Fourier and Hadamard transform spectrometers: a limited comparison

Area of Science:

• Spectroscopy
• Optical Engineering
• Signal Processing

Background:

• Fourier transform spectrometers (FTS) and Hadamard transform spectrometers (HTS) are key spectroscopic instruments.
• Understanding their performance limitations, especially in detector-noise limited scenarios, is crucial for accurate spectral analysis.

Purpose of the Study:

• To establish and compare an encoding figure of merit for detector-noise limited FTS and HTS.
• To analyze the sources of error and performance differences between FTS and HTS systems.

Main Methods:

• Development of an encoding figure of merit for detector-noise limited spectroscopy.
• Comparative analysis of mean square errors for FTS and HTS systems with N spectral densities.
• Evaluation of modulation characteristics and orthonormality of spectral functions in FTS.

Main Results:

• Detector-noise limited FTS exhibits mean square errors twice as large as analogous HTS systems for N measurements.
• FTS limitations stem from cosine-squared modulation and non-orthonormal spectral functions.
• Fellgett's advantage for Michelson interferometers is (N/8)(1/2), not (N/2)(1/2) as commonly cited.

Conclusions:

• HTS offers superior performance over FTS in detector-noise limited conditions.
• The theoretical performance limit for spectral analysis, while high, may be unachievable with current practical FTS instruments.