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From: Tim Wescott on 20 Jul 2010 18:25
On 07/20/2010 02:45 PM, Pawel wrote:
> On Jul 20, 10:22 pm, Tim Wescott<t...(a)seemywebsite.com> wrote:
>> On 07/20/2010 01:50 PM, Pawel wrote:
>>
>>
>>
>>> On Jul 20, 9:30 pm, Tim Wescott<t...(a)seemywebsite.com> wrote:
>>>> On 07/20/2010 01:04 PM, Pawel wrote:
>>
>>>>> Dear All,
>>
>>>>> Maybe one of You Gurus would be able to shed to confirm (or criticise)
>>>>> my way of thinking. I am not DSP expert I am just trying to use DSP
>>>>> methods in other fields of science.
>>>>> My question relate to an all-pass filter structure: H(z)=z^-N[A(z^-1)/
>>>>> A(z)]. The problem that I am trying to solve do not easily tolerate N
>>>>> sample delay z^-N.
>>>>> 1. So I thought that it might be possible to express all-pass function
>>>>> as a cascade of anti-causal FIR filter A(z^-1) and causal IIR 1/A(z)
>>>>> effectively eliminating the excess delay z^-N that it is there to only
>>>>> make A(z^-1) causal. My application is offline so it can be
>>>>> implemented as a backward filtering?
>>>>> 2. Therefore my H_1=A(z^-1)/A(z), now how do I find inverse of it
>>>>> H_1inv=A(z)/A(z^-1). 1/A(z^-1) will be unstable since its poles will
>>>>> be inside the unit circle (A(z^-1) zeros where inside the circle),
>>>>> however I can resort again to the noncausal filtering and treat
>>>>> A(z^-1) as causal that would give me a stable filter?
>>>>> 3. How do I invert filter H_1?
>>>>> 4. What would be pitfalls when I would try to implement this in
>>>>> reality for offline processing � I know that there might be some
>>>>> problems with initial conditions in forward filtering?
>>
>>>> What are you really trying to do?
>>
>>>> By definition, a useful all-pass filter has delay, at least at some
>>>> frequencies. It must, because a slope in the phase of a filter's
>>>> transfer function in the frequency domain translates to a delay (or
>>>> prediction) in the time domain. So the best that you can do is to run
>>>> your all-pass filter in just one direction, then shift the answer (which
>>>> you can do because you're running off line) to get the least overall
>>>> delay, or the least delay "where it matters".
>>
>>>> Note that this problem with allpass vs. phase delay is evinced a bit if
>>>> you think about your question 2: Work out the math, and you'll find
>>>> that H_1 * H_1inv is the only no-delay allpass filter in existence: H = 1.
>>
>>>> --
>>
>>>> Tim Wescott
>>>> Wescott Design Serviceshttp://www.wescottdesign.com
>>
>>>> Do you need to implement control loops in software?
>>>> "Applied Control Theory for Embedded Systems" was written for you.
>>>> See details athttp://www.wescottdesign.com/actfes/actfes.html
>>
>>> Tim,
>>
>>> Foremost thanks a lot for reply.
>>> My application is in the area of microwave engineering and microwave
>>> filter design. I am working on an algortihm that allows to include
>>> some defficiences of the transmission lines of which filter is built
>>> to be included in the design procedure and their influence
>>> eliminated.
>>
>>> Anyway, I do not want zero delay but I want to retain only nonlinear
>>> portion of the all-pass phase shift - z^-N is an artifact coming from
>>> the fact that usually one is interested in the causal filter. Would my
>>> concept as outlined in the question 1 retain this nonlinear phase
>>> shift or I am misinterpreting something?
>>
>> What is A(z) in your definition above? I ask, because the "control
>> engineer's formulation" of an IIR filter would have
>>
>> A(z) = a_N * z^N + ... + a_1 * z + a_0,
>>
>> while the "DSP engineer's formulation" of an IIR filter would have
>>
>> A(z) = a_N / z^N + ... + a_1 / z + a_0.
>>
>> In the former case, A(1/z) would have a be perfectly causal, if delayed.
>> In the latter case A(1/z) would, indeed, have substantial lead that
>> would have to be corrected with the z^-N term. You could leave it off
>> if you're not interested in causality.
>>
>> --
>>
>> Tim Wescott
>> Wescott Design Serviceshttp://www.wescottdesign.com
>>
>> Do you need to implement control loops in software?
>> "Applied Control Theory for Embedded Systems" was written for you.
>> See details athttp://www.wescottdesign.com/actfes/actfes.html
>
> I define A(z) as a_0+a_1z^-1+a_2z^-2+...+a_Nz^-N - which of those
> above mentioned cases would it be - forgive me my stupid question?
>
> Pawel

That's the "DSP Engineer's formulation".

--

Tim Wescott
Wescott Design Services
http://www.wescottdesign.com

Do you need to implement control loops in software?
"Applied Control Theory for Embedded Systems" was written for you.
See details at http://www.wescottdesign.com/actfes/actfes.html
From: Pawel on 20 Jul 2010 18:35
On Jul 20, 11:25 pm, Tim Wescott <t...(a)seemywebsite.com> wrote:
> On 07/20/2010 02:45 PM, Pawel wrote:
>
>
>
> > On Jul 20, 10:22 pm, Tim Wescott<t...(a)seemywebsite.com>  wrote:
> >> On 07/20/2010 01:50 PM, Pawel wrote:
>
> >>> On Jul 20, 9:30 pm, Tim Wescott<t...(a)seemywebsite.com>    wrote:
> >>>> On 07/20/2010 01:04 PM, Pawel wrote:
>
> >>>>> Dear All,
>
> >>>>> Maybe one of You Gurus would be able to shed to confirm (or criticise)
> >>>>> my way of thinking. I am not DSP expert I am just trying to use DSP
> >>>>> methods in other fields of science.
> >>>>> My question relate to an all-pass filter structure: H(z)=z^-N[A(z^-1)/
> >>>>> A(z)]. The problem that I am trying to solve do not easily tolerate N
> >>>>> sample delay z^-N.
> >>>>> 1. So I thought that it might be possible to express all-pass function
> >>>>> as a cascade of anti-causal FIR filter A(z^-1) and causal IIR 1/A(z)
> >>>>> effectively eliminating the excess delay z^-N that it is there to only
> >>>>> make A(z^-1) causal. My application is offline so it can be
> >>>>> implemented as a backward filtering?
> >>>>> 2. Therefore my H_1=A(z^-1)/A(z), now how do I find inverse of it
> >>>>> H_1inv=A(z)/A(z^-1).  1/A(z^-1) will be unstable since its poles will
> >>>>> be inside the unit circle (A(z^-1) zeros where inside the circle),
> >>>>> however I can resort again to the noncausal filtering and treat
> >>>>> A(z^-1) as causal that would give me a stable filter?
> >>>>> 3. How do I invert filter H_1?
> >>>>> 4. What would be pitfalls when I would try to implement this in
> >>>>> reality for offline processing – I know that there might be some
> >>>>> problems with initial conditions in forward filtering?
>
> >>>> What are you really trying to do?
>
> >>>> By definition, a useful all-pass filter has delay, at least at some
> >>>> frequencies.  It must, because a slope in the phase of a filter's
> >>>> transfer function in the frequency domain translates to a delay (or
> >>>> prediction) in the time domain.  So the best that you can do is to run
> >>>> your all-pass filter in just one direction, then shift the answer (which
> >>>> you can do because you're running off line) to get the least overall
> >>>> delay, or the least delay "where it matters".
>
> >>>> Note that this problem with allpass vs. phase delay is evinced a bit if
> >>>> you think about your question 2:  Work out the math, and you'll find
> >>>> that H_1 * H_1inv is the only no-delay allpass filter in existence: H = 1.
>
> >>>> --
>
> >>>> Tim Wescott
> >>>> Wescott Design Serviceshttp://www.wescottdesign.com
>
> >>>> Do you need to implement control loops in software?
> >>>> "Applied Control Theory for Embedded Systems" was written for you.
> >>>> See details athttp://www.wescottdesign.com/actfes/actfes.html
>
> >>> Tim,
>
> >>> Foremost thanks a lot for reply.
> >>> My application is in the area of microwave engineering and microwave
> >>> filter design. I am working on an algortihm that allows to include
> >>> some defficiences of the transmission lines of which filter is built
> >>> to be included in the design procedure and their influence
> >>> eliminated.
>
> >>> Anyway, I do not want zero delay but I want to retain only nonlinear
> >>> portion of the all-pass phase shift - z^-N is an artifact coming from
> >>> the fact that usually one is interested in the causal filter. Would my
> >>> concept as outlined in the question 1 retain this nonlinear phase
> >>> shift or I am misinterpreting something?
>
> >> What is A(z) in your definition above?  I ask, because the "control
> >> engineer's formulation" of an IIR filter would have
>
> >> A(z) = a_N * z^N + ... + a_1 * z + a_0,
>
> >> while the "DSP engineer's formulation" of an IIR filter would have
>
> >> A(z) = a_N / z^N + ... + a_1 / z + a_0.
>
> >> In the former case, A(1/z) would have a be perfectly causal, if delayed.
> >>    In the latter case A(1/z) would, indeed, have substantial lead that
> >> would have to be corrected with the z^-N term.  You could leave it off
> >> if you're not interested in causality.
>
> >> --
>
> >> Tim Wescott
> >> Wescott Design Serviceshttp://www.wescottdesign.com
>
> >> Do you need to implement control loops in software?
> >> "Applied Control Theory for Embedded Systems" was written for you.
> >> See details athttp://www.wescottdesign.com/actfes/actfes.html
>
> > I define A(z) as a_0+a_1z^-1+a_2z^-2+...+a_Nz^-N - which of those
> > above mentioned cases would it be - forgive me my stupid question?
>
> > Pawel
>
> That's the "DSP Engineer's formulation".
>
> --
>
> Tim Wescott
> Wescott Design Serviceshttp://www.wescottdesign.com
>
> Do you need to implement control loops in software?
> "Applied Control Theory for Embedded Systems" was written for you.
> See details athttp://www.wescottdesign.com/actfes/actfes.html

Ok,
Just to summarise - so my idea of filtering as presented in question 1
makes sense (?) and it is possible to implement such formulated filter
as noncausal (fir anticausal and iir causal) to eliminate z^-N or is
it better to filter with causal all-pass filter and later correct for
the delay by backshifting with z^N?

Pawel
From: Tim Wescott on 20 Jul 2010 20:12
On 07/20/2010 03:35 PM, Pawel wrote:
> On Jul 20, 11:25 pm, Tim Wescott<t...(a)seemywebsite.com> wrote:
>> On 07/20/2010 02:45 PM, Pawel wrote:
>>
>>
>>
>>> On Jul 20, 10:22 pm, Tim Wescott<t...(a)seemywebsite.com> wrote:
>>>> On 07/20/2010 01:50 PM, Pawel wrote:
>>
>>>>> On Jul 20, 9:30 pm, Tim Wescott<t...(a)seemywebsite.com> wrote:
>>>>>> On 07/20/2010 01:04 PM, Pawel wrote:
>>
>>>>>>> Dear All,
>>
>>>>>>> Maybe one of You Gurus would be able to shed to confirm (or criticise)
>>>>>>> my way of thinking. I am not DSP expert I am just trying to use DSP
>>>>>>> methods in other fields of science.
>>>>>>> My question relate to an all-pass filter structure: H(z)=z^-N[A(z^-1)/
>>>>>>> A(z)]. The problem that I am trying to solve do not easily tolerate N
>>>>>>> sample delay z^-N.
>>>>>>> 1. So I thought that it might be possible to express all-pass function
>>>>>>> as a cascade of anti-causal FIR filter A(z^-1) and causal IIR 1/A(z)
>>>>>>> effectively eliminating the excess delay z^-N that it is there to only
>>>>>>> make A(z^-1) causal. My application is offline so it can be
>>>>>>> implemented as a backward filtering?
>>>>>>> 2. Therefore my H_1=A(z^-1)/A(z), now how do I find inverse of it
>>>>>>> H_1inv=A(z)/A(z^-1). 1/A(z^-1) will be unstable since its poles will
>>>>>>> be inside the unit circle (A(z^-1) zeros where inside the circle),
>>>>>>> however I can resort again to the noncausal filtering and treat
>>>>>>> A(z^-1) as causal that would give me a stable filter?
>>>>>>> 3. How do I invert filter H_1?
>>>>>>> 4. What would be pitfalls when I would try to implement this in
>>>>>>> reality for offline processing � I know that there might be some
>>>>>>> problems with initial conditions in forward filtering?
>>
>>>>>> What are you really trying to do?
>>
>>>>>> By definition, a useful all-pass filter has delay, at least at some
>>>>>> frequencies. It must, because a slope in the phase of a filter's
>>>>>> transfer function in the frequency domain translates to a delay (or
>>>>>> prediction) in the time domain. So the best that you can do is to run
>>>>>> your all-pass filter in just one direction, then shift the answer (which
>>>>>> you can do because you're running off line) to get the least overall
>>>>>> delay, or the least delay "where it matters".
>>
>>>>>> Note that this problem with allpass vs. phase delay is evinced a bit if
>>>>>> you think about your question 2: Work out the math, and you'll find
>>>>>> that H_1 * H_1inv is the only no-delay allpass filter in existence: H = 1.
>>
>>>>>> --
>>
>>>>>> Tim Wescott
>>>>>> Wescott Design Serviceshttp://www.wescottdesign.com
>>
>>>>>> Do you need to implement control loops in software?
>>>>>> "Applied Control Theory for Embedded Systems" was written for you.
>>>>>> See details athttp://www.wescottdesign.com/actfes/actfes.html
>>
>>>>> Tim,
>>
>>>>> Foremost thanks a lot for reply.
>>>>> My application is in the area of microwave engineering and microwave
>>>>> filter design. I am working on an algortihm that allows to include
>>>>> some defficiences of the transmission lines of which filter is built
>>>>> to be included in the design procedure and their influence
>>>>> eliminated.
>>
>>>>> Anyway, I do not want zero delay but I want to retain only nonlinear
>>>>> portion of the all-pass phase shift - z^-N is an artifact coming from
>>>>> the fact that usually one is interested in the causal filter. Would my
>>>>> concept as outlined in the question 1 retain this nonlinear phase
>>>>> shift or I am misinterpreting something?
>>
>>>> What is A(z) in your definition above? I ask, because the "control
>>>> engineer's formulation" of an IIR filter would have
>>
>>>> A(z) = a_N * z^N + ... + a_1 * z + a_0,
>>
>>>> while the "DSP engineer's formulation" of an IIR filter would have
>>
>>>> A(z) = a_N / z^N + ... + a_1 / z + a_0.
>>
>>>> In the former case, A(1/z) would have a be perfectly causal, if delayed.
>>>> In the latter case A(1/z) would, indeed, have substantial lead that
>>>> would have to be corrected with the z^-N term. You could leave it off
>>>> if you're not interested in causality.
>>
>>>> --
>>
>>>> Tim Wescott
>>>> Wescott Design Serviceshttp://www.wescottdesign.com
>>
>>>> Do you need to implement control loops in software?
>>>> "Applied Control Theory for Embedded Systems" was written for you.
>>>> See details athttp://www.wescottdesign.com/actfes/actfes.html
>>
>>> I define A(z) as a_0+a_1z^-1+a_2z^-2+...+a_Nz^-N - which of those
>>> above mentioned cases would it be - forgive me my stupid question?
>>
>>> Pawel
>>
>> That's the "DSP Engineer's formulation".
>>
>> --
>>
>> Tim Wescott
>> Wescott Design Serviceshttp://www.wescottdesign.com
>>
>> Do you need to implement control loops in software?
>> "Applied Control Theory for Embedded Systems" was written for you.
>> See details athttp://www.wescottdesign.com/actfes/actfes.html
>
> Ok,
> Just to summarise - so my idea of filtering as presented in question 1
> makes sense (?) and it is possible to implement such formulated filter
> as noncausal (fir anticausal and iir causal) to eliminate z^-N or is
> it better to filter with causal all-pass filter and later correct for
> the delay by backshifting with z^N?

I think it'll prove easier to implement the all-pass as a causal filter
-- basically you'll be able to do the usual direct-form (whatever) as a
causal filter, then after the fact you can backshift the data by
whatever amount seems optimal.

--

Tim Wescott
Wescott Design Services
http://www.wescottdesign.com

Do you need to implement control loops in software?
"Applied Control Theory for Embedded Systems" was written for you.
See details at http://www.wescottdesign.com/actfes/actfes.html
From: HardySpicer on 21 Jul 2010 02:30
On Jul 21, 8:04 am, Pawel <prulikow...(a)gmail.com> wrote:
> Dear All,
>
> Maybe one of You Gurus would be able to shed to confirm (or criticise)
> my way of thinking. I am not DSP expert I am just trying to use DSP
> methods in other fields of science.
> My question relate to an all-pass filter structure: H(z)=z^-N[A(z^-1)/
> A(z)]. The problem that I am trying to solve do not easily tolerate N
> sample delay z^-N.
> 1.      So I thought that it might be possible to express all-pass function
> as a cascade of anti-causal FIR filter A(z^-1) and causal IIR 1/A(z)
> effectively eliminating the excess delay z^-N that it is there to only
> make A(z^-1) causal. My application is offline so it can be
> implemented as a backward filtering?
> 2.      Therefore my H_1=A(z^-1)/A(z), now how do I find inverse of it
> H_1inv=A(z)/A(z^-1).  1/A(z^-1) will be unstable since its poles will
> be inside the unit circle (A(z^-1) zeros where inside the circle),
> however I can resort again to the noncausal filtering and treat
> A(z^-1) as causal that would give me a stable filter?
> 3.      How do I invert filter H_1?
> 4.      What would be pitfalls when I would try to implement this in
> reality for offline processing – I know that there might be some
> problems with initial conditions in forward filtering?
>
> Regards
>
> Pawel

The answer is yes you can. This is not a new idea and has been used in
a few papers for more complex systems.
eg here

http://www.springerlink.com/content/617jaakplgmnx3a7/


and many similar papers..


Hardy
From: Pawel on 22 Jul 2010 07:11
On Jul 21, 7:30 am, HardySpicer <gyansor...(a)gmail.com> wrote:
> On Jul 21, 8:04 am, Pawel <prulikow...(a)gmail.com> wrote:
>
>
>
> > Dear All,
>
> > Maybe one of You Gurus would be able to shed to confirm (or criticise)
> > my way of thinking. I am not DSP expert I am just trying to use DSP
> > methods in other fields of science.
> > My question relate to an all-pass filter structure: H(z)=z^-N[A(z^-1)/
> > A(z)]. The problem that I am trying to solve do not easily tolerate N
> > sample delay z^-N.
> > 1.      So I thought that it might be possible to express all-pass function
> > as a cascade of anti-causal FIR filter A(z^-1) and causal IIR 1/A(z)
> > effectively eliminating the excess delay z^-N that it is there to only
> > make A(z^-1) causal. My application is offline so it can be
> > implemented as a backward filtering?
> > 2.      Therefore my H_1=A(z^-1)/A(z), now how do I find inverse of it
> > H_1inv=A(z)/A(z^-1).  1/A(z^-1) will be unstable since its poles will
> > be inside the unit circle (A(z^-1) zeros where inside the circle),
> > however I can resort again to the noncausal filtering and treat
> > A(z^-1) as causal that would give me a stable filter?
> > 3.      How do I invert filter H_1?
> > 4.      What would be pitfalls when I would try to implement this in
> > reality for offline processing – I know that there might be some
> > problems with initial conditions in forward filtering?
>
> > Regards
>
> > Pawel
>
> The answer is yes you can. This is not a new idea and has been used in
> a few papers for more complex systems.
> eg here
>
> http://www.springerlink.com/content/617jaakplgmnx3a7/
>
> and many similar papers..
>
> Hardy

Thank You very much for all Your replies.

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