Coherent covariance analysis of periodic
ARTICLE IN PRESS
Signal Processing 87 (2007) 13–32
www.elsevier.com/locate/sigpro
Coherent covariance analysis of periodically correlated
random processes
I. Javors’kyja,b, I. Isayeva,c,, Z. Zakrzewskib, S.P. Brooksc
a
Karpenko Physico-mechanical Institute of the National Academy of Sciences of Ukraine, Naukova St. 5, 79601, L’viv, Ukraine
Institute of Telecommunications, University of Technology and Agriculture, Al. Prof. S.Kaliskiego, 7, 85-796, Bydgoszcz, Poland
c
Statistical Laboratory, Cambridge University, Wilberforce Road, CB3 0WB, Cambridge, UK
b
Received 6 February 2005; received in revised form 13 April 2006; accepted 13 April 2006
Available online 5 June 2006
Abstract
A coherent method of estimating of periodically correlated random processes (PCRP) is introduced. Properties of
estimates of the mean, covariance function and their Fourier coefficients that are obtained using process values averaging
over the period are investigated. Asymptotic formulae for estimates of bias and variances are obtained and the
relationships of these characteristics to realization length are discussed. The probabilistic structure of one of the simplest
PCRP-based signals is analysed.
r 2006 Elsevier B.V. All rights reserved.
Keywords: Periodically correlated random processes; Mean; Covariance function; Coherent estimation; Bias; Consistency
1. Introduction
Recurrence and stochasticity are characteristic features of many physical processes exhibiting time
changeability [1–3]. Recurrence of process properties could be caused by the influence of external forces (e.g.,
forced oscillations) acting on the system or from internal forces (e.g., autonomous oscillation) that exist within
the system itself. Rhythmic processes occur in many fields including radio physics [4], telecommunications [3],
geophysics [5], oceanography [1,2], meteorology [6–9], vibro-diagnostics [10,11], biology [12,13] and
seismology [14,15], for example. The methodological basis for investigating rhythmic process structure based
on experimental data is the mathematical model. Early investigations into rhythmic processes were based on
deterministic models using periodic and almost-periodic functions. Attempts to describe the stochastic
changeability of oscillations were enhanced by the evolution of probabilistic methods based mainly on models
in the form of stationary random processes. Rhythmic changeability of physical processes is evidenced by the
oscillatory character of the associated correlograms and also in the peak values present in spectral density
estimates. However, these characteristics describe only time-averaged oscillatory properties. They have no
Corresponding author. Cambridge University, Statistical Laboratory, Wilberforce Road, CB3 0WB, Cambridge, UK. Tel.: +44
(0)1223 766925; fax: +44 (0)1223 337956.
E-mail addresses: ihor@statslab.cam.ac.uk, isayev@ipm.lviv.ua (I. Isayev).
0165-1684/$ - see front matter r 2006 Elsevier B.V. All rights reserved.
doi:10.1016/j.sigpro.2006.04.002
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14
Nomenclature
x(t), xk(t), Z(t) stochastic signals
T
period
y
realization length
E
sign of mathematical expectation
m(t)
mean function
mk
mean components
b(t,u) covariance function
Bk(u) covariance components
D
variance
information about the time structure of any oscillations, which might be introduced, at least in idealized form,
by the incorporation of additional deterministic structure, for example. The development of probabilistic
models such as PCRP and their generalizations (bi-, poly- and almost-PCRP) is the natural evolution of these
approaches [1–4,16–18]. These models generalize the idea of recurrence to situations where stochasticity plays
a dominant role in the system process and they enable us to make more detailed and objective descriptions of
stochastic recurrence including, as special cases, the earlier models described above. PCRP models also enable
us to analyse the properties of physical processes without needing to specify special characteristics of the
process a priori, but provide a generic method for identifying characteristics present in the underlying process.
Suppose that we have a time series xðtÞ, t 2 R then this series is said to be periodically stationary with
period T if
Pr½xðt1 þ hÞ 2 A1 ; xðt2 þ hÞ 2 A2 ; . . . ; xðtn þ hÞ 2 An ¼ Pr½xðt1 Þ 2 A1 ; xðt2 Þ 2 A2 ; :::; xðtn Þ 2 An
for h ¼ T40, and no smaller value of h.
PCRP (or cyclo-stationary) models are defined uniquely by the specification of a mean function mðtÞ ¼
ExðtÞ;
t 2 ½0; TÞ and a covariance function bðt; uÞ ¼ E xðtÞ xðt þ uÞ; for t 2 ½0; TÞ and u 2 R, where
xðtÞ ¼ xðtÞ mðtÞ. Analysis of PCRP process models therefore involves estimation of these
functions.
R Ttwo
mðtÞdto1 and
PCRP
means,
m(t)
and
covariance
functions
b(t,
u)
are
periodic
functions
of
time
and
if
0
RT
0 bðt; uÞdt o1, u 2 R, then they can be represented by the Fourier series:
X
mðtÞ ¼
mk eiko0 t
k2Z
¼ m0 þ
X
mck cos ko0 t þ msk sin ko0 t
k2N
and
bðt; uÞ ¼
X
k2Z
Bk ðuÞeiko0 t ¼ B0 ðuÞ þ
X
Bck ðuÞ cos ko0 t þ Bsk ðuÞ sin ko0 t ;
ð1Þ
(2)
k2N
respectively. Here o0 ¼ 2p=T; mk ¼ 12 mck imsk ; Bk ðuÞ ¼ 12 Bck ðuÞ iBsk ðuÞ and both jmk j ! 0 and Bk ðuÞ !
0 as k ! 1. The parameters Bk(u) are referred to as the covariance components [1,2,17], coefficient function
[19] or the cyclic autocorrelation function [3,16]. The Fourier components are also commonly of interest and
their estimation often forms part of the analytic process.
An important part of understanding the properties PCRP’s is the PCRP decomposition into stationary
connected random processes [1–3], or harmonic serial representations [3]
X
xðtÞ ¼
xk ðtÞeiko0 t ,
(3)
k2Z
where the xk(t) denote stationary connected random processes. The characteristics of the signal x(t) can
therefore be derived from the corresponding characteristics of the constituent xk(t) processes.
ARTICLE IN PRESS
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15
Various methods have been proposed for the estimation of the mean and covariance functions. For example
Isayev and Javors’kyj [6] investigates the use of component methods, whereas Javors’kyj et al. [20] develop
approaches based upon least squares estimation. In this paper, we will focus on the so-called coherent method
estimation, proposed by Gudzenko [4]. The coherent method (also known as synchronized averaging [16]) is
based on averaging sample PCRP realization values at the same point across different periods to obtain
empirical estimates of the mean and covariance functions. For example, since y is an integer multiple of the
period, T, the coherent estimate of the mean has the form:
^ ðtÞ ¼
m
1
X
1N
xðt þ nT Þ for t 2 ½0; T Þ.
N n¼0
(4)
Several authors have used the coherent method for analysing PCRP processes. In this paper, we focus on the
properties of the corresponding estimators. We begin in Section 2 by looking at the coherent estimators of the mean
function and the corresponding Fourier series components and demonstrate their unbiasedness and consistency. In
Section 3, we prove the asymptotic unbiasedness and consistency of the covariance function before doing the same
for the corresponding Fourier components in Section 4. In Section 5, we conclude with an example in which we
investigate the properties of the signal corresponding to a particular implementation of the PCRP process.
2. Properties of the PCRP mean function and corresponding component estimators
The coherent mean function estimator in (4) is clearly unbiased since
^ ðtÞ ¼
Em
1
X
1N
mðt þ nT Þ ¼ mðtÞ.
N n¼0
^ ðtÞ ! 0 as N ! 1. We begin by noting that
To prove consistency, we need to show that D½m
X
jnj
1 N1
2
^ ðtÞ ¼ E ½m
^ ðtÞ E m
^ ðtÞ ¼
1
D½m
bðt; nT Þ.
N n¼Nþ1
N
The proof is trivial.
"
#
N1
X
1
n
^ ðtÞ ¼
bðt; 0Þ þ 2
D½m
1
bðt; nT Þ
N
N
n¼1
(5)
(6)
and, if
N
1X
bðt; nT Þ ¼ 0,
N!1 N
n¼1
lim
(7)
^P
then D½m
ðtÞ ! 0 as N ! 1. Thus, the estimator in (4) is also consistent. Note that condition (7) is satisfied if
a
the sum N
n¼1 bðt; nT Þ increases with N no faster than N , where ao1. Note also that condition (7) is evident if
limjuj!1 bðt; uÞ ¼ 0.
Taking into account the foregoing we now formulate the following theorem:
Theorem 1. Statistic (4) is an unbiased and consistent estimate of the PCRP mean if condition (7) is satisfied,
its variance being defined by expression (6).
The variance (6) is a periodic function of time with corresponding Fourier series expansion:
X
^ ðtÞ ¼ g0 þ
D½m
gcl cos lo0 t þ gsl sin lo0 t .
(8)
l2N
From (2), we therefore have that:
"
#
N
1
X
1
n
B0 ð0Þ þ 2
g0 ¼
1
B0 ðnT Þ
N
N
n¼1
(9)
ARTICLE IN PRESS
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16
and
gc;s
l
"
#
N1
X
1
n c;s
c;s
B ð0Þ þ 2
B ðnT Þ .
¼
1
N l
N l
n¼1
(10)
Similarly, from (2) and under the condition in (7), we have that:
N
1X
B0 ðnT Þ ¼ 0;
N!1 N
n¼1
lim
N
1X
Bc;s
l ðnT Þ ¼ 0
N!1 N
n¼1
and lim
(11)
and thus g0 ! 0 and gc;s
l ! 0 as N ! 1.
The zero component g0 defines the average value of the variance of the coherent estimator (8) and depends
only upon the zero covariance component B0(u). The lth harmonic components which define the amplitudes of
^ ðtÞ depend only on the covariance components of the same order. If correlations vanish
the variance D½m
within a time interval less than the period T (i.e., if B0 ðnTÞ ¼ Blc;s ðnTÞ ¼ 0 for all n40), then expressions
(9)–(10) reduce to:
B 0 ð 0Þ
Bc ð0Þ
B s ð 0Þ
and gsl ¼ l .
; gcl ¼ l
N
N
N
^ ðtÞ given in (6) is simply bðt; 0Þ=N.
In this case the time-varying behaviour of the estimator variance D½m
Let us now consider the estimation of the Fourier components corresponding to the mean function given in
^ ðtÞ, is known for all t 2 ½0; TÞ. Then we can create the following
(1). Suppose, that the mean estimate m
statistics:
Z
Z
Z
1 T
1 T
1 T
^0 ¼
^ sl ¼
^ ðtÞ dt; m
^ cl ¼
^ ðtÞ cos lo0 t dt; and m
^ ðtÞ sin lo0 t dt:
m
(12)
m
m
m
T 0
T 0
T 0
g0 ¼
Combining (4) and (12), we obtain:
Z
1 y
^0 ¼
xðtÞ dt;
m
y 0
^ cl ¼
m
2
y
Z
0
y
^ sl ¼
xðtÞ cos lo0 t dt and m
(13)
2
y
Z
y
xðtÞ sin lo0 t dt:
(14)
0
^ ðtÞ ¼ mðtÞ, the estimates in (12) are unbiased for the corresponding components given in (1) i.e.,
Since E m
^ 0 ¼ m0 , E m
^ cl ¼ mcl and E m
^ sl ¼ msl .
Em
To demonstrate consistency of the component estimates, we note that
Z Z
1 y y
^ 0 ¼ 2
bðt; s tÞ dt ds;
(15)
D½m
y 0 0
and
c
4
^l ¼ 2
D m
y
s
4
^l ¼ 2
D m
y
Z
y
y
bðt; s tÞ cos lo0 t cos lo0 s dt ds
(16)
bðt; s tÞ sin lo0 t sin lo0 s dt ds:
(17)
0
0
Z
Z
y
Z
y
0
0
See Appendix A for details. Setting u ¼ s t we obtain:
Z Z
2 y yu
^ 0 ¼ 2
bðt; uÞ dt du;
D½m
y 0 0
c
4
^l ¼ 2
D m
y
Z
0
y
Z
(18)
yu
bðt; uÞ½cos lo0 u þ cos lo0 ð2t þ uÞ dt du
0
(19)
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and
s
4
^l ¼ 2
D m
y
Z
y
0
Z
yu
bðt; uÞ½cos lo0 u cos lo0 ð2t þ uÞ dt du.
(20)
0
Finally, substituting b(t,u) for its Fourier series decomposition in (18) and integrating with respect to t, we
obtain
Z "
X
2 y
u
^ 0 ¼
D½m
1 B0 ðuÞ þ
Bcl ðuÞf cl ð0; y uÞ þ Bsl ðuÞf sl ð0; y uÞ du.
y 0
y
l2N
yuÞlo0
Þlo0 1
where f cl ð0; y uÞ ¼ sin ½ðlo
:
; f sl ð0; y uÞ ¼ cos ½ðyu
lo0 y
0y
c
s
Functions f l ð0; y uÞ and f l ð0; y uÞ are clearly rapidly damped with increasing series length, y. So, for
large y we have
Z
2 y
u
^ 0
D½m
1 B0 ðuÞ du.
(21)
y 0
y
Using similar manipulations, (19) and (20) become
Z
c 2 y h
i
u
^l ¼
1 ½2B0 ðuÞ cos lo0 u Bc2l ðuÞ cos lo0 u þ Bs2l ðuÞ sin lo0 u du,
D m
y 0
y
s 2
^l ¼
D m
y
Z y h
0
1
i
u
2B0 ðuÞ cos lo0 u þ Bc2l ðuÞ cos lo0 u Bs2l ðuÞ sin lo0 u du.
y
If the following conditions
Z
Z
1 y
1 y c
lim
B0 ðuÞ du ¼ 0; lim
Bk ðuÞ du ¼ 0;
y!1 y 0
y!1 y 0
1
and lim
y!1 y
Z
0
(22)
(23)
y
Bsk ðuÞ du ¼ 0
(24)
are satisfied, then the variances in Eqs. (21)–(23) vanish as y ! 1. Thus the estimators in (12) are consistent
as well as asymptotically unbiased.
Taking into consideration the foregoing we now formulate the following theorem:
Theorem 2. The statistics in (12) are unbiased and consistent estimates of the PCRP mean Fourier coefficients
if the conditions in (24) are satisfied, their variances being defined by the expressions in (21)–(23).
It is clear from Eq. (21) that the zero covariance component is the main characteristic defining the variance
of the zeroth-order mean estimate. The variances of higher-order mean
s;c estimates are functions boths;cof the
^ l depends both B0 and on B2l . The
zero component and the components of double their order i.e., D m
constituent sine and cosine components
in
(22)
and
(23)
have
opposite
signs. Therefore, the variance of the
c
s
^ l is independent of higher covariance components, i.e.,
^l þD m
^ l ¼ 14 D m
complex amplitude estimate D½m
Z
2 y
u
^ l ¼
D½m
1 B0 ðuÞ cos lo0 u du.
y 0
y
The dependence of this variance on the value l clearly stems from the frequency of the cosine weight function.
3. Properties of the covariance function estimators
The coherent covariance function estimate is given by
X
1 N1
^ ðt þ nT Þ½xð½t þ nT þ uÞ m
^ ð½t þ nT þ uÞ.
½xðt þ nT Þ m
b^ðt; uÞ ¼
N n¼1
(25)
Suppose that the realization length is such that y ¼ NT þ um , where um denotes the maximum lag of the
^ ðt þ nT Þ ¼
covariance function (see e.g., [1–3]). The mean function is then estimated only for t 2 ½0; T , with m
ARTICLE IN PRESS
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18
^ ðtÞ when te½0; TÞ and n 2 N. Then, taking expectations, we have
m
1
X
1N
^ ðtÞm
^ ðt þ uÞ.
E½xðt þ nT Þxð½t þ nT þ uÞ E m
N n¼0
h
i
Hence the bias, b^ðt; uÞ ¼ E b^ðt; uÞ bðt; uÞ, of our estimator is given by
1
h
i
X
j nj
1 N
^
bðt; uÞ ¼
1
bðt; u þ nT Þ.
N
N n¼Nþ1
^ uÞ ¼
E½bðt;
(26)
(27)
The proof follows directly from a similar expression to that given in (5).
For u ¼ 0 this expression differs from the variance of mean estimate, given in (5), only by the sign. Clearly,
changes in the covariance function directly affect the bias. Therefore we can obtain a Fourier series
^ ðtÞ given in (8) i.e.,
representation similar to that for D½m
h
i
X
b^ðt; uÞ ¼ 0 ðuÞ þ
cl ðuÞ cos lo0 t þ sl ðuÞ sin lo0 t .
l2N
In analogy to Eq. (9) and (10), it is easy to show that
1
X
jnj
1 N
0 ðuÞ ¼
1
B0 ðu þ nT Þ
N
N n¼Nþ1
(28)
and
c;s
l ð uÞ
1
X
jnj c;s
1 N
¼
1
Bl ðu þ nT Þ.
N
N n¼Nþ1
(29)
h
i
^ðt; uÞ ! 0 as N ! 1. Thus, the
Then, if the conditions in (11) are satisfied, 0 ðuÞ ! 0, c;s
ðuÞ
!
0
and
b
l
coherent estimator in (25) is asymptotically unbiased.
Let us obtain the variance of estimate (25) under the assumption that the PCRP is Gaussian (so that third
and higher-order moments can be expressed in terms of first and second-order moments), and then rewrite (25)
in the form
1
X
1N
^ ðtÞ m
^ ðt þ uÞ,
b^ðt; uÞ ¼
xðt þ u þ nT Þ xðt þ nT Þ m
N n¼0
where
^ ðtÞ ¼
m
X
1 N1
xðt þ nT Þ.
N n¼0
Then, since third and higher-order moments can be expressed in terms of first and second-order moments
under the Gaussian assumption, as a first order approximation we get:
h
i
X
j nj
1 N1
^
½bðt; nT Þbðt þ u; nT Þ þ bðt; u þ nT Þbðt; u nT Þ þ oðN 1 Þ.
D bðt; uÞ ¼
(30)
1
N
N n¼Nþ1
The proof follows similarly to that
in Appendix A.
given
Consider the process Zðt; uÞ ¼ xðtÞ xðt þ uÞ with corresponding covariance function
bZ ðt; u1 ; uÞ ¼ E xðtÞ xðt þ uÞ bðt; uÞ xðt þ u1 Þ xðt þ u1 þ uÞ bðt þ u1 ; uÞ
¼ bðt; u1 Þbðt þ u; u1 Þ þ bðt; u1 þ uÞbðt þ u; u1 uÞ.
ð31Þ
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Then, using the fact that bðt; uÞ ¼ bðt u; uÞ it follows from (30) that
1
h
i
X
j nj
1 N
^
1
D bðt; uÞ ¼
bZ ðt; nT; uÞ þ o N 1 .
N n¼Nþ1
N
19
(32)
Theorem 3. The estimate of the covariance function (25) of a Gaussian PCRP is asymptotically unbiased and
consistent if conditions (11) are satisfied and its bias and variance are defined by the formulae in (27) and (32).
Let us represent the function bZ ðt; u1 ; uÞ in terms of the Fourier series decomposition:
X
B~ k ðu1 ; uÞeiko0 t .
bZ ðt; u1 ; uÞ ¼
(33)
k2Z
Then, setting u1 ¼ nT and substituting (33) into (32) we have that
h
i
X
D b^ðt; uÞ ¼ a0 ðuÞ þ
acl ðuÞ cos lo0 t þ asl ðuÞ sin lo0 t þ o N 1
(34)
l2N
where
"
#
N
1
X
1 ~
n ~
B0 ð0; uÞ þ 2
B0 ðnT; uÞ ,
a0 ð uÞ ¼
1
N
N
n¼1
(35)
and
ac;s
l ðuÞ
"
#
N
1
X
1 ~ c;s
n ~ c;s
¼
B ð0; uÞ þ 2
1
B ðnT; uÞ .
N l
N l
n¼1
(36)
Using the earlier decomposition of the covariance function bðt; uÞ into Fourier series, we can directly obtain
expressions for the components B~ l ðu1 ; uÞ. Suppose that we have M covariance components, then we have from
(31) that
8
M
P
>
>
eiqo0 u Bqþk ðu1 þ uÞBq ðu1 uÞ þ Bqþk ðu1 ÞBq ðu1 Þ ; kp0;
>
>
< q¼Mþk
(37)
B~ k ðu1 ; uÞ ¼
Mk
P iqo0 u
>
>
>
ð
u
u
Þ
þ
B
ð
u
ÞB
ð
u
Þ
;
k40
e
B
ð
u
þ
u
ÞB
q
1
qþk
1
q
1
qþk
1
>
:
q¼M
where ‘‘’’ denotes complex conjugation and k 2 ½2M; 2M
Finally, using the definition of B~ k ðu1 ; uÞ in (37), and substituting into Eqs. (35) and (36), Eq. (11) then
implies that
lim ac;s ðuÞ
juj!1 l
¼ 0.
h
i
Thus from (34), we have that lim D b^ðt; uÞ ¼ 0, the condition required to demonstrate the consistency of the
juj!1
estimator in (25).
From (35) it is clear that the average value of the variance in (34), corresponding to component a0 ðuÞ,
depends on the zeroth and all higher order covariance components. Thus, in the case of PCRP processes the
precision of our estimates could not be obtained by looking at stationary processes alone.
lim a0 ðuÞ ¼ 0;
juj!1
4. Properties of the covariance component estimators
Let us now consider the coherent estimators for the covariance components. Suppose that the estimate
b^ðt; uÞ is known for all t 2 ½0; TÞ and for all u 2 ½0; um . Note that, in practice, we truncate our estimator for the
covariance function to lags uoum where um is the truncation point of correlogram [1]. Then for lags in this
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20
range, we can create the following estimates of the covariance components:
Z
1 T ^
bðt; uÞ dt,
B^ 0 ðuÞ ¼
T 0
c
B^ l ðuÞ ¼ T2
RT
0
R
2 T
s
B^ l ðuÞ ¼ T
0
b^ðt; uÞ cos lo0 t dt;
b^ðt; uÞ sin lo0 t dt:
(38)
(39)
Taking into consideration expressions for bias of the covariance function estimates in Eqs. (28) and (29) and
the Fourier decomposition of the covariance function in (2) we have, after integrating, that
1
X
1 N
j nj
B^ 0 ðuÞ ¼
1
B0 ðu þ nT Þ,
N n¼Nþ1
N
1
h c;s i
X
jnj
1 N
B^ l ðuÞ ¼
1
B0 ðu þ nT Þ.
N
N n¼Nþ1
The biases of the covariance component estimates for any order depend only on the covariance components
of the same order. If the conditions in equation (11) are satisfied then these biases clearly converge to zero as
N ! 1. Thus, the estimators in (38) and (39) are asymptotically unbiased.
In order to investigate consistency, we require expressions for the variances of the estimators in Eqs. (38)
and (39). Combining Eqs. (38) and (39), using the definition of b^ðt; uÞ from (25) and the periodic property of
^ ðt þ nT Þ ¼ m
^ ðtÞ, we have that
the mean estimate i.e., m
Z y
1
^ ðtÞ m
^ ðt þ uÞ dt,
B^ 0 ðuÞ ¼
xðtÞ xðt þ uÞ m
y 0
2
c
B^ l ðuÞ ¼
y
Z y
^ ðtÞ m
^ ðt þ uÞ cos lo0 t dt,
xðtÞ xðt þ uÞ m
0
Z
2 y
s
^
^ ð sÞ m
^ ðs þ uÞ sin lo0 s ds.
Bl ðuÞ ¼
x ð s Þ xð s þ uÞ m
y 0
For Gaussian signals we get a first-order approximation:
Z Z
2 y yu1
D B^ 0 ðuÞ ¼
bZ ðt; u1 ; uÞ dt du1 þ oðy1 Þ,
y 0 0
and
(40)
Z Z
h c i
4 y yu1
D B^ l ðuÞ ¼ 2
bZ ðt; u1 ; uÞ½cos lo0 u1 þ cos lo0 ð2t þ u1 Þ dt du1 þ oðy1 Þ,
y 0 0
(41)
Z Z
h s i
4 y yu1
D B^ l ðuÞ ¼ 2
bZ ðt; u1 ; uÞ½cos lo0 u1 cos lo0 ð2t þ u1 Þ dt du1 þ oðy1 Þ.
y 0 0
(42)
h c i 2 Z y
u1 ~
s
c
D B^ l ðuÞ
B0 ðu1 ; uÞ cos lo0 u1 B~ 2l ðu1 ; uÞ cos lo0 u1 þ B~ 2l ðu1 ; uÞ sin lo0 u1 du1 ,
1
y 0
y
(44)
Proof follows similarly to that given in Appendix A.
Substituting the Fourier decomposition of bZ ðt; u1 ; uÞ from (33) into Eqs. (40)–(42) and integrating with
respect to t, we obtain for large y (or, equivalently, N):
Z
2 y
u1 ~
D B^ 0 ðuÞ
(43)
1
B0 ðu1 ; uÞ du1 ;
y 0
y
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h s i 2 Z y
u1 ~
c
s
D B^ l ðuÞ
1
B0 ðu1 ; uÞ cos lo0 u1 þ B~ 2l ðu1 ; uÞcos lo0 u1 B~ 2l ðu1 ; uÞ sin lo0 u1 du1 ,
y 0
y
21
(45)
Clearly, if
1
lim
y!1 y
Z
y
0
1
B~ 0 ðu1 ; uÞ du1 ¼ 0 and lim
y!1 y
Z
0
y
c;s
B~ l ðu1 ; uÞ du1 ¼ 0,
(46)
then the variances in equations (43)–(45) vanish as y ! 1. Hence, we have asymptotic consistency for the
estimators in Eqs. (38) and (39).
Theorem 4. Estimates (38)–(39) for Gaussian PCRP are asymptotically junbiased
and consistent
j s k if the
k
c
c
^
^
conditions in (11) and (46) are satisfied and their biases B0 ðuÞ ¼ 0 ðuÞ, Bk ðuÞ ¼ k ðuÞ, B^ k ðuÞ ¼ sk ðuÞ
and variances are defined by the formulae in (28)–(29) and (43)–(45).
c;s
Since values B~ 0 ðu1 ; uÞ and B~ l ðu1 ; uÞ are evaluated through the PCRP covariance components products, the
conditions (46) hold if and only if the conditions in (24) hold true. The proof is trivial. The expressions in Eqs.
(43)–(45) are similar to those for the variances of mean components in Eqs. (22) and (23). The difference is that
we use the Fourier components of the function bZ ðt; u1 ; uÞ in the first case and covariance components in the
second. It follows from (37) that B~ 0 ðu1 ; uÞ depends upon all covariance components. Hence, the accuracy of
the estimator for the zero covariance component B0(u) cannot be measured without considering the higher
order covariance components. Once again, the variance of our estimators could not be obtained by looking at
stationary processes alone. The variances of the sine and cosine covariance components depend also on the
higher order
h components
i of the function bZ ðt; u1 ; uÞ. However, for the variances of complex-valued estimates
s
1 ^c
^
^
Bl ðuÞ ¼ B ðuÞ iB ðuÞ we have
2
l
l
2
D B^ l ðuÞ
y
Z y
u1 ~
B0 ðu1 ; uÞ cos lo0 u1 du1 .
1
y
0
Thus, the accuracy of the higher-order complex-valued covariance component estimates depend only on
B~ 0 ðu1 ; uÞ.
5. Characteristics of modulated signals—an example
As we saw in Section 1, Eq. (3), we can decompose our signal into a series of stationary connected subsignals (harmonic serial representations) as follows:
X
xðtÞ ¼
xk ðtÞeiko0 t ;
k2Z
where the xk(t) denote stationary random processes.
There are several common forms of constituent process. For example, if we set xk ðtÞ ¼ ak þ Zk ðtÞ, where the
Zk ðtÞ are uncorrelated, then we get the additive model
X
X
xðtÞ ¼
ak eiko0 t þ
Zk ðtÞeiko0 t ¼ f ðtÞ þ ZðtÞ,
k2Z
k2Z
where f(t) is a periodic function and Z(t) is a stationary random process. Such models are commonly used for
simple processes with stochastic variation around a periodic mean.
If we set xk ðtÞ ¼ ak ZðtÞ, then we get the multiplicative PCRP model, where
X
xðtÞ ¼ ZðtÞ
ak eiko0 t ¼ f ðtÞZðtÞ.
(47)
k2Z
This model is commonly used to describe more complicated processes than the simple additive model above
and allows for interaction between the stochastic errors process and the mean function.
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22
A third model is obtained if we x1 ðtÞ ¼ 1=2½xc ðtÞ ixs ðtÞ and x1 ðtÞ ¼ x1 ðtÞ, with the remaining stationary
processes all set equal to zero i.e., xk ðtÞ 0 for jkja1. In this case, we get the quadrature model that allows for
interactions between the stationary and quadrature components.
xðtÞ ¼ xc ðtÞ cos o0 t þ xs ðtÞ sin o0 t.
In this section, we take a specific example of a multiplicative model and investigate the properties of the
signal for a particular form of periodic function and associated stationary random processes. We shall assume
that f ðtÞ ¼ cos o0 t in (47) and, using the properties of elementary circular functions, we obtain the following
mean and covariance functions
mðtÞ ¼ m cos o0 t and bðt; uÞ ¼ B0 ðuÞ þ Bc2 ðuÞ cos 2o0 t þ Bs2 ðuÞ sin 2o0 t,
where
m ¼ EZðtÞ; B0 ðuÞ ¼ Bc2 ðuÞ ¼ 1=2RZ ðuÞ cos o0 u and Bs2 ðuÞ
¼ 1=2RZ ðuÞ sin o0 u
ð48Þ
and RZ ðuÞ denotes the covariance function of stationary process ZðtÞ. In this case, and from (8) and (9), the bias
of the mean equals zero and we get the following variance for the mean:
^ ðtÞ ¼ g0 þ gc2 cos 2o0 t þ gs2 sin 2o0 t,
D½m
where
and
"
#
N
1
X
1
n
R Z ð 0Þ þ 2
1
g0 ¼ gc2 ¼
RZ ðnT Þ ,
2N
N
n¼1
gs2 ¼ 0.
Now, suppose that we set RZ ðuÞ ¼ Deajuj , then we have that
g0 ¼ gc2 ¼
D
½1 þ 2SðaT; N Þ,
2N
where
SðaT; N Þ ¼
N1
X
1
n¼1
n aTn
eaT
1 eaNT
1
¼
e
N
1 eaT
N ð1 eaT Þ
(49)
and, for large N, we obtain
D 1 þ eaT
ð1 þ cos 2o0 tÞ.
(50)
2N 1 eaT
Fig. 1 plots the variance of the mean estimate as N and t vary. This plot clearly demonstrates the rapid
^ ðtÞ as N increases. Note also that the variance in (50) vanishes at the points
decrease in the variance D½m
tk ¼ ð2k þ 1ÞT=4, k 2 Z. This implies that the mean at the points tk must be known and is a consequence of
the definition of m(t) as m(t) ¼ m cos o0t which clearly equals zero at the points tk.
As we saw in Section 2, the mean components can be defined using a Fourier transform of the mean
function. Here, we have only one component l ¼ 1 and so, from (21)–(23), it is simple to show that
^ ðtÞ ¼
D½m
^ 0 ¼
D½m
D
P1 ðaT; N Þ;
N
^ c ¼
D½m
D
P2 ðaT; N Þ;
N
^ s ¼
D ½m
D
½2P0 ðaT; N Þ þ P2 ðaT; N Þ,
N
where
1
Pl ðaT; N Þ ¼
T
"
#
Z y
u ajuj
1
a2 T 2 4l 2 p2
aNT
1e
aT
1 e
cos lo0 u du ¼
.
y
a2 T 2 þ 4l 2 p2
N a2 T 2 þ 4l 2 p2
0
(51)
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23
0.8
0.6
0.4
0.2
0
0
10
0
20
5
10
30
15
20
N
40
50
t
Fig. 1. Variance of mean estimate depending on period numbers N and time t for o0 ¼ 0.2, a ¼ 0.1 and D ¼ 1.
zero
cos
sin
0.6
0.5
0.4
0.3
0.2
0.1
0
0
5
10
15
20
^ 0 , D½m
^ c D½m
^ s for o0 ¼ 0.2, a ¼ 0.1, D ¼ 1.
Fig. 2. Variance of mean components estimate variances D½m
The product aT denotes the rate of damping of the estimate variances with N. Fig. 2 provides a plot of these
variances as N changes.
We now consider the properties of the estimate of the covariance function, b(t,u). Eqs. (27)–(29) suggest the
following result for the bias:
j
k
b^ðt; uÞ ¼ 0 ðuÞ þ c2 ðuÞ cos 2o0 t þ s2 ðuÞ sin 2o0 t,
where
0 ðuÞ ¼ c2 ðuÞ ¼
and
D
S0 ðaT; N; uÞ cos o0 u;
2N
s2 ðuÞ ¼
D
S 0 ðaT; N; uÞ sin o0 u
2N
jnj ajuþnT j
S0 ðaT; N; uÞ ¼
.
1
e
N
n¼Nþ1
N
1
X
From (49) and for zero lag (i.e., u ¼ 0) we have S0(aT,N,u) ¼ 1+2S(aT,N). So, in analogy to the case for
the components of the bias of the covariance function, expressions for the components e0(0) and ec2(0) differ
from the corresponding Fourier components of the mean estimate variance g0 and gc2 only in sign.
However, the damping rate is
The function S0(aT,N,u) oscillates but its amplitude decreases asju increases.
k
^ðt; uÞ (see Fig. 3) It is relatively easy to
less than that for the covariance.
This
obviously
affects
the
bias
b
h
i
show that the relative value b^ðt; uÞ =bðt; uÞ increases with lag u and, because the numerator is only defined
over a finite range, it is possible to assess the reliability of the covariance function estimate only for lags within
the finite interval [0,um].
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24
We now examine the variance of the covariance function estimate. Using Eqs. (24), (48) and the properties
of elementary circular functions, we obtain the following Fourier components of the function bZ(t,u1,u) in (31):
B~ 0 ðu1 ; uÞ ¼ 18R~ Z ðu1 ; uÞð1 þ cos 2o0 u þ cos 2o0 u1 Þ,
c
B~ ðu1 ; uÞ ¼ 1R~ Z ðu1 ; uÞ½1 þ cos 2o0 u þ cos 2o0 u1 þ cos 2o0 ðu1 þ uÞ,
2
8
s
B~ 2 ðu1 ; uÞ ¼ 18R~ Z ðu1 ; uÞ½sin 2o0 u þ sin 2o0 u1 þ sin 2o0 ðu1 þ uÞ,
s
c
B~ ðu1 ; uÞ ¼ 1R~ Z ðu1 ; uÞ cos 2o0 ðu1 þ uÞ; and; B~ ðu1 ; uÞ
4
4
8
¼ 18R~ Z ðu1 ; uÞ sin 2o0 ðu1 þ uÞ,
where R~ Z ðu1 ; uÞ ¼ R2Z ðu1 Þ þ RZ ðu þ u1 ÞRZ ðu u1 Þ. Similarly, from (34)–(36) we obtain the following expression
for the variance of the covariance function estimate b(t,u):
j
k
X
acl ðuÞ cos lo0 t þ asl ðuÞ sin lo0 t
(52)
D b^ðt; uÞ ¼ a0 ðuÞ þ
l¼2;4
where
D2 ~
S ðaT; N; uÞð2 þ cos 2o0 uÞ,
8N
D2 ~
ac2 ðuÞ ¼
S ðaT; N; uÞð1 þ cos 2o0 uÞ,
4N
D2 ~
as2 ðuÞ ¼
S ðaT; N; uÞ sin 2o0 u,
4N
2
D ~
ac4 ðuÞ ¼
S ðaT; N; uÞ cos 2o0 u,
8N
D2 ~
as4 ðuÞ ¼
S ðaT; N; uÞ sin 2o0 u,
8N
a0 ð uÞ ¼
and
S~ ðaT; N; uÞ ¼ 1 þ e2ajuj þ 2S ð2aT; N Þ þ 2S1 ðaT; N; uÞ,
N
1
X
n aðjuþnT jþjunT jÞ
1
.
S1 ðaT; N; uÞ ¼
e
N
n¼1
Components a0(u), ac2 (u), ac4 (u) are even functions of lag u and as2 (u), ac4 (u) are odd functions. For u ¼ 0,
the components a0 (u), ac2 (u), ac4 (u) are bounded above, whilst as2 (u) and ac4 (u) take the value zero. Thus
0.1
0
-0.1
-0.2
50
30
10
0 5
10 15
20 -50
N
-10
-30
u
Fig. 3. Bias of covariance function estimate depending on averaging period numbers N and lag u for o0 ¼ 0.2, a ¼ 0.1, D ¼ 1.
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25
Eq. (52) reduces to
h
i D2
½1 þ 2S ð2aT; N Þð3 þ 4 cos 2o0 t þ cos 4o0 tÞ
D b^ðt; 0Þ ¼
4N
j
k
in this case. The variance of the covariance function estimate is plotted in Fig. 4. Again, the value D b^ðt; 0Þ
equals zero at the points tk ¼ ð2k þ 1ÞT=4 which can be explained by noting that bðt; 0Þ ¼ D=2ð1 þ cos 2o0 tÞ.
The behaviour of all of the components take the form of damped oscillations for
j small
k lags. However, for
^
larger lags, the components converge to periodic functions. Thus, the variances D bðt; uÞ do not vanish with
increasing lag, but exhibit damped oscillatory behaviour. The amplitudes of these oscillations decrease with N
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
h
i
and the relative mean-square error D b^ðt; uÞ =bðt; uÞ grows rapidly with increasing lag. Thus, we use a
simple process of correlogram cutting to obtain reliable estimates. Note also that the maximum truncation
point um increases with N.
It is clear from Eqs. (37) and (52) that the undamped periodic tail of the variance for covariance function
estimate is caused by non-stationary characteristics of the underlying signal i.e., the second covariance
components Bc2(u) and Bs2(u). From (37), the stationary contribution in a0(u) depends only on the stationary
approximation of the covariance function B0(u) and has the form
D2
½2ð1 þ 2Sð2aT; N ÞÞ þ ð1 þ cos 2o0 uÞ e2ajuj þ 2S 1 ðaT; N; uÞ
8N
(see Appendix B).
For the non-stationary part, which depends on the higher-order component, we have
að0sÞ ðuÞ ¼
D2 2ajuj
e
þ cos o0 u þ 2S ð2aT; N Þcos 2o0 u þ 2S1 ðaT; N; uÞ.
8N
Obviously a0 ðuÞ ¼ að0sÞ ðuÞ þ að0nÞ ðuÞ and, for u ¼ 0 we obtain
að0nÞ ðuÞ ¼
D2
½1 þ 2Sð2aT; N Þ;
2N
For large u we also have that
að0sÞ ð0Þ ¼
að0nÞ ð0Þ ¼
D2
½1 þ 2Sð2aT; N Þ.
4N
D2
D2
½1 þ 2Sð2aT; N Þ; að0nÞ ðuÞ ¼
½1 þ 2Sð2aT; N Þ cos 2o0 u
4N
8N
Clearly, að0nÞ ð0Þ=að0sÞ ð0Þ ¼ 0:5 and the same limit is reached by the equivalent ratio for large lag values. Thus, the
non-stationarity component plays a significant role in the analysis of estimation reliability. Elimination of the
non-stationary components can lead to significant error for all possible lags.
Finally, we investigate the properties of the covariance components estimates. Biases in these estimates have
the same form as those for the corresponding Fourier components of the covariance function estimate. For the
að0sÞ ðuÞ ¼
0.3
0.2
0.1
60
40
0
0
10
N
20
20
30
u
0
Fig. 4. Variance of the covariance function estimate for o0 ¼ 0.2, a ¼ 0.1, D ¼ 1.
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26
variances, Eqs. (43)–(45) give us the following results:
D2
P0 ð2aT; N Þ þ P~ 0 ðaT; N; uÞ
D B^ 0 ðuÞ ¼
4N
ð1 þ cos 2o0 uÞ þ P2 ð2aT; N Þ þ P~ 2 ðaT; N; uÞ
h c i
D2
P0 ð2aT; N Þ þ P~ 0 ðaT; N; uÞ þ 2 P2 ð2aT; N Þ þ P~ 2 ðaT; N; uÞ
D B^ 2 ðuÞ ¼
4N
ð1 þ cos 2o0 uÞ þ P4 ð2aT; N Þ þ P~ 4 ðaT; N; uÞ
and
h s i
D2
D B^ 2 ðuÞ ¼
P0 ð2aT; N Þ þ P~ 0 ðaT; N; uÞ ð1 cos 2o0 uÞ
4N
þ 2 P2 ð2aT; N Þ þ P~ 2 ðaT; N; uÞ
ð1 þ cos 2o0 uÞ þ P4 ð2aT; N Þ þ P~ 4 ðaT; N; uÞ ,
where
1
P~ l ðaT; N; uÞ ¼
T
Z
NT
1
0
ð53Þ
ð54Þ
ð55Þ
u1 aðju1 þujþju1 ujÞ
cos lo0 u1 du1 :
e
y
Fig. 5 illustrates the cyclic variations in the functions given in Eqs. (53)–(55). These variations are caused by
the periodic non-stationarity of the underlying signal.
For u ¼ 0 the functions P~ l ðaT; N; uÞ and Pl ð2aT; N Þ are equal: i.e., P~ l ðaT; N; 0Þ ¼ Pl ð2aT; N Þ. There are
three distinct components to the expressions in Eqs. (53)–(55). The first does not depend on lag u, the second
component is damped with increasing lag and the third changes periodically with the lag. For the variance of
the zeroth covariance component when u ¼ 0, we have
D2
½2P0 ð2aT; N Þ þ P2 ð2aT; N Þ
D B^ 0 ð0Þ ¼
2N
and for large lags we obtain
D2
½P0 ð2aT; N Þð1 þ cos 2o0 uÞ þ P2 ð2aT; N Þ
(56)
D B^ 0 ðuÞ ¼
4N
since P~ l ðaT; N; uÞ ! 0 as u ! 1.
Cyclic
variations in the function given in (56) have large amplitude and substantially change the variance
D B^ 0 ðuÞ (see Fig. 5). These variations are caused by the presence of periodic stationarity in the signal.
The variance in (41) can be split into two components (following similarly to the procedure shown in
Appendix B): a stationary component;
D2
P0 ð2aT; N Þ þ P~ 0 ðaT; N; uÞ cos 2o0 u þ P2 ð2aT; N Þ þ P~ 2 ðaT; N; uÞ
DðsÞ B^ 0 ðuÞ ¼
4N
and a non-stationary component;
(57)
D2
½P0 ð2aT; N Þ cos 2o0 u þ P~ 0 ðaT; N; uÞ .
DðnÞ B^ 0 ðuÞ ¼
(58)
4N
Expression (57) contains functions that do not depend on the lag u, and also functions, which are damped as
u increases. Expression (58) contains damped and periodic functions. The ratio of these two expressions
defines the influence of signal non-stationarity on the reliability of estimating the covariance function, B0(u).
For zero lag (i.e., u ¼ 0) we get from (51) that
D2
½P0 ð2aT; N Þ þ P2 ð2aT; N Þ,
DðsÞ B^ 0 ð0Þ ¼
2N
D2
P0 ð2aT; N Þ.
DðnÞ B^ 0 ð0Þ ¼
2N
(59)
(60)
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0.15
0.1
0
10
0.05
20
0
0 2 4
6 8
10
N
(a)
30
40
u
50
0.2
0
0.1
10
20
0
0 2
4 6 8
10
N
(b)
30
40
u
50
0.1
0
0.05
10
20
0
0 2 4
6 8 10
N
(c)
30
40
u
50
h s i
h c i
Fig. 5. Variance of covariance components estimates: (a) D B^ 0 ðuÞ ; (b) D B^ 2 ðuÞ ; (c) D B^ 2 ðuÞ ) for o0 ¼ 0.2, a ¼ 0.1, D ¼ 1.
Thus, DðnÞ B^ 0 ð0Þ =DðsÞ B^ 0 ð0Þ X0:5 and the elimination of signal non-stationarity leads to significant error
in calculating the variance of our estimate. For large u, the stationary component is independent of lag, i.e.,
D2
½P0 ð2aT; N Þ þ P2 ð2aT; N Þ,
DðsÞ B^ 0 ðuÞ ¼
4N
(61)
whereas the non-stationary component is a cyclical function of lag i.e.,
D2
DðnÞ B^ 0 ðuÞ ¼
P0 ð2aT; N Þ cos 2o0 u.
4N
(62)
The value of (61) is half as great as the starting value DðsÞ B^ 0 ð0Þ given in (59) and the
of the cyclical
amplitude
oscillations of the non-stationary part in (62) is also half as great as that for DðnÞ B^ 0 ð0Þ given in (60). Thus,
the signal non-stationarity significantly changes the variance of the estimate and also its as we change the lag,
u.
It also follows from expressions (54)–(55), that the zeroth and second-order covariance components
define the properties of the variances of the second covariance component estimates. For zero lag (i.e., u ¼ 0)
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28
we have
h c i D2
½2P0 ð2aT; N Þ þ 4P2 ð2aT; N Þ þ P4 ð2aT; N Þ,
D B^ 2 ð0Þ ¼
2N
(63)
h s i D2
½2P2 ð2aT; N Þ þ P4 ð2aT; N Þ
D B^ 2 ð0Þ ¼
2N
(64)
h c i D2
½½P0 ð2aT; N Þ þ 2P2 ð2aT; N Þð1 þ cos 2o0 uÞ þ P4 ð2aT; N Þ,
D B^ 2 ðuÞ ¼
4N
(65)
h s i D2
½P0 ð2aT; N Þð1 cos 2o0 uÞ þ 2P2 ð2aT; N Þð1 þ cos 2o0 tÞ þ P4 ð2aT; N Þ.
D B^ 2 ðuÞ ¼
4N
(66)
and for large lags, we have
Variances (65) and (66) decrease by a factor of two relative to the initial values of the constant part given in
(63) and the cyclic part in (64), respectively.
h s i
j c k
As the lag increases, the rate of decrease in the variances D B^ 0 ðuÞ , D B^ 2 ðuÞ and D B^ 2 ðuÞ is less than the
corresponding decreasing rate for the covariance components B0 ðuÞ, Bc2 ðuÞ and Bs2 ðuÞ. This means that as the
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ffi
lag increases, so do the corresponding relative mean-squares errors
D B^ 0 ðuÞ =B0 ðuÞ and
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
h c;s i
D B^ 2 ðuÞ =Bc;s
c ðuÞ . As before this limits the consistency of estimating the covariance component for lag
values beyon0d the interval [0,um], though we note that the maximum lag value um grows as N increases.
6. Conclusions
One of the most significant tasks in the statistical analysis framework is the analysis of the reliability of
estimators for quantitative indices of interest. Here, we investigate this task for the analysis of periodic
stationary random signals. Obtaining quantitative indices makes it possible to justify the choice of process
parameters. This is necessary for avoiding errors, understanding the model and for model verification. In this
paper, the reliability of the so-called coherent process is investigated in terms of biases and variances of the
non-stationary mean, covariance function and the corresponding Fourier components. The relationships we
obtain show a dependence of both bias and variance on realization length and on the signal covariance
components. On the basis of formulae for the covariance components for the partial PCRP cases, it is possible
to investigate in detail the properties of these estimates and to obtain relationships between parameters of the
stationary processes, which form the PCRP. The variances of the covariance characteristic estimates take the
form of undamped oscillations as the lag increases. The amplitude of these oscillations is defined by
characteristics of the signal non-stationarity and their average values are defined by the characteristics of the
signal stationary approach. Such behaviour in the variance as the lag changes indicates that the reliable
estimation of the covariance characteristics is possible only for lags within the range [0,um]. Within this
interval, the variances of the estimates take the form of damped oscillations. Specific value for um can be
calculated by means of the formulae we derive here.
Appendix A
Proof of consistency of the component estimates given in Eqs. (12).
^ 0 ¼ E ½m
^ 0 Em
^ 0 2 ¼ E ½m
^ 0 m0 2
D½m
Z y
2
1
¼E
xðtÞ dt
y 0
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29
Z Z
1 y y
E xðtÞxðsÞ ds dt
y2 0 0
Z Z
Z Z
1 y y
1 y yt
¼ 2
bðt; s tÞ ds dt ¼ 2
bðt; uÞ du dt.
y 0 0
y 0 t
¼
After changing the order of integration, we have
^ 0 ¼
D½m
1
y2
Z
0
y
Z
y
bðt; uÞ dt du þ
u
Z
y
0
Z
yu
0
bðt; uÞ dt du .
Changing u for u in the first term and using the covariance function property bðt; uÞ ¼ bðt u; uÞ, we obtain
1
^ 0 ¼ 2
D½m
y
Z y
Z
y
bðt u; uÞ dt þ
u
0
Z
yu
bðt; uÞ dt du.
0
Substituting here s for tu we receive
Z
Z yu
Z yu
1 y
^ 0 ¼ 2
D½m
bðt; uÞ dt du
bðs; uÞ ds þ
y 0
0
0
Z Z
2 y yu
¼ 2
bðt; uÞ dt du.
y 0 0
Substituting here the Fourier decomposition of covariance function b(t,u), we have
^ 0 ¼
D½m
2
y2
Z
y
0
X
Bk ðuÞ
k2Z
Z
yu
0
eiko0 t dt du:
If we define the function
1
f l ð0; y uÞ ¼
y
Z
yu
e
ilo0 t
dt ¼
0
(
1 uy ;
ilo ðyuÞ
i
0
1 ;
lo0 y e
l ¼ 0;
la0:
then f l ð0; y uÞ ¼ f l ð0; y uÞ and the even and odd parts of f l ð0; y uÞ have the following representation
for l6¼0:
f cl ð0; y uÞ ¼
sin lo0 ðy uÞ
;
lo0 y
f sl ð0; y uÞ ¼
cos lo0 ðy uÞ 1
.
lo0 y
Thus, we have
#
Z y "
1
1
X
X
u
Bl ðuÞf l ð0; y uÞ du
Bl ðuÞf l ð0; y uÞ þ
1 B0 ðuÞ þ
y
0
l¼1
l¼1
#
Z "
X
2 y
u
1 B0 ðuÞ þ
¼
Bl ðuÞf l ð0; y uÞ þ Bl ðuÞf l ð0; y uÞ du.
y 0
y
l2N
2
^ 0 ¼
D½m
y
Recall that Bl ðuÞ ¼ 12 Bcl ðuÞ iBsl ðuÞ . After substituting into the previous formula and cancelling, we obtain
#
Z "
X
1 y
u
c
s
c
s
^ 0 ¼
D½m
1 B0 ðuÞ þ
Bl ðuÞf l ð0; y uÞ þ Bl ðuÞf l ð0; y uÞ du.
y 0
y
l2N
c;s
^ l is similar.
The proof for D m
ARTICLE IN PRESS
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30
Appendix B
From (37) we have
B~ 0 ðu1 ; uÞ ¼
X
q¼2;0;2
eiqo0 u Bq ðu1 þ uÞBq ðu1 uÞ þ Bq ðu1 ÞBq ðu1 Þ .
The ‘‘stationary part’’ of this equation depends only on zero covariance component and ‘‘non-stationary
part’’ depends on covariance components B2 ðuÞ and B2 ðuÞ ¼ B2 ðuÞ:
eðsÞ ðu1 ; uÞ ¼ B0 ðu1 þ uÞB0 ðu1 ; uÞ þ B0 ðu1 ÞB0 ðu1 Þ.
B
0
eðnÞ ðu1 ; uÞ ¼ e2io0 u B2 ðu1 þ uÞB2 ðu1 uÞ þ B2 ðu1 ÞB2 ðu1 Þ
B
0
þ e2io0 u B2 ðu1 þ uÞB2 ðu1 uÞ þ B2 ðu1 ÞB2 ðu1 Þ .
Taking into account (48) we obtain
eðsÞ ðu1 ; uÞ ¼ 1 RZ ðu1 þ uÞ
B
0
4
1
RZ ðu1 uÞ cos o0 ðu1 uÞ cos o0 ðu1 þ uÞ þ R2Z ðu1 Þcos2 o0 u1 .
4
i
2io0 u h
2io0 u
2
eðnÞ ðu1 ; uÞ ¼ e
B
R
ð
u
þ
u
ÞR
ð
u
u
Þe
þ
R
ð
u
Þ
Z
1
Z
1
1
0
Z
16
i
2io0 u
e
þ
RZ ðu1 þ uÞRZ ðu1 uÞe2io0 u þ R2Z ðu1 Þ ,
16
h
i
eðnÞ ðu1 ; uÞ ¼ 1 RZ ðu1 þ uÞRZ ðu1 uÞ þ R2 ðu1 Þ cos 2o0 u1 .
B
0
Z
8
Suppose that we set for the stationary covariance function RZ ðuÞ ¼ Deajuj , then we have
2
eðsÞ ðu1 ; uÞ ¼ D eaðju1 þujþju1 ujÞ cos o0 ðu1 þ uÞcos o0 ðu1 uÞ þ e2aju1 j cos2 o0 u1
B
0
4
D2 aðju1 þujþju1 ujÞ
e
ðcos 2o0 u þ cos 2o0 u1 Þ
¼
8
þ e2aju1 j ð1 þ cos 2o0 u1 Þ ,
D2 aðju1 þujþju1 ujÞ
ðn Þ
B~ 0 ðu1 ; uÞ ¼
e
þ e2aju1 j cos 2o0 u1 .
8
From (26) we have
"
#
N
1
X
1 e
n e
B0 ð0; uÞ þ 2
1
B0 ðnT; uÞ .
a0 ð uÞ ¼
N
N
n¼1
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31
Taking into account above expressions we obtain
"
D2 2ajuj
ðsÞ
e
ðcos 2o0 u þ 1Þ
a0 ð uÞ ¼
8N
þ2þ2
n¼1
að0nÞ ðuÞ ¼
þ2
D2 2ajnT j
e
þ cos 2o0 u
8N
N
1
X
n¼1
#
n aðjnTþujþjntujÞ
1
ðcos 2o0 u þ 1Þ þ 2e2ajnT j .
e
N
"
N
1
X
#
n 2ajuj
aðjnTþujþjntujÞ
1
cos 2o0 u þ e
e
.
N
And finally,
D2
2ð1 þ 2Sð2aT; N ÞÞ þ e2ajuj þ 2S1 ðaT; N; uÞ ð1 þ cos 2o0 uÞ ,
8N
D2 2ajuj
e
að0nÞ ðuÞ ¼
þ cos o0 u þ 2S ð2aT; N Þ
8N
cos 2o0 u þ 2S1 ðaT; N; uÞ ,
að0sÞ ðuÞ ¼
where
SðaT; N Þ ¼
N1
X
1
n¼1
S1 ðaT; N; uÞ ¼
N
1
X
n aTn
e
,
N
1
n¼1
n aðjuþnT jþjunT jÞ
e
.
N
References
[1] Ya. Dragan, I. Javors’kyj, Rhythmics of Sea Waving and Underwater Acoustic Signals, Naukova dumka, Kijev, 1982 (in Russian).
[2] Ya. Dragan, V. Rozhkov, I. Javors’kyj, Methods of Probabilistic Analysis of Oceanological Rhythmics, Gidrometeoizdat, Leningrad,
1987 (in Russian).
[3] W.A. Gardner, Cyclostationarity in Communications and Signal Processing, IEEE Press, New York, 1994.
[4] L. Gudzenko, On periodically nonstationary random processes, Radiotekh. Elektron, 6 (1959) 1062–1064 (in Russian).
[5] E.R. Kanasewich, Time Sequence Analysis in Geophysics, The University of Alberta Press. Edmonton, Canada, 1981.
[6] I. Yu. Isayev, I. N. Javors’kyj, Component analysis for time series with rhythmical structure, Izvestija VUZov ‘‘Radioelektronika’’,
No.1, 1995, pp. 34–45 (in Russian).
[7] Y. Kwang-Y, Statistical prediction of cyclostationary processes, J. Climate 13 (6) (1999) 1098–1115.
[8] D. Reynolds, T.H. Vonder Haar, A bispectral method for cloud parameter determination, Mon. Weather Rev. 105 (1977) 446–457.
[9] D.J. Thomson, Time series analysis of Holocene climate data, Philos. Trans. R. Soc. Lond. A. 330 (1990) 601–616.
[10] A.C. Mccormick, A.K. Nandi, Cyclostationarity in rotating machine vibrations, Mech. Systems Signal Process. 12 (2) (1998)
114–131.
[11] P.D. Mcfadden, M.M. Toozhy, Application of synchronous averaging to vibration monitoring of rolling element bearings, Mech.
Systems Signal Process. 14 (6) (2000) 891–906.
[12] M.J. Cassidy, W.D. Penny, Bayesian nonstationary autoregressive models for biomedical signal analysis, IEEE Trans. Biomed. Eng.
49 (10) (2002) 1142–1152.
[13] B. Upshaw, M. Rangoussi, T. Sinkjaer, Detection of human nerve signals using higher-order statistics, in: Proceedings of the 1996 8th
IEEE Signal Processing Workshop on Statistical Signal and Array Processing, SSAP’96, Cortir, Greece, 1996, pp. 186–189.
[14] K. Bullen, B. Bolt, An Introduction to the Theory of Seismology, Cambridge Press, Cambridge, UK, 1985.
[15] J. Park, C.R. Lindberg, F.L. Vernon, Multitaper spectral analysis of high-frequency seismograms, J. Geophys. Res. 92 (1987)
12675–12684.
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[16] W.A. Gardner, The spectral correlation theory of cyclostationary time-series, Signal Processing 11 (1986) 13–36.
[17] E. Gladyshev, Periodically and almost periodically correlated random processes with continuous time, Probabilistic Theory and its
Applications, vol. 3, 1963, pp. 184–189 (in Russian).
[18] O. Koronkievych, The linear dynamic systems under action of the random forces Nauk. Zap. Lvivskogo Univ. 44 (1957) 175–183
(in Ukrainian).
[19] H.L. Hard, Nonparametric time series analysis for periodically correlated processes, IEEE Trans. Inform Theory IT-35 (1989)
350–359.
[20] I. Javors’kyj, V. Mykhailyshyn, O. Zabolotnyj, Least squares method for statistical analysis of polyrhythmics, Appl. Math. Lett. 16
(8) (2003) 1217–1222.
Signal Processing 87 (2007) 13–32
www.elsevier.com/locate/sigpro
Coherent covariance analysis of periodically correlated
random processes
I. Javors’kyja,b, I. Isayeva,c,, Z. Zakrzewskib, S.P. Brooksc
a
Karpenko Physico-mechanical Institute of the National Academy of Sciences of Ukraine, Naukova St. 5, 79601, L’viv, Ukraine
Institute of Telecommunications, University of Technology and Agriculture, Al. Prof. S.Kaliskiego, 7, 85-796, Bydgoszcz, Poland
c
Statistical Laboratory, Cambridge University, Wilberforce Road, CB3 0WB, Cambridge, UK
b
Received 6 February 2005; received in revised form 13 April 2006; accepted 13 April 2006
Available online 5 June 2006
Abstract
A coherent method of estimating of periodically correlated random processes (PCRP) is introduced. Properties of
estimates of the mean, covariance function and their Fourier coefficients that are obtained using process values averaging
over the period are investigated. Asymptotic formulae for estimates of bias and variances are obtained and the
relationships of these characteristics to realization length are discussed. The probabilistic structure of one of the simplest
PCRP-based signals is analysed.
r 2006 Elsevier B.V. All rights reserved.
Keywords: Periodically correlated random processes; Mean; Covariance function; Coherent estimation; Bias; Consistency
1. Introduction
Recurrence and stochasticity are characteristic features of many physical processes exhibiting time
changeability [1–3]. Recurrence of process properties could be caused by the influence of external forces (e.g.,
forced oscillations) acting on the system or from internal forces (e.g., autonomous oscillation) that exist within
the system itself. Rhythmic processes occur in many fields including radio physics [4], telecommunications [3],
geophysics [5], oceanography [1,2], meteorology [6–9], vibro-diagnostics [10,11], biology [12,13] and
seismology [14,15], for example. The methodological basis for investigating rhythmic process structure based
on experimental data is the mathematical model. Early investigations into rhythmic processes were based on
deterministic models using periodic and almost-periodic functions. Attempts to describe the stochastic
changeability of oscillations were enhanced by the evolution of probabilistic methods based mainly on models
in the form of stationary random processes. Rhythmic changeability of physical processes is evidenced by the
oscillatory character of the associated correlograms and also in the peak values present in spectral density
estimates. However, these characteristics describe only time-averaged oscillatory properties. They have no
Corresponding author. Cambridge University, Statistical Laboratory, Wilberforce Road, CB3 0WB, Cambridge, UK. Tel.: +44
(0)1223 766925; fax: +44 (0)1223 337956.
E-mail addresses: ihor@statslab.cam.ac.uk, isayev@ipm.lviv.ua (I. Isayev).
0165-1684/$ - see front matter r 2006 Elsevier B.V. All rights reserved.
doi:10.1016/j.sigpro.2006.04.002
ARTICLE IN PRESS
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14
Nomenclature
x(t), xk(t), Z(t) stochastic signals
T
period
y
realization length
E
sign of mathematical expectation
m(t)
mean function
mk
mean components
b(t,u) covariance function
Bk(u) covariance components
D
variance
information about the time structure of any oscillations, which might be introduced, at least in idealized form,
by the incorporation of additional deterministic structure, for example. The development of probabilistic
models such as PCRP and their generalizations (bi-, poly- and almost-PCRP) is the natural evolution of these
approaches [1–4,16–18]. These models generalize the idea of recurrence to situations where stochasticity plays
a dominant role in the system process and they enable us to make more detailed and objective descriptions of
stochastic recurrence including, as special cases, the earlier models described above. PCRP models also enable
us to analyse the properties of physical processes without needing to specify special characteristics of the
process a priori, but provide a generic method for identifying characteristics present in the underlying process.
Suppose that we have a time series xðtÞ, t 2 R then this series is said to be periodically stationary with
period T if
Pr½xðt1 þ hÞ 2 A1 ; xðt2 þ hÞ 2 A2 ; . . . ; xðtn þ hÞ 2 An ¼ Pr½xðt1 Þ 2 A1 ; xðt2 Þ 2 A2 ; :::; xðtn Þ 2 An
for h ¼ T40, and no smaller value of h.
PCRP (or cyclo-stationary) models are defined uniquely by the specification of a mean function mðtÞ ¼
ExðtÞ;
t 2 ½0; TÞ and a covariance function bðt; uÞ ¼ E xðtÞ xðt þ uÞ; for t 2 ½0; TÞ and u 2 R, where
xðtÞ ¼ xðtÞ mðtÞ. Analysis of PCRP process models therefore involves estimation of these
functions.
R Ttwo
mðtÞdto1 and
PCRP
means,
m(t)
and
covariance
functions
b(t,
u)
are
periodic
functions
of
time
and
if
0
RT
0 bðt; uÞdt o1, u 2 R, then they can be represented by the Fourier series:
X
mðtÞ ¼
mk eiko0 t
k2Z
¼ m0 þ
X
mck cos ko0 t þ msk sin ko0 t
k2N
and
bðt; uÞ ¼
X
k2Z
Bk ðuÞeiko0 t ¼ B0 ðuÞ þ
X
Bck ðuÞ cos ko0 t þ Bsk ðuÞ sin ko0 t ;
ð1Þ
(2)
k2N
respectively. Here o0 ¼ 2p=T; mk ¼ 12 mck imsk ; Bk ðuÞ ¼ 12 Bck ðuÞ iBsk ðuÞ and both jmk j ! 0 and Bk ðuÞ !
0 as k ! 1. The parameters Bk(u) are referred to as the covariance components [1,2,17], coefficient function
[19] or the cyclic autocorrelation function [3,16]. The Fourier components are also commonly of interest and
their estimation often forms part of the analytic process.
An important part of understanding the properties PCRP’s is the PCRP decomposition into stationary
connected random processes [1–3], or harmonic serial representations [3]
X
xðtÞ ¼
xk ðtÞeiko0 t ,
(3)
k2Z
where the xk(t) denote stationary connected random processes. The characteristics of the signal x(t) can
therefore be derived from the corresponding characteristics of the constituent xk(t) processes.
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15
Various methods have been proposed for the estimation of the mean and covariance functions. For example
Isayev and Javors’kyj [6] investigates the use of component methods, whereas Javors’kyj et al. [20] develop
approaches based upon least squares estimation. In this paper, we will focus on the so-called coherent method
estimation, proposed by Gudzenko [4]. The coherent method (also known as synchronized averaging [16]) is
based on averaging sample PCRP realization values at the same point across different periods to obtain
empirical estimates of the mean and covariance functions. For example, since y is an integer multiple of the
period, T, the coherent estimate of the mean has the form:
^ ðtÞ ¼
m
1
X
1N
xðt þ nT Þ for t 2 ½0; T Þ.
N n¼0
(4)
Several authors have used the coherent method for analysing PCRP processes. In this paper, we focus on the
properties of the corresponding estimators. We begin in Section 2 by looking at the coherent estimators of the mean
function and the corresponding Fourier series components and demonstrate their unbiasedness and consistency. In
Section 3, we prove the asymptotic unbiasedness and consistency of the covariance function before doing the same
for the corresponding Fourier components in Section 4. In Section 5, we conclude with an example in which we
investigate the properties of the signal corresponding to a particular implementation of the PCRP process.
2. Properties of the PCRP mean function and corresponding component estimators
The coherent mean function estimator in (4) is clearly unbiased since
^ ðtÞ ¼
Em
1
X
1N
mðt þ nT Þ ¼ mðtÞ.
N n¼0
^ ðtÞ ! 0 as N ! 1. We begin by noting that
To prove consistency, we need to show that D½m
X
jnj
1 N1
2
^ ðtÞ ¼ E ½m
^ ðtÞ E m
^ ðtÞ ¼
1
D½m
bðt; nT Þ.
N n¼Nþ1
N
The proof is trivial.
"
#
N1
X
1
n
^ ðtÞ ¼
bðt; 0Þ þ 2
D½m
1
bðt; nT Þ
N
N
n¼1
(5)
(6)
and, if
N
1X
bðt; nT Þ ¼ 0,
N!1 N
n¼1
lim
(7)
^P
then D½m
ðtÞ ! 0 as N ! 1. Thus, the estimator in (4) is also consistent. Note that condition (7) is satisfied if
a
the sum N
n¼1 bðt; nT Þ increases with N no faster than N , where ao1. Note also that condition (7) is evident if
limjuj!1 bðt; uÞ ¼ 0.
Taking into account the foregoing we now formulate the following theorem:
Theorem 1. Statistic (4) is an unbiased and consistent estimate of the PCRP mean if condition (7) is satisfied,
its variance being defined by expression (6).
The variance (6) is a periodic function of time with corresponding Fourier series expansion:
X
^ ðtÞ ¼ g0 þ
D½m
gcl cos lo0 t þ gsl sin lo0 t .
(8)
l2N
From (2), we therefore have that:
"
#
N
1
X
1
n
B0 ð0Þ þ 2
g0 ¼
1
B0 ðnT Þ
N
N
n¼1
(9)
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16
and
gc;s
l
"
#
N1
X
1
n c;s
c;s
B ð0Þ þ 2
B ðnT Þ .
¼
1
N l
N l
n¼1
(10)
Similarly, from (2) and under the condition in (7), we have that:
N
1X
B0 ðnT Þ ¼ 0;
N!1 N
n¼1
lim
N
1X
Bc;s
l ðnT Þ ¼ 0
N!1 N
n¼1
and lim
(11)
and thus g0 ! 0 and gc;s
l ! 0 as N ! 1.
The zero component g0 defines the average value of the variance of the coherent estimator (8) and depends
only upon the zero covariance component B0(u). The lth harmonic components which define the amplitudes of
^ ðtÞ depend only on the covariance components of the same order. If correlations vanish
the variance D½m
within a time interval less than the period T (i.e., if B0 ðnTÞ ¼ Blc;s ðnTÞ ¼ 0 for all n40), then expressions
(9)–(10) reduce to:
B 0 ð 0Þ
Bc ð0Þ
B s ð 0Þ
and gsl ¼ l .
; gcl ¼ l
N
N
N
^ ðtÞ given in (6) is simply bðt; 0Þ=N.
In this case the time-varying behaviour of the estimator variance D½m
Let us now consider the estimation of the Fourier components corresponding to the mean function given in
^ ðtÞ, is known for all t 2 ½0; TÞ. Then we can create the following
(1). Suppose, that the mean estimate m
statistics:
Z
Z
Z
1 T
1 T
1 T
^0 ¼
^ sl ¼
^ ðtÞ dt; m
^ cl ¼
^ ðtÞ cos lo0 t dt; and m
^ ðtÞ sin lo0 t dt:
m
(12)
m
m
m
T 0
T 0
T 0
g0 ¼
Combining (4) and (12), we obtain:
Z
1 y
^0 ¼
xðtÞ dt;
m
y 0
^ cl ¼
m
2
y
Z
0
y
^ sl ¼
xðtÞ cos lo0 t dt and m
(13)
2
y
Z
y
xðtÞ sin lo0 t dt:
(14)
0
^ ðtÞ ¼ mðtÞ, the estimates in (12) are unbiased for the corresponding components given in (1) i.e.,
Since E m
^ 0 ¼ m0 , E m
^ cl ¼ mcl and E m
^ sl ¼ msl .
Em
To demonstrate consistency of the component estimates, we note that
Z Z
1 y y
^ 0 ¼ 2
bðt; s tÞ dt ds;
(15)
D½m
y 0 0
and
c
4
^l ¼ 2
D m
y
s
4
^l ¼ 2
D m
y
Z
y
y
bðt; s tÞ cos lo0 t cos lo0 s dt ds
(16)
bðt; s tÞ sin lo0 t sin lo0 s dt ds:
(17)
0
0
Z
Z
y
Z
y
0
0
See Appendix A for details. Setting u ¼ s t we obtain:
Z Z
2 y yu
^ 0 ¼ 2
bðt; uÞ dt du;
D½m
y 0 0
c
4
^l ¼ 2
D m
y
Z
0
y
Z
(18)
yu
bðt; uÞ½cos lo0 u þ cos lo0 ð2t þ uÞ dt du
0
(19)
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17
and
s
4
^l ¼ 2
D m
y
Z
y
0
Z
yu
bðt; uÞ½cos lo0 u cos lo0 ð2t þ uÞ dt du.
(20)
0
Finally, substituting b(t,u) for its Fourier series decomposition in (18) and integrating with respect to t, we
obtain
Z "
X
2 y
u
^ 0 ¼
D½m
1 B0 ðuÞ þ
Bcl ðuÞf cl ð0; y uÞ þ Bsl ðuÞf sl ð0; y uÞ du.
y 0
y
l2N
yuÞlo0
Þlo0 1
where f cl ð0; y uÞ ¼ sin ½ðlo
:
; f sl ð0; y uÞ ¼ cos ½ðyu
lo0 y
0y
c
s
Functions f l ð0; y uÞ and f l ð0; y uÞ are clearly rapidly damped with increasing series length, y. So, for
large y we have
Z
2 y
u
^ 0
D½m
1 B0 ðuÞ du.
(21)
y 0
y
Using similar manipulations, (19) and (20) become
Z
c 2 y h
i
u
^l ¼
1 ½2B0 ðuÞ cos lo0 u Bc2l ðuÞ cos lo0 u þ Bs2l ðuÞ sin lo0 u du,
D m
y 0
y
s 2
^l ¼
D m
y
Z y h
0
1
i
u
2B0 ðuÞ cos lo0 u þ Bc2l ðuÞ cos lo0 u Bs2l ðuÞ sin lo0 u du.
y
If the following conditions
Z
Z
1 y
1 y c
lim
B0 ðuÞ du ¼ 0; lim
Bk ðuÞ du ¼ 0;
y!1 y 0
y!1 y 0
1
and lim
y!1 y
Z
0
(22)
(23)
y
Bsk ðuÞ du ¼ 0
(24)
are satisfied, then the variances in Eqs. (21)–(23) vanish as y ! 1. Thus the estimators in (12) are consistent
as well as asymptotically unbiased.
Taking into consideration the foregoing we now formulate the following theorem:
Theorem 2. The statistics in (12) are unbiased and consistent estimates of the PCRP mean Fourier coefficients
if the conditions in (24) are satisfied, their variances being defined by the expressions in (21)–(23).
It is clear from Eq. (21) that the zero covariance component is the main characteristic defining the variance
of the zeroth-order mean estimate. The variances of higher-order mean
s;c estimates are functions boths;cof the
^ l depends both B0 and on B2l . The
zero component and the components of double their order i.e., D m
constituent sine and cosine components
in
(22)
and
(23)
have
opposite
signs. Therefore, the variance of the
c
s
^ l is independent of higher covariance components, i.e.,
^l þD m
^ l ¼ 14 D m
complex amplitude estimate D½m
Z
2 y
u
^ l ¼
D½m
1 B0 ðuÞ cos lo0 u du.
y 0
y
The dependence of this variance on the value l clearly stems from the frequency of the cosine weight function.
3. Properties of the covariance function estimators
The coherent covariance function estimate is given by
X
1 N1
^ ðt þ nT Þ½xð½t þ nT þ uÞ m
^ ð½t þ nT þ uÞ.
½xðt þ nT Þ m
b^ðt; uÞ ¼
N n¼1
(25)
Suppose that the realization length is such that y ¼ NT þ um , where um denotes the maximum lag of the
^ ðt þ nT Þ ¼
covariance function (see e.g., [1–3]). The mean function is then estimated only for t 2 ½0; T , with m
ARTICLE IN PRESS
I. Javors’kyj et al. / Signal Processing 87 (2007) 13–32
18
^ ðtÞ when te½0; TÞ and n 2 N. Then, taking expectations, we have
m
1
X
1N
^ ðtÞm
^ ðt þ uÞ.
E½xðt þ nT Þxð½t þ nT þ uÞ E m
N n¼0
h
i
Hence the bias, b^ðt; uÞ ¼ E b^ðt; uÞ bðt; uÞ, of our estimator is given by
1
h
i
X
j nj
1 N
^
bðt; uÞ ¼
1
bðt; u þ nT Þ.
N
N n¼Nþ1
^ uÞ ¼
E½bðt;
(26)
(27)
The proof follows directly from a similar expression to that given in (5).
For u ¼ 0 this expression differs from the variance of mean estimate, given in (5), only by the sign. Clearly,
changes in the covariance function directly affect the bias. Therefore we can obtain a Fourier series
^ ðtÞ given in (8) i.e.,
representation similar to that for D½m
h
i
X
b^ðt; uÞ ¼ 0 ðuÞ þ
cl ðuÞ cos lo0 t þ sl ðuÞ sin lo0 t .
l2N
In analogy to Eq. (9) and (10), it is easy to show that
1
X
jnj
1 N
0 ðuÞ ¼
1
B0 ðu þ nT Þ
N
N n¼Nþ1
(28)
and
c;s
l ð uÞ
1
X
jnj c;s
1 N
¼
1
Bl ðu þ nT Þ.
N
N n¼Nþ1
(29)
h
i
^ðt; uÞ ! 0 as N ! 1. Thus, the
Then, if the conditions in (11) are satisfied, 0 ðuÞ ! 0, c;s
ðuÞ
!
0
and
b
l
coherent estimator in (25) is asymptotically unbiased.
Let us obtain the variance of estimate (25) under the assumption that the PCRP is Gaussian (so that third
and higher-order moments can be expressed in terms of first and second-order moments), and then rewrite (25)
in the form
1
X
1N
^ ðtÞ m
^ ðt þ uÞ,
b^ðt; uÞ ¼
xðt þ u þ nT Þ xðt þ nT Þ m
N n¼0
where
^ ðtÞ ¼
m
X
1 N1
xðt þ nT Þ.
N n¼0
Then, since third and higher-order moments can be expressed in terms of first and second-order moments
under the Gaussian assumption, as a first order approximation we get:
h
i
X
j nj
1 N1
^
½bðt; nT Þbðt þ u; nT Þ þ bðt; u þ nT Þbðt; u nT Þ þ oðN 1 Þ.
D bðt; uÞ ¼
(30)
1
N
N n¼Nþ1
The proof follows similarly to that
in Appendix A.
given
Consider the process Zðt; uÞ ¼ xðtÞ xðt þ uÞ with corresponding covariance function
bZ ðt; u1 ; uÞ ¼ E xðtÞ xðt þ uÞ bðt; uÞ xðt þ u1 Þ xðt þ u1 þ uÞ bðt þ u1 ; uÞ
¼ bðt; u1 Þbðt þ u; u1 Þ þ bðt; u1 þ uÞbðt þ u; u1 uÞ.
ð31Þ
ARTICLE IN PRESS
I. Javors’kyj et al. / Signal Processing 87 (2007) 13–32
Then, using the fact that bðt; uÞ ¼ bðt u; uÞ it follows from (30) that
1
h
i
X
j nj
1 N
^
1
D bðt; uÞ ¼
bZ ðt; nT; uÞ þ o N 1 .
N n¼Nþ1
N
19
(32)
Theorem 3. The estimate of the covariance function (25) of a Gaussian PCRP is asymptotically unbiased and
consistent if conditions (11) are satisfied and its bias and variance are defined by the formulae in (27) and (32).
Let us represent the function bZ ðt; u1 ; uÞ in terms of the Fourier series decomposition:
X
B~ k ðu1 ; uÞeiko0 t .
bZ ðt; u1 ; uÞ ¼
(33)
k2Z
Then, setting u1 ¼ nT and substituting (33) into (32) we have that
h
i
X
D b^ðt; uÞ ¼ a0 ðuÞ þ
acl ðuÞ cos lo0 t þ asl ðuÞ sin lo0 t þ o N 1
(34)
l2N
where
"
#
N
1
X
1 ~
n ~
B0 ð0; uÞ þ 2
B0 ðnT; uÞ ,
a0 ð uÞ ¼
1
N
N
n¼1
(35)
and
ac;s
l ðuÞ
"
#
N
1
X
1 ~ c;s
n ~ c;s
¼
B ð0; uÞ þ 2
1
B ðnT; uÞ .
N l
N l
n¼1
(36)
Using the earlier decomposition of the covariance function bðt; uÞ into Fourier series, we can directly obtain
expressions for the components B~ l ðu1 ; uÞ. Suppose that we have M covariance components, then we have from
(31) that
8
M
P
>
>
eiqo0 u Bqþk ðu1 þ uÞBq ðu1 uÞ þ Bqþk ðu1 ÞBq ðu1 Þ ; kp0;
>
>
< q¼Mþk
(37)
B~ k ðu1 ; uÞ ¼
Mk
P iqo0 u
>
>
>
ð
u
u
Þ
þ
B
ð
u
ÞB
ð
u
Þ
;
k40
e
B
ð
u
þ
u
ÞB
q
1
qþk
1
q
1
qþk
1
>
:
q¼M
where ‘‘’’ denotes complex conjugation and k 2 ½2M; 2M
Finally, using the definition of B~ k ðu1 ; uÞ in (37), and substituting into Eqs. (35) and (36), Eq. (11) then
implies that
lim ac;s ðuÞ
juj!1 l
¼ 0.
h
i
Thus from (34), we have that lim D b^ðt; uÞ ¼ 0, the condition required to demonstrate the consistency of the
juj!1
estimator in (25).
From (35) it is clear that the average value of the variance in (34), corresponding to component a0 ðuÞ,
depends on the zeroth and all higher order covariance components. Thus, in the case of PCRP processes the
precision of our estimates could not be obtained by looking at stationary processes alone.
lim a0 ðuÞ ¼ 0;
juj!1
4. Properties of the covariance component estimators
Let us now consider the coherent estimators for the covariance components. Suppose that the estimate
b^ðt; uÞ is known for all t 2 ½0; TÞ and for all u 2 ½0; um . Note that, in practice, we truncate our estimator for the
covariance function to lags uoum where um is the truncation point of correlogram [1]. Then for lags in this
ARTICLE IN PRESS
I. Javors’kyj et al. / Signal Processing 87 (2007) 13–32
20
range, we can create the following estimates of the covariance components:
Z
1 T ^
bðt; uÞ dt,
B^ 0 ðuÞ ¼
T 0
c
B^ l ðuÞ ¼ T2
RT
0
R
2 T
s
B^ l ðuÞ ¼ T
0
b^ðt; uÞ cos lo0 t dt;
b^ðt; uÞ sin lo0 t dt:
(38)
(39)
Taking into consideration expressions for bias of the covariance function estimates in Eqs. (28) and (29) and
the Fourier decomposition of the covariance function in (2) we have, after integrating, that
1
X
1 N
j nj
B^ 0 ðuÞ ¼
1
B0 ðu þ nT Þ,
N n¼Nþ1
N
1
h c;s i
X
jnj
1 N
B^ l ðuÞ ¼
1
B0 ðu þ nT Þ.
N
N n¼Nþ1
The biases of the covariance component estimates for any order depend only on the covariance components
of the same order. If the conditions in equation (11) are satisfied then these biases clearly converge to zero as
N ! 1. Thus, the estimators in (38) and (39) are asymptotically unbiased.
In order to investigate consistency, we require expressions for the variances of the estimators in Eqs. (38)
and (39). Combining Eqs. (38) and (39), using the definition of b^ðt; uÞ from (25) and the periodic property of
^ ðt þ nT Þ ¼ m
^ ðtÞ, we have that
the mean estimate i.e., m
Z y
1
^ ðtÞ m
^ ðt þ uÞ dt,
B^ 0 ðuÞ ¼
xðtÞ xðt þ uÞ m
y 0
2
c
B^ l ðuÞ ¼
y
Z y
^ ðtÞ m
^ ðt þ uÞ cos lo0 t dt,
xðtÞ xðt þ uÞ m
0
Z
2 y
s
^
^ ð sÞ m
^ ðs þ uÞ sin lo0 s ds.
Bl ðuÞ ¼
x ð s Þ xð s þ uÞ m
y 0
For Gaussian signals we get a first-order approximation:
Z Z
2 y yu1
D B^ 0 ðuÞ ¼
bZ ðt; u1 ; uÞ dt du1 þ oðy1 Þ,
y 0 0
and
(40)
Z Z
h c i
4 y yu1
D B^ l ðuÞ ¼ 2
bZ ðt; u1 ; uÞ½cos lo0 u1 þ cos lo0 ð2t þ u1 Þ dt du1 þ oðy1 Þ,
y 0 0
(41)
Z Z
h s i
4 y yu1
D B^ l ðuÞ ¼ 2
bZ ðt; u1 ; uÞ½cos lo0 u1 cos lo0 ð2t þ u1 Þ dt du1 þ oðy1 Þ.
y 0 0
(42)
h c i 2 Z y
u1 ~
s
c
D B^ l ðuÞ
B0 ðu1 ; uÞ cos lo0 u1 B~ 2l ðu1 ; uÞ cos lo0 u1 þ B~ 2l ðu1 ; uÞ sin lo0 u1 du1 ,
1
y 0
y
(44)
Proof follows similarly to that given in Appendix A.
Substituting the Fourier decomposition of bZ ðt; u1 ; uÞ from (33) into Eqs. (40)–(42) and integrating with
respect to t, we obtain for large y (or, equivalently, N):
Z
2 y
u1 ~
D B^ 0 ðuÞ
(43)
1
B0 ðu1 ; uÞ du1 ;
y 0
y
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I. Javors’kyj et al. / Signal Processing 87 (2007) 13–32
h s i 2 Z y
u1 ~
c
s
D B^ l ðuÞ
1
B0 ðu1 ; uÞ cos lo0 u1 þ B~ 2l ðu1 ; uÞcos lo0 u1 B~ 2l ðu1 ; uÞ sin lo0 u1 du1 ,
y 0
y
21
(45)
Clearly, if
1
lim
y!1 y
Z
y
0
1
B~ 0 ðu1 ; uÞ du1 ¼ 0 and lim
y!1 y
Z
0
y
c;s
B~ l ðu1 ; uÞ du1 ¼ 0,
(46)
then the variances in equations (43)–(45) vanish as y ! 1. Hence, we have asymptotic consistency for the
estimators in Eqs. (38) and (39).
Theorem 4. Estimates (38)–(39) for Gaussian PCRP are asymptotically junbiased
and consistent
j s k if the
k
c
c
^
^
conditions in (11) and (46) are satisfied and their biases B0 ðuÞ ¼ 0 ðuÞ, Bk ðuÞ ¼ k ðuÞ, B^ k ðuÞ ¼ sk ðuÞ
and variances are defined by the formulae in (28)–(29) and (43)–(45).
c;s
Since values B~ 0 ðu1 ; uÞ and B~ l ðu1 ; uÞ are evaluated through the PCRP covariance components products, the
conditions (46) hold if and only if the conditions in (24) hold true. The proof is trivial. The expressions in Eqs.
(43)–(45) are similar to those for the variances of mean components in Eqs. (22) and (23). The difference is that
we use the Fourier components of the function bZ ðt; u1 ; uÞ in the first case and covariance components in the
second. It follows from (37) that B~ 0 ðu1 ; uÞ depends upon all covariance components. Hence, the accuracy of
the estimator for the zero covariance component B0(u) cannot be measured without considering the higher
order covariance components. Once again, the variance of our estimators could not be obtained by looking at
stationary processes alone. The variances of the sine and cosine covariance components depend also on the
higher order
h components
i of the function bZ ðt; u1 ; uÞ. However, for the variances of complex-valued estimates
s
1 ^c
^
^
Bl ðuÞ ¼ B ðuÞ iB ðuÞ we have
2
l
l
2
D B^ l ðuÞ
y
Z y
u1 ~
B0 ðu1 ; uÞ cos lo0 u1 du1 .
1
y
0
Thus, the accuracy of the higher-order complex-valued covariance component estimates depend only on
B~ 0 ðu1 ; uÞ.
5. Characteristics of modulated signals—an example
As we saw in Section 1, Eq. (3), we can decompose our signal into a series of stationary connected subsignals (harmonic serial representations) as follows:
X
xðtÞ ¼
xk ðtÞeiko0 t ;
k2Z
where the xk(t) denote stationary random processes.
There are several common forms of constituent process. For example, if we set xk ðtÞ ¼ ak þ Zk ðtÞ, where the
Zk ðtÞ are uncorrelated, then we get the additive model
X
X
xðtÞ ¼
ak eiko0 t þ
Zk ðtÞeiko0 t ¼ f ðtÞ þ ZðtÞ,
k2Z
k2Z
where f(t) is a periodic function and Z(t) is a stationary random process. Such models are commonly used for
simple processes with stochastic variation around a periodic mean.
If we set xk ðtÞ ¼ ak ZðtÞ, then we get the multiplicative PCRP model, where
X
xðtÞ ¼ ZðtÞ
ak eiko0 t ¼ f ðtÞZðtÞ.
(47)
k2Z
This model is commonly used to describe more complicated processes than the simple additive model above
and allows for interaction between the stochastic errors process and the mean function.
ARTICLE IN PRESS
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22
A third model is obtained if we x1 ðtÞ ¼ 1=2½xc ðtÞ ixs ðtÞ and x1 ðtÞ ¼ x1 ðtÞ, with the remaining stationary
processes all set equal to zero i.e., xk ðtÞ 0 for jkja1. In this case, we get the quadrature model that allows for
interactions between the stationary and quadrature components.
xðtÞ ¼ xc ðtÞ cos o0 t þ xs ðtÞ sin o0 t.
In this section, we take a specific example of a multiplicative model and investigate the properties of the
signal for a particular form of periodic function and associated stationary random processes. We shall assume
that f ðtÞ ¼ cos o0 t in (47) and, using the properties of elementary circular functions, we obtain the following
mean and covariance functions
mðtÞ ¼ m cos o0 t and bðt; uÞ ¼ B0 ðuÞ þ Bc2 ðuÞ cos 2o0 t þ Bs2 ðuÞ sin 2o0 t,
where
m ¼ EZðtÞ; B0 ðuÞ ¼ Bc2 ðuÞ ¼ 1=2RZ ðuÞ cos o0 u and Bs2 ðuÞ
¼ 1=2RZ ðuÞ sin o0 u
ð48Þ
and RZ ðuÞ denotes the covariance function of stationary process ZðtÞ. In this case, and from (8) and (9), the bias
of the mean equals zero and we get the following variance for the mean:
^ ðtÞ ¼ g0 þ gc2 cos 2o0 t þ gs2 sin 2o0 t,
D½m
where
and
"
#
N
1
X
1
n
R Z ð 0Þ þ 2
1
g0 ¼ gc2 ¼
RZ ðnT Þ ,
2N
N
n¼1
gs2 ¼ 0.
Now, suppose that we set RZ ðuÞ ¼ Deajuj , then we have that
g0 ¼ gc2 ¼
D
½1 þ 2SðaT; N Þ,
2N
where
SðaT; N Þ ¼
N1
X
1
n¼1
n aTn
eaT
1 eaNT
1
¼
e
N
1 eaT
N ð1 eaT Þ
(49)
and, for large N, we obtain
D 1 þ eaT
ð1 þ cos 2o0 tÞ.
(50)
2N 1 eaT
Fig. 1 plots the variance of the mean estimate as N and t vary. This plot clearly demonstrates the rapid
^ ðtÞ as N increases. Note also that the variance in (50) vanishes at the points
decrease in the variance D½m
tk ¼ ð2k þ 1ÞT=4, k 2 Z. This implies that the mean at the points tk must be known and is a consequence of
the definition of m(t) as m(t) ¼ m cos o0t which clearly equals zero at the points tk.
As we saw in Section 2, the mean components can be defined using a Fourier transform of the mean
function. Here, we have only one component l ¼ 1 and so, from (21)–(23), it is simple to show that
^ ðtÞ ¼
D½m
^ 0 ¼
D½m
D
P1 ðaT; N Þ;
N
^ c ¼
D½m
D
P2 ðaT; N Þ;
N
^ s ¼
D ½m
D
½2P0 ðaT; N Þ þ P2 ðaT; N Þ,
N
where
1
Pl ðaT; N Þ ¼
T
"
#
Z y
u ajuj
1
a2 T 2 4l 2 p2
aNT
1e
aT
1 e
cos lo0 u du ¼
.
y
a2 T 2 þ 4l 2 p2
N a2 T 2 þ 4l 2 p2
0
(51)
ARTICLE IN PRESS
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23
0.8
0.6
0.4
0.2
0
0
10
0
20
5
10
30
15
20
N
40
50
t
Fig. 1. Variance of mean estimate depending on period numbers N and time t for o0 ¼ 0.2, a ¼ 0.1 and D ¼ 1.
zero
cos
sin
0.6
0.5
0.4
0.3
0.2
0.1
0
0
5
10
15
20
^ 0 , D½m
^ c D½m
^ s for o0 ¼ 0.2, a ¼ 0.1, D ¼ 1.
Fig. 2. Variance of mean components estimate variances D½m
The product aT denotes the rate of damping of the estimate variances with N. Fig. 2 provides a plot of these
variances as N changes.
We now consider the properties of the estimate of the covariance function, b(t,u). Eqs. (27)–(29) suggest the
following result for the bias:
j
k
b^ðt; uÞ ¼ 0 ðuÞ þ c2 ðuÞ cos 2o0 t þ s2 ðuÞ sin 2o0 t,
where
0 ðuÞ ¼ c2 ðuÞ ¼
and
D
S0 ðaT; N; uÞ cos o0 u;
2N
s2 ðuÞ ¼
D
S 0 ðaT; N; uÞ sin o0 u
2N
jnj ajuþnT j
S0 ðaT; N; uÞ ¼
.
1
e
N
n¼Nþ1
N
1
X
From (49) and for zero lag (i.e., u ¼ 0) we have S0(aT,N,u) ¼ 1+2S(aT,N). So, in analogy to the case for
the components of the bias of the covariance function, expressions for the components e0(0) and ec2(0) differ
from the corresponding Fourier components of the mean estimate variance g0 and gc2 only in sign.
However, the damping rate is
The function S0(aT,N,u) oscillates but its amplitude decreases asju increases.
k
^ðt; uÞ (see Fig. 3) It is relatively easy to
less than that for the covariance.
This
obviously
affects
the
bias
b
h
i
show that the relative value b^ðt; uÞ =bðt; uÞ increases with lag u and, because the numerator is only defined
over a finite range, it is possible to assess the reliability of the covariance function estimate only for lags within
the finite interval [0,um].
ARTICLE IN PRESS
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24
We now examine the variance of the covariance function estimate. Using Eqs. (24), (48) and the properties
of elementary circular functions, we obtain the following Fourier components of the function bZ(t,u1,u) in (31):
B~ 0 ðu1 ; uÞ ¼ 18R~ Z ðu1 ; uÞð1 þ cos 2o0 u þ cos 2o0 u1 Þ,
c
B~ ðu1 ; uÞ ¼ 1R~ Z ðu1 ; uÞ½1 þ cos 2o0 u þ cos 2o0 u1 þ cos 2o0 ðu1 þ uÞ,
2
8
s
B~ 2 ðu1 ; uÞ ¼ 18R~ Z ðu1 ; uÞ½sin 2o0 u þ sin 2o0 u1 þ sin 2o0 ðu1 þ uÞ,
s
c
B~ ðu1 ; uÞ ¼ 1R~ Z ðu1 ; uÞ cos 2o0 ðu1 þ uÞ; and; B~ ðu1 ; uÞ
4
4
8
¼ 18R~ Z ðu1 ; uÞ sin 2o0 ðu1 þ uÞ,
where R~ Z ðu1 ; uÞ ¼ R2Z ðu1 Þ þ RZ ðu þ u1 ÞRZ ðu u1 Þ. Similarly, from (34)–(36) we obtain the following expression
for the variance of the covariance function estimate b(t,u):
j
k
X
acl ðuÞ cos lo0 t þ asl ðuÞ sin lo0 t
(52)
D b^ðt; uÞ ¼ a0 ðuÞ þ
l¼2;4
where
D2 ~
S ðaT; N; uÞð2 þ cos 2o0 uÞ,
8N
D2 ~
ac2 ðuÞ ¼
S ðaT; N; uÞð1 þ cos 2o0 uÞ,
4N
D2 ~
as2 ðuÞ ¼
S ðaT; N; uÞ sin 2o0 u,
4N
2
D ~
ac4 ðuÞ ¼
S ðaT; N; uÞ cos 2o0 u,
8N
D2 ~
as4 ðuÞ ¼
S ðaT; N; uÞ sin 2o0 u,
8N
a0 ð uÞ ¼
and
S~ ðaT; N; uÞ ¼ 1 þ e2ajuj þ 2S ð2aT; N Þ þ 2S1 ðaT; N; uÞ,
N
1
X
n aðjuþnT jþjunT jÞ
1
.
S1 ðaT; N; uÞ ¼
e
N
n¼1
Components a0(u), ac2 (u), ac4 (u) are even functions of lag u and as2 (u), ac4 (u) are odd functions. For u ¼ 0,
the components a0 (u), ac2 (u), ac4 (u) are bounded above, whilst as2 (u) and ac4 (u) take the value zero. Thus
0.1
0
-0.1
-0.2
50
30
10
0 5
10 15
20 -50
N
-10
-30
u
Fig. 3. Bias of covariance function estimate depending on averaging period numbers N and lag u for o0 ¼ 0.2, a ¼ 0.1, D ¼ 1.
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25
Eq. (52) reduces to
h
i D2
½1 þ 2S ð2aT; N Þð3 þ 4 cos 2o0 t þ cos 4o0 tÞ
D b^ðt; 0Þ ¼
4N
j
k
in this case. The variance of the covariance function estimate is plotted in Fig. 4. Again, the value D b^ðt; 0Þ
equals zero at the points tk ¼ ð2k þ 1ÞT=4 which can be explained by noting that bðt; 0Þ ¼ D=2ð1 þ cos 2o0 tÞ.
The behaviour of all of the components take the form of damped oscillations for
j small
k lags. However, for
^
larger lags, the components converge to periodic functions. Thus, the variances D bðt; uÞ do not vanish with
increasing lag, but exhibit damped oscillatory behaviour. The amplitudes of these oscillations decrease with N
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
h
i
and the relative mean-square error D b^ðt; uÞ =bðt; uÞ grows rapidly with increasing lag. Thus, we use a
simple process of correlogram cutting to obtain reliable estimates. Note also that the maximum truncation
point um increases with N.
It is clear from Eqs. (37) and (52) that the undamped periodic tail of the variance for covariance function
estimate is caused by non-stationary characteristics of the underlying signal i.e., the second covariance
components Bc2(u) and Bs2(u). From (37), the stationary contribution in a0(u) depends only on the stationary
approximation of the covariance function B0(u) and has the form
D2
½2ð1 þ 2Sð2aT; N ÞÞ þ ð1 þ cos 2o0 uÞ e2ajuj þ 2S 1 ðaT; N; uÞ
8N
(see Appendix B).
For the non-stationary part, which depends on the higher-order component, we have
að0sÞ ðuÞ ¼
D2 2ajuj
e
þ cos o0 u þ 2S ð2aT; N Þcos 2o0 u þ 2S1 ðaT; N; uÞ.
8N
Obviously a0 ðuÞ ¼ að0sÞ ðuÞ þ að0nÞ ðuÞ and, for u ¼ 0 we obtain
að0nÞ ðuÞ ¼
D2
½1 þ 2Sð2aT; N Þ;
2N
For large u we also have that
að0sÞ ð0Þ ¼
að0nÞ ð0Þ ¼
D2
½1 þ 2Sð2aT; N Þ.
4N
D2
D2
½1 þ 2Sð2aT; N Þ; að0nÞ ðuÞ ¼
½1 þ 2Sð2aT; N Þ cos 2o0 u
4N
8N
Clearly, að0nÞ ð0Þ=að0sÞ ð0Þ ¼ 0:5 and the same limit is reached by the equivalent ratio for large lag values. Thus, the
non-stationarity component plays a significant role in the analysis of estimation reliability. Elimination of the
non-stationary components can lead to significant error for all possible lags.
Finally, we investigate the properties of the covariance components estimates. Biases in these estimates have
the same form as those for the corresponding Fourier components of the covariance function estimate. For the
að0sÞ ðuÞ ¼
0.3
0.2
0.1
60
40
0
0
10
N
20
20
30
u
0
Fig. 4. Variance of the covariance function estimate for o0 ¼ 0.2, a ¼ 0.1, D ¼ 1.
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26
variances, Eqs. (43)–(45) give us the following results:
D2
P0 ð2aT; N Þ þ P~ 0 ðaT; N; uÞ
D B^ 0 ðuÞ ¼
4N
ð1 þ cos 2o0 uÞ þ P2 ð2aT; N Þ þ P~ 2 ðaT; N; uÞ
h c i
D2
P0 ð2aT; N Þ þ P~ 0 ðaT; N; uÞ þ 2 P2 ð2aT; N Þ þ P~ 2 ðaT; N; uÞ
D B^ 2 ðuÞ ¼
4N
ð1 þ cos 2o0 uÞ þ P4 ð2aT; N Þ þ P~ 4 ðaT; N; uÞ
and
h s i
D2
D B^ 2 ðuÞ ¼
P0 ð2aT; N Þ þ P~ 0 ðaT; N; uÞ ð1 cos 2o0 uÞ
4N
þ 2 P2 ð2aT; N Þ þ P~ 2 ðaT; N; uÞ
ð1 þ cos 2o0 uÞ þ P4 ð2aT; N Þ þ P~ 4 ðaT; N; uÞ ,
where
1
P~ l ðaT; N; uÞ ¼
T
Z
NT
1
0
ð53Þ
ð54Þ
ð55Þ
u1 aðju1 þujþju1 ujÞ
cos lo0 u1 du1 :
e
y
Fig. 5 illustrates the cyclic variations in the functions given in Eqs. (53)–(55). These variations are caused by
the periodic non-stationarity of the underlying signal.
For u ¼ 0 the functions P~ l ðaT; N; uÞ and Pl ð2aT; N Þ are equal: i.e., P~ l ðaT; N; 0Þ ¼ Pl ð2aT; N Þ. There are
three distinct components to the expressions in Eqs. (53)–(55). The first does not depend on lag u, the second
component is damped with increasing lag and the third changes periodically with the lag. For the variance of
the zeroth covariance component when u ¼ 0, we have
D2
½2P0 ð2aT; N Þ þ P2 ð2aT; N Þ
D B^ 0 ð0Þ ¼
2N
and for large lags we obtain
D2
½P0 ð2aT; N Þð1 þ cos 2o0 uÞ þ P2 ð2aT; N Þ
(56)
D B^ 0 ðuÞ ¼
4N
since P~ l ðaT; N; uÞ ! 0 as u ! 1.
Cyclic
variations in the function given in (56) have large amplitude and substantially change the variance
D B^ 0 ðuÞ (see Fig. 5). These variations are caused by the presence of periodic stationarity in the signal.
The variance in (41) can be split into two components (following similarly to the procedure shown in
Appendix B): a stationary component;
D2
P0 ð2aT; N Þ þ P~ 0 ðaT; N; uÞ cos 2o0 u þ P2 ð2aT; N Þ þ P~ 2 ðaT; N; uÞ
DðsÞ B^ 0 ðuÞ ¼
4N
and a non-stationary component;
(57)
D2
½P0 ð2aT; N Þ cos 2o0 u þ P~ 0 ðaT; N; uÞ .
DðnÞ B^ 0 ðuÞ ¼
(58)
4N
Expression (57) contains functions that do not depend on the lag u, and also functions, which are damped as
u increases. Expression (58) contains damped and periodic functions. The ratio of these two expressions
defines the influence of signal non-stationarity on the reliability of estimating the covariance function, B0(u).
For zero lag (i.e., u ¼ 0) we get from (51) that
D2
½P0 ð2aT; N Þ þ P2 ð2aT; N Þ,
DðsÞ B^ 0 ð0Þ ¼
2N
D2
P0 ð2aT; N Þ.
DðnÞ B^ 0 ð0Þ ¼
2N
(59)
(60)
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27
0.15
0.1
0
10
0.05
20
0
0 2 4
6 8
10
N
(a)
30
40
u
50
0.2
0
0.1
10
20
0
0 2
4 6 8
10
N
(b)
30
40
u
50
0.1
0
0.05
10
20
0
0 2 4
6 8 10
N
(c)
30
40
u
50
h s i
h c i
Fig. 5. Variance of covariance components estimates: (a) D B^ 0 ðuÞ ; (b) D B^ 2 ðuÞ ; (c) D B^ 2 ðuÞ ) for o0 ¼ 0.2, a ¼ 0.1, D ¼ 1.
Thus, DðnÞ B^ 0 ð0Þ =DðsÞ B^ 0 ð0Þ X0:5 and the elimination of signal non-stationarity leads to significant error
in calculating the variance of our estimate. For large u, the stationary component is independent of lag, i.e.,
D2
½P0 ð2aT; N Þ þ P2 ð2aT; N Þ,
DðsÞ B^ 0 ðuÞ ¼
4N
(61)
whereas the non-stationary component is a cyclical function of lag i.e.,
D2
DðnÞ B^ 0 ðuÞ ¼
P0 ð2aT; N Þ cos 2o0 u.
4N
(62)
The value of (61) is half as great as the starting value DðsÞ B^ 0 ð0Þ given in (59) and the
of the cyclical
amplitude
oscillations of the non-stationary part in (62) is also half as great as that for DðnÞ B^ 0 ð0Þ given in (60). Thus,
the signal non-stationarity significantly changes the variance of the estimate and also its as we change the lag,
u.
It also follows from expressions (54)–(55), that the zeroth and second-order covariance components
define the properties of the variances of the second covariance component estimates. For zero lag (i.e., u ¼ 0)
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28
we have
h c i D2
½2P0 ð2aT; N Þ þ 4P2 ð2aT; N Þ þ P4 ð2aT; N Þ,
D B^ 2 ð0Þ ¼
2N
(63)
h s i D2
½2P2 ð2aT; N Þ þ P4 ð2aT; N Þ
D B^ 2 ð0Þ ¼
2N
(64)
h c i D2
½½P0 ð2aT; N Þ þ 2P2 ð2aT; N Þð1 þ cos 2o0 uÞ þ P4 ð2aT; N Þ,
D B^ 2 ðuÞ ¼
4N
(65)
h s i D2
½P0 ð2aT; N Þð1 cos 2o0 uÞ þ 2P2 ð2aT; N Þð1 þ cos 2o0 tÞ þ P4 ð2aT; N Þ.
D B^ 2 ðuÞ ¼
4N
(66)
and for large lags, we have
Variances (65) and (66) decrease by a factor of two relative to the initial values of the constant part given in
(63) and the cyclic part in (64), respectively.
h s i
j c k
As the lag increases, the rate of decrease in the variances D B^ 0 ðuÞ , D B^ 2 ðuÞ and D B^ 2 ðuÞ is less than the
corresponding decreasing rate for the covariance components B0 ðuÞ, Bc2 ðuÞ and Bs2 ðuÞ. This means that as the
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ffi
lag increases, so do the corresponding relative mean-squares errors
D B^ 0 ðuÞ =B0 ðuÞ and
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
h c;s i
D B^ 2 ðuÞ =Bc;s
c ðuÞ . As before this limits the consistency of estimating the covariance component for lag
values beyon0d the interval [0,um], though we note that the maximum lag value um grows as N increases.
6. Conclusions
One of the most significant tasks in the statistical analysis framework is the analysis of the reliability of
estimators for quantitative indices of interest. Here, we investigate this task for the analysis of periodic
stationary random signals. Obtaining quantitative indices makes it possible to justify the choice of process
parameters. This is necessary for avoiding errors, understanding the model and for model verification. In this
paper, the reliability of the so-called coherent process is investigated in terms of biases and variances of the
non-stationary mean, covariance function and the corresponding Fourier components. The relationships we
obtain show a dependence of both bias and variance on realization length and on the signal covariance
components. On the basis of formulae for the covariance components for the partial PCRP cases, it is possible
to investigate in detail the properties of these estimates and to obtain relationships between parameters of the
stationary processes, which form the PCRP. The variances of the covariance characteristic estimates take the
form of undamped oscillations as the lag increases. The amplitude of these oscillations is defined by
characteristics of the signal non-stationarity and their average values are defined by the characteristics of the
signal stationary approach. Such behaviour in the variance as the lag changes indicates that the reliable
estimation of the covariance characteristics is possible only for lags within the range [0,um]. Within this
interval, the variances of the estimates take the form of damped oscillations. Specific value for um can be
calculated by means of the formulae we derive here.
Appendix A
Proof of consistency of the component estimates given in Eqs. (12).
^ 0 ¼ E ½m
^ 0 Em
^ 0 2 ¼ E ½m
^ 0 m0 2
D½m
Z y
2
1
¼E
xðtÞ dt
y 0
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29
Z Z
1 y y
E xðtÞxðsÞ ds dt
y2 0 0
Z Z
Z Z
1 y y
1 y yt
¼ 2
bðt; s tÞ ds dt ¼ 2
bðt; uÞ du dt.
y 0 0
y 0 t
¼
After changing the order of integration, we have
^ 0 ¼
D½m
1
y2
Z
0
y
Z
y
bðt; uÞ dt du þ
u
Z
y
0
Z
yu
0
bðt; uÞ dt du .
Changing u for u in the first term and using the covariance function property bðt; uÞ ¼ bðt u; uÞ, we obtain
1
^ 0 ¼ 2
D½m
y
Z y
Z
y
bðt u; uÞ dt þ
u
0
Z
yu
bðt; uÞ dt du.
0
Substituting here s for tu we receive
Z
Z yu
Z yu
1 y
^ 0 ¼ 2
D½m
bðt; uÞ dt du
bðs; uÞ ds þ
y 0
0
0
Z Z
2 y yu
¼ 2
bðt; uÞ dt du.
y 0 0
Substituting here the Fourier decomposition of covariance function b(t,u), we have
^ 0 ¼
D½m
2
y2
Z
y
0
X
Bk ðuÞ
k2Z
Z
yu
0
eiko0 t dt du:
If we define the function
1
f l ð0; y uÞ ¼
y
Z
yu
e
ilo0 t
dt ¼
0
(
1 uy ;
ilo ðyuÞ
i
0
1 ;
lo0 y e
l ¼ 0;
la0:
then f l ð0; y uÞ ¼ f l ð0; y uÞ and the even and odd parts of f l ð0; y uÞ have the following representation
for l6¼0:
f cl ð0; y uÞ ¼
sin lo0 ðy uÞ
;
lo0 y
f sl ð0; y uÞ ¼
cos lo0 ðy uÞ 1
.
lo0 y
Thus, we have
#
Z y "
1
1
X
X
u
Bl ðuÞf l ð0; y uÞ du
Bl ðuÞf l ð0; y uÞ þ
1 B0 ðuÞ þ
y
0
l¼1
l¼1
#
Z "
X
2 y
u
1 B0 ðuÞ þ
¼
Bl ðuÞf l ð0; y uÞ þ Bl ðuÞf l ð0; y uÞ du.
y 0
y
l2N
2
^ 0 ¼
D½m
y
Recall that Bl ðuÞ ¼ 12 Bcl ðuÞ iBsl ðuÞ . After substituting into the previous formula and cancelling, we obtain
#
Z "
X
1 y
u
c
s
c
s
^ 0 ¼
D½m
1 B0 ðuÞ þ
Bl ðuÞf l ð0; y uÞ þ Bl ðuÞf l ð0; y uÞ du.
y 0
y
l2N
c;s
^ l is similar.
The proof for D m
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30
Appendix B
From (37) we have
B~ 0 ðu1 ; uÞ ¼
X
q¼2;0;2
eiqo0 u Bq ðu1 þ uÞBq ðu1 uÞ þ Bq ðu1 ÞBq ðu1 Þ .
The ‘‘stationary part’’ of this equation depends only on zero covariance component and ‘‘non-stationary
part’’ depends on covariance components B2 ðuÞ and B2 ðuÞ ¼ B2 ðuÞ:
eðsÞ ðu1 ; uÞ ¼ B0 ðu1 þ uÞB0 ðu1 ; uÞ þ B0 ðu1 ÞB0 ðu1 Þ.
B
0
eðnÞ ðu1 ; uÞ ¼ e2io0 u B2 ðu1 þ uÞB2 ðu1 uÞ þ B2 ðu1 ÞB2 ðu1 Þ
B
0
þ e2io0 u B2 ðu1 þ uÞB2 ðu1 uÞ þ B2 ðu1 ÞB2 ðu1 Þ .
Taking into account (48) we obtain
eðsÞ ðu1 ; uÞ ¼ 1 RZ ðu1 þ uÞ
B
0
4
1
RZ ðu1 uÞ cos o0 ðu1 uÞ cos o0 ðu1 þ uÞ þ R2Z ðu1 Þcos2 o0 u1 .
4
i
2io0 u h
2io0 u
2
eðnÞ ðu1 ; uÞ ¼ e
B
R
ð
u
þ
u
ÞR
ð
u
u
Þe
þ
R
ð
u
Þ
Z
1
Z
1
1
0
Z
16
i
2io0 u
e
þ
RZ ðu1 þ uÞRZ ðu1 uÞe2io0 u þ R2Z ðu1 Þ ,
16
h
i
eðnÞ ðu1 ; uÞ ¼ 1 RZ ðu1 þ uÞRZ ðu1 uÞ þ R2 ðu1 Þ cos 2o0 u1 .
B
0
Z
8
Suppose that we set for the stationary covariance function RZ ðuÞ ¼ Deajuj , then we have
2
eðsÞ ðu1 ; uÞ ¼ D eaðju1 þujþju1 ujÞ cos o0 ðu1 þ uÞcos o0 ðu1 uÞ þ e2aju1 j cos2 o0 u1
B
0
4
D2 aðju1 þujþju1 ujÞ
e
ðcos 2o0 u þ cos 2o0 u1 Þ
¼
8
þ e2aju1 j ð1 þ cos 2o0 u1 Þ ,
D2 aðju1 þujþju1 ujÞ
ðn Þ
B~ 0 ðu1 ; uÞ ¼
e
þ e2aju1 j cos 2o0 u1 .
8
From (26) we have
"
#
N
1
X
1 e
n e
B0 ð0; uÞ þ 2
1
B0 ðnT; uÞ .
a0 ð uÞ ¼
N
N
n¼1
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31
Taking into account above expressions we obtain
"
D2 2ajuj
ðsÞ
e
ðcos 2o0 u þ 1Þ
a0 ð uÞ ¼
8N
þ2þ2
n¼1
að0nÞ ðuÞ ¼
þ2
D2 2ajnT j
e
þ cos 2o0 u
8N
N
1
X
n¼1
#
n aðjnTþujþjntujÞ
1
ðcos 2o0 u þ 1Þ þ 2e2ajnT j .
e
N
"
N
1
X
#
n 2ajuj
aðjnTþujþjntujÞ
1
cos 2o0 u þ e
e
.
N
And finally,
D2
2ð1 þ 2Sð2aT; N ÞÞ þ e2ajuj þ 2S1 ðaT; N; uÞ ð1 þ cos 2o0 uÞ ,
8N
D2 2ajuj
e
að0nÞ ðuÞ ¼
þ cos o0 u þ 2S ð2aT; N Þ
8N
cos 2o0 u þ 2S1 ðaT; N; uÞ ,
að0sÞ ðuÞ ¼
where
SðaT; N Þ ¼
N1
X
1
n¼1
S1 ðaT; N; uÞ ¼
N
1
X
n aTn
e
,
N
1
n¼1
n aðjuþnT jþjunT jÞ
e
.
N
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