Effects of Noise on the Interpolation Accuracy for Apodized FFT Spectra of Time-Domain Damped Signals

Abstract

Interpolation formulas for the apodized magnitude-mode fast Fourier transformed (FFT) spectra determine accurately the frequency, damping constant, and amplitude of time-domain damped signals. However, additive noise causes a large amount of error in interpolation. In this paper, we obtain, theoretically, the frequency-domain signal-to-noise (S/N) ratio due to windowing by the function of sin^α(X) and quantization with finite bit-length analog-to-digital (A/D) converters. Then, with the use of the squared ratios between three magnitudes nearest to the peak maximum on the apodized FFT spectrum, we derive the relationship equation between the frequency error and the S/N ratio. The results obtained by computer simulation of experimental conditions (i.e., sampling, quantization, windowing, FFT, and interpolation) for the Hanning window (α = 2) agree well with the theoretical calculations; the frequency errors decrease with increasing bit-length of the A/D converter. These observed errors are unavoidable because A/D converters are indispensable for measurements with Fourier transform spectrometers. Furthermore, as shown theoretically, the observed accuracy of interpolation is inversely proportional to the S/N ratio, provided that the S/N ratio is below the value due to quantization and windowing.