title
Shannon Nyquist Sampling Theorem

description
Follow on Twitter: @eigensteve Brunton's website: https://eigensteve.com This video discusses the famous Shannon-Nyquist sampling theorem, which discusses limits on signal reconstruction given how fast it is sampled and the frequency content of the signal. For original papers: Shannon, 1948: http://people.math.harvard.edu/~ctm/home/text/others/shannon/entropy/entropy.pdf Nyquist, 1928: https://bayes.wustl.edu/Manual/CertainTopicsInTelegraphTransmissionTheory.pdf

detail
{'title': 'Shannon Nyquist Sampling Theorem', 'heatmap': [{'end': 613.674, 'start': 518.759, 'weight': 0.703}], 'summary': 'Covers the shannon-nyquist sampling theorem, its significance in signal processing and control systems, and the evolution of the theorem over 70-90 years, emphasizing its applications, including audio sampling at 44 khz, and its potential in reconstructing audio signals due to sparsity in the frequency domain.', 'chapters': [{'end': 247.555, 'segs': [{'end': 50.814, 'src': 'embed', 'start': 6.923, 'weight': 0, 'content': [{'end': 7.403, 'text': 'Welcome back.', 'start': 6.923, 'duration': 0.48}, {'end': 13.067, 'text': "So I'm really excited today to tell you about the Shannon-Nyquist sampling theorem,", 'start': 8.284, 'duration': 4.783}, {'end': 17.19, 'text': 'which is one of the most important results in all of information theory.', 'start': 13.067, 'duration': 4.123}, {'end': 20.833, 'text': "You're going to see it everywhere now that I'm going to tell you about it.", 'start': 17.891, 'duration': 2.942}, {'end': 30.44, 'text': "And it's really important, especially when we think about signal processing control systems, especially compressed sensing and sparsity,", 'start': 21.333, 'duration': 9.107}, {'end': 33.582, 'text': 'and kind of the recent trends in applied mathematics.', 'start': 30.44, 'duration': 3.142}, {'end': 38.045, 'text': "So I'm going to jump in and I'm going to tell you all about this Shannon Nyquist sampling theorem,", 'start': 34.022, 'duration': 4.023}, {'end': 45.87, 'text': "which basically tells you if you have some signal you're measuring, let's say it's some signal that's oscillating and doing something,", 'start': 38.045, 'duration': 7.825}, {'end': 50.814, 'text': 'how fast you have to measure it to perfectly represent and reconstruct that signal.', 'start': 45.87, 'duration': 4.944}], 'summary': 'The shannon-nyquist sampling theorem is vital for signal processing, control systems, and compressed sensing, determining the required sampling rate for perfect signal reconstruction.', 'duration': 43.891, 'max_score': 6.923, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/FcXZ28BX-xE/pics/FcXZ28BX-xE6923.jpg'}, {'end': 116.5, 'src': 'embed', 'start': 82.257, 'weight': 5, 'content': [{'end': 84.939, 'text': 'Nyquist is a Swedish-American mathematician.', 'start': 82.257, 'duration': 2.682}, {'end': 90.383, 'text': 'And both of them developed a lot of their best work working at Bell Labs,', 'start': 85.38, 'duration': 5.003}, {'end': 93.466, 'text': 'which is kind of this famous think tank where a lot of great ideas came out of.', 'start': 90.383, 'duration': 3.083}, {'end': 105.753, 'text': 'Shannon, in particular, did a lot of wartime research during World War II in encryption and code breaking and just kind of.', 'start': 95.187, 'duration': 10.566}, {'end': 109.256, 'text': "you can kind of think of him as America's counterpart of Alan Turing.", 'start': 105.753, 'duration': 3.503}, {'end': 116.5, 'text': "And in fact they were contemporaries and Claude Shannon loved Turing's work because it complimented his so well.", 'start': 109.316, 'duration': 7.184}], 'summary': "Shannon and nyquist did significant work at bell labs, with shannon contributing to wartime research and being considered america's counterpart to alan turing.", 'duration': 34.243, 'max_score': 82.257, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/FcXZ28BX-xE/pics/FcXZ28BX-xE82257.jpg'}, {'end': 247.555, 'src': 'embed', 'start': 154.305, 'weight': 1, 'content': [{'end': 160.828, 'text': 'How much can you compress information and expect to get a faithful decompression kind of downstream on the other end?', 'start': 154.305, 'duration': 6.523}, {'end': 163.971, 'text': 'And of course we already had lots of knowledge about this.', 'start': 161.408, 'duration': 2.563}, {'end': 170.279, 'text': 'There was already Morse code and some kind of redundancy built into communications in telegraphing.', 'start': 163.991, 'duration': 6.288}, {'end': 173.323, 'text': 'But this really put this on a firm mathematical footing.', 'start': 170.68, 'duration': 2.643}, {'end': 182.929, 'text': 'Claude Shannon in particular, is oftentimes called the father of kind of information theory.', 'start': 174.604, 'duration': 8.325}, {'end': 187.831, 'text': 'And a lot, a lot, a lot of what we have in information theory came from Claude Shannon.', 'start': 183.73, 'duration': 4.101}, {'end': 193.073, 'text': 'So he also took ideas from thermodynamics, entropy, and introduced that into information theory.', 'start': 187.871, 'duration': 5.202}, {'end': 199.716, 'text': "So he introduced, you know, information entropy when you're sending signals over the ocean, okay, across the Atlantic.", 'start': 193.133, 'duration': 6.583}, {'end': 203.677, 'text': 'Nyquist was really pivotal in control theory.', 'start': 200.696, 'duration': 2.981}, {'end': 209.019, 'text': 'So a lot of our classical control theory comes from Nyquist and a lot of things are named after Nyquist.', 'start': 203.737, 'duration': 5.282}, {'end': 218.363, 'text': 'So these are two titans that kind of came together in this idea of sampling for the sampling theorem.', 'start': 209.399, 'duration': 8.964}, {'end': 219.823, 'text': 'Okay, good.', 'start': 218.703, 'duration': 1.12}, {'end': 230.227, 'text': "And I'll just also point out that Shannon, you know, also was very deeply interested in kind of this theory of secrecy and the way he said it is,", 'start': 220.604, 'duration': 9.623}, {'end': 234.949, 'text': 'that kind of communication and cryptography are inseparable.', 'start': 230.227, 'duration': 4.722}, {'end': 238.891, 'text': "You can't study one without inherently analyzing the other.", 'start': 235.009, 'duration': 3.882}, {'end': 247.555, 'text': "You can't talk about communication theory without the theory of kind of encoding and coding and cryptography.", 'start': 238.931, 'duration': 8.624}], 'summary': 'Claude shannon and nyquist laid the foundation for information theory and control theory, integrating ideas from thermodynamics, entropy, and cryptography.', 'duration': 93.25, 'max_score': 154.305, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/FcXZ28BX-xE/pics/FcXZ28BX-xE154305.jpg'}], 'start': 6.923, 'title': 'Shannon-nyquist sampling theorem, information theory, and code breaking', 'summary': 'Introduces the shannon-nyquist sampling theorem, emphasizing its importance in various fields including signal processing, control systems, and applied mathematics. it also explores the connections between code breaking and information theory, showcasing the foundational contributions of claude shannon and harry nyquist in information theory and control theory.', 'chapters': [{'end': 50.814, 'start': 6.923, 'title': 'Shannon-nyquist sampling theorem', 'summary': 'Introduces the shannon-nyquist sampling theorem, emphasizing its importance in signal processing, control systems, compressed sensing, sparsity, and applied mathematics. it discusses the necessity of measuring a signal at a certain speed for perfect representation and reconstruction.', 'duration': 43.891, 'highlights': ['The Shannon-Nyquist sampling theorem is introduced as one of the most important results in information theory, with applications in signal processing, control systems, compressed sensing, sparsity, and applied mathematics.', 'It emphasizes the necessity of measuring a signal at a certain speed to perfectly represent and reconstruct the signal.']}, {'end': 130.294, 'start': 51.675, 'title': 'Connection between code breaking and information theory', 'summary': 'Explores the seminal papers by claude shannon and harry nyquist, both mathematicians at bell labs, and their work on communication theory and code breaking, showcasing the intriguing connections between code breaking and information theory.', 'duration': 78.619, 'highlights': ['Claude Shannon and Harry Nyquist were mathematicians at Bell Labs who authored two seminal papers on communication theory.', "Shannon's wartime research during World War II in encryption and code breaking is noteworthy, reflecting the intriguing connection between code breaking and information theory.", 'Nyquist and Shannon developed their best work at Bell Labs, a renowned think tank where many innovative ideas originated.']}, {'end': 247.555, 'start': 130.854, 'title': 'Information theory and communication', 'summary': 'Discusses the foundational contributions of claude shannon and nyquist in information theory and control theory, addressing the challenge of compressing and transmitting information over long distances, and their impact on communication and cryptography.', 'duration': 116.701, 'highlights': ["Claude Shannon's role in laying the mathematical foundation for information theory and its application in communication and cryptography, drawing from ideas in thermodynamics and introducing the concept of information entropy.", "Nyquist's pivotal contributions to control theory and the sampling theorem, which have had a significant impact on signal processing and communication.", 'The challenge of compressing and transmitting information over long distances, addressing the question of faithful decompression, building on existing knowledge such as Morse code and redundancy in communications.']}], 'duration': 240.632, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/FcXZ28BX-xE/pics/FcXZ28BX-xE6923.jpg', 'highlights': ['The Shannon-Nyquist sampling theorem is introduced as one of the most important results in information theory, with applications in signal processing, control systems, compressed sensing, sparsity, and applied mathematics.', "Claude Shannon's role in laying the mathematical foundation for information theory and its application in communication and cryptography, drawing from ideas in thermodynamics and introducing the concept of information entropy.", "Nyquist's pivotal contributions to control theory and the sampling theorem, which have had a significant impact on signal processing and communication.", 'It emphasizes the necessity of measuring a signal at a certain speed to perfectly represent and reconstruct the signal.', "Shannon's wartime research during World War II in encryption and code breaking is noteworthy, reflecting the intriguing connection between code breaking and information theory.", 'Nyquist and Shannon developed their best work at Bell Labs, a renowned think tank where many innovative ideas originated.', 'The challenge of compressing and transmitting information over long distances, addressing the question of faithful decompression, building on existing knowledge such as Morse code and redundancy in communications.']}, {'end': 809.769, 'segs': [{'end': 308.645, 'src': 'embed', 'start': 275.945, 'weight': 0, 'content': [{'end': 277.025, 'text': 'again measured in hertz.', 'start': 275.945, 'duration': 1.08}, {'end': 283.008, 'text': "So if you have a function, and I'm actually gonna say this Another way, I want to flip it also.", 'start': 277.565, 'duration': 5.443}, {'end': 292.034, 'text': 'If you have a function and you want to perfectly represent that function, you want to perfectly resolve all of its frequency content perfectly,', 'start': 284.009, 'duration': 8.025}, {'end': 296.337, 'text': 'then you have to sample that function at twice its highest frequency.', 'start': 292.034, 'duration': 4.303}, {'end': 298.319, 'text': "Okay, so I'll say this a couple ways.", 'start': 296.357, 'duration': 1.962}, {'end': 302.121, 'text': "And we're used to thinking about functions like an audio signal.", 'start': 298.379, 'duration': 3.742}, {'end': 308.645, 'text': "You can take its Fourier transform and it's got a power spectrum that tells you where in that signal.", 'start': 302.141, 'duration': 6.504}], 'summary': 'To perfectly represent a function, sample it at twice its highest frequency.', 'duration': 32.7, 'max_score': 275.945, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/FcXZ28BX-xE/pics/FcXZ28BX-xE275945.jpg'}, {'end': 347.453, 'src': 'embed', 'start': 319.737, 'weight': 1, 'content': [{'end': 325.844, 'text': 'If you want to perfectly represent that information and you have a highest frequency you care about,', 'start': 319.737, 'duration': 6.107}, {'end': 328.588, 'text': 'you have to sample at twice that highest frequency.', 'start': 325.844, 'duration': 2.744}, {'end': 335.73, 'text': "Okay, and so that establishes what's known now as the Nyquist rate, which is two omega measured in hertz.", 'start': 329.068, 'duration': 6.662}, {'end': 340.991, 'text': "or alternatively, it says if you care about a frequency omega, if that's the highest frequency you care about,", 'start': 335.73, 'duration': 5.261}, {'end': 347.453, 'text': 'you have to sample at least as fast as a delta T of one over two omega in seconds.', 'start': 340.991, 'duration': 6.462}], 'summary': 'Nyquist rate requires sampling at least twice the highest frequency.', 'duration': 27.716, 'max_score': 319.737, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/FcXZ28BX-xE/pics/FcXZ28BX-xE319737.jpg'}, {'end': 613.674, 'src': 'heatmap', 'start': 518.759, 'weight': 0.703, 'content': [{'end': 523.282, 'text': "So that 2 omega is critical because otherwise you'll get this phenomenon.", 'start': 518.759, 'duration': 4.523}, {'end': 526.025, 'text': "This is called aliasing, and I'm going to write that up here.", 'start': 523.323, 'duration': 2.702}, {'end': 528.667, 'text': "So you'll get this aliasing phenomenon.", 'start': 526.065, 'duration': 2.602}, {'end': 536.781, 'text': 'which basically says that as far as your sampling is concerned, these curves are aliases of one another.', 'start': 530.014, 'duration': 6.767}, {'end': 543.188, 'text': 'They might as well be the same person or the same thing as far as this blue sampling is concerned.', 'start': 536.861, 'duration': 6.327}, {'end': 550.735, 'text': "So if you sample below the Nyquist rate, you're gonna get aliasing and you're gonna lose the high frequency components that you might care about.", 'start': 543.788, 'duration': 6.947}, {'end': 559.058, 'text': "Now there's a really cool plot that you can make that kind of shows this idea of aliasing, and sometimes this is called frequency folding.", 'start': 551.656, 'duration': 7.402}, {'end': 568.021, 'text': 'So if you look at the power spectral density versus frequency, and remember, omega is the frequency I care about.', 'start': 559.098, 'duration': 8.923}, {'end': 575.873, 'text': "That's where all of my frequency content Actually, I'm gonna say this a little differently.", 'start': 568.061, 'duration': 7.812}, {'end': 578.715, 'text': "Let's say here omega is what I'm actually sampling at.", 'start': 576.073, 'duration': 2.642}, {'end': 579.916, 'text': 'So I sample at omega.', 'start': 578.755, 'duration': 1.161}, {'end': 582.818, 'text': "This is kind of all that I'm able to sample.", 'start': 580.296, 'duration': 2.522}, {'end': 588.661, 'text': "But let's say that my signal has frequency content up here at 1.5 omega.", 'start': 583.538, 'duration': 5.123}, {'end': 601.572, 'text': "Okay, so if my signal has frequency content at 1.5 omega, but I sample at omega, I'm actually going to what I measure to my measurement at this.", 'start': 592.15, 'duration': 9.422}, {'end': 603.632, 'text': 'frequency is going to look like this', 'start': 601.572, 'duration': 2.06}, {'end': 609.813, 'text': "So it's going to look like an aliased version that's folded over in the frequency domain.", 'start': 604.612, 'duration': 5.201}, {'end': 613.674, 'text': "I'm going to say this again because this is a little tricky and a little subtle, okay?", 'start': 610.374, 'duration': 3.3}], 'summary': 'Sampling below nyquist rate leads to aliasing, losing high frequency components.', 'duration': 94.915, 'max_score': 518.759, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/FcXZ28BX-xE/pics/FcXZ28BX-xE518759.jpg'}, {'end': 726.691, 'src': 'embed', 'start': 698.818, 'weight': 2, 'content': [{'end': 701.679, 'text': 'then this camera might actually cause there to be aliasing.', 'start': 698.818, 'duration': 2.861}, {'end': 706.001, 'text': "so as i walk around from pixel to pixel, you'd almost see it like sparkling.", 'start': 701.679, 'duration': 4.322}, {'end': 712.024, 'text': "that's a weird camera effect that you get sometimes, and it's because of this idea of aliasing.", 'start': 706.001, 'duration': 6.023}, {'end': 714.065, 'text': 'so you also see this.', 'start': 712.024, 'duration': 2.041}, {'end': 716.607, 'text': 'this is a picture of a curtain.', 'start': 714.065, 'duration': 2.542}, {'end': 726.691, 'text': 'i believe this is at um Heathrow Airport, I think, and you can see these kind of these kind of more patterns that you see sometimes in optics.', 'start': 716.607, 'duration': 10.084}], 'summary': 'Camera may cause aliasing, seen as sparkling effect in pixels and patterns in optics.', 'duration': 27.873, 'max_score': 698.818, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/FcXZ28BX-xE/pics/FcXZ28BX-xE698818.jpg'}], 'start': 247.855, 'title': 'Nyquist sampling theorems', 'summary': 'Discusses the shannon nyquist sampling theorem, emphasizing the importance of sampling at twice the highest frequency for complete signal determination. it also explains the nyquist rate and its application in audio sampling at 44 khz, highlighting the repercussions of undersampling and aliasing with practical examples.', 'chapters': [{'end': 319.237, 'start': 247.855, 'title': 'Shannon nyquist sampling theorem', 'summary': 'Discusses the shannon nyquist sampling theorem, stating that a function can be completely determined by sampling at twice its highest frequency, particularly relevant in interpreting and reconstructing audio signals.', 'duration': 71.382, 'highlights': ['The sampling theorem states that a function containing no frequency higher than omega measured in hertz is completely determined by sampling that function at two omega, measured in hertz.', 'To perfectly represent a function and resolve all of its frequency content, it must be sampled at twice its highest frequency.', 'The discussion emphasizes the relevance of the sampling theorem in interpreting and reconstructing audio signals through Fourier transform, power spectrum, compression, and decompression.']}, {'end': 809.769, 'start': 319.737, 'title': 'Nyquist sampling theorem', 'summary': 'Explains the nyquist rate as the minimum sampling frequency required to perfectly represent a signal, illustrated by the example of audio sampling at 44 khz due to human hearing capabilities, and discusses the consequences of undersampling and aliasing, with practical examples and applications.', 'duration': 490.032, 'highlights': ['The Nyquist rate establishes the minimum sampling frequency to perfectly represent a signal, such as audio sampling at 44 kHz due to human hearing capabilities. Audio signals are sampled at 44 kHz because humans can hear up to about 20 kHz, and doubling that frequency is required for perfect fidelity reconstruction during decompression.', 'Undersampling below the Nyquist rate can lead to aliasing, causing the loss of high frequency components in the signal. Undersampling below the Nyquist rate can result in aliasing, leading to the loss of high frequency components in the signal, demonstrated through practical examples and applications in signal processing and optics.', 'The consequences of aliasing are illustrated through examples such as the visual phenomena in cameras and the artistic rendition of aliasing patterns. Various practical examples of aliasing are illustrated, including the visual phenomena in cameras and artistic renditions of aliasing patterns in curtains, and the effects observed in the downsampling of the discrete Fourier transform matrix.']}], 'duration': 561.914, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/FcXZ28BX-xE/pics/FcXZ28BX-xE247855.jpg', 'highlights': ['To perfectly represent a function and resolve all of its frequency content, it must be sampled at twice its highest frequency.', 'The Nyquist rate establishes the minimum sampling frequency to perfectly represent a signal, such as audio sampling at 44 kHz due to human hearing capabilities.', 'The consequences of aliasing are illustrated through examples such as the visual phenomena in cameras and the artistic rendition of aliasing patterns.']}, {'end': 1036.542, 'segs': [{'end': 839.58, 'src': 'embed', 'start': 810.41, 'weight': 0, 'content': [{'end': 816.137, 'text': "Now, what's interesting about this, and I'm gonna kind of close the lecture on where we're at now.", 'start': 810.41, 'duration': 5.727}, {'end': 817.739, 'text': 'Now fast forward.', 'start': 816.157, 'duration': 1.582}, {'end': 823.165, 'text': 'you know 70, 80, 90 years past Shannon and Nyquist to today.', 'start': 817.739, 'duration': 5.426}, {'end': 833.176, 'text': 'Advances in applied math and optimization and statistics are starting to change how we think about the Shannon Nyquist sampling theorem.', 'start': 824.79, 'duration': 8.386}, {'end': 839.58, 'text': 'So technically speaking, all of the results from Shannon and Nyquist are for broadband signals.', 'start': 833.756, 'duration': 5.824}], 'summary': 'Advances in math and stats are changing shannon nyquist theorem for broadband signals.', 'duration': 29.17, 'max_score': 810.41, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/FcXZ28BX-xE/pics/FcXZ28BX-xE810410.jpg'}, {'end': 941.702, 'src': 'embed', 'start': 856.531, 'weight': 2, 'content': [{'end': 862.836, 'text': 'That is definitely the then the Shannon Nyquist sampling theorem absolutely applies to those dense broadband signals.', 'start': 856.531, 'duration': 6.305}, {'end': 869.622, 'text': "You can't beat the Nyquist sampling theorem and have full perfect signal reconstruction.", 'start': 863.137, 'duration': 6.485}, {'end': 880.97, 'text': 'But if your signal is not broadband or dense, if it is only a couple of frequencies that are high frequencies, you technically can in fact sometimes,', 'start': 870.582, 'duration': 10.388}, {'end': 885.533, 'text': 'under some conditions, beat the the nyquist sampling frequency on average.', 'start': 880.97, 'duration': 4.563}, {'end': 891.437, 'text': "and i'm going to talk about this more in the context of compressed sensing and reconstructing audio signals.", 'start': 885.533, 'duration': 5.904}, {'end': 894.96, 'text': "but for now i'm just going to walk you through, at a very high level, what we expect to see.", 'start': 891.437, 'duration': 3.523}, {'end': 903.164, 'text': 'So here we have a signal F, which is the sum of two sine waves at 73 Hz and 531 Hz.', 'start': 895.82, 'duration': 7.344}, {'end': 912.489, 'text': 'And so what the Shannon Nyquist sampling theorem would say is that to fully resolve this F, I would have to sample twice the highest frequency,', 'start': 904.005, 'duration': 8.484}, {'end': 917.512, 'text': 'or twice 531, which is 1062 samples per second.', 'start': 912.489, 'duration': 5.023}, {'end': 921.054, 'text': "That's how fast I'd have to sample to perfectly reconstruct this signal.", 'start': 917.732, 'duration': 3.322}, {'end': 924.295, 'text': "And here's the power spectrum, here's the signal.", 'start': 922.254, 'duration': 2.041}, {'end': 930.017, 'text': 'Now, that is technically only true for broadband signals, but this is not a broadband signal.', 'start': 925.015, 'duration': 5.002}, {'end': 933.439, 'text': 'This signal is very sparse in the frequency domain.', 'start': 930.037, 'duration': 3.402}, {'end': 937.5, 'text': "There's only two frequency peaks, only two tones making up this function.", 'start': 933.459, 'duration': 4.041}, {'end': 941.702, 'text': "And so you can get away, I'm going to zoom in here.", 'start': 938.28, 'duration': 3.422}], 'summary': 'Nyquist sampling theorem applies to dense broadband signals, but sparse signals may require lower sampling frequency.', 'duration': 85.171, 'max_score': 856.531, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/FcXZ28BX-xE/pics/FcXZ28BX-xE856531.jpg'}, {'end': 1010.523, 'src': 'embed', 'start': 967.267, 'weight': 1, 'content': [{'end': 973.529, 'text': 'but if I do it randomly, not regularly or uniformly, then results in compressed sensing.', 'start': 967.267, 'duration': 6.262}, {'end': 980.891, 'text': 'say that you can actually faithfully reconstruct the sparse vector in the Fourier domain, the sparse power spectrum, and then you can inverse,', 'start': 973.529, 'duration': 7.362}, {'end': 985.012, 'text': 'Fourier, transform and reconstruct your full signal, essentially with no aliasing.', 'start': 980.891, 'duration': 4.121}, {'end': 989.734, 'text': 'So this is a really cool result in the field of compressed sensing.', 'start': 985.612, 'duration': 4.122}, {'end': 992.695, 'text': "I don't know why this is spelled wrong.", 'start': 989.754, 'duration': 2.941}, {'end': 996.437, 'text': 'It should be compressed sensing or compressive sampling.', 'start': 993.035, 'duration': 3.402}, {'end': 1001.219, 'text': 'And the basic idea is that if I sample randomly in time,', 'start': 997.117, 'duration': 4.102}, {'end': 1008.502, 'text': 'I can get away with a much lower average sampling rate than predicted by Shannon Nyquist if my signal is sparse.', 'start': 1001.219, 'duration': 7.283}, {'end': 1010.523, 'text': "So I'm going to tell you a lot more about this.", 'start': 1008.962, 'duration': 1.561}], 'summary': 'Compressed sensing allows faithful signal reconstruction with lower average sampling rate, based on sparsity.', 'duration': 43.256, 'max_score': 967.267, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/FcXZ28BX-xE/pics/FcXZ28BX-xE967267.jpg'}], 'start': 810.41, 'title': 'Evolution of nyquist sampling theorem', 'summary': 'Discusses the evolution of shannon nyquist sampling theorem over 70-90 years, its applicability to broadband signals, exceptions for non-broadband or dense signals, compressed sensing, reconstructing audio signals, and the potential to sample at lower frequencies due to sparsity in the frequency domain.', 'chapters': [{'end': 880.97, 'start': 810.41, 'title': 'Relevance of shannon nyquist sampling theorem today', 'summary': 'Discusses the evolution of the shannon nyquist sampling theorem over 70-90 years, highlighting the applicability to broadband signals and exceptions for non-broadband or dense signals.', 'duration': 70.56, 'highlights': ['Shannon Nyquist theorem applies to dense broadband signals, ensuring perfect signal reconstruction (quantifiable: broad frequency content).', 'Advances in applied math, optimization, and statistics are reshaping the understanding of the Shannon Nyquist sampling theorem (quantifiable: technological advancements).', 'Exceptions to the Shannon Nyquist theorem exist for non-broadband or dense signals with only a few high frequencies (quantifiable: limited frequency content).']}, {'end': 941.702, 'start': 880.97, 'title': 'Nyquist sampling theorem', 'summary': 'Explains how under certain conditions, it is possible to beat the nyquist sampling frequency on average, focusing on compressed sensing and reconstructing audio signals, with an example of a signal consisting of two sine waves at 73 hz and 531 hz, demonstrating the potential to sample at lower frequencies due to sparsity in the frequency domain.', 'duration': 60.732, 'highlights': ['The Shannon Nyquist sampling theorem states that to fully resolve the given signal, it would have to be sampled at 1062 samples per second, given the highest frequency of 531 Hz.', 'The signal is sparse in the frequency domain, containing only two frequency peaks, allowing for sampling at lower frequencies than dictated by the Nyquist theorem.', 'The potential to sample at lower frequencies is demonstrated through the example of a signal consisting of only two sine waves at 73 Hz and 531 Hz, challenging the traditional Nyquist sampling theorem.']}, {'end': 1036.542, 'start': 941.702, 'title': 'Compressed sensing: sampling below nyquist rate', 'summary': 'Discusses the concept of compressed sensing, highlighting the ability to faithfully reconstruct sparse signals below the nyquist sampling rate, with a specific emphasis on the advantage of random sampling over uniform sampling and the potential for reconstruction with much lower average sampling rate than predicted by shannon nyquist.', 'duration': 94.84, 'highlights': ['Compressed sensing allows faithful reconstruction of sparse signals below the Nyquist sampling rate, particularly with random sampling over uniform sampling.', 'Random sampling at an average of 128 samples per second can result in compressed sensing, enabling reconstruction of the sparse power spectrum with no aliasing.', 'The concept of compressed sensing allows for a much lower average sampling rate than predicted by Shannon Nyquist when dealing with sparse signals.']}], 'duration': 226.132, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/FcXZ28BX-xE/pics/FcXZ28BX-xE810410.jpg', 'highlights': ['Advances in applied math, optimization, and statistics are reshaping the understanding of the Shannon Nyquist sampling theorem (technological advancements).', 'Compressed sensing allows faithful reconstruction of sparse signals below the Nyquist sampling rate, particularly with random sampling over uniform sampling.', 'The potential to sample at lower frequencies is demonstrated through the example of a signal consisting of only two sine waves at 73 Hz and 531 Hz, challenging the traditional Nyquist sampling theorem.', 'Exceptions to the Shannon Nyquist theorem exist for non-broadband or dense signals with only a few high frequencies (limited frequency content).', 'The signal is sparse in the frequency domain, containing only two frequency peaks, allowing for sampling at lower frequencies than dictated by the Nyquist theorem.', 'Random sampling at an average of 128 samples per second can result in compressed sensing, enabling reconstruction of the sparse power spectrum with no aliasing.', 'Shannon Nyquist theorem applies to dense broadband signals, ensuring perfect signal reconstruction (broad frequency content).', 'The Shannon Nyquist sampling theorem states that to fully resolve the given signal, it would have to be sampled at 1062 samples per second, given the highest frequency of 531 Hz.']}], 'highlights': ['The Shannon-Nyquist sampling theorem is introduced as one of the most important results in information theory, with applications in signal processing, control systems, compressed sensing, sparsity, and applied mathematics.', 'Advances in applied math, optimization, and statistics are reshaping the understanding of the Shannon Nyquist sampling theorem (technological advancements).', 'Compressed sensing allows faithful reconstruction of sparse signals below the Nyquist sampling rate, particularly with random sampling over uniform sampling.', 'The potential to sample at lower frequencies is demonstrated through the example of a signal consisting of only two sine waves at 73 Hz and 531 Hz, challenging the traditional Nyquist sampling theorem.', 'The Nyquist rate establishes the minimum sampling frequency to perfectly represent a signal, such as audio sampling at 44 kHz due to human hearing capabilities.', 'The Shannon Nyquist sampling theorem states that to fully resolve the given signal, it would have to be sampled at 1062 samples per second, given the highest frequency of 531 Hz.', 'It emphasizes the necessity of measuring a signal at a certain speed to perfectly represent and reconstruct the signal.']}