IIR filter stability issues

Hi,
It would very helpful if someone could tell me how to test for the
stability of an IIR filter.(i.e the stability problems related to the
implementation-due to availability of only finite precision)

thanks
srikar

srikar.ebe@gmail.com wrote:

> Hi,
> It would very helpful if someone could tell me how to test for the
> stability of an IIR filter.(i.e the stability problems related to the
> implementation-due to availability of only finite precision)
> 
> thanks
> srikar
> 
There are two problems that you need to check for: coefficient 
quantization and data quantization.

If it's a fixed-coefficient filter the coefficient quantization happens 
at design time, so you just take your as-designed coefficients and check 
the pole locations (better yet, check the transfer function and make 
sure it's close enough -- you can have a stable filter that's still not 
acceptable).  If it's a variable-coefficient filter then play around 
with the math to get an idea of the filter's sensitivity to coefficient 
quantization and use that as a guide for setting your precision.

Data quantization can be handled by treating the quantization as a noise 
process.  Make a block diagram of your filter as implemented, plug in 
your coefficients, and calculate the transfer function from each 
quantization point to the output.  The conservative analyist will find 
the worst-case gain and take that for the overall quantization gain.

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

Thanks a lot for the solutions Tim. I am working on a filter with
pre-set coefficients for which I need to verify if the filter is stable
in a given precision implementation. What inputs should I feed for such
a filter to check whether the filter is stable?( Here, the poles are
inside the unit circle but close to it and hence the problem with the
implementation in finite precision.)

Thanks
Srikar

PS: as a newbie in DSP, I would be extremely glad if you could relate
the stability in these filters to the stability of systems in general.
i.e What exactly do I mean when I conclude that a filter is stable? (In
general systems studied until now it is FIFO stability- how do I assure
that the given filter is stable by FIFO was perhaps a better question)

Thanks a lot for the solutions Tim. I am working on a filter with
pre-set coefficients for which I need to verify if the filter is stable
in a given precision implementation. What inputs should I feed for such
a filter to check whether the filter is stable?( Here, the poles are
inside the unit circle but close to it and hence the problem with the
implementation in finite precision.)

Thanks
Srikar

PS: as a newbie in DSP, I would be extremely glad if you could relate
the stability in these filters to the stability of systems in general.
i.e What exactly do I mean when I conclude that a filter is stable? (In
general systems studied until now it is FIFO stability- how am I
assured that the given filter is stable by FIFO was perhaps a better
question)

Srikar wrote:

> Thanks a lot for the solutions Tim. I am working on a filter with
> pre-set coefficients for which I need to verify if the filter is stable
> in a given precision implementation. What inputs should I feed for such
> a filter to check whether the filter is stable?( Here, the poles are
> inside the unit circle but close to it and hence the problem with the
> implementation in finite precision.)

OK, in my last post I told you to take the filter coefficients _as 
implemented_, i.e. _after quantization_ and check the filter poles -- is 
that not what you're doing?  If you feel you must do it experimentally 
just give it a step input, and hope that what you've coded doesn't have 
any bugs that's making what should be stable unstable or visa versa 
(fixing a coding bug only to uncover a serious system definition bug is 
_not_ a nice thing to have happen).

It is good to be worried about poles close to the unit circle creeping 
outwards when the coefficients are quantized -- those little buggers can 
move an astounding distance for relatively small changes in the 
coefficients.
> 
> Thanks
> Srikar
> 
> PS: as a newbie in DSP, I would be extremely glad if you could relate
> the stability in these filters to the stability of systems in general.
> i.e What exactly do I mean when I conclude that a filter is stable? (In
> general systems studied until now it is FIFO stability- how do I assure
> that the given filter is stable by FIFO was perhaps a better question)
> 
The filter is a system, just like any other system.  I assume you mean 
BIBO stability (bounded-input, bounded-output)?  At any rate, a filter 
follows all the same rules for stability that any other linear 
shift-invariant system would follow.

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

If I see no instability in the step response, I can only say that the
filter is stable for the step input, and I assume it doesnt mean that
the filter has to be stable for all inputs( please correct me if I am
wrong) because there is no theorem/rule that validates the stability of
the filter in case of it being stable to step input.( I have searched
for a long time on the net, and I think I saw somewhere white noise
being given as an input to the filter to test its stability.)
Unfortunately I lost that site. Please let me know how this test is
implemented.

Srikar wrote:

> If I see no instability in the step response, I can only say that the
> filter is stable for the step input, and I assume it doesnt mean that
> the filter has to be stable for all inputs( please correct me if I am
> wrong) because there is no theorem/rule that validates the stability of
> the filter in case of it being stable to step input.

Theoretically correct, but you'd have to go out of your way to construct 
an unstable filter that wouldn't have troubles with the step input.

> ( I have searched
> for a long time on the net, and I think I saw somewhere white noise
> being given as an input to the filter to test its stability.)
> Unfortunately I lost that site. Please let me know how this test is
> implemented.
> 
Matlab has, I believe, a white noise generator.  If you're checking a 
physical system you can get various noise generators.

You seem reluctant to do this with theoretical means.  Is it that you 
have already verified it in theory and want to double check, that you 
don't trust the theory, or that you don't want to make the effort?  The 
first is perfectly valid, the second -- well, linear systems theory is 
pretty solid and will save you time (but a check of the real system is a 
good idea).  In the last case -- just do the check!!

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

Srikar wrote:
> Thanks a lot for the solutions Tim. I am working on a filter with
> pre-set coefficients for which I need to verify if the filter is stable
> in a given precision implementation. What inputs should I feed for such
> a filter to check whether the filter is stable?( Here, the poles are
> inside the unit circle but close to it and hence the problem with the
> implementation in finite precision.)
>
> Thanks
> Srikar
>
> PS: as a newbie in DSP, I would be extremely glad if you could relate
> the stability in these filters to the stability of systems in general.
> i.e What exactly do I mean when I conclude that a filter is stable? (In
> general systems studied until now it is FIFO stability- how am I
> assured that the given filter is stable by FIFO was perhaps a better
> question)

Would it not be a good idea as a practical test, to use a very large
i.e. full scale input step function (to check for overflow issues)  as
well as a very small  one (to check for limit cycle issues) ?


Mark



Mark

Mark wrote:

> 
> Srikar wrote:
> 
>>Thanks a lot for the solutions Tim. I am working on a filter with
>>pre-set coefficients for which I need to verify if the filter is stable
>>in a given precision implementation. What inputs should I feed for such
>>a filter to check whether the filter is stable?( Here, the poles are
>>inside the unit circle but close to it and hence the problem with the
>>implementation in finite precision.)
>>
>>Thanks
>>Srikar
>>
>>PS: as a newbie in DSP, I would be extremely glad if you could relate
>>the stability in these filters to the stability of systems in general.
>>i.e What exactly do I mean when I conclude that a filter is stable? (In
>>general systems studied until now it is FIFO stability- how am I
>>assured that the given filter is stable by FIFO was perhaps a better
>>question)
> 
> 
> Would it not be a good idea as a practical test, to use a very large
> i.e. full scale input step function (to check for overflow issues)  as
> well as a very small  one (to check for limit cycle issues) ?
> 
Probably -- and if it's a resonant filter it'd be a good idea to check 
it through the neighborhood of the resonant frequency as well.  Here 
again I find that doing a block diagram and investigating the 
sensitivity of the thing (in this case the internal states vs. the 
input) is a good idea.

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

Thanks Tim, and ya.. its for the second reason that I have to check the
filter stability. I hav already checked them in theory and the problem
has been that some of these filters turned up with noise when tested
practically. Hence the effort to check them before I implement them in
hardware.
Srikar

PS: Mark could you please elucidate further, the issue of limit cycle
testingand also  With a large step function what should i look for in
the output?