By Haykin S., Totterdill P.
At a degree compatible for graduate classes on adaptive sign processing, this textbook develops the mathematical concept of assorted realizations of linear adaptive filters with finite-duration impulse reaction, and likewise offers an introductory therapy of supervised neural networks. various laptop experiments illustrate the underlying conception and functions of the LMS (least mean-square) and RLS (recursive-least-squares) algorithms, and difficulties finish every one bankruptcy.
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Extra info for Adaptive Filter Theory
The DFf can effectively be computed by Fast Fourier Transform (FFf) algorithms, which were developed by Cooley and Tuckey in 1965 and subsequently improved and refined by other researchers. 4 The Relationship Between Time-Continuous Sampled and Discrete-Time Signals in the Frequency- and Time-Domain In order to thoroughly understand the operation of sampled-data systems, it is necessary to determine the influence of the sampling frequency CJs = 27T/T on the frequency spectra of the sampled signals, to compare them with the frequency spectra of the corresponding analog signals and also to investigate how they are related to the original analog signal.
15. The above consideration leads directly to the statement of the Sampling Theorem attributed to Shannon: * A band-limited [ow-pass signal x ( t) with X (j CJ) =0 for all I CJ I > CJg is uniquely and completely described by a set of sample values x (n T) taken at unifonnly spaced time instants separated by T = 7T / CJg seconds or less. Thus, proper sampling requires the knowledge of samples at a rate of at least twice the highest frequency present inx (t), which means CJs ~ 2CJg • The sampling rate CJs /27T = liT = CJg /7T is called the Nyquist rate.
The sum above is a geometric progression and it converges for / z -I / < 1; hence, the ROC is/z/>1. (ii) Exponential function x (t) =U (t)e -at with sampling period T. Thus, X (z ) can be written in the form of the following power series: z z - e -aT with ROC: I z / > e -aT • 20 Fundamentals of Sampled-Data Systems Note that the same result can be obtained immediately from the z -transform of the unit step function and Property (4). This result may be compared with the Laplace transform of the corresponding continuous signal x ( t) = u ( t) e -at.