Fourier Transforms from a Weighted Trace Map

The class of generalised Hadamard transforms includes the Fourier, generalised, discrete Fourier, Walsh-Hadamard, complex Hadamard and reverse jacket transforms. The generalised Hadamard transforms may by partly classified by signal length, by group of entries in the transform matrix and by a recently introduced third parameter, the jacket width of the transform matrix. Here we introduce a weighted trace map, which realises the Fourier transform as an exponential weighted sum of Galois ring traces. We give examples of Fourier transforms with jacket width 0, jacket width 1 and maximum jacket width (half the signal length). We show the Fourier transforms of length 4^k with entries in {plusmn1, plusmni} obtained using the weighted trace map from the Galois ring GR(4,k) have jacket width 2^k-1