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The
constant
Jnoise is chosen, for a realis-tic dataset, to capture most of the trend energy in swtai2 . Regardless of the
(1,n)
trend distribution between swtdi2 and swtai2 , the vector swtdi2 has vir-
(1,n)
tually no contribution from c. Consequently, the energy in swtdi2 is only
(1,n)
w. Assuming w to be Gaussian white noise, the energy level in swtdi2 is representative for the contribution of w across all scales. Using the windowed RMS, a w energy estimate across time is made.
(1,n)
g.w(n)= wrms[swtdi2 , n, Lw] (6.37)
The component w is then assumed to be the coe.cients in swtdi2 with absolute value less than Tw. As w is assumed to be white, the same threshold is applied across all scales.
j,n j,n
swtdj,n = swtd.(|swtd| < .(n).Tw(6.38)
wi2 i2 gw)
Standard de-noising usually consists of setting the smallest swtd coe.-cients to zero before reconstructing. As the purpose of this exercise is to capture the noise w rather than the signal c, the largest swtdi2 coe.cients and all of swtai2 are zeroed out before reconstruction.
w.= idwt(0, swtdj,n ,ψ)
w (6.39)
The edge component is based on the b. calculated above, but corrected so that its initial value is the initial value of the dataset minus w and s.

6.5. NON-PARAMETRIC PROGRESSION ANALYSIS
.
b(n)= .b.(n)+ i2(0) . w(0) (6.40)
The trend component is the remaining data after s, w and b has been removed.
c.(n)= i(n) . s.(n) . w.(n) . .b(n) (6.41)
Figure
6.26
shows
the
entire
separation
process
as
a
.ow
chart,
with
input,
output, and constants.

6.5.4 Trend Analysis
As already stated, it is in the value of c and the gain of w that are of interest to uncover mechanical faults. Even though this algorithm manages to split the four components making up i, it does not produce a parametric model whose parameters can be evaluated to understand the behavior of the data. It is thus necessary to perform an additional parametrization step, in the form of a trend analysis of c as well as g.w from
(Eq.
6.37).

The HUMS acquires data during .ight from each sensor at regular in-tervals, so that the spacing between each indicator value, in .ight time, is relatively uniform. All methods discussed here assumes uniform spacing. For datasets where this is not the case, with for instance missing data due to sensor problems etc., the indicator series must be interpolated with a smoothing-function and re-sampled.

 

 

6.6. CALIBRATION
Trend analysis of a signal x is performed by the CWT using the the Haar wavelet
[19].
This
corresponds
to
a
sliding
window
linear
regression.
Window
size is given by the scale parameter j so that the size of the window in which the linear regression is performed equals 2j. Consequently, a small value for j will capture rapid .uctuations, while large values for j captures longer trends. In order to detect the increasing and decreasing trends associated with mechanical degradation, it is necessary to use several values for j. This produces a coe.cient matrix a(cj)(n) with dimension N by J.
　
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