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This article is bullshit. It fails to mention the many limitations of analog hardware. > Analog signals are continuous waveforms that vary smoothly over time,

by nayuki 2y ago

This article is bullshit. It fails to mention the many limitations of analog hardware. > Analog signals are continuous waveforms that vary smoothly over time, capturing every nuance of the original sound. > The continuous nature of analog signals means they can theoretically capture an infinite amount of detail. When you play an analog record, for example, the sound you hear is a continuous representation of the original performance. Analog signals get corrupted by noise everywhere. Noise of the read head. Wow-and-flutter in the speed variations of the phonograph or tape player. Noise of the wire. Crosstalk between wires. Noise of the amplifier. Noise of the mixer. Linearity and harmonic distortion. And so on. Digital systems can tightly control the noise and regenerate the signal perfectly. > digital signals are still approximations of the original analog waveform And analog waveforms are approximations of the original analog sound wave. > These [analog] components interact with the signal in real-time, providing an infinite resolution No, it's limited by noise and bandwidth. For example, an op-amp might only be good for 1 MHz. > For example, even a simple analog circuit like a passive 1-pole low-pass filter (see figure above) can have output values where digital emulations can't be sure if the result is correct. This non-computability presents a fundamental limitation in achieving perfect digital emulation. They fail to elaborate on this. Maybe this is a reference to chaos theory and the sensitivity to initial conditions? But that wouldn't imply non-computability. > As technology evolves, the industry will continue to grapple with the balance between the precision of digital systems and the irreplaceable warmth and character of analog sound. Don't tell me this is the same argument as "increasing density will change the character of the neighborhood"... Anyway, the article also fails to mention ideas like: * It is hard to control the variance of analog devices. This knob might behave noticeably differently from that knob, this transistor different from that one, and so on. If you have two analog guitar amps of the same model, it is legitimately possible for one to behave so differently from the other that you would hate to swap them. * Digital algorithms are easier to design, implement, and debug. They are reproducible and modular. You can inspect any part of the signal processing chain without disrupting the result. This is not true in the analog domain, because for example, probing a circuit can change the behavior of the circuit. * A must-see introductory video on digital sampling by Xiph.Org's Monty Montgomery: https://www.youtube.com/watch?v=cIQ9IXSUzuM https://www.youtube.com/watch?v=cIQ9IXSUzuM [2013], https://xiph.org/video/vid2.shtml https://xiph.org/video/vid2.shtml

regularfry 2y ago
This totally misses the point. The article is trying to get at the fact that even if you try to emulate the analogue system including all those noise sources and bandwidth limitations, you still can't do it because of limitations on computation itself. >> For example, even a simple analog circuit like a passive 1-pole low-pass filter (see figure above) can have output values where digital emulations can't be sure if the result is correct. This non-computability presents a fundamental limitation in achieving perfect digital emulation. > They fail to elaborate on this. Maybe this is a reference to chaos theory and the sensitivity to initial conditions? But that wouldn't imply non-computability. They have not failed to elaborate on it, you have failed to follow the reference they give. Let me help: https://www.researchgate.net/publication/360859184_On_non-detectability_of_non-computability_and_the_degree_of_non-computability_of_solutions_of_circuit_and_wave_equations_on_digital_computers https://www.researchgate.net/publication/360859184_On_non-de... The argument for an RC circuit is basically that because it involves exponentiation, you can't calculate its exact output in finite time. And unfortunately this is somewhere that matters: as soon as you get feedback in the system or any other way for errors to accumulate over time, your simulated output is going to diverge from the theoretical. Whether that matters aesthetically isn't at issue. It's not identical, therefore it's not a match for the analogue system, and can't be. > Anyway, the article also fails to mention ideas like: Neither of these matter in the context of the point the article is making. Variance within the emulation is trivial, and digital algorithms being easier to debug don't matter if the thing you're trying to emulate isn't emulable.
- nayuki 2y ago
  > even if you try to emulate the analogue system including all those noise sources and bandwidth limitations > Variance within the emulation is trivial I pointed out those limitations not as a target for digital systems to emulate, but to refute the repeated insinuations that "Analog signals are continuous waveforms that vary smoothly over time, capturing every nuance of the original sound", "The continuous nature of analog signals means they can theoretically capture an infinite amount of detail". > because it involves exponentiation, you can't calculate its exact output in finite time That hasn't stopped the exp() function from existing. https://en.cppreference.com/w/cpp/numeric/math/exp https://en.cppreference.com/w/cpp/numeric/math/exp > as soon as you get feedback in the system or any other way for errors to accumulate over time, your simulated output is going to diverge from the theoretical But then the same argument can be made against analog circuits. Two physical copies of a circuit necessarily cannot compute exp() exactly, exactly the same. So they will diverge too. So the digital version should be no worse than analog versions. It's like you're arguing that a digital double pendulum (a well-known chaotic system) diverges from a theoretical mathematical version of it due to rounding and discretization errors. Like yeah, but any analog double pendulum is also an approximation of the mathematical ideal, and no two copies of the device will behave the same. One difference is that it is relatively easy to increase or decrease the precision of a digital system - just increase the bit depth or sampling rate. Whereas increasing the precision of analog systems requires nontrivial engineering, and is sometimes impossible due to noise floors and other physical limitations.

by nayuki 2y ago

regularfry 2y ago

This totally misses the point. The article is trying to get at the fact that even if you try to emulate the analogue system including all those noise sources and bandwidth limitations, you still can't do it because of limitations on computation itself. >> For example, even a simple analog circuit like a passive 1-pole low-pass filter (see figure above) can have output values where digital emulations can't be sure if the result is correct. This non-computability presents a fundamental limitation in achieving perfect digital emulation. > They fail to elaborate on this. Maybe this is a reference to chaos theory and the sensitivity to initial conditions? But that wouldn't imply non-computability. They have not failed to elaborate on it, you have failed to follow the reference they give. Let me help: https://www.researchgate.net/publication/360859184_On_non-detectability_of_non-computability_and_the_degree_of_non-computability_of_solutions_of_circuit_and_wave_equations_on_digital_computers https://www.researchgate.net/publication/360859184_On_non-de... The argument for an RC circuit is basically that because it involves exponentiation, you can't calculate its exact output in finite time. And unfortunately this is somewhere that matters: as soon as you get feedback in the system or any other way for errors to accumulate over time, your simulated output is going to diverge from the theoretical. Whether that matters aesthetically isn't at issue. It's not identical, therefore it's not a match for the analogue system, and can't be. > Anyway, the article also fails to mention ideas like: Neither of these matter in the context of the point the article is making. Variance within the emulation is trivial, and digital algorithms being easier to debug don't matter if the thing you're trying to emulate isn't emulable.

nayuki 2y ago
> even if you try to emulate the analogue system including all those noise sources and bandwidth limitations > Variance within the emulation is trivial I pointed out those limitations not as a target for digital systems to emulate, but to refute the repeated insinuations that "Analog signals are continuous waveforms that vary smoothly over time, capturing every nuance of the original sound", "The continuous nature of analog signals means they can theoretically capture an infinite amount of detail". > because it involves exponentiation, you can't calculate its exact output in finite time That hasn't stopped the exp() function from existing. https://en.cppreference.com/w/cpp/numeric/math/exp https://en.cppreference.com/w/cpp/numeric/math/exp > as soon as you get feedback in the system or any other way for errors to accumulate over time, your simulated output is going to diverge from the theoretical But then the same argument can be made against analog circuits. Two physical copies of a circuit necessarily cannot compute exp() exactly, exactly the same. So they will diverge too. So the digital version should be no worse than analog versions. It's like you're arguing that a digital double pendulum (a well-known chaotic system) diverges from a theoretical mathematical version of it due to rounding and discretization errors. Like yeah, but any analog double pendulum is also an approximation of the mathematical ideal, and no two copies of the device will behave the same. One difference is that it is relatively easy to increase or decrease the precision of a digital system - just increase the bit depth or sampling rate. Whereas increasing the precision of analog systems requires nontrivial engineering, and is sometimes impossible due to noise floors and other physical limitations.