# Detecting stop-start motion in
trajectories

#### Jim McLean

#### 2023-11-28

If a trajectory contains periodic stop-start motion, what sampling
frequency is required to detect it? A first assumption might be that the
Nyquist rate is
adequate. However, that is not the case, since when we work with
trajectories, we are not measuring speed directly, but instead
approximating it by measuring the mean speed between pairs of
points.

To see why, let’s use an example. We will simulate a trajectory of an
animal that repatedly stops for 0.5 secs then walks at 1 m/s for 0.5
seconds, so speed forms a square wave with frequency 1 Hz. We will will
consider any speed slower than 0.1 to be stopped.

If we create a trajectory by sampling at the Nyquist rate of
`signal frequency * 2 = 2 Hz`

, this is the result, which
looks good:

But it turns out we were “lucky” here, and this frequency is only
adequate if the sampling is in phase with the trajectory. If we phase
shift the sampling times by 1/4 of the signal wavelength without
altering the sampling rate, we suffer from *signal aliasing* and
no longer detect any variation in speed! Remember that we are actually
sampling displacement, and then deriving speed from the sampled
displacement.

To reliably detect stopping, we need to sample (at least) twice
within a stopped (or moving) period. In our example, the animal stops
for 0.5 seconds at a time. To ensure we sample stopped periods twice
regardless of phase shifts, we need to sample at a minimum frequency of
`3 / stopping duration = 6 Hz`

. Now we are certain to obtain
the correct frequency, regardless of phase shifts in sampling, although
stopping and moving durations and acceleration are still very
inaccurate.

Furthermore, if the speed is actually closer to a sine wave than a
square wave, detecting stop-start motion is harder still.

## Conclusion

If you are trying to detect the frequency of stop-start motion in a
trajectory, the highest frequency that it is theoretically possible to
detect reliably is 1/4 the sampling frequency. However, what actually
matters is the number of samples within a stopped or moving period,
rather than the frequency of stopping and starting. So let’s just sample
at a very high frequency!