Help With Interpretation of Pole Locations in Z-Plane

I am modeling a system using a conventional linear predictor, in which
the present output sample is predicted as a linear combination of past
samples.  Having obtained the transfer function of the analysis filter
from those prediction coefficients, I am formulating the synthesis
filter from the inverse of the analysis filter.

All standard stuff, and my question is not about any aspect of the
linear predictor.  Instead, my question is about interpretation of the
results.  I am attempting to use the poles of the synthesis filter to
determine an estimate of the system natural frequencies.  From first
principles I know that z = exp[sT] = exp[sigma + j(omega T)], so I can
determine omega directly from the angle between the poles and the real
axis.  However, I also know that if a complex pole pair is viewed as a
damped second-order system, then the resulting natural frequency is
not only determined by the angle between the poles and the real axis,
but also by the distance from the origin to the poles: z = exp[-(D
omega T) +/- j(omega (sqrt(1-D^2)) T)], where D is damping factor.
Under these circumstances, which is the more appropriate
interpretation?

Thanks,
Greg

"Greg Berchin" <gberchin@sentientscience.com> wrote in message
news:1185992733.658190.286920@19g2000hsx.googlegroups.com...
> I am modeling a system using a conventional linear predictor, in which
> the present output sample is predicted as a linear combination of past
> samples.  Having obtained the transfer function of the analysis filter
> from those prediction coefficients, I am formulating the synthesis
> filter from the inverse of the analysis filter.
>
> All standard stuff, and my question is not about any aspect of the
> linear predictor.  Instead, my question is about interpretation of the
> results.  I am attempting to use the poles of the synthesis filter to
> determine an estimate of the system natural frequencies.  From first
> principles I know that z = exp[sT] = exp[sigma + j(omega T)], so I can
> determine omega directly from the angle between the poles and the real
> axis.  However, I also know that if a complex pole pair is viewed as a
> damped second-order system, then the resulting natural frequency is
> not only determined by the angle between the poles and the real axis,
> but also by the distance from the origin to the poles: z = exp[-(D
> omega T) +/- j(omega (sqrt(1-D^2)) T)], where D is damping factor.
> Under these circumstances, which is the more appropriate
> interpretation?

Hi Greg,

My understanding is the LPC analysis gives you a MSE model of spectrum.
Therefore there is generally no direct relation between the pole locations
of the synth. filter and the modes of the system. As for your question, in
those special cases where there is such relation, the damping should be
accounted for.

Vladimir Vassilevsky
DSP and Mixed Signal Consultant
www.abvolt.com

On Aug 2, 1:10 am, "Vladimir Vassilevsky" <antispam_bo...@hotmail.com>
wrote:

> My understanding is the LPC analysis gives you a MSE model of spectrum.
> Therefore there is generally no direct relation between the pole locations
> of the synth. filter and the modes of the system.

I have been struggling with that issue, too.  Frankly, none of the
reference material I have read addresses the subject directly.
However, I do know that, when the measured data consist of pure
sinusoids, the poles of the synthesis filter match the sinewave
frequencies extremely well.  However, such test signals do not answer
the damping question, because pure sinewaves fall on the jw axis where
damping is nonexistent.

> As for your question, in
> those special cases where there is such relation, the damping should be
> accounted for.

I have been leaning toward that conclusion myself, based upon my
experimental results.  It's always good to get a second opinion,
however.

Thank you, Vladimir,
Greg

On Aug 2, 7:12 am, Greg Berchin <gberc...@sentientscience.com> wrote:
> On Aug 2, 1:10 am, "Vladimir Vassilevsky" <antispam_bo...@hotmail.com>
> wrote:
>
> > My understanding is the LPC analysis gives you a MSE model of spectrum.
> > Therefore there is generally no direct relation between the pole locations
> > of the synth. filter and the modes of the system.
>
> I have been struggling with that issue, too.  Frankly, none of the
> reference material I have read addresses the subject directly.
> However, I do know that, when the measured data consist of pure
> sinusoids, the poles of the synthesis filter match the sinewave
> frequencies extremely well.  However, such test signals do not answer
> the damping question, because pure sinewaves fall on the jw axis where
> damping is nonexistent.

You should expect to get good results with pure sinusoid inputs, as
they are both deterministic and periodic. The linear predictor can
exactly predict future samples based on past values; all it needs is a
single tap with a value of unity at the delay line that corresponds to
the sinusoid's period. For more complicated signals, however, as
Vladimir pointed out, all you get is a MSE model of the system, which
doesn't necessarily relate to the system's modes.

Jason

On Aug 2, 7:40 am, cincy...@gmail.com wrote:

> You should expect to get good results with pure sinusoid inputs, as
> they are both deterministic and periodic.

For that matter, so are the signals that I am characterizing ... at
least it's a reasonable assumption over some short time periods, and
those are the regions in which I am interested.  There is, of course,
also some noise.

> The linear predictor can
> exactly predict future samples based on past values; all it needs is a
> single tap with a value of unity at the delay line that corresponds to
> the sinusoid's period.

The period can be considerably longer than the predictor, in my case.

Thanks,
Greg