Walsh Functions Obtained from the Frame Multiplication of the Split-Extension of Hypercomplex Numbers

Abstract

Communication theory was founded on the system of sine-cosine functions in 1927. However, the theory of Walsh functions is slowly replacing the theory of sine-cosine functions. Walsh functions use sequency while sine-cosine functions use frequency. The concept of frequency is a consequence of these functions, since frequency is defined as a parameter f in sin 2ᴨft and cos 2ᴨft. So far, Walsh functions are the only known functions with desirable features comparable to sine-cosine functions for use in communication. Walsh functions were first introduced in Mathematics by Joseph L. Walsh in 1923. Since then, various researchers have constructed Walsh functions using various techniques. In this project we constructed Walsh functions from the frame multiplication of the split-extension of hyper complex numbers. The multiplication tables for the basis elements of complex split extension, quaternion split extension and octonion split extension were considered.We then obtained matrices using the signs of the values from each of the tables.The rows in each of the matrices obtained were investigated to determine its properties.It is from these rows of each matrix that we constructed Walsh functions and determined the distinct number of the functions