Weighted cross-correlation functions

There are, however, limits in the resolution of heterogeneous populations. In the particular case of multiple diffusing fluorescent species, for example in binding studies, the diffusion times must differ by a factor of 1.6-2.0 to be resolved, corresponding to a 4-8-fold increase in molecular weight. This limitation is alleviated in dual-color fluorescence cross-correlation spectroscopy (DC-FCCS) [116], in which the fluorescence fluctuation is simultaneously recorded in the same observation volume for two different species labeled with distinguishable fluorophores. The fraction of double-labeled and hence associated species is derived from the cross-correlation function between the two fluorescences (fig. 3.4B). 

The weighting function of the system g(9) can be estimated with Eq. (31) from the cross correlation function of the PRBS tracer experiment. The weighting function can be used for the validation of the model and the determination of the model parameters. Generally it is sufficient to measure for two PRBS periods. Stationary conditions are adjusted dining the first period. The measured tracer output concentrations of the second period are used for the calculation of the cross correlation function. 

The adjustment of measurements to compensate for random errors involves the resolution of a constrained minimization problem, usually one of constrained least squares. Balance equations are included in the constraints these may be linear but are generally nonlinear. The objective function is usually quadratic with respect to the adjustment of measurements, and it has the covariance matrix of measurements errors as weights. Thus, this matrix is essential in the obtaining of reliable process knowledge. Some efforts have been made to estimate it from measurements (Almasy and Mah, 1984 Darouach et al., 1989 Keller et al., 1992 Chen et al., 1997). The difficulty in the estimation of this matrix is associated with the analysis of the serial and cross correlation of the data. 

The experiment observes a distorted spectral function (11.11) characterised by the experimental orbital v) (q) (11.15). The summed cross sections for the a manifold are characterised by the normalised orbital i/)a(q), in view of (11.17). We say that the orbital is split among the states I/) of the manifold by the ion correlations. The weight of each state in the manifold sum is S/(a). The differential cross sections for the states of the manifold are in the ratios of these weights. 

---