Author:  Ling, Wingkuen 
Title:  Behaviors of digital filters with two's complement arithmetic 
Degree:  Ph.D. 
Year:  2003 
Subject:  Hong Kong Polytechnic University  Dissertations Digital filters (Mathematics) 
Department:  Dept. of Electronic and Information Engineering 
Pages:  xlii, 252 leaves : ill. ; 30 cm 
Language:  English 
InnoPac Record:  http://library.polyu.edu.hk/record=b1732939 
URI:  http://theses.lib.polyu.edu.hk/handle/200/3736 
Abstract:  Interesting nonlinear behaviors have been found for digital filters with two's complement arithmetic. Some necessary conditions relating the set of initial conditions, periodic and admissible properties of symbolic sequences and the trajectory behaviors have been reported in the existing literature for the direct form autonomous systems. Our work covers a more comprehensive range of results including (1) necessary and sufficient conditions; (2) behaviors of some forced response systems, such as step response systems and sinusoidal response systems; (3) effects of other realizations, such as cascade realization and parallel realization. Our methodologies employed are based on representing a nonlinear system as a linear system with the symbolic sequence as an extra 'input', and then develop a modified affine transformation to obtain various relationships among the properties of symbolic sequences, trajectory behaviors and the set of initial conditions. Based on these derived relationships, some novel and counterintuitive results are found. By representing a nonlinear system as a linear system with the symbolic sequence as an extra 'input' and developing a modified affine transformation, we obtain a set of necessary and sufficient conditions relating the trajectory behaviors, the periodic properties of the symbolic sequences and the corresponding set of initial conditions. The derived necessary and sufficient conditions are so simple that we can determine the trajectory behaviors from the initial conditions directly without running the simulations. Using this methodology, we discover that for secondorder digital filters with two's complement arithmetic: (1) even though the eigenvalues of the system matrix are stable: (i) the phase trajectories may in some situations converge to some fixed points which are not the origin, (ii) in some other cases, they may exhibit polygonal fractal patterns; (2) even though the eigenvalues of the system matrix are unstable: (i) overflow may not occur and (ii) the state trajectory may converge to some fixed points or periodic orbits for arbitrary initial conditions; (3) for the step response case: (i) a single elliptical trajectory may be exhibited on the phase plane even though overflow occurs, (ii) overflow may occur in certain situations even though the input step size is small, and (iii) overflow may not occur in some situations even though the input step size is large; (4) for the sinusoidal response case: several ellipses may be exhibited on the phase plane even though overflow does not occur, and these portraits are similar in appearance to some of the portraits for the autonomous and step response cases in which overflow occurs. Besides, this method also provides the relationship between the periodic properties of the symbolic sequences and the set of initial conditions for some well known trajectory behaviors, such as limit cycle and convergence behaviors, which had not been reported by other researchers before. This methodology has also been successfully applied to thirdorder digital filters with two's complement arithmetic realized in either the cascade form or the parallel form. When the digital filter is realized in the cascade form, the trajectories will converge to a horizontal phase plane and give visual patterns similar to those of the secondorder digital filter case. When the digital filter is realized in the parallel form, we find that the output and the delayed output may exhibit a pattern resulting from overlapping of corresponding secondorder phase portraits. The most important practical implication of the above results is the provision of a theoretical basis for a designer to (1) select the initial condition and the filter parameters so that chaotic behaviors can be avoided; (2) select the parameters to generate chaos for certain applications, such as chaotic communications, encryption and decryption, fractal coding, etc. Besides, we have developed a new technique based on the Shannon entropy of the state variables and symbolic sequences for detecting special chaotic and periodic trajectories, and some novel and counterintuitive results are thus found, Also, we have found some new results on a secondorder digital filter with saturationtype or quantizationtype nonlinearities. In addition, we have explored some possible applications of digital filter systems with certain types of nonlinearities, such as effective encryption via a digital filter bank system. 
Files  Size  Format 

b17329395.pdf  17.95Mb 


As a bona fide Library user, I declare that:  


By downloading any item(s) listed above, you acknowledge that you have read and understood the copyright undertaking as stated above, and agree to be bound by all of its terms. 