Convolution and the Fourier Series

How the Fourier Transform Works, Lecture 6 | Convolution and the Fourier Series
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howthefouriertransformworks.com/
What is Convolution? What does it have to do with the Fourier Transform? Have you ever wondered what the Fourier Transform equation meant? Why is my mischievous little 2-year-old son hiding my slippers? I hope that by the end of this lecture, I will have explained the process of Convolution to you clearly enough that you'll understand how Fourier's revolutionary idea actually works... and yes, the slippers are really part of the explanation.
This is the eighth and final video lecture that I am releasing from the course for the time being. More videos are in preparation. While I am producing the rest of the videos, the second half of the course can be found in blog form at: howthefouriertransformworks.com/
Please help me finish filming the course by supporting me on Patreon: www.patreon.com/MarkNewman
A transcript of this lecture can be downloaded from the following link:
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#learning #mathematics #math #convolution
Thanks for watching the video How the Fourier Transform Works, Lecture 6 | Convolution Signals and Systems

[@vishnukartha47]

I cant believe, how much work this guy put for making us understand the whole concept that easy. Hats off. 👍👍👍

[@MarkNewmanEducation]

It was a labour of love.

[@vishnukartha47]

@@MarkNewmanEducation love from India😍😍😍😍😍

[@MarkNewmanEducation]

I've wanted to understand the Fourier Transform for years, but the maths was always beyond me until one day I was playing around with Sines and Cosines in Excel and discovered that if I added a Sine and a Cosine at the same frequency but different amplitudes together, it caused a phase shift in the resultant signal, all of a sudden I finally understood what the whole thing of complex numbers meant. Then I wanted to make sure I had understood it properly. The best way, I find, to really understand something is to try and teach it. Hence, the course.

[@vishnukartha47]

@@MarkNewmanEducation, every time i watch the video, i feel wonder, because how could one spend this much effort and sincerity with free of cost. In India we call people like you as Sanyasi, the one who is doing any activity with out expecting a profit. All they do is for the betterment of whole world. 🙏.

[@usmanzafar4751]

Yours is the most clearest explanation of Fourier Transforms that I found Online. I can say that Today I finally understood Fourier Transform. Thanks...

[@martonkristofpalinkas7936]

Certainly a hidden gem, and yet the one that shines the brightest! Thank you, Mark, you are great!

[@MarkNewmanEducation]

You are most welcome. Thank you.

[@kdg66]

Sir, Thank you for the brilliant six videos so far on the Fourier transform. The build up to this video was a great refresher for me, but the section here on convolution is a real eye opener to me as never understood it and thus how the FT comes about. I studied Chemical Engineering (1985 - 1988) with modules on Laplace transforms for use in Chemical Process Control. I never understood it intuitively. Now 28 years later, having a bit more time to re-visit this, I'm slowly joining up the pieces, (I have an average brain). Your work has been a tremendous help to me. I've not fully actualised the matter yet, but am getting there. I've got to get my "layers of mathematics" in the correct order & down on paper. I hope your efforts pay-off with your career, as your clearly a natural tutor and educator of difficult subject matter. Warm regards from UK.

[@MarkNewmanEducation]

Thank you for your kind words. I do think that mathematical concepts are generally explained in a very specific way which I have always found really hard to understand. Maths is sometimes thought of as its own world quite independent of our own. I like to see it as a language to describe the physical processes of our real-life world. That's what I like about the Fourier Transform. It has so many real world applications and I think it best to demonstrate it via those applications such as sound as I do in the course.

[@ZenoDiac]

Beautiful description of Fourier Series. Truely more students need to watch this.

[@MarkNewmanEducation]

Thank you. Please help more students see the video by sharing it. See also howthefouriertransformworks.com/ for more on the Fourier Transform.

[@Amine-gz7gq]

Fourier, Laplace and the mathematician who played a major role in the development of wavelets (Yves Meyer) are all French! I finally understood the Laplace transform only when I understood what "e" really is and the function e(growth rate * growth time) thanks to the book "Better Explained" by Kalid Azad and after watching a video by Zach Star (KZbin channel) where he talks about the Laplace transform. For a better visualization of the Fourier transform, I can recommend William cox (gallamine channel). Your video nailed it Mark even though I only watched a few minutes and I'm not exaggerating! I now know that the wavelet is another tool to analyze a function and you explained in one word and clearly what the Laplace transform is, congratulations! When I was a student I had classes on control theory and I wish I had this kind of content, I was so frustrated that I didn't have a clear view of "e" (which is just as important as pi), Fourier and Laplace transforms and how these mathematical tools relate to system controls (differential equations), now that I'm 32 I can finally say that I understand these abstract topics that have bothered and frustrated me for years. Educational systems suck (and Wikipedia too which I used to consult a lot when I was young and it didn't help me at all to see clearly), I think a lot of teachers don't understand these things clearly and the emphasis is not on explaining the origin of the topics taught and what exactly they are for and visualization but rather to use them stupidly (calculate integrals, etc...), if I don't understand the nature of the tool and I can't visualize it, I can't use it.

[@MarkNewmanEducation]

I completely agree with you and thanks for the list of resources. I am always scouring KZbin for my research into the topics of my videos so the resources you mentioned will be very useful to me. Thank you.

[@thefitnesstraits7529]

I am literally in tears .. of joy of course !! I am an analog design engineer this is a pure joy... after watching 100s of videos brought me here to understand the basis of Laplace transform.

[@JustinNinan]

Such beauty to Math! Such clarity in these videos and WHAT quality to these animations!!! Thank you so much Mark Newman, THANK YOU, THANK YOU, THANK YOU! and if you ever stop making such videos, I will find you and I will twist your arm! Because, by doing so you will be depriving humanity from LOVING MATH! :D

[@littleKingSolomon]

Dear Prof. Mark Newman, you make learning so much fun. This fourier transform series is very enlightening and educative and yet very entertaining. Been on this somewhat hard-to-grasp concepts for about 3 hours and I still want more. Can't thank you enough.

[@jpunditreviewsandtutorials6878]

I’ve been searching for a video like this for months. This video should have more views!

[@maheswarapudixit4345]

This was the best explanation I could ever encounter to understand Fourier ! Thank you so much . We all indebted for this incredible and lucid explanation . Looking forward for much videos .

[@samuelmilki4181]

these lectures are just amazing!!!

[@MarkNewmanEducation]

Thanks. I really enjoy making them. I'm currently looking for funding to try and help me complete the course. These lectures took me 4 years to produce as I can only work on them in my free time and I keep having these hugely complicated ideas which I love making into reality, but it takes AGES!!

[@vibinjacob4610]

kzbin.info/www/bejne/gHeZpJ6cacqUepY

[@victorpan1531]

this is the best video i've watched and i watched it in normal speed! i've watched all other videoes in 2x speed. i especially like the treadmill you used. lol. you are a genius.

[@lazyboy9089]

Your guidance and patience have made learning this complex topic a joyful journey, knowing it's all thanks to your incredible teaching.

[@MarkNewmanEducation]

Wow, thank you!

[@curtpiazza1688]

This is wonderful! I'm beginning to understand this subject! Thanx! 😊

[@hygenz26]

Thanks for the lecture! Your explanation about fourier series and fourier transform is the very intuitive and comprehensive, especially your explanation about the fourier series formula. Nicely done!

[@MarkNewmanEducation]

Thank you.

[@rauldurand]

This is amazing. How this lecture has no more views?

[@MarkNewmanEducation]

I haven't got the hang of KZbin's algorithm yet. 😥

[@acluster3411]

The whole presentation of the idea with visualization and animation is done to the perfection. You have figured out the art of story telling of the math. Hope you take up few more complex topic and make those videos to help generations of people for eons to come! Thank you.

[@DasKishor]

Simply brilliant!!!

[@rajabali9701]

Awesome work

[@praveenthomas8919]

Thats the best video I ever saw explained in the most effective way on the subject..Thank you very much..

[@MarkNewmanEducation]

So nice of you, thanks. Glad it was helpful.

[@anandbavkar8572]

Thank you very much Mark!! First time seeing the light when it comes to Fourier Transform...👏 Keep up the great work❤

[@MarkNewmanEducation]

Amazing. Check out my channel for more videos like this.

[@mohsengolzardian3497]

You are absolutely creative dear Mark you make me enjoy and appreciate it I need more connections way with you, I have got many ideas in mind for long time and need to share them with you Thanks.

[@sybrpnk]

Absolutely incredible, these lectures are so thoughtful and well put together. I thought I might never really understand how this transform really worked, but you made it so clear and intuitive. KZbin algorithm sent me here so hopefully you get the millions of views you deserve soon!!!

[@MarkNewmanEducation]

Thank you so much for your kind words. More are videos are on the way. I've just completed filming, and I'm now in the editting stage of the first of a 3 video set which explore the changes which Derichelet made to the Fourier Series to change it into the Fourier Transform.

[@sirousmohseni4]

I live your style of presentation. Many thanks

[@jacobmller5278]

Beautiful video work and clever explanations!

[@MarkNewmanEducation]

Many thanks!

[@christopheracob3689]

Thanks Mark, may God give you more wisdom and merry Christmas! Your presentation, kind of a great art, enlightened me as you had the mastery of the subject and clear delivery!

[@MarkNewmanEducation]

Thank you for your kind words and good wishes. I hope you enjoyed the festive period.

[@yf1177]

Holy Fourier this is a brilliant video!

[@MuhammadGhufran5]

THANK YOU SO VERY MUCH FOR THESE LECTURES PROF. NEWMAN!!!!!!! THESE ARE THE BEST LECTURES EVER1!

[@MarkNewmanEducation]

You are very welcome, although I am no professor. Just a humble electronics engineer who needed to understand the Fourier Transform for work. I'm making more lectures, but the videos just take ages to produce. The unproduced lectures currently exist as blog posts. These posts will form the basis for the scripts for the videos. See howthefouriertransformworks.com/ for the whole course in its current form.

[@marcioaugusto4942]

Nice, very nice job. It's unique! Congratulations and a lot of thanks. You've inspired me, not only because I found the lessons that I'm looking for... but because you teached me how to teach. I believe that anyone can understand anything, but students have no motivation when they can't undestand a new concept at first look... this is because the teachers don't prepare his lessons, they have no empaty, they don't care about the different ways to explain the concepts... You are a good teacher of teachers! My best regards, from Brazil.

[@gavinpeters9531]

This is great! Well explained.

[@MarkNewmanEducation]

Glad it was helpful!

[@julians7785]

You deserve way more subs and views! Great content

[@bradzepfan]

extremely well done. the best. you are a blessing

[@MarkNewmanEducation]

Wow, thank you!

[@ethandrood]

Wonderful.

[@kishorsahu4271]

Your teaching style is amazing and simple

[@MarkNewmanEducation]

Thank you.

[@carlemilhansen5531]

Thank you so much!

[@Murphyalex]

I had so many 'aha' moments watching this video. I think I first tried to learn about the Fourier Transform about 5-6 years ago and the mechanics have always been a mystery. But now, for example, knowing the whole sine-and-cosine-component trick, used to arrive directly at the phase value via inverse tangent of the dot products, now when I visualise some of the equations it finally means something more than just memorised symbols. + The slippers example was a nice metaphor :) Really appreciative of these videos, Mark.

[@MarkNewmanEducation]

I'm so happy. Thanks for writing. Yes, those aha moments... that's what made me want to do the course. My aha moment, the moment when the missing bit of the Fourier puzzle fell into place for me was that day I was playing with a Sine and Cosine wave in Excel and I noticed that when I added them together, so long as they both had the same frequency, they changed the phase of the summed wave. The moment I saw that... I suddenly, finally understood what Fourier was doing. Everything fell into place. Why he used complex numbers, what convolution was, how he used it to find the frequencies in a signal. It is amazing how just one missing link can make or break one's understanding of a subject.

[@markmcflounder15]

This is brilliant

[@afonsogomes2903]

Very nice way of explaining what convolution means, congrats!

[@MarkNewmanEducation]

Thank you. I enjoyed using our little boy for the analogy. He played his part very well.

[@hemas5663]

Really awesome lecture series.

[@MarkNewmanEducation]

Thanks. I really enjoyed making it and am making more... it just takes time as the process is slow. I love it though and I only wish I could work on this full time.

[@cayezara8110]

Beautiful and awesome explanation. The illustration is fascinating and visually crystal clear that helps me understand the history and application of complex numbers in relation to advanced mathematics which is the Fourier Transform, Lapalce Transform, and the likes. I had been watching video lectures from MIT Professors but it wasn't presented as clear as what Mark Newman did. I may call him now Dr. Mark Newman or Professor Newman. With the aid of this video, I believe it is easy for me now to understand the Signal and Systems. PTL.

[@ffhashimi]

Very informative and entertaining Video; it's the best explanation about this subject I ever found; surely I will buy the full course when it finished(hope very soon); the new site is clear and more organized than the old blog; it's very good work you are doing, thanks

[@MarkNewmanEducation]

Wow thanks.

[@MarkNewmanEducation]

I still have some work to do on the site. I moved the blog over from the old site and not everything is working yet, but I'll have it sorted soon.

[@GimbertLane]

Brilliant

[@findLuxuryHub]

I wasted 5 year learning electrical engineering in college till i get your video, הודה רבה אחי

[@MarkNewmanEducation]

בכיף. אני שמח שהצלחתי לעזור.

[@bernardrevenaz896]

Great job, maybe you could explain a little more how the real amplitude is calculated from the obtained score. I learn Laplace and Fourier transform at school : It is interesting to talk about the common points betwin these two wondeful tools. Bernard. France.

[@sergten]

This is a lot easier to understand for mere mortals like myself than 3blue1brown. What I was unclear about is now clear: his winding and tracing of the center of mass corresponds to multiplication and calculation of the area. And I was missing the connection that sine and cosine of different amplitudes cause a phase shift. Looking forward to the next videos.

[@MarkNewmanEducation]

Wow, thank you so much for the compliment. I was always so frustrated when I was learning about the Fourier Transform at university. I could see that it was something amazing I couldn't quite understand how it worked. Now that I do, I really want to help others understand this most amazing of tools. I'm so glad I was able to help you on the point you mentioned in your comment. Thanks for your support.

[@sriprakhar5561]

I was watching this video after passing out the static signals ..but funny thing is I watched this video and completely watching the circle of fifth in piano... You are an brilliant guy..hats off!!🤗🤗🤗

[@MarkNewmanEducation]

Thank you for your kind words. To my mind there is a lot of maths in music. The logic that sits at the heart of musical theory is very mathematical.

[@rolandgerard6064]

Thanks Mark.

[@MarkNewmanEducation]

You're most welcome

[@abdulkhalidabdulmukti8861]

Really great explanation❤Keep up the great works really need this😂

[@MarkNewmanEducation]

Making more videos all the time. I love working on them and I love the feedback people give. It's really nice to know that these videos are helping people. Please share.

[@subasinidas1711]

🙏 Sir, please please please please make a video on discrete fourier transform , laplace transform and how fourier transform is the special type of laplace transform?

[@MarkNewmanEducation]

Hi, thank you for your suggestions. More videos are on the way. All of the ideas you suggested are videos I want to make and are in the pipeline. I am currently working on a video on how to interpret the output of an FFT. Subscribe and set the bell notification to get notified when new videos come out, but I am aiming to release a new video every month, which is how long the production process takes me at the moment.

[@juliensorel7057]

Best explanation ever thanks! This deserves millions of views

[@MarkNewmanEducation]

Wow, thanks!

[@juliensorel7057]

I have been trying to understand Fourier transform for more than a year and explanations always start by showing the integral which doesn't give any intuitive sense of what is happening.

[@MarkNewmanEducation]

That is precisely my problem and why I wanted this course to be the way it is. Maths is a language, but not everyone understands it. A picture is worth a thousand words.

[@musichat741]

Sir could you please explain how you derived the amplitude of a sine component of a particular frequency in the actual wave??

[@markawbolton]

Mark many THX. The quality of your exposition is extremely good. I am surprised you have so few subs. There seems to be no rhyme or reason to it since some really crappy site have more subs. I will make a point of passing your link on.

[@MarkNewmanEducation]

Thanks ever so much. I've been concentrating a lot more on the content than on the marketing of these videos until now. Now, that I finally have a body of work out there (about half of the course), I am turning my attention to try and getting them more widely viewed, as I need to begin generating an income from them in order to be able to continue production on the course.

[@markawbolton]

@@MarkNewmanEducation I really enjoyed Trade School and had a gifted Maths teacher. As Engineers we tended to use crude tools to get the maths done or to rely on conceptual heuristics to understand what a machine is doing. Despite getting a good grade at Trade School i wasn't as much immersed in the Maths as I would like to have been , I was too busy with everything else. You need the content first before you can think about marketing. You are probably at about that stage now. You are obviously enjoying all the graphics and Green Screen stuff. It's kinda goofy but in good taste.

[@dosomething3]

Amazing.

[@MarkNewmanEducation]

Thank you! Cheers!

[@MuhammadGhufran5]

EAGERLY WAITING FOR THE REMAINING LECTURES

[@MarkNewmanEducation]

Working on it. The videos just take ages to produce. The unproduced lectures currently exist as blog posts. These posts will form the basis for the scripts for the videos. See howthefouriertransformworks.com/ for the whole course in its current form.

[@gok_dogan]

19:00 💡 That is the key point of 👉 Difference of between the convolution and fourier. What Joseph Fourier add with his famous formula.

[@Hi-om9bl]

Dear sir, Thank you so much for these visual and intuitive series of videos explaining Fourier analysis- they have been a life saver for my undergraduate degree and are a testament to your education! One thing I can’t seem to wrap my head around is the analogy of vectors and matrixes to the Fourier series that I hear of in the occasional lecture. For example, orthogonal signals, which I suspect explain the shape of our multiplied graph area addition (aka a single sin or cos component of our signal). I am aware of how much effort and time goes into these videos- perhaps a video explanation is not needed. Any ideas? Is this to do with the complex plane too? How do vectors relate to matrixes and is this in 3D or 2D?! (Our lecture uses a 3D analogy but I fail to understand this in terms of sin cos and what else?) thank you

[@r.bhargavram3546]

What is the difference between this frequency matching mechanism in convolution and similarity identification in correlation?

[@muckabeaga2536]

Hi Mark, your sliding test signal explanation is very intuitive, but I'm struggling with the shortcut method. I understand that IQ signals of varying amplitudes produce a phase shifted signal, but it's not immediately obvious how convolving the I and Q somehow "discovers" the amplitudes of the IQ signals required to produce the high score signal. Excellent series by the way, very intuitive. Thankyou.

[@jwsincla99999]

Every math, science, and engineering department of every institution of higher learning should make this series required viewing

[@MarkNewmanEducation]

Wow... Thanks. I'm flattered.

[@hassanomer7777]

First of all thank you for this incredible work I tried to apply what I have learned in this lecture on a function f(x) = sin(x+90) First I multiplied my function by sinx and took the integral between 0 and 2pi secondly I multiplied the function with cosx and took the integral between 0 and 2pi and then I used the inverse tangent rule which gave me 88 as phase shift but the phase shift in my function is 90 degree what is wrong with my calculation sincerely Sincerely

[@lauralhardy5450]

I don't understand how you obtained the score values mentioned at 10:25. You say its the area under the sinusoidal (I understand that) but how did you get the numerical values of 76.07 etc ? Obviously you must have integrated the values on the X and Y axes but can you provide a worked example for say the value of 180, mentioned at 10:33 ?

[@sammyapsel1443]

Thx for the video!: Small question : at 18:00 , Why is the 2nd method(by multiplying the signal by non-shifted cosine and sine) of calculating the score equivalent to the 1st method(by shifting a sine wave of the frequency of interest and calculating the score for every phase shift and taking the maximum) ?

[@MarkNewmanEducation]

Because when you add a cosine wave and a sine wave together with the same frequency but different amplitudes, they produce the phase shifted sine wave that fits the signal at that frequency. Instead of searching for the correct phase by convolving the test wave with the signal, you can calculate it in one go by this method.

[@ranam]

Telling the application of convolution in terms of signal processing may help a part of people in the telecommunication and eee engineers but when convolution meets signal processing it also is used as a filter but my question is I have read convolution even it's useful for mechanical engineers such that in resonance when a failure occurs not due to massive force hitting an object results in large deformation could cause failure or an large impulsive force acting on it for a duration of time could cause failure but there is an another phenomenon where the natural frequency of any object is reached the energy builds in it very high and could cause a failure in this manner a small disturbance which accumulates over a time and causes a high energy to build in the system due to energy very high it causes stress and the system collapses this is highly different from stability perspective of control system being not stable does not mean it's accumulating energy inside it but in case of amplifier there is an capacitor or inductance device which causes the attenuation in the electrical signal and filters some frequencies but in other perspective amplifier amplifies the signal such that it stack ques and scales the signal but I don't know this is done by capacitor or am inductor but convolution is useful to both mechanical civil eee ece and every applied scientist and engineers hence it's used as a filter in an circuit or used to amplify but even transistor amplifies the signal without an capacitor or an inductor I guess also mechanical engineers can use it to model resonance hence the energy inside the system build high by periodic accumulation of the system reaching its natural frequency which leads to failure and I can also tell you that when amplifier filter or amplifies the signal it used convolution hence it's useful to every applied scientist and engineers but not to mention the pure Mathematicians use it of convolution of kernels thankyou guys some of my inference could be wrong if somebody or the author of the video is familiar with it please correct the above and educate me thank you for the wonderful video sir

[@saththiyambharathiyan8175]

what is the intuition to find the amplitude of the sine wave in the signal by dividing the test score with half of the period of the signal.................

[@emiliepons6801]

Great course. Dirichlet would love to have his name said and spelt correctly in the course subtitles. Thanks for him

[@hintergedankee]

I believe great minds would not care about small errors like these.

[@bijoydas6044]

🙏 sir , please make a video on laplace transform and how fourier transform is the special type of laplace transform?

[@MarkNewmanEducation]

Will do. Thanks for the idea.

[@mohanm7864]

This is not a lecture ,we feel this like a most entertaining movie directed by Oscar award director. Great Thank you so much for such a great entertainment with mathematics.

[@MarkNewmanEducation]

Ha ha... Thank you. Yes, I always dreamed of getting into the movies. I wish I knew how to do half of the special FX or had a studio big enough to move around in more. I love with quote by Einstein: “Most of the fundamental ideas of science are essentially simple, and may, as a rule, be expressed in a language comprehensible to everyone.” If I can make these explanations entertaining as well along the way, then that would be a great thing to do. Get people to enjoy Maths, not fear it.

[@surjasekharchakraborty210]

Sir how to determine the score? Just one more doubt: Whether contribution of non phase shifted sine wave and cosine wave multiplication have mistakenly reversed? Can you please check and confirm.

[@omridrori3286]

Hey from israel I think your content is really amazing i so like the Fourier analysis stuff But more than that i do really love visualizations of that i think is the beautiful things that is saw I think you can use more kind of intutive visualizations to describe this process of shifting test wave with the visualization of kind 3blue1brown It’s more intuitve than to describe it with two diffrent graph for the cos and for the sin instead you take the center of mass of warped graph … And also Fourier is amazing when describing with epicycles summing really cycles with different frequency and amplitudes to approximate things is so cool But again your content is amazing And i also really wait for more videos from you In the series ? When you release the rest ? And by the way Look like we are from the same Jewish family so it is also make me happy Thanks a lot

[@MarkNewmanEducation]

Thanks for your kind words and encouragement. I am working on more videos. So far, there are 3 new videos available as part of the 2nd half of the course. This part of the course is currently available to people who are supporting me via Patreon. Their support is helping me devote the time required to produce the videos. Please see howthefouriertransformworks.com/patreon for more details.

[@vedantshah6313]

How does difference in intensity of harmonics very with respect to beat frequency when they are analysed through Fourier transform?? Any idea ?

[@MarkNewmanEducation]

I'm sorry, I don't know.

[@stejarelstejarel]

Could you please tell us what software did you used to graph and animate the sine waves?

[@MarkNewmanEducation]

I wrote the graph animation in JavaScript. I found that no ready made software did all the things I wanted it to do for these videos so I wrote my own in JavaScript. I love JavaScript. It is SO versatile.

[@calvinluiramo5514]

I wish someone would explain the green's function as good and simple as what Mark you did with the Fourier, have been watching too many trying to explain the green's function and try to simplify it in a laymen's view but non come any closer, they either too engulf with the mathematics fabrics of it or they thought every one sees what saw in their mental mind

[@bijoydas6044]

🙏 sir , i have a doubt at 12:40 that how to calculate the score or area under the multiplied graph curve "test wave * signal" limit 0→360 is 78 score i guess that limit 0°→360° ∫ cos(2θ)*(0.5cos(2θ−30°)+0.4cos(3θ−60°)) dθ am i correct? but i can't get 78 please give me the exact mathematical expression to get the score or area under the multiplied graph curve

[@ZenoDiac]

14:42 I don't quite understand the term "Relative Contribution". It's not percentage, because all the relative contributions 90 and 72 don't add to 100, so what is it? I also don't understand how dividing by P/2 gives the actual amplitude. Can someone explain the link that I'm missing, please?

[@MarkNewmanEducation]

I've been trying to understand this myself. I know it works, but I am not 100% sure why. It is definitely something to do with the fact that the integral of cos^2(x) from 0 to the Period is equal to Period/2, but I have been struggling to understand why. If you look at some examples of the Fourier series, they sometimes include a 1/PI term before the integral to scale the result. PI being half of the period of a cosine wave with a frequency of 1 if you are working in radians. I've just had to accept it as a truth for the moment.... but I am unsure why. I didn't really investigate it any further in the video as this scaling factor is more often employed in the inverse Fourier Series which I haven't covered yet.

[@lgalina997]

What practical value or useful application, if any, does this have for day-to-day life?

[@MarkNewmanEducation]

Oooh where should I start? Your phone uses the Fourier Transform to compress the speech data before it is transmitted to the base station. JPG files use a two dimensional version of the Fourier transform to give you really sharp pictures for often under a megabyte of data. Any voice recognition or speech recognition app uses the Fourier transform to identify the frequencies in your voice and understand what you are saying. That is just a few of millions of applications of the Fourier Transform. All this from a man who lived 200 years before computers were even invented.

[@markawbolton]

@@MarkNewmanEducation IIRC Fourier first arrived at the application by considering the best rate at which to fire cannons without them over heating? I also believe they are used in Seismic Surveys. The explosive shot provides a stepped input and the return signal by way of Geofantasy can be resolved into useful spatial data. The uses are stunningly multitudinous and varied.

[@MarkNewmanEducation]

@Mark Bolton I heard something about that from a friend but couldn't find any source for it. Fascinating. I've been pulled up on historical innacuracies in my videos before so I wanted a source I could quote before telling the story.

[@markawbolton]

@@MarkNewmanEducation It always struck me as spurious too. Well I got it from Dr David Marr who lectured us in Electrical Maths III at Trade School in '83. He had a dual doctorate in Education and Maths. We asked him why he didn't tech at University. He would say " Any schmuck can teach smart kids to do Maths. If I can teach you dumb buggers to do calculus then no greater pedagogic feat exists on this Earth. You guys are my Mt Everest." I always wondered about that too. I couldn't see a military situation where you would be timing the firing of cannons based on heating. You would be firing the damn things as fast as you could load them or not at all. Also I couldnt really see how it could be rendered as a Fourier type mathematical model. Science History is replete with these little myths. If I get to the bottom of it I will let you know.

[@roygalaasen]

Mark Bolton it does sound like a theoretical problem though. I am imagining that he wanted to find the most efficient fire rate because he wanted to fire as fast as possible without the guns blowing up because of over heating?

[@longluo1788]

Sir, Thank you very much. But I have a doubt on 15: 10 - 15:30, when the freq is 3Hz, the score is 72, why the amplitude is 72/180, why is 72/180, not 72/360, or 72/120? We integral it from 0° to 360°？ Can you explain that. Thank you.

[@MarkNewmanEducation]

This fact confused me for ages, and I still didn't understand it when I made the video a few years ago. However, now I understand why. Here's the reason: You should indeed divide by 360 degrees, but when you do so, you discover that only half of the energy is contained in the frequency you just found. It baffled me for ages where the other half was. The answer is in the mirror negative frequency. If you ever look at the magnitude output of an FT, you notice it is mirrored around 0Hz. Only half of the energy for each frequency is contained in the positive frequencies. The rest is in the negative frequencies. Negative frequencies are a mathematical requirement for the original signal to be totally real (with no imaginary component). It's a bit like the concept of multiplying by the complex conjugate to get rid of the imaginary component. I will be explaining this further in a video that I am currently working on.

[@hassanomer7777]

how the area in 9:44 is not infinity ?

[@MarkNewmanEducation]

Because it is bounded by limits. In the Fourier Series, you only measure the area between 2 angles. Not over the whole of the sine wave which goes on for ever. What you have noticed, by the way, is a potential problem when you come to the Fourier Transform. See my blog post entitled "From Fourier Series to Fourier Transform" at howthefouriertransformworks.com/2020/09/20/from-fourier-series-to-fourier-transform-part-1/ for more information.

[@venkatanagasasidharjaldu5282]

Can anybody explain my doubt So my doubt is instead of convoluting the test signal through out our signal we are using non shifted sin and non shifted cosine .so these two test waves are not actually sweeping through signal just convoluted as it is . Is that it? Can someone explain how we can actually make shifted signal using non shifted cosine and sine?

[@NN-sp9tu]

You're correct (I think). Using non shifted sin and cosine and multiplying them by the signal x(t) before taking the area under each curve, the phase can be calculated by taking the inverse tan of these areas. This phase is equivalent to the phase landed on by sweeping across the signal and stopping when the maximum area value is found, so the sweeping is not needed. It was explained in the previous videos that the sum of non-shifted sin and cos waves with respective amplitudes a and b but the same freq result in a phase shifted wave of amplitude (sqrt(a^2+b^2)) and phase of tan^-1(a/b). Why exactly this relationship holds for the "scores" is still magic to me though. It was not fully explained so I think your understanding is on point as far as these videos go.

[@venkatanagasasidharjaldu5282]

Thanks @@NN-sp9tu so if you understand that magical moment (that instead of using shifted signal he has used non shifted sine and cosine and also he didn't sweep those signals across the main signal ) could you please explain me that if you understand in further lectures of him he told that he would continue his work on explaining after Fourier transform

[@velmuruganr9321]

My lord....eyes are filling with waters....😍😘

[@motitsizling7370]

היי מארק איך אפשר ליצור איתך קשר?

[@mikimauseontheway]

yours are grey, wifes are purple :) sir you said reverse.. but amazing videos opened my mind to learn the whole concept.. thanks a lot ..

[@sergten]

So essentially in order to get the set of frequencies in a given signal one has to know the candidates? It sounds like the algorithm works like "try 1Hz, then try 2Hz, and so on until the highest frequency you think is present there", as opposed to "give me a set of frequencies that are present". For example, if I have a signal that is 3.5Hz and I probe 1, 2, 3, ... - will the result be some combination of my probes rather than a single value of 3.5Hz? Also, how does the algorithm work for high frequencies, where the gap between probes is large (e.g. 1GHz, 2GHz, with lots of values in between)?

[@sergten]

I suppose a reasonable assumption can be made about the set of candidate frequencies in the following way. If I know the time interval over which the data was collected, say T, then I can start with the wave that has the period of that interval T, and end with the wave that has the period of T/N, N being chosen depending on how accurately do I want to model the signal. Is that a reasonable approach?

[@sergten]

I just realized that in the pure mathematical context it doesn't really matter because we're taking an integral - meaning we are going through all the frequencies, not a set of specific ones. It's when one tries to compute components the candidates matter. Sorry for the blabbing.

[@sergten]

Found the author's blog post on the topic, it's all clear now: www.themobilestudio.net/the-fourier-transform-part-7

[@MarkNewmanEducation]

That's exactly how it works. You don't know which frequencies are present in the signal so you have to try everyone. The Fourier Transform tries an infinite number of intermediate frequencies too.

[@johntasker7322]

Well , a great effort ! thanks so much for sharing the goodness That was hard to say if you know what I mean haha, but I wish you a happy life in the one we are now and in the second from Occupied Palestine ! Peace شكرا جزيلا لجهدك المحترم في هذه السلسلة ، من فلسطين المحتلة

[@MarkNewmanEducation]

Thank you. I am very happy to share the knowledge.

[@grounded9623]

This is W'HY the French invented the Guillotine :)

[@junghh21]

I think. This stupid cap is better than 3Blue1Brown. (though that is good too)

[@tahmid4377]

what is in your head

[@T0NYD1CK]

Fourier Transforms DO work with non-repetitive signals. As a simple example, think of a cosine wave. Its Fourier transform is an impulse (or two if you count negative frequencies). That also works the other way round. The Fourier transform of that impulse (or pair of impulses) is a cosine wave. And Fourier does not use convolution to find the phase, either. it uses two sine waves with a phase lag of 90° (or π/2 radians). The results can then be combined to determine the phase - or left as they are and can be quoted as real and imaginary parts.