Scale-Shift and Harmonic analysis approach to the Mellin transform for Discrete-time signals

We investigate the scale-shift operator for discrete-time signals via action of hyperbolic Blaschke group. Practical implementation issues are discussed and given any arbitrary scale, in framework very classical linear filtering. Our group theoretical standpoint leads to a purely harmonic analysis definition Mellin transform signals. Explicit analytical expressions atoms Fourier-Mellin decomposition provided along with simple algorithm their computation. The so-defined also allows us establish mathematical equivalence between wavelet coefficients signal corresponding Voice-transform generated by well-chosen unitary representation Hyperbolic group, Hardy space unit disc.