By Tokunbo Ogunfunmi
Specializes in method id functions of the adaptive tools offered. yet that can even be utilized to different purposes of adaptive nonlinear techniques. Covers fresh examine ends up in the realm of adaptive nonlinear method id 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)
In fact every Volterra series has a unique Wiener model representation. This model gives us a good eigenvalue spread of autocorrelation matrix (which is a requirement for convergence of gradient-based adaptive filters as discussed in chapter 5), and also allows us to represent a complicated Volterra series without over-parameterization with only a few coefficients. It is interesting to note that most of the linear properties of adaptive algorithms are still preserved. By using this nonlinear model, a detailed adaptation performance analysis can be done.
Dx n 34 Chapter 2 The generating function of the Hermite polynomial is ∞ H n ( x)t n n! n=0 e 2tx−t = ∑ 2 It can be shown that ∞ ∫e −∞ − x2 ⎧⎪0, m ≠ n H m ( x) H n ( x)dx = ⎨ n ⎪⎩2 n ! π m = n We note that H n ( x) is even when n is even and H n ( x) is odd when n is odd. Hermite polynomials form a complete orthogonal2 set on the interval −∞ < x < +∞ with respect to the weighting function e − x . By using this orthogonality, a piece-wise continuous function f ( x) can be expressed in terms of Hermite polynomials: ⎧ f ( x) where f ( x) is continuous ⎪ = C H x ( ) ⎨ f ( x− ) + f ( x+ ) ∑ n n at dis-continuous points n =0 ⎪ 2 ⎩ ∞ where Cn = 1 2 n!
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. 5 QR and Inverse QR Matrix Decompositions QR matrix decomposition is frequently used in RLS-type adaptive algorithms. This QR decomposition technique can be obtained by using the GramSchmidt procedure (Brogam 1991). , QTQ = I) and an upper-triangular m × m matrix R. The Gram-Schmidt procedure is one way of determining Q and R such that P = QR.
Adaptive Nonlinear System Identification: The Volterra and Wiener Model Approaches (Signals and Communication Technology) by Tokunbo Ogunfunmi