By Maurice Bellanger
This article emphasizes the complex dating among adaptive filtering and sign research - highlighting stochastic methods, sign representations and houses, analytical instruments, and implementation equipment. This moment version comprises new chapters on adaptive ideas in communications and rotation-based algorithms. It offers useful functions in info, estimation, and circuit theories.
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Additional info for Adaptive Digital Filters (Signal Processing and Communications)
7 7 5 0 .. .. .. where ð2:86Þ ÀaN 0 ÁÁÁ For real signals, the ﬁrst ðN þ 1Þ ACF terms are obtained from the equation 3 2 3 2 rð0Þ dð0Þ 6 rð1Þ 7 6 dð1Þ 7 7 6 7 2 0 À1 6 ð2:87Þ 6 .. 7 ¼ e ½A þ A 6 .. 7 4 . 5 4 . 5 rðNÞ dðNÞ In summary, the procedure to calculate the ACF of an ARMA signal from the generating ﬁlter coefﬁcients is as follows: TM Copyright n 2001 by Marcel Dekker, Inc. All Rights Reserved. 1. 2. 3. 4. 82). Compute the auxiliary variables dðpÞ for 0 4 p 4 N. 87). 68) to derive rðpÞ when p 5 N þ 1.
TM Copyright n 2001 by Marcel Dekker, Inc. All Rights Reserved. The set of coefﬁcients can also be a multidimensional signal. 11. The N time-varying ﬁlter coefﬁcients ai ðnÞ are obtained as the outputs of N ﬁxed-coefﬁcient ﬁlters fed by independent white noises with same variances. A typical choice for the coefﬁcient ﬁlter transfer function is the ﬁrst-order low-pass function Hi ðzÞ ¼ 1 ; 1 À zÀ1 0(
<1 ð2:158Þ whose time constant is ¼ 1 1À ð2:159Þ For close to unity, the time constant is large and the ﬁlter coefﬁcients are subject to slow variations.
Linear prediction is useful for classifying signals and, particular, distinguishing between deterministic and random processes. 9. PREDICTABLE SIGNALS A signal xðnÞ is predictable if and only if its prediction error power is null: Z 1 Ea ¼ jAðe j! Þj2 Sðe j! Þ d! ¼ 0 ð2:121Þ 2 À or, in the time domain, TM Copyright n 2001 by Marcel Dekker, Inc. All Rights Reserved. xðnÞ ¼ 1 X ai xðn À iÞ ð2:122Þ i¼1 which means that the present value xðnÞ of the signal can be expressed in terms of its past values.
Adaptive Digital Filters (Signal Processing and Communications) by Maurice Bellanger