Ordering of Walsh Functions

The construction of Walsh functions is derived by means of the concepts of "symmetric copy" and "shift copy." Recursive relations based on the Kronecker product of matrices are also deduced from these concepts. There is a fourth useful ordering of the Walsh functions, called here X-ordering, in addition to the three known orderings usually referred to as Walsh-, Paley-, and Hadamardordering. An X-ordering function has the following features: lower order numbers of X-ordering correspond to even functions; higher order numbers of X-ordering correspond to odd functions; even order numbers of X-ordering correspond to lower sequencies; odd number of X-ordering correspond to higher sequencies. Finally, relations between the four orderings are given.