Approximation Methods in Multidimensional Filter Design.

Abstract

Abstract : First, new stability tests were developed for multidimensional recursive digital filters and any double-ended n-dimensional noncausal linear processor which is said to be stable if its impulse response decreases exponentially in all 2-n directions. It was than shown that the impulse response operator for a 2-D discrete Hilbert transformer, though not by itself sum-separable, becomes so after appropriate classification. Subsequently it was proved that the multiplicative complexity of computation of a 2-D DHT is not greater than twice the sum of multiplicative complexities of two 1-D DHT's. Subsequently, the 1-D matrix Pade approximation problem via a three-term recursive computation scheme was tackled as a prelude to the solution of 2-D and n-D cases. Specifically, given a 1-D matrix power series, it was shown that a recurrence relation relates the ((L+1)/(M+1)), (L/M), ((L-1)/(M-1)) order Pade approximants, which are guaranteed to exist provided a certain rank condition is satisfied by characterizing matrices possessing block-Hankel structure. Attention to stability, algebraic computational complexity and approximation were necessary because efficient implementation of stable recursion is desired. (Author)