The Haar-Recursive transform and its consequence to the Walsh-paley spectrum and the autocorrelation function - Rafiq, Hashum M.; Siddiqi, Mohammad Umar (IJERA Publication, 2016)
The Walsh and Haar spectral transforms play a crucial part in the analysis, design, and testing of digital devices. They are most suitable for analysis and synthesis of switching or Boolean functions (BFs). It is well known that, the connection between the two spectral domains is given in terms of the Walsh-Paley transform. This paper derives an alternative expression of the Walsh-Paley transform in terms of the Haar transform. The work demonstrates the possibility of obtaining both the Haar spectrum and the Walsh-Paley spectrum using only the Haar transform domain. The paper introduces a new Haar-based transform algorithm (Haar-Paley-Recursive Transform, HPRT) in the form of a recursive function along with its fast transform version. The new algorithm is then explored in its interpretation of the Walsh-Paley transform and its connection to the Autocorrelation function (ACF) of a BF. The connection is given analogously in terms of the Haar-Paley power spectrum via the Wiener-Khintchine theorem. The paper then presents the simulation results on the execution times of both derived algorithms in comparison to the existing Walsh benchmark. The work shows the advantages of using the Haar transform domain in computing the Walsh-Paley spectrum and in effect the ACF.