Jani Huhtanen wrote:> Andor wrote: > > >>Jani Huhtanen wrote: >> >> >>>Andor wrote: >>> >>>>robert bristow-johnson wrote: >>>> >>>>... >>>> >>>>>there are three things that i can understand being what you are >>>>>refering to. first of all, there are no *exactly* linear phase IIR >>>>>filters. >>>> >>>>You meant: There are no *exactly* linear phase _causal_ IIR filters. >>>> >>> >>>You meant: There are no *exactly* linear phase _causal_ and _stable_ IIR >>>filters. ;) >> >>My claim: >> >>An (non-zero) IIR filter has linear-phase => the filter is acausal. >> >>(By acausal I mean there exists no number N such that h(n) = 0 for all >>n < N, where h(n) is the impulse response of the filter). >> >>Stability of the filter is a property which is independent of this >>claim. >> >>Regards, >>Andor > > > > You claimed that there are no *exactly* linear phase _causal_ IIR filters > (right?). I claim that there exists *exactly* linear phase _causal_ IIR > filters but those filters are not _stable_. For example, > > H(z) = 1/(1 - z^-1) > > is _causal_ and it has linear phase. >In this sort of usage a stable filter is assumed. For a filter with poles outside of the unit circle to be stable it must be non-causal, because those portions of the filter with poles outside the unit circle must be executed in 'reverse time' while those portions with poles inside the unit circle must be executed in 'normal time'. -- Tim Wescott Wescott Design Services http://www.wescottdesign.com Posting from Google? See http://cfaj.freeshell.org/google/ "Applied Control Theory for Embedded Systems" came out in April. See details at http://www.wescottdesign.com/actfes/actfes.html
Linear phase IIR filter
Started by ●May 7, 2006
Reply by ●May 9, 20062006-05-09
Reply by ●May 9, 20062006-05-09
Andor wrote:> The necessary and sufficient condition for linear-phase FIR > filters is well known > (http://groups.google.ch/group/comp.dsp/msg/9be6c8f2861d1d3a). > Do you know some similar condition for linear-phase filters with > infinite impulse response?This paper talks about causal infinite linear-phase sequences, but their Fourier transforms are not rational functions: Clements, Pease -- Causal Linear Phase IIR Digital Filters http://0xdc.com/paper.pdf Martin -- Quidquid latine scriptum sit, altum viditur.
Reply by ●May 9, 20062006-05-09
Martin Eisenberg wrote:> Andor wrote: > >> The necessary and sufficient condition for linear-phase FIR >> filters is well known >> (http://groups.google.ch/group/comp.dsp/msg/9be6c8f2861d1d3a). >> Do you know some similar condition for linear-phase filters with >> infinite impulse response? > > This paper talks about causal infinite linear-phase sequences, but > their Fourier transforms are not rational functions: > > Clements, Pease -- Causal Linear Phase IIR Digital Filters > http://0xdc.com/paper.pdf > > > Martin >Intresting paper, thanks. Altough, in the paper the anti-symmetric case for linear-phase filters seems to be ignored. Shouldn't equation (3) be x(d + t) = b * h(d - t) and likewise the equations (6) and (7)? Or have I misunderstood something? -- Jani Huhtanen Tampere University of Technology, Pori
Reply by ●May 9, 20062006-05-09
Jani Huhtanen said the following on 09/05/2006 18:48:> Martin Eisenberg wrote: > >> Andor wrote: >> >>> The necessary and sufficient condition for linear-phase FIR >>> filters is well known >>> (http://groups.google.ch/group/comp.dsp/msg/9be6c8f2861d1d3a). >>> Do you know some similar condition for linear-phase filters with >>> infinite impulse response? >> This paper talks about causal infinite linear-phase sequences, but >> their Fourier transforms are not rational functions: >> >> Clements, Pease -- Causal Linear Phase IIR Digital Filters >> http://0xdc.com/paper.pdf >> > > Intresting paper, thanks. Altough, in the paper the anti-symmetric case for > linear-phase filters seems to be ignored.Strictly speaking, anti-symmetric FIR filters aren't phase-linear. Phase-linear means that: phi(w) = k.w (where k is some real constant), as it is for a symmetric FIR. However, for an anti-symmetric FIR: phi(w) = k.w + pi/2 This is an affine equation rather than a linear equation, hence these may be called "affine-phase" filters. -- Oli
Reply by ●May 9, 20062006-05-09
Oli Filth wrote:> Jani Huhtanen said the following on 09/05/2006 18:48: > >> Martin Eisenberg wrote: >> >>> Andor wrote: >>> >>>> The necessary and sufficient condition for linear-phase FIR >>>> filters is well known >>>> (http://groups.google.ch/group/comp.dsp/msg/9be6c8f2861d1d3a). >>>> Do you know some similar condition for linear-phase filters with >>>> infinite impulse response? >>> >>> This paper talks about causal infinite linear-phase sequences, but >>> their Fourier transforms are not rational functions: >>> >>> Clements, Pease -- Causal Linear Phase IIR Digital Filters >>> http://0xdc.com/paper.pdf >>> >> >> Intresting paper, thanks. Altough, in the paper the anti-symmetric >> case for >> linear-phase filters seems to be ignored. > > > Strictly speaking, anti-symmetric FIR filters aren't phase-linear. > Phase-linear means that: > > phi(w) = k.w > > (where k is some real constant), as it is for a symmetric FIR. > However, for an anti-symmetric FIR: > > phi(w) = k.w + pi/2 > > This is an affine equation rather than a linear equation, hence these > may be called "affine-phase" filters.A plot of phase vs. frequency is a straight line. That's linear enough for me. :-) y = ax + b is generally called a linear (as opposed to quadratic) equation. Jerry -- Engineering is the art of making what you want from things you can get. �����������������������������������������������������������������������
Reply by ●May 9, 20062006-05-09
Jerry Avins said the following on 09/05/2006 20:41:> Oli Filth wrote: > >> Jani Huhtanen said the following on 09/05/2006 18:48: >> >>> Martin Eisenberg wrote: >>> >>>> Clements, Pease -- Causal Linear Phase IIR Digital Filters >>>> http://0xdc.com/paper.pdf >>>> >>> Intresting paper, thanks. Altough, in the paper the anti-symmetric >>> case for >>> linear-phase filters seems to be ignored. >> >> Strictly speaking, anti-symmetric FIR filters aren't phase-linear. >> Phase-linear means that: >> >> phi(w) = k.w >> >> (where k is some real constant), as it is for a symmetric FIR. >> However, for an anti-symmetric FIR: >> >> phi(w) = k.w + pi/2 >> >> This is an affine equation rather than a linear equation, hence these >> may be called "affine-phase" filters. > > A plot of phase vs. frequency is a straight line. That's linear enough > for me. :-) y = ax + b is generally called a linear (as opposed to > quadratic) equation. >Yes, you're right :). I guess it depends which meaning of "linear" you're using. I suppose I should have said "linear transform" rather than "linear equation". What I meant was, y = ax + b isn't linear in the sense that it doesn't obey superposition or homogeneity... -- Oli
Reply by ●May 9, 20062006-05-09
Oli Filth wrote: ...> I suppose I should have said "linear transform" rather > than "linear equation". What I meant was, y = ax + b isn't linear in > the sense that it doesn't obey superposition or homogeneity...Yes. The realization was a bitter pill for me to swallow once. Jerry -- Engineering is the art of making what you want from things you can get. �����������������������������������������������������������������������
Reply by ●May 9, 20062006-05-09
Jani Huhtanen wrote:> Martin Eisenberg wrote: > >> Andor wrote: >> >>> The necessary and sufficient condition for linear-phase FIR >>> filters is well known >>> (http://groups.google.ch/group/comp.dsp/msg/9be6c8f2861d1d3a). >>> Do you know some similar condition for linear-phase filters >>> with infinite impulse response? >> >> This paper talks about causal infinite linear-phase sequences, >> but their Fourier transforms are not rational functions: >> >> Clements, Pease -- Causal Linear Phase IIR Digital Filters >> http://0xdc.com/paper.pdf >> >> >> Martin >> > > Intresting paper, thanks. Altough, in the paper the > anti-symmetric case for linear-phase filters seems to be > ignored. Shouldn't equation (3) be > > x(d + t) = b * h(d - t) > > and likewise the equations (6) and (7)? Or have I misunderstood > something?Maybe you could redevolp the contents for antisymmetric responses, replacing Re with Im in eqs. 1 and 4, but I haven't tried. Martin -- Quidquid latine scriptum sit, altum viditur.
Reply by ●May 9, 20062006-05-09
Jerry Avins wrote:> A plot of phase vs. frequency is a straight line. That's linear enough > for me. :-) y = ax + b is generally called a linear (as opposed to > quadratic) equation.Except that such an equation is not a linear system unless b=0. :-) Ciao, Peter K.
Reply by ●May 9, 20062006-05-09
Peter K. wrote:> Jerry Avins wrote: > > >>A plot of phase vs. frequency is a straight line. That's linear enough >>for me. :-) y = ax + b is generally called a linear (as opposed to >>quadratic) equation. > > > Except that such an equation is not a linear system unless b=0. :-)It may not be a linear system, but it's a linear equation. That was my point. Jerry -- Engineering is the art of making what you want from things you can get. �����������������������������������������������������������������������
