On 22.06.2014, at 20:24, robert bristow-johnson <r...@audioimagination.com>
wrote:
> On 6/22/14 1:20 PM, Urs Heckmann wrote:
>> On 22.06.2014, at 19:04, robert bristow-johnson<r...@audioimagination.com>
>> wrote:
>>
>>>> 2. Get the computer to crunch numbers by iteratively predicting,
>>>> evaluating and refining values using the actual non-linear equations until
>>>> a solution is found.
>>> perhaps in analysis. i would hate to see such iterative processing in
>>> sample processing code. (it's also one reason i stay away from terminology
>>> like "zero-delay feedback" in a discrete-time system.)
>> We're doing this a lot. It shifts the problem from implementation to
>> optimisiation. With proper analysis of the system behaviour, for any kind of
>> circuit we've done (ladder, SVF, Sallen-Key) one can choose initial
>> estimates that converge with results after an average of just over 2
>> iterations. Therefore it's not even all that expensive, and oversampling
>> makes the initial guesses easier
>
> so, if you can guarantee that you're limited to 2 iterations (or 3 or 4, but
> fix it), doesn't this mean you can "roll out" the looped iterations into a
> "linear" process? then, at the bottom line, you still have y[n] as a
> function of y[n-1], y[n-2], ... and x[n], x[n-1], x[n-2]... isn't that the
> case? you *don't* have y[n] as a function of y[n] (because you didn't define
> y[n] yet). no?
No, if we take a typical equation like
y = g * ( x - tanh( y ) ) + s
We form it into
Y_result = g * ( x - tanh( Y_estimate ) ) + s
as x, g and s are known, we simply use an iterative root finding algorithm
(Newton's method, bisection method etc.) over Y_estimate until Y_estimate and
Y_result become nearly equal. During that process g, x and s do not change.
Hence, as a result we have found y as a function of itself.
An alternative notation would be
error = g * ( x - tanh( y ) ) + s - y
where the root finding algorithm optimises y to minimize the error, like this
while( abs( error ) > 0.0001 )
{
y = findABetterGuessUsingMethodOfChoice( y, error );
error = g * ( x - tanh( y ) ) + s - y;
}
This way we hadn't had to find an actual mathematical solution, we let the
computer crunch it. Therefore the implementation is very simple (even
trivial...), and all it boils down to is finding a good method to come up with
an Y_estimate that's as close as possible to Y_result.
Cheers,
- Urs
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