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

Abstract

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.

Detection of Chaos in Some Local Regions of Phase Portraits Using Shannon Entropies

“…Similar behaviors may also occur in third-order digital filters with two's complement arithmetic [Chua, 1990b;Ling, 2002Ling, , 2003d.…”

“…To analyze the chaotic behaviors of the second-order digital filters with two's complement arithmetic, the periodicity and admissibility of the symbolic sequences were investigated [Chua, 1988[Chua, , 1990aGalias, 1992;Wu, 1993;Kocarev, 1993Kocarev, , 1996Yu, 2001;Ling, 2003d.…”

“…Since the state variables will converge to zero and no overflow will occur if both the eigenvalues of the secondorder digital filters with two's complement arithmetic are real and inside the unit circle, the Shannon entropies of the symbolic sequences are therefore zero. When the eigenvalues are complex and inside or on the unit circle, three types of trajectories are exhibited on the phase portraits [Chua, 1988[Chua, , 1990aGalias, 1992;Wu, 1993;Kocarev, 1993Kocarev, , 1996Yu, 2001;Ling, 2002Ling, , 2003aLing, , 2003d The organization of this letter is shown as the following. The system is first described in section 2, followed by the simulation results in section 3.…”