Behaviors of digital filters with two's complement arithmetic

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 eignevalues 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 eignevalues 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. (Abstract shortened by UMI.)...