>> Two points here. The last thing you said is not actually true -- each
>> interpolation scheme has an associated convolution function, which can
>> be calculated by imagining what the interpolation would look like for
>> a single sample whose value was 1.0 surrounded by zeroes everywhere
>> else. This 4-point piecewise function can be used to write four
>> samples in its immediate vicinity the same way that the sinc does in
>> the csound example.
>
> Meaning, there is also a convolution kernel for linear interpolation?
> How would it look like? Ah, it would be a simple dirac delta, but the
> point is, the kernel can be applied time-shifted with fractional
> delay, matching the fraction in the index. By the way, this also holds
> for sinc-interpolated resampling as described by Julius O. Smith: a
> linear interpolation in the sinc-table to make the result more
> precise. Interpolating the interpolation kernel...
>
The kernel for linear interpolation is a linear ramp from 0.0 to 1.0
and then back down to 0.0. This means that if one were to do linear
interpolation in the same way the csound opcode applies the sinc, you
would center the peak of this ramp at the fractional index into the
table, multiply it by the incoming sample, and then write the values
of the ramp at the intersection points of the two nearby samples in
the table to those samples.
So, say the index is 126.78 and the incoming sample value was -0.45.
You'd in effect ramp down from 0.0 to -0.45 over indices 125.78 to
126.78, and then back up to 0.0 from 126.78 to 127.78. As desired,
this would intersect with samples at index 126 and 127, and the values
would be -0.099 at 126 and -0.351 at 127.
So, theoretically you could do the same with Pd's cubic lagrangian
interpolation kernel, writing 4 points in the table for every incoming
sample. This should work fine with a long interpolation kernel, but I
foresee a lot of problems with doing this just for 4 points.
Like, imagine you're going to write samples 0, 1, and 2 from the
incoming signal to indices 2.3, 45.7, and 89.1. Doing it the way I
described with a 4-point interpolator you'd put some values at indices
1-4, 44-47, and 88-91, and leave everything else at 0. Imagine you
continued like this for a long time -- you'd have the original signal,
each sample of which would be interpolated into 4 points, and each of
these little spots would be, separated by long strings of zeroes. This
isn't going to play back as "the same sound but lower," unless you ran
it through a low-pass filter that smoothed this signal over the
intervening samples, which I think would be pretty much like using the
sinc for interpolating in the first place.
I should mention that any time I've ever heard of moving the write
head to make a variable delay, it's always with a big interpolation
kernel. I'm going to look at a couple of other places - something like
csound's spat3d opcode, or Richard Furse's old program "vspace" would
be candidates for getting ideas.
Hope this is all clear, and sorry if it's all obvious; sometimes it
helps me to think about things to write out my thoughts.
Matt