Behaviors of digital filters with two's complement arithmetic

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 counter-intuitive 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 second-order 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 third-order 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 second-order 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 second-order 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 counter-intuitive results are thus found, Also, we have found some new results on a second-order digital filter with saturation-type or quantization-type 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.