A new approach for generating function sequences—Hybrid Bridge functions derived from Walsh functions

This paper reviews the essence of the Walsh and Bridge functions from the point of view of the recursive relationship; it also unifies their expressions. A new kind of function—the hybrid Bridge function—is constructed from the definition of the hybrid matrix, in which the row vectors are taken from the Walsh and Bridge function matrices. We also propose a new approach for generating function sequences; one that uses the column vectors of the hybrid Bridge function matrix as the new function sequences. These sequences are able to adjust the number of zeroes flexibly; thus, resolving the constraint on the application of excessive zeroes in Bridge function sequences, while at the same time maximizing the research field for function sequences. Through strict mathematical analysis, it is proved that when the initial matrix order of the parent matrices and the similarity of the hybrid matrices meet certain conditions, the hybrid function sequences exhibit good orthogonal characteristics. This provides a theoretical basis for its further applications in communication systems.