where e is a small positive number introduced to pre-
vent division-by-zero exception.
As the characteristic signals are periodical and the
white noise has no periodicity, NLMS algorithm is used
to separate the white noise and periodical signals. But
the difficult point is to find reference signals d(n), as the
signals measured on the gearbox surface contains some
signals of the same components of the noise and charac-
teristic signals but with different amplitude values. So,
this article uses the Self-Adaptive Noise Cancellation
(SANC) method
13
to separate the periodical signal and
white noise. The SANC method which is using a finite
number delay signal as the reference signal, and combi-
nation with the NLMS, it is effective to deal with the
signals. The signal processing procedure is expressed in
Figure 2.
KLMS adaptive filter
By taking advantage of the reproducing property of
reproducing kernel Hilbert space (RKHS), the KLMS
adaptive filter can implement a nonlinear transforma-
tion to transform the input signal u(n) into high-
dimensional feature-space signal as u(u(n)), and then,
the filtering and adaptation operations can be
performed in RKHS.
14
For the difference in dimen-
sionality of u and u(u), v
T
u(u) is a much more power-
ful model than w
T
u. So, finding v through stochastic
gradient descent may prove as an effective way of non-
linear filtering.
15
The calculation process can be
expressed as follows
v 0ðÞ= 0 ð6Þ
enðÞ= dnðÞf
n1
u nðÞðÞ ð7Þ
v n
ðÞ
= v n 1
ðÞ
+ hen
ðÞ
u u n
ðÞðÞ
= h
X
n
j = 1
ej
ðÞ
u u j
ðÞðÞ
ð8Þ
The output of the system to a new input u
0
(n) can be
expressed as
f
n1
u nðÞðÞ= h
X
n1
j = 1
ejðÞu u jðÞðÞ
"#
u u
0
nðÞðÞ
= h
X
n1
j = 1
ejðÞu u jðÞðÞu u
0
nðÞðÞ½
= h
X
n1
j = 1
ejðÞk u jðÞ, u
0
nðÞðÞ
ð9Þ
enðÞ= dnðÞh
X
n1
j = 1
ejðÞk u jðÞ, u
0
nðÞðÞð10Þ
After N step training, the final input–output is
v nðÞ= h
X
N
j = 1
ejðÞu u jðÞðÞ ð11Þ
ynðÞ= h
X
N
j = 1
ejðÞk u jðÞ, u NðÞðÞð12Þ
where k(u(j)) is the kernel function. The commonly used
kernels include the Gaussian kernel and the polynomial
kernel among many others. In this article, we use the
Gaussian kernel to establish the nonlinear mapping of
the input signal. The Gaussian kernel is defined as
k u, u
0
ðÞ= exp a u u
0
kk
2
ð13Þ
The proposed scheme and numerical
validation
This section describes how SANC and KLMS algo-
rithms were applied to a signal simulated from gearbox
frequencies.
The gearbox applied to the DFIG wind turbine has
two main kind structures, one consists of a planetary
gear and two fixed-axis gears and the other consists of
two planetary gears and one fixed-axis gear. Faults on
the fixed-axis gears are easy to diagnose; this article
mainly focuses on the vibration signals of planetary
gear fault. The structure of a planetary gearbox is
shown in Figure 3; it is composed of planetary gear,
ring gear, sun gear and carrier. When rotating, the ring
gear is fixed, and the carrier rotates with its planetary
gear. In Figure 3, the vibration accelerometer transdu-
cer, mounted on the surface of planetary gear, experi-
ences a periodic variation in vibration amplitudes as
the meshing position periodic changes, and then, the
measured vibration signals have strong nonlinearity.
5
Besides, the different transmission paths would lead the
vibration signals of different measurement points to
nonlinear relationship.
Figure 2. Signal processing procedure of SANC.
Tian and Qian 3