robert bristow-johnson писал 2016-03-28 23:31:
This statement implies the LTI case, where the concept of the
transfer
function exists.
i didn't say that. i said "applying ... to the same integrator."
about each individual "transfer function" that looks like "s^(-1)"
You were talking about bilinear transform, which is a thing applied to
the transfer function, therefore I thought you implied that.
i didn't say anything about a "transfer function", until this post. i
am saying that the trapezoidal rule for modeling integrators is
replacing those integrators (which by themselves are LTI and *do*
happen to have a transfer function of "s^(-1)") with whatever "T/2 *
(1 + z^(-1)) / (1 - z^(-1))" means. and that's the same as the
trapezoid rule. (the stuff connected in-between might be neither L
nor TI.)
No argument about that.
a "transfer function" is not enough information to fully define a
system unless other assumptions are made (like "observability" or
"controllability"). that's why the general state-variable system
(Hal's title is a little bit of a misnomer), ya know with the A, B, C,
D matrices, exists to generalize it. i'm pretty confident that any
*linear* circuit (but possibly time-varying) you toss up there, with
either gooder or badder emulation of the reactive elements, will come
out as this generalized state-variable system.
To the best of my knowledge a transfer function fully defines the
"outside" behavior of an LTI system (treated as black box) regardless of
observability or controllability. As for the state-space form, being a
set of explicitly written differential or difference equations, it
contains full information about the time-varying behavior of the system.
but with some coefficients that can vary. like in
http://control.ucsd.edu/mauricio/courses/mae280a/lecture8.pdf . i
couldn't easily find a discrete-time version on the web. you might
notice that there *is* a concept of a time-variant impulse response
h(t, tau) (if it were LTI, h(t, tau) = h(t-tau)). it's the impulse
response, h(t), responding to a unit impulse applied at time tau. fix
tau and you have an h(t). if you have an h(t), then you also have an
H(s) (or in general an H(s,tau)) and i might call that a "transfer
function". but it's a time-varying transfer function and if it varies
wildly, you can't use Fourier analysis at all. but if it varies
slowly enough, you can use Fourier analysis, at least to the point of
discussing frequency response and the behavior of the system for short
periods of time.
Tau is the time of the application of the impulse, not the time, when we
measure the output signal. This time therefore has no connection to
filter's parameters at some given time moment (like asking, what is the
filter's formal impulse response at the given time, when the filter's
parameters are set to a given value). That is, the formal transfer
function obtained from such impulse response will be dependent on the
specific way the filter's parameters are changing over the entire time
axis, starting with tau, and cannot be attributed to a specific moment
in time when "the cutoff knob is set to a given position".
But I believe we strayed too far from the original topic. I believe the
purpose of asking the question of handling the filter's state had to do
simply with nice filter's behavior under time-varying conditions, and
that question is easily answered in the ZDF domain (or by explicit
trapezoidal integration of differential equations). One possible reason
for continuous-time filters behaving nicely under parameter modulation
is that the cutoff variation in continuous time is equivalent to a
(monotone) warping of time axis, so there is an equivalent LTI case for
each cutoff curve.
Of course other ways of dealing with the problem are not excluded, but
if the reason for asking is purely practical, there is an easy answer.
--
Vadim Zavalishin
Reaktor Application Architect | R&D
Native Instruments GmbH
+49-30-611035-0
www.native-instruments.com
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