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/0a4a1f563461c283/d= 2368b77ac8c674f?#d2368b77ac8c674f A simple example of such a frequency response is T1(w) =3D 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=B0. A slightly more complex variant is T2(w) =3D 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
Minimum-Phase Systems Characteristics
Started by ●November 20, 2007
Reply by ●November 20, 20072007-11-20
On 20 Nov, 16:17, Andor <andor.bari...@gmail.com> 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) =3D 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=B0. A > slightly more complex variant is > > T2(w) =3D 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?As far as I can see, both your T1 and T2 functions are irrational in the sense that they can *not* be expressed in terms of ratios of finite-degree polynomials, T(w) =3D D(w)/N(w). Hence, it is not obvious how to evaluate the poles and zeros of T, which in turn makes it difficult to make a useful statement about properties like "minimum phase." Basically, before you can discuss the details you need to establish that the term "minimum phase" makes sense for irrational transfer functions. Rune
Reply by ●November 20, 20072007-11-20
I wrote: ...> These frequency responses don't really have poles or zeros,... I should have written that they don't have poles or zeros of integer order.
Reply by ●November 20, 20072007-11-20
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/0a4a1f563461c283/d2368b77ac8c674f?#d2368b77ac8c674f > > 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?If you can implement it, measure what you get. If you can't, who cares? Suppose you knew the amplitude and phase responses. Could you recognize minimum phase? Jerry -- Engineering is the art of making what you want from things you can get. �����������������������������������������������������������������������
Reply by ●November 20, 20072007-11-20
On Nov 20, 7:17 am, Andor <andor.bari...@gmail.com> 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) =3D 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=B0. A > slightly more complex variant is > > T2(w) =3D 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?Try looking at the complex cepstrums of your frequency responses. Are you looking for an ideal answer or a working approximation? IMHO. YMMV. -- rhn A.T nicholson d.0.t C-o-M
Reply by ●November 20, 20072007-11-20
On 20 Nov., 17:32, Jerry Avins <j...@ieee.org> 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) =3D 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=B0. A > > slightly more complex variant is > > > T2(w) =3D 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? > > If you can implement it, measure what you get. If you can't, who cares?What do you mean "measure what you get". Get what?> > Suppose you knew the amplitude and phase responses. Could you recognize > minimum phase?I guess that's my question (obviously, I know amplitude and phase response of my systems): how to recognize minimum-phase? Regards, Andor
Reply by ●November 20, 20072007-11-20
On 20 Nov, 17:39, Andor <andor.bari...@gmail.com> wrote:> On 20 Nov., 17:32, Jerry Avins <j...@ieee.org> 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) =3D 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=B0. =A> > > slightly more complex variant is > > > > T2(w) =3D 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? > > > If you can implement it, measure what you get. If you can't, who cares? > > What do you mean "measure what you get". Get what? > > > > > Suppose you knew the amplitude and phase responses. Could you recognize > > minimum phase? > > I guess that's my question (obviously, I know amplitude and phase > response of my systems): how to recognize minimum-phase?In the usual AR applications, one estimates a power spectrum and proceed from there to find the parameters of a minimum phase system. The only attempt I have seen that resembles your question (to determine exactly what phase model applies to a set of measured data) was done by Ursin and Porsani: "Estimation of an optimal mixed-phase inverse filter" Geophysical Prospecting, Volume 48, Number 4, July 2000 They used a Genetic Algorithm to test which zeros were located outside the unit circle. As far as I remember, they computed the impulse response of each model and compared it to the measured data. Rune
Reply by ●November 20, 20072007-11-20
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/0a4a1f563461c283/d2368b77ac8c674f?#d2368b77ac8c674f > > 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, > AndorIIRC for a minimum-phase system the phase is the Hilbert transform of the amplitude -- anything more makes it not minimum phase. Perhaps this is the ceptstrum that Ron N was talking about -- I need to hunt that term down. -- Tim Wescott Wescott Design Services http://www.wescottdesign.com Do you need to implement control loops in software? "Applied Control Theory for Embedded Systems" gives you just what it says. See details at http://www.wescottdesign.com/actfes/actfes.html
Reply by ●November 20, 20072007-11-20
On Nov 21, 4:17 am, Andor <andor.bari...@gmail.com> 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) =3D 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=B0. A > slightly more complex variant is > > T2(w) =3D 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, > AndorWhere does such a system originate? Or is this a theoretical problem. Hardy
Reply by ●November 21, 20072007-11-21
On 20 Nov., 17:35, "Ron N." <rhnlo...@yahoo.com> wrote:> On Nov 20, 7:17 am, Andor <andor.bari...@gmail.com> 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) =3D 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=B0. A > > slightly more complex variant is > > > T2(w) =3D 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? > > Try looking at the complex cepstrums of your frequency responses. > Are you looking for an ideal answer or a working approximation?It's a yes/no question: is T(w) a minimum-phase system? I don't see how I can approximate an answer (0.5 * yes? :-). How will the cepstrum help? Regards, Andor
