Highly Accurate Frequency Interpolation of Apodized FFT Magnitude-Mode Spectra

Abstract

“Generalized” interpolation (called GIα here) of fast Fourier transform (FFT) spectra apodized by a family of sinα ( X) windows has previously been proposed. The GIα gives the highly accurate interpolated frequency by calculating the simple formula of frequency determination with the use of two squared ratios between three magnitudes nearest to the peak maximum on the apodized FFT spectrum. Although the value of window parameter α, limited to integer values, has been used for the GIα, we show in the present paper that the GIα with a real α value also gives an extremely good estimate of the true frequency from the sinα ( X)-apodized spectra. Thus, we intend to apply the GIα with the optimal values of α to FFT spectra apodized by any other window functions that are often used in Fourier spectroscopy. Simulation results show that the GIα is easier and more accurate than the KCe interpolation, which uses a family of interpolating functions [ KCe(ω) = ( aω2 + bω + c)e] proposed by Keefe and Comisarow. Finally, in the presence of noise we examine effects of damping and windowing on the frequency interpolation of FFT spectra. Because damping and windowing reduce the signal-to-noise ratio (SNR), we define anew the relative SNR by the ratio of the SNR of the apodized spectrum of a damped sinusoid to the SNR of the unapodized spectrum of an undamped sinusoid. Numerical calculation shows that the relative SNR varies, owing to damping rather than windowing. In fact, the observed frequency error roughly increases as the damping ratio increases for any window functions, as is expected from our previous investigation that the frequency error based upon the GIα is inversely proportional to the SNR. However, no obvious differences between the various window functions are observed in the presence of noise.