View Full Version : Delay and wave-forms

07-14-2005, 12:19 AM
Dear readers

With an eye on the previous discussions of *electromechanical delay*
(EMD) some comments, in term of a few simple examples, may be due
concerning possible definitions of *delay*.

For specificity let's discuss signals av the form

(1) S_i(t) = t*exp(-t/t_i) (for t > 0 and = 0 for t < 0).

These start at t = 0, reach a maximum at t = t_i ("characteristic
time"), and then die off. E.g. muscle twitches are often modelled using
(1). If we have two signals S_1 and S_2 with characteristic times t_2 >
t_1, they both start at the same time (t = 0) but S_2 will reach its
peak later then S_1. Thus, there is a peak-to-peak delay = t_2 - t_1,
yet there is no start delay (latency) since both start at t = 0. It is
this form (mathematically speaking) of peak-to-peak delay that may
dominate, say, the EMG-to-force *delay* (100 ms or larger). The so
called EMD seems to be associated with the start-up-delay (typically
around 10 ms) whose estimation indeed may be sensitive to how one
chooses the threshold for defining the beginning of a signal (force,
EMG). The smaller slope of the signal vs time, the more sensitive is the
obtained threshold-crossing time on the threshold level and the *noise*.

If one calculates the cross-correlation C(t) for S_1 and S_2 (t_2 > t_1)
above then it does not in general reach maximum for t = t_2 -t_1 but for
(in case i got it right ...)

t = (t_2/(t_1 + t_2))*(t_2 - t_1).

This example shows how peak-to-peak delay may differ from the delay
based on a cross-correlation analysis. The results of such methods
depend on the typical wave-forms of the signals. As an example, we may
have two signals, one being of the form of a series of up-slope ramps,
the other of the form of down-slope ramps. Although they may start and
end at the same times, cross-correlation may suggest a non-zero delay,
or phase shift, up to about 40% of the ramp length. Another,
physiological example, is where one has to compare a burst-like
(*phasic*) EMG signal with a more continuous (*tonic*) EMG wave-form.

Any of this does of course not mean that delays etc are just
*artifacts*, only that the range of variations of the delays, and their
interpretations, depend on the chosen method of calculation (also
emphasizes the continuing need to develop and refine biosignal analysis
methods). For instance, there may arise the following question: is the
change in the *delay*, observed during altered conditions, due to a
change in the latency (of sort) or perhaps mainly an effect of a change
in the wave-forms?


Frank Borg

Biosignals Project

tel. (06) 8294 268,
gsm 040 - 8448 376

Jyväskylä University, Chydenius Institute
Långbrogatan 1-3
67100 Karleby


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