sufficient condtion for linear phase IIR

Hello, everyone
I met a problem about the linear phase IIR. I saw a lot of discussion
about linear phase FIR but really knew little about  linear phase IIR.

Having a constant group delay seems to be its necessary condition, but
what is its sufficient condtion?

Any suggestion is welcome
Thanks in advance!

jacobus

hzzc1...@gmail.com wrote:
>
> I met a problem about the linear phase IIR. I saw a lot of discussion
> about linear phase FIR but really knew little about  linear phase IIR.

strictly speaking, there is no linear phase IIR, because to be linear
phase, there has to be some point in the middle of the impulse response
where the IR is perfectly symmetrical, and since the IIR is both causal
(no IR before t=0) and infinite (in the +t direction), there can be no
symmetry going out forever.  there was discussion of "Truncated IIR"
filters  (TIIR) to do phase-linear filtering, and i would recommend
looking it up with Google and Google Groups.

> Having a constant group delay seems to be its necessary condition, but
> what is its sufficient condtion?

constant group delay or constant phase delay are both precisely
equivalent to phase linear.  it's not simply a "sufficient condtion",
it *is* the same thing.  that is why sometimes people want phase-linear
is so that the delay at every frequency is the same.

r b-j

hzzc1012@gmail.com wrote:

> Hello, everyone
> I met a problem about the linear phase IIR. I saw a lot of discussion
> about linear phase FIR but really knew little about  linear phase IIR.
> 
> Having a constant group delay seems to be its necessary condition, but
> what is its sufficient condtion?

The non-trivial IIR will never have the linear phase. Just because it is 
IIR. IIRs can only approximate the linear phase response with more or 
less accuracy.


Vladimir Vassilevsky

DSP and Mixed Signal Design Consultant

http://www.abvolt.com

robert bristow-johnson wrote:
> hzzc1...@gmail.com wrote:
> >
> > I met a problem about the linear phase IIR. I saw a lot of discussion
> > about linear phase FIR but really knew little about  linear phase IIR.
>
> strictly speaking, there is no linear phase IIR, because to be linear
> phase, there has to be some point in the middle of the impulse response
> where the IR is perfectly symmetrical, and since the IIR is both causal
> (no IR before t=0) and infinite (in the +t direction), there can be no
> symmetry going out forever.

An IIR filter does not have to be causal - for example, a sampled sinc
function to generate a lowpass, with symmetry about 0 (we aren't yet
talking about technical realizability).

There is that paper that pops up here periodically (just like the
linear-phase IIR topic), [1], which discusses phase-linearity
conditions for infinitely long sequences. A theorem in that paper
states:

an IIR is linear-phase <=> there exists a point of symmetry in the
sinc-interpolated impulse response.

I think we also discussed how this theorem probably uses a limited
interpretation of "linear-phase", because antisymmetric impulse
responses are also "linear-phase" by the widely used definition (ie.
have a factorization into a purely real amplitude response multiplied
by a linear phase term), but are not covered in the theorem.


> there was discussion of "Truncated IIR"
> filters  (TIIR) to do phase-linear filtering, and i would recommend
> looking it up with Google and Google Groups.

You are too modest:
http://groups.google.com/group/comp.dsp/msg/18640683ae047541

Regards,
Andor

[1] Clements, M A and Pease, W J:
"On Causal Linear Phase IIR Digital Filters",
IEEE Transactions on Acoustics, Speech, and Signal Processing, vol. 37,
No 4., April 1989.

robert bristow-johnson wrote:
> hzzc1...@gmail.com wrote:
>> I met a problem about the linear phase IIR. I saw a lot of discussion
>> about linear phase FIR but really knew little about  linear phase IIR.
> 
> strictly speaking, there is no linear phase IIR, because to be linear
> phase, there has to be some point in the middle of the impulse response
> where the IR is perfectly symmetrical, and since the IIR is both causal
> (no IR before t=0) and infinite (in the +t direction), there can be no
> symmetry going out forever.  there was discussion of "Truncated IIR"
> filters  (TIIR) to do phase-linear filtering, and i would recommend
> looking it up with Google and Google Groups.
> 
>> Having a constant group delay seems to be its necessary condition, but
>> what is its sufficient condtion?
> 
> constant group delay or constant phase delay are both precisely
> equivalent to phase linear.  it's not simply a "sufficient condtion",
> it *is* the same thing.  that is why sometimes people want phase-linear
> is so that the delay at every frequency is the same.

Robert,

For stored signals, the notion of causality loses much of its meaning. 
All IIR filters are linear-phase when they are used twice, once with the 
data running forward, and once with time reversed. Then the overall 
response then extends from - to + infinity, and is symmetric about what 
we usually call time zero (even though it may in fact be many years 
after Caruso actually sang).

Of course you knew that. You probably also knew that hzzc1012 wouldn't 
be interested.

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

Vladimir Vassilevsky wrote:
> hzzc1012@gmail.com wrote:
>
> > Hello, everyone
> > I met a problem about the linear phase IIR. I saw a lot of discussion
> > about linear phase FIR but really knew little about  linear phase IIR.
> >
> > Having a constant group delay seems to be its necessary condition, but
> > what is its sufficient condtion?
>
> The non-trivial IIR will never have the linear phase. Just because it is
> IIR. IIRs can only approximate the linear phase response with more or
> less accuracy.

You probably had finite order filters with rational transfer functions
in mind. There are other IIR filters, and some of them have an exactly
linear phase response.

Regards,
Andor

Andor wrote:


>>The non-trivial IIR will never have the linear phase. Just because it is
>>IIR. IIRs can only approximate the linear phase response with more or
>>less accuracy.
> 
> 
> You probably had finite order filters with rational transfer functions
> in mind. There are other IIR filters, and some of them have an exactly
> linear phase response.

The IIR has the word "infinite" in the name, and I think if the IIR is 
casual then the phase is nonlinear. Please correct me if there is a mistake.

Vladimir Vassilevsky

DSP and Mixed Signal Design Consultant

http://www.abvolt.com

Vladimir Vassilevsky wrote:
> Andor wrote:
>
>
> >>The non-trivial IIR will never have the linear phase. Just because it is
> >>IIR. IIRs can only approximate the linear phase response with more or
> >>less accuracy.
> >
> >
> > You probably had finite order filters with rational transfer functions
> > in mind. There are other IIR filters, and some of them have an exactly
> > linear phase response.
>
> The IIR has the word "infinite" in the name, and I think if the IIR is
> casual then the phase is nonlinear.

There are causal and non-causal IIRs. Obviously, there exist non-causal
IIRs that are linear-phase (the ideal linear-phase lowpass, for
instance) - these already provide counter examples to your statement:
"The non-trivial IIR will never have the linear phase". Thinking about
it, any linear-phase FIR filter can be trivially made into an IIR
filter - just time shift the impulse response by a non-integer amount.

Additionally (and rather surprisingly), there exist (discrete time)
causal, linear-phase IIRs. As it turns out, their impulse responses,
after sinc-interpolation, have a point of symmetry, which makes these
filters linear-phase.

> Please correct me if there is a mistake.

You should read that paper, it's quite enlightening.

Regards,
Andor

Andor wrote:

>    ... there exist (discrete time)
> causal, linear-phase IIRs. As it turns out, their impulse responses,
> after sinc-interpolation, have a point of symmetry, which makes these
> filters linear-phase.

I see described a filter whose response is zero before t = zero and 
extends to t = infinity. If there is a time about which the response is 
symmetric (or antisymmetric), that time must be infinity/2. In what way 
is my sight clouded?

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

Jerry,

In the theory, it is possible to drink beer using colander. You only 
need enough patience to wait for the cold winter. In the practice, 
nobody does that.

VLV


Jerry Avins wrote:

> Andor wrote:
> 
>>    ... there exist (discrete time)
>> causal, linear-phase IIRs. As it turns out, their impulse responses,
>> after sinc-interpolation, have a point of symmetry, which makes these
>> filters linear-phase.
> 
> 
> I see described a filter whose response is zero before t = zero and 
> extends to t = infinity. If there is a time about which the response is 
> symmetric (or antisymmetric), that time must be infinity/2. In what way 
> is my sight clouded?
> 
> Jerry