Kramers–Kronig relations generalized: on dispersion relations for finite frequency intervals A spectrum-restoring filter

Abstract

When the real (imaginary) part of the transfer function of a causal linear system is known for all frequencies, the imaginary (real) part may be calculated for all frequencies from the Kramers–Kronig relations. For physical systems, the real part could be the refractive index or the amplitude, and the imaginary part could be the extinction coefficient or the phase. Experimentally these quantities are known only for limited-frequency intervals. This paper presents generalized Kramers–Kronig relations, from which the real and imaginary parts may be calculated for all frequencies from knowledge of these parts for at least partly overlapping frequency intervals. When the procedure is applied to experimental data, errors are introduced. Certain types of errors of the known real and imaginary parts completely destroy the possibility of calculating the unknown parts, whereas others give negligible errors. The existence of a filter with the property of allowing the restoration of a truncated spectrum is established. The transfer function and the impulse response function of this filter are given.