On Cyclic Autocorrelation and the Walsh-Hadamard Transform

It is shown that the complete set of circular shift invariants called the Q-spectrum, of the Walsh-Hadamard transform (WHT) of a periodic sequence is related to the cyclic autocorrelation of the given sequence through the Hadamard matrices. It is also shown that the modified WHT (MWHT) of the cyclic autocorrelation yields the Q-spectrum within some scale factors. This is analogous to the discrete Fourier transform (DFT) case, i.e., the DFT of the autocorrelation of a sequence yields the shift invariant power spectrum. The Q-spectrum can be computed efficiently using the MWHT rather than the WHT. A physical interpretation for the Q-spectrum is also provided. The motivation for this study is to show that while the WHT is inherently associated with the notion of dyadic time shifts, it does have analogous properties with respect to cyclic time shifts.