**Introduction**

HWTs
are used to detect significant changes in signal values. In a previous post, I discussed 1D Haar Wavelet up-down spikes that can be used to detect changes in signals. Specifically, four types of spikes were 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 define down-up spikes.

**Dow-Up Spikes**

**Fig**.

**1**shows down-up triangle and trapezoid spikes. In this figure, the lower graphs represent the possible values of the corresponding Haar wavelets at a chosen scale

*k*.

The decline segment of the 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

The values
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;
and
are
the *k*-th scale wavelet coefficient ordinates at and respectively.*k*-th scale wavelet coefficient ordinates at and respectively.

For a trapezoid down-up spike, the flat segment is
characterized byand
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. The values
and
are the

*k*-th scale wavelet coefficients corresponding to and respectively.Fig. 1. 1D Haar wavelet down-up spikes |

A down-up spike describe signals that first decrease and then, after
an optional flat segment, increase, unlike an up-down spike where a decline segment follows a climb segment. The climb and decline angles are measured in the same way as for up-down spikes.