On 6/23/14 1:18 AM, Andrew Simper wrote:
On 23 June 2014 12:37, robert bristow-johnson<r...@audioimagination.com> wrote:
Andy and Urs, i have been making consistent and clear points and challenges
and the response is not addressing these squarely.
let's do the Sallen-Key challenge, Andy. that's pretty concrete.
With respect Robert, I have really tried to address your points,
please go and read all the links I've posted so that you get a better
picture of what I am saying.
no, i'm letting *you* make the case. i'm not accepting deflection to
someone or something else.
i'm the one that's being clear and concrete so we can make a
"falsifiable" test that can go one way or the other and we can all see
which way it goes.
you pick the circuit (i suggested the one at wikipedia) so we have a common
reference. then you pick an R1, R2, C1, C2 (or an f0 and Q, i don't care).
let's leave the DC gain at 0 dB. then we'll have a common and unambiguous
H(s).
In the LTI case, with enough precision, then any implementation
structure (if done properly) will result in the same output.
no it won't! digital filters implemented from analog prototypes using,
say, impulse invariant will *not* result in the same output as those
implemented from the same analog prototype using bilinear transform.
There is no question here, I have already state this any number of times.
is that what you said here?:
"please show me the derivation for a 2 pole Sallen Key using the
bi-linear transform, then I'll show you the difference between using
trapezoidal integration ... and the bi-linear transform used by engineers ..."
you've consistently said that trapezoidal integration results in
something different than bilinear, and i've been saying consistently
that, *for* *LTI*, they result in the same transfer function and
frequency response. you limit or hedge your claim conceding they're the
same for the "1-pole case":
"In the one pole case they come down to the same thing (there is only
one capacitor), but the bi-linear transform is actually derived from
the time domain trapezoidal rule."
but the fact is they're the same for 2-pole or 3-pole or N-pole filters,
if you consistently replace each continuous-time integrator with the
trapezoidal rule, you will get precisely the same filter transfer
function as what you would get with bilinear transform without
prewarping any frequency specs.
I'M the one making the tougher, more restrictive claim. it's a claim
that can be refuted with a single counter-example. you, Andy, made an
offer of a counter example that i want to take up. but i am not gonna
do all the work regarding it. *you* have to spell out the difference
equations you get from the Sallen-Key circuit and applying the trapezoid
rule to both integrators (and you get to assert that it will come out
different than bilinear), and *i* get to show that it will come out to
be the same transfer function that comes from the same H(s) using
bilinear (without prewarping).
If we want to now look at how things differ in the time varying case, which
is what I was talking about,
not all the time.
and i have been consistently hedging about time-varying. i was clear
that, without clipping or nonlinearities, and *in* the steady state,
that, given the same analog filter, the frequency response (and
"transfer function") of the filter you derive replacing integrators with
the trapezoid rule is precisely the same as the frequency response
obtained using bilinear transform.
we are all clear that different topologies will have different behavior
with clipping, quantization noise, and transients when the knob is
twisted. i am yet unconvinced that the these issues are not adequately
disposed of using lattice or normalized-ladder filters.
then we can continue. Otherwise I have a
feeling that people reading this thread will be getting bored with the
repetition of these posts (I know I am).
fine. but you have come nowhere close to showing that for linear
filters that modeling a Sallen-Key with trapezoidal rule for integration
comes up with anything new or novel.
and Urs has come nowhere close to showing that the iterative recursive
non-linear function evaluation comes out differently than the properly
set up "linear process" (and, by "linear process", i do *not* mean
anything LTI, i mean program flow or code that starts with y[n-1].
y[n-2]... and x[n], x[n-1], x[n-2]... and defines explicitly y[n] from
those given values). we get that "linear process" by unrolling the loop
a finite number of times, because it was claimed that a finite (and
small) number of iterations was sufficient.
--
r b-j r...@audioimagination.com
"Imagination is more important than knowledge."
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