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 correspondence 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