By R. S. Bucy, José M. F. Moura (auth.), Richard S. Bucy, José M. F. Moura (eds.)
This quantity corresponds to the invited lectures and complex learn papers offered on the NATD complicated learn Institute on Nonlinear Stochastic issues of emphasis on id, sign Processing, keep an eye on and Nonlinear Filtering held in Algarve (Portugal), on could 1982. The publication is a mix of theoretical matters, algorithmic implementation features, and alertness examples. in lots of components of technological know-how and engineering, there are difficulties that are intrinsically nonlinear 3nd stochastic in nature. transparent examples come up in id and mOdeling, sign processing, nonlinear filtering, stochastic and adaptive conLrol. The assembly used to be equipped since it was once felt that there's a want for dialogue of the tools and philosophy underlying those varied components, and with the intention to converse these methods that experience confirmed to be powerful. because the computational know-how progresses, extra basic methods to a few difficulties that have been handled formerly through linearization and perturbation tools develop into possible and rewarding.
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Extra info for Nonlinear Stochastic Problems
Example text
To avoid notational complexity in our derivations, we shall consider here only the cornrnon case where R itself satisfies the conditions on the variat ion of R. Hhen R is constrained to have particular numerical values, one can treat the problem by similar methods but using a reduced dimensionality. The Algorithrn. The iterative process that we consider here we shall term inverse iteration. At any stage, we begin with an approximation RK and a new approximation RK+ l is determined as follows: 1.
In some of the chosen examples below, we alrea:iy "know" what the best answer should be. In other cases, one may not be so certain that the derived answer is the best. In this latter case, use of the variational principle may end up changing one I s intuition about what answers do make the most sense. The Unconstrained Case The simplest problem to solve is when the R matrix is not constrained. Then the variation of R is arbitrary and the gradient must be identically zero. Then we have from equation (5) that -1 -1 -1 R S R - R = O.
If S(v)dv denotes that fraction of the received power returned by scatterers whose velocity components in the direction of the radar lie in the increment dv, then we may write S(f)df = S(v)dv where it applies. llle of is understood that the Doppler relation f = 2V/A This shows that the power spectrum S(f) does indeed picture of the radial velocity distribution within the scatterers of interest. In [2], it is shown that the power spectra due to various sources of radar clutter (ground targets such as wooded hills, sea echo, rain clouds) may be closely approximated by an all-pole rational function.