Large symmetric π transformations for Hadamard transforms

Many multiplexing instruments utilize the fast Hadamard transform (FHT) to demultiplex the signal. In the past, the FHT includes the π1 and π2 transformations to reorder vectors before and after a Sylvester-type Hadamard transform. Although the computational effort involved in the π1 and the Sylvester-type Hadamard transform scales as n log2n, calculating the π2 transformation (which only has to be done once) scales as n2. Recently Gunson (1980) has suggested a method by which the π transformations are symmetric, that is π2 = π1. We have calculated a complete set of symmetric π transformations for FHT of sizes 23 to 230. Special emphasis has been placed on the phase of the π transformation so as to have the correct phase in the demultiplexed signal.