COMPOSITE STOCHASTIC MODEL FOR EMI-IDENTIFICATION

Asst Prof Dr. D P Roy, VSM

FEL, MCEME, Trimulgherry PO, Secunderabad - 500015

INTRODUCTION

Detection of EM1 in a radar can be found from an adaptive technique. Stochastic parameters are analytically derived for EM1 complex quantity. The pdf of the amplitude and phase are double bounded processes which can be determined theoretically. Instantaneous phases and amplitudes of this complex signal are related by Hilbert transform pair. Auto correlation of the amplitude is evaruated. An adaptive process based on auto correlation is used to recover a desired signal in the EM1 environment.

DOUBLE BOUNDED P.D.F [ 1 ]

EM1 corresponding to the two consecutive sweep of radar antenna can be considered as two sets of EM1 data. The individual double bounded pdfs for the two sets of data can be developed viz p (ZI) and p (22) with the help of the first four moments (i) Mean (ii) Standard deviation (iii) Skewness coefficient and (iv) Kurtosis 'coefficient. These four parameters are calculated with the help of first four moments (i.e m l = x m

where x = (Z - Zmin 1 (Zmax - Zmin )

m2 =xm* ; m3 = xm3 ; m4 =xm 4

Z = r . v of process (Z)

Zmin = Lower bound of random

Zmax = Upper bound of random variables m i = xtmean) = Xm

variables

Fo = Cumulative probability of Zmin ; S = Standard deviation

= d(m2 - m.12); * I

g1 m l +2m13)/S3

= Skewness coefficient = (m3 - 3m2

92 = Kurtosis coefficient = (m4 - 4m3 m l + 6m2 m12 - 3 m j 4 )Is4

a, b = parameters

F(Z) = cumulative probability of Z

m l , s , g1 , 92 , are the four parameters help to calculate Zmin Zmax Fo I 'a' and 'b' of this distribuition,

Let a = 1 and b is expressed as (4 $ - 1) / (1 - 4 ) with C = S(Zm) dz, (Zm = mean and S(Zm) = Standard deviation) and is expressed as :

X m = (1 - Fo ) / (1 + b) and

p.r.p N pulse repetation period & Zmax = zm /xm nth moment of

D.B. P.D.F. can be calculated

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and p.d.f. = (1 - Fo ). a.b. (x)a-l (1 - xa )b-1

The probability density function for a double-bounded random variable is expressed as :

p(z) = (1 -Fo)ab. za-l (1 -za)b-l where a and b are parameter and Fo is cumulative probability.

JOINT P.D.F. FOR DOUBLE BOUNDED DISTRIBUTION: [ 2 ]

To get the correlation function of these two sets of data, we require to calculate the joint density function from the marginal density function of p(z1) and p(z2). The joint P.D.F. (h(z1 ,z2) ) in terms of these two marginal p.d.fs is expressed as : [ 3 ]

where, k ((zl ,z2) is a functioin of z l and 22 provided the following condition is satisfied : [ 4 ]

h(zllz2)=P(zl) P(Z2) (1 - Wl IZ2))

The joint p.d.f. is expressed as

where, A=( 1 -Fozl )( 1 -Foz2)azl az2 bz l bZ2

CORRELATIONFUNCTION: [ 2 J

The correlation function is evaluated in terms of this joint p.d.f. :

This correlation function is useful in the unified detection and estimation scheme (for likelihood ratio test). Variation of upper and lower limit will effect the a.c.f and (as z = (x - xmin)/ (xmax - xmin), the a.c.v.f. as well.

In the composite model of EM1 it is possible to use this correlation function to model a time series with the same correlation function, using a recursive (LMS operated) adaptive shceme. [ 5 ]

The amplitude of these €MI field obeys the two bounddlimit in magnitude. Both these p.d.fs are are independent, thus a joint p.d.f. is developed to suit the stochastic behaviour of a homogenous random elementary EMI.

Nm of the mth EM1 obey Poisson p.d.f. i.e. p(Nm)=(Nm Nm/Nm!).exp (-Nm) and its amplitude 'z' obeys the double bounded p.d.f i.e.

p(z) = (1 -Fo)ab. za-l (1 -za)b-1

As the question of amplitude arises only when the N occurs and this leads to a conditional probability. The conditional probability is f(", z)/f(z) where the joint p.d.f. is the f(", z ) and f(z) is the d.b. density function. The joint density function can be expressed as :

(Nm,z)=p(Nm).P(z).[l -KWm,z) where, p(") is the Poisson distribution and p(z) is the double bounded p.d.f. and let K(Nm,z) = Nm(z -Zm), which meets the required condition for the joint p.d.f. The conditional p d f becomes

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pz(z/Nm) = A . Z ~ - ~ ( I - Z ~ ) ( ~ - ~ ) . [ I - K ( N ~ , Z ) ] Further we can take the effect of background or internal noise contribution and this can be taken as complex Gaussian variables. This can also be expressed in terms of Rayleigh amplitude distribution. This is independent of the conditional p.d.f. Thus the effective p.d.f. can be expressed as :

exP[-zb2/21 pz(zb,z/") = _-___-_-------- .A.za-l .(l-za)b-l

[I-"@ - zm)] where, zb is due to background noise, Gaussian distributed.

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Correlation function of such distribution is useful to evaluate the imaginary part of the complex correction function, taking into account the effect of background or internal noise contribution as complex Gaussian variables, where the internal noise is independent of the double bounded p.d.f based EMI.

To get the readily available stochastic informabn on the EM1 is

functions. This likelihood ratio test results into voltage, which defines the threshold for the radar detection in EM1 environment. Computer evaluation of this scheme is found to be suitable for pulse tracking radar, operatihg in EM1 environment.

Reference :

1. Kumaraswamy, P. : 'A generalized probability density function for double bounded random process', Journal of Hydrology, 46, (1 980), pp 79-88.

2. Study of Radar Signal Processing Ph.D Thesis, Roy DP ,1990, I.I.T, Bombay.

3. Al-Hussain, E.K. and Add - El- Hakim, N.S. : 'Bivariate inverse Gaussain Distribution, Ann. Inst, of State. Maths (Japan), Vol-33, Part A, (1981) pp 57-58.

4. Tweedie, M.C.K. (1957): 'Statistical properties of inverse Gaussian distribution', I. Ann. Math. Statist. 28, pp 362-77.

on an auto correlation function with an LMS adaptive scheme. The LMS algorithm adjusts the weights of the adaptive transversal filter depending on the difference of the auto correlation function of the p.d.f based sequence and that of the time series. When the algorithm converges, the corresponding weight factors are stored.

Later this information on the wieght factors are used to identify an unknown EMI. Further to this correlation information can be used to develop a likelihood ratio test based on covariance functions. This likelihood ratio test based on covariance

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