[sdiy] Approximating sine with plain integer math

[Richie Burnett]

Tom wrote:

> Could someone tell me what's the advantage of these numerical approaches 
> over a simple table look-up?

Numerical sinewave calculation can be faster on a lot of platforms.  For 
instance, on a dsPIC there are wait-states when you access ROM via Program 
Space Visibility or table read instructions.  The same thing can happen on 
other platforms where external memory has wait states.  Sure you can copy 
LUTs into RAM for faster access during interpolation but "fast memory" is a 
limited resource.

Routines like the magic circle algorithm (state-variable filter) also 
produce a *very* pure sinewave for only a couple of instructions.  You still 
need quite a large sinewave LUT to get comparable performance using linear 
interpolation, or else go to higher order interpolation to get comparable 
THD.

> About the waveshaping approach suggested by John: doesn't it introduce 
> aliasing at high frequencies?

No.  This is an instance of wave-shaping removing harmonics from a waveform 
rather than adding them.

> What happens to the shape of the output if the input sawtooth is 
> band-limited?

You get a messed up sinewave with ripples in it!  You have to start with the 
naive ramp straight out of the phase accumulator and then manipulate it to 
approximate your sinewave.

Olivier, have you read about the DPW differentiated-parabolic-wave method of 
synthesising analogue waveforms with reduced aliasing?  The sine-like 
waveform that the OP is generating has a 18dB spectral tilt so can play 
quite high in pitch before you get much energy aliasing.  You can then 
differentiate this to get a "reduced-aliasing" triangle wave, and 
differentiate again to get a nice reduced-aliasing square wave.  His 
waveform is really just the first step of the DPW method.

I hope that explanation helps,

-Richie,