Frequency Interpolation of Discrete, Apodized, Magnitude Lineshapes

Abstract

All Fourier spectrometers have a residual error in frequency measurement arising from the discrete nature of the experimental Fourier spectrum. This residual error is a systematic error which has a maximum value of half the channel spacing in the discrete spectrum. This systematic error can be reduced by interpolation of values on the discrete lineshape. The residual error remaining after interpolation has not yet been determined for apodized Fourier spectra. In this work, a systematic study of frequency interpolation of discrete, apodized, magnitude-mode lineshapes is reported. Absolute maximum frequency errors as a percentage of the discrete channel spacing are reported in graphical and tabular form as a function of the type of apodization window, the type of function used for three-point frequency interpolation, the number of zero-fillings, and ( T/ r), the ratio of the acquisition time to the relaxation time of the time domain signal. The results allow independent choice of the window function most appropriate for the dynamic range of the spectrum and the interpolating function/zero-filling level which optimizes the accuracy of frequency measurement. General observations are (1) that the interpolation error is reduced by an order of magnitude for each additional level of zero-filling and (2) that the interpolation error is essentially independent of T/r. For the Hanning window, the Hamming window, the three-term Blackman-Harris window, and the Kaiser-Bessel window, the parabola is the interpolating function of choice.