Laser Doppler Data Processing

Techniques - Forward-Backward (Inter-)Arrival-Time Weighting

If the inter-arrival times as used for the calculation of the mean or the variance ($w_i=t_i-t_{i-1}$) are used as weighting factors for the estimation of the correlation function or that of the power spectral density, there will be a correlation between the respective time lags in the corresponding correlation function and the inter-arrival times. The reason for that is that a cross-product of velocities contributing to the correlation function requires that the two samples have exactly the respective inter-arrival time. If $u_i$ and $u_j$ are two velocity samples at times $t_i$ and $t_j$, where $t_i<t_j$, then the inter-arrival weight of the second sample $w_j=t_j-t_{j-1}$ interferes with the time lag $t_j-t_i$ of the correlation function. Therefore, the usual inter-arrival times cannot be used directly as weighting factors for the correlation or the spectrum estimation. Instead, a cross-product of two velocity samples $u_i$ and $u_j$ can use the two inter-arrival weights

where $u_i$ is assumed to be the first and $u_j$ is the second sample ($t_i<t_j$). In that case, the arrival times are ordered as $t_{i-1}<t_i<t_j<t_{j+1}$, and the two weights are independent of the inter-arrival time $t_j-t_i$ of the two samples.

However, also here, this weighting scheme works efficiently only at high enough data densities of the order of ten samples per integral timescale or larger. Therefore, Transit-Time Weighting should generally be preferred.

original papers (combined with local normalization and fuzzy slotting):

• Nobach H (1999): Processing of Stochastic Sampled Data in Laser Doppler Anemometry. Proc. of the 3rd International Workshop on Sampling Theory and Applications, 11-14 August 1999, Loen, Norway
• Nobach H (2002): Local time estimation for the slotted correlation function of randomly sampled LDA data. Experiments in Fluids 32, 337-345

adapted to the direct spectral estimation:

• Nobach H (2015): Corrections to the direct spectral estimation for laser Doppler data. Experiments in Fluids 56:109

overview and comparison:

• Damaschke N, Kühn V, Nobach H (2018): A Fair Review of Non-parametric Bias-free Autocorrelation and Spectral Methods for Randomly Sampled Data in Laser Doppler Velocimetry. Digital Signal Processing, 76, 22-33