CHAPTER 18 OPTIMAL S IGNAL P ROCESSING IN C AVITY R ING -D OWN S PECTROSCOPY Kevin K. Lehmann and Haifeng Huang Contents 1. Introduction 624 2. The Model 625 3. Generalized Least Squares Fit with Correlated Data 628 4. Weight Matrix for Model of Cavity Ring-Down Data 630 5. Detector Noise Limited Cavity Ring-Down Data 633 5.1. To average and then fit, or fit each decay and average the fit results? 638 6. Linearization of the Fit in Cavity Ring-Down Spectroscopy 638 7. Determination of Ring-Down Rate by Fourier Transform Method 641 8. The Successive Integration Method for Exponential Fitting 643 9. Analog-Detected Cavity Ring-Down 646 9.1. Phase shift method 646 9.2. Gated integrator method 649 9.3. Logarithm-differentiator method 650 10. Shot Noise Limited Cavity Ring-Down Data 651 11. Effect of Residual Mode Beating in the Ring-Down Decay 655 12. Conclusions 657 Acknowledgments 657 References 657 Abstract In this chapter, the authors present a systematic statistical analysis of cavity ring-down signal extraction. The traditional uncorrelated least squares fit can be generalized to the situation with data correlation (e.g. caused by data filtering, which is essential to minimize noise). If the data is sufficiently highly sampled, the effect of the data correlation can be included by introducing an effective variance of the data. This correction has substantial influence on the estimation of the standard error of the fit parameters for correlated data, especially for the fitted decay rate k 0 , because this determines the final sensitivity. For both the white noise dominated and the shot noise dominated situations, the sensitivity limit is given. The authors found that the bias of k in the white noise situation is normally very small and can be neglected. The authors also compared several commonly used alternative algorithms in cavity ring- down community. These mathods include linearized weighted least squares fit, deter- mining the decay rate by Fourier transform, corrected successive integration (CSI) 623