HWTs are used to detect significant changes in signal values. In
this post, I formalize the claim that some changes can be characterized as signal
spikes. Specifically, four types of spikes are proposed: up-down
triangle, up-down trapezoid, down-up triangle, and down-up trapezoid.
The difference between up-down and down-up spikes is the relative
positions of the climb and decline segments. In trapezoid spikes,
flat segments are always in between the climb and decline segments,
regardless of their relative positions. In this post, I will formalize up-down spikes and leave a similar formalization of down-up spikes for my next post.
Fig. 1 shows up-down triangle and trapezoid spikes. In this figure, the
lower graphs represent the possible values of the corresponding Haar
wavelets at a chosen scale k.
 |
| Fig. 1. Up-down spikes. |
Up-down spikes describe signals that first increase and then, after
an optional flat segment, decrease. Formally, a spike is a nine element tuple whose elements are real numbers in Fig. 2.
 |
| Fig. 2. Formal characterization of a spike. |
The first two elements,
and
,
are the abscissae of the beginning and end of the spike’s climb
segment, respectively, when the wavelet coefficients of the 1D HWT
increase. If
and
are
the k-th scale wavelet coefficient ordinates at
and
respectively, then the climb segment of the spike is measured by the
angle
The decline angle of a spike is characterized by
and
where
and
are
the abscissae of the beginning and end of the spike’s decline
segment, respectively, when the wavelet coefficients decrease. If
and
are the k-th scale wavelet coefficient ordinates at
and
respectively, then the decline segment of the spike is measured by
the angle
.
and,
are the abscissae of the beginning and end of the spike’s climb
segment, respectively, when the wavelet coefficients of the 1D HWT
increase. If
and
are
the k-th scale wavelet coefficient ordinates at
and
respectively, then the climb segment of the spike is measured by the
angle
The decline angle of a spike is characterized by
and
where
andare
the abscissae of the beginning and end of the spike’s decline
segment, respectively, when the wavelet coefficients decrease. If
and
are the k-th scale wavelet coefficient ordinates at
and
respectively, then the decline segment of the spike is measured by
the angle .
For a trapezoid up-down spike, the flat segment
is characterized by
and
where
and
are
the abscissae of the beginning and end of the spike’s flat segment,
respectively, over which the wavelet coefficients either remain at
the same ordinate or have minor ordinate fluctuations. If
and
are the k-th scale wavelet coefficients corresponding to
and
respectively, the spike’s flatness angle is
.
The absolute values of
are
close to 0.
and
where
and
are
the abscissae of the beginning and end of the spike’s flat segment,
respectively, over which the wavelet coefficients either remain at
the same ordinate or have minor ordinate fluctuations. If
and
are the k-th scale wavelet coefficients corresponding to
and
respectively, the spike’s flatness angle is
.
The absolute values of
are
close to 0.
