The Fourier-transform extracted exponential decay time constant as a function of frequency. The dashed curve is calculated according to Kirchner et al. (Ref. 1), the dotted curve is calculated according to Mazurenka et al. (Ref. 2), and the solid curve is calculated according to the procedure (DFT-1) described in the text. The inset shows the behavior as the curves approach the low-frequency limit.
The dependence of the fitting time on the number of data points in the exponential waveform. The solid curve is for the Levenberg–Marquardt, the dotted curve is calculated according to the DFT algorithm, and the dashed curve is calculated according to the linear regression of the sum. The inset shows the nonmonotonic behavior of the DFT algorithm at highly factorizable values of .
Histograms of evaluated from noisy simulated data using various algorithms. The upper figure is for data sets with 50 points, the lower figure for data sets 2048 points. Note the different axes. For all curves the noise level is 1%, the true value of is , and the total length of the data waveform is . The curve for FFT-FPR is not shown in the lower figure as it would obscure the curve for DFT-1.
The standard deviation 10 000 separate estimates of from 1000-point waveforms of varying length. The noise level is 1%. The inset shows the minimum of the curves in detail. Note that different methods have different optimal record lengths.
An example of the ring-down data used to test the speed and quality of the two high-speed data reduction techniques detailed in this paper. A single ring-down signal was digitized by eight separate ADCs within a single acquisition card at .
The results of real-time acquisition and fitting of the eight-channel ring-down signals over the course of about 500 laser shots . The DFT-1 routine is the top panel, while the LRS procedure is the lower panel. The fits are not to the same 500 laser shots. The solid lines in each figure are linear fits to the time dependent ring-down loss data.
The histograms of the results presented in Fig. 6. The top panel is for the DFT-1 data reduction procedure, while the bottom panel is the LRS method. Clearly, both obtain the same result, within error and the long-term drift of the system noted above. Both also clearly produce normally distributed loss data that can be expected to improve with signal averaging.
Abbreviations and brief descriptions of algorithms discussed in the text.
The average and standard deviation of recovered from various fitting methods. 10 000 separate 10 000-point raw data waveforms with 1% noise were analyzed.
Weighting coefficients for the frequency components for the evaluation of from a weighted average of the first five frequency components of the FFT.
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