Re: Minimum-Phase Systems Characteristics

On Nov 21, 12:44 am, Rune Allnor <all...@xxxxxxxxxxxx> wrote:
On 21 Nov, 09:29, Andor <andor.bari...@xxxxxxxxx> wrote:
...
On 20 Nov., 18:45, Tim Wescott <t...@xxxxxxxxxxxxxxxx> wrote:

Andor wrote:
Friends,

I have a bunch of frequency responses (of continuous-time systems),
and I want to know whether they are minimum-phase. These frequency
responses don't really have poles or zeros, which makes things tricky.
A similar question was asked by Matt a while back:

http://groups.google.com/group/comp.dsp/browse_frm/thread/0a4a1f56346....

A simple example of such a frequency response is

T1(w) = 1 / sqrt(i w), w > 0.

It has "half a pole" at zero (for w < 0 make it Hermitan symmetric to
get a real impulse response) and a constant phase response of -45°.. A
slightly more complex variant is

T2(w) = 1 / sqrt(i w) (1+ G exp(- L sqrt(i w))) / (1 - G exp(- L
sqrt(i w))).

G and L are real parameters, G ranges in [-1, 1], and L > 0. Are T1(w)
or T2(w) minimum-phase?

Regards,
Andor

IIRC for a minimum-phase system the phase is the Hilbert transform of
the amplitude -- anything more makes it not minimum phase.

That's what Wikipedia says as well. I remember reading a post of r b-
j's outlining a proof for that claim. One of his conditions was that
the system have a rational transfer function. I can immagine that the
statement is also true for other transfer functions, but I'm not
sure.

Proakis & Manolakis have an unusal definition of a minimum-phase
system: H(w) is minimum-phase iff arg(H(0)) = arg(H(pi)). If I have a
continuous-time system, does this definition translate to

"H(w) is minimum-phase iff arg(H(0)) = arg(H(infinity))"

?

If so, then the two systems I gave above would both be minimum-phase.

I can't really see how that would be helpful? For discrete systems,
the convention is that arg(H(0)) = 0. The DFT of a discrete sequence
has a spectrum coefficient located at z=-1, meaning that arg(H(pi))=0
no matter what, provided we deal with real-valued time-domain data.

One can concoct some sort of academic argument about these
values being expressed modulo 2pi,

arg(H(0)) = 0+n*2pi
arg(H(pi))=0+m*2pi

and P&M being right, demanding that n=m. But that isn't all
that helpful in practice.

Rune

and:

On Nov 21, 8:19 am, "rich_158" <eecod...@xxxxxxxxxxx> wrote:
a minimum phase signal starting at t = 0 has the property
that its average value of the slope of its unwrapped FT
phase spectrum is zero.

I know what a mimimum phase signal (or should that be think?)
is but i cannot understand why this statement is
true can someone enlighten me ?

We had a similar discussion in comp.dsp way back in Feb 2004
regarding the minimum phase in the discrete case, and how it
related to poles and zeros in the Z-plane, under the title
"FIR roots and frequency response".

For the discrete case:

Phase increases when traveling counter-clockwise around a
zero, and decreases when traveling counter-clockwise circle
around a pole. Let say that all poles are inside the unit
circle. A unit delay can be implemented by a pole at the
origin. If you travel in a complete circle around a pole,
the phase will decrease by 2 * pi. Increase the number of
unit delays, and this will decrease the phase change by a
higher multiple of 2 * pi. No unit delays is equivalent to
no decrease in absolute phase after one trip around the unit
circle. Any zero inside the unit circle will cancel out the
phase of a pole inside the unit circle after one full traverse
of the unit circle. A zero outside the unit circle will not
cancel out the phase delay of a pole, because a trip around
the unit circle will not travel around zero that's outside.
So the closest you can get to the minimum number of unit
delays, just by moving zeros, is by placing all the zeros
inside the unit circle to cancel out the phase from as
many poles as possible. I think this is the minimum phase
situation for a given number of poles and zeros.

If the phase joins itself after one trip around the unit
circle, then the derivative of the phase integrated over one
trip around the unit circle should be zero. This corresponds
to the average of the slope of the phase being zero.

Can a similar hand-waving case be made for the continuous
case?





IMHO. YMMV.
--
rhn A.T nicholson d.0.t C-o-M
http://www.nicholson.com/rhn/dsp.html
.

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