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.
This letter shows some counter-intuitive simulation results that the symbolic sequences and the state variables of a digital filter with two's complement arithmetic and arbitrary initial conditions and order will be eventually zero when all the filter parameters are even numbers, no matter the system matrix of the filter is stable or not.
SUMMARYIn 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. However, we seldom design a 'digital filter' operating in the marginally stable region or in the unstable region. Practically, we usually design digital filters operating in the stable region. In this paper, chaotic and related behaviors of stable second-order digital filters with two's complement arithmetic are investigated.
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