Data-path accuracy in IIR filters?

Sat Jul 31, 2010 11:20 pm   

On 31 Jul, 21:03, spop...@speedymail.org (Steve Pope) wrote:

Sure. You know that. I know that. But is that konwledge
wide-spread? Would you trust users to depend on knowing
these things?


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

FIRs are unconditionally stable, at the outset.

The lattice structure represents a dobule obfuscation in that it

1) Places restrictions on FIR filter stability
2) Depends on zero locations

Ano one of those restrictions would mess up the amateur's mind;
the two together would play havoc with anyone in two seconds flat.

Remember, the days when people actually read up on DSP before
attempting to use the techniques are long since gone. You have
to deal with the "Matlab does all the thinking" (TM) generation.


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.
In effect the N'th order lattice filter does the computations in
N stages, with cross-copleing between each stage: The output after
*both* n'th stage filters are fed (with different scaling) as
input to *both* the n+1'th stages in the lattice. As there are
the same number of stages as there are poles (IIR) / zeros (FIR),
the IIR part will be unconditionally unstable if there are two
zeros on or outside the unit circle. Concequently, the FIR
will be unstable, as input from one M order unstable IIR will
be used as input to the FIR computations somewhere in the lattice.

The only way I can see where one might get away ith this, is if
there is exactky one unstable zero of the IIR (reflection coefficient

its output is not used as input to the FIR.

If you think I am wrong, you are welcome to provide proofs to show
that the lattice structure is unconditionally stable.

Rune

Sat Jul 31, 2010 11:49 pm   

Rune Allnor <allnor_at_tele.ntnu.no> replies to my post,


Yes, it's as widespread as any stability criteria for any
other filter topology.


The same is true for a lattice topology, and for any other common
topologies.


I have NO idea what you are talking about here.


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


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.

There is no "obfuscation" much less "double obfuscation". This
is a perfectly normal, everyday, widely used filter with better
stability behavior than most.


Steve

Sun Aug 01, 2010 6:10 am   

Steve Pope <spope33_at_speedymail.org> wrote:


Actually, there is a somewhat better Mathworks document on the
subject here:

http://www.mathworks.com/access/helpdesk_r13/help/toolbox/filterdesign/propref7.html#20164

In my experience, the most useful and well behaved forms of lattice
filters are termed as follows in the above:

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


Steve

Sun Aug 01, 2010 10:34 am   

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

The other topologies only matter as the established baseline.
We are focusing on the lattice topology here.


The pleas prove this statement mathematically. 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?


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.


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


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

Rune

Sun Aug 01, 2010 10:35 am   

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

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.

Rune

Sun Aug 01, 2010 10:50 am   

Rune Allnor <allnor_at_tele.ntnu.no> 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.


Internal states may saturate and stay there. Typically.


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.)


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

Sun Aug 01, 2010 10:53 am   

Rune Allnor <allnor_at_tele.ntnu.no> wrote:


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

Sun Aug 01, 2010 12:53 pm   

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

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

Sun Aug 01, 2010 1:54 pm   

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

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.


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

Sun Aug 01, 2010 4:34 pm   

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

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

r b-j

Sun Aug 01, 2010 9:42 pm   

Rune Allnor <allnor_at_tele.ntnu.no> wrote:


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.


Ha!


Good, finally some information.

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

Steve

Sun Aug 01, 2010 9:58 pm   

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

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.


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


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

Tue Aug 03, 2010 12:49 am   

Rune Allnor <allnor_at_tele.ntnu.no> wrote:

[lattice filters]


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