

The easiest and the concise explanation on Fourier transform. Thanks.

[@iain_explains]

Thanks for your comment. I'm glad you found it useful.

[@hongkyulee9724]

Thank you :D Love from South Korea!

[@iain_explains]

I'm glad you found the video helpful.

[@professorinfinity7075]

Oops! I got distracted by your videos. Looks like i have to watch all your videos. Probably the best explanation on Fourier Transform. Thank you Sir.

[@iain_explains]

I'm glad yo like them!

[@bounmysoumountha9165]

I really like your presentation of the meaning of fourier transform. Please make a video on introduction to wavelets and wavelet transforms starting with using Haar wavelets. Thank you.

[@iain_explains]

Thanks for the suggestion. I've added it to my "to do" list (although it's getting to be quite a long list so I may not get to it for some time, sorry).

[@tendi.rutsate]

Wow thank you so much! I finally understand

[@iain_explains]

Glad it helped!

[@zvargulis1988]

ripper video mate!

[@iain_explains]

Thanks. Glad you found it helpful.

[@transientmatter6088]

really, simple in a piece of paper with pen, you did better than 3d animations, matlab sims,,,if someone explaine something easy ans short, it means they have highest understanding of what they talk,,,,,i love this clip

[@iain_explains]

Glad you found it helpful, and I'm glad you appreciate the approach I take to explaining things. Animations and simulations can be helpful, but more often true understanding comes from careful thinking and visualising for yourself.

[@sadegh5429]

amazing! thanks!

[@iain_explains]

Glad you liked it.

[@vallurirajesh]

Thank you very much. I was wondering if you can explain how this equation came about in the first place, it would be a great addition to our knowledge. I mean, you have explained the components of the equation and how it helps in transforming a signal, but how did we arrive at this equation. Excellent video by the way.

[@iain_explains]

You'd have to ask Jean-Baptiste Joseph Fourier how he came up with it - except he died in 1830, so I guess we'll never know.

[@acluster3411]

Fourier knew Correlation and Euler formula, he realized that if you take the real signal and do correlation with the theoretical signal of a given frequency then he will able to uncover all the frequencies with in a mixed frequency signal as long as he add all the multiplication result between certain range. If the signal is similar then all the positive number of signal X(n) multiplied by ideal signal e^-2PI()Kn/N will produce the positive . Also the negative number of signal X(n) multiplied by ideal signal e^-2PI()Kn/N negative number will produce the positive number then adding them together will produce larger positive number. But if the frequencies are not same then they will not have exact matching positive number or exact matching negative number so the result will be smaller. So he just put together Euler equation with Correlation in a integral form and that was his brilliance. But I do not know what use he had in his day of his own equation!!

[@ahmetserdr2920]

İ think he wondered the summation of cosine and sine wave. That is all. İf you speak turkish you can watch Fuat Serkan Orhans videos😀

[@vaughanwilliamson173]

@@acluster3411 Concerning the use and motivation to find and extract single frequency elements from a signal by Fourier, the story I was told was that he was interested in music and how different instruments and voices can produce the unique but harmonious sounds that they do. Sounds a good story to me. I'll believe it!

[@milenark]

Thanks so much, incredibly helpful! Could you maybe do some examples of how this all looks when you transform a non-periodical signal?

[@iain_explains]

Thanks for the suggestion. I've just realised that almost all of the examples I've used in videos so far have been periodic signals. I'll add an example of a non-periodic signal to my "to do" list.

[@T0NYD1CK]

The trick to understanding non-periodic signals is to consider them repeating at infinity! The other thing you need to know is that if you stretch a time domain function it results in the frequency domain function being compressed. So, if you stretch out a sine wave, say, until it is infinitely long the spectrum will collapse to the origin. In other words, the infinitely long sine wave will look like a steady level so the spectrum will be an impulse at the origin. Like all symmetrical (even) functions, that works backwards as well so an impulse in the time domain will give a steady spectrum at all frequencies. That explains how people can determine the acoustics of a building by firing a gun. The bang provides a time domain impulse which creates all frequencies.

[@sofeabalqis7415]

Thank you for the straight-to-the-point explanation. Of all the content ive watched and read, this is the first where i actually understand the link between the application and the maths! though some things that were still a bit unclear to me, where did the 1/2pi come from? and if this is the IFT, using this same example, what would the FT formula be?

[@iain_explains]

The 1/2pi comes from the fact that radial frequency omega = 2pi f For the relationship between FT and IFT see: "Fourier Transform and Inverse Fourier Transform: What's the difference?" kzbin.info/www/bejne/hGm1h2mNiZmmirs

[@sofeabalqis7415]

@@iain_explains ohh! okay, thanks a bunch!

[@ethancooper4154]

Wow this is actually an exceptionally simple and effective explanation. Too bad my college SS profs never watched these videos...

[@iain_explains]

I'm glad you like them. And I sympathise with your comment about your college profs - you're not the only one who has (or had) similar experiences - it's one of the main motivations for me making my videos! Thanks for watching.

[@iain_explains]

... and I'm presuming you are talking about your Signals and Systems professors, and not some form of German SS professors from the 1940's. 😉

[@chrismaguire8165]

u r so intelligent ..u r a boss.

[@iain_explains]

I'm so glad you like the videos.

[@chrismaguire8165]

@@iain_explains Sir , we are not using T1/E1 in modern system , could you please make video on current backhaul please?

[@iain_explains]

Sorry, I don't see the context in your question. I don't mention T1/E1 or backhaul in this video. Which aspect of "current backhaul" are you referring to?

[@chrismaguire8165]

@@iain_explains Sorry i did not explain myself better. Yes ,agreed that this video has nothing to do with backhaul or old T1/E1. I am requesting for entirely new video on current setup of backhaul /VOIP if time permits please.

[@modestyblaise6125]

Thank you very much for the explanation of Fourier Transform Equation. I did not quite understand the magnitude plot (vertical axis). Could you please elaborate how you got it?Thank you very much.

[@iain_explains]

Hopefully this video helps (not sure if this is what you're asking, or not): "Fourier Transform of Cos" kzbin.info/www/bejne/g5SshYGcpJh_gcU

[@modestyblaise6125]

Thank you very much for the reply. However, I was expecting delta functions for |X(jw)| v.s w plot, therefore my confusion. Why and how did you get |X(jw)| to look like "positive part of a sine wave" ? Thank you in advance.

[@physicsconcepts6213]

Your videos are very helpful and..Please explain the part as u mentioned..for symmetric X(jw) function how the final value of the integral will be real.

[@iain_explains]

To be precise, I should have said "complex conjugate symmetric". It's a bit hard to explain in these comments, but I've added your question to my "to do" list, and I'm planning to make a video to explain.

[@physicsconcepts6213]

@@iain_explains thank u.

[@amahbubul85]

Thanks for this video. As you have said, complex sin parts get cancelled when we take negative omega domain of the integration However, even after that, X(jw) remains there with cos. Is X(jw) always real? It has to be that to make X(t) a real one.

[@iain_explains]

No, X(jw) is not always real. In fact, the _only_ cases where it is real is if x(t) only contains cos(wt) terms with no phase offsets. All other signals have complex valued Fourier transforms. Note that all real signals (ie. signals that have real values in the time domain) have Fourier Transforms that are complex-conjugate-symmetric around the zero frequency.

[@daopengliu6138]

Sir, you explained it well. But may I ask why there is a 1/2pi?

[@iain_explains]

Because I wrote the formula in terms of the angular frequency, omega (not the real frequency f). Where, omega = 2pi f

[@RAJIBLOCHANDAS]

good

[@ahmetserdr2920]

Sir please correct me. If we can find the frequency that is make cos(0) the one of part complex integral, we can say the frequency is the one of term the signal. But i dont understand why we use complex exponantial?

[@iain_explains]

I'm sorry, I don't understand what you are asking about. But if I'm right, then you might find these videos helpful in getting an answer: "Orthogonal Basis Functions in the Fourier Transform" kzbin.info/www/bejne/pGPOlqaCmLWMbdE , "Fourier Transform Equation Explained" kzbin.info/www/bejne/boeZeZxjoLVse6c , and "Fourier Trfm and Inv FT: What's the difference?" kzbin.info/www/bejne/hGm1h2mNiZmmirs

[@htchtc203]

Sir, would you mind elaborate notation, please. As a coefficient funtion X is in respect of w (omega), why notation comes together with j, as X(jw). Is that just to indicate X as a complex coefficient? I mean, would it be posible to use notation as "X(w), where X is complex" ?

[@iain_explains]

This video explains it: "Transform Notation" kzbin.info/www/bejne/r2mUgGywataheKc

[@htchtc203]

@@iain_explains Thank you Sir. That was good clarification.

[@TheSpazzattack95]

Hey im trying to tune my 4 subs, ive been learning all things physics, sound engineering, and electrical engineering. I have 2 10 inch dvc subs, wired to 2 ohms on its own amp in the same box as my other pair of 4ohm dvc subs wired to 1 ohm on its own amp. Soo that being said am i to use the function in order to make my subs hit as close to the same as i can?

[@TheSpazzattack95]

I have a huge understanding of it all, but i am still learning. Tho i have too much knoledge and not enough at the same time as to, i get how the in phase pair will be playing its just im haveing troubles getting my two ohm pair to even be audible..

[@TheSpazzattack95]

I know the higher ohm pair will need more noise so i gotta up the sub level inorder to achieve that but!!! I am confused on what more noise is as well.. if im upping my sub level im gunna want to drop my gain right?

[@TheSpazzattack95]

These are all questions i rlly need to understand as to i "get/understand" eqautions/functions, ive been using sin to "tune" my subs. So i do have "understandings" i just honestly need sumone to help me get it.. as to i know wayyy tooo much on this stuff but little things that i dont know i try to ask everyone that i think does know, but they all say thats ideal its wrong.. ive been seeing that no! Its not wrong! Its time and effort through errors we are trying to work around/with. But i rlly i need a litlle helpp.. i dont have the money to go to school for it ive been self teaching myself for months now. I got pages full of what i learnt in my "journal". And of my system in my vehicle/ to stuff about my box as to i built my first box a ported 28hrtz tuned box thats 5.0cf pretty sure it was after displacements tooken out

[@drsimons]

I have watched a bunch of your lectures but can't find any explanation of the syntax X(jw1). This should be a complex-valued function of a real-valued frequency, no? Is that what the "j" is supposed to mean? Because it's written like it's part of the function's argument.

[@iain_explains]

Good question. This video explains it: "Transform Notation" kzbin.info/www/bejne/r2mUgGywataheKc

[@prasanthr3875]

It is a combination of different frequencies . The frequencies can.be represented as pure sine waves or cos waves. Or frequncies with phaseshifted cos or sine waves. Is that correct

[@iain_explains]

A signal at a particular exact frequency _is_ a sinusoidal waveform. That his how _frequency_ is defined.

[@kamilbudagov9335]

Do magnitude and phase for particular frequency in continuous frequency spectrum represent exact magnitude and phase of sinusoid with this frequency? Thanks in advance!

[@iain_explains]

I'm not sure what you're asking. Yes, the values are exact.

[@kamilbudagov9335]

Just suppose we have got sinc function for continuous frequency spectrum and we have certain magnitude and phase for particular frequency, do these magnitude and phase tell that we have exactly this sinusoid or magnitude shows relatively this magnitude for this frequency is higher than another one i mean relatively

[@iain_explains]

The Fourier transform actually gives what you might call a "frequency density" function. So any specific _exact_ frequency has "zero voltage", unless there is a delta function at that frequency (which has infinite "height"). Hopefully this video provides the intuition you're looking for: "What are the Units of the Fourier Transform?" kzbin.info/www/bejne/r2ennKKCa9d7fZI

[@user-qm3xg4gu8c]

Hello teacher, I would like to ask why the image at 5:35 is symmetrical about the coordinate axis? What is the basis for this?🤔

[@iain_explains]

This video should hopefully help: "What is Negative Frequency?" kzbin.info/www/bejne/nauZcn6NYrdpb9U

[@user-qm3xg4gu8c]

@@iain_explains thanks!

[@ahmetserdr2920]

Sir can we say fourier transform pull the signal spectrum to DC frequency and integrate the constant value because cos(0) is 1?

[@iain_explains]

I'm sorry, I don't understand your question.

[@rafalzasada8826]

I am beginner in this so maybe I am asking a wrong question but what about resolution of frequencies (step size) we are adding up?

[@cams1365]

Integrals add in what are essentially infinitely small steps, so it should be an infinitely high resolution

[@ashutoshsingh-et7vm]

so area under X(jw) vs w (frequency) and x vs t is same and whats the unit of X(jw) consider x is pascal in x vs t

[@iain_explains]

This video should help: "What are the Units of the Fourier Transform?" kzbin.info/www/bejne/r2ennKKCa9d7fZI

[@braineedly7543]

So if we want to detect what are frequencies involved in our signal for example audio, then we should do inverse of this formula?

[@iain_explains]

Yes, that's right. This video may shed more light: "Fourier Transform and Inverse Fourier Transform: What's the difference?" kzbin.info/www/bejne/hGm1h2mNiZmmirs

[@braineedly7543]

@@iain_explains thank you so much, great explanation

[@alexlim1509]

Is he starting out with an Inverse Fourier transform for the example and then turning it into a normal fourier transform?

[@iain_explains]

Sorry, I should have been more clear about this in the video. I like to start by thinking about the time domain signal (ie. the actual signal that we observe in the real world), because time is one of the main dimensions that we observe, and we are familiar with signals in this domain. I wrote the equation for the time domain signal in terms of the frequency components (which is actually the Inverse Fourier Transform). Once you see the signal from this perspective (ie. as being made up of a sum of individual single-frequency waveforms), then you will understand that there are two ways of viewing the same signal (ie. time-domain and frequency-domain), and that they are related by the Fourier Transform and the Inverse Fourier Transform. That's the important point. This video will hopefully give more insights into the relationship between the FT and the IFT, and point out that they are almost identical transforms: "Fourier Transform and Inverse Fourier Transform: What's the difference?" kzbin.info/www/bejne/hGm1h2mNiZmmirs

[@SalmanKhan-er2ic]

There is a "Like" option in KZbin. There is a "Dislike" option in KZbin. I wish there is a "Love This Video" option.

[@iain_explains]

Great suggestion. Thanks for your vote! 😁

[@brettclark3885]

What exactly is the function X(jw) though?

[@iain_explains]

Here's a video that explains the notation: "Transform Notation" kzbin.info/www/bejne/r2mUgGywataheKc

[@nedimtutkun]

It seems harmonic order n is missing in the 1st equation.

[@iain_explains]

I think you might be thinking about the Fourier Series. That's not what this video is about. This video is about the Fourier Transform. For some explanation of the various transforms, see: "How are the Fourier Series, Fourier Transform, DTFT, DFT, FFT, LT and ZT Related?" kzbin.info/www/bejne/aJywhH-ndsd_oJY

[@mattdamutt5681]

It's always weird to me when I see an imaginary number used to figure out real math. I don't think I'll ever know how that works.

[@iain_explains]

Perhaps these videos will help: "How do Complex Numbers relate to Real Signals?" kzbin.info/www/bejne/in26dmZuba-KfdU and "Visualising Complex Numbers with an Example" kzbin.info/www/bejne/nonPZqiOa76mnpI

[@bobbaberson3654]

I didn't get the point about having a symmetric function @4:42?

[@iain_explains]

To be more accurate, I should have said "complex conjugate symmetric". If the Fourier transform is a complex conjugate symmetric function, then (X(-jw))' = X(jw) where ' indicates the complex conjugate, and therefore (X(-jw))'e^(-jwt) = X(jw)e^(jwt) , and so when you perform the integral from -inf to inf, the Imaginary components of the values for the negative frequencies will cancel with the Imaginary components of the values for the positive frequencies, leaving you with a real-valued function.

[@pradumnanayak494]

Namaste ji.

[@iain_explains]

You're welcome.

[@user-bv6hg2cx8o]

Still not clear if sin component is negative why it cancels out…

[@iain_explains]

This video should help: "How do Complex Numbers relate to Real Signals?" kzbin.info/www/bejne/in26dmZuba-KfdU

[@RadoslavFicko]

dE(k)= F(x).e^(ikx)dx. dF(x)=(1/2π).E(k).e^(-ikx)dk. where k=[1/m], x=[m], E=[kg.m^2/s^2] - electric field energy, F=[kg.m./s^2] - gravitational force

[@almiknacademy3626]

what is X(jw)?

[@iain_explains]

This video explains it: kzbin.info/www/bejne/r2mUgGywataheKc

[@davidjohn2534]

i get confused with this and fourier series

[@iain_explains]

Hopefully this video will help: "How are the Fourier Series, Fourier Transform, DTFT, DFT, FFT, LT and ZT Related?" kzbin.info/www/bejne/aJywhH-ndsd_oJY

[@ted2231]

The only problem I have with this video is that it came out AFTER my undergrad😂

[@sarasramdas7770]

I am so glad I found this! Preparing for a job interview and this tremendously helped me brush up on my EE basics. Thank you!

[@iain_explains]

That's great to hear. I'm so glad the video was helpful.

[@AbdulHadi-hs1uf]

I don't understand

[@iain_explains]

Perhaps this video might help: "What is the Fourier Transform?" kzbin.info/www/bejne/fWiXpWiXr5uDgtE

[@joliver81]

Good will hunting?? ……anyone???😂

[@T0NYD1CK]

I prefer to write my transforms using 2πf rather than ω because it gets rid of those annoying π terms outside the integrals.

[@iain_explains]

But then you'll have 2π appearing in all the exponentials. It's really just a matter of personal preference, I guess.

[@T0NYD1CK]

@@iain_explains Yes, that's true but at least you know where it goes, then. There are many versions of the transform pair with either a 2π in one or the other or even a square root in some and, anyway, I find frequency more intuitive than radians per second. Maybe that's because I find it easier to understand that Concert Pitch "A" is 440Hz and not approximately 2,764.601535159018049847126177286 radians per second. ;) Nearly forgot: It also makes both the forward and inverse transforms the same apart from the change in sign of the exponential. I find that easier to remember but, as you say, it comes down to personal choice.

[@iain_explains]

Sure, but once you start thinking about digital sampling, I think you'll change your mind, and start to prefer using omega. Because after you digitally sample something, you lose the time aspect, because all the neighbouring samples are separated by 1 (in "digital time"), for all digital signals, no matter what the sampling rate was you just get a sequence of discrete samples (... then what does the frequency f mean?). There are lots of sampling videos on my channel, if you're interested. iaincollings.com

[@T0NYD1CK]

@@iain_explains What you say is true. However, if you ever need to explain forward and inverse Fourier transforms mathematically I find it helps if you can have the most consistent set of formulae. There are multiple options if you use ω. If you use f then you have, basically, just one consistent formula to remember but with an optional minus sign to change from inverse to direct. For information, I did start to think about digital sampling but that was in the late 1970s.

[@iain_explains]

Sure. You're right, it is good to dot the i's and cross the t's. But from a fundamental understanding point of view, I'm not too fussed about that. The forward and inverse transforms are essentially the same, in my view. At least conceptually. "Fourier Transform and Inverse Fourier Transform: What's the difference?" kzbin.info/www/bejne/hGm1h2mNiZmmirs

[@timliu1916]

Great explanation. Thank you!

[@iain_explains]

Glad you liked it!