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## Main Question or Discussion Point

Hi all,

Suppose I have a system which can be described using something like:

[tex]y(t) = a_1 x(t) + a_2 x^2(t) + \dots + a_p x^p(t)[/tex]

I want to find the coefficients using samples from x(t) and y(t) (pairwise taken at same times) using as fewest samples as possible.

Clearly this is linear in parameters and can be recast in a LS problem.

However, despite the problem being linear, the underlying process is nonlinear in this results in the problem that the linear part "overshades" the non-linear parts (or lower terms higher terms). For example, using few samples, a1 usually gets a good estimate while the error on the nonlinear parts is much higher.

Is it proven that LS will always give the best performance in such scenario or is there hope that the error on the non-linear (or higher order) coefficients can be improved (at the cost of slightly larger error on the linear (or low order) coefficients)?

I think LS should attend the CRLB (and is therefore optimum), right?

On the other hand, I have some ideas

Thank you,

divB

Suppose I have a system which can be described using something like:

[tex]y(t) = a_1 x(t) + a_2 x^2(t) + \dots + a_p x^p(t)[/tex]

I want to find the coefficients using samples from x(t) and y(t) (pairwise taken at same times) using as fewest samples as possible.

Clearly this is linear in parameters and can be recast in a LS problem.

However, despite the problem being linear, the underlying process is nonlinear in this results in the problem that the linear part "overshades" the non-linear parts (or lower terms higher terms). For example, using few samples, a1 usually gets a good estimate while the error on the nonlinear parts is much higher.

Is it proven that LS will always give the best performance in such scenario or is there hope that the error on the non-linear (or higher order) coefficients can be improved (at the cost of slightly larger error on the linear (or low order) coefficients)?

I think LS should attend the CRLB (and is therefore optimum), right?

On the other hand, I have some ideas

*why*I think that we can do better but I didn't find a concrete way to do it. I'll wait with this to keep the initial posting short.Thank you,

divB

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