On 2013-05-01, Fons Adriaensen wrote:
The point is that companders such as Dolby-A and Telcom are linear at least over a short time span - they do not introduce distortion.
I started thinking about that as well. Yeah, if you put in all possible spectra, in all combinations, at all speeds of change, you can characterize a system like this rather perfectly and prolly emulate it. It's just that the amount of data is stupendous and the problem you're trying to solve is probably harder than implementing the thing by hand to begin with.
But they are certainly not time-invariant - they would be useless if they were.
As a technical nitpick, they are too: if you just shift a signal in time, the output will be a shifted version of the original. I.e. they do have memory and if you deside to linearize them and then talk about their instantaneous system function, yes, that one changes in a regular, weakly nonlinear way based on what they've seen in the past. Still, all in all if you only do time shifts, they don't mind. The same is not true even of tremolo as a still-obviously-linear system, because it has an independent oscillator in it; by definition time-variance means the system is sensitive to absolute delay, not only relative,, which is not the case for any dynamics processor I know of. (It is for certain reverbs though, which use internal LFO's to break up resonances.)
But yes, there's usually little point in treating homogeneity, additivity and time/shift invariance as separate things within the audio field. They interact strongly enough so that usually you either take the whole LTI package or go back to the basics.
The difficult part in writing any software emulation of the Dolby-A or similar systems is modelling the dynamic behaviour of the compander, not the actual audio processing. Such systems will have 'designed' and documented attack/release times, but analog electronics being what they are, the dynamic behaviour of the compressors and expanders will very probably not fit to any simple equations.
Yes. Or let's say, the sidechain where they do all of their psychoacoustical analysis and control functions. That stuff is pretty complex too, at least when we go to systems like SR. (A is somewhat easier, because it's apparently just a multichannel compander. Though, I don't seem to have access to the fundamental papers anymore. Care to slip me a copy of at least the AES ones?)
Nor does it have to: the way Dolby-A and Telcom switch between encoding and decoding (by making the decoding algorithm the mirror image of the encoding) guarantees that the two will cancel each other, whatever they are.
Yeah, Dolby apparently has been doing that stuff for ages: they derive the inverse of their designs very much like you'd turn an IIR filter to its FIR inverse, and vice versa. They essentially sandwich the encoder within a negative servo loop in the decoder, be the result nonlinear or not. That approach is pretty general and simple too, if you know what you're doing: even if you want to formally show that it converges, you don't need much besides asymptotically limited memory in the encoder and a bunch of monotonicity assumptions. (I don't think they've ever shown this formally, though.)
To me the funkiest thing is that for the longest time Pro Logic was one of their bigger sources of income, yet that one broke the precedent: it was an open loop design. Only in Pro Logic II did they go back to their old feedback balance network ways. I'm guessing that's because there the problem is underdetermined, which makes stability and self-synchronizing tracking considerably more difficult to guarantee.
Be as it may, that architecture makes the precise nonlinear dynamics difficult to characterize and emulate, except by direct translation of the circuit.
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