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### Highly Accurate Frequency Interpolation of Apodized FFT Magnitude-Mode Spectra

*Volume 52, Number 1 (Jan. 1998) Page 134-138*

Goto, Yutaka

''Generalized'' interpolation (called GI alpha here) of fast Fourier transform (FFT) spectra apodized by a family of sin alpha ( X ) windows has previously been proposed. The GI alpha 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 alpha , limited to integer values, has been used for the GI alpha , we show in the present paper that the GI alpha with a real alpha value also gives an extremely good estimate of the true frequency from the sin alpha ( X )-apodized spectra. Thus, we intend to apply the GI alpha with the optimal values of alpha to FFT spectra apodized by any other window functions that are often used in Fourier spectroscopy. Simulation results show that the GI alpha is easier and more accurate than the KCe interpolation, which uses a family of interpolating functions [ KCe ( omega ) = ( a omega2 + b omega + 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 GI alpha is inversely proportional to the SNR. However, no obvious differences between the various window functions are observed in the presence of noise.