# [music-dsp] Name of delay-free loop technique

Mon May 25 07:12:04 EDT 2009

```> I just meant that it is a desirable terminological property.
> Thanks though for pointing out that state update is only possible
> once the output is known whatever the procedure -- I overlooked
> that detail ;) My current favorite is "direct implementation" of
> instantaneity, as opposed to reconfiguring the graph. As you say,
> there aren't many alternatives to exclude.

That's my favorite as well. Anyway, I think both approaches are formally
equivalent.

>
>> What do you also think about "instantaneously unstable" (or is
>> it instable?) term then?
>
> ...
>
> For instance, if we close a loop with feedback gain g around an
> integrator, the overall analog transfer function is Ha(s) =
> 1/(s-g) so stability requires g < 0. BLT'ing the integrator with
> a scale parameter of c gives a digital pole at z = (c+g)/(c-g)
> and the loop gain is k = g/c, so divergence is immediate if g =
> 1/c. Now, this is positive by definition of c and hence makes Ha
> unstable, but the response is still defined for all time.
> *However* (and I just noticed this), positive g is outside the
> trapezoidal rule's so-called stability region which only
> comprises the left half s-plane!
>
> ...
>

I think what I was referring to is slightly different. I do want to include
unstable systems into the analysis (e.g. self-oscillating filters). However,
there are unstable cases where the solution of the feedback equation
produces a "wrong" result. In fact the solution of the feedback equation
yields the stability point, but it's not guaranteed that this stability
point is to be reached. This is what I referred to in my article as
"non-zero impedance approach". I believe you could explicitly obtain the
same result with the approach of the Harma's article you pointed me to,
except instead of a unit delay running at a very high sampling rate in the
feedback path you need to use a lowpass filter with a very high cutoff
running at a very high sampling rate. I once did the same in the continuous
time domain, obtaining similar results.

Usually you don't run into this problem. However e.g. for the ladder filter
model (which is a particular case of the structure in Fig.3 in my article)
you run into this case at k<-1. Interestingly, this also corresponds to
having more than one solution for the nonlinear case, for a typical shape of
the nonlinearity (linear around zero, then clipping/saturating).

So, while 1+gk is not zero, the straightforward solution of the feedback
equation is wrong. Instead, the system reaches infinitely high level (unless
there is saturation) within a single sample.

Regards,

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