FIR Kramers–Kronig transformers for relaxation data conversion

It is shown that relaxation data conversion by the Kramers–Kronig (KK) relations can be treated as a filtering problem of band-unlimited relaxation signals in the Mellin transform domain. Based on this concept, KK relations are implemented in the form of FIR filters with the logarithmic sampling. It is demonstrated that KK transformers have sampling rate dependent impulse and frequency responses and only calculation of the imaginary part from the real part can be implemented by a computationally realisable filter. The performance of different types of transformers is studied. Approximately inversely proportional relationship is established between the error and the frequency range of input signal used for computing an output sample. It is found that transformers with even number of coefficients provide better performance than those having an odd number if input data are available within a frequency range wider than four decades. Usage of additional transformers with the shifted or shortened impulse responses is investigated for eliminating shortening of usable output sequence due to filter delay. The simulation results are provided for an elementary relaxation system (a single Debye relaxation) and systems described by the Havriliak–Negami dispersion relation. Some known numerical KK transform techniques are analysed in the functional filtering context.