title

The more general uncertainty principle, regarding Fourier transforms

description

The meaning of the uncertainty principle in the context of Fourier transforms
Help fund future projects: https://www.patreon.com/3blue1brown
An equally valuable form of support is to simply share some of the videos.
Special thanks to these supporters: http://3b1b.co/uncertainty-thanks
For more on quantum mechanical wave functions, I highly recommend this video by udiprod:
https://youtu.be/p7bzE1E5PMY
Minute physics on special relativity:
https://youtu.be/1rLWVZVWfdY
Main video on the Fourier transform
https://youtu.be/spUNpyF58BY
Louis de Broglie thesis:
http://aflb.ensmp.fr/LDB-oeuvres/De_Broglie_Kracklauer.pdf
More on Doppler radar:
Radar basics: https://www.eetimes.com/document.asp?doc_id=1278808
There's a key way in which the description I gave of the trade-off in Doppler radar differs from reality. Since the speed of light is so drastically greater than the speed of things being detected, the Fourier representation for pulse echoes of different objects would almost certainly overlap unless it was played for a very long time. In effect, this is what happens, since one does not send out a single pulse, but a whole bunch of evenly spaced pulses as some pulse repetition frequency (or PRF).
This means the Fourier representation of all those pulses together can actually be quite sharp. Assuming a large number of such pulses, it will look like several vertical lines spaced out by the PRF. As long as the pulses are far enough apart that the echoes of multiple objects on the field from different targets don't overlap, it's not a problem for position determinations that the full sequence of pulses occupies such a long duration. However, the trade-off now comes in choosing the right PRF. See the above article for more information.
Music by Vincent Rubinetti:
https://vincerubinetti.bandcamp.com/album/the-music-of-3blue1brown
------------------
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detail

{'title': 'The more general uncertainty principle, regarding Fourier transforms', 'heatmap': [{'end': 269.422, 'start': 249.738, 'weight': 1}], 'summary': "Delves into the uncertainty principle in wave phenomena, illustrating the trade-off between frequency and duration and its impact on frequency information confidence, with examples from sound waves, doppler radar, and particles, as well as covering the application of fourier transform in analyzing frequencies and in doppler radar to measure object parameters, and exploring louis de broglie's wave-particle theory and the heisenberg uncertainty principle.", 'chapters': [{'end': 153.072, 'segs': [{'end': 123.248, 'src': 'embed', 'start': 93.891, 'weight': 0, 'content': [{'end': 99.636, 'text': 'So that certainty about the frequency information required an observation spread out over time.', 'start': 93.891, 'duration': 5.745}, {'end': 109.01, 'text': 'And this trade-off right here between how short your observation can be and how confident you can feel about the frequency is an example of the general uncertainty principle.', 'start': 100.456, 'duration': 8.554}, {'end': 111.546, 'text': 'Similarly, think of a musical note.', 'start': 110.086, 'duration': 1.46}, {'end': 116.327, 'text': 'The shorter it lasts in time, the less certain you can be about what its exact frequency is.', 'start': 111.846, 'duration': 4.481}, {'end': 123.248, 'text': 'In the extreme, I could ask you what the pitch of a clap or a shockwave is, and even someone with perfect pitch would be unable to answer.', 'start': 117.007, 'duration': 6.241}], 'summary': 'Observation over time affects certainty of frequency. uncertainty principle applies to musical notes.', 'duration': 29.357, 'max_score': 93.891, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/MBnnXbOM5S4/pics/MBnnXbOM5S493891.jpg'}, {'end': 158.876, 'src': 'embed', 'start': 132.61, 'weight': 2, 'content': [{'end': 137.951, 'text': 'it would be a little more accurate here to say that the short signal correlates highly with a wider range of frequency.', 'start': 132.61, 'duration': 5.341}, {'end': 146.848, 'text': 'and that the signal correlating strongly with only a narrow range of frequencies must last for a longer time.', 'start': 141.465, 'duration': 5.383}, {'end': 153.072, 'text': "Here, that's the kind of phrase that's made a little bit clearer when we bring in the actual math.", 'start': 149.61, 'duration': 3.462}, {'end': 158.876, 'text': "So let's turn now to talking about the Fourier transform, which is the relevant construct for analyzing frequencies.", 'start': 153.533, 'duration': 5.343}], 'summary': 'Short signal correlates highly with wider frequency range, while strong correlation with narrow range must last longer. fourier transform is relevant for analyzing frequencies.', 'duration': 26.266, 'max_score': 132.61, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/MBnnXbOM5S4/pics/MBnnXbOM5S4132610.jpg'}], 'start': 3.755, 'title': 'Uncertainty principle in wave phenomena', 'summary': 'Explains the uncertainty principle as a general trade-off in wave phenomena, demonstrating how the interplay between frequency and duration affects confidence in frequency information, with examples from sound waves, doppler radar, and particles.', 'chapters': [{'end': 153.072, 'start': 3.755, 'title': 'Uncertainty principle in wave phenomena', 'summary': 'Explains the uncertainty principle as a general trade-off in wave phenomena, demonstrating how the interplay between frequency and duration affects confidence in frequency information, with examples from sound waves, doppler radar, and particles.', 'duration': 149.317, 'highlights': ['The uncertainty principle is a general trade-off in wave phenomena, showcasing the interplay between frequency and duration, impacting confidence in frequency information.', 'Observation spread out over time increases confidence in frequency information, illustrating the trade-off between the duration of observation and the confidence in frequency.', 'Shorter duration of a signal correlates with a wider range of frequency, while a longer duration signal strongly correlates with a narrow range of frequencies.']}], 'duration': 149.317, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/MBnnXbOM5S4/pics/MBnnXbOM5S43755.jpg', 'highlights': ['Observation spread out over time increases confidence in frequency information, illustrating the trade-off between the duration of observation and the confidence in frequency.', 'The uncertainty principle is a general trade-off in wave phenomena, showcasing the interplay between frequency and duration, impacting confidence in frequency information.', 'Shorter duration of a signal correlates with a wider range of frequency, while a longer duration signal strongly correlates with a narrow range of frequencies.']}, {'end': 624.988, 'segs': [{'end': 278.839, 'src': 'heatmap', 'start': 237.776, 'weight': 3, 'content': [{'end': 241.378, 'text': 'So the whole weight of the graph is kind of off-center, so to speak.', 'start': 237.776, 'duration': 3.602}, {'end': 249.738, 'text': 'The idea behind the Fourier transform is that if you follow the center of mass of the wound up graph with frequency f,', 'start': 242.697, 'duration': 7.041}, {'end': 255.559, 'text': 'the position of that center of mass encodes the strength of that frequency in the original signal.', 'start': 249.738, 'duration': 5.821}, {'end': 261.019, 'text': 'The distance between that center of mass and the origin captures the strength of that frequency.', 'start': 256.579, 'duration': 4.44}, {'end': 264.441, 'text': "And this is something I didn't really talk about in the main video,", 'start': 261.901, 'duration': 2.54}, {'end': 269.422, 'text': 'but the angle of that center of mass off the horizontal corresponds to the phase of the given frequency.', 'start': 264.441, 'duration': 4.981}, {'end': 278.839, 'text': "Now, one way to think of this whole winding mechanism is that it's a way to measure how well your signal correlates with a given pure frequency.", 'start': 271.877, 'duration': 6.962}], 'summary': 'The fourier transform measures the strength and phase of frequencies in a signal.', 'duration': 41.063, 'max_score': 237.776, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/MBnnXbOM5S4/pics/MBnnXbOM5S4237776.jpg'}, {'end': 348.689, 'src': 'embed', 'start': 320.907, 'weight': 2, 'content': [{'end': 326.112, 'text': "is the Fourier transform's way of telling us that the dominant frequency of the signal is 5 beats per second.", 'start': 320.907, 'duration': 5.205}, {'end': 330.836, 'text': "And, equally importantly, the fact that it's a little bit spread out around.", 'start': 326.773, 'duration': 4.063}, {'end': 337.883, 'text': 'that 5 is an indication that pure sine waves near 5 beats per second also correlate pretty well with the signal.', 'start': 330.836, 'duration': 7.047}, {'end': 341.786, 'text': 'And that last idea is key for the uncertainty principle.', 'start': 339.465, 'duration': 2.321}, {'end': 348.689, 'text': 'What I want you to do is think about how this spread changes as the signal persists longer or shorter over time.', 'start': 342.286, 'duration': 6.403}], 'summary': 'Fourier transform shows dominant frequency of 5 beats per second, with spread indicating correlation with signal; crucial for uncertainty principle.', 'duration': 27.782, 'max_score': 320.907, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/MBnnXbOM5S4/pics/MBnnXbOM5S4320907.jpg'}, {'end': 456.78, 'src': 'embed', 'start': 424.428, 'weight': 0, 'content': [{'end': 428.231, 'text': 'And one other place where this comes up in a really tangible way is Doppler radar.', 'start': 424.428, 'duration': 3.803}, {'end': 429.572, 'text': 'So with radar.', 'start': 428.971, 'duration': 0.601}, {'end': 434.498, 'text': 'the idea is you send out some radio wave pulse and the pulse might reflect off of objects,', 'start': 429.572, 'duration': 4.926}, {'end': 439.984, 'text': 'and the time that it takes for this echo signal to return to you lets you deduce how far away those objects are.', 'start': 434.498, 'duration': 5.486}, {'end': 446.552, 'text': 'And you can actually take this one step further and make deductions about the velocities of those objects using the Doppler effect.', 'start': 440.765, 'duration': 5.787}, {'end': 449.214, 'text': 'Think about sending out a pulse with some frequency.', 'start': 447.313, 'duration': 1.901}, {'end': 456.78, 'text': 'If this gets reflected off an object moving towards you, then the beats of that wave get kind of smushed together,', 'start': 449.835, 'duration': 6.945}], 'summary': 'Doppler radar uses radio waves to measure distance and velocity of objects.', 'duration': 32.352, 'max_score': 424.428, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/MBnnXbOM5S4/pics/MBnnXbOM5S4424428.jpg'}, {'end': 532.196, 'src': 'embed', 'start': 502.966, 'weight': 1, 'content': [{'end': 508.83, 'text': 'What follows is meant to be a distilled, if somewhat oversimplified description of the Fourier trade-off in this setup.', 'start': 502.966, 'duration': 5.864}, {'end': 518.677, 'text': 'The salient fact is that time and frequency of that echo signal correspond respectively to the position and the velocity of the object being measured,', 'start': 509.51, 'duration': 9.167}, {'end': 524.081, 'text': 'which is what makes this example much more closely analogous to the quantum mechanical Heisenberg uncertainty principle.', 'start': 518.677, 'duration': 5.404}, {'end': 532.196, 'text': 'You see, there is a very real way in which a radar operator faces a dilemma where the more certain you can be about the positions of things,', 'start': 524.801, 'duration': 7.395}], 'summary': 'The fourier trade-off in radar relates time and frequency to object position and velocity, similar to the heisenberg uncertainty principle.', 'duration': 29.23, 'max_score': 502.966, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/MBnnXbOM5S4/pics/MBnnXbOM5S4502966.jpg'}], 'start': 153.533, 'title': 'Fourier transform and doppler radar', 'summary': 'Covers the understanding of fourier transform and its application in analyzing frequencies, as well as the use of fourier transform in doppler radar to measure position and velocity of objects, emphasizing the trade-off between time and frequency in determining parameters.', 'chapters': [{'end': 423.147, 'start': 153.533, 'title': 'Understanding fourier transform', 'summary': 'Explains the fourier transform and its relevance for analyzing frequencies, illustrating how it encodes the strength and phase of different frequencies in a signal, while demonstrating the uncertainty principle and its relation to the spread of the signal in time.', 'duration': 269.614, 'highlights': ['The Fourier transform encodes the strength and phase of different frequencies in a signal, with the winding frequency matching the signal frequency resulting in peaks and valleys lining up, and the position of the center of mass encoding the strength of that frequency.', 'The spread of the Fourier transform around the dominant frequency indicates the correlation of pure sine waves near that frequency with the signal, which is key for the uncertainty principle, demonstrating that a signal concentrated in time must have a spread out Fourier transform.', "The relationship between the persistence of the signal over time and the balancing out around the circle, as well as the change in winding frequency affecting the signal's balance, demonstrates the uncertainty principle in the language of Fourier transforms."]}, {'end': 624.988, 'start': 424.428, 'title': 'Doppler radar and fourier transform', 'summary': 'Explains how doppler radar uses fourier transforms to measure the position and velocity of objects, highlighting the trade-off between time and frequency in determining their parameters.', 'duration': 200.56, 'highlights': ['Doppler radar uses Fourier transforms to measure the position and velocity of objects, illustrating the trade-off between time and frequency in determining their parameters.', 'The Fourier transform of the original signal provides information about the frequencies, and the Doppler shifted echo allows deducing the velocity of the object based on the shift in the Fourier transform.', 'The trade-off between time and frequency in determining the position and velocity of objects in Doppler radar is closely analogous to the quantum mechanical Heisenberg uncertainty principle.', 'The Fourier trade-off dictates that a radar operator faces a dilemma where the more certain they are about positions, the less certain they are about velocities, due to the overlapping echoes and noise in the field.']}], 'duration': 471.455, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/MBnnXbOM5S4/pics/MBnnXbOM5S4153533.jpg', 'highlights': ['Doppler radar uses Fourier transforms to measure position and velocity of objects', 'The trade-off between time and frequency in determining parameters is emphasized', 'The spread of the Fourier transform around the dominant frequency indicates correlation of pure sine waves', 'The Fourier transform encodes the strength and phase of different frequencies in a signal']}, {'end': 1063.097, 'segs': [{'end': 672.578, 'src': 'embed', 'start': 644.957, 'weight': 0, 'content': [{'end': 650.099, 'text': 'in his 1924 PhD thesis he proposed that all matter has wave-like properties.', 'start': 644.957, 'duration': 5.142}, {'end': 660.464, 'text': 'And more than that he concluded that the momentum of any moving particle is going to be proportional to the spatial frequency of that wave.', 'start': 652.628, 'duration': 7.836}, {'end': 663.711, 'text': 'How many times that wave cycles per unit distance.', 'start': 661.085, 'duration': 2.626}, {'end': 672.578, 'text': "Okay, now that's the kind of phrase that can easily fly into one ear and out the other.", 'start': 668.516, 'duration': 4.062}], 'summary': '1924 phd thesis proposed wave-like properties of matter, momentum proportional to wave frequency.', 'duration': 27.621, 'max_score': 644.957, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/MBnnXbOM5S4/pics/MBnnXbOM5S4644957.jpg'}, {'end': 776.343, 'src': 'embed', 'start': 745.618, 'weight': 4, 'content': [{'end': 751.763, 'text': 'with all of these weights oscillating up and down in sync and with most of the mass concentrated towards a single point.', 'start': 745.618, 'duration': 6.145}, {'end': 758.351, 'text': 'The energy of these oscillating weights is meant to be a metaphor for the energy of a particle.', 'start': 753.328, 'duration': 5.023}, {'end': 762.314, 'text': 'specifically, the E equals mc squared style energy residing in its mass.', 'start': 758.351, 'duration': 3.963}, {'end': 768.258, 'text': 'And de Broglie emphasized how the conception he had in mind involves the particle being dispersed across all of space.', 'start': 762.954, 'duration': 5.304}, {'end': 776.343, 'text': 'The whole premise he was exploring here is that the energy of a particle might have to do with something that oscillates over time,', 'start': 768.998, 'duration': 7.345}], 'summary': "De broglie's metaphorical oscillating weights represent particle energy.", 'duration': 30.725, 'max_score': 745.618, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/MBnnXbOM5S4/pics/MBnnXbOM5S4745618.jpg'}, {'end': 870.855, 'src': 'embed', 'start': 841.932, 'weight': 2, 'content': [{'end': 845.434, 'text': 'velocity should have anything to do with spatial frequency.', 'start': 841.932, 'duration': 3.502}, {'end': 852.059, 'text': 'And the basic line of reasoning here is if mass is the same as energy via E equals mc squared,', 'start': 846.534, 'duration': 5.525}, {'end': 858.164, 'text': 'and if that energy was carried as some kind of oscillating phenomenon, similar to how it is for photons,', 'start': 852.059, 'duration': 6.105}, {'end': 865.29, 'text': 'then this sort of relativistic Doppler effect means changes to how that mass moves corresponds to changes in the spatial frequency.', 'start': 858.164, 'duration': 7.126}, {'end': 870.855, 'text': 'So what does our general Fourier trade-off tell us in this case?', 'start': 867.432, 'duration': 3.423}], 'summary': 'Velocity affects spatial frequency via relativistic doppler effect and fourier trade-off.', 'duration': 28.923, 'max_score': 841.932, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/MBnnXbOM5S4/pics/MBnnXbOM5S4841932.jpg'}, {'end': 1022.695, 'src': 'embed', 'start': 996.82, 'weight': 3, 'content': [{'end': 1005.404, 'text': 'So, when one of these waves is concentrated near a point, what that actually means is that we have a higher probability of finding it near that point,', 'start': 996.82, 'duration': 8.584}, {'end': 1007.485, 'text': 'that we are more certain of its location.', 'start': 1005.404, 'duration': 2.081}, {'end': 1010.206, 'text': 'And just to beat this drum one more time.', 'start': 1008.425, 'duration': 1.781}, {'end': 1017.009, 'text': 'since that concentration implies a more spread out Fourier transform, then the wave describing its momentum would also be more spread out,', 'start': 1010.206, 'duration': 6.803}, {'end': 1022.695, 'text': "So you wouldn't be able to find a narrow range of momenta that the particle has a high probability of occupying.", 'start': 1017.669, 'duration': 5.026}], 'summary': 'Concentrated waves imply higher probability and spread-out momentum, affecting particle occupation.', 'duration': 25.875, 'max_score': 996.82, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/MBnnXbOM5S4/pics/MBnnXbOM5S4996820.jpg'}, {'end': 1063.097, 'src': 'embed', 'start': 1037.146, 'weight': 1, 'content': [{'end': 1042.75, 'text': "When I think of the Heisenberg uncertainty principle, what makes it fascinating is not so much that it's a statement about randomness.", 'start': 1037.146, 'duration': 5.604}, {'end': 1047.633, 'text': 'I mean, yes, that randomness is very thought-provoking and contentious and just plain weird.', 'start': 1043.271, 'duration': 4.362}, {'end': 1057.6, 'text': "But to me equally fascinating is that underpinning Heisenberg's conclusion is that position and momentum have the same relationship as sound and frequency.", 'start': 1048.194, 'duration': 9.406}, {'end': 1063.097, 'text': "As if a particle's momentum is somehow the sheet music describing how it moves through space.", 'start': 1058.32, 'duration': 4.777}], 'summary': 'Heisenberg uncertainty principle links position and momentum like sound and frequency.', 'duration': 25.951, 'max_score': 1037.146, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/MBnnXbOM5S4/pics/MBnnXbOM5S41037146.jpg'}], 'start': 625.688, 'title': 'Quantum mechanics principles', 'summary': "Explores louis de broglie's wave-particle theory, highlighting the relationship between momentum and spatial frequency, and delves into the heisenberg uncertainty principle, emphasizing the trade-off between wave concentration and frequency representation.", 'chapters': [{'end': 841.932, 'start': 625.688, 'title': "Louis de broglie's wave-particle theory", 'summary': "Discusses louis de broglie's 1924 phd thesis proposing all matter having wave-like properties, and the relationship between momentum and spatial frequency of waves, exploring the concept through a metaphor of oscillating weights and its connection to einstein's theory of relativity.", 'duration': 216.244, 'highlights': ['Louis de Broglie proposed in his 1924 PhD thesis that all matter has wave-like properties, and the momentum of any moving particle is proportional to the spatial frequency of that wave.', "De Broglie's metaphor of oscillating weights hanging from springs illustrates the energy of a particle and its dispersion across space, resembling the E=mc^2 style energy residing in its mass.", 'The relationship between momentum and spatial frequency involves special relativity, where the perception of simultaneous events in one reference frame may differ in a different reference frame, as explained by de Broglie.']}, {'end': 1063.097, 'start': 841.932, 'title': 'Heisenberg uncertainty principle', 'summary': 'Discusses how the heisenberg uncertainty principle is a trade-off between the concentration of a wave and its frequency representation, impacting the probability of finding a particle and its momentum, illustrating the fundamental nature of particles as waves.', 'duration': 221.165, 'highlights': ['The Heisenberg uncertainty principle is a trade-off between the concentration of a wave and its frequency representation, impacting the probability of finding a particle and its momentum, illustrating the fundamental nature of particles as waves.', 'The spread of a wave over space and momentum is fundamental to the nature of a particle, analogous to how a musical note being spread out over time is fundamental to its meaning.', "The probability of finding a particle is proportional to the strength of the wave in that region, and a concentrated wave implies a more spread out Fourier transform, affecting the certainty of the particle's location and momentum.", "The uncertainty principle reflects the relationship between position and momentum, akin to the relationship between sound and frequency, portraying a particle's momentum as the 'sheet music' describing its movement through space.", 'The Heisenberg uncertainty principle is not merely about randomness, but also about the relationship between position and momentum, analogous to sound and frequency.']}], 'duration': 437.409, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/MBnnXbOM5S4/pics/MBnnXbOM5S4625688.jpg', 'highlights': ['Louis de Broglie proposed in his 1924 PhD thesis that all matter has wave-like properties, and the momentum of any moving particle is proportional to the spatial frequency of that wave.', 'The Heisenberg uncertainty principle is a trade-off between the concentration of a wave and its frequency representation, impacting the probability of finding a particle and its momentum, illustrating the fundamental nature of particles as waves.', 'The relationship between momentum and spatial frequency involves special relativity, where the perception of simultaneous events in one reference frame may differ in a different reference frame, as explained by de Broglie.', "The probability of finding a particle is proportional to the strength of the wave in that region, and a concentrated wave implies a more spread out Fourier transform, affecting the certainty of the particle's location and momentum.", "De Broglie's metaphor of oscillating weights hanging from springs illustrates the energy of a particle and its dispersion across space, resembling the E=mc^2 style energy residing in its mass."]}], 'highlights': ['Doppler radar uses Fourier transforms to measure position and velocity of objects', 'Observation spread out over time increases confidence in frequency information, illustrating the trade-off between the duration of observation and the confidence in frequency', 'The uncertainty principle is a general trade-off in wave phenomena, showcasing the interplay between frequency and duration, impacting confidence in frequency information', 'Louis de Broglie proposed in his 1924 PhD thesis that all matter has wave-like properties, and the momentum of any moving particle is proportional to the spatial frequency of that wave', 'The Heisenberg uncertainty principle is a trade-off between the concentration of a wave and its frequency representation, impacting the probability of finding a particle and its momentum, illustrating the fundamental nature of particles as waves']}