coherent estimates of correlation characteristics of interconnected periodically correlated random...

Coherent Estimates of Correlation Characteristics

of Interconnected Periodically Correlated

Random Processes

I. N. Yavorskyj,1,2

R. Yuzefovych,1

I. Y. Matsko,1

and Z. Zakrzewski2

1Karpenko Physico-Mechanical Institute of NASU, Lviv, Ukraine

2Institute of Telecommunications of The University of Technology and Life Sciences (UTP), Bydgoszcz, Poland

Received in final form July 17, 2012

Abstract—Brief analysis of correlation and spectral characteristics that describe interconnections

between two periodically correlated random processes is conducted. Properties of coherent estimates of

signals’ mutual correlation function and estimates of mutual correlation components are studied.

Expressions for biases and dispersions of estimates are specified for amplitude modulated signals.

DOI: 10.3103/S0735272712090038

INTRODUCTION

Use of mathematical model of signals in the form of periodically correlated random processes (PCRP)

leads to more efficient solutions of problems connected with their conversion and processing [1–3]. During

analysis of both single- and multi-channel data transmission systems, identification of propagation paths of

random signals, determination of way of finding their sources we come to the necessity of analyzing of

signals interconnection. In this paper within the boundaries of second order theory properties of mutual

characteristics of two periodically correlated random processes are briefly described, and analysis of

coherent estimates of their mutual correlation characteristics is presented.

Expected values of PCRP are given by m t E t� �( ) ( )� and m t E t� �( ) ( )� , E denotes averaging with

respect to distribution and their autocorrelation functions

b t u E t t u� � �( , ) ( ) ( )� ��

, b t u E t t u� � �( , ) ( ) ( )� ��

,

� � � ��

�

( ) ( ( )t m t� � , � � � ��

�

( ) ( ( )t m t� �

are periodical functions of time

m t m t T� �( ) ( )� � , m t m t T� �( ) ( )� � ,

b t u b t T u� �( , ) ( , )� � , b t u b t T u� �( , ) ( , )� �

and in the case of absolute integrability on the interval [ , ]0 T

| ( )|m t t

T

� d

0

�, | ( )|m t t

T

� d

0

�,

405

ISSN 0735-2727, Radioelectronics and Communications Systems, 2012, Vol. 55, No. 9, pp. 405–417. © Allerton Press, Inc., 2012.

Original Russian Text © I.N. Yavorskyj, R. Yuzefovych, I.Y. Matsko, Z. Zakrzewski, 2012, published in Izv. Vyssh. Uchebn. Zaved., Radioelektron., 2012, Vol. 55,

No. 9, pp. 26–36.

� �b t u t

T

�( , ) d

0

�, � �b t u t

T

�( , ) d

0

�

u��

may be represented in the form of Fourier series:

m t m ek

k t

k

��

( )( )

�

�

�i 0

�

, m t m ek

k t

k

��

( )( )

�

�

�i 0

�

,

b t u B u ek

k t

k

��

( , ) ( )( )

�

�

�i 0

�

, b t u B u ek

k t

k

��

( , ) ( )( )

�

�

�i 0

�

.

Zeroth correlation components B u0

( )( )

�and B u

0

( )( )

�and their Fourier transform—zeroth spectral

components—describe properties of stationary approximations of PCRP. Structure of stochastic

repeatability of signals is described by correlation and spectral components of higher orders. Similar values

also may be used during analysis of stochastic interconnection of two PCRP.

PROPERTIES OF CORRELATION AND SPECTRAL CHARACTERISTICS

Let’s assume that mutual correlation function b t u E t t u�� ( , ) ( ) ( )� ��

also periodically changes in

time: b t T u b t u�� ( , ) ( , )� � . Random processes for which this condition holds true are called connected

PCRP. Then

b t u B u ek

k t

k

��

( , ) ( )( )

�

�

�i 0

�

, (1)

where B u

T

b t u e tk

k tT

( )( ) ( , )

��

��

�

10

0

id .

Since

b t u��( , ) � � � � � � �E t t u E t u t b t u u� � � � ��

� � � �

( ) ( ) ( ) ( ) ( , ),

then

B u e B u ek

k t

k Z

k

k t u

k Z

( ) ( ) ( )( ) ( )

�� i i0 0

�

�

�

� �� .

Hence for mutual correlation components we have

B u B u ek k

k u( ) ( )( ) ( )

��

�i 0 . (2)

For zeroth correlation component

B u B u0 0

( ) ( )( ) ( )

�� .

RADIOELECTRONICS AND COMMUNICATIONS SYSTEMS Vol. 55 No. 9 2012

406 YAVORSKYJ et al.

Signals �( )t and �( )t may be represented in the form of series [4, 5]:

� ��

( ) ( )t t ek

k t

k Z

�

�

�i 0 ,

� ��

( ) ( )t t ek

k t

k Z

�

�

�i 0 ,

where� k t( )and �k t( )are stationary connected random processes. Based on these representations we obtain

b t u e R u ek t

k Z

q k q

q u

q Z

��

( , ) ( ),

( )�

��

�

� �i i0 0 , (3)

where R E t t upq p q

( )( ) ( )

��

, � ��

� �

p p pt t m( ) ( )( )

� � , � ��

� �

q q qt t m( ) ( )( )

� � , “–” denotes the conjugation

operand. Comparing (1) and (3), we conclude that

B u R u ek q k q

q u

q Z

( )

,

( )( ) ( )

��

��

�i 0 . (4)

Random processes � k t( ) and �k t( ), which describe amplitude and phase modulation of harmonic

components are directly connected with information on signal sources, hence based on mutual correlation

function R upq

( )( )

��one may judge about connections between them. If there are no correlation connections

between stationary components � p t( ) and �q t( ), i. å. R u R upq pp pq

( ) ( )( ) ( )

�� , � pq is Kronecker symbol,

then all mutual correlation components except for zeroth equal zero.

Mutual correlation function in this case does not change it time, which means that two PCRP are

stationary connected. Obviously when R upp

( )( )

��0 signals will be non-correlated.

Let’s assume that function b t u��( , ) t T� [ , ]0 is absolutely integrable

� �b t u u��( , ) d

��

�

�.

Then for each t there exists a Fourier transform

f t b t u e uu

��

��

( , ) ( , )��

��

�

1

2

id , (5)

which we’ll call mutual spectral density of signals �( )t and �( )t . This quantity is complex:

f t f t f t�� ( , ) Re ( , ) Im ( , )� � i .

Let’s represent b t�� ( , ) as a sum of even and odd components:

b t u b t u b t u�� ( , ) ( , ) ( , )� ��

. (6)

Obviously,


COHERENT ESTIMATES OF CORRELATION CHARACTERISTICS 407

� �b t u b t u b t u u��

� � �( , ) ( , ) ( , )1

2,

� �b t u b t u b t u u��

� � �( , ) ( , ) ( , )1

2.

Considering (6), we obtain

Re ( , ) ( , )cosf t b t u u u��

��

�

1

0

d , (7)

Im ( , ) ( , )sinf t b t u u u��

��

�

1

0

d . (8)

The real part of mutual spectral density is a cosine transform of even component of mutual correlation

function, while the imaginary part is given by a sine transform of odd component.

As follows from formulas (7) and (8),

Re ( , ) Re ( , )f t f t�� ,

Im ( , ) Im ( , )f t f t�� ,

i. å. f t f t�� ( , ) ( , )� � � . Considering the last equalities we obtain:

b t u f t u��

�

� ( , ) Re ( , )cos2

0

d ,

b t u f t u��

�

� ( , ) Im ( , )sin2

0

d .

The value of mutual correlation function when u �0is determined by integration in the frequency domain

of mutual spectral density:

b t f t�� ( , ) Re ( , )0 2

0

�

�

d .

Substituting series (1) into (5) yields

f t f ek

k t

k Z

��

� �( , ) ( )( )

�

�

�i 0 ,

where

f B u e uk k

u( ) ( )( ) ( )

��

��

�

��

�

1

2

id . (9)



Quantities fk

( )( )

�� , which are Fourier coefficients of mutual spectral density are called mutual spectral

components. Zeroth spectral component determines average with respect to time value of instantaneous

spectral density.

Since B u B uk k�

�( ) ( )

( ) ( )��

, then

f B u e u fk k

u

k�

�

��

�

� � �( ) ( ) ( )

( ) ( ) ( )��

��

�1

2

id .

Considering (2), we obtain

f B u e uk k

u( ) ( )( ) ( )

��

��

�

��

�

1

2

id

� � ��

��

�

1

2

0

0�

� ��

B u e u f kk

k u

k

( ) ( ) ( )( ) ( )

id .

Thus

f f f kk k�

� � � �( ) ( ) ( )

( ) ( ) ( )��

� � � �0 ,

specifically, f f f0 0 0

( ) ( ) ( )( ) ( ) ( )

�� .

Substituting expression (4) into (9) yields

f f qk q k q

k Z

( )

,

( )( ) ( )

��

��

� 0 ,

where f pq

( )( )

�� denotes mutual spectral densities of stationary components of processes �k t( ):

f R u e upq pq

u( ) ( )( ) ( )

��

��

�

��

�

1

2

id .

Zeroth mutual spectral component is determined by mutual spectral densities of stationary modulating

processes with the equal numbers, while kth mutual spectral component is determined by mutual spectral

densities of modulating processes whose numbers are shifted by k.

ESTIMATION OF MUTUAL CORRELATION FUNCTION

Coherent [4] and component [5] methods, least squares method [6] and linear filtering methods [7, 8]

may be used for estimating characteristics of PCRP. During initial study of signals’ structure based on

experimental measurements it is reasonable to use coherent method, which requires knowledge only of

period T. If this value is unknown, it may be determined using procedures specifically developed for this

purpose [9].

During coherent estimation of mutual correlation function the following statistics may be used:



� �� ( , ) ( ) � ( )b t u

N

t nT m t nT

n

N

��

�

�

�1

0

1

� �� ( ) � ( )t u nT m t u nT� � � � � , (10)

� ( , )b t u��

�

�

�1

0

1

N

t nT t u nT m t m t u

n

N

� � � �( ) ( ) � ( ) � ( ), (11)

� ( , )b t u��

�

�

�1

0

1

N

t nT t u nT m t nT m t u nT

n

N

� � � �( ) ( ) � ( ) � ( ) , (12)

where

� ( ) ( )m t

N

t nT

n

N

� ��

�

�

�1

0

1

,

� ( ) ( )m t

N

t nT

n

N

� ��

�

�

�1

0

1

, (13)

and N denotes number of averaging periods. Formulas (11), (12) differ in the ways of centering by the

estimated values.

Let’s prove validity of the statement:

THEOREM 1

Estimates (10)–(12) when the condition

lim ( , )| |u

b t u��

�� 0 , t T� [ , ]0 (14)

is satisfied are asymptotically unbiased estimates of mutual correlation function of connected PCRP.

To prove the theorem let’s first biases of estimates �� ( , ) � ( , ) ( , )b t u Eb t u b t u� � . After transforms

for estimate (10) considering (13) we obtain

�� ( , )b t u � � � �

�

��

�

��

��

�

�1 1

1

1

N

b t u

N

n b t u nT

n N

N

�� ( , ) | | ( , ) , (15)

while for estimates (11), (12) we get

�� ( , )

| |( , )b t u

N

n

N

b t u nT

n N

N

� � ��

�

!

"# �

��

�

�1

1

1

1

. (16)

As follows from expressions (14), (15) during fading of correlation connections with increasing offset,

i.e. when the boundary condition (14) is true �� ( , )b t u � 0 if N � 0, for all statistics. The theorem is

proved.

Quantities (15) and (16) in the first approximation differ in the fact that bias of estimates (11) and (12)

contains additional components that are determined by the values of mutual correlation function for offsets

u u nTn � � . Higher infinitesimal order component have equals absolute value, but have opposite signs. If



signals’ correlation connections quickly converge to zero on the interval shorter than period T, then the bias

values (15) and (16) will be very close.

When calculating dispersions of estimates (10)–(12) we’ll assume that mutual probability density of

signals �( )t and �( )t is Gaussian. In this case central moments of third order equal zero, while fourth order

moments are determined by correlation functions of all possible couples:

E t t t t� � � ��

( ) ( ) ( ) ( )1 2 3 4 � � �b t t t b t t t�� ( , ) ( , )1 2 1 3 4 3

� � �b t t t b t t t� �( , ) ( , )1 3 1 2 4 2 � � �b t t t b t t t�� ( , ) ( , )1 4 1 2 3 2 .

In the first approximation for estimates (10)–(12) we obtain

� � $ �D b t u

N

b t m n T u

m n

N� ( , ) ,( ) ,

,

�� %� �

�

�

�1

20

1

, (17)

where b t s t u%( , , )� is correlation function of process % � �( , ) ( ) ( )t u t t u� ��

b t s t u E t t u b t u s s% �� ( , , ) ( ) ( ) ( , ) ( ) (� � � ��

��

�

��

� � � �

u b s u) ( , )��

��

�

��

� � � � � � � � �b t s t b t u s t b t s t u b s t s u� � �� ( , ) ( , ) ( , ) ( , ).

This function is determined by products of auto- and mutual correlation functions of signals�( )t and �( )t ,

hence if s t u� � 1 changes periodically in time t and may be represented by Fourier series

b t u u B u u er

r t

r Z

%�

( , , )~

( , )1 10�

�

�i

,

where

~( , ) ( ) ( ) (

( ) ( ) ( )B u u B u B u e B ur q r q

q u

q r1 1 1 10� � ��

��

� � � ��iu B u u eq

q u

q

) ( )( )��

��

��

��

�

� 10 1i

�

. (18)

Since b t s t u b s t s u% %( , , ) ( , , )� � � , then

~( , )

~( , )B u u B u u er r

r u� �

�1 1

0 1i �. (19)

Let’s represent dispersion (17) by

� �D b t u� ( , )�� & & � & �0 0 0( ) [ ( )cos ( )sin ]u u l t u l tl

c

l

s

l

� �

�

��

, (20)

Considering expression (19), derivations for coefficients of the series (20) yield:

& 0 0 0

1

11

0 2 1( )~

( , )~

( , )u

N

B un

N

B nT u

n

N

� � ��

�

!

"#

�

��

�

��

�

�

�n

N

�

�

�0

1

, (21)



& l

c

l

c

l

c

n

N

u

N

B un

N

B nT u( )~

( , )~

( , )� � ��

�

!

"#

�

��

�

�

�1

0 2 1

1

1 �

��

�

�

�n

N

0

1

, (22)

& l

s

l

s

l

s

n

N

u

N

B un

N

B nT u( )~

( , )~

( , )� � ��

�

!

"#

�

��

�

�

�1

0 2 1

1

1 �

��

�

�

�n

N

0

1

, (23)

while � �~( , )

~( , )

~( , )B u u B u u B u ul l

c

l

s

1 1 1

1

2� � i . As follows from condition (14) and expression (18),

coefficients~

( , )B u u0 1 and~

( , ),

B u ul

c s

1 fade infinitely when u1 � �. This leads to satisfying the boundary

equalities

lim ( )N

u��

�& 0 0, lim ( ),

Nl

c su

��& 0, � �lim � ( , )

N

D b t u��

�� 0.

Based on the considerations given above we derive the following theorem.

THEOREM 2

For connected Gaussian PCRP, if condition (14) is satisfied, estimates (10)–(12) are consistent and their

dispersions in the first approximation are given by expressions (20)–(23).

ESTIMATES OF MUTUAL CORRELATION COMPONENTS

Let’s represent mutual correlation components B uk

( )( )

��, k ' 0 by

� �B u C u S uk k k

( ) ( ) ( )( ) ( ) ( )

��

1

2,

and form statistics

� ( ) � ( , )( )

B u

T

b t u t

T

0

0

1�� d , (24)

� ( ) � ( , )cos( )

C u

T

b t u k t tk

T��

�� 2

0

0

d ,

� ( ) � ( , )sin( )

S u

T

b t u k t tk

T��

�� 2

0

0

d . (25)

When estimating correlation function further we’ll assume that expected value estimates are determined

only for t T� [ , ]0 , while for other instants in time we assume � ( ) � ( )m t nT m t� �� , � ( ) � ( )m t nT m t� �� , n Z� .

In this case estimates (10)–(12) coincide and are given by (11).

Biases of estimates (24) and (25) are determined from expressions

� ��

� ( )| |

( )( ) ( )

B u

N

n

N

B u nT

n N

N

0 0

1

11

1� � ��

�

!

"# �

��

�

� , (26)



� ��

� ( )| |

( )( ) ( )

C u

N

n

N

C u nTk k

n N

N

� � ��

�

!

"# �

��

�

�1

1

1

1

, (27)

� ��

� ( )| |

( )( ) ( )

S u

N

n

N

S u nTk k

n N

N

� � ��

�

!

"# �

��

�

�1

1

1

1

. (28)

Obviously, they depend only of quantities to be estimated. If condition (14) is satisfied � ��

� ( )( )

B u0

0� ,

� ��

� ( )( )

C uk

� 0, � ��

� ( )( )

S uk

� 0, when N � �.

For dispersions of estimates (24) and (25) when speaking of Gaussian PCRP we obtain

� �D B u b t s t u s t� ( ) ( , , )( )

0 2

00

1��%

((

(� � d d ,

� �D C uk

� ( )( )��

� �4

20 0

00(

� �%

((

b t s t u k t k s t s( , , )cos cos d d ,

� �D S uk

� ( )( )��

� �4

20 0

00(

� �%

((

b t s t u k t k s t s( , , )sin sin d d ,

where( � NT. After making a substitution u s t1 � � , changing integration order and transforms considering

representation (18) in the first approximation we have

� �D B uu

B u u u� ( )~

( , )( )

0

10 1 1

0

21

��(

( (� �

�

�

!

"# d , (29)

� �D C uk

� ( )( )��

�

�

!

"#

21 2

10 1 0 1

0( (

�

(u

B u u k u~

( , )cos

�� ~

( , )cos~

( , )sinB u u k u B u u k u uk

c

k

s

2 1 0 1 2 1 0 1 1� � d , (30)

� � �D S uu

B u u k uk

� ( )~

( , )cos( )��

(

( (��

�

�

!

"#

21 2

10 1 0 1

0

�� ~

( , )cos~

( , )sinB u u k u B u u k u uk

c

k

s

2 1 0 1 2 1 0 1 1� � d . (31)

As follows from expressions (29)–(31), given that (14) is true dispersions of estimates (24) and (25)

decrease infinitely with increasing length of realization interval (.

Thus, we conclude the following theorem.



THEOREM 3

For Gaussian connected PCRP estimates of mutual correlation components (24) and (25) provided that

conditions (14) are satisfied are asymptotically unbiased and consistent, while their biases and dispersion in

the first approximation are given by expressions (26)–(28) and (29)–(31) respectively.

As follows from (14) zeroth component~

( , )B u u0 1 is a function of all correlation components of the

analyzed signals, hence calculation of estimation error for zeroth mutual correlation component conducted

with standard time averaging cannot be performed without accounting for higher correlation components.

Dispersions of estimates of cosine and sine mutual correlation components depend on both zeroth and higher

order components of function b t u u�( , , )1 . For dispersion of estimate

� �� ( ) � ( ) � ( )( ) ( ) ( )

B u C u S uk k k

��

1

2,

considering (30) and (31) we obtain

� �D B uk

� ( )( )��

��

1

4D C u D S u

k k

� ( ) � ( )( ) ( )��

� ��

�

!

"#

21

10 1 0 1 1

0( (

�

(u

B u u k u u~

( , )cos d .

Obviously, this quantity depends only on zeroth correlation component of process � �$ ��$ �( , )t u t t u� ��

.

Dispersions of higher order component estimates differ in the oscillation frequency of the weight function.

MUTUAL CORRELATION ANALYSIS OF AMPLITUDE-MODULATED SIGNALS

Amplitude-modulated signals� � �( ) ( )cost t t� 0 , � ) �( ) ( )cost t t� 0 , where�( )t and )( )t denote stationary

connected processes, are simplest special cases of PCRP. Their expected values and correlation functions

are given by

m t m t� � �( ) cos� 0 , m t m t� ) �( ) cos� 0 ,

b t u B u C u t S u� ��

�,

( , ) ( , ) ( , )( , ) ( ) ( )cos ( )s� � �

0 2 0 22 in2 0� t,

b t u B u C u t S u��

� �( , ) ( ) ( )cos ( )sin( ) ( ) ( )

� � �0 2 0 2

2 2 0t,

where

B u C u R u u0 2 0

1

2

( , ) ( , )

,( ) ( ) ( )cos� � � �

� ) �� , S u R u u2 0

1

2

( , )

,( ) ( )sin� �

� ) �� ,

B u C u R u u0 2 0

1

2

( ) ( )( ) ( ) ( )cos

�� ) �� , S u R u u

2 0

1

2

( )( ) ( )sin

��) �� ,

considering

m E t� �� ( ), m E t) )� ( ),

R u E t t u� � �( ) ( ) ( )� ��

, R u E t t u) ) )( ) ( ) ( )� ��

, R u E t t u�) � )( ) ( ) ( )� ��

,



� � �

�

( ) ( )t t m� � , ) ) )

�

( ) ( )t t m� � .

Bias of mutual correlation function’s estimate may be represented by

� �� ( , )b t u �� 0 2 0 2 02 2( ) ( )cos ( )sinu u t u t

c s� � .

If R u Deu

�)&

( )| |

��

, then

� � & �0 2 0 02

( ) ( ) ( , , )cosu uD

N

S N u uc

� � � ,

� & �2 0 02

su

D

N

S N u u( ) ( , , )sin� � , (32)

where S N un

N

eu nT

n N

N

0

1

1

1( , , )| | | |

&&

� ��

�

!

"#

� �

��

�

� .

Function S N u0( , , )& decreases with increasing offset u, but its decrease rate is less then the fading rate of

correlation components when the offset grows. Hence relative bias| [ � ( , )]|

| ( , )|

� ��

��

b t u

b t u

will increase with

increasing offset. Thus, mutual correlation function may be calculated with required precision only for some

limited finite interval [ , ]0 um .

Dispersion of mutual correlation function’s estimate is given by

� �D b t u� ( , )�� & & � & �0 0 0

2 4

( ) [ ( )cos ( )sin ]

,

u u l t u l tl

c

l

s

l

� �

�

� . (33)

Coefficients & 0( )u and &l

c su

,( ), if R u D e

u

�&

( )| |

��

11 , R u D e

u

)&

( )| |

��

22 are given by

& �0 0

1

82 2( ) ( , )( cos )u

N

M N u u� � , (34)

& �2 0

1

41 2

cu

N

M N u u( ) ( , )( cos )� � , & �2 0

1

42

su

N

M N u u( ) ( , )sin� � , (35)

& �4 0

1

82

cu

N

M N u u( ) ( , )cos� , & �4 0

1

82

su

N

M N u u( ) ( , )sin� � , (36)

where

� � � �M N u D D S N D e S N uu

( , ) ( , )~

( , , )| |

� � � � ��

1 2 1 2

21 2 2& & &

&,

S Nn

N

enT

n

N

( , )| | ( )

& && &

1 2

1

1

1 1 2� � ��

�

!

"#

� �

�

�

� ,



$ �~( , , )

| | | | | |S N u

n

N

eu nT u nT

n

N

&&

� ��

�

!

"#

� � � �

�

�

� 1

1

1

.

Coefficients (34)–(36) for initial offsets are represented by fading oscillations. If u � �, then~

( , , )S N u& � 0, which means that for large offsets dispersive properties are determined only by

autocorrelation functions of signals �( )t and �( )t . Then coefficients (34)–(36) are periodic functions. Thus,

dispersion D b t u[ � ( , )]�� does not converge to zero with increasing offset, but rather is given by non-fading

oscillations. Amplitude of these oscillations decreases if the number of averaging periods N increases. This

important property of dispersion (33) should be accounted for during signal processing. Since correlation

connections of signals fade to zero, their root-mean-square error| [ � ( , )]|

| ( , )|

/D b t u

b t u

��

��

1 2

rapidly grows with

increasing offset, which leads to necessity of truncating the correlation curve.

Expressions for estimate biases of mutual correlation components coincide with those that determine the

corresponding Fourier coefficients of mutual correlation function bias, i. e. (32). The dispersion of zeroth

mutual correlation component considering the approximation given above is represented by

� � � ��D B u D D r u r� ( ) ( )( cos ) ( )( )

0 1 2 0 1 2 0 2 1 2

1

41 2

��& & � & &� � � � �

� �� D r u u r u2

0 0 21 2~

( , )( cos )~

( , )& � & , (37)

where

r

T

ue l u ul

uNT

( ) cos&(

�&

� ��

�

!

"#

�

11 0

0

d ,

~( , )r ul & � �

�

�

!

"#

� � � �

11 1 1

0

0T

ue l u u

u u u uNT

(�

& (| | | |)cos d .

From formula (37) it is possible to extract a component, which is independent of offset and is given by an

oscillating function of offset, as well as a component that decreases with increasing offset. The forth is

determined only by autocorrelation functions of signals, while the latter depends on their mutual correlation

function. In this case the constant component is caused by only stationary signals’ approximations, while the

variable one is a consequence of periodic non-stationarity. These two components determine behavior of

dispersion (37) for large offsets.

Estimate dispersions of second mutual correlation components possess similar properties. Thus, periodic

non-stationarity of signals significantly changes not only the value of dispersion, but their behavior with

increasing offset as well. This again point out importance of choosing signal’s model during systematic

analysis.

CONCLUSIONS

Stochastic dependence of two PCRP in second order theory is described by mutual correlation function,

mutual spectral density and their Fourier coefficients representing mutual correlation and spectral

components. Harmonic signal representation plays an important role when finding the connection structure.

Mutual characteristics of signals may be calculated based on experimental data using averaging of

samples chosen every second period—the so-called coherent method. Coherent estimates of mutual

correlation function as well as mutual correlation components are asymptotically unbiased and consistent if

correlation properties of signals fade with increasing offset.



Expressions obtained in the paper for biases and estimate dispersions provide a possibility to conduct

analysis of statistical estimation error dependence on realization interval length and on signal parameters.

These quantities serve as a theoretical basis for well-grounded choice of processing parameters.

Particularization of expressions for amplitude-modulated signals reveals significant influence of signals’

non-stationarity characteristics on estimation precision.

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