Abstract
A new family of highly efficient interpolating functions, the <i>KCe</i> functions, <i>KCe</i>(ω) = (<i>a</i>ω<sup>2</sup> + <i>b</i> ω + <i>c</i>)<sup>e</sup>, where <i>e</i> is the exponent, is developed for three-point frequency interpolation of discrete, magnitude-mode, apodized Fourier transform spectra. The family is characterized by high interpolation accuracy and ease of implementation. Various members of the family can be generated by varying the exponent. Prior work from this laboratory indicated that the parabola is the interpolating function of choice for interpolation of discrete, apodized magnitude spectra. We show here that, compared to parabolic interpolation, <i>KCe</i> interpolation typically gives residual systematic errors which are lower by between one and two orders of magnitude. These systematic errors are analytically derived and the efficacy of interpolation is rigorously examined as a function of the <i>KCe</i> exponent, the number of zero-fillings, the amount of damping in the transient, and the window function used to apodize the spectrum. For Hanning-apodized spectra, the <i>KC</i>5.5 function gives the lowest residual systematic errors, which are typically 15 times less than those remaining after parabolic interpolation. Similarly, the <i>KC</i>6.6 function is optimal for Hamming-apodized spectra (22 times better than parabolic interpolation) and the <i>KC</i>9.5 function is optimal for Blackman-Harris-apodized spectra (80 times better than parabolic interpolation). By extrapolation from other optimal <i>KCe</i> functions, we estimate that the optimal <i>KCe</i> function for interpolation of Kaiser-Bessel-apodized spectra is <i>KC</i>12.5. Analytical formulae for propagation of random errors in spectral intensity into random errors in interpolated frequency are derived for parabolic interpolation and for <i>KCe</i> interpolation. These error propagation formulae give random errors which are inversely proportional to the SNR of the spectrum. These formulae are evaluated with the appropriate <i>KCe</i> exponent for each of the Hanning, the Hamming, and the Blackman-Harris windows. In all cases we find that the random error is essentially independent of both window type and interpolation scheme. While zero-filling prior to interpolation reduces the residual systematic frequency interpolation error, it <i>increases</i> the random frequency error. The increase in random error with higher levels of zero-filling is explained. Because the random errors are proportional to noise level, the optimal number of zero-fillings varies with SNR. If the apodizing window is chosen to match the dynamic range of the spectrum, as we have previously recommended, then the systematic error for <i>KCe</i> interpolation of non-zero-filled spectra is so low that the overall error is dominated by the random error. In this case, <i>KCe</i> interpolation is, for all intents and purposes, exact. Since the random error is minimized by no zero-filling, the lowest overall error will be achieved by a combination of no zero-filling and <i>KCe</i> interpolation. In constrast, the minimum total error for parabolic interpolation is achieved by interpolation of the once-zero-filled spectrum. A further advantage of <i>KCe</i> interpolation, over and above its lower total error, is that <i>KCe</i> interpolation obviates the need for zero-filling.
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