Further investigation on chaos of real digital filters

This Letter displays, via the numerical simulation of a real digital filter, that a finite-state machine may behave in a near-chaotic way even when its corresponding infinite-state machine does not exhibit chaotic behavior

Detection of chaos in some local regions of phase portraits using Shannon entropies

This letter demonstrates the use of Shannon entropies to detect chaos exhibited in some local regions on the phase portraits. When both the eigenvalues of the second-order digital filters with two’s complement arithmetic are outside the unit circle, the Shannon entropies of the state variables are independent of the initial conditions and the filter parameters, except for some special values of the filter parameters. At these special values, the Shannon entropies of the state variables are relatively small. The state trajectories corresponding to these filter parameters either exhibit random-like chaotic behaviors in some local regions or converge to some fixed points on the phase portraits. Hence, by measuring the Shannon entropies of the state variables, these special state trajectory patterns can be detected. For completeness, we extend the investigation to the case when the eigenvalues of the second-order digital filters with two’s complement arithmetic are complex and inside or on the unit circle. It is found that the Shannon entropies of the symbolic sequences for the type II trajectories may be higher than that for the type III trajectories, even though the symbolic sequences of the type II trajectories are periodic and have limit cycle behaviors, while that of the type III trajectories are aperiodic and have chaotic behaviors

Chaotic behaviors of stable second-order digital filters with two’s complement arithmetic

In this paper, the behaviors of stable second-order digital filters with two’s complement arithmetic are investigated. It is found that even though the poles are inside the unit circle and the trajectory converges to a fixed point on the phase plane, that fixed point is not necessarily the origin. That fixed point is found and the set of initial conditions corresponding to such trajectories is determined. This set of initial conditions is a set of polygons inside the unit square, whereas it is an ellipse for the marginally stable case. Also, it is found that the occurrence of limit cycles and chaotic fractal pattern on the phase plane can be characterized by the periodic and aperiodic behaviors of the symbolic sequences, respectively. The fractal pattern is polygonal, whereas it is elliptical for the marginally stable case

Autonomous response of a third-order digital filter with two’s complement arithmetic realized in cascade form

In this letter, results on the autonomous response of a third-order digital filter with two’s complement arithmetic realized as a first-order subsystem cascaded by a second-order subsystem are reported. The behavior of the second-order subsystem depends on the pole location and the initial condition of the first-order subsystem, because the transient behavior is affected by the first-order subsystem and this transient response can be viewed as an excitation of the original initial state to another state. New results on the set of necessary and sufficient conditions relating the trajectory equations, the behaviors of the symbolic sequences, and the sets of the initial conditions are derived. The effects of the pole location and the initial condition of first-order subsystem on the overall system are discussed. Some interesting differences between the autonomous response of second-order subsystem and the response due to the exponentially decaying input are reported. Some simulation results are given to illustrate the analytical results

Nonlinear behaviors of second-order digital filters with two’s complement arithmetic

The main contribution of our work is the further exploration of some novel and counter-intuitive results on nonlinear behaviors of digital filters and provides some analytical analysis for the account of our partial results. The main implications of our results is: (1) one can select initial conditions and design the filter parameters so that chaotic behaviors can be avoided; (2) one can also select the parameters to generate chaos for certain applications, such as chaotic communications, encryption and decryption, fractal coding, etc; (3) we can find out the filter parameters when random-like chaotic patterns exhibited in some local regions on the phase plane by the Shannon entropies

Autonomous response of a third-order digital filter with two’s complement arithmetic realized in parallel form

This paper investigates the output and state trajectories of a third-order digital filter with two’s complement arithmetic realized in parallel form. Although the output of the third-order digital filter seems to behave randomly, some regular patterns can be displayed on the plot of versus , where those regular patterns are similar to the second-order case. When the first-order subsystem is operated at the marginally stable points, the output of the third-order system is still mainly dependent on the behaviors of the corresponding second-order digital filter, even though overflow occurs. Explicit equations relating the trajectories of the system to the filter parameters and the initial conditions provide further insights into the behaviors of the system