Functional conversion of signals in the study of relaxation phenomena

Digital signal processing for relaxation data conversion

“…Therefore, integral transform (1) describing an ideal functional filter has digital counterpart -a digital functional filter (DFF), which from a logarithmically sampled input sequence x(r 0 q n ) produces a new logarithmically sampled output sequence y(r 0 q m ) [8]: where q is a ratio of geometric progression determining the rate of the logarithmic sampling, r 0 is an arbitrary normalization constant, h[n] is impulse response containing theoretically infinite number of coefficients, which shall be restricted to finite number N = N1 + N2 + 1 in practice. DFF (2) has a periodic frequency response [8] H ðe jl Þ ¼…”

“…In its turn, the formal theory of relaxing systems describing the formal causal relationships between input and output of the dynamic systems having monotonic decaying impulse responses breeds from the similar mathematics of the physical relaxation theories, such as dielectric [12,13], magnetic [14] and mechanic [11,12] ones, as well as from the general theory of signals and systems [10]. The bole includes the theory and practice of the digital functional filtering with logarithmic sampling [8] providing the tools for relaxation data conversion. Several branches grow up from the bole representing digital functional filters for the three main tasks of relaxation data conversion, i.e.…”

“…Several branches grow up from the bole representing digital functional filters for the three main tasks of relaxation data conversion, i.e. : (i) spectrum analysis and synthesis for data conversion from the time-domain representations to the frequency-domain representations, and vice versa [16], (ii) relaxation analysis for determination of distribution of relaxation and retardation times (DRRT) [8,17], (iii) the KramersKronig transform for computing the real and imaginary parts of a complex frequency function and verification of data and models for the causal consistency.…”

“…With the target to exploit a potential of the advanced signal processing, research has been carried out at the Institute of Polymer Mechanics of the University of Latvia resulting in developing a unified model-free approach [8] for the relaxation data transformations without employing extrapolation and numerical integration, which is based on the up-to-date signal processing concepts [9,10] and takes into account the specific relaxation behaviour [11][12][13][14]. The objective of the presented paper is to highlight the origins, philosophy and some basic practical aspects of digital signal processing of relaxation data.…”

“…[2][3][4][5][6][7], as special cases of digital filters implemented in various -mainly non-classical ways. Thus, basic idea of the approach is to execute relaxation data conversion in the logarithmically transformed argument domain by digital filters, which due to their specific application ideology have been named as functional filters [8].…”