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From: robert bristow-johnson on 5 Mar 2010 13:30
On Mar 5, 6:35 am, VV <vanam...(a)netzero.net> wrote:
> On Mar 5, 3:42 pm, Rune Allnor <all...(a)tele.ntnu.no> wrote:
>
> > So if the proof in the book is based on the supposition
> > that a causal filter response must have positive group
> > delay everywhere, the proof is wrong.

APF adds another condition.

> The allpass is filter is causal and stable and its group delay is
> always positive.  There is nothing wrong with the proof.  Assume the
> pole is at r exp(j x), where r < 1.  The group delay for a first order
> allpass is (1-r^2)/|1-r exp(j(x-w))|^2, which is always positive.  If
> the filter is not allpass, causal and stable filters can give rise to
> an expression for group delay that goes negative for some w, which is
> also well-know, I guess.  (On negative group delay, it may be of
> interest to some people to see Morgan Mitchell and Raymond Y. Chiao:
> “Causality and Negative Group Delays in a simple band-pass amplifier”,
> American Journal of Physics, Vol. 66 no. 1, January 1998).

is your problem with rigor that it's only a 1st-order APF? if so,
imagine two 1st-order APFs in series, one with the pole/zero rotated
by w0 and the other rotated by -w0. because the two poles and two
zeros are complex conjugate, it's still a real APF (and 2nd-order).

otherwise, i do not understand your problem with the rigor. the
normal way that i ever prove that some function is monotonic is to
show that the derivative of that function never changes sign - always
non-negative for monotonically increasing, always non-positive for
monotonically decreasing.

after Newton and Leibniz, how else do we prove monotonicity?

r b-j
From: robert bristow-johnson on 5 Mar 2010 14:30
On Mar 5, 9:54 am, Jerry Avins <j...(a)ieee.org> wrote:
>
>  From what I read here, the proof in the book assumes that an allpass
> filter has positive group delay everywhere. I don't have the book, so I
> can't check that.
>

Jerry, it's pretty easy to see in the s-plane, why APFs have
monotonically decreasing phase. you can see that geometrically, but
you can prove it analytically for a single pole and zero.

extending it for higher order APFs is a matter of translating where
"f=0" goes (and adding the phase results). extending that to the z-
plane can be done by using the bilinear transform and recognizing that
the frequency warping that results is also monotonic.

r b-j
From: Jerry Avins on 5 Mar 2010 14:38
robert bristow-johnson wrote:
> On Mar 5, 9:54 am, Jerry Avins <j...(a)ieee.org> wrote:
>> From what I read here, the proof in the book assumes that an allpass
>> filter has positive group delay everywhere. I don't have the book, so I
>> can't check that.
>>
>
> Jerry, it's pretty easy to see in the s-plane, why APFs have
> monotonically decreasing phase. you can see that geometrically, but
> you can prove it analytically for a single pole and zero.
>
> extending it for higher order APFs is a matter of translating where
> "f=0" goes (and adding the phase results). extending that to the z-
> plane can be done by using the bilinear transform and recognizing that
> the frequency warping that results is also monotonic.

Sure. What I don't know about Vaidayanthan is whether or not he limits
his proof to APFs, or allows it to be overextended to all filters


Jerry
--
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to Azazel. The critical turning was having been chosen to participate.
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