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The same is also true if you drop down to a single dimension - audio data is a discrete sampling of a continuous signal, and again, it is commonly misrepresente

The same is also true if you drop down to a single dimension - audio data is a discrete sampling of a continuous signal, and again, it is commonly misrepresented as a step function (a staircase) which is very misleading and leads to many mistakes when considering signal processing.

IIAOPSW 4y ago
If you try to sample continuously and eliminate that discretization error, I have some bad news for you about the nature of reality itself...
- chriswarbo 4y ago
  I think you're confusing 'discrete' with 'stepped': reality can only be sampled at a bunch of discrete points; however, we can decide how to interpolate 'in-between' those samples in whatever way we like. We could use a 'stepped' interpolation (i.e. nearest-neighbour/voronoi cells); however, the result will sound pretty crap. That's because 'steps' are a very unrealistic model: sound is made of pressure waves, which vary smoothly (at least, at the resolutions we tend to sample at); sound waves do not instantaneously jump between flat levels (again, ignoring microscopic effects like phonons, etc.). A better approximation is to interpolate smoothly between the points, e.g. using sine functions (i.e. Fourier series). That's a much better approximation of the way air actually moves (and also ear-drums, loudspeakers, etc.); whilst it's still completely discrete. As the parent says: "discrete" does not mean "stepped".
  - IIAOPSW 4y ago
    I think you're not familiar with quantum mechanics. If you try to sample to infinite precision, reality itself is stepped and in fact does instantaneously jump between flat levels. Its called wave function collapse. Your implicit premise that there's a continuous, real valued, sound wave form which can be sampled to arbitrary precision is false. Discretization isn't just an artifact of the machine, its present in the underlying reality too!
    - chriswarbo 4y ago
      > reality itself is stepped and in fact does instantaneously jump between flat levels That's why I said "at least, at the resolutions we tend to sample at", and "ignoring microscopic effects like phonons" > reality itself is stepped and in fact does instantaneously jump between flat levels. Its called wave function collapse. That's pretty-much right, but that phenomenon is "quantisation". "Wave-function collapse" is a different (but related) thing. Values like energy are "quantised", meaning they can only occur in certain amounts/steps (for energy these are called "energy levels"). A classic example is a "particle in a box" ( https://en.wikipedia.org/wiki/Particle_in_a_box https://en.wikipedia.org/wiki/Particle_in_a_box ), which acts a bit like a guitar string: each "energy level" is like a resonant frequency of a string. These are discrete, since (a) frequency is related to wavelength, and (b) resonance only occurs when a half-integer multiple of the wavelength fits inside-the-box/along-the-string (i.e. half a wavelength; a full wavelength; one and a half wavelengths; etc.). Guitar strings can also vibrate in more complicated ways; which can be described as adding together several of the resonant frequencies (possibly with different amplitudes). This "adding together" creates a "superposition", where the different waves can interfere, to produce the complicated vibrations we see/hear. The same is true for quantum systems: their behaviour can be a complicated adding-together and interference-between multiple energy levels (possibly with different amplitudes). SPOILER ALERT: Describing a complicated wave by adding together a bunch of simple, discrete waves is called a Fourier series; and it's exactly what the parent was talking about for audio sampling! Wave-function collapse is a separate thing: this adding-up and interference perfectly describes the behaviour of quantum systems; but we've never actually measured a system to be in such a mixed-state. Instead, every measurement shows a particular energy level (or, more generally, "eigenstate of the measurement operator"). We don't know why, but one explanation is that mixtures "collapse" when measured, with the probability of each outcome being the square of its amplitude. (Although that can't be the full picture, since it's observer-dependent; e.g. see https://en.wikipedia.org/wiki/Wigner%27s_friend https://en.wikipedia.org/wiki/Wigner%27s_friend ) > I think you're not familiar with quantum mechanics I have a Masters degree in Physics ;) --- Regarding the issue of discretely sampling a sound: you still seem to be conflating "discrete" with "stepped". Let's stick with the "particle in a box" example above, which is actually a good model of sound waves in a solid (where the particles are called "phonons" https://en.wikipedia.org/wiki/Phonon https://en.wikipedia.org/wiki/Phonon ). This system has discrete energy levels; meaning that two neighbouring states have no states "in-between". Yet each of those energy levels is a smooth, continous function over space; in fact, they're perfect sine waves! This is the heart of the confusion: if we sample a system at time/position 0 (let's call that sample S0), and we sample it at time/position 1 (to get S1), we can model that system using any function f we like, as long as f(0) = S0 and f(1) = S1. (NOTICE: I'm only using two samples, not "sampling to infinite precision") A "stepped" function (AKA piece-wise interpolation, AKA nearest neighbour interpolation, AKA Voronoi cells) would be something like: f(t) = (t > 0.5)? S1 : S0 That fits the constraints, but it's a terrible model of a sound wave (and a terrible model of a quantum state), since it doesn't vary smoothly like the real thing. Its 2D equivalent, f(x, y) = ..., is the "little squares" model of pixels. A better model would approximate the smoothly-varying shape of the sound wave (or quantum system). For example, we can eliminate the sudden jumps using linear interpolation: f(t) = (S1-S0)*t + S0 Notice that both of these definitions only use two samples (S0 and S1). Yet we can sample these functions at any point t we like. This is how we can go from "input pixels" (point samples S0 and S1) to "output pixels". Say we want to drive a loudspeaker, whose position can be adjusted at evenly spaced times like -1.0, -0.9, -0.8, ..., 1.9, 2.0. We can choose the position of the loudspeaker at each time by using our function f, i.e. f(-1.0), f(-0.9), ..., f(1.9), f(2.0) NOTE: The values of t will almost-always be translated/shifted by some amount. For example, if we record some audio then play it back, the time 't=0' of the recorded samples is different to that of the playback samples, since the playback occurs at a later time. Likewise, if we sample some light at position (x, y) on our camera sensor in Hawaii, those will be different to the (x, y) positions on our computer display in New York. We may also decide to stretch/compress the scale, although that's less common for audio! If we use the "stepped" definition of f, our loudspeaker will be stuck at position S0 for a while, then quickly move to position S1, then stay stuck there for a while. If we use the "linear" definition of f, our loudspeaker will start at position -S1 + 2S0, then gradually move to position 2S1 - S0 (passing through position S0 at time t=0, and passing through position S1 when time t=1). The linear definition only takes two samples into account, so it doesn't work well if we have many samples; we hit a "sharp corner" as we pass through each sampled point. In the loudspeaker example, its speed will jump (AKA it has a discontinuous derivative). We can smooth-out such corners by using an equation involving a few more samples, e.g. higher-degree polynomials, or sine/cosine waves (AKA Fourier series, which the parent was alluding to), etc. For nice visuals see https://en.wikipedia.org/wiki/Interpolation#Example https://en.wikipedia.org/wiki/Interpolation#Example > SIDE NOTE: In normal Quantum Mechanics, mixed states can have any Real numbers as their amplitudes. Hence we can transition from one energy state to another in a smooth, continuous way; e.g. starting at 100% A + 0% B, smoothly decreasing/increasing the amplitudes of A/B, and ending up at 0% A + 100% B. That's unrelated to this discussion though. Also it requires that no measurements occur during the process; e.g. see https://en.wikipedia.org/wiki/Quantum_Zeno_effect https://en.wikipedia.org/wiki/Quantum_Zeno_effect
tialaramex 4y ago
As I understand it the popular Audacity software fixed this, if you zoom in you'll see that nope, it's always a sine wave. But yes, lots of older software acted as though there was "really" a step function here.
djmips 4y ago
Really good video on the topic from xiph.org https://youtu.be/cIQ9IXSUzuM https://youtu.be/cIQ9IXSUzuM
wittjeff 4y ago
I recall reading (again, a hard-to-google topic!) that before Edison invented the (cylinder) phonograph, there was a debate over whether sound could even be meaningfully mapped to one wave, as it was known that in any real environment there are actually multiple independent sound waves overlapping from different angles.

IIAOPSW 4y ago

If you try to sample continuously and eliminate that discretization error, I have some bad news for you about the nature of reality itself...

chriswarbo 4y ago
I think you're confusing 'discrete' with 'stepped': reality can only be sampled at a bunch of discrete points; however, we can decide how to interpolate 'in-between' those samples in whatever way we like. We could use a 'stepped' interpolation (i.e. nearest-neighbour/voronoi cells); however, the result will sound pretty crap. That's because 'steps' are a very unrealistic model: sound is made of pressure waves, which vary smoothly (at least, at the resolutions we tend to sample at); sound waves do not instantaneously jump between flat levels (again, ignoring microscopic effects like phonons, etc.). A better approximation is to interpolate smoothly between the points, e.g. using sine functions (i.e. Fourier series). That's a much better approximation of the way air actually moves (and also ear-drums, loudspeakers, etc.); whilst it's still completely discrete. As the parent says: "discrete" does not mean "stepped".
- IIAOPSW 4y ago
  I think you're not familiar with quantum mechanics. If you try to sample to infinite precision, reality itself is stepped and in fact does instantaneously jump between flat levels. Its called wave function collapse. Your implicit premise that there's a continuous, real valued, sound wave form which can be sampled to arbitrary precision is false. Discretization isn't just an artifact of the machine, its present in the underlying reality too!
  - chriswarbo 4y ago
    > reality itself is stepped and in fact does instantaneously jump between flat levels That's why I said "at least, at the resolutions we tend to sample at", and "ignoring microscopic effects like phonons" > reality itself is stepped and in fact does instantaneously jump between flat levels. Its called wave function collapse. That's pretty-much right, but that phenomenon is "quantisation". "Wave-function collapse" is a different (but related) thing. Values like energy are "quantised", meaning they can only occur in certain amounts/steps (for energy these are called "energy levels"). A classic example is a "particle in a box" ( https://en.wikipedia.org/wiki/Particle_in_a_box https://en.wikipedia.org/wiki/Particle_in_a_box ), which acts a bit like a guitar string: each "energy level" is like a resonant frequency of a string. These are discrete, since (a) frequency is related to wavelength, and (b) resonance only occurs when a half-integer multiple of the wavelength fits inside-the-box/along-the-string (i.e. half a wavelength; a full wavelength; one and a half wavelengths; etc.). Guitar strings can also vibrate in more complicated ways; which can be described as adding together several of the resonant frequencies (possibly with different amplitudes). This "adding together" creates a "superposition", where the different waves can interfere, to produce the complicated vibrations we see/hear. The same is true for quantum systems: their behaviour can be a complicated adding-together and interference-between multiple energy levels (possibly with different amplitudes). SPOILER ALERT: Describing a complicated wave by adding together a bunch of simple, discrete waves is called a Fourier series; and it's exactly what the parent was talking about for audio sampling! Wave-function collapse is a separate thing: this adding-up and interference perfectly describes the behaviour of quantum systems; but we've never actually measured a system to be in such a mixed-state. Instead, every measurement shows a particular energy level (or, more generally, "eigenstate of the measurement operator"). We don't know why, but one explanation is that mixtures "collapse" when measured, with the probability of each outcome being the square of its amplitude. (Although that can't be the full picture, since it's observer-dependent; e.g. see https://en.wikipedia.org/wiki/Wigner%27s_friend https://en.wikipedia.org/wiki/Wigner%27s_friend ) > I think you're not familiar with quantum mechanics I have a Masters degree in Physics ;) --- Regarding the issue of discretely sampling a sound: you still seem to be conflating "discrete" with "stepped". Let's stick with the "particle in a box" example above, which is actually a good model of sound waves in a solid (where the particles are called "phonons" https://en.wikipedia.org/wiki/Phonon https://en.wikipedia.org/wiki/Phonon ). This system has discrete energy levels; meaning that two neighbouring states have no states "in-between". Yet each of those energy levels is a smooth, continous function over space; in fact, they're perfect sine waves! This is the heart of the confusion: if we sample a system at time/position 0 (let's call that sample S0), and we sample it at time/position 1 (to get S1), we can model that system using any function f we like, as long as f(0) = S0 and f(1) = S1. (NOTICE: I'm only using two samples, not "sampling to infinite precision") A "stepped" function (AKA piece-wise interpolation, AKA nearest neighbour interpolation, AKA Voronoi cells) would be something like: f(t) = (t > 0.5)? S1 : S0 That fits the constraints, but it's a terrible model of a sound wave (and a terrible model of a quantum state), since it doesn't vary smoothly like the real thing. Its 2D equivalent, f(x, y) = ..., is the "little squares" model of pixels. A better model would approximate the smoothly-varying shape of the sound wave (or quantum system). For example, we can eliminate the sudden jumps using linear interpolation: f(t) = (S1-S0)*t + S0 Notice that both of these definitions only use two samples (S0 and S1). Yet we can sample these functions at any point t we like. This is how we can go from "input pixels" (point samples S0 and S1) to "output pixels". Say we want to drive a loudspeaker, whose position can be adjusted at evenly spaced times like -1.0, -0.9, -0.8, ..., 1.9, 2.0. We can choose the position of the loudspeaker at each time by using our function f, i.e. f(-1.0), f(-0.9), ..., f(1.9), f(2.0) NOTE: The values of t will almost-always be translated/shifted by some amount. For example, if we record some audio then play it back, the time 't=0' of the recorded samples is different to that of the playback samples, since the playback occurs at a later time. Likewise, if we sample some light at position (x, y) on our camera sensor in Hawaii, those will be different to the (x, y) positions on our computer display in New York. We may also decide to stretch/compress the scale, although that's less common for audio! If we use the "stepped" definition of f, our loudspeaker will be stuck at position S0 for a while, then quickly move to position S1, then stay stuck there for a while. If we use the "linear" definition of f, our loudspeaker will start at position -S1 + 2S0, then gradually move to position 2S1 - S0 (passing through position S0 at time t=0, and passing through position S1 when time t=1). The linear definition only takes two samples into account, so it doesn't work well if we have many samples; we hit a "sharp corner" as we pass through each sampled point. In the loudspeaker example, its speed will jump (AKA it has a discontinuous derivative). We can smooth-out such corners by using an equation involving a few more samples, e.g. higher-degree polynomials, or sine/cosine waves (AKA Fourier series, which the parent was alluding to), etc. For nice visuals see https://en.wikipedia.org/wiki/Interpolation#Example https://en.wikipedia.org/wiki/Interpolation#Example > SIDE NOTE: In normal Quantum Mechanics, mixed states can have any Real numbers as their amplitudes. Hence we can transition from one energy state to another in a smooth, continuous way; e.g. starting at 100% A + 0% B, smoothly decreasing/increasing the amplitudes of A/B, and ending up at 0% A + 100% B. That's unrelated to this discussion though. Also it requires that no measurements occur during the process; e.g. see https://en.wikipedia.org/wiki/Quantum_Zeno_effect https://en.wikipedia.org/wiki/Quantum_Zeno_effect

tialaramex 4y ago

As I understand it the popular Audacity software fixed this, if you zoom in you'll see that nope, it's always a sine wave. But yes, lots of older software acted as though there was "really" a step function here.

djmips 4y ago

Really good video on the topic from xiph.org https://youtu.be/cIQ9IXSUzuM https://youtu.be/cIQ9IXSUzuM

wittjeff 4y ago

I recall reading (again, a hard-to-google topic!) that before Edison invented the (cylinder) phonograph, there was a debate over whether sound could even be meaningfully mapped to one wave, as it was known that in any real environment there are actually multiple independent sound waves overlapping from different angles.