Introduction to the Fourier Transform (Part 1)

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This video is an introduction to the Fourier Transform. I try to give a little bit of background into what the transform does and then I go step by step through explaining the Inverse Transform in detail. I meant to cover the entire topic in this video but I ran of time so now there will be a part 2 which cover the Forward Fourier Transform. Also, I realized that I might have gone too quickly through the end because it I think it's a little hard to follow. 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!
Errata:
9:18 The last of the four terms should be positive [+sqrt(2)/2*sin(2pivt)i] and not negative.
Don't forget to subscribe! Follow me on Twitter @BrianBDouglas!

[@BrianBDouglas]

Thanks for the comment! Sal Khan and the Khan Academy is who I try to model my videos after.

[@confidential303]

What board are you using if I may ask? It looks very organised with those colors.

[@BrianBDouglas]

One month ago a viewer named DrRichardRobinson pointing out that I had a sign error on the last term in the equation and I agreed and said I would put an annotation so nobody else get's confused. Somehow I managed to forget that annotation and have thus confused you with an incorrect equation. I apologize. The last term should be +sqrt(2)/2*sin(2pivt)i. Try that now and see if it works out for you ... in the meantime I need to add an annotation...

[@iron4us]

Sounds like roast turkey on the microwave, that magnathingy likes a good old fashion action, good times, good times

[@darshan7267]

9:23 yup here

[@paraescucharrap]

I was watching videos from MIT to understand this and all they do is use complex language going nowhere... saw your video and understood in 10 min what I was looking for. Congrats master!

[@rohanmandloi7561]

True

[@dimplamen]

Dude, if the FT course I took in college had started with an introduction like this: 1. I would have actually understood the concept before we focused on the math 2. I would have realized how useful FT is before (not after) I finished college 3. I would have probably been more motivated throughout the course Sadly, this does not apply to FT only... try asking someone taking an undergrad course in linear alebra what eigenvalues are and what are they useful for. Chances are most people would be able to make the calculations (often this is the only focus of such a course) but would have no idea about the potential applications. Thanks for the wonderful video!

[@audiocorps2334]

Well, at least you aren't being forced to apply the information in matlab, learning matlab, and learning the information at the same time. Got a shit professor for this summer course...

[@audiocorps2334]

It wouldn't be that bad if it wasn't for the fact that he didn't actually attempt to teach it properly... You normally teach the basics before you make use of the language, he completely jumped the step.

[@saqehi]

+dimplamen My professor on linear algebra said that if he was to explain what we needed linear algebra for it wouldn't make sense to you. That's the motivation that we needed it :{

[@theDgrader]

+dimplamen should'a could'a would'a

[@chiekohoki]

I asked my TA to explain the usefulness of Eigenvalues, and he couldn't explain it to me. He said something about circles and then just that I'd understand later.

[@dabulls1g]

Your lecture, speaking, and writing are so well organized. Your work and channel are tremendously valuable to students of a large variety of engineering disciplines. Thank you!

[@eminbolukbas1277]

exactly

[@CreativeCommons11]

Dude you literally have grey scale posting comments! ($)

[@BatmanButtman]

You have literally saved my degree! Give this man a medal!

[@jari11scb]

Worst thing a university can do: teach fourier transform in terms of mathematics in a math course without having the professor explain the correlation between time and frequency. Our professor just stood up there and derived the transform without explaining what it's use is..

[@jam2347]

😂 !

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@Milan Zeke instablaster ;)

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@Kylen Donovan I really appreciate your reply. I found the site on google and I'm in the hacking process atm. I see it takes a while so I will get back to you later when my account password hopefully is recovered.

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[@BrianBDouglas]

Hi Bal Krishna Parajuli, I get this question a lot. So instead of answering it individually like this I'll put out a real short video this weekend on how I make a video. Hopefully it'll answer all of your questions!

[@mponcardas94]

i like your handwriting. for me, it's a great factor in learning.

[@corydiehl764]

This makes so much more sense now. I watched these videos before I took my circuits 1 class, and now for the second time after, and this clears up all the misconceptions I was having.

[@BrianBDouglas]

Hi Frank, it was covered in the time and frequency video but I promised then that I would go into more depth in a later video. Once I finish these I will definitely cover new topics just like you mention.

[@daviscyl]

Hands down the best control theory lectures you can find anywhere online. Thank you Brian! your videos have helped me so much in school, at work, and during my job interview preparations. If there is ways we can donate money, or purchase your book, let us know.

[@blueridgecoin]

Brian, Wow, what a great explanation of the Transform. I recently had an idea for a project and quickly realized I would need the FFT and the IFFT to make it happen. The math is a little above my head so I started studying. Your explanation has brought the math to my level and helped me to get a better grip on FFT. I just wanted to Thank You. Kevin

[@austinmw89]

DUDE. I finally get it. Thanks! Key part for me was "amplitude, frequency and phase are the only components required for full information of the sinusoid." Now the appearance of these frequency domain signals make sense :)

[@BrianBDouglas]

The real part of F(v) is even and the imaginary part is odd ONLY when you have a real time signal. That is you aren't dealing with imaginary time. Which is almost always the case ... unless you're a theoretical mathematician. See the response from Harry Rickards above for a good explanation why this is so.

[@badmintongo4832]

imaginary time and wick rotation

[@LaszloKorte]

To be clear: the time value itself is never imaginary or complex, but always real. and the frequencies themself are also always real numbers. But in the time domain each (always real) time value maps to a (maybe complex) signal value and in the frequency domain, each (always real) frequency value maps to a (maybe complex) value representing the amplitude(re) and phase(im). For complex functions the definition of even is f(x) = conugate(f(-x)), the definition for odd is: f(x) = -conugate(f(-x)). This leads to: if the values in on domain(eg time) are all real, the function of values in the opposite domain(eg spectrum) is even, if the values in on domain are purely imaginary, the function values in the other domain is odd. if the values in one domain are neither only real nor only imaginary, the function of values in the other domain is neither odd nor even.

[@hoomanmousavinasab8]

God bless you. After two years now, I feel that there is some hope of passing the Automatic Control System exam and Industrial Automation exam. I just reached this part of the playlist. By the way, you are an amazing Drawer.

[@davewalwark]

I love the pace and appreciate the production value. It's hard to skip forward correctly on a slow video, but it's easy to pause a faster video (like this) to catch up.

[@Mammutinc]

This is REALLY REALLY great! Very Nice explanation, clear and concise without any clutter or mumbling, nice illustrations and in a good tempo which does not linger but keep the pace the going. Amazing.

[@nikolaivacko5397]

It is scary how easily you can explain such a complex topic, God bless you!

[@henriquemata5651]

It couldn't be better explained. Thank you

[@chiranjibidhakal5191]

I was roaming too many videos in youtube just to know what is fourier transform ..thanks god I clicked your video ...its outstanding and now I made a mental picture of fourier transform ..thankyou so much for this help.

[@xDe1m0nx]

That was overall brilliant. Your explanations, the visuals, just fantastic

[@jeffbarrett2730]

These videos are excellent! Great explanation, neat handwriting, and at a pace that isn't too fast nor too slow. Great work! I enjoyed this one!

[@sk10200x]

Thanks for making such awesome videos. Watching the entire playlist to brush up my basics!

[@tarunpreetkaur1115]

You have explained something in few lines that others took video to. Amazing, detailed and highly simplified explanation. Cheers !!

[@AtulKumar-xv5zu]

You write and explain beautifully. Thanks!

[@BrianBDouglas]

Hello Dr. Robinson, you are correct! I got carried away with my negative signs. I'll put an annotation in there so other viewers won't be confused. Thanks for pointing that out.

[@WoodenDiamonds]

Never seen such a well organized and structured video which explains a difficult topic so clearly and fast mentioning so many details. It answered almost every question I had after my prof's explanation. I'm really grateful. Thanks for your effort!!

[@dev.regotube]

Quarantined at home. KZbin is my school now. Thanks for the lecture!

[@Hartrusion]

The best explaination for this I ever heard. thank you.

[@irateindividual8086]

The best intro to this subject I have seen, bravo!

[@amitferman4723]

This video clearly explains a great interpretation of the Fourier Transform. Thanks!

[@abhinavsarkar8354]

We were taught fourier transforms in maths but they never taught us in such a simple way which included all the reasons and basics. We were just given those integration formulae and asked to solve questions. Thank you. Hatsoff

[@AkshayRoyal]

Your explanation is crystal clear. Awesome work dude.

[@stsk0001]

Excellent lecture, through a splendid lecturer! E.g. the explanation from time- to frequency domain an v.v. and further WHY!

[@millenjanclarkconsuegra4907]

After rewatching it for three times, I finally understood it. BIG THANKS MAN!

[@suhasamalkar]

Very elegantly explained from first principles. Thanks for posting!

[@anannyauberoi7984]

Wonderful video! I was facing trouble with the Fourier transforms due to a chapter in Digital Image Processing, and had absolutely no background knowledge of Signals and Systems. This was so helpful!

[@lindsaywright2818]

You have beautiful handwriting and you explain things very fluidly. Thanks for helping me get ready for my exam!

[@simontadesse5356]

Your videos are awesome and to the point. I really like your organization and clarification. Thanks for posting.

[@ibrahimadeoti4430]

Happy to see you again. Thanks for your good work.

[@joangonzalvez9865]

Your teaching technique is absolutly great!

[@mahmouderaqi6895]

I'm sure you worked hard to be this organized , thank you ,that was really helpful.

[@belovedge]

After struggling with FT for hours, Google brought me to here. Thank you so much! clear and informative explanation!

[@TannerLooney]

Outstanding work Brian!

[@pizzayolo3563]

For everyone struggling around 11:00: I think it should either be explained or left out completely, it's quite confusing like this. It is important to notice that the green box is only the (+)frequency part, you have to do the (-) part for yourself using the plotted properties of F(v). An Example if F = 1+i: (+): (1+i)(c(f)+is(f)) = c(f)+is(f)+ic(f)-s(f) = (c(f)-s(f))+i(c(f)+s(f)) For the negative equation, (1+i) -> (1-i) and f->-f; (-): (1-i)(c(-f)+is(-f)) = c(-f)+is(-f)-ic(-f)+s(-f) = (c(-f)+s(-f)) +i(s(-f)-c(-f)) = (c(f)-s(f))-i(c(f)+s(f) (using the symmetric properties of cosinus & sinus)

[@jerrywu751]

thanks for clarify that. But I'm wondering why does the phase change sign when you do the negative frequency?

[@pizzayolo3563]

@@jerrywu751 Remind yourself, that F(v) is some amplitude and some phase information. The imaginary part lies in the phase information. A simple phase shift would be defined by exp(i*w*t) which is (cos(w*t) + i*sin(w*t)) but usually it is a sum of multipla (arbitrary number) of cosines and sinuses. What we can say for sure that the complex part only containes the sinuses, and therefore is an odd function (since the superposition of multiple odd functions is odd, and a sinus is odd). This means, the imaginary part of F(v) changes the sign when v changes the sign. Therefore also tan(phi) = Im(F)/Re(F) changes the sign. When the tangens changes the sign, the corresponding angle changes the sign. Also think about this: When something oscillates in the "other direction", the phase shift changes sign: This is pretty intuitive.

[@jerrywu751]

@@pizzayolo3563 thanks a lot!

[@geoffeolson]

I still don't understand something. I got the same positive frequency equation that you show. However your positive frequency equation has one sign that is different from the equation in the green rectangle at the video time 11:00.

[@yassermorsy8481]

This makes more sense... But what does it mean when the amplitude for the positive and negative is only half? I used the original amplitude of sqrt(2)/2 and after finding the -v and +v, I end up with sqrt(2) when adding, which is not the same as the original amplitude of sqrt(2)/2. I get the same general form that he gets, but not the right amplitude.

[@rohitn6910]

Not only individual concepts but their relation is also important ...And this is what you have cleared through these videos ,😇🤘🏾👍🏾thank u

[@abdulmajeedmohameed6398]

Don't now how much I suffered when I took this course , the book I had was writing by our teacher and it was like ** just as the teacher was .. really thank you for what you do 🙏🏽

[@rejathsimon9001]

The best lecture ever on fourier transform....

[@plekkchand]

very well done, much better than some others Ive seen.

[@gyozakeynsianism]

Really great video. Thank you. Helps this economics grad student a lot. Time series stuff is a lot easier now.

[@andreyOMARama]

Simplest Fourier transform explanation ever!!!

[@Muuip]

Great concised information. Thanks for posting.

[@HarveyBirdman7]

Thanks, I thought I was not capable of understanding Fourier Transforms for my mechanical engineering lab class, but this was really clear.

[@markgargon1223]

Thank you for uploading your videos, it really helped me understand more clearly, thank you, please don't stop uploading your lecture videos... Mark from the Philippines...

[@alinapol]

so professional. thank you!

[@mophez]

Great lecture. Please do not stop.

[@BuildThreadsNorway]

Thanks you so much for giving an understandable explaination on this this. By the way, your drawings are really nice!

[@brahmaduttamahapatra9360]

Sir, ur vids are awsome. honestly it filled with conceptual approach. HAPPY NEW YEAR and plz continue showering ur knowldge

[@Alohamora_13]

Thanks a lot Brian! This is brilliant.... and explained very nicely!

[@yousefmorcos6979]

I really appreciate your videos.. thank you so much Brian. this has been so helpful.

[@fukgovernment]

Amazing videos sir.They helped a lot thanks!

[@ShizzleMyChizzle]

Absolutely brilliant explanation

[@wassollderscheiss33]

I am at 9:12. The imaginary sine term of the expanded (green) equation should really be positive because (a+ai)*(cost+isint) gives only one negative term (the one with i^2, which is the real sine term).

[@AdamKlingenberger]

I thought I was going crazy, glad I checked the comments if anyone else caught it.

[@markgross9582]

Same!

[@prateekjaipur]

you have to change (1+i)/2 to (1-i)/2 as well for getting the correct result

[@alchamiwa]

@@AdamKlingenberger i did the same heheh

[@poppyflorist]

Thank u sir. I was going crazy

[@BrianBDouglas]

Hi Adam, go to the "List of trigonometric identities" page in wikipedia and scroll half way down to the "Linear combinations" section. That first equation will show you how to combine them.

[@bot5am]

Thank you. I had never understood what all those integrals meant before. You are brilliant. Greetings from India

[@crysise94]

Amazing Amazing Amazing....I loved it This is how undergrad classes should be

[@shafiraanindita3087]

Soooo good! saved a lot of time, thanks Prof.

[@tretolien1195]

So simple, so easy, so fast, so ... ... Engineer, I love it!

[@adityapimparkar6829]

Just what I wanted. Helpful for the Laplace transforms.

[@jasonott7538]

Earth sciences major with a physics focus here...taking mathematical physics 2 this summer quarter, and honestly have been feeling pretty lost due to rambling explanations of my instructor and the tendency of Boas' textbook to occasionally imply information instead of explicitly writing it out. This video cleared up a great deal of my confusion about Fourier transforms...thanks!

[@SHONSL]

Great location analogy and highly instructive video!

[@juliodemedeiros]

thank you so much, your explanation helped me a lot.

[@kanjilalbaishali]

fantastic explanation. Loved the video

[@curtpiazza1688]

This looks so INTERESTING! 😊

[@andrepedersen8397]

Absolutely brilliant described and told. Subscribed! :)

[@Demonlogist]

Thanks you this video as well. I love it when he sync his drawing speed so you can watch diagrams with the same speed as you speak :P

[@anasal-daghestani6371]

this was very helpful.... im taking signal processing and I havent taken fourier transform in a math course.... THANKS

[@BrianBDouglas]

Hello Waranoa, I probably messed you up by writing 'i' at the end of the imaginary part. When you put the real and imaginary component into the amplitude and phase equations the 'i' doesn't come along. Because we are calculating the length of the line on the real and imaginary axis we're just performing pythagorean theorem. And for that we just need the value of each component. So it's real = sqrt(2)/2 and imaginary = sqrt(2)/2. Hope this helps.

[@captainvlog]

This video has 300K views. It's great to see that much interest in controls engineering.

[@ariearie5054]

You sir, are a master of your craft! Thanks for the amazing video's

[@SheilMehtaYNWA]

Thanks so much! you've been a great help! it was easy to understand even though I'm not at a very high level of math

This is an amazing explaination!

[@SlowMister112]

great introductory class

[@engman6]

Because your videos are brilliant, I would suggest that you compose a one play list that has all the videos in the order of a complete course.

[@bigfactsbroski]

this was a really good video, thanks

[@Winston01]

This is a wonderful unpacking of this operation. But Jeebus! You either need to slow down or break it into smaller chunks at a slower pace. But seriously, thanks for your efforts here.

[@atulsain6170]

Winston thanks for adding this. You made this video more useful.

[@tensor131]

yeah I had to hit the pause button a few times and even rewind once or twice ... thank you video!

[@mohamedradwan388]

Now I finally understand !! Thanks a lot.

[@Infosus]

Holy F***, you sir in just 5 minutes made more sense then my lecturer did in 2 hours. Thank god people like you exist =D

[@abbashoseini9344]

thank you , thank you , and again thank you ... you explained it really really clear

[@BrianBDouglas]

Hello, is it Dmytryj? My keyboard doesn't make those letters :-) I use Photoshop with a Bamboo tablet from Wacom to make the drawings. Although you don't need anything near as powerful as Photoshop. You could probably use something free online like Sumo Paint. I also use the video screen capture from Quicktime to record. Best of luck with your presentation!

[@MrStudent1978]

Very nice, very clear explanation... Thanks

[@ckalas]

these videos are so much more succinct and intuitive than my lecturer's.

[@tensorbundle]

amazing talent you have ..I admire you

[@ap2139]

Just brilliant. Thanks from Italy :)

[@AbhijeetSachdev]

Thanks a lot sir :) Love you Keep ur blessing like that with all us :) You just made me feel fourier uncle :)))))))

[@datakprosto123]

talent explaining! keep it coming!

[@s.platini9390]

Awesome Lecture!

[@igorsabadini]

what a video my friend, i was punched by knowledged. Thanks for the video, it helped a lot.

[@chiruswapnil]

best tutorial ever. Thanks man