Time-domain analysis of cross correlation for time delay estimation with an autocorrelated signal

On the application of cross correlation function to subsample discrete time delay estimation

“…However, estimating Δ becomes non-trivial, if Δ is a fractional multiple of T and an arbitrary precision of Δ is desired. The current technique is to fit a parabola function with points 12 (n 0 − 1), 12 (n 0 ) and 12 (n 0 + 1), and estimate Δ by the peak position of the fitted parabola [9][10][11]. This practice is justified if one can show that the parabola is a good approximation of R 12 (τ ) in the neighborhood of its peak.…”

“…Since υ 1 and υ 2 are uncorrelated, there is (2.6) where N is the length of samples used in calculation. 12 (n) is an approximation of the Dirac sampled value of continuous function R 12 (τ ) at time nT − Δ. The accuracy of 12 (n) is affected by the level of measurement noises υ 1 and υ 2 and the sample length N .…”

“…The three parameters are determined as follows. Let 12 (n 0 − 1) = f g (−1), 12 (n 0 ) = f g (0) and 12 Let 12 (n 0 ) = R 12 (0), 12 (n 0 −1) = R 12 (−1) and 12 (n 0 +1) = R 12 (1), in Fig. 6 we plot true curves of R 12 (τ ) in the neighborhood of its peak together with the fitted curves of R 12 (τ ) by the parabola and Gaussian models.…”

“…Time delay estimation of two analog signals through cross-correlation has been studied by many authors [6][7][8][9][10][11][12][13]. Knapp and Carter gave a maximum likelihood estimator of the relative delay between two continuous signals [6].…”

On Cross Correlation Based Discrete Time Delay Estimation

“…However, this method needs to know the spectra of signals and noise and it applies to analog signals only. For digital systems a popular approach of time delay estimation is to locate the peak of the two discrete signals' CCF [4][5][6]. The delay is generally not an integral multiple of the sampling period.…”