Introduction to the Fourier Transform (Part 2)

Check out this 3B1B video on Fourier Transforms for a fantastic visual explanation: • But what is the Fourie...
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This video is the second part of the introduction to the Fourier Transform. I address an error that I made in the first video concerning the scaling term of the transform. I also try to give you a way to think about the forward Fourier Transform as just a division of the time domain signal and sines and cosines. If you have any questions on it leave them in the comment section below and I'll try my best to answer them.
I will be loading a new video each week and welcome suggestions for new topics. Please leave a comment or question below and I will do my best to address it. Thanks for watching!
Don't forget to subscribe! Follow me on Twitter @BrianBDouglas!

[@AljazJelen1992]

Hi Brian! So as a Mechanical Engineer, we never really dug deep into Fourier. Short overview and then straight to Simulink and FFT libraries... all in discrete domain. However, now I work in Automotive industry on electric vehicles, dealing with topics that i never really studies. Long story short, we often need to define parameters of SW and HW elements, which contribute to different delays and phase shifts (simple LPF). Diving from analog to digital has always been intuitive to me, but I wanted to 100% understand the background of Control Systems, Digital Signal Processing... Recent situation of quarantine "finally" forced me to do a step... and man... I am happy I did. With my "rusted" knowledge I started from scratch and finally got to the point, where I understand the main principle. All this is in 2 afternoons! Thank you for putting an effort. Explanations really feel like they were designed to focus on listener's basis and gradually build up the comprehension of the topic. Best regards, Aljaz

[@bradlane3371]

The analogy of dividing $15 by the $5 bills is great! And then applying it to the F.T. has just made this, make much more sense. I just had a 2 month class on Signals, in which we used a lot of FTs, which i now understan a little more. Thank you!

[@nmbileg]

man this lecture is ten time useful than college class... thank you mr douglas, so appreciated

[@italianodoc8161]

agreed

[@HassanAli-os3py]

Best explanation of the Fourier series/transform I've ever come across! Marvelous!

[@ahmadasadi93]

Both of your videos about Fourier transform are awesome! Thanks a lot, Brian.

[@kashishsheth5310]

Thanks for spoiling my plan to Complete revision in less than 3 days. Each 10-15 mins lecture takes me around 30-50 mins! However, I have never been so happy for having my plan spoiled! Cheers Brian for such an amazing work!

[@alakbakbak]

They way you compared finding the frequencies in a signal like dividing $15 into $5 bills made everything so clear. Thank you!

[@ConnorHofius]

Watching this in 2021 and this still makes so much more sense than what my professor says in class...

[@daviddieken5442]

wow i have memorized the formulas without a clear explanation from professors as to why it works or how it came to be. thank you from the bottom of my heart for taking the time to putting together such amazing videos to help people all over the world learn and grow.

[@absurdsun8580]

I reaalllyyy got goosebumps when you explained the division analogy!!

[@Redant1Redant]

This is best explanation for the laymen I have seen. Thank you! I love the graphic model at the end, that helped me understand how forward and inverse are related. Thanks.

[@Runzolf]

The sequence where you plot the power of the multiplied sines is what did it for me! I'm already graduated but damn how many things I took for granted! Thanks a lot!

[@ShariefSaleh]

Amazingly amazing!! I have been using Fourier Transform for the past 2 years and I have never understood it intuitively as I do now!! THANKS!

[@sikservant206]

This was very helpful in my understanding of the Fourier transform! Thank you.

[@pumpkinhead002]

senior computer engineering student here: I have been struggling with the intuition behind fourier series and transforms for a while now, but the part at 6:00 and specifically at 6:14 just pieced it all together. Thank you!

[@lindsay5985]

That's why there's a negative power; it's just division of e^(i*2*Pi*U*t), but to express it on the top line we negate the power. Waveform divided by sinusoid gives how much of the sinusoid is in the waveform. This is a great video. When I tried to learn it at uni, if you didn't get it, the lecturer just moved on and you didn't get it for the next four years - there was no KZbin!

[@ozomata]

Mind=blown the first part was like magic, yet made lots of sense Thanks

[@looksintolasers]

You've got a gift for explaining these things intuitively. Thank you.

[@am-i-ai]

I understand the Fourier Transform so much better now. Thank you! :)

[@dvmehta46]

Thanks Brian! You have really made the whole concept pretty much clear. I was actually struggling to understand the -j reason in e^-jwt of the Fourier Transformation. Salute to you sir! 👌👍

[@BrianBDouglas]

Forward FT goes from time domain to frequency domain, inverse does the opposite. Fourier series breaks a periodic time signal into the frequency domain. It does the same thing as the FT but the transform works with non-periodic signals. Yes X(v) and F(v) are the same. I'm really bad about using the same variable letter from video to video. Scaling is not amplitude. If you plot the complex output of the FT amplitude is the distance from the origin. Scaling is just a gain part of the equation.

[@MrShutterButter]

really appreciate your videos Brian. Although this one i will need to watch a couple of times to get my head around.

[@BrianBDouglas]

There definitely is a jump, I don't cover everything in them. My goal is to provide supplemental information for people taking a controls class or have taken one. So I assume that the viewer understands how to solve the problem but maybe doesn't understand why it works or how to use it. Let me try to answer you questions in another post.

[@juniorcyans2988]

I’ve been lucky to find your channel. Thank you so much for demystifying it! What I want is not just knowing how to use it, like a monkey imitating humans, but understanding why. Thanks again!

[@isurubandara7177]

You explain this so well, amazing content man!

[@ozanozy93]

great work , I look forward to see new lectures

[@BrianBDouglas]

Great catch San! Starting around 1:15 I accidentally wrote dt instead of dv. I'll put an annotation in there so others aren't confused. Thanks!

[@adel7935]

You nailed it! thanks man for this awesome explanation!

[@malayali_m]

wonderful explanation! :D what i loved the most was the 'dividing by e^jwt to find how much of e^jwt is there in a signal'. Just wonderful :) i was used to the other explanation of 'dot product between two functions to measure the similarity' which is the common explanation. to see an even simpler explanation is just great! thanks a lot!

[@nachobatero]

Thanks for your vids Brian! you must be a veeeery good teacher.

[@hugeride]

Amazing. I understood more in 13 mins of this video then I did by reading 2 books.

[@Fransamsterdam]

This is the best way to use a blackboard on youtube I have ever seen. Congratulations!

[@shekharyadav380]

This video just blew my mind.......amazing....indebted for life !!!!

[@kindjupiter]

Thank you man, this is one of the best video i have seen in FT, :)

[@bozhidaratanasov7800]

This is a more advanced approach, which includes more calculations and fancy words. For a basic understanding i advise watching the 3Blue1Brown video on the topic.

[@fridaperez2363]

Not sure I have a full understanding. It will be long before then but thank you! These videos have helped me so much !

[@user-kt5dz4cr8f]

This video can help me to undertand easier than source other got. The explanation very detailed to find why the base formula get. Thank you so much, may God blessed you 🙂.

[@BKJoe116]

so amazing and beautiful. and it helps a lot. thanks so much!

[@xnr348]

Thank you so much!! This is really helpful to me!

[@Brun69M]

absolutely illuminating, thanks!

[@AbuSayed-er9vs]

Extremely easy,mind blowing!!! Upload more plz....

[@abhilashpatel1809]

simplest explanation of ft ever seen.. thanks lot

[@estebanlopez1701]

Man you are a boss. The analogy of the 5 dollar bill is really exciting!

[@pein0048]

This is an awesome video. Thanks for sharing :)

[@esterxpark]

Omg that analogy of dividing the envelope fn over carrier waves. It makes so much sense now holy shit. I’ve been living a lie, this is ingenious intuition

[@sandeepanpal6975]

Can you explain how is it dividing the envelope over carrier waves? I think it should be dividing Modulated Wave over carrier wave?

[@quincewilliams6058]

incredible job

[@MrSirPockets]

My signals and systems class just got a bit easier! Thank you

[@hklausen]

Thank you for your last remark that i'm still in pretty good shape :-)

[@romyw2851]

really nice teaching, right there!

[@masoudrahman5849]

Thank you for the clear lectures. Very informative. The scaling is still not clear in this one. Talking about a real example that scaling comes into play could be helpful. Thanks.

[@EmreAy1]

Sen var ya sen koskocaman bir adamsın! Thanks for your great lectures.

[@notgate2624]

I love these so much! You rock. Videos like these are amazing and the ending was exactly what I needed to understand the forward Fourier transform. I've never seen such a clear and awesome explanation of it.

[@saharkhawatmi660]

Thanks, nice explanations.

[@tejesdas3896]

Thanks Brian.Both part 1 and part 2 are well presented, albeit bit fast for my pace of understanding.

[@abcxyz4207]

I think they suck, Fourier transform is better explained in my books

[@karthikKarthik-by6ws]

@@abcxyz4207 book name?

[@anandjj7731]

@@abcxyz4207 Its for someone already familiar with these concepts but looking for intuition, he also says it in the ending of this video, 'I hope these videos help demystify fourier transform for you'. Its also really good revision.

[@SatyabrataParida1]

It was awesome.Thanks for the video. I would request you to upload a video explaining the z transform.

[@aliebrahimi1301]

very great job, thanks a lot Mr

[@ahnafsakib]

The 5$ analogy is astonishingly effective yet simple 👌👌👌

[@ahmed017162]

very good work , thanks.

I think my faculty would do way better by just running your videos to us instead of regular signal processing lectures :-D Thank you! This really helped

[@RaedMohsen]

We can call this video " Another approach for dealing with Fourier transform " :).. Nice explanation, also your graphical interpretation is very related to the concept of active and reactive power in Ac Circuits theory.. Looking froward to "Another approach for Laplace" :)

[@scoutjawalcommunicationlea641]

think for this beautiful explanation

[@andrewdavis6191]

dude that was amazing! when you have a print book i'll buy it!

[@Dyslexic-Artist-Theory-on-Time]

Good info!!!

[@leonhans8391]

awesome viedo， ths for sharing

[@ohwaititsbait]

Much appriciated Brian. (Y)

[@SDTricker]

THANK YOU!!!! Can you also do Fourier Series???

[@user-pt-au-hg]

Unless your an actor I can't believe how clear the presentation is, the wording and voice sound so pleasing to learn from (many instructors don't have style, which is important when learning something new. I listen at 1.5 speed while going back at times. Great work, love to listen :)

[@zephyrliu8833]

9:25 the scaling of infinity is blowing my mind

[@candycandy319]

you're a legend

[@kewalbajaj837]

@brian The differential of the integration at 1:15 should be d(Nu). Awesome video.

[@12388696]

it's just beautiful.

[@kunjal96]

great !!!!!!!!if possible than make same sorts of videos on circuits and networks .....

[@mikemoruzzi]

Brilliant

[@Dina-he1uc]

did not understand a single thing but so addicting to watch🤣

[@dongolahmed]

brilliant

[@dn_panic]

Thanks for the great video series! Just wanted to point out that the reason positive i*sin(t) works out just like negative i*sin(t) is because i and -i are completely interchangeable as long as you're consistent about it - the defining equation i^2 = 1 is true for both.

[@esinlayo6894]

panic i^2 = -1; his explanation still works out in the end because doing the integral for cos(x)sin(x) or cos (x)(-sin (x)) both give 0

[@dn_panic]

Err yeah, that's obviously what I meant!

[@GAment_11]

Brian, thanks for these videos. I have taken controls in the past but never quite clicked. So far, these videos have really cleared up concepts I struggled with prior. That said, because I am studying this on my own, do you have an pointers on where I could start in making a controls project? That is, I understand examples like close-loop control given cruise control or running dishwasher until something is clean (your prior examples). But to actually put it all together, that is, applying the Laplace Transform, Bode plots, poles, conversion of time domain to frequency and vice versa, etc, do you have any thoughts as to where I can start implementing this into a project? I am looking for something basic that will prove the concept. I am looking to gain a deeper physical understanding for instance, in how the poles of a Laplace Transform help in fixing the output. What does it physically do to the output? I would like to setup an experiment to gain that hands on understanding of these complex formulas to build upon for projects beyond my first. Perhaps I will have better insight once further into your videos. Thanks again for these, you guys make academics fun.

[@TheMechatronicEngineer]

Great!

[@curtpiazza1688]

Great ! 🎉

[@MrMouse12]

Hey Brian, when the fourier transform of a perfect cosine is performed you show that we get the magnitude times the dirac delta function. But isn't the value of the delta function infinite? Doesn't there need to be an integration of the delta function to get it's value to be 1?

[@jm10oct]

You are the best, any chance of having the lecture as pdf somewhere?

[@HvatrexHamam]

thanks alot

[@bikideka7880]

9:17 Why the amplitude become infinity? please please explain. And it's really explanatory video . thanks for this amazing piece.

[@linzhu9136]

Thanks for your video,your first part about magic is really awesome,but I really confused about sacling ,what it mean,and why multiply dirac delta function?

[@AliJoohy]

what program do you use to make your lectures?

[@yanhuang4911]

at 7:57, when you draw the result of cos(2*pi*v*t) * cos(pi*v*t), the areas concel out in every period, but when time -> inf, the intergral does not exist rather than -> 0 as you said.

[@christophertamina8569]

Awesome

[@k_abhisharma]

awesome

[@solivanvalente2724]

Hi Brian, I started following your channel recently, and I really have to compliment you! Your videos are amazingly didactic! However, I have a question about the meaning of the Fourier transform interpretation. Let me express it with the angular frequency “w”, with which I am used to. Imagine the real signal f(t) is a voltage signal in [V], and time is in seconds [s]. Since F(w) = int f(t).exp(-i.w.t).dt , the sum performed by the integral obviously has [V][s] as unit, which can also be written as [V]/[Hz]. This means F(w) would not represent the actual amplitudes of the sinusoids present in f(t), but rather an “amplitude density per unitary bandwidth”. Another way of looking at it is by recalling that the curve of F(w) is continuous (it has infinite points, representing infinite frequency components). If each point of F(w) represented the actual amplitude of each sinusoid, then the integral sum of the inverse transform would result in infinite amplitudes for f(t). Yet another way of checking this interpretation is by looking at the cosine transform, with dirac deltas “d”: F(w) = (1/2).d(w - w0) + (1/2).d(w + w0). Dirac deltas have infinite amplitudes, and the “1/2” in front of them represent their areas. Exactly what we would expect if F(w) represented an “amplitude density per unitary bandwidth”. Then, the amplitudes of the sinusoids in f(t) would really be represented by the area under the curve of F(w), per unitary bandwidth. Of course, this interpretation does not apply to the Fourier series, because the frequency domain is discrete. Would that make sense for you? Regards, Solivan Valente

[@moosa010101]

GREAT

[@learnwithveer7359]

Sir, could you please elaborate more about Fourier and Laplace transforms....

[@thesameen786]

Can you please explain how can i do the inverse operation on the impulses mentioned in your video

[@kutilkol]

Top!

[@sooryaprakash6390]

@Brian Douglas Loved you Video and 3B1B's video. But I seem totally lost when I relate your video with his video. Can you explain how both are related and what is similar between your video and his video?

[@NilupulBandara]

do an example question pls

[@aniketambhore]

Yes, they are. Laplace transform is a generalised form of fourier transform. s=iw over here(w is taken to be complex.more clarity when you study laplace) and in laplace s = a + iw. What makes it special? 5:23 to 7 explains it better.

[@niets421]

We use Laplace transform time to s-domain. Can we say that we use Fourier transform time domain to phasor domain?

[@MRBthecharger]

I blinked and got lost. It's like my first college physics course all over again. ".. equal and opposite reaction .." *blinks* *board filled with integrals and derivatives* "So, the 15th integral of the.." *nani?!*

[@TheJq32]

8:05 "The green signal has no power at the red signal frequency." What does that mean?

[@safatkhan6839]

What I got from it is that at the frequency of the red signal, the green one sums up to zero (from positive infinity to negative infinity). As such, it has no power. The total area under the curve that's positive equals the total area under the curve that's negative. Having said all that, this is the most confusing video of his that I've seen so far.

[@ChupoCro]

In the real world the energy for running some system is more than zero (the power is being dissipated) only if the force is in the phase with the direction of the motion (W = F*s*cos (theta)). Similarly, electric power is dissipated only if voltage and current are *in phase* (P = U*I*cos (phi)). There could be significant current but if the voltage and the current are out of phase the power is zero, therefore the power counter will not 'count' anything. In that case there is only a reactive component of the power for which you don't pay because you didn't spend anything (reactive component cannot do any useful work). Analogously, the sinusoidal components which are off the frequency of the red signal are out of the phase with the actual signal - therefore their power is zero. Analogy is in that the power is always the area under some function and by mixing the signals which are out of phase the area above and under the x axis cancels out. In other workd, if you are using old style vibrating reed type freqency meter, the reeds corresponding to the frequencies where power is zero wouldn't move.

[@PARAMETATRONIC]

[@kexuchen4288]

just like the relationship of two vertical vectors multiply by each other, the answer is zero.

[@miro.s]

@@safatkhan6839 It is not truth that it sums up to zero as it was said in video. It regularly oscillates between bounds with increasing interval of integration and so the integral doesn't exist, because the limit doesn't exist. But integral of red signals sums up to infinity, so we don't need to care about something oscillating in bounded interval that is incomparably so small regarding to infinity.

[@kevinv2441]

Wow, I found the speed of which the equations and examples are written is very effective at holding my attention. However, the speed at which he is speaking and his monotonous tone takes away from his clarity and my ability to absorb all of the information. (I had to replay and pause multiple times!) Yet, Thanks for the upload!

[@chironjo]

That is the purpose; pause and play, thanks to technology. The other way around is what you don't want; too slow talk and you can't speed it up.

[@nickfleming3719]

you actually can speed it up

[@AgentmetdeFlaporen]

@@nickfleming3719 that wasn't a function on KZbin 4 years ago

[@TheFlP4EvEr]

Can you do some example in part 3?