Adaptive System Identification using Linear Chirp

Hello dear dsp fellows,

i want to identify an unknown acoustical transfer function using an
adaptive LMS system identification set-up. Both the plant (the unknown
acoustical transfer function ) and the model (coefficients of the adaptive
filter) will be fed with the same excitation signal. The difference of
their outputs will form the error, which will be fed to the LMS update
formula. Now, as excitation signal i want to use a linear choip, which
simply consists of cyclic repetitions of a linear chirp. The question is
this:
1)Are there any rules about how long the duration of each harmonic of the
chirp should be?
2)Are there any rules about how many times the chirp must repeat itself in
order for the adaptive filter to converge satisfactorily?

I have done a small research in the web but i have found very few things
about system identification with chirps. The only article is that of John
Burgess (Acoustical Society of America); however it is not relevant to my
questions.

Manolis

Manolis C. Tsakiris wrote:
> Hello dear dsp fellows,
> 
> i want to identify an unknown acoustical transfer function using an
> adaptive LMS system identification set-up. Both the plant (the unknown
> acoustical transfer function ) and the model (coefficients of the adaptive
> filter) will be fed with the same excitation signal. The difference of
> their outputs will form the error, which will be fed to the LMS update
> formula. Now, as excitation signal i want to use a linear choip, which
> simply consists of cyclic repetitions of a linear chirp. The question is
> this:
> 1)Are there any rules about how long the duration of each harmonic of the
> chirp should be?
> 2)Are there any rules about how many times the chirp must repeat itself in
> order for the adaptive filter to converge satisfactorily?
> 
> I have done a small research in the web but i have found very few things
> about system identification with chirps. The only article is that of John
> Burgess (Acoustical Society of America); however it is not relevant to my
> questions.

Why has everything except the practicality been set in concrete before 
the work starts? Is this a homework question?

I assume when you say LMS, you will really be using some form of NLMS?

To make LMS adapt to what most people would consider to be *the* correct 
answer (one which works for any stimulus), rather than what books tend 
to call a correct answer (which may only work for a single stimulus), 
you need to stimulate the system with a representative mix of stimuli. 
That means covering the whole band.

A wide enough chirp will do that, as long as you don't adapt too 
quickly. If you do, the adaption may run through a set of solutions 
which suit only the current pitch. You can store just a few sweeps 
(enough to average out the error/noise in the responses) and play them 
through the algorithm as many times as it takes to get a clean answer.

Gaussian noise is probably best stimulus for most uses. You can adapt 
quite fast without unpleasant side effects, as the adaption will not 
follow short term effects very much.

My son's Yamaha digital piano does its adaption to room acoustics by 
emitting what sounds like a crazed harpsichordist thrashing across the 
keyboard several times.

Basically anything which covers the band should eventually home to a 
generic solution, as long as you don't adapt too fast.

Regards,
Steve

This is not a homework question, this is a research question.

I am familiar with the explanation that you gave Steve, and this is the
way i have been working so far. The problem is that i cannot find any
papers on this subject.

More explicitely, the parameters that will determine the quality and speed
of convergence, say of the LMS, are basically three (assuming that the
number of taps of the adaptive filter are enough and the whole band of
interest is covered by the chirp):

*the step-size
*the duration of each harmonic in the chirp (how many samples a single
*harmonic is present before the next one appears)
 the number of chirp repetitions

So, for example given a step-size and the number of chirp repetitions,
what is the optimal duration of each harmonic?
Or given a step-size and the duration of each harmonic what is the optimal
number of chirp repetitions?

Manolis

On May 19, 1:06 am, "Manolis C. Tsakiris" <el01...@mail.ntua.gr>
wrote:
> This is not a homework question, this is a research question.
>
> I am familiar with the explanation that you gave Steve, and this is the
> way i have been working so far. The problem is that i cannot find any
> papers on this subject.
>
> More explicitely, the parameters that will determine the quality and speed
> of convergence, say of the LMS, are basically three (assuming that the
> number of taps of the adaptive filter are enough and the whole band of
> interest is covered by the chirp):
>
> *the step-size
> *the duration of each harmonic in the chirp (how many samples a single
> *harmonic is present before the next one appears)
>  the number of chirp repetitions
>
> So, for example given a step-size and the number of chirp repetitions,
> what is the optimal duration of each harmonic?
> Or given a step-size and the duration of each harmonic what is the optimal
> number of chirp repetitions?
>
> Manolis

Get a copy of Schouken's paper "Survey of Excitation Signals for FFT
Based Signal Analyzers" from the September 1988 issue of IEEE
Transactions on Instrumentation and Measurement.  It should answer
your question completely.

Dan