Re: Interpolation on WaveShaperNode?

FWIW, in Gecko, I'm implementing linear interpolation by default.

--
Ehsan
<http://ehsanakhgari.org/>


On Sat, Apr 6, 2013 at 6:12 PM, robert bristow-johnson <
rbj@audioimagination.com> wrote:

> On 4/6/13 12:09 AM, Chris Rogers wrote:
>
>
>>
>>
>> On Thu, Apr 4, 2013 at 6:57 PM, robert bristow-johnson <
>> rbj@audioimagination.com <mailto:rbj@audioimagination.**com<rbj@audioimagination.com>>>
>> wrote:
>>
>>     On 4/4/13 7:06 PM, Chris Rogers wrote:
>>
>>         Another aspect of the WaveShaperNode is anti-aliasing.  In
>>         certain cases it would be great to be able to up-sample the
>>         signal before applying the shaping, then down-sampling.  This
>>         is to avoid the extremely harsh aliasing that can occur in
>>         applications such as guitar amp simulations.  Once again we
>>         could have an attribute .upsample ("none", "2x", "4x") or
>>         something like that.  Then the default value for that would be
>>         "none" I think.
>>
>>
>>     just lurking, and i haven't looked at the code at all, but thought
>>     i might mention a couple of things that might be applicable.
>>
>>     if you can get away from table lookup and implement the waveshaper
>>     by use of a pure polynomial if finite order, you can get a solid
>>     handle on aliasing.  a finite-order polynomial is not as general
>>     and a general lookup table, but for the purposes of distortion (or
>>     "warmth" or whatever) in audio, it might be closer to what you
>>     want anyway.  you can fit polynomials to tube curves and the sort
>>     pretty well.
>>
>>     the images generated is no higher in frequency than the order of
>>     the polynomial (let's call that M) times the highest frequency.
>>      if that highest frequency is potentially Nyquist, then upsampling
>>     by a factor of N means that the highest frequency is the *new*
>>     Nyquist/N.  that makes the highest frequency image (M/N)*Nyquist.
>>      you can allow aliases as long as they don't get back into your
>>     original baseband which is below the new Nyquist/N.  that means
>>
>>         2*Nyquist - (M/N)*Nyquist > Nyquist/N
>>
>>     or
>>
>>         2*N - M > 1
>>
>>     or
>>
>>         M <  2*N - 1
>>
>>     if you upsample by 2x, you can have a 3rd-order polynomial.  if
>>     you upsample by 4x, you can have a 7th-order polynomial.
>>
>>     then a decent brick-wall LPF with cutoff at Nyquist/N to kill the
>>     images and aliases.   then downsample by factor of N and you have
>>     output.  you will get the distortion components you were meant to
>>     get (harmonics) and no non-harmonic components which are the
>>     tell-tales of aliasing and cheezy distortion.
>>
>>     you can do this with table lookup if you make sure the table ain't
>>     defined to wildly (like if it's implementing a Mth-order
>>     polynomial), have enough points in the table (memory is cheap),
>>     and at least linearly interpolate between points.  how many points
>>     you need (based on what interpolation is done between points) in
>>     the table is something that i had done some analysis about long
>>     ago, but i might be able to find notes.  if computational burden
>>     is no problem, i might suggest implementing this as a polynomial
>>     and use Horner's rule.
>>
>>     just an idea.
>>
>>
>> Thanks Robert, this is really valuable information.  I'd still like to
>> support general shaping curves and bit-crushing applications.  But I'd
>> really like to get the highest quality sound and best general purpose
>> approach that is possible, especially for these "warming" applications.
>>
>>
> probably, for generality and for efficiency regarding speed, you might
> want to implement the non-linear function simply with table lookup and
> linear interpolation.  that is less computation than computing a 7th-order
> polynomial directly using the Horner method.  but i might suggest that what
> goes *into* that table are the points of a polynomial of limited order.
>  neglecting the error from linear interpolation (which can be very, very
> small if there are a decent number of points in the table), then they are
> mathematically equivalent, and since it's usually the case that memory is
> cheap, you may as well do this with a table.
>
> but if you define points in the table that would be the same as evaluating
> the limited-order polynomial, then you still enjoy the benefits of limited
> frequency to the images, and with enough upsampling (and downsampling on
> the other end), you can totally lose the non-harmonic aliasing (from
> foldover around Nyquist) that makes some digital distortion algs sound
> cheesy.  again, you can do a 7th-order polynomial with no aliasing if you
> upsample 4x and, at the output, LPF well and downsample 4x.  and you can
> make a 7th-order polynomial fit practically any tube curve to a very good
> fit.
>
> whether that 7th-order polynomial is implemented directly or is
> implemented with a decently large lookup table and linear interpolation,
> that shouldn't matter.  but if your table implements some wild-and-crazy
> function with discontinuities or with amazing slopes and corners, that
> might generate harmonics that fold over and cause aliases that you can't
> get rid of.
>
>
> --
>
> r b-j                  rbj@audioimagination.com
>
> "Imagination is more important than knowledge."
>
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