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Although the Spectral Correlation is one of the most versatile spectral tools to analyze cyclostationary signals (i.e. signals comprising hidden periodicities or repetitive patterns), its use in condition monitoring has so far been hindered by its high computational cost. The Cyclic Modulation Spectrum (the Fourier transform of the spectrogram) stands as a much faster alternative, yet it suffers from the uncertainty principle and is thus limited to detect relatively slow periodic modulations. This paper fixes the situation by proposing a new fast estimator of the spectral correlation, the Fast Spectral Correlation, based on the short-time Fourier transform (STFT). It proceeds from the property that, for a cyclostationary signal, the STFT evidences periodic flows of energy in and across its frequency bins. The Fourier transform of the interactions of the STFT coefficients then returns a quantity which scans the Spectral Correlation along its cyclic frequency axis. The gain in computational cost as compared to the conventional estimator is like the ratio of the signal length to the STFT window length and can therefore be considerable. The validity of the proposed estimator is demonstrated on non trivial vibration signals (very weak bearing signatures and speed varying cases) and its computational advantage is used to compute a new quantity, the Enhanced Envelope Spectrum.
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Fast computation of the spectral correlation
Jérôme Antoni, Ge Xin
, Nacer Hamzaoui
Laboratoire Vibrations Acoustique, Univ Lyon, INSA-Lyon, LVA EA677, F-69621 Villeurbanne, France
article info
Article history:
Received 16 May 2016
Received in revised form 6 December 2016
Accepted 11 January 2017
Keywords:
Cyclostationarity
Cyclostationary signals
Cyclic spectral analysis
Spectral correlation
Cyclic modulation spectrum
Fast spectral correlation
Condition monitoring
Bearing diagnosis
Nonstationary regime
abstract
Although the Spectral Correlation is one of the most versatile spectral tools to analyze
cyclostationary signals (i.e. signals comprising hidden periodicities or repetitive patterns),
its use in condition monitoring has so far been hindered by its high computational cost. The
Cyclic Modulation Spectrum (the Fourier transform of the spectrogram) stands as a much
faster alternative, yet it suffers from the uncertainty principle and is thus limited to detect
relatively slow periodic modulations. This paper fixes the situation by proposing a new fast
estimator of the spectral correlation, the Fast Spectral Correlation, based on the short-time
Fourier transform (STFT). It proceeds from the property that, for a cyclostationary signal,
the STFT evidences periodic flows of energy in and across its frequency bins. The Fourier
transform of the interactions of the STFT coefficients then returns a quantity which scans
the Spectral Correlation along its cyclic frequency axis. The gain in computational cost as
compared to the conventional estimator is like the ratio of the signal length to the STFT
window length and can therefore be considerable. The validity of the proposed estimator
is demonstrated on non trivial vibration signals (very weak bearing signatures and speed
varying cases) and its computational advantage is used to compute a new quantity, the
Enhanced Envelope Spectrum.
Ó2017 Elsevier Ltd. All rights reserved.
Conventions
Whereas the SC S
x
ð
a
;fÞis a theoretical quantity, the ACP S
ACP
x
ð
a
;fÞ, the CMS S
CMS
x
ð
a
;fÞ, and the Fast-SC S
Fast
x
ð
a
;fÞare three
different estimators of the SC.
The connections between the spectral quantities handled in the paper are schemed in Fig. 1.
Definitions of the SC found in the literature may differ in the measurement units. The definition given here is such that for
a signal with measurement units U, the SC has units U
2
=Hz. It is a one dimensional density of variable f. As a consequence, the
particular case
a
¼0 returns the power spectral density, S
x
ð0;fÞS
x
ðfÞ. Another definition of the SC is actually as a two
dimensional density of variables fand
a
, with units U
2
=Hz
2
[1]. The power spectral density is then evaluated as
S
x
ðfÞ¼lim
B!0
R
B
B
S
x
ð
a
;fÞd
a
.
1. Introduction
Whether of mechanical or electrical nature, rotating machine signals are perfectly modelled by cyclostationary processes.
The reason is that, due to the inherent operation of a machine, signals are produced by some periodic – or cyclic – mechanisms.
http://dx.doi.org/10.1016/j.ymssp.2017.01.011
0888-3270/Ó2017 Elsevier Ltd. All rights reserved.
Corresponding author.
E-mail addresses: jerome.antoni@insa-lyon.fr (J. Antoni), ge.xin@insa-lyon.fr (G. Xin).
Mechanical Systems and Signal Processing 92 (2017) 248–277
Contents lists available at ScienceDirect
Mechanical Systems and Signal Processing
journal homepage: www.elsevier.com/locate/ymssp
The cyclostationary class defines processes whose statistics are periodic. It encompasses most of the processes usually
encountered in machines as particular cases, be they deterministic or random, e.g. periodic signals, stationary signals,
periodically-modulated signals, repetitive transients, etc. This makes cyclostationarity a preferred framework in vibration-
based condition monitoring. Because of its ability to perfectly describe the statistical behavior of faults in the form of
symptomatic modulations or repetition of transients, it provides optimal tools for their detection, their identification, and
possibly their quantification. One central tool for the ‘‘cyclic spectral analysis” of machine signals is the Spectral Correlation
(SC) which displays at once, in the form of a bi-spectral map, the whole structure of modulations and carriers in a signal
[2–6]. Although the demonstration of the capabilities of cyclic spectral analysis in condition monitoring has been undertaken
in several research works [7–13], its practice is still not as systematic as it deserves. Many methods are constantly published
Nomenclature
SC Spectral Correlation
ACP Averaged Cyclic Periodogram
CMS Cyclic Modulation Spectrum
Fast-SC Fast Spectral Correlation
DFT Discrete Fourier Transform
FAM FFT Accumulation Method
FFT Fast Fourier Transform
STFT Short-Time Fourier Transform
SES Squared Envelope Spectrum
EES Enhanced Envelope Spectrum
OF Order-Frequency
CPU Central Processing Unit
xðtnÞsignal of interest
w½ndata window (function of time index n)
Xwði;fÞGabor coefficient at time index iand frequency f
XSTFT ði;fÞSTFT coefficient at time index iand frequency f
Lsignal length
Nwwindow length in STFT
N0central time index of window
Rblock shift in STFT
Ktotal number of blocks used in spectral estimates
Fssampling frequency
tnn-th discrete time instant (in s)
stime-lag (in s)
Tcyclic period of a cyclostationary signal (in s)
acyclic (or modulation) frequency (in Hz)
amax maximum scrutinizable cyclic frequency (in Hz)
fspectral (or carrier) frequency (in Hz)
fkk-th discrete frequency (in Hz)
Dacyclic frequency resolution in a(in Hz)
Dffrequency resolution in f(in Hz)
pindex of STFT frequency closest to a given cyclic frequency a
Pindex of STFT frequency closest to amax
Rxðtn;sÞinstantaneous autocorrelation function of signal x
RwðaÞdiscrete Fourier transform of jw½nj2
Sxða;fÞSpectral Correlation of signal x
cxða;fÞSpectral Coherence of signal x
SACP
xða;fÞAveraged Cyclic Periodogram of signal x
SCMS
xða;fÞCyclic Modulation Spectrum of signal x
Sxða;f;pÞScanning Spectral Correlation of signal x
SFast
xða;fÞFast Spectral Correlation of signal x
cFast
xða;fÞFast Spectral Coherence of signal x
SSES
xðaÞSquared Envelope Spectrum of signal x
SEES
xðaÞEnhanced Envelope Spectrum of signal x
CCMS computational complexity of Cyclic Modulation Spectrum
CACP computational complexity of Averaged Cyclic Periodogram
CFast computational complexity of Fast Spectral Correlation
J. Antoni et al. / Mechanical Systems and Signal Processing 92 (2017) 248–277 249
that painfully solve a diagnostic problem with sophisticated ad hoc tools where, instead, cyclostationarity would easily suc-
ceed. Two reasons might explain why a wider resort to cyclostationarity has been hindered so far. One is that it involves
advanced theory of stochastic processes. Efforts have been spent to make this aspect as transparent as possible [14], but they
are probably still insufficient. The second reason is that signal processing tools dedicated to cyclostationary analyses are so
far computationally intensive. In particular, the SC may be extremely costly to compute in some situations, a fact which is
likely to prevent its use by non-experts or for quick trouble shooting tasks. Alternatives have been proposed to the SC, like
the Cyclic Modulation Spectrum (CMS) [15]. Being essentially a waterfall of envelope spectra at the output of a filterbank, it
is fast to compute while intending to return similar information as the SC. The CMS has been recently formalized in Ref. [16]
as an estimator of the SC. Although the CMS proves to be a valid diagnostic tool in many situations, it has limited perfor-
mance in general: being constrained to the uncertainty principle, it cannot detect periodic patterns other than in the form
of modulations whose frequencies are necessarily lower than the frequency resolution. As shown latter in this paper, the
CMS is also not properly calibrated to quantify modulation depth. Besides, computationally efficient algorithms for the esti-
mation of the SC have been proposed early in the nineteens in Refs. [3,17], of which the FFT Accumulation Method (FAM) is
the fastest. As far as the authors known, the FAM is still recognized as the most computationally efficient algorithm in the
specialized literature. Unfortunately, its computational advantage comes at the price of a degradation of the statistical
performance of the estimator: the cyclic frequency resolution and variance are non-uniform, meaning that estimation errors
can be locally very high; this is probably unacceptable in the kind of applications targeted by this paper.
The aim of this paper is to introduce a fast algorithm to estimate the SC, the ‘‘Fast Spectral Correlation” (Fast-SC), which
essentially proceeds from the Short-Time Fourier Transform (STFT). It may be seen as a correction of the CMS such as to make
it approach the ideal SC. Most of the computational effort is required for the calculation of the STFT – for which many effi-
cient implementations now exist in commercial software. This makes the proposed algorithm weakly intrusive and of low
complexity. For all these reasons, the approach proposed in this paper should participate in making the SC a more widely
spread tool in condition monitoring.
Another contribution of this paper is to maintain a simple vision of cyclostationarity. While the CMS is simply interpreted
as the detection of periodic flows of energy in frequency bands, the Fast-SC extends it to the detection of periodic flows across
different frequency bands. This should help to make cyclostationarity easier to interpret while not sacrifying the usage of its
most performant tools.
2. Background on cyclic spectral analysis
The object of this section is to resume the definitions and main properties of the SC and CMS; these will serve as a basis to
discuss their pros and cons and will motivate the introduction of the Fast-SC in Section 3. Emphasis is also put on the two
different visions entailed by the SC and the CMS in cyclic spectral analysis. The section opens with a presentation of the
notations and main quantities used in the paper.
2.1. Prerequisites
Let denote the signal of interest xðt
n
Þwhere t
n
¼n=F
s
refers to time instants acquired with sampling frequency F
s
. When-
ever convenient, the stream of samples xðt
n
Þ;n¼0;...;L1, will be simply denoted by x½n. The STFT of signal xðt
n
Þover a
time interval of length N
w
=F
s
is defined as
X
STFT
ði;f
k
Þ¼ X
N
w
1
m¼0
x½iR þmw½me
j2
p
m
fk
Fs
ð1Þ
with discrete frequencies f
k
¼k
D
f;k¼0;...;N
w
1, frequency resolution
Fig. 1. Connections between the spectral quantities handled in the paper.
250 J. Antoni et al. / Mechanical Systems and Signal Processing 92 (2017) 248–277
D
f¼F
s
N
w
;ð2Þ
a symmetric data window w½mwith central time index N
0
such that w½N
0
þn¼w½N
0
n(N
0
¼N
w
=2ifN
w
is even and
N
0
¼ðN
w
þ1Þ=2 if it is odd), and time shift Rbetween consecutive windows – see Fig. 2.
In the subsequent analyses, the phase of the signal will play a fundamental role. It is therefore necessary to correctly
reference it to the beginning of the signal, at the time origin t
n
¼0. This leads to the phase-corrected STFT (sometimes also
presented as the Gabor transform), denoted here by
X
w
ði;f
k
Þ¼X
L1
n¼0
x½nw½niRe
j2
p
n
fk
Fs
¼X
STFT
ði;f
k
Þe
j2
p
iR
fk
Fs
:ð3Þ
The interpretation of X
w
ði;f
k
Þis the ‘‘complex envelope” of signal x½nin a narrow frequency band of bandwidth
D
fcentered
on f
k
and sampled at time instants iR=F
s
. Its squared magnitude, jX
w
ði;f
k
Þj
2
, thus reflects the energy flow in the frequency
band [14]. The collection of squared coefficients jX
w
ði;f
k
Þj
2
’s for all time indices iR and frequencies f
k
defines the spectro-
gram. From now on, signal xðt
n
Þis assumed to be cyclostationary on the second-order. This means that its instantaneous
autocorrelation function,
R
x
ðt
n
;
s
Þ¼Efxðt
n
Þxðt
n
s
Þ
g;ð4Þ
(where Estands for the ensemble average operator and
for the complex conjugate) is a periodic function of time t
n
with
some period T,
R
x
ðt
n
;
s
Þ¼R
x
ðt
n
þT;
s
Þ:ð5Þ
Intuitively, the periodicity of the autocorrelation function evidences the presence of a repetitive statistical behavior in the
signal, for instance due to the occurrence of a fault in the form of a series of impulses or in the form of periodic modulations,
but not only. The cyclostationary framework is actually rather large and includes many types of signals produced by periodic
mechanisms.
The characterization of cyclostationary signals in the frequency domain usually provides more insight. This introduces
the SC and its degraded version, the CMS.
2.2. The Spectral Correlation
The SC is defined as the double discrete Fourier transform of the instantaneous autocorrelation function (actually a
Fourier series in time tand a Fourier transform in time-lag
s
when continuous time is considered),
S
x
ð
a
;fÞ¼lim
N!1
1
ð2Nþ1ÞF
s
X
N
n¼N
X
1
m¼1
R
x
ðt
n
;
s
m
Þe
j2
p
n
a
Fs
e
j2
p
m
f
Fs
;t
n
¼n
F
s
;
s
m
¼m
F
s
:ð6Þ
In the case of a second-order cyclostationary signal, the SC displays a characteristic signature continuous in frequency f
and discrete in cyclic frequency
a
,
S
x
ð
a
;fÞ¼ S
k
x
ðfÞ;
a
¼k=T
0;elsewhere
(ð7Þ
where the S
k
x
ðfÞ’s, k¼0;1;2;...are ‘‘cyclic spectra”. In words, the SC is a two-dimensional representation (a bi-frequency
map) made of a collection of parallel cyclic spectra at the discrete cyclic frequencies
a
¼k=T. The line at
a
¼0 returns the
classical power spectral density.
The alignment of non-zero values at a given cyclic frequency
a
of the SC therefore indicates the existence of a sinusoidal
modulation in the signal at that frequency
a
, which envelops a carrier characterized by the cyclic spectrum S
k
x
ðfÞ(seen as a
Fig. 2. Illustration of quantities N
w
;R, and N
0
used in the Short-Time Fourier Transform.
J. Antoni et al. / Mechanical Systems and Signal Processing 92 (2017) 248–277 251
function of f). The SC may therefore be understood as a decomposition of the signal with respect to the ‘‘modulation fre-
quency”
a
and the ‘‘carrier frequency” f.
A popular estimator of the SC is obtained from the so-called ‘‘time-smoothed cyclic periodogram” [2,18] or, equivalently,
the Averaged Cyclic Periodogram (ACP) [1] which is an extension of Welch’s method (also known in spectral analysis as the
‘‘Weighted-Overlapped-Segment-Averaging” method) to cyclostationary signals. The ACP is defined as
S
ACP
x
ð
a
;fÞ¼ 1
Kkwk
2
F
s
X
K1
i¼0
X
w
ði;fÞX
w
ði;f
a
Þ
;ð8Þ
kwk
2
¼X
N
w
1
n¼0
jw½nj
2
;
where X
w
ði;fÞis defined in Eq. (3) and K¼ðLN
w
þRÞ=Ris the total number of N
w
-long windows shifted by Rsamples in a
L-long signal. One has [2]
lim
N
w
!1
lim
K!1
S
ACP
x
ð
a
;fÞ¼S
x
ð
a
;fÞ ð9Þ
where lim denotes the probability limit taken in the mean-square sense.
One advantage of formula (8) is to express the SC as a measure of cross-correlation with respect to time between the
complex envelopes at frequencies fand f
a
. This highlights the property of a cyclostationary signal to be characterized
by non-zero correlations between spectral components spaced apart by
a
. In other words, spectral components at different
frequencies are recruited synchronously in order to produce periodic modulations in the time domain.
The SC has been shown to be an ideal tool in condition monitoring because of its high capability to unwrap complicated
signals onto a two-dimensional map that clearly reveals the presence of modulations and makes easy the identification of
fault frequencies. It actually provides a high-resolution version of the envelope spectrum – an everlasting tool in condition
monitoring – since the scrutinized carrier frequency band can be made arbitrarily small (the bandwidth is
D
f¼F
s
=N
w
) while
still maintaining a very fine cyclic frequency resolution on the order of
D
a
¼F
s
=L(with Lthe signal length). Despite all its
advantages, the use of the SC may be hindered in practice by its high computational cost when large cyclic frequency ranges
have to be explored (estimator (8) is computed in a loop over cyclic frequencies
a
k
¼k
D
a
). This may be detrimental for unex-
perienced users, in particular when trials and errors approaches are followed.
In order to illustrate the estimation of the SC with the ACP, an example is given here of the cyclic spectral analysis of a
simple cyclostationary signal. The signal is the response to white noise of an oscillator with a resonance frequency
f
0
¼250 Hz and damping ratio of 7%, which is further modulated by a square wave with 90%dead time and frequency
a
0
¼1:8 Hz. The sampling frequency is F
s
¼1000 Hz and the signal length is L¼10
5
samples. The spectral correlation is
computed from the ACP estimator of Eq. (8) with N
w
¼2
8
, a Hann window, and R¼26. This involves K¼3853 windows
and produces a frequency resolution
D
f¼4 Hz and a cyclic frequency resolution
D
a
¼0:01 Hz. The estimated SC is displayed
in Fig. 3(a) as a colormap. It is seen that the vertical lines are correctly identified at
a
0
and its multiples in a frequency band
Fig. 3. (a) Spectral correlation estimated from the Averaged Cyclic Periodogram,S
ACP
x
ða;fÞ, with N
w
¼2
8
(
D
f¼4 Hz;
D
a¼0:01 Hz) and (b) its evaluation at
f¼250 Hz together with the theoretical envelope of the peaks (dotted line) as obtained from a square modulation.
252 J. Antoni et al. / Mechanical Systems and Signal Processing 92 (2017) 248–277
around the resonance at 250 Hz. The SC evaluated at the resonance frequency, S
ACP
x
ð
a
;f
0
Þ, is displayed in Fig. 3(b): it is
expected to show the Fourier spectrum of the square modulation whose theoretical envelope (the cardinal sine function)
is indicated by the black dotted line. A slight underestimation is seen due to the picket fence effect (the multiples of
a
0
do not fall exactly on the cyclic frequency grid
a
k
¼k
D
a
). It is worth noting that the computation of the ACP took more than
15 min on a laptop computer (i7-4810MQ Processor 2.80 GHz) which is detrimental as compared to other methods – more
will be said on this matter in Section 4.
2.3. The Cyclic Modulation Spectrum
The CMS takes a different look at cyclostationary signals. It intends to track periodic flows of energy in frequency bands by
evaluating the Fourier transform of the squared envelope at the output of a filter bank [15,16]. It is thus interpreted as a
waterfall of envelope spectra. The CMS is efficiently computed as the Discrete Fourier Transform (DFT) of the spectrogram,
i.e.
S
CMS
x
ð
a
;fÞ¼ 1
Kkwk
2
F
s
X
K1
i¼0
jX
STFT
ði;fÞj
2
e
j2
p
ðiRþN
0
Þ
a
Fs
¼1
Kkwk
2
F
s
DFT
i!
a
fjX
STFT
ði;fÞj
2
ge
j2
p
N
0
a
Fs
;ð10Þ
where it is reminded that jX
STFT
ði;fÞj
2
is an evaluation of the energy flow in a band centered on frequency fand of bandwidth
D
fand where the notation DFT
i!
a
indicates that variable iis transformed into
a
. Note that the presence of N
0
in Eq. (10) is to
evaluate the DFT at time instants ðiR þN
0
Þ=F
s
which corresponds to the centers of the windows of the STFT (see Fig. 2).
The computational cost of the CMS is essentially dictated by the computation of the STFT, for which very efficient imple-
mentation are nowadays available in most software. This makes it a very valuable tool in practice. Unfortunately, the CMS is
a biased estimator of the SC and its approximation error increases with the cyclic frequency. It is shown in Appendix A that,
provided that N
w
is longer than the correlation length of the signal (i.e. the extent of the instantaneous autocorrelation
function along the
s
axis),
EfS
CMS
x
ð
a
;fÞg ’ S
x
ð
a
;fÞR
w
ð
a
Þ
R
w
ð0Þð11Þ
with
R
w
ð
a
Þ¼ X
N
w
1
n¼0
jw½nj
2
e
j2
p
ðnN
0
Þ
a
Fs
ð12Þ
[U2/Hz]
(a)
(b)
Fig. 4. (a) Cyclic Modulation Spectrum S
CMS
x
ða;fÞof the signal analyzed in Fig. 3 computed with N
w
¼2
8
and R¼26 (a
max
D
f¼4 Hz) and (b) its evaluation
at f¼250 Hz together with the theoretical envelope of the peaks (dotted line) as obtained from a square modulation. The limit a
max
is indicated by the
vertical blue dotted line. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
J. Antoni et al. / Mechanical Systems and Signal Processing 92 (2017) 248–277 253
and R
w
ð0Þ¼kwk
2
. Eq. (11) means that the CMS ‘‘sees” the SC through kernel R
w
ð
a
Þ=R
w
ð0Þ. As shown in Fig. 6, the latter
applies a low-pass weighting in the
a
direction with a cutoff frequency on the order of
a
max
F
s
N
w
¼
D
f:ð13Þ
In words, the STFT windows should not undersample the modulations in the signal.
As explained in Ref. [14], the fact that the highest cyclic frequency seen by the CMS is bounded upward by
D
fis a direct
consequence of the uncertainty principle, T
D
fP1, which affects the STFT. With T¼1=
a
, the latter reads
a
6
D
f:ð14Þ
Thus, modulations which are faster than allowed by the frequency resolution
D
fof the STFT are not detected. This subjects
the CMS to a compromise between a fine resolution
D
fto accurately analyze the spectral content of the carrier and a coarse
D
fto enlarge the range
a
max
of detectable modulation frequencies.
The limitation of the CMS is hereafter illustrated on the example of the previous subsection. Using the same parameters,
the CMS is thus limited to detect modulations lower than
a
max
D
f¼4 Hz. This is evidenced in Fig. 4 where the cyclic upper
limit is indicated by a vertical blue dotted line (the evaluation of the CMS at the resonance frequency, S
CMS
x
ð
a
;f
0
Þ, is shown in
Fig. 7(b)). Obviously the limit could be pushed up by decreasing the value of N
w
at the detriment of frequency resolution
D
f.
However, it is advised to keep the latter reasonably fine in general in order to correctly identify the spectral content of the
carrier in particular when several modulation patterns coexist in the ð
a
;fÞplane as is commonplace with complex vibration
signals. Note that the computation of the CMS took only 1 s.
1/f0
f0 + f0 -
f
f
f
t
(a)
(b)
(c)
B
B
f
f0
f0 + f0 - f
(d)
f0
f
X(f)
X(f- )
Fig. 5. (a) Example of a cyclostationary signal composed of a sinusoidal amplitude modulation on a narrow-band carrier and (b) its frequency spectrum. The
CMS requires that
D
f>aþBin order to fully capture the periodic beatings in time, which inevitably prevents a fine description of the spectral density. (c)
Spectrum of a similar signal with faster modulation and presence of noise; the condition
D
f>aþBimplies that a large amount of noise is fatally absorbed
in the analysis band.
254 J. Antoni et al. / Mechanical Systems and Signal Processing 92 (2017) 248–277
2.4. Narrow-band modulations
This subsection elaborates more on the type of cyclostationary patterns which the CMS can hardly characterize. Let con-
sider the elementary modulated signal (chosen for the sake of simplicity),
xðt
n
Þ¼aðt
n
Þðt
n
Þ;ð15Þ
where
ðt
n
Þstands for a stationary, narrow-band carrier with bandwidth Bcentered on frequency f
0
and
aðt
n
Þ¼1þcosð2
pa
t
n
Þis a periodic modulation with frequency
a
. An example of such a signal and its corresponding spec-
trum are displayed in Fig. 5(a) in the case where
a
>B.
Let us consider the Fourier transform XðfÞof signal xðt
n
Þover a very long but finite time interval. It is easy to show that
XðfÞ¼EðfÞþ
1
2
Eðf
a
Þþ
1
2
Eðfþ
a
Þwith EðfÞthe Fourier transform of
ðt
n
Þ. For the CMS to detect the periodic modulation, its
resolution
D
fmust be large enough to contain the component EðfÞand its two shifted version Eðf
a
Þor Eðfþ
a
Þ(located
around f
0
þ
a
and f
0
a
, respectively, in Fig. 5(b)) whose combination produces the characteristic periodic ‘‘beating” in the
time domain. As seen in Fig. 5(b), this condition requires that
D
f>2
a
þB, i.e. a frequency resolution
D
ffor the CMS neces-
sarily coarser than the bandwidth Bof the narrow-band carrier. As a result, the CMS is therefore unable to analyze accurately
the spectral content of the carrier. The situation is particularly critical with fast modulations,
a
B, which imply that
D
fB. Such a condition is difficult to satisfy a priori when the signal to be analyzed is unknown. In addition, setting
D
f
becomes particularly hazardous when the cyclic frequencies varies in time, a situation which is investigated in details in
Section 6.3.
A last drawback of having to set
D
fexaggeratedly large as compared to the carrier bandwidth Bis to decrease the overall
signal-to-noise ratio in the band and thus to lower the capability of detecting the presence of cyclostationarity. This is illus-
trated in Fig. 5(c) and further discussed in Section 6.1.3.
In contrast, the SC does not suffer from these limitations since it does not try to detect a periodic modulation of the signal
energy in a band of width
D
f, but rather searches for correlation between two frequency components at fand ðf
a
Þ. The
frequency resolution
D
fcan therefore be made arbitrarily fine to analyze accurately the spectral content of the carrier while
still detecting the modulation. This is illustrated in Fig. 5(d).
3. The Fast Spectral Correlation
This section explains how the concept of the CMS can be extended, at moderate computational cost, to closely approach
the ideal SC. The idea is first introduced on an intuitive ground and is next formally proved.
3.1. An intuitive approach
Let us start with the ACP estimator (8) of the SC and see how it can be expressed in terms of the STFT. The first constraint
is to evaluate the spectral components at frequencies fand ðf
a
Þat multiples of
D
f¼F
s
=N
w
. Assuming that the constraint is
satisfied for f, viz f¼f
k
¼k
D
f, an approximation is to be made for ðf
k
a
Þ. It starts with the polar representation of the
‘‘wave packet”
X
w
ði;f
k
a
Þ¼jX
w
ði;f
k
a
Þje
j/
i
ðf
k
a
Þ
ð16Þ
by means of a magnitude and a phase. Each of these components is now going to be approximated in a different way. First, a
rough approximation is allowed for the magnitude which smoothly envelopes the wave packet. Setting
a
¼p
D
fþdð17Þ
where p
D
fis the closest frequency bin to
a
and dthe residue, it comes jX
w
ði;f
k
a
Þj ’ jX
w
ði;f
kp
Þj. Second, a more accurate
approximation is required for the phase since its derivative contains high frequency oscillations of the wave packet. Using a
first-order Taylor expansion, /
i
ðf
k
a
Þ’/
i
ðf
k
p
D
fÞd/
0
i
ðf
k
p
D
fÞwhere /
0
i
denotes the group delay of the wave packet.
cyclic frequency f0
f
P
f
Rw( ) Rw(f)p=0Rw(pf)/P
P
Fig. 6. Kernel function R
w
ðaÞ, its shifted versions R
w
ðap
D
fÞ;p¼1;...;P, and their sum (scaled by 1=P) used to equalize the Fast-SC in Eq. (24). The
effective half-bandwidth of the aggregated kernel is about P
D
f¼F
s
=ð2RÞ.
J. Antoni et al. / Mechanical Systems and Signal Processing 92 (2017) 248–277 255
Since, by construction, the latter is located around time instant ðiR þN
0
Þ=F
s
, it reads /
0
i
ðf
k
p
D
fÞ’2
p
ðiR þN
0
Þ=F
s
(see
Fig. 2 and recall that, by definition, the group delay returns the time delay of a group of waves localized around a given
frequency). Collecting all terms,
X
w
ði;f
k
a
Þ’jX
w
ði;f
kp
Þje
j/
i
ðf
k
p
D
fÞ
e
j2
p
d
Fs
ðiRþN
0
Þ
¼X
w
ði;f
kp
Þe
j2
p
d
Fs
ðiRþN
0
Þ
¼X
w
ði;f
kp
Þe
j2
p
ð
a
Fs
p
Df
Fs
ÞðiRþN
0
Þ
:ð18Þ
Now, inserting into Eq. (8) and using the correspondence (3) with the STFT, one arrives at
S
x
ð
a
;f
k
;pÞ¼ 1
Kkwk
2
F
s
X
K1
i¼0
X
w
ði;f
k
ÞX
w
ði;f
kp
Þ
e
j2
p
a
Fs
p
Nw
ðÞ
ðiRþN
0
Þ
¼1
Kkwk
2
F
s
X
K1
i¼0
X
STFT
ði;f
k
ÞX
STFT
ði;f
kp
Þ
e
j2
p
a
Fs
ðiRþN
0
Þ
e
j2
p
pN0
Nw
¼1
Kkwk
2
F
s
DFT
i!
a
fX
STFT
ði;f
k
ÞX
STFT
ði;f
kp
Þ
ge
j2
p
N
0
a
Fs
p
Nw
ðÞ ð19Þ
where the notation S
x
ð
a
;f
k
;pÞis used to make the difference with the ACP estimator of Eq. (8) and to highlight the
introduction of the frequency shift p
D
f. The quantity S
x
ð
a
;f
k
;pÞwill be hereafter coined the ‘‘Scanning Spectral Correlation”
for a reason to become clear shortly.
The above equation is a fundamental step towards the results of this paper. It suggests that the SC can be estimated sim-
ply from the Discrete Fourier Transform of the interactions between the STFT coefficients in frequency bins not necessarily
spaced apart by exactly
a
(as required in the ACP). As a matter of fact, the CMS happens to be a particular case of S
x
ð
a
;f;pÞ
with p¼0. It now remains to find how the quantity S
x
ð
a
;f;pÞis related to the theoretical SC and how a versatile and well
calibrated estimator can be constructed from it.
3.2. Scanning cyclic frequencies
Taking the expected value of S
x
ð
a
;f;pÞ, it is shown in Appendix A that
EfS
x
ð
a
;f;pÞg ’ S
x
ð
a
;fÞR
w
ð
a
p
D
fÞ
R
w
ð0Þð20Þ
where R
w
ð
a
Þis defined in Eq. (12). Kernel R
w
ð
a
p
D
fÞ=R
w
ð0Þhas effective bandwidth 2
D
fand is now centered on p
D
f(see
Fig. 6).
The Scanning Spectral Correlation S
x
ð
a
;f;pÞscans a cyclic frequency zone roughly delimited by interval
½ðp1Þ
D
f;ðpþ1Þ
D
f. This is illustrated in Fig. 7(b) when p¼4 which is to be compared to Fig. 7(a) in the case p¼0 corre-
sponding to the CMS. It is seen that scanning provides a solution to scrutinize the cyclic frequency axis arbitrarily high with-
out being limited any longer by the
a
max
D
flimit imposed to the CMS (see Section 2.3). By scanning the cyclic frequency
axis with several values of pand ‘‘merging” the corresponding intervals ½ðp1Þ
D
f;ðpþ1Þ
D
f, the full SC can be reconstructed
over the whole cyclic frequency axis, as illustrated in Fig. 7(c). This will define the ‘‘Fast Spectral Correlation”, introduced
hereafter in Section 3.4. Before proceeding further, the physical meaning of expression (20) is first discussed.
3.3. Physical interpretation: periodic energy flow between STFT bins
Eq. (19) is the DFT, evaluated at cyclic frequency ð
a
=F
s
p=N
w
Þ, of the product of the STFT coefficient X
STFT
ði;f
k
Þand its
frequency shifted version X
STFT
ði;f
kp
Þ
. As a particular case, the CMS is returned for p¼0, which reflects the fact that for
a cyclostationary signal with cycle T, the energy jX
STFT
ði;f
k
Þj
2
flows periodically in band ½f
k
D
f=2;f
k
þ
D
f=2with frequency
a
¼1=T. Now, for p0, the interaction X
w
ði;fÞX
w
ði;f
kp
Þ
measures the energy flow between bands ½f
k
D
f=2;f
k
þ
D
f=2and
½f
kp
D
f=2;f
kp
þ
D
f=2. Using Eq. (7) one can show that (see Appendix A), on the average,
EfX
STFT
ði;f
k
ÞX
STFT
ði;f
kp
Þ
g’F
s
X
l
S
l
x
f
ðÞ
R
w
ð
a
l
p
D
fÞe
j2
p
l
TFs
ðN
0
þiRÞ
e
j2
p
pN0
Nw
ð21Þ
with S
l
x
fðÞas defined in Eq. (7). This proves that the interaction X
STFT
ði;f
k
ÞX
STFT
ði;f
kp
Þ
contains a periodic function of time
with frequency
a
¼1=T. As a result, for a cyclostationary signal, the energy flows periodically between two STFT bins, with
cyclic frequency
a
¼1=T.
This is illustrated in Fig. 8 by a cyclostationary signal composed of a periodic amplitude modulation on a broad-band car-
rier. The real parts of the STFT coefficients in five adjacent frequency bins are displayed when the signal is ‘‘sampled” by a
256 J. Antoni et al. / Mechanical Systems and Signal Processing 92 (2017) 248–277
short window w½n(Fig. 8(b)) and a long window (Fig. 8(c)). The first case is representative of
D
f>
a
and visually evidences
the periodicity of the energy flow (squared STFT coefficients) in each subband. The second case is representative of
D
f<
a
and, as a consequence, hardly evidences periodicity of the energy flow. However, the presence of cyclostationarity is encoded
in the correlation between the STFT coefficients coming from different subbands.
(a)
(b)
(c)
[U2/Hz][U2/Hz][U2/Hz]
Fig. 7. (a) Cyclic Modulation Spectrum S
CMS
x
ða;fÞof the signal analyzed in Fig. 3, (b) scanning spectral correlation S
x
ða;f;pÞ;p¼4, and (c) Fast Spectral
Correlation S
Fast
x
ða;fÞall evaluated at f¼250 Hz. The blue shaded areas indicate the effective cyclic range of each quantity. The black dotted line in (c) is the
theoretical envelope of the peaks as obtained from a square modulation. (For interpretation of the references to color in this figure legend, the readeris
referred to the web version of this article.)
-200
0
200
-400
-200
0
200
400
-400
-200
0
200
400
(a)
(b)
(c)
Fig. 8. (a) Example of a cyclostationary signal composed of a sinusoidal amplitude modulation on a broad-band noise carrier. (b) Real parts of the STFT
coefficients X
STFT
ði;f
k
Þin five adjacent frequency bins f
k
;k¼1;...;5, in the case of a short window w½n(red dotted shape), such that
D
f>a. (c) Real parts
of STFT coefficients in five adjacent frequency bins in the case of a long analysis window w½n(red dotted shape), such that
D
f<a. In (b), the presence of
cyclostationarity is essentially reflected by the periodicity of the energy flow jX
STFT
ði;f
k
Þj
2
in each subband, whereas in (c) it is encoded in the interaction
X
STFT
ði;f
k
ÞX
STFT
ði;f
l
Þ
between STFT coefficients from different subbands kl. (For interpretation of the references to color in this figure legend, the reader is
referred to the web version of this article.)
J. Antoni et al. / Mechanical Systems and Signal Processing 92 (2017) 248–277 257
3.4. Definition of the Fast Spectral Correlation
The fact that quantity S
x
ð
a
;f;pÞscans the SC in a cyclic frequency interval ½ðp1Þ
D
f;ðpþ1Þ
D
fnaturally suggests its
aggregation for several values of pso as to reconstruct the SC over the whole cyclic frequency range. Given the sampling
period R=F
s
of the STFT in Eq. (1), the maximum cyclic frequency that can be scanned is now
a
max
F
s
2R;ð22Þ
which corresponds to the maximum value of p,
P¼F
s
=ð2RÞ
D
f

¼N
w
2R

;ð23Þ
where ½xstands for the nearest whole number rounded down. This defines the ‘‘Fast Spectral Correlation” (Fast-SC)
S
Fast
x
ð
a
;fÞ¼ P
P
p¼0
S
x
ð
a
;f;pÞ
P
P
p¼0
R
w
ð
a
p
D
fÞR
w
ð0Þ:ð24Þ
Note that the division by P
P
p¼0
R
w
ð
a
p
D
fÞ– which can be readily computed for a given data window w½n– equalizes the
estimator from the effect of kernel R
w
ð
a
Þ. It is proved in Appendix A that
EfS
Fast
x
ð
a
;fÞg ’ Sð
a
;fÞ ð25Þ
with asymptotic equality when K!1and N
w
!1. In addition, it is proved in Appendix C that the variance of the Fast-SC is
Var S
Fast
x
ð
a
;fÞ
no
1
KX
k2K
K
w
ð
a
k
;
a
ÞS
k
x
ðfÞS
k
x
ðf
a
Þ
;ð26Þ
almost everywhere, where K
w
ð
a
k
;
a
Þbehaves like R
w
ð
a
k
Þ
2
=ðN
2
w
w½N
0
4
Þ. Therefore, combining the above two results, it holds
that
lim
N
w
!1
lim
K!1
S
Fast
x
ð
a
;fÞ¼S
x
ð
a
;fÞ ð27Þ
(where the probability limit is taken in the mean-square sense) as for the ACP in Eq. (9). It is noteworthy that the expression
of the variance of the Fast-SC has the same structure as that of the quadratic estimators of the SC (including the ACP) inves-
tigated in Ref. [1]. It is also proved in Appendix B that the frequency resolution of the Fast-SC is
ð
D
fÞ
Fast
F
s
N
w
ð28Þ
for the carrier frequency and
ð
D
a
Þ
Fast
F
s
Lð29Þ
for the cyclic frequency, which is again in perfect accordance with other quadratic estimators of the SC (including the ACP)
reported in [1].
The above results prove that the Fast-SC is an asymptotically convergent (unbiased and nil variance) estimator of the SC
with similar statistical performance as the ACP. Its main advantage is that the computational effort is considerably alleviated
as compared to the ACP (8) since it essentially relies on calculating the FFT of STFT products. It will be shown in the next
section (Section 4) that it is also faster than the FAM [18], yet being closer to it than to the ACP in terms of computational
complexity. However, contrary to the FAM, the Fast-SC does not suffer from non-uniform frequency resolution and has a uni-
formly bounded variance in the ðf;
a
Þplane.
Proceeding with the same example, the Fast-SC is now computed with the same parameters as for the ACP and CMS; it is
displayed in Fig. 9 together with its evaluation at 250 Hz. A comparison in three dimensions with the ACP and CMS is also
displayed in Fig. 10. The computation took less than 5 s, which is a considerable gain as compared to the ACP (15 min). It is
seen that the maximum cyclic frequency has been pushed up much higher than the theoretical limit
a
max
19 Hz of the CMS
given by Eq. (22).
Fig. 7(c) shows that the theoretical magnitude of the peaks is accurately estimated and even slightly better than in the
ACP of Fig. 3(b). This is because the Fast-SC is so fast as compared to the latter that zero-padding could be used in evaluating
S
x
ð
a
;f;pÞby the DFT so as to increase the numerical resolution of the cyclic frequency
a
and thus reduce the picket fence
effect.
258 J. Antoni et al. / Mechanical Systems and Signal Processing 92 (2017) 248–277
[U2/Hz]
(a)
(b)
Fig. 9. (a) Fast Spectral Correlation S
Fast
x
ða;fÞof the signal analyzed in Fig. 3 computed with N
w
¼2
8
;R¼26 (a
max
D
f¼4 Hz) and 100%zero-padding and
(b) its evaluation at f¼250 Hz together with the theoretical envelope of the peaks (dotted line) as obtained from a square modulation.
Fig. 10. 3D comparison of estimates of the SC obtained from (a) the ACP, (b) the CMS and (c) the Fast-SC.
J. Antoni et al. / Mechanical Systems and Signal Processing 92 (2017) 248–277 259
3.5. The Fast-SC algorithm
The block-diagram of the Fast-SC algorithm is given in Fig. 11. It is seen that the implementation of the Fast-SC is easily
amenable to parallel computing, which allows further potential speedup. A publicly available version of the algorithm coded
in Octave/Matlab has been posted at the following address: https://fr.mathworks.com/matlabcentral/fileexchange/60561.
4. Computational cost
The computational cost of the Fast-SC is now addressed and compared to that of the ACP, the CMS and the FAM. For the
sake of simplicity and because it will make possible a direct comparison with the results published in Ref. [18], the compu-
tational cost is addressed in terms of the number of complex multiplications. In addition, the same complex-valued FFT algo-
rithm is considered in all cases (independently of whether the processed sequences are actually real or complex), i.e. with a
fixed computational complexity on the order of Nlog
2
Nfor a N-long sequence.
4.1. Computational cost of the Fast-SC
For the Fast-SC, the count starts with data tapering of Kblocks of N
w
samples each, that is a complexity KN
w
. Next, the
calculation of the STFT involves the DFT of Kblocks of data of length N
w
. Assuming that the Fast Fourier Transform (FFT)
is used (N
w
is a power of two), the complexity is like KN
w
log
2
ðN
w
Þ. Next, there are Kproducts of ðN
w
=2pÞSTFT coefficients
repeated for p¼0;...;P(only positive frequencies are considered due to symmetry of the FFT and shifted frequencies out-
side this range are not considered to avoid aliasing), which amounts to a complexity KN
w
=2þðN
w
=21Þþþð
ðN
w
=2PÞÞ ¼ KððPþ1Þ=2ÞðN
w
PÞ. Finally, the products of ðN
w
=2pÞSTFT coefficients are Fourier transformed over K
Computaon of the STFT
(, )
STFT k
Xif
input parameters
[], ,
w
xn N R
Computaon of the scanning cyclic spectrum
*
(, ;) DFT (, ) (, )
kSTFTkSTFTk
i
Sfp X ifX ifpf
0
2
(, ) (, ) (, ; )
sw
p
jN
FN
kkk
Sf SfSfpe
output = STFT coefficients
summaon of scans with phase calibraon
(, ;)
xk
Sfp
Loop for p= 0..P
2
(0)
(, ) (, ) ( ) || ||
w
kk
ws
R
Sf Sf
RKwF
Computaon of the squared
window spectrum
2
() DFT|[]|
wn
Rwn
() () ( )
www
RRRpf
() 0
w
R
(, ) 0
k
Sf
End loop
magnitude calibraon
Oponal: computaon of spectral
coherence
12
1
0
() | (,)|
K
x k STFT k
Ki
Sf X if
(, )
(, ) ()
STFT k
STFT k
xk
Xif
Xif Sf
power spectrum
equalizaon
Fig. 11. Block-diagram of the Fast-SC algorithm.
260 J. Antoni et al. / Mechanical Systems and Signal Processing 92 (2017) 248–277
blocks for p¼0;...;P, leading to a complexity on the order of ððPþ1Þ=2ÞðN
w
PÞKlog
2
ðKÞ. Ignoring other computations such
as calibration, phase correction and the equalization in Eq. (24), the complexity is
C
Fast
KN
w
1þlog
2
ðN
w
ÞþPþ1
21P
N
w

1þlog
2
ðKÞðÞ

:ð30Þ
Considering that KL=Rand PN
w
=ð2RÞ, the complexity eventually reads
C
Fast
LN
w
2R2þ2log
2
ðN
w
Þþ N
w
2Rþ1

11
2R

1þlog
2
L
R

:ð31Þ
Of concern is now to compare this result with the complexity of other estimators.
4.2. Computational cost of the CMS
The complexity of the CMS comes as a particular case of Eq. (30) with P¼0, that is
C
CMS
LN
w
2R3þ2log
2
ðN
w
Þþlog
2
L
R

:ð32Þ
Therefore, the computational gain of the CMS over the Fast-SC is, asymptotically,
C
Fast
C
CMS
!
LR
ðPþ1Þ1P
N
w

Pþ1ð33Þ
which is upper bounded by Pþ1.
4.3. Computational cost of the ACP
By comparison, the ACP in Eq. (8) computed over the same cyclic frequency range requires
a
max
=
D
a
¼
ðF
s
=ð2RÞÞ=ðF
s
=LÞL=ð2RÞcyclic frequencies to be processed. The calculation of each cyclic frequency involves the tapering
of the original signal and of its frequency shifted version for a complexity 2KN
w
and Kproducts of N
w
=2 STFT coefficients
for a complexity KN
w
=2. Next comes the FFTs of Kblocks of N
w
samples for the original signal and of its frequency shifted
version for a complexity of 2KN
w
log
2
ðN
w
Þ. Ignoring other computations (e.g. multiplication by complex exponentials for
the phase correction), the leading term in the overall complexity of the ACP is C
ACP
N
w
ðL=ð2RÞÞ
2
ð5þ4log
2
ðN
w
ÞÞ. Therefore,
the computational gain is
C
ACP
C
Fast
L
2R
5þ4log
2
ðN
w
Þ
2þlog
2
ðN
w
Þþ
N
w
2R
þ1

1
1
2R

1þlog
2L
R

!
|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}
q
!
LR
L
2R:ð34Þ
Since factor
q
in the above equation is mainly located between 0:5 and 2 for wide ranges of values of N
w
and L(e.g.
2
3
6N
w
62
12
and 2
10
6L62
20
with R¼N
w
=4), the asymptotic result C
ACP
=C
Fast
L=2Ractually returns a very good
approximation of the computational gain.
4.4. Computational cost of the FAM
The computational cost of the FAM has been worked out in Ref. [18]. Using the notation of the present paper and consid-
ering a complex-valued FFT algorithm, it reads
C
FAM
LN
w
2R4þlog
2
ðN
w
ÞþN
w
21þlog
2
L
R

:ð35Þ
The computational gain of the Fast-SC over the FAM is found bounded as
1<C
FAM
C
Fast
<Rð36Þ
with asymptotic behavior
C
FAM
C
Fast
!
LR
R
1þ
2R
N
w

1
1
2R

:ð37Þ
For a fixed ratio R=N
w
(as actually advocated in Ref. [18]), this is found to be on the order of R.
J. Antoni et al. / Mechanical Systems and Signal Processing 92 (2017) 248–277 261
4.5. Discussion
The former results indicate the following ranking of computational complexities,
C
CMS
6C
Fast
<C
FAM
<C
ACP
ð38Þ
with computational gains which can be roughly summarized as
C
Fast
C
CMS
Pþ1;C
FAM
C
Fast
R;and C
ACP
C
Fast
L
2Rð39Þ
when LR.
Several conclusions are noteworthy at this juncture.
Without surprise, the CMS has the lowest complexity, yet the Fast-SC comes close to it when the cyclic frequency range to
scan is limited – i.e. when Pis small. The Fast-SC boils down to the CMS when P¼0.
The Fast-SC has a lower complexity than the FAM and its computational gain with respect to latter is found always greater
than one. In particular, by taking the condition R¼N
w
=4 which is recommended for the FAM in Ref. [18], it can be verified
from Eq. (37) that C
FAM
=C
Fast
N
w
=6.
The computational gain of the Fast-SC over the ACP grows proportionally with the signal length L, which is the most con-
siderable gain of all.
Keeping in mind that Ris a fraction of N
w
, the gain over the ACP is seen directly related to the variance reduction factor
L=N
w
, a quantity that reflects the quality of the SC estimation [1,14]. In practice, the latter should be as large as possible
(e.g. typically more than a few tens of hundreds). This proves that in any scenario where the SC is to be estimated with a
small estimation variance, the Fast-SC will have a computational advantage over the ACP.
One exception where the ACP estimator remains advantageous is when the SC is estimated for one or a few cyclic fre-
quencies
a
only.
Some of these conclusions are now verified by means of a numerical experiment. Fig. 12 compares the CPU (Central
Processing Unit) time (minimum on 100 runs) required to compute the CMS, the Fast-SC and the ACP on a laptop computer
(i7-4810MQ Processor 2.80 GHz). The signal is arbitrarily generated a hundred times (the computation time is supposed not
to depend on the signal structure) and processed for different lengths Lranging from 2
10
to 2
21
. The window length is set
arbitrarily to N
w
¼2
6
, which is small enough to make possible the calculation of the ACP of long signals. A Hann window
signal length log 2(L)
10 12 14 16 18 20 22
log2(CPU)
-15
-10
-5
0
5
10
15
20
CMS
STFT
FAST
Fast
FAM
ACP
CMS
STFT ACP
Fig. 12. CPU time in seconds (minimum value over 100 runs) required to compute the STFT, the Cyclic Modulation Spectrum (CMS), the Fast Spectral
Correlation (Fast-SC) and the Averaged Cyclic Periodogram (ACP) for different signal lengths Lwith N
w
¼2
6
and R¼10. The dotted black line shows the
theoretical CPU time of the ACP.
262 J. Antoni et al. / Mechanical Systems and Signal Processing 92 (2017) 248–277
is used with shift R¼10 which corresponds to nearly 85%overlap. Also shown is the CPU time required to compute the STFT.
For the CMS and Fast-SC, care is taken to use the next power of two for Kin formula (19) in order to benefit from the fastest
version of the FFT. In addition, the time required to compute the denominator in Eq. (24) (used for calibration) is not
accounted for – this factor can actually be pre-calculated.
Comparison of the results with the theoretical trends shows good agreement for signal lengths larger than 2
11
.Itis
believed that the departure for smaller values of Lis due to the time used for memory allocation (not accounted for in
the above results) which prevails on short signals. The asymptotic behavior for long signals is yet very good. It is seen that
the gain in computational time becomes considerable when the signal length increases: for instance when L¼2
17
, the CPU
time is about 0.3 s for the Fast-SC whereas it is 10 min for the ACP; when L¼2
20
(which is not uncommon in practice), the
CPU time is 3.7 s for the Fast-SC whereas it is about 17:7 hours for the ACP (not computed in the experiment!).
5. Practical recommendations
Just as for any spectral quantity, the estimation of the SC requires careful setting of some parameters. In the case of the
Fast-SC, the number of parameters to tune can be reduced to two: the maximum cyclic frequency to scrutinize,
a
max
, and the
frequency resolution,
D
f, of the carrier. According to formulas (2) and (22), they correspond to setting N
w
and R, respectively.
The investigation of the computational cost undertaken in the former section will now help to provide strait guidelines to set
the values of Rand N
w
.
5.1. Setting the block shift R
The value of the block shift Rgoverns the maximum cyclic frequency
a
max
that can be scrutinized according to Eq. (22).At
the same time, Rshould not be taken too small since computational cost was found inversely proportional to it in Eq. (31).
Yet, it has been shown in Ref. [1] that a significant fraction of overlap should be provisioned to avoid cyclic leakage
(a recommended value is at least 75% with classical windows, that is R0:25N
w
). Therefore, a safe guideline is to choose
the greatest value which complies with the constraint
R6min 0:25N
w
;F
s
2
a
max

:ð40Þ
5.2. Setting the window length N
w
The value of the window length N
w
governs the frequency resolution
D
faccording to Eq. (28). There are two reasons while
N
w
should be taken as small as possible while not sacrifying the required spectral resolution. First, the computational com-
plexity has been found proportional to the square of N
w
in Eq. (31). Second, it has been shown in Ref. [1] that in order to
control estimation errors, any linear estimator of the SC should satisfy
D
f
D
a
, that is N
w
L.
6. Examples of application
This section illustrates the use of the Fast-SC on several experiments concerned with the diagnostics of rolling element
bearings. Several of the signals actually happen to be too long to be analyzed with the ACP in a reasonable time, a situation
that only makes possible the use of the CMS and the Fast-SC.
As explained in Refs. [9,11,19], a distinctive symptom of faulty rolling element bearings is to produce cyclostationary sig-
nals, both in the incipient and the advanced stages. This is materialized by an alignment of non-zero components in the SC at
the corresponding fault frequency,
a
¼f
fault
. The magnitude and the number of harmonics linked to the incriminated cyclic
frequency may serve as an indicator of severity of the fault. As advocated in Ref. [11], the Spectral Coherence is computed
instead of the SC. The Spectral Coherence
c
x
ð
a
;fÞ¼ S
x
ð
a
;fÞ
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
S
x
ðfÞS
x
ðf
a
Þ
pð41Þ
with S
x
ðfÞS
x
ð0;fÞ(as obtained from the time average of the spectrogram) is a normalized version of the SC with magnitude
normalized within 0 and 1. It may be directly interpreted as the ‘‘depth” of a modulation with frequency
a
and carrier f. The
Spectral Coherence may also be interpreted as the SC of the whitened signal, which tends to equalize regions with very dif-
ferent energy levels and thus to magnify weak cyclostationary signals. In the following, the Spectral Coherence will serve as a
basis to define the Squared Envelope Spectrum (SES) [9],
S
SES
x
ð
a
Þ¼ Z
f
2
f
1
c
x
ð
a
;fÞdf
;ð42Þ
J. Antoni et al. / Mechanical Systems and Signal Processing 92 (2017) 248–277 263
measured in a given frequency band ½f
1
;f
2
and, a newly proposed spectral quantity, the ‘‘Enhanced Envelope Spectrum”
(EES)
S
EES
x
ð
a
Þ¼Z
f
2
f
1
c
x
ð
a
;fÞ
jj
df :ð43Þ
The SES and the EES have the squared units of the signal. The integral in Eqs. (42) and (43) are replaced by discrete sums
over frequencies f
k
¼k
D
fwhen its comes to estimators.
It is noteworthy that S
SES
x
ð
a
Þ6S
EES
x
ð
a
Þin general. The EES is expected to better enhance non-zero cyclic components than
the SES because the latter integrates complex values, a process which may possibly converge towards zero in the case of fast
rotating phases as demonstrated in Section 6.1.3.
The connection between the SC, the Spectral Coherence, the SES and the EES is schemed in Fig. 13. These quantities are
now compared on case studies.
6.1. Bearing signatures in fans
This first experiment deals with the detection of rolling element signatures in small fans in a production line. Being brand
new, the rolling element bearings are not expected to be seriously damaged, yet small defects due to possible mishandling or
forces exerted during mounting are to be detected. This presents several challenges. First, the rolling element bearing signa-
ture is expected to be weak. Second, it happens to occur in very localized frequency bands, which asks for a fine frequency
resolution
D
f. Third, the number of harmonics has to be assessed accurately as it provides an indication of the fault severity;
this requires scrutinizing a wide cyclic frequency range
a
max
. As explained in Section 2.3, these constraints are difficult to be
satisfied conjointly by the CMS.
Vibration signals are recorded on fans hanged by tensioners in order to meet free-free boundary conditions. The sampling
frequency is F
s
= 131,072 Hz in case 1 and F
s
= 51,200 Hz in cases 2 and 3. The recording time is 5 s. All signals are processed
in the same way. First, the harmonics of the shaft rotation are removed – as advocated in Ref. [19] – with a combfilter. Next,
the cyclic range
a
max
is chosen so as to include at least 3 harmonics of the highest fault frequency, viz the Ball Pass Frequency
on Inner Race (BPFI). This fixes the window length N
w
to be used with the CMS according to formula (13). The window length
in the Fast-SC is set to N
w
¼2
9
in order to achieve a frequency resolution
D
f¼100 Hz. The block shift Ris set according to
formula (22) in order to reach the required cyclic frequency range
a
max
. A Hann window is used. The parameter settings are
reported in Table 1 for the three tested fans. In all cases, the SES and the EES are computed over the full frequency range
½f
1
¼0;f
2
¼F
s
=2and hence referred to as wide-band envelope spectra.
6.1.1. Case 1
The first tested fan evidences a marked signature of the rolling elements. Fig. 14(a) and (b) show excerpts of the Spectral
Coherences based on the CMS and the Fast-SC in a frequency band between 44 kHz and 49 kHz. They display alignments of
Spectral Correlation
(, )
x
Sf
Spectral Coherence
(, )
x
f
Squared Envelope Spectrum
()
SES
x
S
Enhanced Envelope Spectru
()
EES
x
S
Fig. 13. Connection between the Spectral Correlation and the envelope spectra handled in the paper.
Table 1
Parameter settings used in the experiments of tested fans.
Case 1 Case 2 Case 3
Sampling frequency F
s
(kHz) 131.072 51.2 51.2
Duration (s) 5
a
max
in CMS 1093 622 909
N
w
in CMS 46 82 144
N
w
in Fast-SC 2
9
Rin Fast-SC 6 12 21
Rotation frequency – f
rot
(Hz) 66.6 39.9 80.1
Ball pass frequency on outer race – BPFO (Hz) 139.9 101.4 168.2
Ball pass frequency on inner race – BPFI (Hz) 259.6 177.9 312.1
Ball spin frequency – BSF (Hz) 101.1 67.4 121.6
Cage frequency – FTF (Hz) 23.3 14.5 28.0
264 J. Antoni et al. / Mechanical Systems and Signal Processing 92 (2017) 248–277
non-zero values at cyclic frequencies of the Ball Spin Frequency (BSF) and its multiples up to 900 Hz. It is seen that the fre-
quency resolution allowed by the CMS is not sufficient to reveal some fine details in the Spectral Coherence. In particular, the
sidebands due to modulation by the shaft rotation (marked by blue dotted vertical lines) are ‘‘erased” in the CMS-based
coherence. The same observation is made on the envelope spectra shown in Fig. 15. It is also noteworthy that the
Fast-SC-based EES is able to show 9 harmonics of the BSF in the scanned cyclic frequency band, whereas the CMS-based
EES can hardly disclose more than 5 harmonics due to the low pass effect described by Eq. (11). A final remark is that the
Fast-SC-based EES displays a flat baseline due to the equalization in Eq. (24), which would make easier the design of thresh-
old in statistical tests.
6.1.2. Case 2
Excerpts of the Spectral Coherences based on the CMS and the Fast-SC of the second tested fan are displayed in Fig. 16 in
the cyclic frequency range [0;250] Hz. They show quite unusual pictures which, at first sight, look more random than struc-
tured (i.e. with expected marked vertical lines) despite the variance reduction factor being considerable (
D
f=
D
a
= 3122 for
the CMS and 500 for the Fast-SC). However, a closer inspection of the Fast-SC in Fig. 16(b) reveals higher ‘‘densities of points”
along some vertical lines. This is better evidenced, after integration, by the Fast-SC-based EES displayed in Fig. 17(c), which is
able to detect the signatures of the outer race (BPFO), the inner race (BPFI), the cage (FTF), and even the rolling elements
(BSF) (note that a wider cyclic frequency range than in the excerpts of Fig. 16 is used here). Similar detection is much more
difficult with the SES and the CMS-based EES shown in Fig. 17(a) and (b).
Several remarks are in order here. First, the apparent randomness of the CMS and Fast-SC is probably due to a very weak
signal produced by the rolling element bearing (this seems to be consolidated by the fact that vertical lines slowly build up
when increasing the record length); actually, the bearing might not be faulty at all, but just noisier than expected due to
possible mounting imprecision. Second, the CMS-based coherence appears unable to detect this weak cyclostationary signal,
probably because it is smeared by a too coarse frequency resolution (as illustrated in Fig. 5(c)).
6.1.3. Case 3
This last case resembles very much the previous one, yet with only the signature of the outer race. Note that the BPFO is
found significantly lower than the expected value, around
a
¼160:1 Hz instead of 168:1 Hz. Here again, both the Spectral
Coherences based on the CMS and the Fast-SC (not shown) display a high degree of randomness and only the Fast-SC-
based EES can unveil the rolling element bearing signature – see Fig. 18. One possible reason of this superiority is that,
0 100 200 300 400 500 600 700 800 900
44
45
46
47
48
49
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
0.5
spectral frequency f (kHz)
44
45
46
47
48
49
spectral frequency f (kHz)
(a)
(b)
Fig. 14. Case 1. Excerpts of the Spectral Coherence based on (a) the Cyclic Modulation Spectrum S
CMS
x
ða;fÞ(
D
f¼1113 Hz,
D
a¼0:2 Hz) and (b) the Fast-SC
S
Fast
x
ða;fÞ(
D
f¼100 Hz; Note: the initial cyclic frequency resolution
D
a¼0:2 Hz has been decreased to
D
a2 Hz after smoothing the images in the
horizontally direction in order to match the screen resolution).
J. Antoni et al. / Mechanical Systems and Signal Processing 92 (2017) 248–277 265
(a)
(b)
(c)
1xBSF
2xBSF
3xBSF
4xBSF
5xBSF
6xBSF
7xBSF
8xBSF
9xBSF
[U2]
[U2]
[U2]
frot
Fig. 15. Case 1. (a) Squared Envelope Spectrum S
SES
x
ðaÞ, (b) CMS-based Enhanced Envelope Spectrum S
EES
x
ðaÞand (c) Fast-SC-based Enhanced Envelope
Spectrum S
EES
x
ðaÞin full band ½0;F
s
=2with
D
a¼0:2 Hz.
(a)
(b)
Fig. 16. Case 2. Excerpts of the Spectral Coherence based on (a) the Cyclic Modulation Spectrum S
CMS
x
ða;fÞ(
D
f¼624 Hz,
D
a¼0:2 Hz) and (b) the Fast-SC
S
Fast
x
ða;fÞ(
D
f¼100 Hz,
D
a¼0:2 Hz).
266 J. Antoni et al. / Mechanical Systems and Signal Processing 92 (2017) 248–277
1xBPFO
1xBPFI
FTF
2xBSF
(a)
(b)
(c)
2xBPFO
3xBPFO
4xBPFO
2xBPFI
frot
frot
[U2]
[U2]
[U2]
Fig. 17. Case 2. (a) Squared Envelope Spectrum S
SES
x
ðaÞ, (b) CMS-based Enhanced Envelope Spectrum S
EES
x
ðaÞand (c) Fast-SC-based Enhanced Envelope
Spectrum S
EES
x
ðaÞin full band ½0;F
s
=2with
D
a¼0:2 Hz.
1xBPFO
(a)
(b)
(c)
frot 2xBPFO 3xBPFO 4xBPFO 5xBPFO 6xBPFO
[U2]
[U2]
[U2]
Fig. 18. Case 3. (a) Squared Envelope Spectrum S
SES
x
ðaÞ, (b) CMS-based Enhanced Envelope Spectrum S
EES
x
and (c) Fast-SC-based Enhanced Envelope
Spectrum S
EES
x
ðaÞin full band ½0;F
s
=2with
D
a¼0:2 Hz.
J. Antoni et al. / Mechanical Systems and Signal Processing 92 (2017) 248–277 267
due to a finer frequency resolution, the numerical evaluation of integral (43) involves more frequency bins for the Fast-SC-
based EES than for the CMS-based EES and therefore results in a better averaging of estimation noise.
This situation may be mathematically understood with the help of the following phenomenological model. From inspec-
tion of Figs. 16 and 18, the estimated Spectral Coherence seems to behave like
c
c
x
ð
a
;fÞS
x
ð
a
Þð
a
;fÞ ð44Þ
where
ð
a
;fÞis a zero-mean random variable and S
x
ð
a
Þis the theoretical envelope spectrum. According to this model, the
complex average
1
k
2
k
1
þ1X
k
2
k¼k
1
c
c
x
ð
a
;f
k
Þ
!0ð45Þ
– i.e. converges to zero as ðk
2
k
1
þ1Þgrows – whereas the absolute average
1
k
2
k
1
þ1X
k
2
k¼k
1
jc
c
x
ð
a
;f
k
Þj ! jS
x
ð
a
Þj ð46Þ
converges to the correct magnitude of the envelope spectrum. Referring back to Eqs. (42) and (43), the SES corresponds to the
extreme case represented by the complex average in Eq. (45) and therefore tends to converge destructively as seen in
Figs. 17(a) and 18(a), whereas the Fast-SC-based EES corresponds to the extreme case represented by the absolute average
in Eq. (46) and therefore tends to converge constructively as seen in Figs. 17(c) and 18(c). The CMS-based EES corresponds to
an intermediate case where some of the complex spectral components are first averaged in a large frequency bin of width
D
f
(this is implicit to the use of the STFT
1
) before the absolute average is considered; although this intermediate position is clear in
Fig. 15(b), the overall effect is rather destructive in Figs. 17(b) and 18(b).
6.2. Performance evaluation in a benchmark database
The database of the Case Western Reserve University (CRWU) Bearing Data Center [20] has become a standard benchmark
against which newly proposed techniques are often tested. Ref. [21] provides a valuable description of the vibration signals
found in the database together with the identification of the difficult cases which are worth consideration when trying to
improve upon results obtained from state-of-the-art methods, such as the SES with possible prewhitening [22] and optimal
prefiltering with the kurtogram [23]. The Fast-SC and the EES have been systematically computed for all cases investigated in
Ref. [21]. The conclusion was that the Fast-SC never performed worse than the reference methods, but could improve the
diagnosis in some difficult cases. Only one example is reported here which illustrates quite well the general observation.
It relates to record 277DE, a case with an inner-race fault, denoted as ‘‘partially successful” for all the 3 methods tested
in Ref. [21] (see Table B4 therein). This is a difficult case because the accelerometric sensor is located on the drive-end
bearing and is therefore far from the faulty bearing located on the fan-end, on the other side of a large – and possibly noisy
– electrical motor.
The parameter settings are given in Table 2.
Fig. 19(a) and (b) compare excerpts of the Spectral Coherences based on the CMS and the Fast-SC in the band ½0;F
s
=2kHz.
Although the fundamental of the fan-end inner-race fault is detected in the CMS at 142.3 Hz, it is much less resolved than in
the Fast-SC probably because of the coarse frequency resolution. Fig. 19(c) shows a zoomed view of the Fast Spectral
Table 2
Parameter settings used in the experiment of record 277DE.
Data set 277DE
Sampling frequency F
s
(kHz) 12
Duration (s) 10
a
max
(Hz) 900
N
w
in CMS 16
N
w
in Fast-SC 256
Rin Fast-SC 9
Rotation frequency f
rot
(Hz) 28.9
Ball pass frequency on inner race (fan end) – BPFI (Hz) 142.9
Ball pass frequency on inner race (drive end) – BPFI (Hz) 156.4
1
Let XLðfÞdenote the Fourier transform of signal xðtnÞover the full record length n¼0;...;L1; then the STFT at time index iR is related to XLðfÞas
X
STFT
ði;fÞ¼Z
þF
s
=2
F
s
=2
X
L
ðfuÞWðuÞe
j2
p
iRu
du Z
D
f=2
D
f=2
X
L
ðfuÞWðuÞe
j2
p
iRu
du
where WðuÞstands for the Fourier transform of w½n.
268 J. Antoni et al. / Mechanical Systems and Signal Processing 92 (2017) 248–277
(a)
(b)
(c)
Fig. 19. Excerpts of the Spectral Coherence based on (a) the Cyclic Modulation Spectrum S
CMS
x
ða;fÞ(
D
f¼900 Hz,
D
a¼0:1 Hz) and (b) the Fast-SC S
Fast
x
ða;fÞ
in full band ½0;F
s
=2and (c) in selected band ½4:3;5:5kHz (
D
f¼50 Hz,
D
a¼0:1 Hz).
(a)
(b)
(c)
1xBPFI
frot
2xBPFI
3xBPFI
3xBPFI
Fig. 20. (a) Squared Envelope Spectrum S
SES
x
ðaÞ, (b) Fast-SC-based Enhanced Envelope Spectrum S
EES
x
ðaÞin full band ½0;F
s
=2and (c) in selected band
½4:3;5:5kHz with
D
a¼0:1 Hz.
J. Antoni et al. / Mechanical Systems and Signal Processing 92 (2017) 248–277 269
Coherence in band ½4:3;5:5kHz which seems to maximize the SNR. The detection of the fault is further demonstrated by
means of the EES computed in this band in Fig. 20. The Fast-SC-based EES clearly evidences the dominant harmonics of
the BPFI with sidebands at the shaft rotation, contrary to the classical SES and the CMS-based EES which have a poorer SNR.
It is noted that the fact that the BPFI (¼4:947 f
rot
) happens to fall close to the fifth multiple of the shaft speed might be
troublesome for diagnostics. Here, the very fine cyclic frequency resolution
D
a
¼0:1 Hz in addition to the detection of higher
order harmonics of the fault prevent us from such a confusion.
Incidentally, the Spectral Coherence in Fig. 19(b) also shows a high interference at
a
¼156:4 Hz which dominates in the
band ½1:5;3:5kHz. The Fast-SC-based EES computed in this band is shown in Fig. 21. It displays the signature of an inner-
race fault in the drive-end bearing with marked side-bands at twice the rotation speed. Although no such fault is reported in
the literature for the configuration relating to record 277DE, it is believed that the accelerometric sensor (which is close to
drive-end bearing) sees a misalignment of the drive-end bearing due to the numerous dismantling operations carried out in
the experiment – a loose misaligned inner-race would then have a potential signature at the BPFI with modulations at twice
the rotation speed.
6.3. Diagnostics of bearing under variable regime with the Order-Frequency Fast-SC
This section illustrates how the proposed Fast-SC can be easily extended to analyze machine signals recorded under vary-
ing regime. The consideration of variable operating conditions is not only a necessity in applications where the machine can
hardly be operated at constant regime, but it is also apt to provide more diagnostic information than could be obtained
otherwise [24,25]. The class of cyclostationary signals has recently been enlarged to account for this situation. Namely,
‘‘angle-time” cyclostationarity embodies processes whose statistics are periodic with respect to the angle of rotation of
the machine (thus invariant under speed variations) while maintaining a structural description of the carrier that is constant
in time [26] (see also Ref. [27] for other generalizations of cyclostationarity). An important statistical quantity for describing
angle-time cyclostationary processes is the angle-time autocorrelation function (given here with continuous variables)
R
x
ðh;
s
Þ¼EfxðtðhÞÞxðtðh
s
ÞÞ
g ð47Þ
which is periodic in h;R
x
ðh;
s
Þ¼R
x
ðhþ
H
;
s
Þ. Its double Fourier transform with respect to hand
s
then defines the Order-
Frequency Spectral Correlation (OF-SC), S
x
ð
a
;fÞ, where the cyclic frequency
a
– now expressed in order – truly reflects
the kinematic information of the machine whereas the spectral frequency f– expressed in Hz – reflects the dynamic infor-
mation about the propagation medium (e.g. vibration modes). Estimators of the OF-SC have been proposed in Refs. [28–30]
based on the ACP. The Fast-SC happens to be very well suited to estimate the OF-SC with the expected gain in computational
time discussed in Section 4. The methodology essentially relies on resampling the STFT from the time to the angular domain,
while maintaining a constant spectral bandwidth:
1. compute the STFT coefficients X
STFT
ði;fÞof the signal in the time domain,
2. convert the STFT coefficients to their phase-corrected counterpart X
w
ði;fÞ(the Gabor coefficients) using Eq. (3):
X
w
ði;f
k
Þ¼X
STFT
ði;f
k
Þe
j2
p
t
i
f
k
with t
i
¼iR=F
s
,
3. resample X
w
ði;f
k
Þfrom time to angle: t
i
!h
n
¼n
D
h,
4. convert back to the STFT coefficients: X
STFT
ðn;f
k
Þ¼X
w
ðn;f
k
Þe
j2
p
tðh
n
Þf
k
with tðh
n
Þthe time instant corresponding to angle h
n
,
5. apply the Fast-SC to X
STFT
ðn;f
k
Þ.
The reason of resampling the Gabor coefficients rather than the STFT coefficients is because the former are much
smoother in time than the latter (e.g. see Fig. 7), which follows the idea elaborated in Ref. [31].
1xBPFI
2xBPFI
3xBPFI
2x frot
2x frot 2x frot
2x frot 2x frot
2x frot 2x frot
Fig. 21. Fast-SC-based Enhanced Envelope Spectrum S
EES
x
ðaÞin selected band ½1:5;3:5kHz with
D
a¼0:1 Hz.
270 J. Antoni et al. / Mechanical Systems and Signal Processing 92 (2017) 248–277
The methodology is now illustrated for the diagnostics of rolling element bearings under varying regime. For the sake of
comparison with previously published works, the data provided as supplementary material in Ref. [32] are analyzed. They
consist of measurements taken by a laser vibrometer on a small test-rig with three pre-fabricated bearing faults whose fault
orders are reported in Table 3. The machine speed is manually varied between 10 Hz and 20 Hz and measured with an enco-
der mounted on the drive shaft. Sampling frequency is F
s
¼50 kHz and record duration is 21 s (see Ref. [32] for a full descrip-
tion of the experimental protocol).
All signals were processed with the Fast OF-SC as explained above, using a Hann window of length N
w
¼2
11
with 90%
overlap. This returned frequency resolutions of
D
f¼37 Hz and
D
a
¼0:004 order. The STFT was resampled from time to
angle by using cubic splines interpolation. The computational time of the STFT was about 2 s and that of the Fast OF-SC about
26 s including 6 s for the resampling process. By comparison, Eq. (34) indicates that about 3 h would have been necessary to
obtain the same results with the ACP.
The results are displayed in Figs. 22(a), 23(a) and 24(a) for the outer-race fault, the inner-race fault, and the ball fault
respectively. The Fast-OF coherence clearly evidences the presence of angle-time cyclostationarity with several vertical lines
at orders that correspond to the shaft rotation, the bearing faults, their harmonics, and sidebands due to other modulations.
They are mainly located in a frequency range below 5 kHz, which most probably corresponds to the achievable passband
returned by the laser velocity measurements. Close-ups of the Fast-OF coherence show that the bearing faults are correctly
identified at the expected orders (see Table 3), except for the ball-fault which has a signature at the cage order (a more
typical signature would have been at the ball spin order with sidebands at the cage order). It is also observed that several
resonances are excited up to 5 kHz which differ with the type of fault. Note that the Fast-OF coherence preserved a very fine
resolution in both the spectral frequency and the order domains, in spite of the speed fluctuations.
The EES was next computed by integrating the OF coherence according to formula (43) in a band below 5 kHz with max-
imal signal-to-noise ratio and further multiplied with the complementary gain of a sharp combfilter in order to remove har-
monics of the shaft orders. Results are displayed in Figs. 22(b), 23(b) and 24(b). They demonstrate typical signatures of the
bearing faults with few ambiguity for diagnostics. The outer-race fault shows distinctly the BPOO (ball-pass order on the
outer race) and its harmonics with marked sidebands at twice the shaft rotation which might be due to a slight ovalization
(a)
(b)
BPOO
2xBPOO
BPOO+2
BPOO-2
2xBPOO-2
2xBPOO+2
3xBPOO-2
Fig. 22. (a) Spectral Coherence c
Fast
x
ða;fÞ(
D
f¼37 Hz,
D
a¼0:004 order) and Enhanced Envelope Spectrum S
EES
x
ðaÞin the band ½0;3kHz. Outer-race fault
found at order 3.543.
Table 3
Bearing fault characteristics (orders).
Ball pass order on the outer race (BPOO) 3.592
Ball pass order on the inner race (BPOI) 5.409
Ball spin order (BSO) 2.376
Fundamental train (cage) order (FTO) 0.399
J. Antoni et al. / Mechanical Systems and Signal Processing 92 (2017) 248–277 271
(a)
(b)
BPOI
BPOI-1
BPOI-2
BPOI-3
BPOI-4
BPOI-5
BPOI+1
BPOI+2
Fig. 23. (a) Spectral Coherence c
Fast
x
ða;fÞ(
D
f¼37 Hz,
D
a¼0:004 order) and Enhanced Envelope Spectrum S
EES
x
ðaÞin the band ½0;5kHz. Inner-race fault
found at order 5.398.
(a)
(b)
FTO
2xFTO
3xFTO
4xFTO
5xFTO
6xFTO
7xFTO
Fig. 24. (a) Spectral Coherence c
Fast
x
ða;fÞ(
D
f¼37 Hz,
D
a¼0:004 order) and Enhanced Envelope Spectrum S
EES
x
ðaÞin the band ½0;5kHz. Ball fault found at
order 0.396.
272 J. Antoni et al. / Mechanical Systems and Signal Processing 92 (2017) 248–277
of the cage. The inner-race fault shows distinctly the BPOI (ball-pass order on the inner race) with several sidebands at the
shaft speed typically due to rotation of the fault in a static but non-homogeneous load distribution. As previously mentioned,
the BSO (ball spin order) is missing in the ball fault signature, yet the presence of several harmonics of the cage order is
symptomatic of a fault related to it.
By comparison, the approach followed in Ref. [32] was to first to resample the signals in the angular domain, then to
denoise them with orthogonal wavelets and to perform a short-term running synchronous average. The processed signal
were then bandpass filtered and enveloped in a target band corresponding to 1800–2400 Hz. As mentioned in Ref. [32], this
posed a problem since the bandpass filter could not be hold constant in Hertz when performed in the angular domain.
Another mentioned problem is that angular resampling ‘‘compressed or elongated” the impulse response of the bearing sig-
nal so that ‘‘too much variation in speed could not be considered”.
It is emphasized that no such limitations arise here with the OF-SC. First, no pre-processing is necessary and the signals
are analyzed in their entirety. Second, the frequency decomposition of the STFT is applied in the time domain, thus without
distortions as would happen if done in the angular domain. Finally, the OF-SC and the corresponding Enhanced Envelope
Order Spectrum clearly evidence the fault signature with maximum peaks at the expected fault orders (Ref. [32] introduced
further post-processing because the maximum peak in the envelope order spectrum of the inner-race fault did not coincide
with the fault order).
7. Conclusions
This paper has introduced a new algorithm to compute the Spectral Correlation. Compared to the classical approach based
on the Averaged Cyclic Periodogram (ACP), the proposed estimator – coined Fast Spectral Correlation (Fast-SC) – offers a sub-
stantial computational gain which makes it very practical when it comes to analyse long records over a wide cyclic frequency
range. The gain may be explained by an analogy with the computation of an auto/cross correlation function by means of the
FFT algorithm instead of a loop on time-lags. Besides, the Fast-SC is easily amenable to parallel implementation which would
still allow further speedup. The Fast-SC has been shown to have similar statistical performance as the ACP, contrary to other
fast estimators which are either biased (the Cyclic Modulation Spectrum) or have a non-uniformly bounded variance (the FFT
Accumulation Method).
At the same time, the principle of the Fast-SC provides a new interpretation of (second-order) cyclostationarity in terms
of periodic correlations between frequency bins of the Short-Time Fourier Transform. The use of the Fast-SC has been illus-
trated on several vibration signals in order to detect rolling bearing signatures and faults. Obviously, the same procedure can
be followed in other domains of applications. The increased computational efficiency of the Fast-SC has also led to the pro-
posal of an Enhanced Envelope Spectrum – an improved version of the envelope spectrum based on the integration of the
magnitude of the Fast Spectral Coherence over frequencies. Eventually, it has been shown how the Fast-SC can be easily
extended to angle-time cyclostationary signals, for instance when analysing vibration signals of rotating machines captured
under nonstationary regimes.
Acknowledgments
This work was performed within the framework of the Labex CeLyA of University of Lyon, operated by the French
National Research Agency.
Appendix A. Proof of Eqs. (11) and (20)
Let us start by expressing the STFT
X
STFT
ði;f
k
Þ¼X
m2Z
x½iR þmw½me
j2
p
m
fk
Fs
ðA:1Þ
(where the summation is now over an infinite number of samples) by means of the spectral decomposition of a discrete
stochastic process [33],x½n¼RdXð
m
Þe
j2
p
n
m
=F
s
and of the data window, w½n¼RWð
m
Þe
j2
p
ðnN
0
Þ
m
=F
s
d
m
. Inserting the former for-
mulas into the expression of the STFT, one has
X
STFT
ði;fÞ¼ZZ dXð
m
ÞWð
m
0
ÞX
m2Z
e
j2
p
ðiRþmÞ
m
Fs
e
j2
p
ðmN
0
Þ
m
0
Fs
e
j2
p
m
f
Fs
d
m
0
:ðA:2Þ
Let us now evaluate the expected value C
i
ðf
1
;f
2
Þ¼EfX
STFT
ði;f
2
ÞX
STFT
ði;f
1
Þ
g:
Ciðf1;f2Þ¼ZZZZ EfdXðm2ÞdXðm1ÞgWðm0
2ÞWðm0
1ÞX
m;m
0
2Z
2
ej2
p
ðiRþmÞ
m
2
Fs
ej2
p
ðiRþm
0
Þ
m
1
Fs
ej2
p
ðmN
0
Þ
m
0
2
Fs
ej2
p
ðm
0
N
0
Þ
m
0
1
Fs
ej2
p
m
f2
Fs
ej2
p
m
0f1
Fs
dm0
2dm0
1:
ðA:3Þ
J. Antoni et al. / Mechanical Systems and Signal Processing 92 (2017) 248–277 273
The above equation involves the quantity EfdXð
m
2
ÞdXð
m
1
Þ
gwhich, for a cyclostationary stochastic process with cyclic fre-
quencies f
a
k
g
k2K
, is equal to P
k2K
S
k
x
ð
m
2
Þdð
m
2
m
1
a
k
Þd
m
2
d
m
1
[4]. Therefore, using the property of the Dirac,
C
i
ðf
1
;f
2
Þ¼ZZZ X
k
S
k
x
ðm
1
þ
a
k
ÞWðm
0
2
ÞWðm
0
1
Þ
X
m;m
0
2Z
2
e
j2
p
ðiRþmÞ
ð
m
1þakÞ
Fs
e
j2
p
ðiRþm
0
Þ
m
1
Fs
e
j2
p
ðmN
0
Þ
m
0
2
Fs
e
j2
p
ðm
0
N
0
Þ
m
0
1
Fs
e
j2
p
m
f2
Fs
e
j2
p
m
0f1
Fs
dm
0
2
dm
0
1
dm
1
:
ðA:4Þ
The next step is to recognize that P
m2Z
e
j2
p
m
m
=F
s
¼F
s
dð
m
Þfor j
m
j<F
s
=2. Therefore,
C
i
ðf
1
;f
2
Þ¼F
s
ZX
k
S
k
x
ð
m
1
þ
a
k
ÞWðf
2
m
1
a
k
ÞWðf
1
m
1
Þ
d
m
1
e
j2
p
ðiRþN
0
Þ
a
k
Fs
e
j2
p
N
0
ðf2f1Þ
Fs
:ðA:5Þ
Now, if S
k
x
ðfÞis assumed smooth enough in fto be almost constant as compared to WðfÞ,
ZS
k
x
ð
m
1
þ
a
k
ÞWðf
2
m
1
a
k
ÞWðf
1
m
1
Þ
d
m
1
S
k
x
ðf
2
ÞZWðf
2
m
1
a
k
ÞWðf
1
m
1
Þ
d
m
1
¼S
k
x
ðf
2
ÞR
w
ð
a
k
f
2
þf
1
Þ ðA:6Þ
with R
w
ðfÞ(an even function) as defined in Eq. (12). Finally, setting f
2
¼fand f
1
¼fp
D
fin C
i
ðf
1
;f
2
Þ, one arrives at
EfX
STFT
ði;fÞX
STFT
ði;fp
D
fÞ
g’e
j2
p
N
0
p
Df
Fs
F
s
X
k
S
k
x
ðfÞR
w
ð
a
k
p
D
fÞ
|fflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}
c
k
e
j2
p
ðiRþN
0
Þ
a
k
Fs
:ðA:7Þ
The latter expression shows that e
j2
p
N
0
p
D
f
Fs
F
1
s
EfX
STFT
ði;fÞX
STFT
ði;fp
D
fÞ
ghas a Fourier series with Fourier coefficients
c
k
¼S
k
x
ðfÞR
w
ð
a
k
p
D
fÞ. Therefore,
1
KF
s
X
K1
i¼0
EfX
STFT
ði;fÞX
STFT
ði;fp
D
fÞ
ge
j2
p
ðiRþN
0
Þ
a
k
Fs
e
j2
p
N
0
p
Df
Fs
ðA:8Þ
is an evaluation of c
k
. The latter expression is recognized as the second line of Eq. (19), thus proving result (20).Result (11)
comes as a particular case when p¼0.
Appendix B. Frequency resolution
The frequency resolution of the Fast-SC is evaluated by comparing its expression to the theoretical SC when the signal of
interest is made of a pure complex exponential, xðt
n
Þ¼e
j2
pa
0
t
n
. In this case, the SC is actually made of a product of a contin-
uous Dirac and a discrete Dirac,
S
x
ð
a
;fÞ¼F
1
s
dðf
a
0
Þd½
a
;ðB:1Þ
(where d½
a
¼1if
a
¼0 and 0 otherwise) as found from direct application of definition (6).
From Eq. (24), the expected value of the Fast-SC then reads
ES
Fast
x
ð
a
;fÞ
no
¼P
P
p¼0
ES
x
ð
a
;f;pÞ
fg
P
P
p¼0
R
w
ð
a
p
D
fÞR
w
ð0Þ;ðB:2Þ
where the quantity
ES
x
ð
a
;f;pÞ
fg
¼1
Kkwk
2
F
s
X
K1
i¼0
EX
STFT
ði;fÞX
STFT
ði;fp
D
fÞ
fg
e
j2
p
a
Fs
ðiRþN
0
Þ
e
j2
p
pN0
Nw
ðB:3Þ
appearing in the above equation has been addressed in Eq. Appendix A. In the case of a pure complex exponential, it comes
dXð
m
Þ¼dð
m
a
0
Þd
m
and therefore EdXð
m
2
ÞdXð
m
1
Þ
fg¼dð
m
2
a
0
Þdð
m
1
a
0
Þd
m
2
d
m
1
in Eq. (A.3). Hence,
ES
x
ð
a
;f;pÞ
fg
¼1
Kkwk
2
F
s
Wðf
a
0
ÞWðf
a
0
p
D
fÞ
X
K1
i¼0
e
j2
p
a
Fs
ðiRþN
0
Þ
ðB:4Þ
¼1
Kkwk
2
F
s
Wðf
a
0
ÞWðf
a
0
p
D
fÞ
D
K
a
F
s

e
j2
p
a
Fs
ðN
0
þRðK1Þ=2Þ
ðB:5Þ
274 J. Antoni et al. / Mechanical Systems and Signal Processing 92 (2017) 248–277
where D
K
ðxÞ¼sinð
p
KxÞ=sinð
p
xÞstands for the Dirichlet kernel. Eq. (B.2) then becomes
ES
Fast
x
ð
a
;fÞ
no
¼Wðf
a
0
ÞD
K
a
F
s

P
P
p¼0
Wðf
a
0
p
D
fÞ
P
P
p¼0
R
w
ð
a
p
D
fÞ
R
w
ð0Þ
Kkwk
2
F
s
e
j2
p
a
Fs
ðN
0
þRðK1Þ=2Þ
:ðB:6Þ
Now, using the facts that R
w
ð0Þ¼kwk
2
;P
P
p¼0
Wðf
m
0
þ
a
k
p
D
fÞ¼w½N
0
N
w
=F
s
and P
P
p¼0
R
w
ð
a
p
D
fÞ¼jw½N
0
j
2
N
w
=F
s
in
the frequency range of interest, it finally comes
ES
Fast
x
ð
a
;fÞ
no
¼F
1
s
Wðf
a
0
Þ
w½N
0
|fflfflfflfflfflfflffl{zfflfflfflfflfflfflffl}
!
Nw!1
dðf
a
0
Þ
D
K
a
F
s

Ke
j2
p
a
Fs
ðN
0
þRðK1Þ=2Þ
|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}
!
K!1
d½
a
:ðB:7Þ
Comparison of Eq. (B.7) to the theoretical SC in Eq. (B.1) shows that
1. the frequency resolution
D
fis governed by the bandwidth of WðfÞ, that is
D
fF
s
=N
w
,
2. the cyclic frequency resolution
D
a
is governed by the bandwidth of D
K
ð
a
=F
s
Þ, that is
D
a
F
s
=L.
These results are similar to those of the classical estimators of the SC, for instance as discussed in Ref. [1]. They also prove
that the frequency resolution of the Fast-SC is independent of frequency f, contrary to the computationally efficient estimator
introduced in Ref. [18].
Appendix C. Variance of the FastSC
The variance of the Fast-SC is calculated here for R¼1. This choice not only returns the minimum achievable variance
which is closely approached by the high fraction of overlap advocated in Section 5, but is also significantly simplifies the
calculations. The variance reads
Var S
Fast
x
ð
a
;fÞ
no
¼R
w
ð0Þ
P
P
p¼0
R
w
ð
a
p
D
fÞ
2
Var X
P
p¼0
S
x
ð
a
;f;pÞ
() ðC:1Þ
where
Var X
P
p¼0
S
x
ð
a
;f;pÞ
()
¼X
p;p
0
EfS
x
ð
a
;f;pÞS
x
ð
a
;f;p
0
Þ
gEfS
x
ð
a
;f;pÞgEfS
x
ð
a
;f;p
0
Þg
:ðC:2Þ
Substituting S
x
ð
a
;f;pÞfor its expression in Eq. (19) and assuming that the STFT coefficients are Gaussian distributed,
2
it is
found that
3
Var X
P
p¼0
S
x
ð
a
;f;pÞ
()
¼
j
2
X
i;i
0
X
p;p
0
EfX
STFT
ði;fÞX
STFT
ði
0
;fÞ
gEfX
STFT
ði;fp
D
fÞ
X
STFT
ði
0
;fp
0
D
fÞg
þEfX
STFT
ði;fÞX
STFT
ði
0
;fp
0
D
fÞgEfX
STFT
ði;fp
D
fÞ
X
STFT
ði
0
;fÞ
ge
j2
p
ðii
0
ÞR
a
Fs
e
j2
<