title
Coding Challenge #130.1: Drawing with Fourier Transform and Epicycles
description
In this coding challenge, I implement the Discrete Fourier Transform algorithm in JavaScript and render a drawing using epicycles derived from the transform. Code: https://thecodingtrain.com/challenges/130-drawing-with-fourier-transform-and-epicycles
p5.js Web Editor Sketches:
đšī¸ Fourier Transform: https://editor.p5js.org/codingtrain/sketches/RfrZibjfL
đšī¸ Fourier Transform - user-drawn path: https://editor.p5js.org/codingtrain/sketches/jawHqwfda
đšī¸ Fourier Transform - complex numbers: https://editor.p5js.org/codingtrain/sketches/ldBlISrsQ
đšī¸ Fourier Transform - user-drawn path with complex numbers: https://editor.p5js.org/codingtrain/sketches/sPvZsg2w4
Other Parts of this Challenge:
đē Part 2: https://youtu.be/n9nfTxp_APM
đē Part 3: https://youtu.be/7_vKzcgpfvU
đĨ Previous video: https://youtu.be/X8bXDKqMsXE?list=PLRqwX-V7Uu6ZiZxtDDRCi6uhfTH4FilpH
đĨ Next video: https://youtu.be/0j86zuqqTlQ?list=PLRqwX-V7Uu6ZiZxtDDRCi6uhfTH4FilpH
đĨ All videos: https://www.youtube.com/playlist?list=PLRqwX-V7Uu6ZiZxtDDRCi6uhfTH4FilpH
References:
đ Coding Train Logo: https://twitter.com/TomFevrier/status/1079437780466520065
đž p5.FFT: https://p5js.org/reference/#/p5.FFT
đž FFT on Algorithm Archive: https://www.algorithm-archive.org/contents/cooley_tukey/cooley_tukey.html
đ Discrete Fourier transform on Wikipedia: https://en.wikipedia.org/wiki/Discrete_Fourier_transform
đ Complex Number on Wikipedia: https://en.wikipedia.org/wiki/Complex_number
Videos:
đĨ But what is the Fourier Transform? A visual introduction: https://youtu.be/spUNpyF58BY
đĨ Fourier Analysis For The Rest Of Us: https://youtu.be/2hfoX51f6sg
đĨ Epicycles, complex Fourier series and Homer Simpson's orbit: https://youtu.be/qS4H6PEcCCA
đ´ Code Train Live 165: https://youtu.be/0b3R8oWffkw?t=2657s
Related Coding Challenges:
đ #125 Fourier Series: https://youtu.be/Mm2eYfj0SgA
Timestamps:
0:00 Introducing today's topic: a Fourier transform drawing machine
1:29 Fourier transform algorithm
3:08 Signal processing
5:49 Make a signal array
12:09 Euler's formula
15:34 Write the discrete Fourier transform function
21:50 What do we need for a circular epicycle?
22:47 Calculate the amplitude, frequency, and phase
26:30 Calculate amount of time to move per frame of animation
29:44 Draw an arbitrary path for both x and y
31:07 Write an epiCycles() function
34:30 Have epiCycles() return a vector with an x and y
38:59 Grab the path of the Coding Train logo
41:38 Render the epiCycles by order of amplitude
43:57 Conclusion and next steps
Editing by Mathieu Blanchette
Animations by Jason Heglund
Music from Epidemic Sound
đ Website: http://thecodingtrain.com/
đž Share Your Creation! https://thecodingtrain.com/guides/passenger-showcase-guide
đŠ Suggest Topics: https://github.com/CodingTrain/Suggestion-Box
đĄ GitHub: https://github.com/CodingTrain
đŦ Discord: https://thecodingtrain.com/discord
đ Membership: http://youtube.com/thecodingtrain/join
đ Store: https://standard.tv/codingtrain
đī¸ Twitter: https://twitter.com/thecodingtrain
đ¸ Instagram: https://www.instagram.com/the.coding.train/
đĨ Coding Challenges: https://www.youtube.com/playlist?list=PLRqwX-V7Uu6ZiZxtDDRCi6uhfTH4FilpH
đĨ Intro to Programming: https://www.youtube.com/playlist?list=PLRqwX-V7Uu6Zy51Q-x9tMWIv9cueOFTFA
đ p5.js: https://p5js.org
đ p5.js Web Editor: https://editor.p5js.org/
đ Processing: https://processing.org
đ Code of Conduct: https://github.com/CodingTrain/Code-of-Conduct
This description was auto-generated. If you see a problem, please open an issue: https://github.com/CodingTrain/thecodingtrain.com/issues/new
#fouriertransformsquarewave #discretefouriertransform #epicyclesdrawing #javascript #p5js
detail
{'title': 'Coding Challenge #130.1: Drawing with Fourier Transform and Epicycles', 'heatmap': [{'end': 1496.472, 'start': 1438.063, 'weight': 0.926}, {'end': 1717.977, 'start': 1574.383, 'weight': 0.891}], 'summary': 'Explores creating a fourier transform drawing machine through the discrete fourier transform algorithm, implementing the algorithm in javascript, and drawing paths with epicycles by computing arbitrary signals and sorting epicycles by amplitude, resulting in the visualization of rotating paths.', 'chapters': [{'end': 305.577, 'segs': [{'end': 39.847, 'src': 'embed', 'start': 16.853, 'weight': 0, 'content': [{'end': 24.94, 'text': 'But this very, very long video, if you can stand to watch it, has as part of, at the end, this.', 'start': 16.853, 'duration': 8.087}, {'end': 26.32, 'text': 'This is the end result.', 'start': 25.26, 'duration': 1.06}, {'end': 35.945, 'text': "I am using an algorithm called Fourier transform to take an arbitrary signal, in this case a sequence of x's and a sequence of y's,", 'start': 26.721, 'duration': 9.224}, {'end': 37.986, 'text': 'the path of the coding train logo.', 'start': 35.945, 'duration': 2.041}, {'end': 39.847, 'text': 'Thank you to Tom Fevrier.', 'start': 38.286, 'duration': 1.561}], 'summary': 'Using fourier transform to process signal data for the coding train logo path.', 'duration': 22.994, 'max_score': 16.853, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/MY4luNgGfms/pics/MY4luNgGfms16853.jpg'}, {'end': 116.881, 'src': 'embed', 'start': 89.509, 'weight': 1, 'content': [{'end': 92.189, 'text': 'I somewhat conflated these in my previous video.', 'start': 89.509, 'duration': 2.68}, {'end': 94.57, 'text': 'The idea of the Fourier transform.', 'start': 92.449, 'duration': 2.121}, {'end': 96.71, 'text': 'Now where I know this algorithm,', 'start': 94.61, 'duration': 2.1}, {'end': 103.191, 'text': 'from where I learned about this algorithm first in learning about coding and creative coding and new media and sound and video,', 'start': 96.71, 'duration': 6.481}, {'end': 104.912, 'text': 'is with the terminology FFT.', 'start': 103.191, 'duration': 1.721}, {'end': 115.04, 'text': "And actually, if you go into the p5 sound library, you'll see there is a class or function called p5.fft.", 'start': 106.152, 'duration': 8.888}, {'end': 116.881, 'text': "I don't remember exactly what it's called, but something like that.", 'start': 115.08, 'duration': 1.801}], 'summary': 'The transcript discusses learning about the fft algorithm in the context of coding and new media.', 'duration': 27.372, 'max_score': 89.509, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/MY4luNgGfms/pics/MY4luNgGfms89509.jpg'}, {'end': 266.513, 'src': 'embed', 'start': 237.759, 'weight': 2, 'content': [{'end': 240.342, 'text': 'The three blue, one brown video.', 'start': 237.759, 'duration': 2.583}, {'end': 242.345, 'text': 'but what is the Fourier transform?', 'start': 240.342, 'duration': 2.003}, {'end': 250.816, 'text': 'will give you a infinitely better understanding of what the Fourier transform algorithm is and what it does, and even how it works,', 'start': 242.345, 'duration': 8.471}, {'end': 254.561, 'text': 'way better than my rambling trying to ramble through that over on the whiteboard.', 'start': 250.816, 'duration': 3.745}, {'end': 261.408, 'text': 'I would also highly recommend this gold-plated goof video Fourier Analysis for the Rest of Us,', 'start': 255.602, 'duration': 5.806}, {'end': 266.513, 'text': 'which goes through the details of the math in a much deeper way.', 'start': 261.408, 'duration': 5.105}], 'summary': 'Understanding the fourier transform is crucial; recommended videos provide in-depth insight.', 'duration': 28.754, 'max_score': 237.759, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/MY4luNgGfms/pics/MY4luNgGfms237759.jpg'}], 'start': 1.397, 'title': 'Creating a fourier transform drawing machine', 'summary': "Covers the development of a fourier transform drawing machine through the discrete fourier transform algorithm to trace a logo's path with rotating circles, and includes references to signal processing and audio signal processing, along with recommendations for additional learning resources.", 'chapters': [{'end': 305.577, 'start': 1.397, 'title': 'Fourier transform drawing machine', 'summary': 'Discusses the creation of a fourier transform drawing machine using the discrete fourier transform algorithm, which is used to trace the path of a logo through a sequence of rotating circles, with references to signal processing and audio signal processing. the video also provides recommendations for further learning resources on the fourier transform algorithm.', 'duration': 304.18, 'highlights': ['The creation of a Fourier transform drawing machine using the discrete Fourier transform algorithm to trace the path of a logo through a sequence of rotating circles The algorithm called Fourier transform is used to trace the path of the logo through a sequence of rotating circles, referred to as epicycles, using the discrete Fourier transform algorithm.', 'References to signal processing and audio signal processing in relation to the Fourier transform algorithm The video mentions the use of the Fourier transform algorithm in signal processing and audio signal processing, highlighting its application in filtering out specific sound wave patterns.', "Recommendations for further learning resources on the Fourier transform algorithm The video provides recommendations for further learning resources on the Fourier transform algorithm, including references to 'The three blue, one brown video' and 'Fourier Analysis for the Rest of Us' video."]}], 'duration': 304.18, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/MY4luNgGfms/pics/MY4luNgGfms1397.jpg', 'highlights': ['Creation of a Fourier transform drawing machine using the discrete Fourier transform algorithm to trace the path of a logo through rotating circles', 'References to signal processing and audio signal processing in relation to the Fourier transform algorithm', "Recommendations for further learning resources on the Fourier transform algorithm, including references to 'The three blue, one brown video' and 'Fourier Analysis for the Rest of Us' video"]}, {'end': 541.186, 'segs': [{'end': 391.652, 'src': 'embed', 'start': 332.604, 'weight': 1, 'content': [{'end': 336.988, 'text': "This is really like all of a bunch of y values that I'm calculating.", 'start': 332.604, 'duration': 4.384}, {'end': 341.932, 'text': "So let's make an array called something like y.", 'start': 337.268, 'duration': 4.664}, {'end': 345.991, 'text': 'And this is going to be the signal.', 'start': 344.71, 'duration': 1.281}, {'end': 347.592, 'text': 'This is my signal.', 'start': 346.351, 'duration': 1.241}, {'end': 355.416, 'text': 'This could be audio, it could be Y positions, any arbitrary digital signal slash array of numbers.', 'start': 347.792, 'duration': 7.624}, {'end': 366.246, 'text': 'Then what I want to do is I want to have the Fourier transform of that particular signal.', 'start': 356.036, 'duration': 10.21}, {'end': 378.283, 'text': 'So I want to say Fourier y equals like Fourier transform, or maybe like DFT, discrete Fourier transform of, the y value.', 'start': 366.626, 'duration': 11.657}, {'end': 379.424, 'text': 'So this is the idea.', 'start': 378.464, 'duration': 0.96}, {'end': 385.688, 'text': 'The first thing that I need to do is compute the discrete Fourier transform of an arbitrary signal.', 'start': 379.704, 'duration': 5.984}, {'end': 387.049, 'text': 'Now, I need some kind of signal.', 'start': 385.708, 'duration': 1.341}, {'end': 391.652, 'text': "So I think what I'm going to absurdly do is hard code the signal.", 'start': 387.569, 'duration': 4.083}], 'summary': 'Compute fourier transform of arbitrary signal for analysis.', 'duration': 59.048, 'max_score': 332.604, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/MY4luNgGfms/pics/MY4luNgGfms332604.jpg'}, {'end': 440.35, 'src': 'embed', 'start': 415.87, 'weight': 0, 'content': [{'end': 422.455, 'text': "And then we'll do some interesting things that we might maybe all try like a Perlin noise signal or just a sine wave signal.", 'start': 415.87, 'duration': 6.585}, {'end': 423.976, 'text': "We'll try different things, random numbers.", 'start': 422.475, 'duration': 1.501}, {'end': 424.756, 'text': "Let's see what that does.", 'start': 423.996, 'duration': 0.76}, {'end': 431.121, 'text': 'So then this actual code here can largely stay the same.', 'start': 425.157, 'duration': 5.964}, {'end': 440.35, 'text': 'Because in theory, the difference is now instead of following the specific Fourier series for the square wave.', 'start': 431.461, 'duration': 8.889}], 'summary': 'Exploring signal generation using perlin noise and sine wave, experimenting with random numbers.', 'duration': 24.48, 'max_score': 415.87, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/MY4luNgGfms/pics/MY4luNgGfms415870.jpg'}, {'end': 523.359, 'src': 'embed', 'start': 495.919, 'weight': 3, 'content': [{'end': 500.783, 'text': "The idea is that I'm going to return the discrete Fourier transform of those values.", 'start': 495.919, 'duration': 4.864}, {'end': 503.444, 'text': 'So a couple of things.', 'start': 501.283, 'duration': 2.161}, {'end': 503.965, 'text': 'One is.', 'start': 503.625, 'duration': 0.34}, {'end': 515.113, 'text': 'I highly recommend if you want to pause this video right now and read this particular article on the Algorithm Archive by James Schloss or the Leo IOS YouTube channel.', 'start': 503.965, 'duration': 11.148}, {'end': 523.359, 'text': 'This is a really nice article about Fourier transforms and the discrete Fourier transform algorithm and this particular algorithm for FFT.', 'start': 515.573, 'duration': 7.786}], 'summary': 'Return discrete fourier transform of values. recommended article on algorithm archive by james schloss or leo ios youtube channel.', 'duration': 27.44, 'max_score': 495.919, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/MY4luNgGfms/pics/MY4luNgGfms495919.jpg'}], 'start': 305.737, 'title': 'Computing and implementing discrete fourier transform', 'summary': 'Focuses on computing the discrete fourier transform of an arbitrary signal to obtain amplitudes, frequencies, and phases, and discusses implementing the algorithm to analyze various signals, referencing external resources for guidance.', 'chapters': [{'end': 391.652, 'start': 305.737, 'title': 'Computing discrete fourier transform', 'summary': 'Focuses on computing the discrete fourier transform of an arbitrary signal to obtain amplitudes, frequencies, and phases, using an array of y values representing the signal.', 'duration': 85.915, 'highlights': ['The first step is to create an array representing the signal, which could be audio, Y positions, or any arbitrary digital signal.', 'The next step involves computing the discrete Fourier transform (DFT) of the signal to obtain the amplitudes, frequencies, and phases.']}, {'end': 541.186, 'start': 392.092, 'title': 'Implementing discrete fourier transform', 'summary': 'Discusses implementing the discrete fourier transform algorithm to analyze different types of signals such as square waves, perlin noise, and sine waves, while referencing external resources for algorithm guidance.', 'duration': 149.094, 'highlights': ['The chapter discusses implementing the discrete Fourier transform algorithm The main focus of the chapter is on implementing the discrete Fourier transform algorithm to analyze different types of signals.', 'Analyzing different types of signals such as square waves, Perlin noise, and sine waves The chapter explores analyzing various signals including square waves, Perlin noise, and sine waves to observe their behavior under the Fourier transform.', 'Referencing external resources for algorithm guidance External resources, such as an article on the Algorithm Archive and the Leo IOS YouTube channel, are recommended for further understanding of the Fourier transform algorithm.']}], 'duration': 235.449, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/MY4luNgGfms/pics/MY4luNgGfms305737.jpg', 'highlights': ['The chapter explores analyzing various signals including square waves, Perlin noise, and sine waves to observe their behavior under the Fourier transform.', 'The next step involves computing the discrete Fourier transform (DFT) of the signal to obtain the amplitudes, frequencies, and phases.', 'The first step is to create an array representing the signal, which could be audio, Y positions, or any arbitrary digital signal.', 'Referencing external resources, such as an article on the Algorithm Archive and the Leo IOS YouTube channel, for further understanding of the Fourier transform algorithm.', 'The main focus of the chapter is on implementing the discrete Fourier transform algorithm to analyze different types of signals.']}, {'end': 862.297, 'segs': [{'end': 674.285, 'src': 'embed', 'start': 652.108, 'weight': 1, 'content': [{'end': 661.956, 'text': "even if I start with a array of real numbers single numbers I'm going to apply the Fourier transform and get out a complex number.", 'start': 652.108, 'duration': 9.848}, {'end': 671.183, 'text': "What I'm going to return from that function is both A's and B's, otherwise known as the real component, which is often written in code as like RE,", 'start': 662.176, 'duration': 9.007}, {'end': 674.285, 'text': 'and the imaginary component, which is often written as IM.', 'start': 671.183, 'duration': 3.102}], 'summary': "Applying fourier transform to array of real numbers yields complex numbers with a's and b's as real and imaginary components.", 'duration': 22.177, 'max_score': 652.108, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/MY4luNgGfms/pics/MY4luNgGfms652108.jpg'}, {'end': 726.594, 'src': 'embed', 'start': 695.054, 'weight': 0, 'content': [{'end': 700.835, 'text': "We're going to figure out how to translate all of this notation and symbols and stuff into JavaScript code.", 'start': 695.054, 'duration': 5.781}, {'end': 706.637, 'text': "Now, again, it'll be super interesting to go down the rabbit hole of deriving all these formulas in the background for how Fourier transform works.", 'start': 701.075, 'duration': 5.562}, {'end': 707.838, 'text': "But I'm not going to do that.", 'start': 706.937, 'duration': 0.901}, {'end': 714.104, 'text': "If you look in the video's description, there are several excellent videos and resources linked to that will give you that background.", 'start': 708.619, 'duration': 5.485}, {'end': 716.966, 'text': 'But I do want to mention one thing which is quite important,', 'start': 714.464, 'duration': 2.502}, {'end': 726.594, 'text': 'which is that this particular formula on the top here for the discrete Fourier transform uses this symbol e,', 'start': 716.966, 'duration': 9.628}], 'summary': 'Learning to translate notation into javascript. not delving into formula derivation.', 'duration': 31.54, 'max_score': 695.054, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/MY4luNgGfms/pics/MY4luNgGfms695054.jpg'}, {'end': 802.596, 'src': 'embed', 'start': 771.65, 'weight': 3, 'content': [{'end': 778.553, 'text': "What are the components we need to understand? Now really, if this were a math lesson about Fourier transform, we wouldn't be using summation.", 'start': 771.65, 'duration': 6.903}, {'end': 780.754, 'text': 'We would be using integration,', 'start': 778.873, 'duration': 1.881}, {'end': 790.804, 'text': "but because working on a computer and I need to write a loop and I don't have infinity as a tool that I could just use I need to,", 'start': 780.754, 'duration': 10.05}, {'end': 793.187, 'text': 'instead of doing integration, do summation.', 'start': 790.804, 'duration': 2.383}, {'end': 802.596, 'text': "And that's why also this is called discrete Fourier transform, because I'm doing it over the sort of like discrete, Okay, so this means summation.", 'start': 793.207, 'duration': 9.389}], 'summary': 'Explanation of using summation for discrete fourier transform on a computer', 'duration': 30.946, 'max_score': 771.65, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/MY4luNgGfms/pics/MY4luNgGfms771650.jpg'}, {'end': 846.471, 'src': 'embed', 'start': 815.506, 'weight': 4, 'content': [{'end': 817.948, 'text': "And that's kind of oh.", 'start': 815.506, 'duration': 2.442}, {'end': 823.232, 'text': "and then the other thing that's really important is that basically what I get to do is separate out.", 'start': 817.948, 'duration': 5.284}, {'end': 831.138, 'text': 'This is the real component and this is the imaginary component.', 'start': 824.773, 'duration': 6.365}, {'end': 839.765, 'text': "So even though this is all written as one formula, I'm going to sum up all the real components and all the imaginary components together.", 'start': 832.119, 'duration': 7.646}, {'end': 846.471, 'text': 'And by the way, as Simon, who has been watching this live, pointed out to me, there are only complex numbers.', 'start': 839.785, 'duration': 6.686}], 'summary': 'Separating real and imaginary components in complex numbers.', 'duration': 30.965, 'max_score': 815.506, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/MY4luNgGfms/pics/MY4luNgGfms815506.jpg'}], 'start': 541.186, 'title': 'Fourier transform and its implementation in javascript', 'summary': "Delves into computing fourier transform and its translation into javascript, highlighting the importance of understanding complex numbers, and emphasizing the use of euler's formula and discrete fourier transform implementation through summation, with a focus on the real and imaginary components.", 'chapters': [{'end': 674.285, 'start': 541.186, 'title': 'Fourier transform and complex numbers', 'summary': 'Discusses the computation of fourier transform, emphasizing the need to understand complex numbers, where the function returns both real (a) and imaginary (b) components.', 'duration': 133.099, 'highlights': ['The chapter explains the need to understand complex numbers when working with Fourier transforms, highlighting the representation of complex numbers on the complex plane and the return of both real and imaginary components from the Fourier transform function.', 'The function computes the Fourier transform and returns it as an array, with the result comprising both real (A) and imaginary (B) components.']}, {'end': 862.297, 'start': 674.585, 'title': 'Understanding fourier transform in javascript', 'summary': "Explains how to translate fourier transform notation into javascript code, emphasizing the use of euler's formula and the implementation of discrete fourier transform through summation instead of integration, while also addressing the real and imaginary components.", 'duration': 187.712, 'highlights': ["The chapter emphasizes the translation of Fourier transform notation into JavaScript code, focusing on the use of Euler's formula and the implementation of discrete Fourier transform through summation.", 'It highlights the significance of using summation instead of integration in the context of discrete Fourier transform, due to the limitations of working with computers.', 'The explanation includes the separation of real and imaginary components in the Fourier transform formula, emphasizing the distinction between real and complex numbers.']}], 'duration': 321.111, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/MY4luNgGfms/pics/MY4luNgGfms541186.jpg', 'highlights': ["The chapter emphasizes the translation of Fourier transform notation into JavaScript code, focusing on the use of Euler's formula and the implementation of discrete Fourier transform through summation.", 'The chapter explains the need to understand complex numbers when working with Fourier transforms, highlighting the representation of complex numbers on the complex plane and the return of both real and imaginary components from the Fourier transform function.', 'The function computes the Fourier transform and returns it as an array, with the result comprising both real (A) and imaginary (B) components.', 'It highlights the significance of using summation instead of integration in the context of discrete Fourier transform, due to the limitations of working with computers.', 'The explanation includes the separation of real and imaginary components in the Fourier transform formula, emphasizing the distinction between real and complex numbers.']}, {'end': 1199.121, 'segs': [{'end': 999.939, 'src': 'embed', 'start': 915.486, 'weight': 0, 'content': [{'end': 920.908, 'text': 'I want to loop through every k, which is going from 0 all the way up to n, and then also sum up.', 'start': 915.486, 'duration': 5.422}, {'end': 923.549, 'text': 'So k is going to stay constant within each one of these.', 'start': 920.928, 'duration': 2.621}, {'end': 925.97, 'text': 'And k is actually really the frequency.', 'start': 923.609, 'duration': 2.361}, {'end': 929.672, 'text': "We'll see that, the frequency of the particular wave pattern in that slot.", 'start': 926.01, 'duration': 3.662}, {'end': 931.833, 'text': 'OK All right, so let me write the code for this.', 'start': 929.772, 'duration': 2.061}, {'end': 937.253, 'text': 'The first thing that I want to do is Create an empty array.', 'start': 931.933, 'duration': 5.32}, {'end': 939.216, 'text': "This is where I'm going to store all of the results.", 'start': 937.273, 'duration': 1.943}, {'end': 947.225, 'text': 'Then I need to write a loop, which is let k equals 0, k is less than n, k plus plus.', 'start': 939.636, 'duration': 7.589}, {'end': 952.05, 'text': "And then I'm going to be saying Fourier k equals something.", 'start': 948.426, 'duration': 3.624}, {'end': 962.108, 'text': 'So this by the way, I mean I could call this like capital X if I wanted to be, and maybe I will, just to like follow this notation exactly.', 'start': 953.106, 'duration': 9.002}, {'end': 965.489, 'text': 'So this is capital X, this is lowercase x.', 'start': 962.309, 'duration': 3.18}, {'end': 972.091, 'text': "So let's actually, as silly as this might be, let's change this to x so I can use all the same notation as that formula.", 'start': 965.489, 'duration': 6.602}, {'end': 974.352, 'text': "So I'm trying to calculate this.", 'start': 972.911, 'duration': 1.441}, {'end': 981.014, 'text': 'Now in order to do this, I need to, for each one, go through n.', 'start': 974.572, 'duration': 6.442}, {'end': 986.455, 'text': 'So this is where the nested loop comes in.', 'start': 984.028, 'duration': 2.427}, {'end': 999.939, 'text': 'To calculate each each element of capital X, index k, I need to sum up n going from zero to the n.', 'start': 988.741, 'duration': 11.198}], 'summary': 'Loop through k and n to calculate fourier transformation.', 'duration': 84.453, 'max_score': 915.486, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/MY4luNgGfms/pics/MY4luNgGfms915486.jpg'}, {'end': 1199.121, 'src': 'embed', 'start': 1116.178, 'weight': 1, 'content': [{'end': 1126.951, 'text': "because that's where my formula is written is going to say x index n times, cosine of phi.", 'start': 1116.178, 'duration': 10.773}, {'end': 1138.363, 'text': 'boy, this should look Strangely, like a lot of code that I write a lot and then x index n times and this should say cosine times sine of phi.', 'start': 1126.951, 'duration': 11.412}, {'end': 1144.345, 'text': "But one thing you'll notice here is that this, there is a minus here.", 'start': 1139.203, 'duration': 5.142}, {'end': 1148.426, 'text': 'This minus here is not, is quite an important detail.', 'start': 1144.485, 'duration': 3.941}, {'end': 1157.33, 'text': "So what I'm going to do is, I don't know the best way to handle this, but I could just say minus equals, but maybe I'll, yeah, let's say minus equals.", 'start': 1149.227, 'duration': 8.103}, {'end': 1158.77, 'text': "Let's do that, okay.", 'start': 1158.049, 'duration': 0.721}, {'end': 1163.333, 'text': 'And then this is called enhanced object literals, thank you.', 'start': 1158.97, 'duration': 4.363}, {'end': 1166.416, 'text': 'I can just say this.', 'start': 1163.353, 'duration': 3.063}, {'end': 1173.582, 'text': 'So this will give me that and there, okay.', 'start': 1168.998, 'duration': 4.584}, {'end': 1176.905, 'text': 'So all of that explanation, wow.', 'start': 1173.602, 'duration': 3.303}, {'end': 1179.347, 'text': 'And here we are, discrete Fourier transform.', 'start': 1177.125, 'duration': 2.222}, {'end': 1184.32, 'text': 'All right, so thank you to the chat for asking a couple key questions and for pointing out some errors.', 'start': 1180.117, 'duration': 4.203}, {'end': 1187.982, 'text': 'For example, I forgot to actually define what n is, which is x.length.', 'start': 1184.38, 'duration': 3.602}, {'end': 1190.053, 'text': 'You know,', 'start': 1189.793, 'duration': 0.26}, {'end': 1197.259, 'text': "I should probably get in the habit of using the variable declaration const when I know it's something that's going to stay constant like this.", 'start': 1190.053, 'duration': 7.206}, {'end': 1199.121, 'text': 'So I will attempt to use that here.', 'start': 1197.54, 'duration': 1.581}], 'summary': 'Discussed code implementation for discrete fourier transform with some errors and key questions addressed.', 'duration': 82.943, 'max_score': 1116.178, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/MY4luNgGfms/pics/MY4luNgGfms1116178.jpg'}], 'start': 862.717, 'title': 'Fourier and discrete fourier transform', 'summary': 'Explains fourier analysis with nested loops and summation to calculate wave pattern frequency, and constructing discrete fourier transform with real and imaginary components, object literals, and error handling.', 'chapters': [{'end': 999.939, 'start': 862.717, 'title': 'Fourier analysis explained', 'summary': 'Explains the process of calculating fourier analysis with nested loops and nested summation, aiming to create an array of k elements and loop through each k, going from 0 to n, to calculate the frequency of wave patterns.', 'duration': 137.222, 'highlights': ['The process involves creating an array of k elements and looping through each k, going from 0 to n, to calculate the frequency of wave patterns.', 'Nested loops and nested summation are utilized to calculate each element of capital X, index k, by summing up n from 0 to n.', 'The code involves initializing an empty array to store all the results and then executing a loop where Fourier k is calculated for each k, going from 0 to n.']}, {'end': 1199.121, 'start': 999.939, 'title': 'Discrete fourier transform process', 'summary': 'Entails the process of constructing a discrete fourier transform, including the creation of real and imaginary components, the use of object literals, and the handling of important details like variable declaration and errors.', 'duration': 199.182, 'highlights': ['The process involves constructing a discrete Fourier transform by creating real and imaginary components, using object literals, and handling important details like variable declaration and errors.', 'The real component of the Fourier transform is constructed by multiplying x index n with cosine of phi, and the imaginary component is constructed by multiplying x index n with sine of phi, with the important detail of a minus sign to be handled.', 'The value of n is defined as x.length, and the use of const for constant variables is recommended.']}], 'duration': 336.404, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/MY4luNgGfms/pics/MY4luNgGfms862717.jpg', 'highlights': ['The process involves creating an array of k elements and looping through each k, going from 0 to n, to calculate the frequency of wave patterns.', 'The process involves constructing a discrete Fourier transform by creating real and imaginary components, using object literals, and handling important details like variable declaration and errors.', 'Nested loops and nested summation are utilized to calculate each element of capital X, index k, by summing up n from 0 to n.', 'The code involves initializing an empty array to store all the results and then executing a loop where Fourier k is calculated for each k, going from 0 to n.', 'The real component of the Fourier transform is constructed by multiplying x index n with cosine of phi, and the imaginary component is constructed by multiplying x index n with sine of phi, with the important detail of a minus sign to be handled.', 'The value of n is defined as x.length, and the use of const for constant variables is recommended.']}, {'end': 1663.402, 'segs': [{'end': 1243.016, 'src': 'embed', 'start': 1199.441, 'weight': 2, 'content': [{'end': 1201.863, 'text': "These cannot be const because I'm adding them up together.", 'start': 1199.441, 'duration': 2.422}, {'end': 1211.972, 'text': 'Now, another thing that is typically done with discrete Fourier transform is to take this sum and then average its contribution over n.', 'start': 1202.484, 'duration': 9.488}, {'end': 1215.835, 'text': 'So I would also say the real component equals the real component divided by n.', 'start': 1211.972, 'duration': 3.863}, {'end': 1218.577, 'text': 'The imaginary component is the imaginary component y by n.', 'start': 1216.215, 'duration': 2.362}, {'end': 1219.638, 'text': "So I'm going to add that in.", 'start': 1218.577, 'duration': 1.061}, {'end': 1225.843, 'text': "And then someone asked me, well, oh, but this is another question that came in, which was the question, oh, there's i here.", 'start': 1219.958, 'duration': 5.885}, {'end': 1234.57, 'text': "Why don't I see i in your formula, in your code? Where is i in your code? And so i isn't explicitly in here.", 'start': 1226.764, 'duration': 7.806}, {'end': 1240.154, 'text': "But what I'm doing, I'm referencing i by separating out the real and imaginary components.", 'start': 1234.91, 'duration': 5.244}, {'end': 1243.016, 'text': 'So the imaginary component is always paired with i.', 'start': 1240.614, 'duration': 2.402}], 'summary': 'Discrete fourier transform involves summing and averaging components, with i implied in the code.', 'duration': 43.575, 'max_score': 1199.441, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/MY4luNgGfms/pics/MY4luNgGfms1199441.jpg'}, {'end': 1292.507, 'src': 'embed', 'start': 1265.111, 'weight': 0, 'content': [{'end': 1269.273, 'text': 'So I have the y array, which is my hard-coded square wave.', 'start': 1265.111, 'duration': 4.162}, {'end': 1277.898, 'text': 'And if I call DFT on y here in the console, we can see I had 12 values in my signal.', 'start': 1269.614, 'duration': 8.284}, {'end': 1280.26, 'text': 'I get 12 complex numbers back.', 'start': 1278.219, 'duration': 2.041}, {'end': 1284.142, 'text': 'each one with a real and an imaginary component.', 'start': 1281.1, 'duration': 3.042}, {'end': 1292.507, 'text': "So this looks good in the sense that there are numbers here, I don't see an error, no red, no not a number, so hopefully we're in the right place.", 'start': 1284.422, 'duration': 8.085}], 'summary': 'Dft on y produced 12 complex numbers with real and imaginary components.', 'duration': 27.396, 'max_score': 1265.111, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/MY4luNgGfms/pics/MY4luNgGfms1265111.jpg'}, {'end': 1406.595, 'src': 'embed', 'start': 1357.457, 'weight': 1, 'content': [{'end': 1361.919, 'text': 'the amplitude is something and the phase is something.', 'start': 1357.457, 'duration': 4.462}, {'end': 1369.128, 'text': 'And the secret to this lies in the fact that a complex number is like a vector.', 'start': 1363.226, 'duration': 5.902}, {'end': 1371.329, 'text': 'And in fact, here we go.', 'start': 1369.488, 'duration': 1.841}, {'end': 1378.271, 'text': 'The amplitude is the length of that vector, and the phase is the angle of that vector.', 'start': 1372.049, 'duration': 6.222}, {'end': 1381.092, 'text': "Well, so that's amplitude and phase, but what's frequency??", 'start': 1378.552, 'duration': 2.54}, {'end': 1382.013, 'text': 'Well, guess what?', 'start': 1381.293, 'duration': 0.72}, {'end': 1390.376, 'text': "I don't know if there was a clue to this on that Wikipedia page, but the frequency is actually just k.", 'start': 1382.033, 'duration': 8.343}, {'end': 1393.608, 'text': "Let's see, frequency, yeah.", 'start': 1392.227, 'duration': 1.381}, {'end': 1396.589, 'text': 'And discrete frequency components.', 'start': 1394.248, 'duration': 2.341}, {'end': 1405.194, 'text': 'So the whole point of doing this is to take the signal and divide it into a bunch of discrete frequency components zero, one, two, three, four,', 'start': 1396.909, 'duration': 8.285}, {'end': 1405.514, 'text': 'et cetera.', 'start': 1405.194, 'duration': 0.32}, {'end': 1406.595, 'text': 'So here we go.', 'start': 1405.774, 'duration': 0.821}], 'summary': 'Complex numbers relate amplitude, phase, and frequency in signal processing.', 'duration': 49.138, 'max_score': 1357.457, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/MY4luNgGfms/pics/MY4luNgGfms1357457.jpg'}, {'end': 1496.472, 'src': 'heatmap', 'start': 1438.063, 'weight': 0.926, 'content': [{'end': 1440.885, 'text': "I'm going to just reference the code that I wrote before.", 'start': 1438.063, 'duration': 2.822}, {'end': 1443.287, 'text': 'Yes, I got it right, okay.', 'start': 1442.346, 'duration': 0.941}, {'end': 1444.928, 'text': 'So this would be the phase.', 'start': 1443.667, 'duration': 1.261}, {'end': 1450.531, 'text': 'So now, I can say frequency, I can add frequency, amplitude, and phase here.', 'start': 1445.208, 'duration': 5.323}, {'end': 1454.914, 'text': 'And I can refresh this page.', 'start': 1452.032, 'duration': 2.882}, {'end': 1464.62, 'text': "I can say DFTY, and let's take a look at any one of these, and we can see, all right, I've got an amplitude, I've got a frequency, I've got a phase.", 'start': 1455.874, 'duration': 8.746}, {'end': 1468.339, 'text': 'We are ready.', 'start': 1466.798, 'duration': 1.541}, {'end': 1472.243, 'text': 'We are ready to start actually now putting this into our code.', 'start': 1468.72, 'duration': 3.523}, {'end': 1477.788, 'text': 'And the good news is we have the code for drawing these epicycles already.', 'start': 1472.303, 'duration': 5.485}, {'end': 1479.189, 'text': 'I commented it out.', 'start': 1478.088, 'duration': 1.101}, {'end': 1480.27, 'text': 'It was right here.', 'start': 1479.409, 'duration': 0.861}, {'end': 1493.929, 'text': 'So if, in fact, I have this Fourier y array, I have this code from before that was drawing the results of those epicycles.', 'start': 1481.351, 'duration': 12.578}, {'end': 1496.472, 'text': 'So I can comment this all back in.', 'start': 1494.209, 'duration': 2.263}], 'summary': 'Preparing to add frequency, amplitude, and phase to the code for drawing epicycles.', 'duration': 58.409, 'max_score': 1438.063, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/MY4luNgGfms/pics/MY4luNgGfms1438063.jpg'}], 'start': 1199.441, 'title': 'Fourier transform and circular epicycles', 'summary': 'Discusses implementing the discrete fourier transform and processing a hard-coded square wave, resulting in 12 complex numbers with real and imaginary components without errors, and the essential components required for creating circular epicycles using fourier transform including amplitude, frequency, and phase, and their mathematical representations, such as amplitude as the length of a vector, phase as the angle of a vector, and frequency as k, and the challenges encountered in implementing these components.', 'chapters': [{'end': 1292.507, 'start': 1199.441, 'title': 'Discrete fourier transform and complex numbers', 'summary': 'Discusses implementing the discrete fourier transform and processing a hard-coded square wave, resulting in 12 complex numbers with real and imaginary components, without errors.', 'duration': 93.066, 'highlights': ['Implementing the discrete Fourier transform and processing a hard-coded square wave resulted in 12 complex numbers with real and imaginary components, indicating a successful implementation.', "The real component is averaged over 'n' and the imaginary component is also averaged over 'n' to derive the final components of the discrete Fourier transform.", "The chapter explains the absence of 'i' in the code, clarifying that the real and imaginary components are separated, with the imaginary component always paired with 'i'."]}, {'end': 1663.402, 'start': 1292.567, 'title': 'Fourier transform for circular epicycles', 'summary': 'Discusses the essential components required for creating circular epicycles using fourier transform, which includes amplitude, frequency, and phase, and their mathematical representations, such as amplitude as the length of a vector, phase as the angle of a vector, and frequency as k, and the challenges encountered in implementing these components.', 'duration': 370.835, 'highlights': ['The amplitude is the length of the vector, and the phase is the angle of the vector, which are essential components for creating circular epicycles using Fourier transform.', 'Frequency is represented as k, and the process involves dividing the signal into discrete frequency components, and the challenges in implementing these components involve errors in the angle and values, as well as the need for careful consideration of time increments.']}], 'duration': 463.961, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/MY4luNgGfms/pics/MY4luNgGfms1199441.jpg', 'highlights': ['Implementing the discrete Fourier transform and processing a hard-coded square wave resulted in 12 complex numbers with real and imaginary components, indicating a successful implementation.', 'The amplitude is the length of the vector, and the phase is the angle of the vector, which are essential components for creating circular epicycles using Fourier transform.', "The real component is averaged over 'n' and the imaginary component is also averaged over 'n' to derive the final components of the discrete Fourier transform.", "The chapter explains the absence of 'i' in the code, clarifying that the real and imaginary components are separated, with the imaginary component always paired with 'i'.", 'Frequency is represented as k, and the process involves dividing the signal into discrete frequency components, and the challenges in implementing these components involve errors in the angle and values, as well as the need for careful consideration of time increments.']}, {'end': 2756.766, 'segs': [{'end': 1721.84, 'src': 'embed', 'start': 1692.526, 'weight': 0, 'content': [{'end': 1693.507, 'text': 'So you can see, look at this.', 'start': 1692.526, 'duration': 0.981}, {'end': 1694.908, 'text': 'This, there it is.', 'start': 1693.867, 'duration': 1.041}, {'end': 1699.691, 'text': 'This is the crazy set of epicycles to draw these random numbers.', 'start': 1695.288, 'duration': 4.403}, {'end': 1703.894, 'text': 'I could also now do, I could do Perlin noise instead.', 'start': 1700.431, 'duration': 3.463}, {'end': 1708.997, 'text': 'And just have it increased by some arbitrary amount.', 'start': 1706.736, 'duration': 2.261}, {'end': 1711.052, 'text': 'All right, so there we go.', 'start': 1710.171, 'duration': 0.881}, {'end': 1712.473, 'text': "I'm kind of doing nonsense here.", 'start': 1711.092, 'duration': 1.381}, {'end': 1717.977, 'text': 'But the point is, any arbitrary signal that I have, I can now compute the for.', 'start': 1712.693, 'duration': 5.284}, {'end': 1721.84, 'text': 'And you can see, by the way, this is it cycling back to the beginning.', 'start': 1718.017, 'duration': 3.823}], 'summary': 'Demonstrating drawing random numbers using epicycles and perlin noise, cycling back to the beginning.', 'duration': 29.314, 'max_score': 1692.526, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/MY4luNgGfms/pics/MY4luNgGfms1692526.jpg'}, {'end': 1872.371, 'src': 'embed', 'start': 1842.657, 'weight': 1, 'content': [{'end': 1852.48, 'text': 'This loop here is visualizing the results of the Fourier transform as a sequence of rotating circles, epicycles.', 'start': 1842.657, 'duration': 9.823}, {'end': 1858.903, 'text': "So what I want to do, though, is I think what's going to be good is to refactor this, not later, but now.", 'start': 1853.32, 'duration': 5.583}, {'end': 1862.805, 'text': 'And put this in a function.', 'start': 1860.884, 'duration': 1.921}, {'end': 1865.227, 'text': "Let's call it epicycle.", 'start': 1862.865, 'duration': 2.362}, {'end': 1872.371, 'text': 'So the idea is that maybe I would get an x and a y.', 'start': 1865.247, 'duration': 7.124}], 'summary': "Visualizing fourier transform results using rotating circles, aiming to refactor code into a function named 'epicycle'.", 'duration': 29.714, 'max_score': 1842.657, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/MY4luNgGfms/pics/MY4luNgGfms1842657.jpg'}, {'end': 2696.828, 'src': 'embed', 'start': 2668.713, 'weight': 2, 'content': [{'end': 2673.815, 'text': "I'm going to come back, and I'm going to make it so that the user draws something, and then it computes the Fourier transform.", 'start': 2668.713, 'duration': 5.102}, {'end': 2676.036, 'text': 'By the way, you should go do that yourself right now.', 'start': 2673.835, 'duration': 2.201}, {'end': 2677.997, 'text': "So take this code that I've released,", 'start': 2676.256, 'duration': 1.741}, {'end': 2683.219, 'text': 'find the link in the video description and go make a version where the user draws something and then do the Fourier transform.', 'start': 2677.997, 'duration': 5.222}, {'end': 2684.099, 'text': "That's a fun exercise.", 'start': 2683.319, 'duration': 0.78}, {'end': 2685.52, 'text': "I'm going to do that in the next video.", 'start': 2684.319, 'duration': 1.201}, {'end': 2696.828, 'text': "And then I am going to rewrite this so that I have the Fourier transform done with the x and y's together as a complex number.", 'start': 2685.8, 'duration': 11.028}], 'summary': 'Next video will feature user drawing and computing fourier transform. also, plan to rewrite code for complex number fourier transform.', 'duration': 28.115, 'max_score': 2668.713, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/MY4luNgGfms/pics/MY4luNgGfms2668713.jpg'}], 'start': 1663.762, 'title': 'Drawing paths with epicycles', 'summary': 'Covers computing arbitrary signals with 100 random values, drawing paths using fourier transform and epicycles, and sorting epicycles by amplitude, leading to the visualization of rotating paths with the fourier transform.', 'chapters': [{'end': 1782.493, 'start': 1663.762, 'title': 'Computing arbitrary signal', 'summary': 'Discusses computing arbitrary signals with 100 random values, creating epicycles to draw these random numbers, and the realization of a sawtooth wave pattern, leading to the next steps in the process.', 'duration': 118.731, 'highlights': ['The chapter discusses computing arbitrary signals with 100 random values, demonstrating the process of creating epicycles to draw these random numbers.', 'It highlights the realization of a sawtooth wave pattern through the process, showcasing the progression in the exploration of signals and mathematical concepts.', 'The presenter acknowledges the initial mislabeling of the wave pattern as a triangle wave, emphasizing the importance of precision and accuracy in terminology.']}, {'end': 2471.298, 'start': 1782.813, 'title': 'Fourier transform and drawing paths', 'summary': 'Discusses how to draw an arbitrary path with x and y coordinates using fourier transform, visualizing the results as rotating circles (epicycles) and ultimately computing the fourier transform for a known path (coding train logo).', 'duration': 688.485, 'highlights': ["The chapter explains the process of drawing an arbitrary path with x and y coordinates using Fourier transform, refactoring the visualization process into a function called 'epicycle', and applying the Fourier transform to both x and y signals, resulting in a sequence of rotating circles (epicycles).", "The speaker demonstrates the transformation of an arbitrary signal into the Fourier transform for x's and y's, and then computes the Fourier transform for a known path, the Coding Train logo, by iterating through the coordinates and skipping points to improve performance.", 'The chapter also includes the use of Perlin Noise to enhance the path drawing, manipulation of the skip variable to optimize the drawing performance, and the successful visualization of the known path, although the aesthetics could be further improved.']}, {'end': 2756.766, 'start': 2472.298, 'title': 'Sorting epicycles by amplitude', 'summary': "Discusses sorting epicycles in order of amplitude using javascript's sort function with a callback, resulting in a rotating path drawn by the fourier transform.", 'duration': 284.468, 'highlights': ["The epicycles are sorted in order of amplitude using JavaScript's sort function with a callback, resulting in a visually appealing rotating path. The speaker utilizes JavaScript's sort function with a callback to sort the epicycles in order of amplitude, resulting in a visually appealing rotating path.", 'The speaker plans to create a version of the code where the user can draw something and then compute the Fourier transform, encouraging the audience to try it as well. The speaker plans to create a version of the code that allows the user to draw something and then compute the Fourier transform, encouraging the audience to attempt this as well.', 'The audience is encouraged to create a version of the code that renders random drawings from the quick draw data set and share their variations on the coding challenge. The audience is encouraged to create a version of the code that renders random drawings from the quick draw data set and share their variations on the coding challenge.']}], 'duration': 1093.004, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/MY4luNgGfms/pics/MY4luNgGfms1663762.jpg', 'highlights': ['The chapter covers computing arbitrary signals with 100 random values and drawing paths using Fourier transform and epicycles.', "The epicycles are sorted in order of amplitude using JavaScript's sort function with a callback, resulting in a visually appealing rotating path.", 'The speaker plans to create a version of the code that allows the user to draw something and then compute the Fourier transform, encouraging the audience to attempt this as well.']}], 'highlights': ['Creation of a Fourier transform drawing machine using the discrete Fourier transform algorithm to trace the path of a logo through rotating circles', 'The chapter explores analyzing various signals including square waves, Perlin noise, and sine waves to observe their behavior under the Fourier transform', "The chapter emphasizes the translation of Fourier transform notation into JavaScript code, focusing on the use of Euler's formula and the implementation of discrete Fourier transform through summation", 'The process involves creating an array of k elements and looping through each k, going from 0 to n, to calculate the frequency of wave patterns', 'Implementing the discrete Fourier transform and processing a hard-coded square wave resulted in 12 complex numbers with real and imaginary components, indicating a successful implementation', 'The chapter covers computing arbitrary signals with 100 random values and drawing paths using Fourier transform and epicycles']}