Fault Feature Extraction of Diesel Engine Based on Bispectrum Image Fractal Dimension
Zhang et al. Chin. J. Mech. Eng.
Fault Feature Extraction of Diesel Engine Based on Bispectrum Image Fractal Dimension
Jian Zhang 0
ChangW‑en Liu 0
Feng‑Rong Bi 0
Xiao‑Bo Bi 0
Xiao Yang 0
0 State Key Laboratory of Engines, Tianjin University , Tianjin 300072 , China
Fault feature extraction has a positive effect on accurate diagnosis of diesel engine. Currently, studies of fault feature extraction have focused on the time domain or the frequency domain of signals. However, early fault signals are mostly weak energy signals, and time domain or frequency domain features will be overwhelmed by strong background noise. In order consistent features to be extracted that accurately represent the state of the engine, bispectrum estimation is used to analyze the nonlinearity, non‑ Gaussianity and quadratic phase coupling (QPC) information of the engine vibration signals under different conditions. Digital image processing and fractal theory is used to extract the fractal features of the bispectrum pictures. The outcomes demonstrate that the diesel engine vibration signal bispectrum under different working conditions shows an obvious differences and the most complicated bispectrum is in the normal state. The fractal dimension of various invalid signs is novel and diverse fractal parameters were utilized to separate and characterize them. The value of the fractal dimension is consistent with the non‑ Gaussian intensity of the signal, so it can be used as an eigenvalue of fault diagnosis, and also be used as a non‑ Gaussian signal strength indicator. Consequently, a symptomatic approach in view of the hypothetical outcome is inferred and checked by the examination of vibration signals from the diesel motor. The proposed research provides the basis for on‑ line monitoring and diagnosis of valve train faults.
Engine fault diagnosis; Bispectrum image processing; Fractal; Signal processing
Vibration signals are widely utilized for health condition
assessment and fault diagnosis in diesel engine and
frequently transfer active data from mechanical elements.
Many advanced techniques have been employed to detect
and extract features from vibration signals [
conventional fault diagnosis techniques based on
vibration signals extract the characteristic quantities from the
time domain and frequency domain, statistical indexes,
such as peak amplitude, root mean square amplitude,
kurtosis and frequency components . Obviously, these
indexes simplify the description of the machine
condition, but the selection of index directly affects the
pattern recognition. Moreover, most frequency spectrums
have similar characteristics, which make misjudgements
in detecting machine faults [
]. Fourier transforms (FT)
and fast Fourier transform (FFT) analysis can be utilized
to examine the signal in the regularity area. The
application premise of discrete Fourier is that it is supposed
that the signal is both stationary and periodic [
However, the diesel engine is a composite system in which the
combined shock is excited by a complicated mechanism
motion and combustion. The vibration signal of the diesel
engine is not linear and stationary, under steady
operation. Intervallic behavior may or may not occur and the
presumption of stationarity may not hold when the flag is
recorded for long spans or the fundamental instruments
of flag era change. In this manner, the blame vibration
highlights of diesel motor are not completely be reflected
by time-area waveform and spectral examination.
As of late, some new strategies for blame flag
determination with non-stationary signs have been proposed,
such as wavelet transform (WT), wavelet packet
transform (WPT), ensemble empirical mode
decomposition (EEMD) and higher order spectral (HOS) analysis
]. Of these new techniques, WT has been proved
to be more appropriate than FT for the vibration signal
examination due to its high time–frequency resolution.
WPT method that evolved from the wavelet can divide
the signal into an entire frequency band. However, both
method is unable to excerpt non-linear connections
within the signal or time sequence and will lose phase
information between frequency components. Moreover,
both methods is subject to the selection of the
wavelet basis function. Once the wavelet basis is set,
decomposition and reconstruction will no longer change and
will no longer be adaptive to signal analysis [
investigation has specific benefits for describing
signals at instantaneous frequency. An intelligent diagnosis
method based on better EEMD and SVM with a small
number of training sets was proposed by Zhang et al. [
to solve the problems of the poor decomposition
accuracy for the short signal. Based on EMD, Shi et al. [
presented a novel procedure to improve the precision of
signal decomposition. EEMD is a powerful tool for
nonstationary signal analysis, but this method is frequently
confined to experiments and applied research. Moreover,
low computational efficiency limits its application for
online detection. HOS allows characterization of signals at
various confinement levels in time and capacity in
signal handling, design acknowledgment, seismology, and
machine fault analysis.
HOS examination can be utilized as an intense device
for the non-straight dynamical investigation of the
engine vibration signals. Firstly, this theoretical approach
can suppress Gaussian noise. If a non-Gaussian signal
contains Gaussian noise, the noise will be eliminated by
calculation of the HOS. Secondly, HOS preserves phase
information. Thirdly, HOS can assume a key part in
identifying and portraying the sort of non-linearity in a
framework from its signals [
]. HOS hypothesis would
be a superior approach than conventional time-area and
recurrence space strategies for analysis of the engine
vibration signals, particularly for the weak and noisy
signals. The appliance of HOS for health state
assessment and fault judgement of diesel engine has not been
widespread. Gu et al. [
] made some excellent work for
motor fault diagnosis by analysis of the HOS features of
electrical motor current. Other researchers used theory
and experiments to analyze gear fault diagnosis using
analysis of the bispectrum of vibration signals [
et al. [
] used bispectrum analysis technique in fault
trait mining of the diesel engine piston-pin. He
considered the bispectral feature frequency face as the feature
parameter and then used a neural network for pattern
recognition. However, the feature frequency face may
lose fault information.
Bispectrum is the most extensively used approach
and the lowest order of all HOS analysis methods. The
bispectrum plan provides supplementary data of the
relationships among the dissimilar frequency
mechanisms. These plans can be used to distinguish diverse
conditions of engine, and the features derived from these
plans will guide the identification of different kinds of
signals. Extracting fit features from the bispectrum plan
to provide decision support for fault diagnoses is an
important and difficult research topic. The bispectral
characteristic plane in frequency field was used to
characterize the data [
]. By bispectral analysis of the
vibration data and searching the field of bispectral modulus,
characteristic planes in data were determined. The fault
characteristics of crankshaft bearing can be extracted
efficiently. Bispectrum estimation based on a AR model
which is not Gaussian was used to describe the
nonlinear and non-Gaussian features of the cylinder vibration
response signal [
]. The diesel engine valve train fault
type was identified by extracting the maximum peak
frequency and amplitude from the main cross slice of
the bispectrum. However, the main diagonal slice and
the characteristic frequency surface contain part of the
bispectrum information, limiting the practical effect of
the fault features.
The color spatial distribution is considered to be the
texture features of the 2D bispectrum plot and reflects
the distribution and intensity of the additional
information in the dual frequency domain, and exhibits its
self-similarity characteristic. Therefore, it can be
characterized by the fractal dimension (FD). FD is broadly
utilized as a part of numerous ranges of science, permitting
quantitative estimation for fractal qualities of nonlinear
frameworks and space-filling limit estimation for signals.
Numerous techniques are utilized to evaluate FD [
this investigation, FD of the bispectrum was assessed by
the crate checking strategy.
Whatever remains of this paper is organized as takes
after. Section 2 presents the essential hypothesis of the
bispectrum, picture preparing, and fractal
measurement. In Section 3, the blame recreation framework and
information procurement framework are displayed. The
exploratory outcomes and investigation are appeared in
Section 4. At long last, the conclusions are given.
2 Bispectrum Calculation
2.1 Definition of Bispectrum
HOS is a spectral representation of greater order
cumulants of an arbitrary process. For a cyclostationary,
discrete stochastic process x(n), the nth-order cumulants
of x(n) is denoted by cn,x(m1, m2, …, mn − 1). First-order
cumulants of a stationary process is known to be the
c1,x(0) = E(x(n)),
where E(•) is mathematical expectation. The second and
third-order cumulants of a motionless procedure are
demarcated by Eq. (1) as follows:
c2,x(m) = E(x(n)x(n + m)),
c3,x(m1, m2) = E(x(n)x(n + m1)x(n + m2)).
HOS are demarcated as FT of the equivalent
higherorder cumulants arrays. Actually, the Fourier transform
of a second-order cumulant is the conventional power
c2,x(m) exp −jωm .
The third-order spectrum is the FT of the third-order
cumulants, which is also called the bispectrum [
S3,x(ω1, ω2) =
× exp −j(ω1m1 + ω2m2) .
Note that in the expressions above, these spectra are
given by products of the deterministic time-domain
signals Fourier transforms. The bispectrum is a
representation of the correlation of the spectrum values and two
frequency component, reflecting the skewness of signal
features. For linear systems, the bispectrum amplitude is
zero when all three are independent. For a nonlinear
system, there will be strong correlation at some frequencies.
In this way, it can distinguish stage coupling between two
frequencies which show up as a third recurrence as the
total or distinction of the initial two with a stage that is
likewise the aggregate or contrast of the initial two [
This marvel is called quadratic stage coupling (QPC) [
Generally, a power range is utilized to break down the
signs into a progression of recurrence segments. In any
case, a power range can’t decide if crests at agreeably
related positions are stage coupled or not on account of
the power range utilizes just the size of the Fourier
segments and stage data is ignored. HOS, as bispectrum, are
equipped for recognizing stage coupling by utilizing stage
data. In this manner the bispectrum can give extra
recurrence data that the established power range can’t give.
The upsides of utilizing HOS examination can be
condensed as takes after [
(1) HOS can be used to suppress the influence of the
additive colored noise.
(2) HOS can identify the cause and effect,
non-minimum phase system or reconstruction
non-minimum phase signal.
(3) HOS can provide additional information due to the
(4) HOS can be used to detect and characterize the
nonlinearity of signals and identify the nonlinear
(5) Additional QPC information is available.
2.2 Calculation of Bispectrum
There exist a few methods for bispectrum approximation,
including models with our without parameters. Each
model includes direct and indirect methods. Although
the parametric model can provide higher resolution and
more phase information using less signal data samples,
there may be loss in important feature information, and
the method cannot reflect the characteristics of the fault
signal. Nonparametric bispectrum estimation usually
uses more data samples, but this approach can reduce the
estimation variance and improve accuracy. In the
present work, the straight method of nonparametric model
approximation was accepted and the calculation
algorithm process is as follows.
(1) Let x = (x1, x2,…, xn) be the zero mean
observation samples and fs is the sampling frequency. In
the domain of the bispectrum, the sampling value
of the frequency number is N0 and the segment of
frequency sampling is Δ0 = fs/N0. Next, segment
the data into k possibly overlapping records, where
each subgroup contains M notation samples.
(2) Remove the average of each subgroup.
(3) Perform the Fourier transformation for each set of
xi( ) =
where ω1 = 2π fs 1/N0, ω2 = 2π fs 2/N0.
2.3 Bispectrum Image Processing
The picture processing handling was created utilizing
Matlab. The schematic portrayal of the picture handling
technique is delineated in Figure 1.
As appeared in Figure 1, the transformation of RGB
pictures into grayscale pictures and the ensuing
background amendment ought to be done first. Next, the
grayscale pictures were improved by histogram evening
out and the commotion was evacuated by low-pass
filtering. In this progression, work histeq and fft2 gave by
Matlab picture handling tool compartment were connected;
the low-pass filter was modified freely. A versatile limit
determination named OSTU method [
], which can
break down picture histograms consequently and acquire
the best edge esteem, was utilized to get paired pictures
after the computerized pictures were upgraded. This
technique was performed by calling capacity graythresh
from the Matlab picture handling tool stash. Next,
double pictures were furthermore prepared by picture
disintegration, calling capacity strel and imerode. At long last,
with the capacity edge, shapes of the pictures were gotten
by Candy administrator.
2.4 Fractal Dimension Calculation
In fractal geometry, the FD is a measurable amount to
demonstrate how totally a fractal seems to fill space
]. FD is a compelling parameter to evaluate fractal
In this paper, FD of the bispectrum were approximated
by the container checking technique as indicated by
] with some modification. All the more
decisively, each electronic picture was overlaid by a
progression of matrices of square boxes of the span of 1–1024
pixels. For a progression of boxes of side length s pixels,
the quantity of boxes meeting the shapes of picture by
the set (N) was tallied. Fractal structures comply with the
power law connection over a scope of length scales, such
N (s) = cs−DB ,
where DB is the case checking fractal measurement, N(s)
is the aggregate number of boxes of side length s that
cross forms of the bispectrum picture, and c is a
consistent. DB is evaluated as the negative slope of a relapse line
through the direct piece of the plot of lgN(s) against lg s,
for a succession of scales s:
lg N (s) = lg c + (−DB) lg s.
3 Test Setup and Data Acquisition
In this paper, an inline 6-cylinder diesel engine was
tested. The engine works under 1600 r/min (40% loads).
As shown in Figure 2, the experimental test fix comprises
of the six-barrel in-line turbo-charged diesel motor,
dynamometer, LMS SCADA III multi-analyzer
framework (piezoelectric accelerometer vibration sensors and
an information securing framework with 25.6 kHz
examining recurrence) and a PC.
The experiment comprises engine valve clearance
normal and fault conditions. At the intake and exhaust
valve of the first cylinder, vibration signals were collected
under preset working fault. The acceleration transducer
was installed between the intake valve and exhaust valve.
Table 1 shows the fault parameter of the conditions. After
the experimental work, the fault and normal vibration
signals were recorded by 9 accelerometer sensors using
a data acquisition system. The positions of the trembling
sensors are shown in Figure 3. The sample time was 30 s.
10 fragments were cut from the collected signals; each
fragment lasts 1 s: 2‒3 s, 5‒6 s, 7‒8 s, 10‒11 s, 13‒14 s,
16‒17 s, 19‒20 s, 22‒23 s, 24‒25 s, 27‒28 s. Each
fragment contains 26.67 working cycle.
4 Fault Feature Extraction
The change of valve clearance can result changes of
engine mechanism or the gas exchange process.
Changing the engine mechanism will transform the whole
character of the engine system. In this case, with the change
of valve chain’s dynamic character, driving force such as
valve-seating impact and impact between valve stems
and rocker will also be different. Thus the vibration
response will change. Changing the gas exchange process
will influence the combustion process. The cylinder
pressure will change accordingly and the vibration response
of cylinder head, cylinder block, piston and other engine
parts may also be different. These changes impact the
whole integrated engine system, changing the nonlinear
character of the engine vibration response.
Quadratic phase coupling (QPC) is the additive
relationship between different components’ frequencies
and phases. If the harmonic components are irrelative,
the third-degree cumulant is equal to 0. If QPC exists
in harmonic components, the third-degree cumulant is
not equal to 0. The bispectrum image has a peak value
when analyzing the vibration signal with bispectrum of
third-order cumulant. This indicated that QPC exists in
vibration signals. At least in theory, the bispectrum can
perfectly suppress the Gaussian components, and explain
the distribution and strength of non-Gaussian
components in the dual frequency domain, and also the QPC
information. Fractal dimension shows the complexity
of images, reflecting the quantification of the character
of images. By using the fractal dimension, a
quantificational description can be made, allowing evaluation of
the engine condition and fault diagnosis. The Signal-flow
diagram demonstration of the engine culpability
diagnosis scheme is shown in Figure 4.
4.1 Bispectrum Analysis
Figure 5 shows the vibration signals in Locations 3 of
the third cylinder under the five different valve clearance
conditions. From the diagram, we can see that the surface
of the cylinder trembling signal is composed of an array
of shock response signals in a certain sequence,
indicating the cycle stationary signal under the steady speed.
Each signal under different conditions varied slightly in
the time domain waveform because in the different states
of valve mechanism, the mechanical stimulation
including the impact of the valve stem, the rocker arm and valve
seating are different. At the same time the difference of
the intake and exhaust cause a change in the
combustion status, leading to a change in the combustion
excitation. Furthermore, for the diesel engine work process,
there is fluctuation in the working cycle and the adjacent
cycle. All of these contribute to the vibration signal, and
the cylinder head vibration response in the time domain
waveform experiences fluctuation.
The nonlinear characteristics of the system have a
significant impact on the output signal. Determining
frequency relationship between the system and its output
signal can identify the nonlinear characteristics. Because
the signal bispectrum is highly sensitive to its
non-Gaussian components, when these non-Gaussian
characteristics of the system signal changes, the bispectrum features
will change too. Here, the parametric bispectrum
estimation was used to analyze the cylinder head vibration
signal. The size of information is N= 25,600, the request
of every subsection perception tests M = 1024, the
superpose level of every subsection information is half, and
the quantity of subsection is K = 2N/M − 1 = 49. Next,
figure the bispectrum of the vibration motion in
Location 3 of the third barrel under the five conditions. The
amplitudes and shapes are appeared in Figure 6.
Normalizing frequency f1 and f2 based on the maximum
frequency, the frequency f1 and f2 (0‒0.5 Hz) correspond
to 0‒12,800 Hz, and the frequencies below are
In Figure 6, the cylinder head trembling signal for
different states of the spectrum and amplitude are not zero,
indicating that the cylinder trembling signal is a
nonlinear and non-Gaussian signal. The bispectrum under
the same condition is stable, but under various working
conditions, the bispectrum shows obvious separability.
Compared with the normal state (Fault 3), the
bispectrum peak and phase distribution in other fault status
is different, and the bispectrum distribution of the
normal state is more complex. Using the normal state as a
benchmark, from fault 1 to fault 3, the phase
distribution of the bispectrum became gradually complicated,
and from fault 3 to fault 5, the phase distribution of state
varied conversely. From the bispectrum amplitudes
figure, we can see that the spectral peak amplitude of fault
3 (normal state) is lower than the other four kinds of fault
states, because the valve clearance led to changes in the
cylinder head trembling of nonlinear features. The degree
of nonlinearity of the cylinder head system changes
constantly with the deterioration of the condition (due to
wear and deformation of the working parts).
4.2 Fractal Dimension
To determine the effect of valve clearance fluctuations
in the fractal size of the vibration signal bispectrum, the
bispectrum image was converted into a digital image
and equivalent images were treated to evaluate the
fractal size. Along with the clearance increase (before fault
3), the peak frequency became larger and the frequency
components became complex and then (after fault 3)
became simple (Figure 7a). The contours of the
corresponding images increased and then decreased near fault
3 (Figure 7b).
For the five different valve clearance diesel engine
conditions, the bispectrum contour maps of fractal
dimension show obvious differences. The fault 3 fractal
dimension is the largest. From state 1 to state 3 (normal
condition), the fractal dimension increased as the valve
clearance increased. This is likely because the valve opens
earlier and closes loosely due to heat expansion under
state 1, causing cylinder leakage and a drop in
combustion pressure. Under this condition, quadratic phase
coupling of vibration signal stimulated by cylinder pressure
(including both low frequency excitation and high
frequency oscillation) decreases. Compared to the normal
condition (state 3), the plot of the bispectrum was more
simple with less side-band and center band components.
The fractal dimension is also smaller. Under state 5 with
too big valve clearance, the valve timing changed with
both the intake and exhaust valve open, and the valve
lift was also reduced. Therefore the gas exchange
process is compressed, resulting in insufficient intake, partial
exhaust, and incomplete combustion. The cylinder
pressure also drops under incomplete combustion, causing
a reduction in quadratic phase coupling and a simpler
bispectrum of vibration signal.
-0.25 0 0.25 0.5
Figure 6 Vibration bispectrum at Location 3 of the third cylinder for
the five conditions
Compared to the vibration signal acquired at 1600 r/
min with 40% load at 9 locations, the signal at location
3 was preferable. In order to accurately acquire the fault
feature, 10 sets of data were intercepted respectively from
the different time domain vibration signal under each
fault condition. Each fragment contained 26.67
working cycle and lasted 1 s: 2−3 s, 5−6 s, 7−8 s, 10−11 s,
13−14 s, 16−17 s, 19−20 s, 22−23 s, 24−25 s, 27−28 s.
Next, the corresponding signals were processed to
calculate the fractal dimensions. The results are shown in
Figure 8 shows the bispectrum fractal dimensions of
10 sets of vibration signal for each fault condition. The
bispectrum fractal dimensions of the vibration signal
under different valve clearance conditions showed
obvious differences. Under the same condition, the
fractal dimensions were similar and almost in the specific
range. The small difference of the bispectrum fractal
dimension under the same conditions were caused by
changes in the non-Gaussian and nonlinear vibration
signal with the fluctuations of the diesel engine working
cycle and cycle-to-cycle working conditions. This
underlying problem cannot be thoroughly solved, even if
signals from more cycles are selected to reduce the error.
Fractal dimensions under different fault states are in
different ranges. When the diagnosis of conditions is
consistent with the certain condition, the fractal dimension
will be within the range of the specific conditions,
allowing fault diagnosis.
(1) The vibration signal of engine is a non-linear,
nonGaussian and cyclostationary signal. By using the
bispectrum image and fractal dimension,
characteristics that can reflect fault condition can be
extracted from the vibration signals.
(2) The bispectrum distribution of the normal state is
more complex. Using the normal state as a
benchmark, from fault 1 to fault 3, the phase distribution
of the bispectrum became gradually complicated,
and from fault 3 to fault 5, the phase distribution of
state varied conversely.
(3) Under different states, the bispectrums of vibration
signal have different fractal dimensions, and these
dimensions are in different ranges. Thus the fractal
dimension can describe the working condition of
engines. Additionally, the fractal dimension is
consistent with the non-Gaussian characteristic of the
signal. Therefore, the fractal dimension can serve as
an index of the Gaussianity.
(4) The fractal dimension of the bispectrum increases
at first then decreases as the valve clearance
increased. Considering the discrepancy of engines,
the fractal dimension under the same conditions
can be different. Building a database could help to
solve this problem.
F‑RB was in charge of the whole trial; JZ wrote the manuscript; C‑ WL, X‑BB,
XY assisted with sampling and laboratory analyses. All authors have read and
approved the final manuscript.
Jian Zhang, born in 1983, is currently a PhD candidate at State Key Laboratory
of Engine, Tianjin University, China. He received his master degree from
Kunming University of Science and Technology, China, in 2012. His research interests
include engine noise and vibration signal processing. Tel: +86‑17720531162;
Chang‑ Wen Liu, born in 1963, is currently a professor at State Key
Laboratory of Engine, Tianjin University, China. His research interest is engine’s
electronic control technology. Tel: +86‑13602057489; E‑mail: liuchangwen@
Feng‑Rong Bi, born in 1965, is currently a professor at State Key Laboratory
of Engine, Tianjin University, China. He received his PhD degree from Tianjin
University, China, in 2003. His research interests include engine noise and vibra‑
tion control, automobile dynamics, etc. Tel: +86‑13802167153; E‑mail: fr_bi@
Xiao‑Bo Bi, born in 1978, is currently a PhD candidate at State Key
Laboratory of Engine, Tianjin University, China. He received his master degree from
Hebei University of Technology, China, in 2014. His research interests include
diesel engine fault diagnosis. Tel: +86‑13820411609; E‑mail: 1014201001@tju.
Xiao Yang, born in 1993, is currently a master candidate at State Key
Laboratory of Engines, Tianjin University, China. He received his bachelor degree from
Tianjin University, China, in 2015. His research interests include engine noise
control. Tel: +86‑18920965767; E‑mail: .
The authors declare that they have no competing interests.
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 G Gelle , M Colas , C Serviere . Blind source separation: a tool for rotating machine monitoring by vibration analysis . Journal of Sound & Vibration , 2001 , 248 ( 5 ): 865 - 885 .
 K Shibata , A T Takahashi . Fault diagnosis of rotating machinery through visualization of sound signals . Mechanical Systems and Signal Processing , 2000 , 14 ( 14 ): 229 - 241 .
 Z M Geng , J Chen , J B Hull . Analysis of engine vibration and design of an applicable diagnosis approach . International Journal of Mechanical Sciences , 2013 , 45 ( 8 ): 1391 - 1410 .
 W Q Wang , I Fathy , M F Golnaraghi . Assessment of gear damage monitoring techniques using vibration measurements . Mechanical Systems and Signal Processing , 2001 , 15 ( 5 ): 905 - 922 .
 H Z Gao , L Liang , X G Chen , et al. Feature extraction and recognition for rolling element bearing fault utilizing short‑time Fourier transform and non‑negative matrix factorization . Chinese Journal of Mechanical Engineer - ing , 2015 , 28 ( 1 ): 96 - 105 .
 J P Shao , H J Jia. Feature extraction of vibration signals based on wavelet packet transform . Chinese Journal of Mechanical Engineering , 2004 , 17 ( 1 ): 25 - 27 .
 Y G Lei , Z J He , Y Y Zi , et al. New clustering algorithm‑based fault diagnosis using compensation distance evaluation technique . Mechanical System and Signal Processing , 2008 , 22 ( 2 ): 419 - 435 .
 X G Chen , L Liang , G H Xu , et al. Feature extraction of kernel regress reconstruction for fault diagnosis based on self‑ organizing manifold learning . Chinese Journal of Mechanical Engineering , 2013 , 26 ( 5 ): 1041 - 1049 .
 J D Wu , J C Chen . Continuous wavelet transform technique for fault signal diagnosis of internal combustion engines . Ndt& E International , 2006 , 39 ( 4 ): 304 - 311 .
 W T Peter , W X Yang , H Y Tam . Machine fault diagnosis through an effective exact wavelet analysis . Journal of Vibration & Acoustics , 2004 , 227 ( 4‒5 ): 1005 - 1024 .
 X Wang , C W Liu , F R Bi , et al. Fault diagnosis of diesel engine based on adaptive wavelet packets and EEMD‑fractal dimension . Mechanical Systems and Signal Processing , 2013 , 41 ( 1 ): 581 - 597 .
 B Liang , S D Iwnicki , Y Zhao . Application of power spectrum, higher order spectrum and neural network analyses for induction motor fault diagnosis . Mechanical Systems and Signal Processing , 2013 , 39 ( 1-2 ): 342 - 360 .
 M Li , J H Yang , X J Wang. Fault feature extraction of rolling bearing based on an improved cyclical spectrum density method . Chinese Journal of Mechanical Engineering , 2015 , 28 ( 6 ): 1240 - 1247 .
 M J Zhang , J Tang , X M Zhang , et al. Intelligent diagnosis of short hydraulic signal based on improved EEMD and SVM with few low‑ dimensional training samples . Chinese Journal of Mechanical Engineering , 2016 , 29 ( 2 ): 396 - 405 .
 K J Shi , S L Liu, C Jiang , et al. Rolling bearing feature frequency extraction using extreme average envelope decomposition . Chinese Journal of Mechanical Engineering , 2016 , 29 ( 5 ): 1029 - 1036 .
 F S Gu , A Naid , N Q Hu , et al. Electrical motor current signal analysis using modified bispectrum for fault diagnosis of reciprocating compressors . Condition Monitoring & Diagnostic Engineering Management , San Sebastian, Spain, June 9-11, 2009 .
 F A Gu , Y Shao , N Q Hu , et al. Electrical motor current signal analysis using modified bispectrum for fault diagnosis of downstream mechanical equipment . Mechanical Systems and Signal Processing , 2011 , 25 ( 1 ): 360 - 372 .
 G J Shen , S Mclaughlin , Y C Xu , et al. Theoretical and experimental analysis of bispectrum of vibration signals for fault diagnosis of gears . Mechanical Systems and Signal Processing , 2014 , 43 ( s1 ‑2): 76 - 89 .
 F Z Feng , A W Si , H X Zhang. Research on fault diagnosis of diesel engine based on bispectrum analysis and genetic neural network . Procedia Engineering , 2011 , 15 ( 1 ): 2454 - 2458 .
 H M Zhao , C Y Xia , Y K Xiao , et al. Bispectrum analysis for vibration data of crankshaft cearing in diesel engine . Journal of Vibration, Measurement & Diagnosis , 2009 , 29 ( 1 ): 14 - 18 . (in Chinese).
 T Li , S Chen , Q Tang , et al. Fault Diagnosis for valve train of diesel engine based on bispectrum estimation via non‑ gaussian AR model . Chinese Internal Combustion Engine Engineering , 2010 , 31 ( 1 ): 82 - 87 . (in Chinese)
 Y Liu , L Y Chen , H M Wang , et al. An improved differential box‑ counting method to estimate fractal dimensions of gray‑level images . Journal of Visual Communication and Image Representation , 2014 , 25 ( 5 ): 1102 - 1111 .
 J Guo , X Guo , P F Luo. A new method for antomatic target recognition. Proceeding of the IEEE Nation Aerospace and Electronics Conference NAECON, Dayton, OH , Jul 14-18 , 1997 , 2 : 1019 - 1024 .
 D R Brillinger . An introduction to polyspectra . Annals of Mathematical Statistics , 1965 , 36 ( 5 ): 1351 - 1374 .
 D R Brillinger. Some basic aspects and uses of higher order spectra . Signal Process , 1994 , 36 ( 3 ) 239 - 249 .
 N Otsu . A threshold selection method from gray‑level histograms . IEEE Transactions on Systems Man & Cybernetics , 1979 , 9 ( 1 ): 62 - 66 .
 Y Y Duan , L Wang , H Z Chen , Digital image analysis and fractal‑based kinetic modelling for fungal biomass determination in solid‑state fermentation . Biochemical Engineering Journal , 2012 , 67 ( 1 ): 60 - 67 .
 L Zhao , Z D Zhou , Y Yang , et al. Feature extraction of rolling bearing fault based on ensemble empirical mode decomposition and correlation dimension . International Manufacturing Science and Engineering Conference , Detroit, Michigan, USA, June 9-13, 2014 .
 J Theiler . Estimating fractal dimension . Journal of the Optical Society of America A , 1990 , 7 ( 6 ): 1055 - 1073 .
 D P Donnelly , L Boddy , J R Leake. Development, persistence and regeneration of foraging ectomycorrhizal mycelial systems in soil microcosms . Mycorrhiza , 2004 , 14 ( 1 ): 37 - 45 .