Determing whether or not a filter is minimum phase

On 13 Jun., 02:39, Jerry Avins <j...@ieee.org> wrote:
> Vladimir Vassilevsky wrote:
>
> > Andor wrote:
>
> >> I quite like Vladimir's approach (should work for many practical
> >> systems).
>
> > Oh, no... Not so simple.
>
> > In the direct form IIR, the numeric error growth is proportional to the
> > transition bandwidth raised to the power of the denominator. For OP's
> > filter of the 500th order, this means the calculations with the ~1000
> > bit numbers to have enough of accuracy :) Otherwise the result will be
> > rubbish.
>
> Isn't it reasonable to suppose that the 500th-order filter is FIR?

And now, how to show if this FIR is minimum-phase? One option is to
check for the stability of the inverse filter. One (numerically
critical) way to do this is to look at the impulse response of the
purely recursive inverse filter by computing it via the time domain
recurrence equation.

SteveSmith  wrote:
> >... Remember this
> >frequency response:
>
> >1/sqrt(j omega)
>
> >It's minimum phase according to your criterion, but not trivially
> >shown to be so.
>
> Hi Andor,
> Do you have a reference to this?

Hi Steve

If one considers a Neumann boundary condition to a diffusion process
in a semi-infinite media with a bounding plane at x=0 with stationary
inital condition, the state on the surface of the bounding plane
depends on the boundary condition through the

H(j omega) = 1/sqrt(j omega)

operator. This means that if you apply a white noise signal as "flux"
through the bounding plane, the resulting "temperature" signal on the
bounding plane is 1/f noise. This is just down your alley, I thought
it would interest you :-). The fact that the operator H is minimum-
phase was pieced together by rb-j and me here in comp.dsp. I have not
found any reference to the minimum-phase characteristics although
there are published references to the pinking property of diffusion
processes, eg. [1].

Regards,
Andor

[1] Henrik Jeldtoft Jensen, 1/f noise from the linear diffusion
equation,
1991 Phys. Scr. 43 593-595   doi: 10.1088/0031-8949/43/6/009

On 13 Jun, 11:22, Andor <andor.bari...@gmail.com> wrote:
> SteveSmith &#4294967295;wrote:
> > >... Remember this
> > >frequency response:
>
> > >1/sqrt(j omega)
>
> > >It's minimum phase according to your criterion, but not trivially
> > >shown to be so.
>
> > Hi Andor,
> > Do you have a reference to this?
>
> Hi Steve
>
> If one considers a Neumann boundary condition to a diffusion process

What does diffursion processes have to do with DSP?
Isn't the wave equation better suited to deal with DSP
type problems?

> The fact that the operator H is minimum-
> phase was pieced together by rb-j and me here in comp.dsp.

Why don't the two of you write a paper on that?
That's the sort of thing that ought to be sublitted
to IEEE Transactions on Signal Processing.

Whatever opinions one might have about comp.dsp,
the fact that a result was derived in a discussion
here is hardly a guarantee that the result is valid.

Rune

Rune Allnor wrote:
> On 13 Jun, 11:22, Andor <andor.bari...@gmail.com> wrote:
>
> > SteveSmith &#4294967295;wrote:
> > > >... Remember this
> > > >frequency response:
>
> > > >1/sqrt(j omega)
>
> > > >It's minimum phase according to your criterion, but not trivially
> > > >shown to be so.
>
> > > Hi Andor,
> > > Do you have a reference to this?
>
> > Hi Steve
>
> > If one considers a Neumann boundary condition to a diffusion process
>
> What does diffursion processes have to do with DSP?

You have to ask the other way round (if you have to ask at all, I
don't know why DSP suddenly enters into this discussion, barring from
the name of this group).

> Isn't the wave equation better suited to deal with DSP
> type problems?

We were faced with a parameter estimation problem in a distributed
diffusion process. Wave equation would definitely have been the wrong
model.

> > The fact that the operator H is minimum-
> > phase was pieced together by rb-j and me here in comp.dsp.
>
> Why don't the two of you write a paper on that?
> That's the sort of thing that ought to be sublitted
> to IEEE Transactions on Signal Processing.
>
> Whatever opinions one might have about comp.dsp,
> the fact that a result was derived in a discussion
> here is hardly a guarantee that the result is valid.

Sure enough. However, I don't know if finding the minimum-phase
version of the pinking filter really is something to write home (or
the IEEE) about.

Regards,
Andor

Rune Allnor wrote:
...
> > 1/sqrt(j omega)
...
> > If one considers a Neumann boundary condition to a diffusion process
>
> What does diffursion processes have to do with DSP?

The closest we have in DSP to the

H(j omega) = 1/sqrt(j omega)

is the fractional integrator, defined with transfer function

Hd(z) = 1/sqrt(1 - z^-1).

The impulse response can be computed by using a Taylor series
expansion of the transfer functions in z. We get

Hi(z) = 1 + 1/2 z^-1 + 3/8 z^-2 + ....

In general,

h_k = 1/k! Gamma(1/2 + k)/(sqrt(pi) k!).

It is used in statistics to model discrete-time 1/f noise (FARIMA
models). These models are used for detecting long-range correlation in
time series. A nice review is [1].

Regards,
Andor

[1] N.J. Kasdin, Discrete Simulation of Colored Nolise and Stochastic
Processes and 1/f^alpha Power Law Noise Generation, Proc. IEEE, Vol.
83, May 1995.

On 13 Jun, 13:57, Andor <andor.bari...@gmail.com> wrote:
> Rune Allnor wrote:
>
> ...
>
> > > 1/sqrt(j omega)
> ...
> > > If one considers a Neumann boundary condition to a diffusion process
>
> > What does diffursion processes have to do with DSP?
>
> The closest we have in DSP to the
>
> H(j omega) = 1/sqrt(j omega)
>
> is the fractional integrator, defined with transfer function
>
> Hd(z) = 1/sqrt(1 - z^-1).
>
> The impulse response can be computed by using a Taylor series
> expansion of the transfer functions in z. We get
>
> Hi(z) = 1 + 1/2 z^-1 + 3/8 z^-2 + ....
>
> In general,
>
> h_k = 1/k! Gamma(1/2 + k)/(sqrt(pi) k!).
>
> It is used in statistics to model discrete-time 1/f noise (FARIMA
> models). These models are used for detecting long-range correlation in
> time series. A nice review is [1].

I am sure you are right about the technicalities.

The question still remains, as I fail to see how
you would implement this filter of yours. Can you
find a difference equation equivalent for the filter?

If 'no', how would you adjust it in order to implement
it? What are the consequences of these amendments?

And no, I am not pedantic. These questions are as
relevant as inexact representations of numbers, be it
fixed point or floating points. The issue is as
relevant to DSP as the answer to the question
"why doesn't 0.5-0.3-0.2 equal 0?"

You wouldn't be able to publish an your filter in
IEEE Transactions unless you come up with good
answers to such questions, which is why I suggest
you publish there.

Rune

On 13 Jun., 16:02, Rune Allnor <all...@tele.ntnu.no> wrote:
> On 13 Jun, 13:57, Andor <andor.bari...@gmail.com> wrote:
>
>
>
>
>
> > Rune Allnor wrote:
>
> > ...
>
> > > > 1/sqrt(j omega)
> > ...
> > > > If one considers a Neumann boundary condition to a diffusion process
>
> > > What does diffursion processes have to do with DSP?
>
> > The closest we have in DSP to the
>
> > H(j omega) = 1/sqrt(j omega)
>
> > is the fractional integrator, defined with transfer function
>
> > Hd(z) = 1/sqrt(1 - z^-1).
>
> > The impulse response can be computed by using a Taylor series
> > expansion of the transfer functions in z. We get
>
> > Hi(z) = 1 + 1/2 z^-1 + 3/8 z^-2 + ....
>
> > In general,
>
> > h_k = 1/k! Gamma(1/2 + k)/(sqrt(pi) k!).
>
> > It is used in statistics to model discrete-time 1/f noise (FARIMA
> > models). These models are used for detecting long-range correlation in
> > time series. A nice review is [1].
>
> I am sure you are right about the technicalities.
>
> The question still remains, as I fail to see how
> you would implement this filter of yours. Can you
> find a difference equation equivalent for the filter?

Read the Kasdin reference.

>
> If 'no', how would you adjust it in order to implement
> it? What are the consequences of these amendments?

Read the Kasdin reference.

>
> And no, I am not pedantic. These questions are as
> relevant as inexact representations of numbers, be it
> fixed point or floating points. The issue is as
> relevant to DSP as the answer to the question
> "why doesn't 0.5-0.3-0.2 equal 0?"
>
> You wouldn't be able to publish an your filter in
> IEEE Transactions unless you come up with good
> answers to such questions, which is why I suggest
> you publish there.

If you are interested, read the Kasdin reference (published in IEEE).

Regards,
Andor

On 13 Jun, 16:49, Andor <andor.bari...@gmail.com> wrote:
> On 13 Jun., 16:02, Rune Allnor <all...@tele.ntnu.no> wrote:

> > And no, I am not pedantic. These questions are as
> > relevant as inexact representations of numbers, be it
> > fixed point or floating points. The issue is as
> > relevant to DSP as the answer to the question
> > "why doesn't 0.5-0.3-0.2 equal 0?"
>
> > You wouldn't be able to publish an your filter in
> > IEEE Transactions unless you come up with good
> > answers to such questions, which is why I suggest
> > you publish there.
>
> If you are interested, read the Kasdin reference (published in IEEE).

So what is *your* contribution here? You started out by
claiming that you and RBJ have shown that the 1/sqrt(1+jw)
filter is minimum phase. Now you refer everything to this
paper by Kasdin.

Does Kasdin show that the filter is minimum phase?
If 'yes' why do *you* claim the credit of the statement?
If 'no' why is the paper relevant to *your* claim?

In case my previous post was unclear, it is *your* claim
that you can show that the 1/sqrt(1+jw) filter is minimum
phase I have a problem with.

To be blunt, the term 'minimum phase' only applies to
rational functions. The 1/(1+jw) filter is obviously
not rational.

My suggestion remains: Publish the claim and attempted
proof that 1/(1+jw) is minimum phase in some IEEE
transactions.

Rune

Hi Andor,
Thanks-- Very useful!
Steve

On Jun 13, 11:53&#4294967295;am, Rune Allnor <all...@tele.ntnu.no> wrote:
>
> So what is *your* contribution here? You started out by
> claiming that you and RBJ have shown that the 1/sqrt(1+jw)
> filter is minimum phase. Now you refer everything to this
> paper by Kasdin.

i'm not making any claims for discovery.  i don't even know if Andor
is, but i don't speak for him.  as far as i can tell, Andor was
referring to this thread:

http://groups.google.com/group/comp.dsp/tree/browse_frm/thread/25ed457e1273f34b/048112b8690ebeb6

(you might have to unwrap the line)

i think it was about a filter of the form 1/sqrt(jw) not 1/sqrt(1+jw)
and certainly not 1/(1+jw), the latter we *know* what it is and that
it's min phase.

it was a small, self-contained simple topic...

> Does Kasdin show that the filter is minimum phase?
> If 'yes' why do *you* claim the credit of the statement?
> If 'no' why is the paper relevant to *your* claim?
>
> In case my previous post was unclear, it is *your* claim
> that you can show that the 1/sqrt(1+jw) filter is minimum
> phase I have a problem with.
>
> To be blunt, the term 'minimum phase' only applies to
> rational functions. The 1/(1+jw) filter is obviously
> not rational.

it's blunt, but i am not sure it's correct.  i've read lotsa lit in
the audio/acoustics field that essentially defines the meaning of
"minimum phase" in terms of the Hilbert Transform relationship between
the log magnitude and the phase in radians.  no mention of where the
zeros are or even mention of zeros.

in EE class i first learned that "minimum phase" meant having all of
the zeros on the left half-plane which didn't ostensibly sound
equivalent to the Hilbert Transform of the log magnitude thingie.  so,
sometime around 2 or so decades ago i purposed it in my heart to make
the connection and although i had to dig around a bit, i've seen the
connection made in complex variables books, in O&S, and in my never
ending quest to look at things as simply as possible ("but no simpler"
a la Einstein), i finally figured out that, in the s-plane case, if
you can show it's true for a simple 1st-order filter, it's true for
one having many poles and zeros (all in the left half-plane).  and you
can show that it is true for the simple 1st-order filter by showing
that it's true for this pair of functions:

     Hilbert{ 1/(1+w^2) + constant } = -w/(1+w^2)

> My suggestion remains: Publish the claim and attempted
> proof that 1/(1+jw) is minimum phase in some IEEE
> transactions.

why bother.  the potatoes here are very small.  i have no idea if this
weren't "scooped" by someone else long ago in IEEE, IEE, Bell Systems
Journal, whatever.  it's a weird hypothetical filter (which doesn't
stop people from publishing, particularly out of academe).

maybe someday some "signals and systems" LTI systems textbook author
might want to include the 1/sqrt(jw) thing as a weird example problem
for the grad students reading it to say "weird".

r b-j