Nous proposons dans cet article une contribution à la théorie récemment introduite des signaux quasi-cyclostationnaires généralisés. Cette classe de signaux étend la classe des signaux quasi-cyclostationnaires au cas de signaux dont les fréquences cycliques dépendent du temps. Des représentations temps-fréquence de ces signaux sont données en fonction des statistiques cycliques généralisées. Notamment, la distribution de Wigner-Ville et la fonction d'ambiguïté sont examinées en détail. Le problème de l'extraction des caractéristiques des signaux quasi-cyclostationnaires généralisés, basée sur l'estimation d'un seul enregistrement, est également traité

IZZO, Luciano

NAPOLITANO, Antonio

IZZO, LUCIANO

A. Napolitano

Archivio della ricerca - Università degli studi di Napoli Federico II

Time-frequency representations of generalized almost-cyclostationary signals

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SEIZIÈME COLLOQUE GRETSI — 15-19 SEPTEMBRE 1997 — GRENOBLE 663
Time-frequency representations of
generalized almost-cyclostationary signals
Luciano Izzo Antonio Napolitano
Università di Napoli Federico II
Dipartimento di Ingegneria Elettronica
via Claudio 21, I-80125 Napoli, Italy
RÉSUMÉ
Nous proposons dans cet article une contribution à la théorie
récemment introduite des signaux quasi-cyclostationnaires général-
isés. Cette classe de signaux étend la classe des signaux quasi-
cyclostationnaires au cas de signaux dont les fréquences cycliques
dépendent du temps. Des représentations temps-fréquence de ces sig-
naux sont données en fonction des statistiques cycliques général-
isées. Notamment, la distribution de Wigner-Ville et la fonction
d’ambiguité sont examinées en détail. Le problème de l’extraction
des caractéristiques des signaux quasi-cyclostationnaires généralisés,
basée sur l’estimation d’un seul enregistrement, est également traité.
ABSTRACT
A contribution to the theory of very recently introduced generalized
almost-cyclostationary signals is given. Such a class of signals
generalizes that of the almost-cyclostationary signals to the case in
which the cycle frequencies depend on the lag parameter. Time-
frequency representations for these signals are expressed in terms
of generalized cyclic statistics. The Wigner-Ville distribution and the
ambiguity function are examined in detail. For generalized almost-
cyclostationary signals, the problem of signal feature extraction based
on single-record estimation is addressed.
1 Introduction
Very recently, the class of higher-order generalized almost-
cyclostationary (GACS) time-series has been introduced [5],
[6]. Time series belonging to this class are characterized
by multivariate statistical functions that are almost-periodic
functions of time whose Fourier series expansions can exhibit
coefficients and frequencies depending on the lag shifts of
the time series. Moreover, the union over all the lag shifts of
the lag-dependent frequency sets is not necessarily countable.
Almost-cyclostationary (ACS) time-series turn out to be the
subclass of GACS time-series for which the frequencies do
not depend on the lag shifts and the union of the above
mentioned sets is countable. Examples of GACS time-series
not belonging to the subclass of ACS time-series arise from
some linear time-variant transformations of ACS time-series,
such as channels introducing a time-varying delay [6]. Chirp
signals [5] and several time-jittered communication signals are
further examples. For those GACS signals that are not ACS the
second-order wide-sense characterization in terms of cyclic
autocorrelation functions and cyclic spectra is inadequate.
Then, lag-dependent cycle frequencies, generalized cyclic
autocorrelation functions and generalized cyclic spectra must
be introduced.
This paper deals with time-frequency representations of
second-order GACS signals. Specifically, after a brief intro-
duction (Section 2) on GACS signals, in Section 3 the Cohen’s
general class of time-frequency distributions [2] of GACS sig-
nals is considered and it is shown that any representation be-
longing to this class is expressed as sum of two terms. The first
one involves the generalized cyclic statistics of the signal; the
second one is related to the residual term obtained by subtract-
ing to the second-order lag product its almost-periodic compo-
nent (the time-varying autocorrelation function). The Wigner-
Ville distribution is examined in detail. Moreover, the ambigu-
ity function is considered. It is shown that it can be expressed
as sum of impulsive terms related to the generalized cyclic
statistics of the signal and a nonimpulsive component related
to the above mentioned residual term. Furthermore, the sub-
class of ACS signals is examined. In Section 4, the problem
of signal feature extraction is considered. It is shown that the
estimation of the cyclic autocorrelation as a function of cy-
cle frequency and lag parameter allows one to determine the
lag-dependent cycle frequencies and the generalized cyclic au-
tocorrelation functions annihilating the effect of the residual
term when the collect time increases. Then, an estimate of the
time-varying autocorrelation function can be easily derived.
On the contrary, in general, in time-frequency representations
the component related to the residual term cannot be separated
by the components related to the generalized cyclic statistics.
Finally, let us note that the time-frequency distributions and
the ambiguity function were originally defined with reference
to finite-energy signals. Moreover, finite-power signals can be
considered by employing Dirac’s delta functions (see, e.g.,
sine waves and chirp signals in [2]). The approach adopted
in this paper follows this line since GACS time-series exhibit
finite power. However, it is worthwhile to underline that
a different approach is adopted in [3] where for a time-
windowed ACS signal the Wigner-Ville distribution is related
to the cyclic periodograms and the ambiguity function is
recognized to be equal, but for a scale factor, to the cyclic
correlogram.
664
2 Second-order GACS signals
In the fraction-of-time probability framework, a continuous-
time finite-power time-series x.t/ is said to exhibit second-
order wide-sense cyclostationarity with cycle frequency 6D 0
if the cyclic autocorrelation function
Rx . / ,

x.t C =2/x.t Ä =2/ eÄ j2t  (1)
exists and is not zero for some  [3]. In (1),  represents
complex conjugation and hi denotes infinite-time averaging.
Analogously, the time series is said to exhibit second-order
wide-sense conjugate cyclostationarity if the conjugate cyclic
autocorrelation function
Rxx . / ,

x.t C =2/x.t Ä =2/ eÄ j2t 
exists and is not zero for some  . In the following, we will deal
with time series exhibiting cyclostationarity. The considera-
tion of time series exhibiting conjugate cyclostationarity will
require some obvious minor changes.
If the set
A ,

 2 R : Rx . / 6D 0

is countable for each  , then the time series is said to be
second-order generalized almost-cyclostationary in the wide-
sense [5] and the almost-periodic function
Rx.t;  / , E fA g

x.t C =2/x.t Ä =2/
D
X
2A
Rx . / e
j2t ; (2)
where E fA gfg denotes the almost-periodic component extrac-
tion operator [3], is referred to as the time-varying autocorrela-
tion function. Then, the lag product x.tC =2/x.tÄ =2/ can
be expressed as the sum of its almost-periodic component and
a residual term not containing additive sine wave components:
x.t C =2/x.t Ä =2/ , Rx.t;  /C ‘x.t;  / ; (3)
where 
‘x.t;  / e
Ä j2t   0; 8 2 R: (4)
In the case where the set
A ,
[
2R
A
is countable, the time-series x.t/ is said to be second-order
wide-sense ACS.
Almost-cyclostationary time-series can be characterized in
the frequency domain by the cyclic spectra Sx . f / which are
the Fourier transforms of the corresponding cyclic autocorre-
lation functions. However, as shown in [5], such a characteri-
zation is not appropriate for those GACS time-series that are
not ACS, that is, when the set A is not countable. In fact, in
such a case, even if the set A and the time-varying autocor-
relation function Rx.t;  / are continuous functions of  , the
cyclic autocorrelation functions are not necessarily continu-
ous functions of  and then the cyclic spectra can result to be
infinitesimal.
A useful characterization in the frequency domain can be
introduced for those GACS time-series for which the set A
and the function Rx.t;  / are both continuous with respect to
 . In fact, in such a case the support in the .;  / plane of the
cyclic autocorrelation function can be written as
supp

Rx . /

,

.;  / 2 A  R : Rx . / 6D 0

D
[
2W
n
.;  / 2 R R :  D  . /; Rx . / 6D 0
o
;
where W is a countable set and the (reduced-dimension) lag-
dependent cycle frequencies . / are continuous functions of
 . Therefore, if one further assumes that for  0 6D  it results
that
 0. / 6D  . /;  2 RÄD;
where D is a set with zero measure in R that does not contain
cluster points, then the time-varying autocorrelation function
(2) can be expressed as
Rx.t;  / D
X
2W
Rx; . / e
j2 ./t ; (5)
where the functions
Rx; . / , lim
1!0
D
x.t C . C1/=2/
x.t Ä . C1/=2/ eÄ j2 .C1/t  ; (6)
referred to as the generalized cyclic autocorrelation functions,
turn out to be continuous functions also when the cyclic
autocorrelation functions are not. It is useful to point out that
the limit for1 ! 0 is introduced into definition (6) to avoid
discontinuities in Rx; . / in correspondence of those  2 D
such that, for some  0 6D  , one has  0. / D  . /.
It can be easily shown that the cyclic autocorrelation
functions and the generalized cyclic autocorrelation functions
are related by the following relationships:
Rx; . / D lim
1!0
Rx . C1/

D .C1/
;
Rx . / D
X
2W
Rx; . /Ä ./;
where  D 1 for  D 0 and  D 0 for  6D 0.
Let us note that for the ACS time-series the functions
 . / are independent of  and then there exists a one-to-one
correspondence between the elements  of the set W and the
cycle frequencies belonging to the set A. Moreover, for each
 and  such that  . / D , it results that
Rx; . / D Rx . /: (7)
The Fourier transform of the generalized cyclic autocorre-
lation function
Sx; . f / ,
Z
R
Rx; . / e
Ä j2 f  d (8)
is called the generalized cyclic spectrum. In the special case of
ACS time-series, it is coincident with the cyclic spectrum.
665
3 Time-frequency representations of
GACS signals
All time-frequency distributions for a complex-valued time-
series x.t/ can be obtained from
Cx.t; f / D
Z
R3
x.u C =2/x.u Ä =2/
.;  / eÄ j2.tÄu/ eÄ j2 f  du d d
D
Z
R
x.t C =2/x.t Ä =2/ 
t
8.t;  / eÄ j2 f  d; (9)
where
8.t;  / ,
Z
R
.;  / eÄ j2 t d
and 
t
denotes convolution with respect to t . The kernel
function .;  / determines the distribution and its properties
[2].
By substituting (3) into (9) and accounting for (5) and (8),
the expression of the generic time-frequency distribution in
terms of generalized cyclic statistics can be obtained:
Cx.t; f / D
X
2W
Sx; . f / 
f
A8 .t; f /CL8x .t; f / ; (10)
where
A8 .t; f / ,
Z
R
e j2 ./t 
t
8.t;  / eÄ j2 f  d (11)
and
L8x .t; f / ,
Z
R
‘x.t;  / 
t
8.t;  / eÄ j2 f  d: (12)
In other words, all time-frequency distributions of GACS time-
series can be expressed as the sum of two contributions.
The first one is the sum of all generalized cyclic spectra
each spreaded (in the frequency domain) by the time-varying
function A8 .t; f / depending on the corresponding (reduced-
dimension) lag-dependent cycle frequency  . / and the
kernel function. The second one is related to the residual term
‘x.t;  /.
By adopting the kernel function .;  / D 1 in (9), one
obtains the Wigner-Ville distribution [2]
Wx.t; f / ,
Z
R
x.t C =2/x.t Ä =2/eÄ j2 f  d;
which, accounting for (10)-(12), can be expressed as
Wx.t; f / D
X
2W
Sx; . f / 
f
F
! f

e j2 ./t
C F
! f

‘x.t;  /

;
where F
! f
denotes the Fourier transform operator from the 
domain to the f domain.
In the special case of ACS time-series the (reduced dimen-
sion) lag-dependent cycle frequencies are constant and, hence,
(10) reduces to
Cx.t; f / D
X
2A
Sx . f / 
f
9.; f / e j2t CL8x .t; f / ; (13)
where
9.; f / ,
Z
R2
8.t;  / eÄ j2.tC f / dt d ;
and (7) has been accounted for. Moreover, for the Wigner-Ville
distribution 9.; f / D . f / and, hence,
Wx.t; f / D
X
2A
Sx . f / e
j2t C F
! f

‘x.t;  /

; (14)
which is just the result derived in [4] except for the component
depending on ‘x.t;  /. The absence of such a term in [4]
stems from the fact that for ACS signals, in the stochastic
process framework (adopted in [4]), ‘x.t;  / is dropped out
by the statistical expectation operation. However, let us note
that such a residual term is present also in the stochastic
process approach when asymptotically mean ACS (AMACS)
processes [1] are considered. Moreover, it is worthwhile to
underline that the residual term is always present in the
single-record based estimate of the Wigner-Ville distribution
that, for both ACS and AMACS processes, asymptotically
approaches expression (14) when the collect time increases.
Multiple-record estimates of the Wigner-Ville distribution lead
to a zero residual term. However, they can be singled out
only when the signal is cyclostationary (i.e., all the cycle
frequencies are multiple of a fundamental one) and the period
of cyclostationarity is known [7].
The ambiguity function
Ax.;  / ,
Z
R
x.t C =2/x.t Ä =2/eÄ j2t dt ;
for GACS time-series, accounting for (3) and (5), can be
expressed in terms of generalized cyclic statistics:
Ax.;  / D
X
2W
Rx; . / . Ä  . //C F
t!

‘x.t;  /

;
(15)
where ./ denotes Dirac’s delta function. Equation (15) shows
that the ambiguity function of GACS signals is the sum of
some impulsive terms whose supports are curves described
by the lag-dependent cycle frequencies and whose amplitudes
are the generalized cyclic autocorrelation functions and an
aperiodic component that, accounting for (4), does not contain
impulses.
Finally, let us note that, in the special case of ACS time-
series, (15) specializes to
Ax.;  / D
X
2A
Rx . / . Ä /C F
t!

‘x.t;  /

:
4 Signal feature extraction
In problems of signal feature extraction for GACS time-series,
in general, no a priori knowledge exists on the possible cyclo-
stationary nature of the signal. Therefore, single-record esti-
mators of time-frequency distributions and generalized cyclic
statistics must be utilized. The estimators are obtained directly
by applying the definitions where, however, integrals and time
averages are performed over a finite collect-time. Then, they
666
asymptotically approach the theoretical values when the ob-
servation time increases. Once the lag-dependent cycle fre-
quencies and/or the generalized cyclic statistics have been esti-
mated, the time-varying autocorrelation function can be recon-
structed, signal parameters can be estimated, and signals can
be classified on the basis of their different generalized cyclic
statistic characteristics.
It is worthwhile to underline that, in general, in time-
frequency representations of GACS time-series, the compo-
nent related to the residual term ‘x.t;  / cannot be separated
from the component related to the cyclic statistics (see (10)).
In the special case of ACS time-series, however, from (13) it
follows that the component related to the cyclic statistics is al-
most periodic and, hence, algorithms for estimating amplitude
and frequencies of almost-periodic signals embedded in noise
can be exploited to obtain estimates of the cyclic parameters
of interest.
The role played by the residual term can be illustrated by
an example. Specifically, let us consider the signal
x.t/ D
Z
R
h.t; u/s.u/ du;
where
s.t/ D w.t/ exp. j2 f0t/
and
h.t; u/ D .u Ä t C D.t//
is the impulse-response function of a channel introducing a
time-varying delay D.t/. Figure 1 shows the magnitude of the
cyclic autocorrelation function Rx . /, for the signal x.t/, as
a function of  and  , as estimated by 256 samples. It has
been assumed that f0 D 0:04=Ts, where Ts is the sampling
period, and the real signal w.t/ is wide-sense stationary with
power spectral density S0w. f / D .1 C f 2=B2/Ä8 with B D
0:015=Ts. Moreover, a time-varying delay D.t/ D d1t C d2t2
with d1 D 0:25 and d2 D 0:02=Ts has been considered. The
support of Rx . / is constituted by curves described by the lag-
dependent cycle frequencies . /; on each of them, the cyclic
autocorrelation function is just equal to the corresponding
generalized cyclic autocorrelation function Rx; . /. Figure
2 shows the magnitude of the Wigner-Ville distribution for
the same signal. The presence of a component related to the
residual term is evident.
Finally, let us observe that the estimates of the cyclic
autocorrelation function and the ambiguity function differ
only for a scaling factor [3]. Therefore, the above cyclic
autocorrelation function based estimation procedure can also
be interpreted in terms of ambiguity function.
References
[1] R.A. Boyles and W.A. Gardner, “Cycloergodic properties
of discrete-parameter nonstationary stochastic processes”,
IendequationE Trans. Inform. Theory, vol. IT-29, pp. 105-
114, January 1983.
[2] L. Cohen, Time-Frequency Analysis. Prentice Hall, Engle-
wood Cliffs, NJ, 1995.
−20
0
20
−0.1
0
0.1
0
0.005
0.01
   |Rxα(τ)|
   τ/Ts
  αTs
Figure 1.
−0.05
0
0.
−200
20
0
2
4
   |Wx(t,f)|
  f Ts
 t/Ts
Figure 2.
[3] W.A. Gardner, Statistical Spectral Analysis: A Nonproba-
bilistic Theory. Prentice Hall, Englewood Cliffs, NJ, 1988.
[4] P. Gournay and P. Nicolas, “Analyse spectrale cyclique
et analyse temps-fréquence pour l’identification automa-
tique de trasmissions,” in Proc. of Quinzieme Colloque
GRETSI, Juan-les-Pins, F, September 1995.
[5] L. Izzo and A. Napolitano, “Characterization of higher-
order almost-periodically correlated time-series,” in Proc.
of Workshop on Cyclostationary Processes, Noisy Le
Grand, F, July 1996.
[6] L. Izzo and A. Napolitano, “Linear time-variant process-
ing of higher-order almost-periodically correlated time-
series,” in Proc. of Eighth European Signal Processing
Conference (EUSIPCO’96), Trieste, I, September 1996.
[7] D. König and J.F. Böme, “Application of cyclostationarity
and time-frequency analysis to car engine diagnosis”,
in Proc. of IendequationE International Conference on
Acoustics, Speech and Signal Processing, (ICASSP’94),
Adelaide, South Australia, April 1994.


Time-frequency representations of generalized almost-cyclostationary signals

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