[math-fun] Chinese Remainders for Audio & Signal Processing ?
[Totally different thread from family planning!!] I sometimes play with MIDI synthesizers to construct & modify music. The basic operation of a MIDI synthesizer is to *add* sounds to one another to produce chords, etc. In order for the human ear to later pull apart these synthesized sounds, it is essential for this *addition* to obey *superposition* -- i.e., *linearity*: f(a+b)=f(a)+f(b) and f(c*a)=c*f(a). The problem with digital sound is that it typically uses too few bits to be able to preserve superposition/linearity; e.g., some systems use 12-16 bits, while others may use fixed precision up to 24 bits. Newer signal processing systems employ *floating point* arithmetic, which is sometimes a bit better, but still incapable of preserving superposition/linearity 100%. Has anyone (else) considered a digital sound/signal processing system based upon Chinese Remainders? The idea is to represent the digital samples (mod 2), (mod 3), (mod 5), etc., (mod p), until we have enough remainders to completely capture the full *dynamic range* of the music or signals. The wonderful thing about arithmetic (mod p) is that it forms a field, so we can do all of our digital filtering convolutions for each of the primes *separately*, and then reconstruct the final signal for playback using the Chinese Remainder Theorem. Superposition/linearity is faithfully preserved. Such a signal processing system would be useful in certain types of signal processing where the signal is extremely faint in comparison with another signal (or noise). Such systems include "spread spectrum" techniques, where the signal may be 10-20 dB below the noise floor. Linear system techniques *without roundoff errors* would be essential to performing the necessary computations. Video is also escaping from the 8-bit world; new cameras can get to 10-bit, 12-bit and even 14-bit dynamic ranges, so Chinese Remainder techniques might be useful for video, as well. An interesting question is in the final sound/signal *reconstruction*. It may be possible to reconstruct the signal from the Chinese Remainders *out of thin air* -- i.e., by letting the human ear (or eye) put everything back together again. But I'm not sure how to build a speaker or display which produces (mod p) sounds/brightnesses !.
Here's a use of a mod p detector for unbounded dynamic range in a camera: http://web.media.mit.edu/~hangzhao/modulo.html On Fri, Nov 6, 2015 at 4:38 PM, Henry Baker <hbaker1@pipeline.com> wrote:
[Totally different thread from family planning!!]
I sometimes play with MIDI synthesizers to construct & modify music.
The basic operation of a MIDI synthesizer is to *add* sounds to one another to produce chords, etc.
In order for the human ear to later pull apart these synthesized sounds, it is essential for this *addition* to obey *superposition* -- i.e., *linearity*: f(a+b)=f(a)+f(b) and f(c*a)=c*f(a).
The problem with digital sound is that it typically uses too few bits to be able to preserve superposition/linearity; e.g., some systems use 12-16 bits, while others may use fixed precision up to 24 bits.
Newer signal processing systems employ *floating point* arithmetic, which is sometimes a bit better, but still incapable of preserving superposition/linearity 100%.
Has anyone (else) considered a digital sound/signal processing system based upon Chinese Remainders?
The idea is to represent the digital samples (mod 2), (mod 3), (mod 5), etc., (mod p), until we have enough remainders to completely capture the full *dynamic range* of the music or signals.
The wonderful thing about arithmetic (mod p) is that it forms a field, so we can do all of our digital filtering convolutions for each of the primes *separately*, and then reconstruct the final signal for playback using the Chinese Remainder Theorem. Superposition/linearity is faithfully preserved.
Such a signal processing system would be useful in certain types of signal processing where the signal is extremely faint in comparison with another signal (or noise). Such systems include "spread spectrum" techniques, where the signal may be 10-20 dB below the noise floor. Linear system techniques *without roundoff errors* would be essential to performing the necessary computations.
Video is also escaping from the 8-bit world; new cameras can get to 10-bit, 12-bit and even 14-bit dynamic ranges, so Chinese Remainder techniques might be useful for video, as well.
An interesting question is in the final sound/signal *reconstruction*. It may be possible to reconstruct the signal from the Chinese Remainders *out of thin air* -- i.e., by letting the human ear (or eye) put everything back together again.
But I'm not sure how to build a speaker or display which produces (mod p) sounds/brightnesses !.
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