Walsh Function Generators for Minimum Orthogonality Error

The design procedure of a new synchronous counter type of Walsh function generator for the generation of a set of Walsh functions with the least possible error in orthogonality is developed. Each flip-flop of the counter generates synchronously one particular Walsh function in the interval 1 of normalized time during one cycle of counting. The usual design procedures for synchronous counters are not applicable because of the large number of logic variables. Using a two-dimensional plot of the enabling inputs, it turns out that the T flip-flop is the most suitable type. It is shown that by using the symmetry properties of the enabling input patterns for a generator consisting of n T flip-flops, only (log2 n)- 1 standard time sequences, from which the remaining could be derived, need be generated. These time sequences can be easily obtained by decoding the outputs of those flip-flops generating the subset of Rademacher functions. The procedure is illustrated by taking an example of the generation of the first 16 Walsh functions.