By Maurice Bellanger
This article emphasizes the problematic dating among adaptive filtering and sign research - highlighting stochastic strategies, sign representations and homes, analytical instruments, and implementation tools. This moment variation comprises new chapters on adaptive strategies in communications and rotation-based algorithms. It offers useful functions in details, estimation, and circuit theories.
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Additional info for Adaptive Digital Filters (Signal Processing and Communications)
7. 13. NONSTATIONARY SIGNALS A signal is nonstationary if its statistical character changes with time. The fundamental decomposition can be extended to such a signal, and the regular component is xr ðnÞ ¼ 1 X hi ðnÞeðn À iÞ ð2:154Þ i¼0 where eðnÞ is a stationary white noise. The generating ﬁlter impulse response coefﬁcients are time dependent. An instantaneous spectrum can be deﬁned as 2 1 X 2 Àj2fi hi ðnÞe ð2:155Þ Sð f ; nÞ ¼ e i¼0 So, nonstationary signals can be generated or modeled by the techniques developed for stationary signals, but with additional means to make the system coefﬁcients time varying .
But, in general, that does not mean statistical independency, since higher-order dependency can exist. Among the probability laws, the Gaussian law has special importance in signal processing. TM Copyright n 2001 by Marcel Dekker, Inc. All Rights Reserved. 4. GAUSSIAN SIGNALS A random variable x is said to be normally distributed or Gaussian if its probability law has a density pðxÞ which follows the normal or Gaussian law: pðxÞ ¼ 2 2 1 pﬃﬃﬃﬃﬃﬃ eÀðxÀmÞ =2x x 2 ð2:43Þ The parameter m is the mean of the variable x; the variance x2 is the second-order moment of the centered random variable ðx À mÞ; x is also called the standard deviation.
2 h1 6 h2 6 B¼6 . 4 .. 80), and those of the state vector, x^ i ðnÞ are the i-step linear predictions of xðnÞ, deﬁned, for the ARMA signal and as shown later, by TM Copyright n 2001 by Marcel Dekker, Inc. All Rights Reserved. 75). Having presented methods for generating signals, we now turn to analysis techniques. First we introduce some important deﬁnitions and concepts . 8. LINEAR PREDICTION AND INTERPOLATION The operation which produces a sequence eðnÞ from a data sequence xðnÞ, assumed centered and wide-sense stationary, by the convolution eðnÞ ¼ xðnÞ À 1 X ai xðn À iÞ ð2:102Þ i¼1 is called one-step linear prediction error ﬁltering, if the coefﬁcients are calculated to minimize the variance of the output eðnÞ.
Adaptive Digital Filters (Signal Processing and Communications) by Maurice Bellanger