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