Instrumented Usability Analysis for Mobile Devices - Human-Computer Interaction and Innovation in Handheld, Mobile and Wearable Technologies - page 8

Figure 5. A user walking with the device and corresponding acceleration trace. The unfiltered vertical
acceleration signal (rough sinusoid), the filtered signal (smooth sinusoid) and the phase estimate (in
radians) for the signal (saw-tooth).

The algorithms used in this article were de-
veloped in research on synchronisation effects in
nature. The oscillations involved in many natural
systems are often irregular, ruling out simple
strategies. In some cases, such as respiratory
examples, or electrocardiogram data, there are
clear marked events with pronounced peaks in
the time-series which can be manually annotated,
or automatically detected. One practical advantage
of the use of synchronization theory is that often
we have a quite complex nonlinear oscillation,
which might be sensed via a large number of
sensors. The phase angle of that oscillation is
however a simple scalar value, so if we are inves-
tigating the synchronization effects in two complex
systems, the analysis can sometimes be a single
value, the relative phase angle ϕ 1 − ϕ 2 .

The Hilbert transform signal s H ( t ) allows you to
construct the complex signal

i φ t

ς t = s t + is t = A t e

( )

( ) ( ) ( ) ( )

H

where ϕ( t ) is the phase at time t , and A( t ) is the
amplitude of the signal at time t . The Hilbert
transform signal of is t ) is

1
π

s
t

( ) τ

T

→∞ ∫

s ( ) t = lim d τ

H

− τ

T T

Although A(t) and ϕ( t ) can be computed for
an arbitrary s( t ) they are only physically mean-
ingful if is t ) is a narrow-band signal. For the gait
analysis, we therefore filter the data to create a
signal with a single main peak in the frequency
spectrum around the typical walking pace (be-
tween 1 and 3Hz).
This phase plot signal is again shown as the
saw-tooth waveform in Figure 5 and Figure 6
and can be seen to reset at the lowest point in the
signal. This corresponds to the lowest point of the
hand in the oscillation.

The Hilbert Transform

How do we find the phase angle from the data? A
common approach is to use the Hilbert transform
introduced by Gabor in 1946, which gives the
instantaneous phase and amplitude of a signal
s( t ) (Pikovsky, Rosenblum, & Kurths, 2001).