Data-path accuracy in IIR filters?

[Steve Pope]

Rune Allnor <allnor@tele.ntnu.no> wrote:

On 31 Jul, 22:49, spop...@speedymail.org (Steve Pope) wrote:

Other topologies have similar regions of instabilities for
their coefficient; but they are not stated as simply.

Wrong. The IIRs are stable subject to poles staying
strictly inside the unit circle. Zeros might be everywhere,
no restrictions there.

The same is true for a lattice topology,

The please prove this statement mathematically.

Personally I am satisfied with this well-known fact from filter theory,
I feel that the literature is strong enough, and I do not feel on the
hook to come up with a proof.

Up to this point
you have been very persistent in restricting the reflection
coefficients to the range [-1,1]. Could you pelase elaborate
on what happens if the reflection coefficients stray outside
that range?

Internal states may saturate and stay there. Typically.

A lattice implementation fuses the IIR and the FIR into a
common structure. That's why it is used in the AR-type
perdictors: You get *both* the perdicted signal, as computed
by the FIR AR predictor *and* the prediction error (as computed
by the IIR predictor inverse) for a minimum ofcomputations.

One constraint for this to work is that the IIR is stable.

2) Depends on zero locations

Again, you've lost me.  Your statements 1) and 2) are not true,
so far as I know.

"As far as you know." Check it out.

You haven't supported these statements. If there is a lattice
topology whose stability depends upon the zero locations, please
provide a cite for it. (I'm sure such a think might exist.
But it is not a mainstream topology I would think.)

Again, I don't have my books easily available, so with the caveat
that
I'm writing off years-old memories:
The FIR and IIR parts are tightly coupled in the lattice structure.

Please look at the figure on page 11-28 of this document:

http://www.busim.ee.boun.edu.tr/~resources/fdq.pdf

The zero location are controlled by the coefficients v1, v2....
These coefficients do not make the filter unstable.

So why isn't it mentioned in every textbook out there?
Why bother with DF I and II if the lattice works so well?

It is covered in a fair fraction of textbooks. When I was in
grad school, this was standardly taught to all students who took
DSP courses that covered filter design. And, while Mathworks
is not a gold standard or anything, what I regard as the three
most useful lattice topologies, as well as three less useful one,
are among the only sixteen "filter structure" properties they
have defined. That seems a fairly significant representation
-- one third of the filter toplogies they deigned to include
in their suite are lattice filters. That is a fair indicator they
are widely used.


Steve

 

[Steve Pope]

Rune Allnor <allnor@tele.ntnu.no> wrote:

On 1 Aug, 05:10, spop...@speedymail.org (Steve Pope) wrote:

"latticema" -- all-zero filter
"latticear" -- all-pole filter
"latticearma" -- filter with both poles and zeros

Steve

Sorry - I got you wrong from the start. I had you down as knowing
your DSP. This reveals your true guise as a mere matlab user.

Sigh. You're grasping at straws.

The above is a good short reference on these topologies,
useful to any designer, Mathworks-using or otherwise, which
is why I posted the link.

(I probably employ Matlab in fewer than 5% of the project
I work on, and it has never been by my decision...)

Steve

 

[Rune Allnor]

On 1 Aug, 09:53, spop...@speedymail.org (Steve Pope) wrote:

Rune Allnor  <all...@tele.ntnu.no> wrote:

On 1 Aug, 05:10, spop...@speedymail.org (Steve Pope) wrote:
"latticema" -- all-zero filter
"latticear" -- all-pole filter
"latticearma" -- filter with both poles and zeros
Steve
Sorry - I got you wrong from the start. I had you down as knowing
your DSP. This reveals your true guise as a mere matlab user.

Sigh.  You're grasping at straws.

I'm not. Matlab has a very poor repurtation as academic
reference. They used to estimate the length / duration of
the impulse response of IIR filters as the number of
numerator coefficients in the transfer functions (which
works for FIR filters). Their IIR filter design procedures
were screwed up to the point where one needed to 'unteach'
students the matlab blunders before one could teach how
things really were doing. Iterative methods like the
Levinson recursion required users to supply 'true' AR
orders up front, as opposed to supplying a *maximum* order
and then use some order estimator within those constraints.

At least that was the status when I last used the SP
toolbox around 2004, and had been the status in the 15 years
prior to that time. I know the SP toolbox has been reworked
since then, but I doubdt that more than two decades worth
of flaws, blunders and mistakes are corrected in a hurry.

Rune

 

[Rune Allnor]

On 1 Aug, 09:50, spop...@speedymail.org (Steve Pope) wrote:

Rune Allnor  <all...@tele.ntnu.no> wrote:

On 31 Jul, 22:49, spop...@speedymail.org (Steve Pope) wrote:
Other topologies have similar regions of instabilities for
their coefficient; but they are not stated as simply.
Wrong. The IIRs are stable subject to poles staying
strictly inside the unit circle. Zeros might be everywhere,
no restrictions there.
The same is true for a lattice topology,
The please prove this statement mathematically.

Personally I am satisfied with this well-known fact from filter theory,
I feel that the literature is strong enough, and I do not feel on the
hook to come up with a proof.

Again, I obviously had you wrong. This is a bout maths and
engineering,
not emotions. If you don't 'feel' up to substantiating your position,
don't challenge the points made.

2) Depends on zero locations

Again, you've lost me.  Your statements 1) and 2) are not true,
so far as I know.
"As far as you know." Check it out.

You haven't supported these statements.  If there is a lattice
topology whose stability depends upon the zero locations, please
provide a cite for it.  (I'm sure such a think might exist.
But it is not a mainstream topology I would think.)

This is well-known from statistichal DSP. I can't come up
with specifc citations, as my library is is storage and
will remain there for a few weeks to come, but I will
tell you what to look for. I know this is treated in the
Proakis & Manolakis general DSP text:

When dealing with AR models, one can solve the Yule-Walker
equations (or rather, and estimator for these equations)
in any number of ways. Direct solutions through linear algebra
will give you the straight-forward FIR prediction filter; the
Levinson recursion will also give you the reflection coefficients
that go directly into the lattice representation.

As one would expect, there exist conversion formulae between
the FIR and lattice representations for the AR model: Insert a
set of FIR coefficients and crank out a set of lattice coefficients.
And vice versa.

The problem is the implicit constraints. In the AR application
the FIR filter is guaranteed to be minimum phase, so its IIR
inverse is causal stable. This translates directly to the
lattice reflection coefficients being constrained to the
interval <-1,1>.

However, the unsuspecting incompetent user who stumbles across
these conversion formulae and tries to convert a linear phase
FIR to lattice form - don't ask *why* one would want to do this;
I assume the user to be *incompetent* - would end up with a
numerically unstable FIR. Which is a contradiction in terms,
given the entry-level indoctrination matra of DSP.

Which will only come back to haunt the designer, who exposed
the incompetent user to the lattice structure in the first place.

Rune

 

[robert bristow-johnson]

On Aug 1, 5:53 am, Rune Allnor <all...@tele.ntnu.no> wrote:

On 1 Aug, 09:53, spop...@speedymail.org (Steve Pope) wrote:

Rune Allnor  <all...@tele.ntnu.no> wrote:

On 1 Aug, 05:10, spop...@speedymail.org (Steve Pope) wrote:
"latticema" -- all-zero filter
"latticear" -- all-pole filter
"latticearma" -- filter with both poles and zeros
Steve
Sorry - I got you wrong from the start. I had you down as knowing
your DSP. This reveals your true guise as a mere matlab user.

Sigh.  You're grasping at straws.

I'm not. Matlab has a very poor repurtation as academic
reference. They used to estimate the length / duration of
the impulse response of IIR filters as the number of
numerator coefficients in the transfer functions (which
works for FIR filters). Their IIR filter design procedures
were screwed up to the point where one needed to 'unteach'
students the matlab blunders before one could teach how
things really were doing. Iterative methods like the
Levinson recursion required users to supply 'true' AR
orders up front, as opposed to supplying a *maximum* order
and then use some order estimator within those constraints.

At least that was the status when I last used the SP
toolbox around 2004, and had been the status in the 15 years
prior to that time. I know the SP toolbox has been reworked
since then, but I doubdt that more than two decades worth
of flaws, blunders and mistakes are corrected in a hurry.

they could begin to fix the blunder of putting the DC component of the
fft into X(1).

r b-j

 

[Steve Pope]

Rune Allnor <allnor@tele.ntnu.no> wrote:

On 1 Aug, 09:50, spop...@speedymail.org (Steve Pope) wrote:

Personally I am satisfied with this well-known fact from filter theory,
I feel that the literature is strong enough, and I do not feel on the
hook to come up with a proof.

Again, I obviously had you wrong. This is a bout maths and
engineering,
not emotions. If you don't 'feel' up to substantiating your position,
don't challenge the points made.

Sorry, Rune, but it is you who has provided zero backup for your
unsupported negative statements about the lattice filter topology.
All I did was tell the OP it would be useful in his case. You
haven't come close to explaining to us what, if any, problems there
might be with it.

You haven't supported these statements.  If there is a lattice
topology whose stability depends upon the zero locations, please
provide a cite for it.  (I'm sure such a think might exist.
But it is not a mainstream topology I would think.)

This is well-known from statistichal DSP. I can't come up
with specifc citations, as my library is is storage and
will remain there for a few weeks to come,

Ha!

but I will
tell you what to look for. I know this is treated in the
Proakis & Manolakis general DSP text:

When dealing with AR models, one can solve the Yule-Walker
equations (or rather, and estimator for these equations)
in any number of ways. Direct solutions through linear algebra
will give you the straight-forward FIR prediction filter; the
Levinson recursion will also give you the reflection coefficients
that go directly into the lattice representation.

As one would expect, there exist conversion formulae between
the FIR and lattice representations for the AR model: Insert a
set of FIR coefficients and crank out a set of lattice coefficients.
And vice versa.

The problem is the implicit constraints. In the AR application
the FIR filter is guaranteed to be minimum phase, so its IIR
inverse is causal stable. This translates directly to the
lattice reflection coefficients being constrained to the
interval <-1,1>.

However, the unsuspecting incompetent user who stumbles across
these conversion formulae and tries to convert a linear phase
FIR to lattice form - don't ask *why* one would want to do this;
I assume the user to be *incompetent* - would end up with a
numerically unstable FIR. Which is a contradiction in terms,
given the entry-level indoctrination matra of DSP.

Which will only come back to haunt the designer, who exposed
the incompetent user to the lattice structure in the first place.

Good, finally some information.

The OP was designing a sixth-order Butterworth, not a linear
phase FIR, but no matter.

Steve

 

[Rune Allnor]

On 1 Aug, 20:42, spop...@speedymail.org (Steve Pope) wrote:

Rune Allnor  <all...@tele.ntnu.no> wrote:

On 1 Aug, 09:50, spop...@speedymail.org (Steve Pope) wrote:
Personally I am satisfied with this well-known fact from filter theory,
I feel that the literature is strong enough, and I do not feel on the
hook to come up with a proof.
Again, I obviously had you wrong. This is a bout maths and
engineering,
not emotions. If you don't 'feel' up to substantiating your position,
don't challenge the points made.

Sorry, Rune, but it is you who has provided zero backup for your
unsupported negative statements about the lattice filter topology.  
All I did was tell the OP it would be useful in his case.  You
haven't come close to explaining to us what, if any, problems there
might be with it.

You haven't supported these statements.  If there is a lattice
topology whose stability depends upon the zero locations, please
provide a cite for it.  (I'm sure such a think might exist.
But it is not a mainstream topology I would think.)
This is well-known from statistichal DSP. I can't come up
with specifc citations, as my library is is storage and
will remain there for a few weeks to come,

Ha!

Send me your credit card info, and I will order a copy of
P&M - on your expense - to be delivered overnight. Everything
is in there. One only needs to read it.

but I will
tell you what to look for. I know this is treated in the
Proakis & Manolakis general DSP text:

When dealing with AR models, one can solve the Yule-Walker
equations (or rather, and estimator for these equations)
in any number of ways. Direct solutions through linear algebra
will give you the straight-forward FIR prediction filter; the
Levinson recursion will also give you the reflection coefficients
that go directly into the lattice representation.

As one would expect, there exist conversion formulae between
the FIR and lattice representations for the AR model: Insert a
set of FIR coefficients and crank out a set of lattice coefficients.
And vice versa.

The problem is the implicit constraints. In the AR application
the FIR filter is guaranteed to be minimum phase, so its IIR
inverse is causal stable. This translates directly to the
lattice reflection coefficients being constrained to the
interval <-1,1>.

However, the unsuspecting incompetent user who stumbles across
these conversion formulae and tries to convert a linear phase
FIR to lattice form - don't ask *why* one would want to do this;
I assume the user to be *incompetent* - would end up with a
numerically unstable FIR. Which is a contradiction in terms,
given the entry-level indoctrination matra of DSP.

Which will only come back to haunt the designer, who exposed
the incompetent user to the lattice structure in the first place.

Good, finally some information.

....which is utterly trivial to come by if one is even moderately
eductaed on DSP.

The OP was designing a sixth-order Butterworth, not a linear
phase FIR, but no matter.

That was what the OP talekd about, yes. Somebody else started
wining about lattice structures only being explained on a per
application basis. There are very good reasons for that - one
needs to know *exactly* what they are used for and why.

Rune

 

[Steve Pope]

Rune Allnor <allnor@tele.ntnu.no> wrote:

[lattice filters]

but I will
tell you what to look for. I know this is treated in the
Proakis & Manolakis general DSP text:

It is. I'm back in my office today so I have Proakis &
Manolakis right in front of me.

In the summary at the end of Chapter 9 these authors state:

"Finally we mention that lattice and lattice-ladder filter
structures are known to be robust in fixed-point implementations."

The is what I was attempting to commuicate to the OP for
his filter problem, before the discussion got sidetracked.


Steve