Laser Doppler Data Processing  Techniques  Local Normalization 

Local Normalization is a technique, where the correlation coefficient $\varrho $ is derived from the primary correlation estimate by normalization with the two correlation estimates of the appropriate selfproducts $$\varrho \left(\tau \right)=\frac{{\sum}_{i}{\sum}_{j\ne i}\left(\left{t}_{j}{t}_{i}\tau \right<\frac{\Delta \tau}{2}\right){w}_{i}{w}_{j}{u}_{i}{u}_{j}}{\sqrt{\left[{\sum}_{i}{\sum}_{j\ne i}\left(\left{t}_{j}{t}_{i}\tau \right<\frac{\Delta \tau}{2}\right){w}_{i}{w}_{j}{u}_{i}^{2}\right]\left[{\sum}_{i}{\sum}_{j\ne i}\left(\left{t}_{j}{t}_{i}\tau \right<\frac{\Delta \tau}{2}\right){w}_{i}{w}_{j}{u}_{j}^{2}\right]}}$$where ${u}_{i}$ are understood as values with subtracted mean. However, due to noise, the denominator has a bias, while the numerator has not. Therefore, the correlation coefficient should be denormalized by an estimate of the variance, which is also affected by noise. $$R\left(\tau \right)=\varrho \left(\tau \right)\cdot \frac{{\sum}_{i}{w}_{i}{u}_{i}^{2}}{{\sum}_{i}{w}_{i}}$$This estimate is not affected by noise, while the estimation variance decreases compared with the primary correlation estimate. If needed, a renormalization can finally be done, which then is not biased due to noise. $$\varrho \left(\tau \right)=\frac{R\left(\tau \right)}{R\left(0\right)}$$original papers:
combined with fuzzy slotting: combined with fuzzy slotting and forwardbackward arrivaltime weighting:
adapted to direct spectral estimation: 