By Tokunbo Ogunfunmi

ISBN-10: 0387263284

ISBN-13: 9780387263281

ISBN-10: 0387686304

ISBN-13: 9780387686301

Makes a speciality of approach id functions of the adaptive tools provided. yet which may even be utilized to different purposes of adaptive nonlinear approaches. Covers fresh examine ends up in the realm of adaptive nonlinear process identity from the authors and different researchers within the box.

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

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The solutions of this differential equation are referred to as Tchebyshev’s functions of degree n . For non-negative n, n = 0,1, 2, 3,........ , Tchebyshev’s functions are referred to as Tchebyshev’s polynomials, T n ( x) . Tchebyshev’s polynomials T n ( x) can be expressed as (Efunda 2006): T n ( x) = 1 − x2 dn n− 1 (1 − x 2 ) 2 where n = 0,1, 2,3...... 1 dx n Polynomial Models of Nonlinear Systems 35 The generating function of the Tchebyshev’s polynomial is ∞ 1 − tx = ∑ T n ( x)t n . 1 − 2tx + t 2 n=0 It can be shown that 1 ∫ −1 ⎧ ⎪0, m ≠ n ⎪ 1 m n T ( x)T ( x)dx = ⎨π , m = n = 0 1 − x2 ⎪π ⎪ m = n = 1, 2,3.....

13) Polynomial Models of Nonlinear Systems 31 The LMS algorithm is probably the most famous adaptive algorithm because of its simplicity and stable properties and is very suitable to apply to the nonlinear orthonormal filter structure. 14) The whole architecture of nonlinear adaptive filter is shown in figure 2-10. u(n) Nonlinear Orthonormal Expansion gm[x(n)] Coefficients update  d(n) d(n) + e(n) Figure 2-10. Architecture of the DFT-based nonlinear filtering This structure is similar to (Mulgrew 1994) and (Scott 1997), but without the estimated probability density function (PDF)-divider which may cause unexpected numerical instability (Chang 1998b).

PM vectors orthogonal to the pk vector in each stage k and repeat this operation to (M-1)th stage. 5 below (Brogam 1991). 5) (k −1) p(k) − α ki w k i = pi −1) w M = p (M M where p (k) i indicates the ith vector at stage k. Theoretically, identical results and the same computational complexity will be performed with both versions. The only difference is the operational sequence. , M} is ill-conditioned. Therefore the modified Gram-Schmidt procedure has much better numerical stability and accuracy than the Gram-Schmidt procedure.

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