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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 a

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.
schoen 2y ago
Thanks for the clarification. I'm still confused about the > Whether that matters aesthetically isn't at issue. Presumably if I have a hardware random number generator that works by measuring thermal noise (directly or indirectly), I also can't emulate that in software. (If I could, cryptographic key generation would be insecure!) But we know what the output of the hardware random number generator sounds like, subjectively, as audio: it sounds like white noise! (Or maybe some other kind of noise depending on the details of the digital to analog conversion process.) So the idea that you can get an analog audio synthesis process that can't be emulated in software doesn't seem that surprising, but also potentially doesn't seem that interesting, except if someone also cares about something about how that process sounds subjectively to a human listener. But at that point, the theoretical argument about things that hardware can do that software can't emulate no longer applies directly anymore, right? That is, we can't extrapolate this theoretical argument about electronic circuits to something about qualitative human perceptions of the audio outputs of those circuits.

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.

schoen 2y ago

Thanks for the clarification. I'm still confused about the > Whether that matters aesthetically isn't at issue. Presumably if I have a hardware random number generator that works by measuring thermal noise (directly or indirectly), I also can't emulate that in software. (If I could, cryptographic key generation would be insecure!) But we know what the output of the hardware random number generator sounds like, subjectively, as audio: it sounds like white noise! (Or maybe some other kind of noise depending on the details of the digital to analog conversion process.) So the idea that you can get an analog audio synthesis process that can't be emulated in software doesn't seem that surprising, but also potentially doesn't seem that interesting, except if someone also cares about something about how that process sounds subjectively to a human listener. But at that point, the theoretical argument about things that hardware can do that software can't emulate no longer applies directly anymore, right? That is, we can't extrapolate this theoretical argument about electronic circuits to something about qualitative human perceptions of the audio outputs of those circuits.