On 2013-05-20, Fons Adriaensen wrote:
So it seems that a stronger definition of TI is not necesssary.
At the same time there is a definite point to compressors being "kind of time-variant" and only weakly nonlinear. They certainly don't behave like a distortion pedal or anything like that.
I guess one way to look at it would be to consider the difference between the whole system as a mathematical operator on the one hand, and the constituent parts of its implementation on the other. In the case of LTI systems the difference is easily neglected because of the strong circuit invariants, including commutativity. Numerical effects like noise accumulation, representable range and coefficient quantization are pretty much the only thing fundamentally separating the two, there, and they have little to do with time. But as soon as you go to time-variant and especially (memoryful) nonlinear systems, any invariants you might have to aid in the analysis are much weaker, they don't compose easily, and so you can't factor out the internal dynamics of the system the way we do with LTI circuits. Suddenly it does matter whether parts of the system can be locally approximated as, say, slowly time-variant linear systems, like all dynamics processing can.
That sort of thing is especially important when there's nonlinear feedback involved, because then you'll pretty much always be relying on such properties to prove stability and convergence. That goes for Dolby A decoders and Pro Logic II type active steering alike, to mention just two recent topics. Or the other way around, it'll bite you even in the case of fully linear but time-variant circuits with feedback: it's a well known DSP nit that the stability of such filters even under well-behaved coefficient modulation cannot be straight-forwardly deduced from the steady state system function(s), but is intimately tied to the actual circuit topology implementing the filter. So, once you contrast the system and its implementation, suddenly it's no longer generally the case that (approximate) time-invariance of (some of) the parts implies the same of the whole, or the other way around, both properties are still very important for the analysis even if only to quantify how much they're lacking (cf. the analysis of modulation artifacts in compressors), and though the two concepts aren't fully comparable, for the most part applying approximate linearity and/or time-invariance to the exploded circuit constitutes a more fine grained, or stronger, approach.
-- Sampo Syreeni, aka decoy - [email protected], http://decoy.iki.fi/front +358-50-5756111, 025E D175 ABE5 027C 9494 EEB0 E090 8BA9 0509 85C2 _______________________________________________ Sursound mailing list [email protected] https://mail.music.vt.edu/mailman/listinfo/sursound
