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FOREIGN AEROSPACE SCIENCE ANDTECHNOLOGY CENTER

SPECTRAL PROPERTIES OF PERIODICALLYCORRELATED RANDOM PROCESSES

by

Ya. P. Dragan

DTICS ELECTE

FEB 1 6 1993D

bk S E93-02719

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HUMAN TRANSLATION

FASTC-ID(RS)T-0560-92 27 January 1993

SPECTRAL PROPERTIES OF PERIODICALLY CORRELATEDRANDOM PROCESSES

By: Ya. P. Dragan

English pages: 13

Source: Otbor i Peredacha Informatsii, Nr. 30, 1971;

pp. 16-24

Country of origin: USSRTranslated by: Charles T. Ostertag, Jr.

Requester: FASTC/TATE/lLt Douglas E. CoolApproved for public release; Distribution unlimited.

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Block Italic Transliteration . Block Italic Transliteration"Aa A 4 A a PP PP R, r

B 6 $ 4 B, b C e C S s 3 $

m BV1 Vv Tm T, t

r r r G o g yy Y y UuA a4 1 Do d * .6 * 0 F, "E 0 E * Ye, ye; E, q* X X X X Kh' kh)K X Zh, zh U U i v T3, ts3 0 9 * Z z 4 04 V Ch, ch

H V H I' Is i 1W W Uj W Sh, sh

AR a? Y 1 m 3 IN Shch, shch

K K K x K, k b 2, S to

JA 1 7 L; 1 b f 6/ w Y, y

MM M M M m b IbH H FHN N n 3 3 E, e

0 o 0 0 o ) 10 i ) *0 Yu, yu

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RUSSIAN AND ENGLISH TRIGONOMETRIC FUNCTIONS

Russian English Russian English Russian English

sin sin sh stnh arc sh sinh-1cos cos ch cosh arc ch cosh- 1

tg tan th tanh arc th tanh"Ictg cot cth coth arc cth coth-1

sec sec sch sech arc sch sech-Icosec csc csch csch arc csch csch-1

Russian English

rot curllg log

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SPECTRAL PROPERTIES OF PERIODICALLYCORRELATED RANDOM PROCESSES

Dragan, Ya.P.

Physicomechanical Institute, AS UkSSR

Submitted 20 Jun 70

In connection with the importance of the class of periodicallycorrelated (PK) random processes (SP) in the investigation of rhythmicphenomena [1) and the insufficient degree of development of theirtheory, a need arises for a comprehensive study of them. In thisarticle, developing the idea in work [2], the spectral properties ofPKSP are investigated. By periodic correlation we understand theperiodic nonstationarity of the second order.

As it follows from the results of work [2] and the definition ofcurrent spectrum [5, 6], the function of covariation of PKSP

btU)==r(t+ UO Et(t + u)?'(i) is presented in the form

b (t, u)= e'"d F (k. t),( )

in this case the dependence of spectral function on time is periodic,and this function possesses the expansion

F 12 FhQ4A, r, (2)

where T - period of correlation of the process, and the functions Tk(-)are determined by the correlations

B,(u))- • e""'dFa(X), b(1, u) £ Ba(u)i 7 . (3)1 ý Ba (u)- e

Definition 1. As the precise period of a periodic function f(-)we shall call that least non-negative number T, for which at all t theequality f(t+T)= f(t) is valid. The constant variable is a degenerateperiodic function, since it is periodic in the case of any T and in themeaning of definition I it does not possess a precise period which isdifferent from zero.

Definition 2. We will call the functions Bk(•) the correlationcomponents (KK).

Taking into account that the dispersion of the values of theprocess

'()=b (1. 0). (4)

on the basis of what was said it is easy to be certain of the validityof the assertion *.

3 Tr% this article for simplicity w( wil] i -, tii, tu ,f,! 11,, ii-1fI,, 1 i, .1

expectation of the process is equ.al to ze,-o. [end of footnote.]

Theorem 1. The current spectral function and the dispersion ofthe values of PKSP are periodic functions with a period equal to theperiod of correlation of the process. In principle the dispersion maybe a precise period, equal to any part of the period of correlation,and it also may be degenerate.

The first assertion follows directly from formula (1) and the factthat T is the precise period of covariation. For proof of the secondof formulas (3) and (4) we find

_2(M V B.(0)e T'. (5)

Assume with a fixed integer m only the initial values of

KK(Bmp(0).'7 0, p=--u,cx_) will be different from zero. Then

o' t + I B.P (0) e'• r e' T ".

from which, taking into account that a= I , we see that the

expression T, is the precise period of dispersion, i.e.

2

O2 (t ) sO(t). . Dispersion is degenerate when BO(O)*0 and all the

remaining Bk(O)=O. These conditions are necessary and sufficient as aresult of the uniqueness of the Fourier expansion.

Subsequently we will assume that PKSP is harmonizable (optionallyabsolutely in the meaning of Rozanov [4]) and its two-frequencyspectral functional is differentiable in the sense of the theory ofgeneralized functions, i.e.

r (s, J) = 5 7 9*-1Ldd, (I. ~& IA~s (k. p) Adý4.

Then its derivative s( A ,u. ) is a covariation of harmonics withfrequencies A and I [6] and has the form

where fk(-)=F'k(-) and Z'(-) - generalized derivative in the sense ofIto [3] of the spectral measure of the process, entering into itsrepresentation

e ( 5IdZ (k) -5e"JAZ' (X) A. (7)

From the representation of the function of covariation

b (1. u) 5 e"' I(v. %) dvcd). (8)

is valid in the meaning of the theory of generalized functions for anyharmonizable SP, if [21

1 v. v) s(V. 0

from formula (6) we find that the function of covariation of harmonics

3

Figure 1.

Definition 3. We will call the functions fk() the spectralcomponents (SpK). Then from the last equality and formula (2) thevalidity of the theorem follows directly.

Theorem 2. The k-th spectral component and the k-th harmonic ofchange in the spectrum of PKSP coincide.

Theorem 3. Each harmonic with a frequency %0 of expansion (7) ofthe PKSP is correlated only with harmonics of frequencies

r, k= -c•.o , and is not correlated with the remainder. The

coefficient of correlation of the harmonics / 0 and ,A0 +kO2 is

determined by the value f (; 0 ) of the complex-valued k 0 -th SpK(Fig. 1).

For establishing the bond between the spectral components and thepositive and negative indexes we will use the correlation [2] for KK:

Bk (u) = B6 1(- u) e

Then, taking the first of formulas (3) into account, we find

f-k NX fk (x T)k

The sets of frequencies (4 1 do not formpartitioning of the axis of frequencies into classes of equivalence,since the magnitude of correlation depends riot only on k, but also on

104

Partitioning will take place only in the case of PK of white noise,defined (see below) as a process with covariation r(s, t)=g(t)65(s-t),

where g() M ChAItt - periodic function, since for it in formula(6) f k(A)ck, and it possesses stationarily correlated harmonics,i.e. s( /, / )=s(A-1).

Let us note the validity of the general theorem.

Theorem 4. We will present the covariation of any process whichis stationarily correlated in spectrum in the form of the productg(t) 8 (s-t), where g(t)40, and conversely, any nonstationary noise is aharmonizable process with stationarily correlated harmonics.

The last assertion follows directly from the theorem of collationof spectra and the Fourier-presentation of the 6-function. Then

r~, 4' -gQ'6(t -- - f f --40 -

and G(-) - Fourier-image of the function g(".

Since the function

BOWu = lim.5 bQt. Udt=..5)dt dt (10)-0 0

exists and is defined positively, then fo(A ))O and it describes thepowers of the harmomincs of the components of the process. On theother hand, the function, determined by the first of equalities (10),is called the covariation of a stationary approximation to anonstationary process. Everything taken together shows that thetheorem is valid.

Theorem 5. A stationary approximation to PKSP - this is astationary process consisting of the same harmonic components and withthe same powers as PKSP, but that are not correlated. Averaging (10)is equivalent to elimination of the correlation of harmonics, and thepower B0(0) of stationary approximation is equal to the power of thePKSP which is average for the period.

Actually, from formulas (10) and (4) it is evident thatTB0(O) = 44•[t dl ( . In that particular case when b(t, -) is almost a

periodic func.ion, functions Bk(.) will be almost periodic, and if

(X P - 00- - set of indices of a Fourier function b(t, -), then

the function of the frequency covariation

&-a* I=-.

is different from zero only in the points of intersection of the set

{ 2.%_ X} with the set = It follows from here that only

the harmonics, for which . will be correlated. In-2a

particular, for periodic processes [2,6] 1I-- , and all the

harmonics are correlated, and the spectrum is averaged in points

(k--f-, I L" . k' 1 00-c. CO

In a general case the following theorem takes place.

Theorem 6. If the function b(t, -) is periodic, then its periodis a multiple of the period of correlation.

Assume the period of the function b(t, -) is equal to To. Then onthe basis of formula (3) the function possesses a Fourier expansion

b(t. u) = \' • ba1e from which it is easy to see that all the

KK will be periodic with the period TO:

B (u)= "bke F so(.. bAI8e _, ,2

and in this case in conformity with formula (9) •(.) •- bh8 •-- •x

X6 (V--k) 2 From this expression and the identity which is known

from the theory of generalized functions 0" •( '):O it follows that

when I - k-r. k0..---)0 , then f(-V),j ) O0, and

then on the basis of the uniqueness of the Fourier transform fromformula (8) we have the identity b(t, u)•zO, i.e., the PKSP isdegenerated to zero. In the case of non-degeneration the inclusion of

6

lk 12A takes place, since T is the precise period of

correlation. Since both these sets of points form arithmetic2A 2a

progressions with the differences 2n and 2- respectively, then2a 2n

there exists such a natural number p that -=P--- The latter is

equivalent to the equality T0=pT, proving the theorem.

Subsequently it will be convenient to use the followingdefinitions.

Definition 4. In a general representation of PKSP [2]

() (12)

where { ()} stationary stationarily bound processes (stationarycomponents (SK PKSP).

Definition 5. We will designate as periodically correlated whitenoise a PKSP for which in formula (6) 4( )f= 4 for all 2 and k.

From definition 5 it is evident that

r (s. t) g2i Ie 6(s- I) g ()(s-- 0.

i.e., for PK of white noise b(t, u)=g(t)0 (u), where g(-);O - periodicfunction of the period, equal to the period of correlation.

Theorem 7. If SK PKSP - white noises, then it degenerates into PKwhite noise.

ActLilly, using formula [2]

b(t, u) = N, r .TO(-) T(13)I---n I .--.

I (

and the condition of noncorrelation of SKri,j(u)=Al,j 5 (u), and alsothe property f (x). (x)= (0) (x), we obtain

7

b(t. U)~ 2~~ 6 (U)= g Mt a (U).

where g(.) - periodic function of period T. It is non-negative whenAl,j)°-

Theorem 8. If SK are uncorrelated, then PKSP degenerates into astationary SP, and if they degenerate into random variables, then theprocess will be periodic, if these variables are uncorrelated theprocess will be periodic stationary [6].

Proof follows from formulas (9) and (10).

Definition 6. We will designate as periodic white noise a processfor which in formula (11) ckl=ck when j 2n Then its covariation

r'a, )- £ q_ r-

which taking into account the Poisson formula from the theory ofgeneralized functions

w7

giving a representation of the peiodic &-function 8T(x) " 5(x-J,I-

can be written in the form r(s, t)=Tg(t) Tr(t-s), which proves thestatement.

Theorem 9. Periodic noise does not coincide with PK noise.

The following theorems are important for an understanding of thestructure of PKSP.

Theorem 10. The state of being correlated by stationary PKSPcomponents and the state of being correlated by its harmonic componentsare equivalent.

Since SK are correlated, and by definition 4 they are stationarilybound, then their mutual covariation permits the representation

r,. (u) 5 dSt

8

where EdZj(X) YZ &'Sj.j(A)8•-p)d, and ZI(.) - random measure,corresponding to the process in expansion (7). Then fromformulas (3) and (13) we find

B,(U)= ( r,.-_ (u)e

from whichFA 1 .

from which the validity of the theorem follows directly, i.e., thecorrelated state of the harmonics implies the correlated state ofstationary components, and vice versa. In particular, from here thespectral function of the statinary approximation to PKSP

(O N s1.(%--.i")" i.e., is equal to the sum of the displaced

spectral functions of stationary components, and the shift is equal tothe frequency of the correlated state of PKSP multiplied by the SKnumber.

Theorem 11. An invariant ideal band-pass filter separates fromthe PKSP its uncorrelated harmonic components when, and only when, thewidth of its bandwidth is less than the frequency of the correlatedstate of PKSP.

Proof follows from the fact that the spectral density of thetransform of the harmonized process with the help of the invarxantoperator q with the frequency characteristic 4 ( L) is expressedthrough the spectral density of the process by the correlation [2,6]

s•Q.., ) = D (.)i�j-1(D. p),

which taking formula (6) into account takes the form

s~~( (~ )=(X) ' I(X) (D ( .6(iX ~+k

Assume now that the frequency characteristic of the filter •(2.) is

equal to zero outside of the interval I/. -- an, X+a± (aI),

i.e., (0( x) .(%) %I&,(.) , where . (') - characteristic function o' the

9

set A. Then ) 44(AX

CA

Figure 2.

Considering the property of the S-function:A (x). f(x)S(x-a)= A (a)f(a),_ we see that the product of the last three

factors differs from zero only in the case wh-en l••l•< 2n and as a

result of the independence of ýL and 1A from the inequalities

jp--•I•14-I•~ it turns out that the product is different from zero

only when II•< R and l~••.Under these conditions the mentioned

F T

product is equal to .0(-).,and then

From here the correlation function of the transform

i.e., the filtration of the PKSP with a bandwidth less than thefrequency of its correlated state is equivalent to the filtration ofits stationary approximation. The sufficiency of the condition of the

theorem follows from the fact that if A=[; 0 -,8, 0+A I is designated,

then IA 0)

only in the case when the set A x A does not intersect with one of the

straight lines ltr=tin- kf t k=-PoK,oo wh with the exception of the

diagonal = (Figure 2) .

Another proof of this theorem can be obtained if in expansion (12)

of the PKSP each stationary component is replaced by its Cramer -

Kolmogorov expansiondiagona ep=A. (Xiure 2)

10

Then the harmonic expansion (7) for PKSP takes the form

1(0= e"3 [6 C( )--k

from where follows directly the type of correlated state of theharmonics mentioned in theorem 3:

Thus, in the representation of PKSP of type (7)

and in the sense of the theorem of generalized functions

f (v, X) = EZ' (v) Z' (v -- X).

Definition 7. We will call a harmonizable SP degenerate if itsdispersion is constant.

According to theorem 1 such a process may be nonstationary.

Theorem 12. A non-degenerate harmonizable SP with periodicdispersion belongs to the class of almost periodically correlated.

For proof of this assertion from formula (8) we determine thedispersion

as (1) ~ X f (v, X) dvd%. (14)

it is evident from expression (14) that dispersion can be periodicwith a precise period To only when

f (v, X)dv 6(X

11.

The latter in turn is possible only when

where f1 (V):f, (92, ). In this case the following conditions shouldbe fulfilled.

Sf (v)dv C, when X.

0 in the opposite case,

From the last equality and formula (14) it follows that dispersionof the component of a random process, possessing a continuous spectrum

• ( -P, X ), is equal to zero. Since the process is centered, then

this component is identically equal to zero, which also proves thetheorem.

It follows from expression (14) that if z is taken as yo, then theinequality (12) will be fulfilled. For a more rapid convergence offormula (11) when y)1 it is possible to use as the initialapproximation the magnitude

@~ ~ 2 Inxo€ 'Y11W 16 Q

Table ~AH ~ .I~ oil~

xEI(. 0.99991 I 8x10'- 5xO-12 0.16yE|o; 2.7511 2 2x10r-'2 8xl0-' 0,15

YO==z 3 7X-10 4- 5x10-9 0,1Idol <0.17; fIy6l<0.4 4 2×x 10-' iOr- 0,02

xEIO;I---10r2 I I0- 2 6x I0-' 3x 10- 3

YE10;5,0421 2 10-1 0-10 10 o-14

YO=zg 3 10-4 2x10-3 3x10-5616,KO. I; l <OK<0.096

Key: (1) Initial data; (2) Absolute error of formulas.

12

The table gives data on the absolute error of formulas (6) and(11), and also of the iteration formula of the third order

Y(+ I = vt-7.-- 0 I'(y) - x~i ' +TvI(Jze' (16)

in the case of the initial approximation (13) and

Ya ==Z2 zwhen 0 < x< 0.82; (7

Z when x > 0.82.(7

REFERENCES

t. ~ a mm a p C K HI2 R. C. a n~p. CnparioqKxK nporpammuwca. 1. Cyztnpomrm3. 11.. 1963.2. M o 4 UI x 6o. H q s. P.. nlo no. a5. A,-.B KM.: MaTema~m'Iecxoe o~ecneq4eKme 3LBM

H 34xoeKTHSH3aN opraHi43aUHi 8WqHc.,1He.MbMoro n pouecca. 4. (HavKOSI3 AYuxa.- K.. 1969.3. M o Hu m6o axq s. P.. nlo nom a . A. Ropm~oam i imjpmeno

7ammb mx~ 3U.BM~ 4n_ OMi~bs m4 4lp0miHb-MD. (HayKOMa AiYMas. K.. 1969.4. C cI A p r. C.- KMX.: MaTeMamq~ecKoe o~ecne'me~me 3BM m s34~eK~THBMRa opralixan.

UHA BbI'lCAHTeJIb~oro npouecca. 2. tHay~Os8 .IYMK3*. K. . 1967.5. T e c A e p r. C.- B KM.: Ma~te3aTm'4eCgoe o6ecneqemme 3IIBM m 3cW4eTHSMaR OpraHK-

3auIIR 9wP4HCalWie.lYbmoro nlpouecca, 2. tHayxosa AYKP K.. t9.9.6. C e r a Ai B. K.. C e u e ni A a e a K. A. n~kTH3Ha'4mhle ma-reuaTm'ecKme Tra6JiHUm. OH3-

Ma~rits, M.. 1962.7. Ri 1 K e E.. 3 M .A e 40.. AI e w 0. CneII~aanbHwe (OY1K11HN. HayKal. M4.. 1968.8. G o dy WA. J. Math. Comnp.. 1969. 23, 107.9. N e m e t h G 6 z a. A k6zponti lozikal Icutato intizet kaidoi csoportja. 1964. 12. N 6.

10. S t re c ok A. J.. Math. Comnp.. 1968, 22,.101.

13

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