# Monotonicity of allpass phase function

Started by March 5, 2010
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

On 5 Mar, 10:50, vv <vanam...@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. =A0The 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

On 5 Mar, 10:59, Rune Allnor <all...@tele.ntnu.no> wrote:
> On 5 Mar, 10:50, vv <vanam...@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. =A0The 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=3Dno

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

On Mar 5, 3:42=A0pm, Rune Allnor <all...@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:
=93Causality and Negative Group Delays in a simple band-pass amplifier=94,
American Journal of Physics, Vol. 66 no. 1, January 1998).

-vv

Rune Allnor wrote:
> On 5 Mar, 10:59, Rune Allnor <all...@tele.ntnu.no> wrote:
>> On 5 Mar, 10:50, vv <vanam...@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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On Mar 5, 6:35=A0am, VV <vanam...@netzero.net> wrote:
> On Mar 5, 3:42=A0pm, Rune Allnor <all...@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. =A0There is nothing wrong with the proof. =A0Assume the
> pole is at r exp(j x), where r < 1. =A0The group delay for a first order
> allpass is (1-r^2)/|1-r exp(j(x-w))|^2, which is always positive. =A0If
> 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. =A0(On negative group delay, it may be of
> interest to some people to see Morgan Mitchell and Raymond Y. Chiao:
> =93Causality and Negative Group Delays in a simple band-pass amplifier=94=
,
> 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

On Mar 5, 9:54=A0am, Jerry Avins <j...@ieee.org> wrote:
>
> =A0From 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=3D0" 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

robert bristow-johnson wrote:
> On Mar 5, 9:54 am, Jerry Avins <j...@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
--
It matters little to a goat whether it be dedicated to God or consigned
to Azazel. The critical turning was having been chosen to participate.
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