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Adaptive Digital Filters, 2nd Edition (Signal Processing and by Bellanger

By Bellanger

This article emphasizes the complicated courting among adaptive filtering and sign research - highlighting stochastic tactics, sign representations and homes, analytical instruments, and implementation equipment. This moment variation contains new chapters on adaptive innovations in communications and rotation-based algorithms. It offers functional purposes in info, estimation, and circuit theories.

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Extra info for Adaptive Digital Filters, 2nd Edition (Signal Processing and Communications)

Example text

7. MARKOV SIGNALS Markov signals are produced by state variable systems whose evolution from time n to time n þ 1 is governed by a constant transition matrix [8]. The state of a system of order N at time n is defined by a set of N internal variables represented by a vector XðnÞ called the state vector. 7, and the equations are Xðn þ 1Þ ¼ AXðnÞ þ BwðnÞ yðnÞ ¼ Ct XðnÞ þ vðnÞ ð2:96Þ The matrix A is the N Â N transition matrix, B is the control vector, and C is the observation vector [9]. The input sequence is wðnÞ; vðnÞ can be a measurement noise contaminating the output yðnÞ.

HARMONIC DECOMPOSITION The fundamental decomposition is used in signal analysis as a reference for selecting a strategy [11]. As an illustration let us consider the case, frequently occurring in practice, where the signal to be analyzed is given as a set of 2 N þ 1 autocorrelation coefficients rðpÞ with ÀN 4 p 4 N, available from a measuring procedure. To perform the analysis, we have two extreme hypotheses. 66). But another hypothesis is that the signal is essentially deterministic and consists of N sinusoids in noise.

By the conventional expressions for digital filtering: Z 1  jAðe j! Þj2 Sðe j! Þ d! ð2:105Þ Ea ¼ 2 À or, in terms of z-transforms, Z 1 dz AðzÞAðzÀ1 ÞSðzÞ Ea ¼ j2 jzj¼1 z ð2:106Þ 40 Chapter 2 FIG. 8 Linear prediction filter and inverse filter. where AðzÞ is the transfer function of the prediction error filter. The prediction filter coefficients depend only on the input signal, and the error power can be expressed as a function of Sðe j! Þ only. To derive that expression, we must first show that the prediction error filter is minimum phase; in other words, all its zeros are inside or on the unit circle in the complex z-plane.

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