Wavelets Find the Fall

Starting interactive Python…

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Convolution is a sliding area

Lay a kernel $k$||(k||) over the signal $y$||(y||), multiply them point by point, and take the area under the product. Green is where they agree in sign and add; red is where they disagree and subtract. Slide $k$||(k||) along and the areas trace the output: $$(y \star k)(\tau) = \int y(t)\, k(t-\tau)\, dt$$||[(y \star k)(\tau) = \int y(t)\, k(t-\tau)\, dt||]Drag across the plot or press play. A box averages, a bump smooths, an edge kernel fires on steps. Strictly, convolution flips $k$||(k||) first; wavelet transforms slide it unflipped, as here.

Every frequency at once is a scalogram

Pair the cosine with a sine and take the magnitude: the ringing above becomes a smooth envelope. Do that for every $f$||(f||) and stack the rows. The swell and the burst separate cleanly in time and frequency; the line marks your $f$||(f||). More cycles sharpen frequency and blur time.

A fall is a slow wave with a burst at its end

Each trial is the acceleration magnitude from the trunk sensor minus 1 g, 4 s before its largest peak to 2 s after. (IMU 1 stays upright through the hard sit while IMU 2 ends horizontal, so IMU 1 is the trunk.) The wavelet is still yours: around 1 Hz it swells over the whole drop, around 8 Hz it rings only at impact. Compare a fall with the hard sit.

Every trial's scalogram on one colour scale, impacts aligned two seconds from the right edge. Each fall darkens around 1 Hz under its impact, where the body drops; the box marks the band scored below.

Peak acceleration confuses a hard sit with a fall

A threshold on peak acceleration is the usual first detector. Filled dots are falls; the dashed tick is the gentlest fall. On peak acceleration the hard sit and the trick do at least as well; on the largest 0.5–1.5 Hz wavelet amplitude, the trick does.I chose the default band by looking at these same nine trials, from one session, so this is a place to look, not a threshold; the trick was built to fool a detector. The test is the same band on falls from other people. The shape is the opening example again: a slow swell with a burst at its end, and a wavelet scores the two separately.