FPGA Central

Hi All,

I have a following question. First I am not DSP specialist and I have
only general knowledge of DSP, filter design etc. I want to compensate
linear, stable, causal system, so basically find inverse filter that
will compensate this system. I know system frequency response -
magnitude and phase. What else I know is that system can be decomposed
into minimum phase system and all-pass network. As far my limited
knowledge of DSP tells me it is no big deal to compensate minimum-
phase system but what about all-pass part. I know that there are
methods that compensate system group delay on the basis of system
response symmetry (as linear phase system or with a flat group delay
has symmetrical impulse response).
So my idea was to compensate in two steps:
1) find minimum phase system transfer function (from magnitude through
Hilbert transform) in z domain and calculate its inverse,
2) equalize all-pass system with some method i.e. cascading all-pass
sections.
Is this approach correct or is it better to do all in one go
(compensate all-pass and minimum phase with one filter).
What methods would You recommend or what would be a "standard"
approach to such problem.

Kind regards

Pawel

On Sun, 07 Dec 2008 11:04:29 -0800, Pawel wrote:

> Hi All,
>
> I have a following question. First I am not DSP specialist and I have
> only general knowledge of DSP, filter design etc. I want to compensate
> linear, stable, causal system, so basically find inverse filter that
> will compensate this system. I know system frequency response -
> magnitude and phase. What else I know is that system can be decomposed
> into minimum phase system and all-pass network. As far my limited
> knowledge of DSP tells me it is no big deal to compensate minimum- phase
> system but what about all-pass part. I know that there are methods that
> compensate system group delay on the basis of system response symmetry
> (as linear phase system or with a flat group delay has symmetrical
> impulse response).
> So my idea was to compensate in two steps: 1) find minimum phase system
> transfer function (from magnitude through Hilbert transform) in z domain
> and calculate its inverse, 2) equalize all-pass system with some method
> i.e. cascading all-pass sections.
> Is this approach correct or is it better to do all in one go (compensate
> all-pass and minimum phase with one filter). What methods would You
> recommend or what would be a "standard" approach to such problem.
>
> Kind regards
>
> Pawel

You never mentioned what you're compensating _for_, although I gather
that you want to make a composite system with a transfer function as
close to H(z) = 1 as possible.

Is that correct?

If you can stand a system whose transfer function is unity + delay (i.e.
if you can allow H(z) = z^-n), then you'll have a much better chance at
achieving your goals. If you can allow a system that has a poor fit at
higher (and perhaps very low) frequencies then you can do even better yet.

So what exactly are you trying to do?

--
Tim Wescott
Control systems and communications consulting
http://www.wescottdesign.com

Need to learn how to apply control theory in your embedded system?
"Applied Control Theory for Embedded Systems" by Tim Wescott
Elsevier/Newnes, http://www.wescottdesign.com/actfes/actfes.html

On Dec 7, 7:56*pm, Tim Wescott <t...@seemywebsite.com> wrote:
> On Sun, 07 Dec 2008 11:04:29 -0800, Pawel wrote:
> > Hi All,
>
> > I have a following question. First I am not DSP specialist and I have
> > only general knowledge of DSP, filter design etc. I want to compensate
> > linear, stable, causal system, so basically find inverse filter that
> > will compensate this system. I know system frequency response -
> > magnitude and phase. What else I know is that system can be decomposed
> > into minimum phase system and all-pass network. As far my limited
> > knowledge of DSP tells me it is no big deal to compensate minimum- phase
> > system but what about all-pass part. I know that there are methods that
> > compensate system group delay on the basis of system response symmetry
> > (as linear phase system or with a flat group delay has symmetrical
> > impulse response).
> > So my idea was to compensate in two steps: 1) find minimum phase system
> > transfer function (from magnitude through Hilbert transform) in z domain
> > and calculate its inverse, 2) equalize all-pass system with some method
> > i.e. cascading all-pass sections.
> > Is this approach correct or is it better to do all in one go (compensate
> > all-pass and minimum phase with one filter). What methods would You
> > recommend or what would be a "standard" approach to such problem.
>
> > Kind regards
>
> > Pawel
>
> You never mentioned what you're compensating _for_, although I gather
> that you want to make a composite system with a transfer function as
> close to H(z) = 1 as possible.
>
> Is that correct?
>
> If you can stand a system whose transfer function is unity + delay (i.e.
> if you can allow H(z) = z^-n), then you'll have a much better chance at
> achieving your goals. *If you can allow a system that has a poor fit at
> higher (and perhaps very low) frequencies then you can do even better yet..
>
> So what exactly are you trying to do?
>
> --
> Tim Wescott
> Control systems and communications consultinghttp://www.wescottdesign.com
>
> Need to learn how to apply control theory in your embedded system?
> "Applied Control Theory for Embedded Systems" by Tim Wescott
> Elsevier/Newnes,http://www.wescottdesign.com/actfes/actfes.html

Tim,

You never mentioned what you're compensating _for_, although I gather
> that you want to make a composite system with a transfer function as
> close to H(z) = 1 as possible.
>
> Is that correct?

Yes, You are right. The system that I want to compensate is lossy,
dispersive microwave transmission line.

> If you can stand a system whose transfer function is unity + delay (i.e.
> if you can allow H(z) = z^-n), then you'll have a much better chance at
> achieving your goals. If you can allow a system that has a poor fit at
> higher (and perhaps very low) frequencies then you can do even better yet..

I do not mind some extra delay as I believe it is rather inevitable if
I want to compensate group delay.
The final output of my "modeling" should be impulse response of such
derived equalizer.

This is serious stuff - I am wrapping up my PhD in area of microwave
engineering and high speed pulse shaping of pulses below ns time
duration.

Regards

Pawel

Pawel wrote:
> Hi All,
>
> I have a following question. First I am not DSP specialist and I have
> only general knowledge of DSP, filter design etc. I want to compensate
> linear, stable, causal system, so basically find inverse filter that
> will compensate this system. I know system frequency response -
> magnitude and phase. What else I know is that system can be decomposed
> into minimum phase system and all-pass network. As far my limited
> knowledge of DSP tells me it is no big deal to compensate minimum-
> phase system but what about all-pass part.

First, you should limit the compensation to some reasonable frequency
range, unless you are willing to run into the impractical things like
the infinite gain, Q or delay.


> I know that there are
> methods that compensate system group delay on the basis of system
> response symmetry (as linear phase system or with a flat group delay
> has symmetrical impulse response).
> So my idea was to compensate in two steps:
> 1) find minimum phase system transfer function (from magnitude through
> Hilbert transform) in z domain and calculate its inverse,
> 2) equalize all-pass system with some method i.e. cascading all-pass
> sections.
> Is this approach correct or is it better to do all in one go
> (compensate all-pass and minimum phase with one filter).

The approach is correct however doing it all at once may lead to the
smaller solution.

> What methods would You recommend or what would be a "standard"
> approach to such problem.

It depends. What response is been optimized, what is the goal, what
hardware is available.

Vladimir Vassilevsky
DSP and Mixed Signal Design Consultant
http://www.abvolt.com

On Sun, 07 Dec 2008 12:08:43 -0800, Pawel wrote:

> On Dec 7, 7:56Â*pm, Tim Wescott <t...@seemywebsite.com> wrote:
>> On Sun, 07 Dec 2008 11:04:29 -0800, Pawel wrote:
>> > Hi All,
>>
>> > I have a following question. First I am not DSP specialist and I have
>> > only general knowledge of DSP, filter design etc. I want to
>> > compensate linear, stable, causal system, so basically find inverse
>> > filter that will compensate this system. I know system frequency
>> > response - magnitude and phase. What else I know is that system can
>> > be decomposed into minimum phase system and all-pass network. As far
>> > my limited knowledge of DSP tells me it is no big deal to compensate
>> > minimum- phase system but what about all-pass part. I know that there
>> > are methods that compensate system group delay on the basis of system
>> > response symmetry (as linear phase system or with a flat group delay
>> > has symmetrical impulse response).
>> > So my idea was to compensate in two steps: 1) find minimum phase
>> > system transfer function (from magnitude through Hilbert transform)
>> > in z domain and calculate its inverse, 2) equalize all-pass system
>> > with some method i.e. cascading all-pass sections.
>> > Is this approach correct or is it better to do all in one go
>> > (compensate all-pass and minimum phase with one filter). What methods
>> > would You recommend or what would be a "standard" approach to such
>> > problem.
>>
>> > Kind regards
>>
>> > Pawel
>>
>> You never mentioned what you're compensating _for_, although I gather
>> that you want to make a composite system with a transfer function as
>> close to H(z) = 1 as possible.
>>
>> Is that correct?
>>
>> If you can stand a system whose transfer function is unity + delay
>> (i.e. if you can allow H(z) = z^-n), then you'll have a much better
>> chance at achieving your goals. Â*If you can allow a system that has a
>> poor fit at higher (and perhaps very low) frequencies then you can do
>> even better yet.
>>
>> So what exactly are you trying to do?
>>
>> --
>> Tim Wescott
>> Control systems and communications
>> consultinghttp://www.wescottdesign.com
>>
>> Need to learn how to apply control theory in your embedded system?
>> "Applied Control Theory for Embedded Systems" by Tim Wescott
>> Elsevier/Newnes,http://www.wescottdesign.com/actfes/actfes.html
>
> Tim,
>
> You never mentioned what you're compensating _for_, although I gather
>> that you want to make a composite system with a transfer function as
>> close to H(z) = 1 as possible.
>>
>> Is that correct?
>
> Yes, You are right. The system that I want to compensate is lossy,
> dispersive microwave transmission line.
>
>> If you can stand a system whose transfer function is unity + delay
>> (i.e. if you can allow H(z) = z^-n), then you'll have a much better
>> chance at achieving your goals. If you can allow a system that has a
>> poor fit at higher (and perhaps very low) frequencies then you can do
>> even better yet.
>
> I do not mind some extra delay as I believe it is rather inevitable if I
> want to compensate group delay.
> The final output of my "modeling" should be impulse response of such
> derived equalizer.
>
> This is serious stuff - I am wrapping up my PhD in area of microwave
> engineering and high speed pulse shaping of pulses below ns time
> duration.

You make a pair of assertions that I'm not sure are mutually true.

On the one hand, you assert that the system can be broken down into a
minimum phase part and an all-pass network. On the other hand you say
that you're working with transmission lines.

Are you saying, then, that the all-pass network is not necessarily lumped-
constant? I certainly don't believe that you can reduce a continuous-
state system such as a transmission line down to any system with a finite
number of states (although you may be able to approximate it as such).

You can do a not-too-bad job of approximating a system with a mix of
stable and unstable poles with a system that's a mix of an FIR part and
an IIR part. The overall response is delayed, and you do it in such a
way that the FIR part contains all the unstable poles' responses in an
exponentially climbing impulse response, followed by the IIR part that
contains all of the stable poles. Just how you distribute the zeros (and
discard some of the poles' responses) has a big impact on how easy it is
to realize the thing in reality; I'm going to leave that as an exercise
to the reader.

Note that if you have a decent way of implementing FIR filters it may be
best to just find the impulse response of your desired inverse system and
approximating it with a FIR system -- you may find that this is more
reliable than trying to mess around with IIR filters.

Are you intending to do this pulse reconstruction in real time? Are you
constrained to doing it digitally? It would seem that with seven decades
of RADAR behind us there would be a lot of prior art in this area -- I
distinctly recall reading of techniques for doing FIR filtering in real
time using both SAW devices and transmission lines back when I was
working on my Master's degree.

--
Tim Wescott
Control systems and communications consulting
http://www.wescottdesign.com

Need to learn how to apply control theory in your embedded system?
"Applied Control Theory for Embedded Systems" by Tim Wescott
Elsevier/Newnes, http://www.wescottdesign.com/actfes/actfes.html

On Dec 7, 9:17*pm, Tim Wescott <t...@seemywebsite.com> wrote:
> On Sun, 07 Dec 2008 12:08:43 -0800, Pawel wrote:
> > On Dec 7, 7:56*pm, Tim Wescott <t...@seemywebsite.com> wrote:
> >> On Sun, 07 Dec 2008 11:04:29 -0800, Pawel wrote:
> >> > Hi All,
>
> >> > I have a following question. First I am not DSP specialist and I have
> >> > only general knowledge of DSP, filter design etc. I want to
> >> > compensate linear, stable, causal system, so basically find inverse
> >> > filter that will compensate this system. I know system frequency
> >> > response - magnitude and phase. What else I know is that system can
> >> > be decomposed into minimum phase system and all-pass network. As far
> >> > my limited knowledge of DSP tells me it is no big deal to compensate
> >> > minimum- phase system but what about all-pass part. I know that there
> >> > are methods that compensate system group delay on the basis of system
> >> > response symmetry (as linear phase system or with a flat group delay
> >> > has symmetrical impulse response).
> >> > So my idea was to compensate in two steps: 1) find minimum phase
> >> > system transfer function (from magnitude through Hilbert transform)
> >> > in z domain and calculate its inverse, 2) equalize all-pass system
> >> > with some method i.e. cascading all-pass sections.
> >> > Is this approach correct or is it better to do all in one go
> >> > (compensate all-pass and minimum phase with one filter). What methods
> >> > would You recommend or what would be a "standard" approach to such
> >> > problem.
>
> >> > Kind regards
>
> >> > Pawel
>
> >> You never mentioned what you're compensating _for_, although I gather
> >> that you want to make a composite system with a transfer function as
> >> close to H(z) = 1 as possible.
>
> >> Is that correct?
>
> >> If you can stand a system whose transfer function is unity + delay
> >> (i.e. if you can allow H(z) = z^-n), then you'll have a much better
> >> chance at achieving your goals. *If you can allow a system that has a
> >> poor fit at higher (and perhaps very low) frequencies then you can do
> >> even better yet.
>
> >> So what exactly are you trying to do?
>
> >> --
> >> Tim Wescott
> >> Control systems and communications
> >> consultinghttp://www.wescottdesign.com
>
> >> Need to learn how to apply control theory in your embedded system?
> >> "Applied Control Theory for Embedded Systems" by Tim Wescott
> >> Elsevier/Newnes,http://www.wescottdesign.com/actfes/actfes.html
>
> > Tim,
>
> > You never mentioned what you're compensating _for_, although I gather
> >> that you want to make a composite system with a transfer function as
> >> close to H(z) = 1 as possible.
>
> >> Is that correct?
>
> > Yes, You are right. The system that I want to compensate is lossy,
> > dispersive microwave transmission line.
>
> >> If you can stand a system whose transfer function is unity + delay
> >> (i.e. if you can allow H(z) = z^-n), then you'll have a much better
> >> chance at achieving your goals. *If you can allow a system that has a
> >> poor fit at higher (and perhaps very low) frequencies then you can do
> >> even better yet.
>
> > I do not mind some extra delay as I believe it is rather inevitable if I
> > want to compensate group delay.
> > The final output of my "modeling" should be impulse response of such
> > derived equalizer.
>
> > This is serious stuff - I am wrapping up my PhD in area of microwave
> > engineering and high speed pulse shaping of pulses below ns time
> > duration.
>
> You make a pair of assertions that I'm not sure are mutually true.
>
> On the one hand, you assert that the system can be broken down into a
> minimum phase part and an all-pass network. *On the other hand you say
> that you're working with transmission lines.
>
> Are you saying, then, that the all-pass network is not necessarily lumped-
> constant? *I certainly don't believe that you can reduce a continuous-
> state system such as a transmission line down to any system with a finite
> number of states (although you may be able to approximate it as such).
>
> You can do a not-too-bad job of approximating a system with a mix of
> stable and unstable poles with a system that's a mix of an FIR part and
> an IIR part. *The overall response is delayed, and you do it in such a
> way that the FIR part contains all the unstable poles' responses in an
> exponentially climbing impulse response, followed by the IIR part that
> contains all of the stable poles. *Just how you distribute the zeros (and
> discard some of the poles' responses) has a big impact on how easy it is
> to realize the thing in reality; I'm going to leave that as an exercise
> to the reader.
>
> Note that if you have a decent way of implementing FIR filters it may be
> best to just find the impulse response of your desired inverse system and
> approximating it with a FIR system -- you may find that this is more
> reliable than trying to mess around with IIR filters.
>
> Are you intending to do this pulse reconstruction in real time? *Are you
> constrained to doing it digitally? *It would seem that with seven decades
> of RADAR behind us there would be a lot of prior art in this area -- I
> distinctly recall reading of techniques for doing FIR filtering in real
> time using both SAW devices and transmission lines back when I was
> working on my Master's degree.
>
> --
> Tim Wescott
> Control systems and communications consultinghttp://www.wescottdesign.com
>
> Need to learn how to apply control theory in your embedded system?
> "Applied Control Theory for Embedded Systems" by Tim Wescott
> Elsevier/Newnes,http://www.wescottdesign.com/actfes/actfes.html

Tim,

> You make a pair of assertions that I'm not sure are mutually true.
>
> On the one hand, you assert that the system can be broken down into a
> minimum phase part and an all-pass network. On the other hand you say
> that you're working with transmission lines.

Transmission line transfer function can be broken into minimum-phase
all-pass network as any other stable LTI system.
Minimum phase can be derived from magnitude of the transfer function.
Minimum phase comes from resistive and dielectric losses in the line,
that via Kramers-Kronig relations (or Hilbert transform) influence
phase of transfer function.
All-pass behaviour is more complicated - first propagation delay
through the line is frequency dependent. I am working with a
dispersive lines i.e. microstrip line where dispersion (propagation
delay that is presented to the signal at different frequencies) is
quite significant in the broad range of frequencies that I am
interested in (pulse excitation) - ranging from DC up to 20-30GHz or
beyond that. In principle delay increases with frequency, so I have
positive group delay slope, reaching some nominal value at very high
frequencies. I thought that I will equalize my delay to some
prescribed point i.e. maximum delay in band of interest.

> Are you intending to do this pulse reconstruction in real time? Are you
> constrained to doing it digitally? It would seem that with seven decades
> of RADAR behind us there would be a lot of prior art in this area -- I
> distinctly recall reading of techniques for doing FIR filtering in real
> time using both SAW devices and transmission lines back when I was
> working on my Master's degree.

No, real time processing is not required. Yes you are right there is
quite a lot analog FIR solutions with SAW filters. I am trying to
implement something like that but with transmission line. In fact this
modelling and equalization of the line is a part of a design algortihm
that works offline. All is calcualted on a PC in software (Matlab).

Pawel

On Dec 7, 8:39*pm, Vladimir Vassilevsky <antispam_bo...@hotmail.com>
wrote:
> Pawel wrote:
> > Hi All,
>
> > I have a following question. First I am not DSP specialist and I have
> > only general knowledge of DSP, filter design etc. I want to compensate
> > linear, stable, causal system, so basically find inverse filter that
> > will compensate this system. *I know system frequency response -
> > magnitude and phase. What else I know is that system can be decomposed
> > into minimum phase system and all-pass network. As far my limited
> > knowledge of DSP tells me it is no big deal to compensate minimum-
> > phase system but what about all-pass part.
>
> First, you should limit the compensation to some reasonable frequency
> range, unless you are willing to run into the impractical things like
> the infinite gain, Q or delay.
>
> > I know that there are
> > methods that compensate system group delay on the basis of system
> > response symmetry (as linear phase system or with a flat group delay
> > has symmetrical impulse response).
> > So my idea was to compensate in two steps:
> > 1) find minimum phase system transfer function (from magnitude through
> > Hilbert transform) in z domain and calculate its inverse,
> > 2) equalize all-pass system with some method i.e. cascading all-pass
> > sections.
> > Is this approach correct or is it better to do all in one go
> > (compensate all-pass and minimum phase with one filter).
>
> The approach is correct however doing it all at once may lead to the
> smaller solution.
>
> > What methods would You recommend or what would be a "standard"
> > approach to such problem.
>
> It depends. What response is been optimized, what is the goal, what
> hardware is available.
>
> Vladimir Vassilevsky
> DSP and Mixed Signal Design Consultanthttp://www.abvolt.com

Vladimir,

Please see my response to Tim.

Pawel

On Dec 8, 10:56*pm, Pawel <prulikow...@gmail.com> wrote:
> On Dec 7, 8:39*pm, Vladimir Vassilevsky <antispam_bo...@hotmail.com>
> wrote:
>
>
>
> > Pawel wrote:
> > > Hi All,
>
> > > I have a following question. First I am not DSP specialist and I have
> > > only general knowledge of DSP, filter design etc. I want to compensate
> > > linear, stable, causal system, so basically find inverse filter that
> > > will compensate this system. *I know system frequency response -
> > > magnitude and phase. What else I know is that system can be decomposed
> > > into minimum phase system and all-pass network. As far my limited
> > > knowledge of DSP tells me it is no big deal to compensate minimum-
> > > phase system but what about all-pass part.
>
> > First, you should limit the compensation to some reasonable frequency
> > range, unless you are willing to run into the impractical things like
> > the infinite gain, Q or delay.
>
> > > I know that there are
> > > methods that compensate system group delay on the basis of system
> > > response symmetry (as linear phase system or with a flat group delay
> > > has symmetrical impulse response).
> > > So my idea was to compensate in two steps:
> > > 1) find minimum phase system transfer function (from magnitude through
> > > Hilbert transform) in z domain and calculate its inverse,
> > > 2) equalize all-pass system with some method i.e. cascading all-pass
> > > sections.
> > > Is this approach correct or is it better to do all in one go
> > > (compensate all-pass and minimum phase with one filter).
>
> > The approach is correct however doing it all at once may lead to the
> > smaller solution.
>
> > > What methods would You recommend or what would be a "standard"
> > > approach to such problem.
>
> > It depends. What response is been optimized, what is the goal, what
> > hardware is available.
>
> > Vladimir Vassilevsky
> > DSP and Mixed Signal Design Consultanthttp://www.abvolt.com
>
> Vladimir,
>
> Please see my response to Tim.
>
> Pawel

You need a deconvolution filter or equalizer. Lots of papers on these
about. Watch out for non-min phase systems.
You need to consider additive noise as well eg if the channel was an
integrator then the filter would be a pure differentiator - not a good
idea noise-wise.