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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;fÞS

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 jw½nj2

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

þn¼w½N

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

n¼N

X

1

m¼1

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

jw½nj

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

jw½nj

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

fÞd/

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 p–0, 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 k–l. (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

=21Þþþð

ðN

w

=2PÞÞ ¼ KððPþ1Þ=2ÞðN

w

PÞ. Finally, the products of ðN

w

=2pÞSTFT coefﬁcients are Fourier transformed over K

Computaon of the STFT

(, )

STFT k

Xif

input parameters

[], ,

w

xn N R

Computaon of the scanning cyclic spectrum

*

(, ;) DFT (, ) (, )

kSTFTkSTFTk

i

Sfp X ifX ifpf

0

2

(, ) (, ) (, ; )

sw

p

jN

FN

kkk

Sf SfSfpe

output = STFT coeﬃcients

summaon of scans with phase calibraon

(, ;)

xk

Sfp

Loop for p= 0..P

2

(0)

(, ) (, ) ( ) || ||

w

kk

ws

R

Sf Sf

RKwF

Computaon of the squared

window spectrum

2

() DFT|[]|

wn

Rwn

() () ( )

www

RRRpf

() 0

w

R

(, ) 0

k

Sf

End loop

magnitude calibraon

Oponal: computaon 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

equalizaon

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

=LÞL=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

ðfÞS

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

m

()

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

a’2 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

;fÞS

x

ð

a

Þð

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],x½n¼RdXð

m

Þe

j2

p

n

m

=F

s

and of the data window, w½n¼RWð

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

gEfS

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

<