Andor wrote:
> robert bristow-johnson wrote:
> > Andor wrote:
> ...
> > > Ok, I get it. That's a good idea. Why don't you write it up?
> >
> > in a sense, i have here on comp.dsp.
>
> Then make it official. Make it a .... DSP Trick!

those DSP Tricks on Grant's website are brief enough to be read and
understood quickly.  even though my addition to the body of knowledge
is very brief for someone who already is familiar with Wang/Smith TIIR
and Powell/Chau block, filtfilt, and overlap-add; not very many people
are familiar with it, and a little introduction to it might make the
trick work, but i am not ambitious enough for it.

> >  actually i thought of it for the
> > DAFx conference that is coming up and going to be *very* close to me
> > (it will be 90 minutes away in Montreal this autumn)  but it's *such* a
> > small idea and the rest of this linear-phase IIR thing has really been
> > beaten to death.
>
> Yeah well, I've seen papers with way less content.

oh yeah, me too!  but i denounce such papers and would feel like a
hypocrite if i did the same.

r b-j

robert bristow-johnson wrote:
> Andor wrote:
...
> > Ok, I get it. That's a good idea. Why don't you write it up?
>
> in a sense, i have here on comp.dsp.

Then make it official. Make it a .... DSP Trick!

>  actually i thought of it for the
> DAFx conference that is coming up and going to be *very* close to me
> (it will be 90 minutes away in Montreal this autumn)  but it's *such* a
> small idea and the rest of this linear-phase IIR thing has really been
> beaten to death.

Yeah well, I've seen papers with way less content. One of them is in
fact about a (pointless) variant of the PC linear-phase IIR algorithm.

> my regret was not publishing the cookbook at first or doing the
> wavetable synth paper in the 80s or early 90s before
> Serra/Rubine/Dannenberg or Horner papers or Kleczkowski's Group
> Additive Synthesis paper.
> 
> r b-j

Andor

Andor wrote:
> robert bristow-johnson wrote:
> > ... my point
> > is that the Powell/Chau will be theoretically *perfect* and still have
> > no unstable filters if they use (for both forward and reverse filters)
> > stable TIIR filters that are essentially the IIRs they originally
> > designed but with the tail cancelled.  that changes the impulse
> > response and frequency response ever so slightly, but, in the
> > reverse-time filter, the output actually gets to zero theoretically and
> > there will be no click at all.
>
> Ok, I get it. That's a good idea. Why don't you write it up?

in a sense, i have here on comp.dsp.  actually i thought of it for the
DAFx conference that is coming up and going to be *very* close to me
(it will be 90 minutes away in Montreal this autumn) but it's *such* a
small idea and the rest of this linear-phase IIR thing has really been
beaten to death.

my regret was not publishing the cookbook at first or doing the
wavetable synth paper in the 80s or early 90s before
Serra/Rubine/Dannenberg or Horner papers or Kleczkowski's Group
Additive Synthesis paper.

r b-j

Jani Huhtanen wrote:
> Andor wrote:
>
> >
> > Andor wrote:
> >> 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)
> >>
> >> Did you mean x(d + t) = b * x(d - t)?
> >>
> >> >
> >> > and likewise the equations (6) and (7)? Or have I misunderstood
> >> > something?
> >>
> >> Yes, I think you are right. I had the same beef when I first read that
> >> article (see
> >> http://groups.google.com/group/comp.dsp/msg/e72aa24f7d701cf5).
> >
> > Furthermore, if your exmple h(n) = 1, n>=0, is indeed linear-phase, it
> > would be a counter-example to the theorem stated at the bottom of page
> > 1.
>
> Perhaps, but they also claim that h(n) = (-1)^n, where n>=0 seems to be
> linear-phase but it is not bounded in continuous time.

Nice! Another sequence that cannot be reconstructed via
sinc-interpolation. I wasn't aware of the fact that a Nyquist cosine
blows up with sinc-interpolation.


> I tried to "verify"
> the claim experimentally. I summed 10^5 sincs and evaluated the function
> around t=0. I didn't notice any unboundness. Evaluated values were all |x
> <5. Perhaps summing 10^6 sincs / value would have made a difference.

It diverges about as slow as the harmonic numbers H_n = sum_{k=1}^n
1/k. One can show that

H_n < ln(n) + 1/2n + gamma,

where gamma is a constant about 0.577... Therefore the sum of the first
10^6 reciprocals is smaller than 14.4 and the sum of the first 10^9
reciprocals is still smaller than 21.4.

> These quick and dirty checks are not really reliable...

Indeed.

>
> However, intuitevely the claim seems to hold (constructive interference and
> all..).

It fits nicely into the collection of non-reconstructable sequences
that we are building up in comp.dsp :-). The collection essentially
consisted, up to now, only of one sequence, being

{ ... -1, 1, -1, 1, 1, -1, 1, -1, ...}

(note the double 1 in the middle)

which I call Oli's sequence, because Olie Niemtalo first mentioned it.
The critically sampled cosine is now the second sequence in that bag.

Regards,
Andor

Andor wrote:

> 
> Andor wrote:
>> 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)
>>
>> Did you mean x(d + t) = b * x(d - t)?
>>
>> >
>> > and likewise the equations (6) and (7)? Or have I misunderstood
>> > something?
>>
>> Yes, I think you are right. I had the same beef when I first read that
>> article (see
>> http://groups.google.com/group/comp.dsp/msg/e72aa24f7d701cf5).
> 
> Furthermore, if your exmple h(n) = 1, n>=0, is indeed linear-phase, it
> would be a counter-example to the theorem stated at the bottom of page
> 1.

Perhaps, but they also claim that h(n) = (-1)^n, where n>=0 seems to be
linear-phase but it is not bounded in continuous time. I tried to "verify"
the claim experimentally. I summed 10^5 sincs and evaluated the function
around t=0. I didn't notice any unboundness. Evaluated values were all |x
<5. Perhaps summing 10^6 sincs / value would have made a difference. These
quick and dirty checks are not really reliable... 

However, intuitevely the claim seems to hold (constructive interference and
all..).

-- 
Jani Huhtanen
Tampere University of Technology, Pori

Andor wrote:
> 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)
>
> Did you mean x(d + t) = b * x(d - t)?
>
> >
> > and likewise the equations (6) and (7)? Or have I misunderstood something?
>
> Yes, I think you are right. I had the same beef when I first read that
> article (see
> http://groups.google.com/group/comp.dsp/msg/e72aa24f7d701cf5).

Furthermore, if your exmple h(n) = 1, n>=0, is indeed linear-phase, it
would be a counter-example to the theorem stated at the bottom of page
1.

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)

Did you mean x(d + t) = b * x(d - t)?

>
> and likewise the equations (6) and (7)? Or have I misunderstood something?

Yes, I think you are right. I had the same beef when I first read that
article (see
http://groups.google.com/group/comp.dsp/msg/e72aa24f7d701cf5).

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.
> 
> 

Oppenheim (and perhaps others) refer these type of filters as generalized
linear phase filters. He defines four types of FIR linear-phase systems
named 
        Type I:   Odd length and symmetric impulse response,
        Type II:  Even length and symmetric ir,
        Type III: Odd length and antisymmetric ir,
        Type IV:  Even length and antisymmetric ir.

-- 
Jani Huhtanen
Tampere University of Technology, Pori

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.
�����������������������������������������������������������������������

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.