Algebraic Separation of Biexponential Time Constants Using Cross-Ratio Invariance

The extraction of time constants from biexponential signals is a fundamental problem in physics, engineering, and biomedical science. Traditional approaches rely on nonlinear least squares fitting, which suffers from ill-conditioning, multiple local minima, and sensitivity to initial parameter estimates. This paper presents a novel algebraic method for extracting biexponential time constants that is completely independent of the amplitude coefficients. The method exploits the Möbius (linear fractional) structure of the log-derivative function and uses the cross-ratio, a projective invariant, to directly extract the rate difference parameter. A subsequent quadratic equation yields the remaining parameters through closed-form algebra. For noisy measurements, we introduce a manifold projection technique that fits noisy samples to the Möbius curve, achieving robust extraction with sub-2% error at signal-to-noise ratios of 50 dB. The amplitude invariance property is preserved even in the presence of noise. We provide complete mathematical derivations, experimental validation across multiple noise levels and amplitude ratios, and practical implementation guidelines. The method has immediate applications in battery diagnostics, medical imaging, spectroscopy, and any domain requiring reliable separation of overlapping exponential decay processes.