Data-path accuracy in IIR filters?
in [FPGA]
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From: Steve Pope on 1 Aug 2010 03:50 Rune Allnor <allnor(a)tele.ntnu.no> wrote: >On 31 Jul, 22:49, spop...(a)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
From: Steve Pope on 1 Aug 2010 03:53 Rune Allnor <allnor(a)tele.ntnu.no> wrote: >On 1 Aug, 05:10, spop...(a)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
From: Rune Allnor on 1 Aug 2010 05:53 On 1 Aug, 09:53, spop...(a)speedymail.org (Steve Pope) wrote: > Rune Allnor <all...(a)tele.ntnu.no> wrote: > > >On 1 Aug, 05:10, spop...(a)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
From: Rune Allnor on 1 Aug 2010 06:54 On 1 Aug, 09:50, spop...(a)speedymail.org (Steve Pope) wrote: > Rune Allnor <all...(a)tele.ntnu.no> wrote: > > >On 31 Jul, 22:49, spop...(a)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
From: robert bristow-johnson on 1 Aug 2010 09:34
On Aug 1, 5:53 am, Rune Allnor <all...(a)tele.ntnu.no> wrote: > On 1 Aug, 09:53, spop...(a)speedymail.org (Steve Pope) wrote: > > > Rune Allnor <all...(a)tele.ntnu.no> wrote: > > > >On 1 Aug, 05:10, spop...(a)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 |