Introduction to the Fourier Transform (Part 1)
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- Опубликовано: 12 сен 2024
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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.
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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!
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...
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.
+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 :{
+dimplamen should'a could'a would'a
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.
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!
exactly
Dude you literally have grey scale posting comments! ($)
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!
True
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...
Sounds like roast turkey on the microwave, that magnathingy likes a good old fashion action, good times, good times
9:23 yup here
Thanks for the comment! Sal Khan and the Khan Academy is who I try to model my videos after.
What board are you using if I may ask? It looks very organised with those colors.
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!
You have literally saved my degree! Give this man a medal!
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..
😂 !
You probably dont give a damn but does any of you know a method to get back into an instagram account??
I was stupid lost the login password. I love any help you can give me!
@Milan Zeke instablaster ;)
@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.
@Kylen Donovan It worked and I now got access to my account again. I'm so happy:D
Thank you so much you saved my account !
i like your handwriting. for me, it's a great factor in learning.
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)
thanks for clarify that. But I'm wondering why does the phase change sign when you do the negative frequency?
@@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.
@@pizzayolo3563 thanks a lot!
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.
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.
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).
I thought I was going crazy, glad I checked the comments if anyone else caught it.
Same!
you have to change (1+i)/2 to (1-i)/2 as well for getting the correct result
@@AdamKlingenberger i did the same heheh
Thank u sir. I was going crazy
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.
Winston thanks for adding this.
You made this video more useful.
yeah I had to hit the pause button a few times and even rewind once or twice ... thank you video!
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.
imaginary time and wick rotation
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.
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
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 :)
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.
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.
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.
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.
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.
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.
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.
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!!
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.
At 11:18 I am unable to convince myself that the imaginary term cancels out after "negative frequency and sum with positive". With a phase shift, your imaginary part no longer remains an odd function (and your real part no longer remains even). Example; consider a phase shift of Pi/2=90deg, you have F(nu)e^{i*2*Pi*nu*t) = (e^{i*Pi/2})*(e^{i*2*Pi*nu*t}) = e^{i*2*Pi*nu*t + i*Pi/2} = cos(2*Pi*nu*t+Pi/2)+i*sin(2*Pi*nu*t+Pi/2) = sin(2*Pi*nu*t)+i*cos(2*Pi*nu*t). Real part is an odd function and imaginary part is an even function, so now when you take negative frequency and add with positive you keep only the imaginary term. When looking at your example; you have F(nu)*e^{i*2*Pi*nu*t} = (sqrt(2)/2)*(cos(2*Pi*nu*t) - sin(2*Pi*nu*t) + i*cos(2*Pi*nu*t) + i*sin(2*Pi*nu*t). The imaginary term i*(sqrt(2)/2)*(cos(2*Pi*nu*t) + sin(2*Pi*nu*t)) is not an odd function and would not cancel out with F(nu)*e^{i*2*Pi*nu*t} + F(-nu)*e^{i*2*Pi*(-nu)*t}. Not sure if your explanation of why we ignore the imaginary term is incomplete, or if I am missing something. Great series by the way, I am finding it very helpful.
I was having the same problem... But I think I got the answer. When you calculate for negative frequencies you have to change the sign of F(nu) as well... So for negative frequencies it will be 1/√2-i1/√2 . Thereafter things work out well. Its been more than a year... So I dont know if it will be of any help now.
@@souptikde8551 But F(nu) should be the same signal right? we are trying to calculate for the negative frequencies for the same F(nu) = 1/√2+i1/√2 right? Imaginary part signifies the phase and we don't have any change in phase. Why should we change the sign then for F(nu)? I'm stuck at this point.
@@johnknox8655 Negative frequency means the spirals in complex plane move in opposite direction. Thus you need to take conjugate of F(nu).
@@souptikde8551 That makes sense. Thankyou. Need to study about complex planes.
@@souptikde8551 But the real part should be doubled after summation ,right?
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
It could improve clarity if, when presenting Euler's formula, you used e^(i*Theta) instead of e^(i*t). Because the t in that expression actually corresponds to 2*pi*nu*t (which is an angle, e.g., theta) in your expression A(nu)*cos(2*pi*nu*t).
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.
It is scary how easily you can explain such a complex topic, God bless you!
3) The Laplace transform converts a time domain signal or function into the S-domain. This is not only frequency content but also exponential decay and growth. So the s-domain is a two dimensional plane with frequency on one axis and exponentials on the other. Certain math operations are much easier to perform in the S-domain (like solving differential equations, and combining two systems together in series).
I hope that helps, after reading through my answer it seems kind of confusing :(
Quarantined at home. RUclips is my school now. Thanks for the lecture!
You have explained something in few lines that others took video to. Amazing, detailed and highly simplified explanation. Cheers !!
"because they are only wave forms that doesnt change shape when subjected to a linear time invariant system." Where is the link for explanation of this term said at 3:40 ?
I am also looking for the link
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.
You have beautiful handwriting and you explain things very fluidly. Thanks for helping me get ready for my exam!
I followed until @8:49 ; when we break it out into real and imaginary numbers, what are these representing? My intuition stops at this expansion.
+1 i cant follow from there
the A(v)= amplitude , with added phase info becomes= F(v): and Cos (wt)=is replaced by Cos(wt)+isin(wt): eqn becomes F(v)e^iwt. intuitively its just change interms of adding phase and replcing Cos term with e^i term
You just expand the terms and do the algebra. There was an error in the video through: the last term should be (minus) -sqr2/2 sin(wt)
i mean positive ... the other way around
The real part of F(v) is just a summation of cosines, and as you can see from drawing a cosine graph, cos is an even function. Similarly, the imaginary part is a sum of sines, and sin is an odd function so the imaginary part is odd.
First of all, I am blown away by your ability to write so legibly. Second, I didnt get quite what I wanted because of no fault of yours: I came across fourier transforms in my stats class, so I am trying to figure out whats going on there, but all the videos cater to electricians (aka jerks, lol, jk); nonetheless, its apparent to me that your video is awesome and Im going to go home and rewire the computer, television and whatnot.
Wile E. Coyote how much he pay u? lol
Wile E. Coyote Seriously right? And fast as hell!
Wile E. Coyote *twitch* electronic *twitch* engineers!
Wile E. Coyote EE's aren't all jerks. It's not our fault we're smarter than you ;)
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!
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 🙏🏽
Excellent lecture, through a splendid lecturer!
E.g. the explanation from time- to frequency domain an v.v. and further WHY!
11:43 "I'll leave that up to the math department to explain how to go through all that..."
Exactly what my physics professor says all the time! Haha :D
Thanks for all the help, man.
Control systems eh, I'll be taking that paper next semester.
If I had a penny for every time that was said. XD
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!
It couldn't be better explained. Thank you
Hi Brian, firstly thanks a lot for these great videos. I am unable to figure out at 10:17, how the real part is a an odd function and img an even fn. I hv gone thorugh the below comments where harry has said that real part is cosine while img part is sine but it no where seems like that from the equation...It may be I am missing on something small..Please explain.
Try seeing like this :
The function's y-axis values are always positive for all values of x co-ordinate in the real part ;however thats not the case with the imaginary part ! Hope you understand :)
That was overall brilliant. Your explanations, the visuals, just fantastic
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!
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!
The best intro to this subject I have seen, bravo!
You write and explain beautifully. Thanks!
After rewatching it for three times, I finally understood it. BIG THANKS MAN!
Very elegantly explained from first principles. Thanks for posting!
I'm sure you worked hard to be this organized , thank you ,that was really helpful.
Your explanation is crystal clear. Awesome work dude.
this was very helpful.... im taking signal processing and I havent taken fourier transform in a math course.... THANKS
After struggling with FT for hours, Google brought me to here. Thank you so much! clear and informative explanation!
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
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...
Not only individual concepts but their relation is also important ...And this is what you have cleared through these videos ,😇🤘🏾👍🏾thank u
Your teaching technique is absolutly great!
Let's see. Here's how I would view them.
1) TF is the Laplace Transform of the impulse response of a function. It can be considered the relation between IN and OUT also, but it is not in the time domain it's in the S-domain. It allows us to multiply systems together rather than convolve them in the time domain (much harder)
2) The Fourier transform converts any time domain signal into a sum of harmonics (frequency domain). This is just a subset of a Laplace transform (only the jw line)
Thank you. I had never understood what all those integrals meant before.
You are brilliant.
Greetings from India
This is very strange and great approach to start explaining the inverse fourier transform before the forward. Strangely starting with the inverse makes more sense 👏
But I find difficulty internalizing the part from 9:17
Your videos are awesome and to the point. I really like your organization and clarification. Thanks for posting.
The best lecture ever on fourier transform....
So simple, so easy, so fast, so ... ... Engineer, I love it!
Happy to see you again. Thanks for your good work.
Thanks, I thought I was not capable of understanding Fourier Transforms for my mechanical engineering lab class, but this was really clear.
Thanks for making such awesome videos. Watching the entire playlist to brush up my basics!
Hey Brian can you answer these questions (or anyone who knows): Q1. To get the Fourier transform from a Laplace transform function, we make s=jw. We know the Fourier transform of cos(w0*t) is 0.5*[d(w-w0) + d(w+w0)] and we know the Laplace transform of cos(w0*t) is s/(s^2+(w0)^2). If we make s=jw for the Laplace transform of cos(w0*t), can we say 0.5*[d(w-w0) + d(w+w0)] = jw/((jw)^2+(w0)^2)? Q2. This one deals with the discrete time signals and the z-transform. For the Fourier transform, if we need it to handle discrete signals or inputs, all we need to do is change the integral sign to the summation sign. Why can't we do the same for the z transform or Laplace transform. Why can't we change the integral sign of the Laplace transform to the a summation sign to handle discrete signals or inputs, or why can't we change the summation sign of the z transform to the integral sign to handle continuous signals or inputs. In short, why can't we use X(s) = sum {n = infinity to infinity} x[n] e^(-sn) for discrete signals, and why can't we use F(z) = integral {t = 0 to infinity} f(t) z^{-t} dt for continuous signals. If we could do so, we only need to use one transform for both discrete and continuous signals. I have read a lot on both these transforms but I can't seem to find the answer. Am I missing something.
Really great video. Thank you. Helps this economics grad student a lot. Time series stuff is a lot easier now.
There is an error in the video at 9:08: the last term should be (positive) + sqr(2)/2 * sin(2*pi*nu*t). The only term that can result in negative sign is when i * i for the real part.
The best explaination for this I ever heard. thank you.
Usually when I watch some lectures I choose 1.25 play speed. Now I thought shouldn't I lower it to 0.75)
That was nice, fast, and straight to the point
I was about to write the same thing. I feel like we need a "pause and ponder" moment like 3Blue1Brown does. It would let me take in what just happened. I play Douglas' videos at .75x too, so much information. Check out the control theory book, it's really good. Fun little drawings and it's a nice compliment to the standard textbooks too.
the video is great, but I think if it was a bit slower it was much easier to grasp. Pace matters a lot.
At 10:00 you claim that the imaginary component of F(v) is always odd, but at 8:31 you write 'F(v) = sqrt(2) / 2 + i*sqrt(2) / 2', which is even with respect to v. What gives?
8:41 How does that become the function f(v)*e^(2pi*i*v*t)
I think I figured it out, it's wrong, he just messed up trying not to mention negative frequencies yet. It's A(v)cos(2πvt) = [F(v)/2]e^2πivt + [F(-v)/2]e^-2πivt
Dope video, the only part which confused me for a while was at 9:10 when you expanded F(freq)e^(2i(pi)(freq)t) and wrote the expansion, I wasn't fully sure what you expanded to get the answer, I got it now
(i·sqrt(2)/2 + sqrt(2)/2)(cos(2t(pi)(freq))+isin(2t(pi)(freq))
Maybe an annotation would be helpful?
or maybe it was just me who got confused here, lol
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
Amazing Amazing Amazing....I loved it
This is how undergrad classes should be
very well done, much better than some others Ive seen.
Thanks you so much for giving an understandable explaination on this this. By the way, your drawings are really nice!
can i just comment on how nice the writing is even with a computer mouse, I am impressed!
I'm still having trouble understanding how you expanded F(v)e^(2pivit). :/
Memo Pony Here: F(v)=√2/2+i*√2/2 and from Euler's formula you know that e^it=cost+isint. Putting things together lead to: (√2/2+i*√2/2)*(cos(2pivt)+isin(2pivt)) which expanded gives you the result Brian pointed out. Hope to have adressed your problem
Wouldn't the sin term in the imaginary part of time expand out to be plus instead of minus? Or am I missing something here?
Anybody tell me the explanation for the solution for negative frequency please !
9:09 yeah, should've been positive
yes there is a correction actually. its a + .
This video has 300K views. It's great to see that much interest in controls engineering.
Many people on you tube use this technique for teaching and lectures. I also would like to find out if this is video or software for a writing tablet. Brian can you explain? thanks Joe
+Joseph Novotka I explain it all in my video called "How I make a control systems lecture video". You should be able to find it if you search for it on my channel. Also +Richard G posted a link to another explanation: www.teachthought.com/uncategorized/how-to-screencast-like-the-khan-academy/
+Brian Douglas Excellent video. I'm afraid I couldn't understand what you meant by solving F(v)e^(2*pi*i*v*t) for negative frequency and adding it to the positive ones (10:40 onward). Once you expand F(v)e^(2*pi*i*v*t) and classify it into real and imaginary parts, what exactly do you mean by 'solving' for positive and negative frequencies?
Sir, ur vids are awsome. honestly it filled with conceptual approach. HAPPY NEW YEAR and plz continue showering ur knowldge
Just what I wanted. Helpful for the Laplace transforms.
This video clearly explains a great interpretation of the Fourier Transform. Thanks!
Simplest Fourier transform explanation ever!!!
Yikes! That is a fantastic question and one that would be hard to answer in text here. If I can squeeze it into the next video I will. I tried to write it out here but it would have been too confusing.
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.
Thanks a lot sir :)
Love you
Keep ur blessing like that with all us :)
You just made me feel fourier uncle :)))))))
I kinda got lost at the polar coordinates and amplitude part, but picked back up at the euler's. Not my field so it is understandable - very helpful video. Thanks for the post.
Just awesome! If you read this! Please comment back! I have never seen such an awesome explaination! With such technology!!!!!!! I just love it!!!!!
Great lecture. Please do not stop.
these videos are so much more succinct and intuitive than my lecturer's.
Outstanding work Brian!
so professional. thank you!
Thank you finally someone who speaks english.