Generalized phase spectrum

Abstract

The generalized discrete transform (GT) <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">r</inf> and its modified version (MGT) <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">r</inf> have been defined and their properties have been developed. The phase or position spectra and the shift-invariant power spectra for these transforms have been developed. For the (GT) <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">r</inf> family, these transforms range from the Walsh-Hadamard transform (WHT) to the discrete Fourier transform (DFT) whereas for the (MGT) <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">r</inf> group, the range is from the modified WHT (MWHT) to the DFT. The power and phase spectra for these transforms represent groups of frequencies unlike the individual frequency composition characteristic of the DFT. Recently, it was shown that the input data can be recovered from the power and phase spectra of the WHT. This concept is now extended to the generalized transforms and it is shown that the original signal can be reconstructed from the phase and power spectra of any of the members of the (GT) <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">r</inf> family. Also an efficient algorithm for evaluating the generalized phase spectrum for various cyclic shifts of the data sequence is developed.