[sdiy] BLIT/BLEP virtual analogue synthesis

[Olivier Gillet]

> However, for the BLEP, I need to know the sub-sample position of the ramp discontinuity. If the sub-sample position is designated "a", then the accumulator value after a wraparound is (1-a) * increment. I can detect the wraparound and divide the accumulator value by the increment to find the sub-sample position, but this is a division by (probably) a 32-bit number.
>
> Is there a way to do it that avoids the big ugly division?

I assume you're working on an architecture where divisions are done in
software. I haven't thought much about it but I think you could use
prior knowledge about the number of bits in the result to unroll the
loop of the binary division algorithm. You know that the result will
be between 0 and 1, the compiler does not.

Let's say you use a 16 bits phase counter positioned at 64000, with a
phase increment of 2400. The phase parameter of the blep will depend
on ((6400 + 2400) % 65536) / 2400 = 1536 / 2400. That's the division
you're talking about, right?

You won't use the number 1536/2400 anyway since you're working with
integers... You'll probably want to compute something which is more
like 256 * 1536 / 2400. So the division you really have to do is
393216 / 2400. Now there's a bit of knowledge you know and the
standard software implementation of / generated by your compiler
doesn't know: the result will always be 8 bits, all the upper bits
will be 0. So you can unroll the 8 iterations of the binary division
loop to compute each of the bit, instead of going with a loop through
the 32 iterations of the standard software division.

> With reference to my earlier email, I wasn't really suggesting listing to 8KHz square waves (bandlimited or not) - it's just that I couldn't really grasp how such an algorithm would manage to degrade correctly and only generate a couple of harmonics in an extreme case.

Trust the maths :) At high frequency, the ratio between the "period"
of the BLEP ripples and the oscillator frequency will get close to 1.
The 8kHz band limited square wave will be a sine because the ripples
of the BLEPs will overlap each other exactly.

> I thought Olivier's suggestion of using BLEP for low frequencies and something else for highs was insightful. BLIT/BLEP algorithms have a computational cost that increases with frequency. This would make it a good match for something like additive synthesis, which has a computation cost that *decreases* with frequency. So use each for the the part of the spectrum it's best at - it's an interesting idea. Of course, there's extra complications in using two algorithms, and perhaps that outweighs the benefits on specific hardware.

Sadly, you can't do real sync or PWM with additive synthesis.