On the stationary points of the output power versus response zero performance surface in power inversion arrays

Abstract

The stationary points and transient behaviour of using zero tracking based techniques in a power inversion array are investigated in this paper. As a function of the response zeros, the output power performance surface has been found to possess no local minimum, although saddle points may exist at positions where some of the zeros are identical. Since the saddle points are unstable and the inputs are stochastic in nature, steepest descent based zero tracking algorithms will always converge to the global minimum, even though the zeros being adjusted may wander close to the saddle points for an appreciable amount of time. Nevertheless, with only a single asymptotic time constant, the use of zero tracking as discussed has been found to have a transient convergence which, when compared with the LMS algorithm of about the same implementation complexity, is generally much faster and more uniform with respect to the starting positions and the environment. Together with the advantages that the zeros are directly available and that the tracking behaviour is determined by the asymptotic time constant only, zero tracking is an attractive viable alternative for algorithms whose complexity is proportional to the size of the array.