By Maurice Bellanger
This article emphasizes the problematic courting among adaptive filtering and sign research - highlighting stochastic techniques, sign representations and houses, analytical instruments, and implementation equipment. This moment version contains new chapters on adaptive strategies in communications and rotation-based algorithms. It presents sensible functions in info, estimation, and circuit theories.
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Additional info for Adaptive Digital Filters (Signal Processing and Communications)
All Rights Reserved. 2 rðN À 1Þ rðNÞ .. rðNÞ 6 rðN þ 1Þ 6 6 .. 4 . ÁÁÁ ÁÁÁ .. rð0Þ rð1Þ .. rð2N À 1Þ Á Á Á rðNÞ rð2NÞ 32 3 3 e2 7 6 0 7 7 6 7 7 ¼ b0 bN 6 .. 7 5 4 . 5 0 ÀaN 1 76 Àa1 76 76 .. 54 . 2 ð2:79Þ For p > N, the sequence rðpÞ is again generated recursively from the N preceding terms. 76) gives xðnÞ ¼ N X bi eðn À iÞ þ i¼0 N X i¼1 ai 1 X hj eðn À i À jÞ j¼0 and xðnÞ ¼ N X bi eðn À iÞ þ i¼0 1 X eðn À kÞ N X ð2:81Þ ai hkÀi i¼1 k¼1 Clearly, the impulse response coefﬁcients can be computed recursively: h0 ¼ b0 ; hk ¼ 0 for k < 0 hk ¼ bk þ N X ai hkÀi ; ð2:82Þ k51 i¼1 In matrix form, for the N þ 1 ﬁrst terms we have 32 2 h0 0 0 1 0 0 ÁÁÁ 0 76 h1 6 Àa1 1 0 Á Á Á 0 h 0 0 76 6 7 6 Àa2 6 Àa 1 Á Á Á 0 h h h 1 1 0 76 2 6 6 ..
The only signals which satisfy the above equations are those whose spectrum consists of lines: N X Sðe j! Þ ¼ jSi j2 ð! i Þ ð2:123Þ i¼1 The scalars jSi j2 are the powers of individual lines. The integer N can be arbitrarily large. i Þ ¼ 0 for 1 4 i 4 N satisfy the deﬁnition and are prediction ﬁlters. Conversely, since AðzÞ is a power series, Aðe j! Þ cannot equal zero for every ! 122) can hold only if Sðe j! Þ ¼ 0 everywhere except at a countable set of points. It follows that S ðe j! 123), and AðzÞ has corresponding zeros on the unit circle.
The signal autocorrelation matrix is symmetric, and therefore all eigenvalues are greater than or equal to zero. For N sinusoids without noise, the ðN þ 1Þ Â ðN þ 1Þ autocorrelation matrix has one eigenvalue equal to zero; adding to the signal a white noise component of power e2 results in adding e2 to all eigenvalues of the autocorrelation matrix. Thus, the noise power e2 is the smallest eigenvalue of the signal, and the recursion coefﬁcients are the entries of the associated eigenvector. As shown in the next chapter, the roots of the ﬁlter AðzÞ ¼ 1 À N X ai zÀ1 ð2:146Þ i¼1 called the minimum eigenvalue ﬁlter, are located on the unit circle in the complex plane and give the frequencies of the sinusoids.