.(j,n)
|mean[cwc ]|
bm(n)= i1 (6.32)
wrms[i1, n, Lb]
An edge is a transition which rises clearly above the background noise. The above equation will normalize the edge magnitude by the total dataset energy in a trailing window with size Lb. An edge exists in a point in time n which satis.es the trend signature criteria, while also having a magnitude bm(n) larger than Tm (Eq.
6.33).
btrue(n)=(bp(n) >Tp) ∧ (bm(n) >Tm) (6.33)
A
recursive
equation
(Eq.
6.34)
provides
an
estimate
for
the
edge
process.
This equation always outputs it last value except when a step is detected, in which case it adds the amplitude of the step to its output value.
.(j,n)
.b.(n)= b.(n . 1) + mean[cwc i1 ].btrue(n) (6.34)
The initial value of .b. is zero. A modi.ed version, .b, will be developed later. The initial value of this component will be the initial value of the dataset, after s and w are removed. Another modi.ed dataset, i2, is con-structed without the edge component.
i2(n)= i1(n) . .b.(n) (6.35)
6.5.3 Random Noise Separation
In order to perform a CWT which contains all information about the source signal, the source signal must be analyzed at an in.nite number of scales, making reconstruction impossible. This problem is overcome by the Dis-crete Wavelet Transform (DWT)(Sec. A.3.2)
and
the
Stationary
Wavelet
Transform
(SWT)
(Sec.
A.3.3),
which
expands
the
input
signal
on
a
wavelet
function using an arbitrary number of scales. The remainder of the signal, which can not be expanded using the number of scales chosen, is left in the approximate vector. Consequently, the approximate vector swta and the de-tail matrix swtd will together contain all information in the original signal, making reconstruction possible. While the DWT has applications in signal compression,
the
SWT
(Eq.
6.36)
is
the
preferred
choice
for
de-noising.
[swtai2 , swtdi2 ]= swt(i2, ψ, Jnoise) (6.36)
This study deals only with .nite signals. In order to keep transform output length the same as input length, while reducing transients, signals / coe.cients are padded at start and end. Padding consists of samples having the mean value of the K .rst / last samples. This provides better results than periodic padding, circular convolution, as the start and end of the signals used in this study can have very di.erent amplitudes.
The dataset i2 is
expanded
on
the
db5
wavelet
[19]
using
the
SWT
at
scales 1 through Jnoise (Eq.
6.36).
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