Adaptive Filter Theory By Simon Haykin Pdf To Word

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Hello, I recently purchased the aformentioned book, hoping to find adaptive filter examples implemented in Matlab. However, it contains none. Adaptive Filters Simon Haykin McMaster University Hamilton, Ontario, Canada L8S 4K1 1. Introduction An adaptive filteris defined as a self-designing system that relies for its operation on a recursive algorithm, which makes it possible for the filter to perform satisfactorily in an environment where knowledge of the relevant statistics is.

Adaptive filter theory 5th edition haykin solutions manual • 1. 21 Adaptive Filter Theory 5th Edition Haykin SOLUTIONS MANUAL Full download at: haykin-solutions-manual/ Chapter 2 Problem 2.1 a) Let wk = x + j y p(−k) = a + j b We may then write f =wkp∗ (−k) =(x+ j y)(a − j b) =(ax + by) + j(ay − bx) Letting where f = u + j v u = ax + by v = ay − bx • 22 Hence, ∂u ∂u = a = b ∂x ∂y ∂v ∂v = a ∂y ∂x = −b • 23 PROBLEM 2.1. K From these results we can immediately see that ∂u ∂v = ∂x ∂y ∂v ∂u ∂x = − ∂y In other words, the product term wkp∗ (−k) satisfies the Cauchy-Riemann equations, and so this term is analytic. B) Let Let f =wkp∗ (−k) =(x− j y)(a + j b) =(ax + by) + j(bx − ay) with f = u + jv u = ax + by v = bx − ay Hence, ∂u ∂u =a ∂x ∂y ∂v ∂v =b ∂x ∂y = b = −a From these results we immediately see that ∂u ∂v = ∂x ∂y ∂v ∂u ∂x = − ∂y In other words, the product term w∗ p(−k) does not satisfy the Cauchy-Riemann equations, and so this term is not analytic. • 24 PROBLEM 2.2. Numbers

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D d d Problem 2.2 a) From the Wiener-Hopf equation, we have w0 = R−1 p (1) We are given that 1 0.5 R = 0.5 1 0.5 p = 0.25 Hence the inverse of R is 1 0.5 −1 R−1 = = 0.5 1 1 1 −0.5 −1 0.75 −0.5 1 Using Equation (1), we therefore get 1 1 −0.5 0.5 w0 = 0.75 −0.5 1 0.25 1 0.375 = 0.75 0 0.5 = 0 b) The minimum mean-square error is Jmin =σ2 − pH w0 =σ2 − 0.5 0.25 =σ2 − 0.25 0.5 0 • 25 PROBLEM 2.2. C) The eigenvalues of the matrix R are roots of the characteristic equation: (1 − λ)2 − (0.5)2 = 0 That is, the two roots are λ1 = 0.5 and λ2 = 1.5 The associated eigenvectors are defined by Rq = λq For λ1 = 0.5, we have 1 0.5 q11 = 0.5 q11 0.5 1 q12 q12 Expanded this becomes q11 + 0.5q12 = 0.5q11 0.5q11 + q12 = 0.5q12 Therefore, q11 = −q12 Normalizing the eigenvector q1 to unit length, we therefore have 1 q1 = √ 2 1 −1 Similarly, for the eigenvalue λ2 = 1.5, we may show that 1 q2 = √ 2 1 1 • 26 PROBLEM 2.3. 1 2 i Accordingly, we may express the Wiener filter in terms of its eigenvalues and eigenvectors as follows: 2 1! W0 = X qiqH p λ i i=1 1 1 = q1 qH + λ1 q2qH p λ2 = 1 1 −1 + 1 1 1 1 0.5 −1 3 1 0.25 = 1 −1 + 1 1 1 0.5 −1 1 3 1 1 0.25 4 2 = 3 − 3 0.5 2 4 − 3 3 4 1 0.25 = 6 − 6 1 1 − 3 + 3 0.5 = 0 Problem 2.3 a) From the Wiener-Hopf equation we have w0 = R−1 p (1) We are given 1 0.5 0.25 R = 0.5 1 0.5 0.25 0.5 1 and p = 0.5 0.25 0.125 T • 27 PROBLEM 2.3. D d d T i Hence, the use of these values in Equation (1) yields w0 =R−1 p 1 0.5 0.25 −1 0.5 = 0.5 1 0.5 0.25 0.5 1 0.25 0.125 1.33 −0.67 0 0.5 = −0.67 1.67 −0.67 0.25 0 −0.67 1.33 0.125 w0 = 0.5 0 0 b) The Minimum mean-square error is Jmin =σ2 − pH w0 0.5 =σ2 − 0.5 0.25 0.125 0 0 =σ2 − 0.25 c) The eigenvalues of the matrix R are λ1 λ2 λ3 = 0.4069 0.75 1.8431 The corresponding eigenvectors constitute the orthogonal matrix: −0.4544 −0.7071 0.5418 Q = 0.7662 0 0.6426 −0.4544 0.7071 0.5418 Accordingly, we may express the Wiener filter in terms of its eigenvalues and eigenvectors as follows: 3 1! Billa movie. W0 = X qiqH p λ i i=1 • 28 PROBLEM 2.4. 0.6426 0.5418 0.6426 0.5418 × 0.25 0.5418 0.125.

1 −0.4544 w0 = 0.4069 0.7662 −0.4544 0.7662 −0.4544 −0.4544 1 + 0.75 −0.7071 0 0.7071 −0.7071 0 −0.7071 1 + 1.8431 0.5418 0.5 1 0.2065 −0.3482 0.2065 w0 = 0.4069 −0.3482 0.5871 −0.3482 0.2065 −0.3482 0.2065 0.5 0 −0.51 + 0.75 0 0 0 −0.5 0 0.5 1 0.2935 0.3482 0.2935 0.5 + 0.3482 0.4129 0.3482 × 0.25 0.5 = 0 0 1.8431 0.2935 0.3482 0.2935 0.125 Problem 2.4 By definition, the correlation matrix R = E[u(n)uH (n)] Where u(n) u(n) = u(n − 1) u(0) Invoking the ergodicity theorem, R(N ) = 1 N + 1 NX u(n)uH (n) n=0 • 29 PROBLEM 2.5. Α Likewise, we may compute the cross-correlation vector p = E[u(n)d∗ (n)] as the time average p(N ) = 1 N + 1 NX u(n)d∗ (n) n=0 The tap-weight vector of the wiener filter is thus defined by the matrix product w0(N ) = NX u(n)uH (n) n=0!−1 N!