From: vv on
I have looked at P.P. Vaidayanthan's "Multirate Systems and Filter
Banks" on the topic of the monotonicity of the unwrapped phase
response of an allpass digital filter. The proof is based on the
observation that the group delay is always positive and hence slope of
the phase response is always negative, ergo phi(w) is a decreasing
function. The group delay > 0 is shown for a first order section,
with the statement that the group delay for an Nth order system is the
addition of N monotonically decreasing functions, and hence the result
follows.

Can someone suggest a reference (in DSP literature or math) to an
alternative proof that is perhaps more rigorous? (without getting
sidetracked by whether or not the the proof in PPV's text is rigorous
enough :->). Thanks!

All pass is, of course, H(z) = z^{-N} A(z^-1)/A(z), where A(z) is 1+
a1 z^{-1} + ... +aN z^{-N} and z \in C.

-vv
From: Rune Allnor on
On 5 Mar, 10:50, vv <vanam...(a)netzero.net> wrote:
> I have looked at P.P. Vaidayanthan's "Multirate Systems and Filter
> Banks" on the topic of the monotonicity of the unwrapped phase
> response of an allpass digital filter.  The proof is based on the
> observation that the group delay is always positive and hence slope of
> the phase response is always negative, ergo phi(w) is a decreasing
> function.

There was a discussion here a few years ago where somebody
(I can't remember who - RBJ? Andor?) demonstrated that a
causal filter might in fact have negative group delay in
parts of the frequency band. The effect showed up as a very
short rise time in the impulse response of the filter.

Rune
From: Rune Allnor on
On 5 Mar, 10:59, Rune Allnor <all...(a)tele.ntnu.no> wrote:
> On 5 Mar, 10:50, vv <vanam...(a)netzero.net> wrote:
>
> > I have looked at P.P. Vaidayanthan's "Multirate Systems and Filter
> > Banks" on the topic of the monotonicity of the unwrapped phase
> > response of an allpass digital filter.  The proof is based on the
> > observation that the group delay is always positive and hence slope of
> > the phase response is always negative, ergo phi(w) is a decreasing
> > function.
>
> There was a discussion here a few years ago where somebody
> (I can't remember who - RBJ? Andor?) demonstrated that a
> causal filter might in fact have negative group delay in
> parts of the frequency band. The effect showed up as a very
> short rise time in the impulse response of the filter.
>
> Rune

Found it:

http://groups.google.no/group/comp.dsp/msg/c820aea7bdaf4cb2?hl=no

It was Andor who saw the flaw in the argment that a
causal system must necessarily have positive group delay
everywhere. He found a filter fundtion that was *both*
causal *and* had negative group delay over substantial
parts of the frequency band.

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.

Rune
From: VV on
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.

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).

-vv
From: Jerry Avins on
Rune Allnor wrote:
> On 5 Mar, 10:59, Rune Allnor <all...(a)tele.ntnu.no> wrote:
>> On 5 Mar, 10:50, vv <vanam...(a)netzero.net> wrote:
>>
>>> I have looked at P.P. Vaidayanthan's "Multirate Systems and Filter
>>> Banks" on the topic of the monotonicity of the unwrapped phase
>>> response of an allpass digital filter. The proof is based on the
>>> observation that the group delay is always positive and hence slope of
>>> the phase response is always negative, ergo phi(w) is a decreasing
>>> function.
>> There was a discussion here a few years ago where somebody
>> (I can't remember who - RBJ? Andor?) demonstrated that a
>> causal filter might in fact have negative group delay in
>> parts of the frequency band. The effect showed up as a very
>> short rise time in the impulse response of the filter.
>>
>> Rune
>
> Found it:
>
> http://groups.google.no/group/comp.dsp/msg/c820aea7bdaf4cb2?hl=no
>
> It was Andor who saw the flaw in the argument that a
> causal system must necessarily have positive group delay
> everywhere. He found a filter function that was *both*
> causal *and* had negative group delay over substantial
> parts of the frequency band.
>
> 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.

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
--
Blaise Pascal: Men never do evil so completely and cheerfully
as when they do it from religious conviction.
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