[music-dsp] Non-linear processing/Volterra series?

[robert bristow-johnson]

on 03/01/2006 06:55, antti at smartelectronix.com at antti at smartelectronix.com
wrote:

> On Tue, 28 Feb 2006, robert bristow-johnson wrote:
> 
>> the fundamental definition of the property of time-invariancy of a
>> system (whether linear or not) is that if the same input is applied to
>> the system at a different time, the same output results but translated
>> to that different time.  that is if
> 
> So, what is the time limit in practise?

short enough that no one has turned the knobs or changed the system in the
meantime.  long enough that all of the states have discharged and the system
is "relaxed".

> I realize that in theory it is probably fine if there is some time T that
> is less than infinite where the previous state has no more effect, but
> surely the NLTI methods stop working at some point?

i know of no *general* NLTI method other than Volterra Series, and it's such
a bitch that usually analyzing the circuit and breaking down each component
into simpler defined devices, some that are non-linear but memoryless,
others that are LTI, and still some that are non-linear *with* memory (like
the output transformer), is the way to go.


> Suppose there is a switch that changes say gain if signal in the past X
> seconds has been above a threshold (such as noise gate)?
> 
>> for any input x(t).  this, of course, is not the case if someone or
>> something, not fully dependent on the input signal, is cranking up or
>> down the volume or tone or compression ratio knobs while processing the
>> signal.
> 
> Wouldn't this be just a case of the system having multiple inputs?

yes, it might make in non-linear, but you could take a linear or non-linear
time-variant system, model those time-varying parameters as inputs and
remodel the system as a multiple input time-invariant system.

> It seems to me that a time-variant system would require an internal signal
> generator.

yup, like tremelo.

> Suppose said generator would be reset whenever there was signal
> after a period of silence. Would this count as time-invariant?
> 
>> an expression for the convolution integral for LTV, and there are
>> Volterra series for NLTI.  dunno if there is a nice expression for
> 
> Can Volterra series deal with switching elements?

i'm not sure, do you mean a switching PWM amplifier?

4 years ago i posted something about Volterra to comp.dsp

http://groups.google.com/group/comp.dsp/msg/eacd1a75ca62f5f4?hl=en&fwc=1

it might help explain it a little.  otherwise, Google is your friend.

-- 

r b-j                  rbj at audioimagination.com

"Imagination is more important than knowledge."