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Additional resources for A first course in numerical analysis

Example text

Note that Q need not be square, and QTQ = I. The S matrix is upper-triangular and nonsingular [Brogam91]. It implies that the inverse matrix S-1 is also triangular. 7) -1 where R = S . 7 is the widely used form of QR decomposition. The original column in P can always be augmented with additional vector vn in such a way that matrix [P | V] has n linearly independent columns. The Gram-Schmidt procedure can then be applied to construct a full set of orthonormal vectors {qj | j = 1,…,m} which can be used to find the n× m matrix Q.

The second model is the orthogonal model. In contrast to the GramSchmidt procedure, the idea here is to use some orthonormal bases or orthogonal polynomials to represent the Volterra series. The benefit of the orthogonal model is obvious when LMS-type adaptive algorithms are applied. The orthonormal DFT–based model will be explored in this chapter. More extensions and variations of nonlinear orthogonal Wiener models will be developed in the next few chapters. 1 Nonorthogonal Polynomial Models A polynomial nonlinear system can be modeled by the sum of increasing powers of the input signal, x(n).

We note that the set of Hermite polynomials is an orthogonal set in a statistical sense. This means that the Volterra series can be represented by some linear combination of Hermite polynomials. 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.