Cross-Spectrum Measurement Statistics: Uncertainties and Detection Limit

Area of Science:

• Signal processing
• Statistical analysis
• Noise characterization

Background:

• Measuring signals with multiple instruments introduces combined noise.
• Characterizing unknown signals (red noise) in the presence of instrument noise (white noise) is challenging.
• Existing methods like Karhunen-Loève transform (KLT) have limitations.

Purpose of the Study:

• To develop and validate a novel cross-spectrum method for signal noise estimation.
• To define a reliable estimator for signal characteristics.
• To compare the proposed method with the KLT.

Main Methods:

• Utilizing the real part of the cross-spectrum as an estimator.
• Characterizing the probability density function (pdf) using the Variance-gamma (VG) distribution.
• Applying Bayes' theorem to solve the inverse problem for noise level estimation.
• Performing extensive Monte Carlo simulations for validation.

Main Results:

• The Variance-gamma (VG) distribution accurately models the estimator's pdf.
• The method reliably provides an upper limit for noise levels across various degrees of freedom (DOFs).
• The VG method yields a slightly different signal level upper limit compared to KLT.

Conclusions:

• The cross-spectrum method with VG distribution is a robust tool for noise level estimation.
• VG distribution is highly reliable for signal analysis, validated by simulations.
• KLT may offer advantages by better utilizing available information for signal level estimation.