Cal-sal Walsh-Hadamard transform

Walsh-Hadamard matrices are rearranged such that the first half of the rows represents cal functions in increasing order of sequency whereas the second half represents sal functions in decreasing order of sequency. The transform based on this rearrangement is called the Cal-Sal Walsh-Hadamard transform or (WHT)cs. General expressions for developing the elements of these matrices are developed. These matrices are decomposed into sparse matrix factors which lead directly to the fast algorithms similar to those for other forms of the WHT. The (WHT)csis useful in mapping an even or odd sequence since, in this case, at least one-half of the transform components will be zero.