Extrapolation of band-limited frequency data using an iterative Hilbert-transform method and its application for Fourier-transform phase-modulation fluorometry

We propose a new band-extrapolation method using an iterative Hilbert-transform procedure, which is applicable for band-limited complex-frequency data obtained from a Fourier-transform phase-modulation fluorometer (FT-PMF). Without the band limitation, the real and the imaginary parts of the complex-frequency data obtained from the FT-PMF form a Hilbert-transform pair because of causality. However, the Hilbert-transform relation cannot actually hold because of the inevitable band limitation in the instrument. In order to alleviate this problem, we propose the band-extrapolation method. First, the real part of the complex-frequency data is Hilbert-transformed and the partial frequency data inside the band are replaced by the imaginary ones actually measured while those outside are kept as they are. Next, the corrected imaginary-frequency data are inverse-Hilbert-transformed again and the real-frequency data inside the band are replaced by those measured while those outside are kept. Such a procedure is iterated until changes in shapes of respective frequency waveforms are sufficiently small. Finally, we can obtain a desired time-domain waveform by the inverse Fourier transform of the pair of the extrapolated complex-frequency data. Numerical simulations showed that the proposed method was valid. A good agreement between two fluorescence decay waveforms of fluo-5N in calcium ion buffer, one obtained from the proposed method and the other obtained from the conventional time-correlated single-photon counting method, showed that the proposed method was useful.