Some Notes on the Walsh Functions

The complete set of orthogonal functions of binary variables called Walsh functions can be obtained as direct products of the subclass of these functions known as Rademacher functions. The complete set of Walsh functions can be conveniently represented by a square matrix of l´s and ¿1´s, which, when normalized, is an orthogonal matrix. The correct Rademacher functions to multiply together to obtain a specified Walsh function can be determined without the use of recursion formulas by reference to a simple modification of the reflected binary code. A binary equivalent of the Walsh matrix, consisting of 0´s and 1´s, can be generated directly by multiplying together, modulo 2, two simpler matrices easily obtained from the reflected binary code and natural binary code, respectively. A modified Walsh matrix that has a simple recursive structure can be obtained by rearranging the rows of the fundamental Walsh matrix. A binary equlivalent of thle former can also be generated directly by multiplying together, modulo 2, two simpler matrices easily obtained from the natural binary code alone. A transformation matrix for direct conversion of the Walsh matrix from its fundamental to modified (or modified to fundamental) form can be written by observing how the rows of the natural binary code are rearranged to form the modified reflected binary code.