LMS and Eigenvalue Spread

I have been reading Haykins book again and he goes on about Eigenvalue
spread. Eg for a 2nd order AR (all pole) model LMS converges more slowly if
the eigenvalues of the cov matrix are dispersed eg one is 10 times the other
for a 2nd order system.

My question is this: Can we get a similar large eigenvalue spread for a
purely FIR system (as opposed to Haykins AR mode)? Also I assume the
convergence will be fastest when the reference is white - is that right?
Is this also true of normalised LMS?

The reason I am asking is that he states that when the systems poles are
complex we get this problem. Now we only get this if there are poles - we
cannot get an oscillitory FIR system can we? If this is the case then many
acoustic problems would not be effected since acoustic transfer functions
are FIR. Or is this right..??

M



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Major Misunderstanding wrote:

> I have been reading Haykins book again and he goes on about Eigenvalue
> spread. Eg for a 2nd order AR (all pole) model LMS converges more slowly if
> the eigenvalues of the cov matrix are dispersed eg one is 10 times the other
> for a 2nd order system.

This is just the scientific way to say that the poles are compensated by 
zeroes, and zeroes are compensated by poles. Pole can be compensated by 
pole or zero can be compensated by zero with difficulty and only to the 
certain accuracy.


> My question is this: Can we get a similar large eigenvalue spread for a
> purely FIR system 

Yes of course. You can model any IIR using long enough FIR.

>(as opposed to Haykins AR mode)? Also I assume the
> convergence will be fastest when the reference is white - is that right?

No. It depends of where you are converging to.

> Is this also true of normalised LMS?

It does not matter. NLMS is addressing the different issues.

> 
> The reason I am asking is that he states that when the systems poles are
> complex we get this problem. Now we only get this if there are poles - we
> cannot get an oscillitory FIR system can we? If this is the case then many
> acoustic problems would not be effected since acoustic transfer functions
> are FIR. Or is this right..??

No. Acoustic proplems are IIR as well, and with the effects of 
nonlinearity.


Vladimir Vassilevsky

DSP and Mixed Signal Design Consultant

http://www.abvolt.com