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Least Pth Norm Optimal Method

Started by Kral June 30, 2008
Hello,
I'm interested in papers, articles or books with information on the
application of this method for designing filters. I don't want articles on
using MATLAB. I know how to apply this method using MATLAB. I'm interested
in the underlying theory. Any suggestions? A google search yields only
MATLAB articles.
Thanks


On Jul 1, 2:40 pm, "Kral" <jd_l...@yahoo.com> wrote:
> Hello, > I'm interested in papers, articles or books with information on the > application of this method for designing filters. I don't want articles on > using MATLAB. I know how to apply this method using MATLAB. I'm interested > in the underlying theory. Any suggestions? A google search yields only > MATLAB articles. > Thanks
There are some paper on adaptive filters (LMS) extending ordinary least-mean squares. Widrow looked at this a long time back. K.
Kral wrote:
> Hello, > I'm interested in papers, articles or books with information on the > application of this method for designing filters. I don't want articles on > using MATLAB. I know how to apply this method using MATLAB. I'm interested > in the underlying theory. Any suggestions? A google search yields only > MATLAB articles.
Indeed, an interesting question: why would one chose one "p" over another? Consider the example of "fitting" FIR filters to a target frequency response. For p=2, one can apply a rectangular window to the impulse response. However, the resulting ringing (see the resecnt thread about Gibbs) is usually not acceptable, meaning that p=2 is not an optimal choice. For matching non-noisy target frequency responses, I would use a larger p's, for example p=infinty (minimax filters). However, for fitting noisy frequency responses (for example in non- parametric frequency domain system identification), p=2 combined with a weighting scheme that takes care of the uneven "information distribution" for excitation signals with non-flat frequency response might be a good choice (in fact, it might be the best choice from a statistical point of view). If more is known about the distribution of the frequency domain noise, it might even be sensible to not minimize a p-norm, but some other error functional. A well-known error functional that is not a p-norm is based on minimizing p=2 for small residues and p=1 for large residues. Some of the noise aspects are discussend in Ljung's already classic book [1]. There is a lot of literature on desigining minmax filters (IIR and FIR). If you are interested in the statistical properties of different p's (and other error functionals), look for M-estimators. Regards, Andor [1] Lennart Ljung, System identification: theory for the user (second edition); Prentice-Hall, Englewood Cliffs, NJ, 1999, ISBN 0-13-656695-2.