Laser Doppler Data Processing

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

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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

w i = t i - t i-1 (backward inter-arrival time) w j = t j+1 - t j (forward inter-arrival time)

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):

adapted to the direct spectral estimation: