Time-Domain (Interferogram) and Frequency-Domain (Absorption-Mode and Magnitude-Mode) Noise and Precision in Fourier Transform Spectrometry

Abstract

It is desirable to be able to predict the precision of a repeated measurement, based on the result of a single measurement and knowledge of the noise level in the original (time-domain) data. We have previously shown that precision in determination of time-domain signal parameters (initial magnitude, frequency, and exponential damping constant for a noisy exponentially damped sinusoidal signal) from least-squares fit to a frequency-domain FFT spectrum is directly proportional to frequency-domain peak height-to-noise ratio and to the square root of the number of data points per peak width, with a proportionality constant which depends on the spectral type (absorption mode, magnitude mode) and peak shape (e.g., Lorentzian, Gaussian, etc.). In this paper, we show that precision in determination of those same parameters by least-squares fit to the time-domain signal itself is similarly related to time-domain initial amplitude-to-noise ratio and the square root of the number of data points per damping period. In addition, we show that although magnitude-mode spectral noise is well described by a Rayleigh distribution in the signal-free baseline segments of the spectrum, noise in the vicinity of a magnitude-mode spectral peak is more accurately described by a normal (Gaussian) distribution. We then proceed to show that determination of spectral parameters from a time-domain data set is more precise by a factor of √2 than estimates based on the FT absorption-mode spectrum. We further show that padding of the N-point time-domain data set by another N zeroes before FFT improves frequency-domain absorption-mode precision by the same √2 factor. Additional zero-filling does not improve spectral precision. Zero-filling has no effect on precision of spectral parameters determined from time-domain data, and variable effects on parameters determined from fits to magnitude-mode spectra. The above theoretical predictions are supported by analysis of simulated noisy data.