The optimum running approximation of band-limited signals based on new concept of multi-legged-type signals in a hyper domain
Yuichi Kida, Takuro Kida · Proceedings of SPIE, the International Society for Optical Engineering/Proceedings of SPIE · 2011
The minimization of the error associated with a running approximation by a filter bank is one of the most important problems of the signal processing. In this paper, for a set of vector-signals such that generalized Fourier transforms have weighted norms smaller than a given positive number, we present the extended optimum running approximation that minimizes various continuous worst-case measures of approximation error at the same time. In this discussion, we introduce a new concept of multi-legged-type signal that is a combined-signal of many one-dimensional band-limited signals. Backbone of this multi-legged-type signal is constituted with a series of small separable segments of the above one-dimensional signals that are determined by the proposed running approximation. Based on this concept, we propose an approximation method of the multi-legged-type signals and we prove that this approximation is the optimum. Then, we define measures of error that become the proposed measures of error in the position of the backbone made by the corresponding running approximation and become small about the other errors. Based on these measures of error, we prove that the presented extended optimum approximation minimizes various continuous worst-case measures of the running approximation error at the same time. As an application, multiple-input multiple-output/space division multiplexing system is discussed.