Sinusoidal response of a second-order digital filter with two’s complement arithmetic

In this paper, results of the sinusoidal response case are presented. It is found that the visual appearance of the trajectory of the sinusoidal response case is much richer than that of the autonomous and step response cases. Based on the state space technique, the state vectors to be periodic are investigated. The set of initial conditions and the necessary conditions on the filter parameters are also derived. When overflow occurs, the system is nonlinear. If the corresponding symbolic sequences are periodic, some trajectory patterns are simulated. Since the state space technique is not sufficient to efficiently derive the sets of initial conditions and the necessary conditions on the filter parameters, a frequency-domain technique is employed to figure out the set of initial conditions. When the symbolic sequences are aperiodic, an elliptical fractal pattern or random-like chaotic pattern is found

Edge detection using fuzzy switch

This paper is to switch various edge filters via a fuzzy approach

Effect of non-polynomial input to a switching circuit

In this paper, the validity of the state-space averaging method is analyzed. We assume that the state-space piecewise method is an exact model for a fast switching circuit. Based on this model, we compute the error predicted by the state-space averaging method. It is found that the error for a polynomial input is bounded by two polynomials with the same order as that of the input. And the percentage error is bounded by a constant. Hence, if the acceptable level is within that constant, then the state-space averaging method can be applied. Similar analysis is carried out on a non-polynomial input. A sinusoidal function is chosen because of its wide applications on AC circuits. Although a similar result is obtained, the percentage error for the sinusoidal input is much greater than that of the polynomial input. Hence, the state-space averaging method may not be so good for the AC analysis

Stabilization of (L,M) shift invariant plant

In this paper, a lifting technique is employed to realize a single input single output linear (L,M) shift invariant plant as a filter bank system. Based on the filter bank structure, a controller is designed so that the aliasing components in the control loop are cancelled and the loop gain becomes a time invariant transfer function. Pole placement technique is applied to stabilize the overall system and ensure the causality of the filters in the controller. An example on the control of a linear (L,M) shift invariant plant with simulation result is illustrated. The result shows that our proposed algorithm is simple and effective

Admissibility of unstable second-order digital filter with two’s complement arithmetic

In this letter, we have extended the existing results on the admissible set of periodic symbolic sequences of a second-order digital filter with marginally stable system matrix to the unstable case. Based on this result, the initial conditions can be computed using the symbolic sequences. The truncation error of the representation of an initial condition due to the use of a finite number of symbols is studied

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

Step response of a second-order digital filter with two’s complement arithmetic

It is well known that the autonomous response of a second-order digital filter with two’s complement arithmetic may exhibit chaotic behaviors [1]. In this paper, results of the step response case are presented. Even though in the presence of the overflow nonlinearity, it is found that the step response behaviors can be related to some corresponding autonomous response behaviors by means of an appropriate affine transformation. Based on this method, some differences between the step response and the autonomous response are explored. The effects of the filter parameter and input step size on the trajectory behaviors are presented. Some previous necessary conditions for the trajectory behaviors, initial conditions and symbolic sequences are extended and strengthened to become necessary and sufficient conditions. Based on these necessary and sufficient conditions, some counter-intuitive results are reported. For example, it is found that for some sets of filter parameter values, the system may exhibit the type I trajectory even when a large input step size is applied and overflow occurs. On the other hand, for some sets of filter parameter values, the system will not give the type I trajectory for any small input step size, no matter what the initial conditions are

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

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

Nonlinear behaviors of first and second order complex digital filters with two’s complement arithmetic

For first order complex digital filters with two’s complement arithmetic, it is proved in this paper that overflow does not occur at the steady state if the eigenvalues of the system matrix are inside or on the unit circle. However, if the eigenvalues of the system matrix are outside the unit circle, chaotic behaviors would occur. For both cases, a limit cycle behavior does not occur. For second order complex digital filters with two’s complement arithmetic, if all eigenvalues are on the unit circle, then there are two ellipses centered at the origin of the phase portraits when overflow does not occur. When limit cycle occurs, the number of ellipses exhibited on the phase portraits is no more than two times the periodicity of the symbolic sequences. If the symbolic sequences are aperiodic, some state variables may exhibit fractal behaviors, at the same time, irregular chaotic behaviors may occur in other phase variables