Radiometric errors in complex Fourier transform spectrometry.

Area of Science:

• Spectroscopy
• Optical Engineering
• Signal Processing

Background:

• Fourier transforms of asymmetric interferograms produce complex spectra.
• Understanding spectral noise is crucial for accurate measurements.
• Existing noise relations require rigorous validation in complex spectral analysis.

Purpose of the Study:

• To rigorously derive and validate the root-mean-square (rms) noise in the real part of a complex spectrum.
• To establish a variance equation for real spectra obtained through complex calibration.
• To quantify the impact of uncertainties in reference spectra on the final spectral variance.

Main Methods:

• Fourier transform analysis of asymmetric interferograms.
• Rigorous mathematical derivation of noise and variance equations.
• Complex calibration techniques using hot and cold reference spectra.

Main Results:

• Confirmed the commonly used relation for rms noise in the real part of the spectrum: sigmaR = 2X x NEP/(etaAomega square root(tauN)).
• Derived a new variance equation for real spectra: sigmaL2 = sigmaR2 + sigma(c)2 (Lh - Ls)2/(Lh - Lc)2 + sigma(h)2 (Ls - Lc)2/(Lh - Lc)2.
• The derived equations are valid under conditions where spectral noise and uncertainties are small relative to spectral differences.

Conclusions:

• The study provides a rigorous theoretical foundation for noise analysis in complex spectral data.
• The findings are essential for improving the accuracy and reliability of spectroscopic measurements, particularly in applications involving asymmetric interferograms.
• Accurate characterization of noise and reference spectra uncertainties is critical for precise spectral analysis.