[music-dsp] Article: antialiased sine waveform syncing

[george bezerianos]

On Mon, Jun 8, 2009 at 7:27 PM, robert
bristow-johnson<rbj at audioimagination.com> wrote:
>
> On Jun 8, 2009, at 8:48 AM, george bezerianos wrote:
>
>>
>> http://s1gnals.blogspot.com/2008/12/bloo_6897.html
>>
> ...
>
> On Wed, May 6, 2009 at 2:12 PM, Vadim Zavalishin wrote:
>
>>>
>>> http://www.native-instruments.de/index.php?id=dsparticles
>>>
>>> Comments welcome.
>>>
>
> hi Vadim and George.
>
> i took a look at both the bloo.pfd and the SineSyncRC.pdf (i could not
> extract what i was looking for in the SineSync.pdf).
>
> now in bloo, George is creating a blit, as best as i can tell, and i am not
> sure if you're getting a sinusoid out of it, but if you are, is your BLIT
> going through some resonant filter or something?  i don't see where the
> sin(wt) is coming out of it.
>

Well, if you convolve a BLIT with a sufficiently long sine wave you
will be approximating
a sine fine - and while you can find a multitude of ways to do that
better the actual BLOO
method is supposed to generalize for _arbitrary_ waveforms. The sine
function is used
as an example because of its Cinf discontinuity. As I said in a
previous post while the article
resets the slave oscillator near the sync point, one can get rid of
the harmonics by increasing
the overlapping and thus hard-sync _any_ waveform as the actual
convolution kernel
can be anything. So, the two articles are rather different in terms of
genericity.

The actual interesting bit as i said in a previous post would be to
actually connect the
higher order BLEP method (generalized for _arbitrary_ waveforms) with
the convolution
BLOO suggests. That would actually add some additional theoretical justification
and insight, but numerically the method works fine both in terms of
complexity and numerical
error.

> in SineSyncRC, Vadim is using a quite long polynomial of |x| to approximate
> a sine function over a period (and i presume he's driving that with a
> sawtooth function coming out of a phase-accumulator that defines the phase
> of the waveform).
>
> are my understandings reasonably correct?
>
> about Vadim's method, that's fine and good, but i think you're throwing
> *much* more polynomial at it to get an acceptable synthesized sine wave.
>  the sine function is fully analytic (not the Hilbert transform meaning of
> "analytic").  it is continuous and has continuous derivatives, no matter how
> deep you go.  that lends itself to very good polynomial approximation,
> *particularly* for the 1/2 cycle before the sin() bends around back (and +/-
> pi/2).
>
> i would suggest aiming to approximate
>
>    sin( pi/2 * x )    for    -1 <= x <= +1
>
> sin(x) is a completely odd-symmetry, so any decent polynomial approximation
> would be odd-symmetry (and odd-order).  if you factor out the x, and express
> as
>
>    sin( pi/2 * x ) =  x * p(x^2)
>
> you are left with a simple polynomial that is made into an even-symmetry
> polynomial when x^2 is the argument.
>
>
>    p(x^2) = sin(pi/2*x)/(pi/2*x) * pi/2 = pi/2 * sinc(x/2)     -1 <= x <= +1
>
> or
>
>    p(x) = pi/2 * sinc( sqrt(x)/2 )      for  0 <= x <= +1
>
>
> unlike George, i (and nearly all my EE textbooks) define the sinc(x)
> function to have the pi in it:
>
>    sinc(x) = sin(pi*x)/(pi*x)

Correct, however I use pi explicitly when i utilize "my" sinc definition as you
can see in equation (1). Sorry for the confusion.

>
> that is the polynomial that you want coefficients for (after determining the
> order necessary).  you might be surprized to see how good of a fit you get
> for merely a 2nd or 3rd-order approximation to the sinc(sqrt(x)/2) function.
>  don't use Taylor series, but try to be more optimal in getting the
> coefficients.  least-squares would be good, in this case, but the Remes
> Exchange algorithm would work also.
>
> oh, the other thing is that you need to turn your sawtooth phase function
> into a triangle since the sin(x) function is being fit for only 1/2 period.
>  if your saw goes from -1 to +1, then
>
>    tri(x) = 2*|saw(x)| - 1
>
> that absolute value operation is the only one you'll need.  everything else
> is just plug-and-chug into the polynomial.
>
>
> --
>
> r b-j                  rbj at audioimagination.com
>
> "Imagination is more important than knowledge."
>
>
>
>
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