The definition of non-minimum phase system is the system whose zeros
and poles are outside the unit circle of Z-domain. But what does it
mean? What problem does it cause in terms of non-minimum phase? Thank
you.

Annie Chen wrote:

> The definition of non-minimum phase system is the system whose zeros
> and poles are outside the unit circle of Z-domain. But what does it
> mean? What problem does it cause in terms of non-minimum phase? Thank
> you.

That's not a definition, but a consequence. Minimum-phase systems are
what one commonly builds from passive components (although lattice
four-port structures need not be minimum phase). You might take as a
definition that no system with the same frequency response can have less
phase shift. Their frequency and phase responses are related by
classical Bode plots, in which slopes of 20 dB/decade give rise to phase
shifts of 90 degrees. More technically, the log magnitude and phase
responses form a Hilbert transform pair. Simply put, non-minimum phase
implies more phase shift than that.

Jerry
-- 
Engineering is the art of making what you want from things you can get.
¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯

in article 3...@uni-berlin.de, Jerry Avins at j...@ieee.org wrote
on 11/17/2004 22:37:

> Annie Chen wrote:
> 
>> The definition of non-minimum phase system is the system whose zeros
>> and poles are outside the unit circle of Z-domain. But what does it
>> mean? What problem does it cause in terms of non-minimum phase? Thank
>> you.
> 
> That's not a definition, but a consequence. Minimum-phase systems are
> what one commonly builds from passive components (although lattice
> four-port structures need not be minimum phase). You might take as a
> definition that no system with the same frequency response can have less
> phase shift. Their frequency and phase responses are related by
> classical Bode plots, in which slopes of 20 dB/decade give rise to phase
> shifts of 90 degrees. More technically, the log magnitude and phase
> responses form a Hilbert transform pair. Simply put, non-minimum phase
> implies more phase shift than that.

in my mind, the "essence" of the meaning of non-minimum phase system is that
it is not a minimum phase system and the essence of the minimum phase system
is that, given a particular magnitude response, the min-phase system has
less phase shift than all other systems of the same magnitude response.

for LTI systems with rational transfer functions (those with poles and
zeros), we know that the poles have to be in the left half-plane of the
analog s-plane (inside the unit circle of the z-plane), but the zeros can be
either side.  if the zeros are reflected to the right half-plane, the
magnitude response is the same but the phase shift is more than what would
be the case if all of the zeros were in the left half-plane.

and, it is true (but hard to prove, who can prove it here on comp.dsp??  and
without residue theory?) that for rational LTI systems that are minimum
phase, that the Hilbert transform of the natural log of the magnitude
response is the same as the phase response in radians.

r b-j

In article <c...@posting.google.com>,
Annie Chen <a...@gmail.com> wrote:
>The definition of non-minimum phase system is the system whose zeros
>and poles are outside the unit circle of Z-domain. But what does it
>mean? What problem does it cause in terms of non-minimum phase?

Increased phase results in increased delay thru the system.  So a
non-minimum phase system will (almost by definition) have a larger
phase and thus more delay than would a minimum phase system with the
the identical magnitude frequency response.

Why the phase (especially near the vicinity of the zero) must decrease
when reflecting a zero from outside to inside the unit circle is left
as an exercise for the student.


IMHO. YMMV.
-- 
Ron Nicholson   rhn AT nicholson DOT com   http://www.nicholson.com/rhn/ 
#include <canonical.disclaimer>        // only my own opinions, etc.

a...@gmail.com (Annie Chen) wrote in message
news:<c...@posting.google.com>...
> The definition of non-minimum phase system is the system whose zeros
> and poles are outside the unit circle of Z-domain. 

Wrong. The poles must be inside the unit circle for the system to be 
both causal and stable. It is the zeros that can be inside or outside
the unit circle. If some zeros are outside the unit circle and some are 
inside, one speaks of a "mixed phase" system. If all the zeros are 
outside the unit circle, one speaks of a "maximum phase" system. 

> But what does it mean? 

The minimum phase property has to do with time delays in the impulse 
response. Minimum phase systems have an impulse response where the 
energy is concentrated at an as early time as possible.

> What problem does it cause in terms of non-minimum phase? 

Well, minimum phase systems are easy to handle from a mathemathical 
point of view. Since all the zeros are inside the unit circle, 
the inverse filter is stable. One can do certain things with 
minimum phase systems that one can not do with mixed phase systems.

If the system one investigates is not minimum phase, one must be
very careful with how one conducts the analysis. The "easy" system 
identification methods are based on the assumption of minimum phase. 
If it's sufficient to describe the system in terms of the Power 
Spectral Density, PDF, of its transfer function, minimum phase 
techniques are often good enough regardless of whether the 
system actually is minimum phase, or not. 

If you need to estimate the time-domain impulse response from a 
frequency domain measurement, you might find yourself in big 
trouble if you are only able to estimate the PDF, which does not 
preserve the phase response of the system.

You can estimate the minimum phase impulse response from a PDF. 
You can not estimate mixed phase impulse responses from a PDF. 
So if you estimate a minimum phase impulse response for a system 
that actually is mixed phase, you make a mistake which may or 
may not be important. It depends on the application.

> Thank you.

Y' welcome.

Rune

robert bristow-johnson wrote:

  ...

> and, it is true (but hard to prove, who can prove it here on comp.dsp??  and
> without residue theory?) that for rational LTI systems that are minimum
> phase, that the Hilbert transform of the natural log of the magnitude
> response is the same as the phase response in radians.

If I had known you were coming, I would not only have baked a cake, I
would have held my peace.

Jerry
-- 
Engineering is the art of making what you want from things you can get.
¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯

Rune Allnor wrote:

  ...

> Spectral Density, PDF, ...

"PDF" --> probability density function. The identity isn't obvious.

Jerry
-- 
Engineering is the art of making what you want from things you can get.
¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯

If we consider a simple stable, 1-st order, low-pass, system
(s-z)/(s+p) (with z and p positive), with an "unstable" zero, i.e. in
the right half s plane, or outside the unit disk in the Z domain, then
physically, it translates into a system with an initial response which
is in the opposite direction of the applied stimulus, and of it's long
term response (ex. to a step input).  A practical example which helps
to see this, is in handling a long flexible stick, or beam. A simple,
crude approximation would consist of two rigid sticks, joined by some
elastic, flexible, spring-like, coupling. If one holds one stick at
one end, and applies some torque to change the angular orientation of
the overall stick, the other end will inevitably, initially (i.e. at
t=0+), turn in the opposite direction. We can then see how this will
add to the overall response time, or phase delay in the frequency
response. A "stable" zero, on the other hand, ex. (s+z)/(s+p), which
effectively adds an anticipative component to the response (the
purpose of P in PID-based control), serves to reduce the overall
response time, precisely because of it's anticipative character. Such
"intuitive" characterization of otherwise technical terminology
hopefully serves to get a better "feel" for what's really going on in
a system and it's sub-components, and therefore how to better control
it.

In article <f...@posting.google.com>,
Rune Allnor <a...@tele.ntnu.no> wrote:
>a...@gmail.com (Annie Chen) wrote in message
>news:<c...@posting.google.com>...
>> The definition of non-minimum phase system is the system whose zeros
>> and poles are outside the unit circle of Z-domain. 
>
>Wrong. The poles must be inside the unit circle for the system to be 
>both causal and stable. It is the zeros that can be inside or outside
>the unit circle. 

Well, she didn't specify that the system be causal and stable.  
(for instance, her class project could be to build a hypothetical
time machine).

And what happens if there enough zeros very close to or on top of
a pole outside the unit circle?  Couldn't the system then be stable
or conditionally stable?


IMHO. YMMV.
-- 
Ron Nicholson   rhn AT nicholson DOT com   http://www.nicholson.com/rhn/ 
#include <canonical.disclaimer>        // only my own opinions, etc.

in article 3...@uni-berlin.de, Jerry Avins at j...@ieee.org wrote
on 11/18/2004 15:08:

> robert bristow-johnson wrote:
> 
> ...
> 
>> and, it is true (but hard to prove, who can prove it here on comp.dsp??  and
>> without residue theory?) that for rational LTI systems that are minimum
>> phase, that the Hilbert transform of the natural log of the magnitude
>> response is the same as the phase response in radians.
> 
> If I had known you were coming, I would not only have baked a cake, I
> would have held my peace.

why, on earth, for, Jerry??!  your common sense wisdom, your experience,
etc. are here for us to hear.

i would *still* like to read someone prove that for a continuous-time LTI
system with a rational transfer function (H(s) is a ratio of polynomials of
s, i.e. H(s) has poles and zeros in the s-plane), that having all poles in
the left half-plane (necessary for stability) and all zeros in the left
half-plane results in the phase response (measured in radians) being the
Hilbert Transform of the natural log of the magnitude response.  to do it
without resorting to residue theory of complex integration would make the
explanation more accessible, especially to folks who haven't messed around
with a course in complex variables.

also to show that if any of those zeros get reflected to the right
half-plane, the magnitude response stays the same but there is *more* phase
shift than the system with all zeros in the left half-plane (thus justifying
the term "minimum phase system" for such a system).

i might get to it eventually if no one else does.  but i don't have a lot of
time as of recent.

hey Airy, why don't you try it out?  it might be a good way to rehabilitate
your reputation on comp.dsp .

-- 

r b-j                  r...@audioimagination.com

"Imagination is more important than knowledge."