[sdiy] Walsh Generators

[ASSI]

On Donnerstag, 17. März 2005 22:32, Glen wrote:
> How many Walsh functions would need to be summed together, in order
> to get a saw waveform that reaches a quality level comparable to a
> 16-bit DAC? Would it actually take 65,536 Walsh Functions? Would
> 24-bit audio quality need 16,777,216 functions?

The question should really be: if the lowest frequency you are trying to 
produce is 43.1Hz - how many Walsh functions would you need to produce 
a distinct value for each sample at a sampling rate of 44.1kHz. The 
answer to that is of course 1023 (OK, I doctored the values to get that 
result). How many nonzero coefficients do you need? Just 10 and in fact 
you wouldn't need any Walsh functions at all, because the saw is made 
entirely out of square waves that are binary weighted. Of course you 
need to provide the coefficients with either 16bit or 24bit precision 
and make sure that no precision loss occurs during the accumulation. On 
the other hand, saw waves can benefit from linear interpolation, so the 
answer to the above question would then be a much smaller number, 
depending on how steep you want to have the downslope. Keep in mind 
that the sawtooth we are discussing above is not bandlimited at all, so 
you really wouldn't want to produce such a waveform in the first place.

My personal take is that a good dose of oversampling (like OSR=128) to 
accomodate a good enough anti-aliasing filter is needed and depending 
on various other factors the useful number of Walsh functions to employ 
is between 32 and 256. If you restrict the waveforms to half-cycle 
symmetric (like many wavetable synths), then half as many Walsh 
functions producing just the first half of the waveform are obviously 
better than wasting half of the available coefficients.


Achim.
-- 
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