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Mon Dec 27 11:05:15 EST 2010

Quantization and number theory

This subject keeps coming back really.  Context: Digital control of
analog synthesizers.

This pops up in an electronics project I'm working on.  The idea is to
build a multi-channel DAC for control signals (say up to 200Hz), using
a cheap microcontroller.

The main design idea is that approximation error is OK, as long as it
averages to 0 and it doesn't correlate with the signal or with itself.
This means ordinary PWM doesn't really cut it as it has very strong
periodic content.  Dithered Sigma-Delta seems to be the proper tool.


Next to digital control, I'd also like to explore pure digital
combination of binary modulated signals using logic gates.  See
previous entries, i.e. [5].  The key idea here is to make sure there
is no correlation.  However some remark in [4], section "Multiplying 2
1-bit signals and getting a noise-shaped result" make me think I'm
missing some key point when multiplying: high frequency modulation
noise being modulated down.

Recently I found a link to a seemingly interesting book [1] about
Analytic Number Theory here [2], and a 2005 workshop[3][4][6] dealing
with the subject and some more from Robert Adams[7].  The last one
probably deserves a separate post[8].

[1] http://148.202.11.158/ebooks/mathbooks/Number%20theory/Analytic%20Number%20Theory%20-%20Newman%20D.J..pdf
[2] http://rjlipton.wordpress.com/2010/12/26/unexpected-connections-in-mathematics/
[3] http://www.cscamm.umd.edu/programs/ocq05/
[4] http://www.cscamm.umd.edu/programs/ocq05/adams/adams_ocq05.htm
[5] entry://20100218-104438
[6] http://www.cscamm.umd.edu/programs/ocq05/wolfe_ocq05.htm
[7] http://www.netsoc.tcd.ie/~fastnet/cd_paper/ICASSP/ICASSP_2005/pdfs/0400077.pdf



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