Cyclostationary 1

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372 C. CAPDESSUS E¹ A¸.
Application in real vibration signals measured on an industrial test bench is presented
and discussed. We conclude with a discussion on the possibilities and limitations of that
technique.
2. CYCLOSTATIONARITY: DEFINITION AND PROPERTIES
The theory of periodically correlated processes was introduced by Hurd [18], and
extensively studied by Gardner [19, 20], who suggested many applications including
communications or vibration signals. Cyclostationary processes are speci"c non-stationary
processes characterised by the periodical variation of the statistical moments. In this section,
we recall the main de"nitions and properties of a cyclostationary process.
2.1. DEFINITIONS
Stationarity: A random signal x (t) is said to be stationary at the nth order if its
time-domain nth-order moment does not depend on time t.
Cyclostationarity: A random signal x (t) is said to show cyclostationarity at the nth order if its
time domain nth order moment is a periodical function of time t. The fundamental frequency
of the periodicity is called cyclic frequency of the signal.
In this paper, we will focus on second-order cyclostationarity. Higher-order cyclo-
stationarity has been recently studied in [21].
Cyclostationarity at ,rst-order: The "rst order moment, i.e. the mean, is time periodical:
m (t )"E (x (t))"m (t#¹ ), (1)
where E is the mathematical expectation.
Cyclostationarity at second order: The autocorrelation function of x (t) is de"ned by
(symmetric de"nition)
RV (t, )"E (x (t# /2) x (t! /2)). (2)
The signal x (t) is second-order stationary if RV (t, )"RV ( ), i.e. independent of t. It is
second order cyclostationary if RV (t, )"RV (t#¹, ) for all t, ¹ being the cyclic period,
and "1/¹ the fundamental cyclic frequency. The periodical function can then be ex-
panded into Fourier series:
RV (t, )"
?
R? ( ) e\.LH?R
.
V
(3)
R? ( ) are the Fourier coe$cients and de"ne the cyclic autocorrelation function. For
"V0, it boils down to the stationary autocorrelation function.
The Fourier transform of R? ( ) with respect to the lag (Wiener}Khinchine relationship)
V
gives the SCDF:
S? ( f )" 8 R? ( ) e\.LHDO d .
V
-
\8
V
(4)
Note that for "0, we obtain the power spectral density (PSD).
The SCDF can be also expressed as [20]
S? ( f )"E (X ( f! /2) X* ( f# /2)) .
V
(5)
This expression shows clearly that the SCDF measures the correlation between frequency
lines centred on f and separated by a shift of .
CYCLOSTATIONARY PROCESSES 373
V V
where SV ( f ) is the power spectrum of the signal.
This function lies between 0 and 1. Like the classical coherence function, the normalisa-
tion allows to emphasise the correlation between components of weak energy compared to
those of great energy.
The second one is called the degree of cyclostationarity and was proposed by Zivanovic
and Gardner [22]. It uses the CAF:
DCS?" "R? ( )". d / "R,( )". d .
-
V
-
V
(7)
It is the ratio between the energy for O0 and that for "0 (i.e. stationary case). For
a stationary signal DCS"1.
374 C. CAPDESSUS E¹ A¸.
V
This parameter is also used in our experiments.
2.3. ESTIMATION OF THE SCDF
V
Estimation of the mean function: Since the mean function of a cyclostationary signal is
periodic, it cannot be estimated by a simple time averaging as in the case of stationary and
ergodic signals. The cyclostationary signal is not supposed to be ergodic, but cycloergodic [20,
11]. That is to say that the statistical mean is then replaced by periodical temporal
averaging, with the cyclic period ¹:
m (t)" lim 21 , x (t!n¹ ) .
,8 N Lffi\,
Synchronous averaging allows to estimate this quantity.
(9)