By Tokunbo Ogunfunmi

Makes a speciality of approach id purposes of the adaptive tools provided. yet that could even be utilized to different functions of adaptive nonlinear procedures. Covers fresh learn leads to the world of adaptive nonlinear process identity from the authors and different researchers within the box.

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**Extra info for Adaptive Nonlinear System Identification: The Volterra and Wiener Model Approaches (Signals and Communication Technology)**

**Sample text**

For linear adaptive systems based on the steepest gradient descent methods, it is well known that their rate of convergence depends on the eigenvalue spread of the autocorrelation matrix of the input vector (Haykin 1996, Widrow 1985, Sayed 2003). This is covered in chapter 5. Let us consider if there is a similar effect on nonlinear adaptive systems. For instance, for the second-order, 3-memory Volterra series case, the elements of {x(n), x2(n), x(n-1), x2(n-1), x(n)x(n-1), x(n-2), x2(n-2), x(n-1)x(n-2), x(n)x(n-2)} are not mutually orthogonal even when x(n) is white.

The first type of polynomial model we will discuss is the power series. Power Series The power series (Beckmann 1973, Westwick K 2003) is a simple polynomial representation involving a sum of monomials in the input signal. The sum can be finite or infinite. The power series is typically the Taylor series of some known function. The output yi (or y (n) ) is represented by a weighted sum of monomials of input signal xi (or x(n) ). If y (n) = f ( x(n)) , then we have an example such as: 32 Chapter 2 Q f ( x(n)) = ∑ c ( q ) x q (n) q =0 f ( x(n)) = c (0) + c x(n) + c x (n) + c (3) x3 (n) + .......

M h(n) Figure 2-3. General nonlinear model It is also possible to have other models such as a multiplicative system, which consists of LTI subsystems in parallel whose outputs are multiplied together to form the overall nonlinear system output (Rugh WJ 2002). 2 Implementation of Volterra Filters Typically Volterra filters are implemented by interconnections of linear, time-invariant (LTI) subsystems and nonlinear, memory-less subsystems. Polynomial Models of Nonlinear Systems 23 Volterra filters can be implemented in a manner similar to the implementation of linear filters.