Journal of Accident Investigation

JOSEPH GREGOR
USING CURVE-FITTING TECHNIQUES TO
OBTAIN A FINAL TRANSFORM
Figure 8. Overlay of FDR(n) [designated by x’s] with CVR(n) [designated by o’s] with
the x-axis expanded to show the goodness of fit between the two data sets.
The exact form of the transform required to convert from
elapsed FDR time to elapsed CVR time may be found most
generally by performing a simple curve fit on the time-aligned
FDR and CVR data. If we subtract the value for C in Eq. (1) –
obtained using the cross-correlation function – the times found
for each corresponding on-key/off-key event should be identical
to within experimental measurement error. The magnitude
of this error is primarily driven by the 1 Sa/s sample rate 7 of
the FDR on-key/off-key data. The goodness of the resulting
transform may be seen most easily by producing a scatter plot of
the data. Figure 9 shows such a plot for data corresponding to
the simplest case, where the time-base operates at an identical,
constant rate in both units.
Each data point in this figure corresponds to a unique
on-key/off-key event, with elapsed FDR time plotted on the
x-axis and elapsed CVR time plotted on the y-axis. The solid
line corresponds to a minimum least squares fit to the data using
a linear transform corresponding to Eq. (2). The results of this
curve fit yield b = 1.0001 and C = 0.6 7, indicating that the
timebases in both the CVR and the FDR were operating at the
same rate to within a small fraction of a percent. Since the
7 Several other sources of error may apply in setting the error bars for this
calculation, but the sampling error is considered by far the largest source.
data are normalized prior to performing the curve fit, we find
C ≈ 0 ±1 s – well within the known error bar of ±1 Sa. Also
plotted in this figure is a variable, called Delta, representing the
difference between the elapsed CVR time as calculated using
the resulting linear curve fit, and the corresponding elapsed time
actually measured and plotted on the graph. This comparison
is performed for each on-key/off-key event to quantify the error
in calculating elapsed CVR time from the elapsed FDR time
using the calculated transform. The resulting error should fall
within the ±1 s error bar established by the 1 Sa/s sample rate;
the exact distribution is dependent on the relative phase of the
actual on-key/off-key event, the FDR sample acquisition time,
and the CVR sample acquisition time [since the CVR data
were also sampled at a 1 Hz rate to obtain a vector CVR(n)
appropriate for correlating with FDR(n)].
The situation highlighted in figure 10 is typical for a properly
functioning solid-state CVR and FDR. Each unit in this case
is operating from a precisely controlled digital time-base. In
this case, the transform required to move from elapsed FDR
time to elapsed CVR time should take the form of Eq. (1), with
C given by the cross-correlation function. The curve fit in
figure 9 serves as a validation that the data in FDR(n) and
CVR(n) were acquired accurately. Any problem data would
show up in the figure as an outlier, signalling the need for closer
scrutiny and possibly a re-run of the algorithm once the problem
with the data has been resolved.
0 NTSB JOURNAL OF ACCIDENT INVESTIGATION, SPRING 2006; VOLUME 2, ISSUE 1