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7.3. Time of Arrival measurements
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Figure 7.9: Binary images for a weak chirp signal, at two thresholds
threshold (1 and 5 standard deviations above the mean). Figure 7.9 shows the same
thresholds for the weak signal from Figure 7.6.
In the case of the strong signal, the low threshold allows through too much —
the Hough transform would produce many possible lines which are not actually as
strong as the correct line. In the case of the weak signal, the high threshold does
not allow through enough — the signal is so weak that the Hough transform would
not have enough data points to produce any reasonable lines.
In order to fix this, it is possible to take several Hough transforms for each
image, each using a different threshold for the binary conversion. These can then
be summed so that the peak is visible for weak and strong signals.
7.3.3
7.3.3.1
Fourier Transform and Cross Correlation
Cross Correlation
It is possible to search for a chirp signal by cross correlating a known chirp against
the signal[234]. A quadratic chirp can be easily detected through cross correlation,
as no part of its signal is repeated. As a chirp can be easily converted into an
impulse (and vice versa), they can provide an accurate timestamp while remaining
sufficiently long that they can be detected over long distances.[244]
The chirp in the pulse is of a known length c. If we are continuously searching
for an unknown but lengthy period of T samples for a signal, we can split the task