Correlated K-Distributed Clutter Generation for Radar Detection and Track

I. INTRODUCTION Correlated K-Distributed Clutter Generation for Radar Detection Track L. JAMES MARIER, Jr., Member, IEEE United Defense, LP Minneapolis, Minnesota The generation of correlated vectors for non-gaussian clutter is considered for log normal, Weibull, K-probability distributions. Previous results for log normal Weibull distributions are summarized. Expressions for the probability distributions moments of K-distributed clutter of any correlation are derived. Procedures for forming samples of each type of clutter are shown to be equivalent to passing white Gaussian noise through a linear filter followed by a nonlinear operation. Curves of correlation coefficients necessary for the simulation of these vectors are presented for each distribution. Manuscript received July 2, 1992; revised August 31, 1993. IEEE Log No. T-AES/31/2/819. Author s address: United Defense, LP, Armament Systems Division, 48 East River Road, Minneapolis, MN 55421. 18-9251/95/$1. c 1995 IEEE Extensive experimental data suggests that radar clutter probability distributions often are not Rayleigh distributed. For twenty-five years, various non-gaussian probability distributions have been offered as models for the amplitude of both l sea clutter. Log normal distributions have been proposed for sea clutter [1, 2] l clutter [3] Weibull distributions were proposed for sea clutter [4, 5] l clutter [5 7]. A useful summary of these other clutter models with their parameters ranges of applicability is contained in [8]. More recently, the K-distribution has been added to model both sea l clutter [9 13]. The K-distribution may well extend to phenomena other than radar including radiation scattered from turbulent or rough surfaces [14]. With the introduction of non-gaussian clutter models, however, the computation of radar detection false alarm probabilities track performance becomes difficult when clutter correlation properties are considered. Log normal distributions were considered in [15, 16] for noncoherent samples in which the sample amplitude was modeled. Coherent approaches, [17, 18], describe the generation of in-phase quadrature clutter components whose amplitude is log normally distributed. In both noncoherent coherent approaches, correlated Gaussian rom variables (RVs) were passed through a zero memory nonlinearity (ZMNL) to form log normally distributed RVs. The same method was applied to Weibull clutter in [19] for noncoherent, in [2, 21] for coherent samples. Another procedure employs the theory of spherically invariant rom processes (SIRP) [22, 23], for the generation of non-gaussian clutter vectors including those from the Weibull K-distributions. The SIRP approach, totally different from the method used herein, can be used to generate coherent noncoherent clutter processes. One stard procedure [24] for noncoherent samples the one applied here is to express the desired joint probability distribution function (pdf) as an expansion of orthogonal polynomials, with the desired marginal pdf proper correlation properties. Using the orthogonal polynomial approach, it is advantageous, for reasons to be discussed later, to transform correlated Gaussian RVs into the final correlated RVs via a nonlinear operation, [24] describes a means to do this formally for general pdfs. The orthogonal technique was applied in [24] for Gamma Rayleigh distributions, in [19] for Weibull clutter in [25] for K-distributed clutter. The last distribution is valuable plays a key role in this work, although its formulation does not provide for the representation of highly correlated, K-distributed clutter. Other correlated distributions considered 568 IEEE TRANSACTIONS ON AEROSPACE AND ELECTRONIC SYSTEMS VOL. 31, NO. 2 APRIL 1995

Fig. 1. Correlated log normal sample generation. S(!)=spectrum. in [26] were: uniform, Gamma, binary; in [27]: triangle, Rayleigh, Laplace, uniform, exponential. We examine the problem of the generation of correlated non-gaussian clutter vectors for the log normal, Weibull, K-distributions. We summarize previously published results for the generation of noncoherent log normal Weibull clutter distributions provide detailed processing information for them. The new result of this work is the derivation description of a scheme to generate noncoherent, K-distributed clutter vectors of any correlation. In Section II, we discuss our general approach for the generation of correlated clutter vectors. In Sections III IV, we summarize previous results for log normal Weibull distributed clutter generation provide detailed processing correlation information. In Section V, we examine the problem of correlated K-distribution clutter accompanying moments for temporal highly correlated clutter. In Section VI, we define the processing necessary to generate correlated K-distributed vectors. II. CORRELATED VECTOR GENERATION We wish to generate N data samples z 1,z 2,z 3,:::,z N to represent the amplitude of radar clutter returns. The samples are to have, 1) some specified probability distribution, for example, log normal, Weibull, or K, 2) an arbitrary spectrum. In each case, the clutter is represented as white, Gaussian noise passed through a linear filter followed by a ZMNL operation. This process is illustrated in Fig. 1 for the log normal case. First, N samples, v i (i =1,2,3,:::,N), of uncorrelated (white spectrum), Gaussian RVs are passed through the linear filter, H(!), the correlated samples, w i (i =1,2,:::,N) emerge. The pdf of the w i is easily found to be Gaussian with a spectral power density, jh(!)j 2. When these Gaussian samples are passed through the nonlinear exponentiation filter, they assume the desired log normal pdf with spectrum S z (!). Our goal is to achieve an arbitrary spectral shape for S z (!), e.g., Gaussian, by the proper selection of H(!). It may also be possible to generate the desired statistics for z i by beginning the process with non-gaussian distributed v i,thenselectingan appropriate H(!) nonlinear filter. However, starting with Gaussian v i results in comparative computational simplicity for an already complicated problem because then the pdfs for w i remain Gaussian easy to compute after linear filtering. For this reason, a Gaussian v i approach is straightforward computationally appealing. For the same reasons, the v i are assumed to be independent identically distributed, convenient for simulation purposes. Then, the output RVs, w i, have a Toeplitz correlation matrix although the process could also begin with correlated Gaussian RVs providing their correlation matrix was nonnegative definite. Processing similar to Fig. 1, i.e., linear filtering plus a nonlinear operation, is used to generate vectors for all three distributions (log normal, Weibull K) considered in this work. We assume that both the probability distribution function spectrum for z are given. In the three cases, H(!) is found as follows. First, a realization of z i beginning with zero mean, Gaussian distributed RVs is needed. Second, the specified correlation coefficient for z i, s, is computed in terms of the correlation coefficient of w i, ½,where q s =(E[z i z j ] E[z i ]E[z j ])= D 2 [z i ]D 2 [z j ], i,j =1,2,3,:::,N (1) q ½ =(E[w i w j ] E[w i ]E[w j ])= D 2 [w i ]D 2 [w j ] (2) where E[ ] denotes expectation D 2 [ ] denotes the variance. Knowledge of ½ permits the spectrum of w H(!) to be defined according to S w (!)=FfE[w i w j ]g (3) where F is the Fourier transform operator, jh(!)j 2 = S w (!) (4) H(!) can be found. 1 While the expectation variance of the z i are easy to compute for the three pdfs under consideration, the autocorrelation of z, E[z i z j ], is not. The central problem addressed here is the computation of this quantity in terms of the autocorrelation of w, E[w i w j ]. Before proceeding, we cite some minor drawbacks of the clutter generation approaches which follow in this work. First, if the sequence of Gaussian RVs input to the ZMNL is blimited, the output is blimited if only if the nonlinearity is a polynomial [29]. Moreover, for input Gaussian processes, any ZMNL smoothes broadens the output spectrum. However, we can at least partially compensate for this effect by the proper selection of the ½. Thus, if one desires a narrow output spectrum S z (!), for example at the 3 db points, one chooses a filter H(!) with sufficiently narrow bwidth. Second, these approaches do not 1 Subject to the requirements of the Paley Wiener criterion [28] providing that a suitable phase function is found for H(!) toensure that the resulting transfer function is physically realizable. MARIER: CORRELATED K-DISTRIBUTED CLUTTER GENERATION FOR RADAR DETECTION AND TRACK 569

Fig. 2. s(i,j) versus½(i, j) log normal clutter constant sigma-c (db). afford independent control of the marginal pdf correlation function although most combinations can be simulated. That is, parameters of the desired non-gaussian distribution appear in the expressions which relate ½ to s thereby linking the correlation functions final pdf. Third, the most serious drawback is that the correlation matrix at the output of the nonlinearity is not guaranteed to be nonnegative definite a requirement for valid representations of rom processes. In the next two sections, we summarize previously published results for log normal Weibull clutter, provide pdf realizations of these via detailed processing block diagrams, graphs of s versus ½. This is all of the information required to generate log normal or Weibull vectors. III. SUMMARY OF LOG NORMAL RESULTS For the distribution w, w» N(ln¹ c,¾ 2 c ) the nonlinear operation, z =exp(w), results in a two parameter log normal pdf for z, f(z)=1= p 1 2¼¾ c z exp 2¾c 2 fln 2 (z=¹ c )g z>, ¾ c >, ¹ c > : For s ½ as defined in (1) (2), the result from [16] is s = e¾2 c ½ 1 : (5) e ¾2 c 1 This is plotted in Fig. 2 for a family of ¾ c expressed in decibels 2 Fig. 3 shows the processing. In Fig. 3, the RVs u i (i =1,2,:::) have correlation coefficient ½ 2 The voltage mean/median ratio of the returned signal (often used to describe log normal distributions), ½ v,is½ v = expf[:233¾ c (db)] 2 =8g. This is equivalent to ¾ c (db) = 8:684¾ c. See [3]. Fig. 3. Generation of correlated log normal clutter. whichisusedtofindh(!). Setting jg(!)j 2 =F(½) defining a normalized transfer function Á Z 1 1 jh(!)j 2 = jg(!)j 2 jg(!)j 2 d! 2¼ 1 where H(!) is defined except for a phase angle. Fig. 2 also shows that under conditions of small ¾ c (<» 5dB),s» ½ the exponential filter does not appreciably affect the correlation properties of the RV passing through it. This is easily seen analytically by solving (5) for ½, (6) ½ = ln[1 + s (e¾2 c 1)] ¾c 2 (7) exping the logarithm with the approximation for ln(1 + x) forsmallx: ln(1 + x) ¼ x x 2 =2+x 3 =3+, so that ½» s, ½» s (e ¾2 c 1)=¾ 2 c ¾ c < :6» 5dB a condition typical of medium clutter backgrounds. IV. SUMMARY OF WEIBULL RESULTS For the independent identically distributed RVs w 1,i w 2,i (i =1,2,:::) distributed as N(,¾ 2 ), the RV z i =(w 2 1,i + w2 2,i )1=p (8) 57 IEEE TRANSACTIONS ON AEROSPACE AND ELECTRONIC SYSTEMS VOL. 31, NO. 2 APRIL 1995

Fig. 4(a). s(i,j) versus½(i,j) Weibull clutter constant p 1. where 2 F 1 is the Gaussian hypergeometric function [31] ( ) is the Gamma function. 3 The graph of s versus ½ is shown in Figs. 4(a), p 1, 4(b), p 1. The computer computation of s is not difficult. A straightforward evaluation of the infinite series representation for 2 F 1 to one hundred terms provides 6 7 place accuracy while the approximation [32] for the Gamma function is surprisingly accurate to 9 1 places. Fig. 5 shows the processing. The RVs u 1,i u 2,i (i =1,2,:::) have correlation coefficient ½ 4 where, Fig. 4(b). s(i,j) versus½(i,j) Weibull clutter constant p 1. has the two-parameter Weibull distribution, f(z i )=(z i =q) p 1 (p=q)exp[ (z i =q) p ], z i >, p>, q> (9) q =(2¾ 2 ) 1=p : (1) We define the correlation s as in (1) for the Weibull z i the correlation coefficient ½ as ½ = E[w k,i w k,j ] E[w k,i ]E[w k,j q ], k =1,2: (D 2 [w k,i ]D 2 [w k,j ]) (11) Itisshownin[19]that s = f 2 (1 + 1=p)=[ (1 + 2=p) 2 (1 + 1=p)]g [ 2 F 1 ( 1=p, 1=p;1;½ 2 ) 1] (12) ½ = E[u k,i u k,j ] E[u k,i ]E[u k,j ] ¾ uk,i ¾ uk,j, k =1,2 terms of the same sequence fu k,i g are correlated, but terms u 1,i u 2,j drawn from the two different sequences are not. Again, setting jg(!)j 2 =F[½] defining a normalized transfer function, where Á Z 1 1 jh(!)j 2 = jg(!)j 2 jg(!)j 2 d!, 2¼ 1 H(!) is defined except for a phase angle. 3 Because of the presence of the Gaussian hypergeometric function in (12), this equation is not readily invertible to find ½.Givena Weibull correlation function, it is not possible to determine the correlation function of the Gaussian sequence at the input of the nonlinear transformation in terms of s in closed form. A graph of s versus ½ is helpful in illustrating the relationship. 4 The two sequences fv 1,i g fv 2,i g are independent hence, uncorrelated, proceed independently through the filter H(!). Thus the two sequences fu 1,i g fu 2,i g are not correlated. If they were, the resulting sequence at the output of the nonlinearity would no longer be Weibull. This is due to the fact that the nonlinearity belongs to the class of phase-invariant transformations is useful only when a uniform phase is preserved at the input output of the nonlinear transformation. MARIER: CORRELATED K-DISTRIBUTED CLUTTER GENERATION FOR RADAR DETECTION AND TRACK 571

Fig. 5. Generation of Weibull clutter. V. CORRELATED K-DISTRIBUTIONS In [25], a method for the generation of K-distributed, spatially correlated radar clutter pdfs was developed. This model is useful if one wishes to examine the statistical properties of two separate distinct clutter patches. But it has an important limitation: it cannot model two highly correlated clutter returns. In this section, we present a new result: an extension which provides for the representation of K-distributed, highly correlated clutter with correlation coefficients of all values from to 1. This type of clutter could arise, for example, from two overlapping clutter patches or from a single patch, temporally correlated. We first introduce the following notations for probability distribution functions: K[x;a,º]= G[x;a, ]= 2 ³ x ³ x Kº, a (º +1) 2a º+1 a x a +1 ( +1) exp R(x;a)= x a exp μ x 2 2a μ x a, x> º> 1 (13a) x> > 1 (13b), x> (13c) where [ ] is the Gamma function, K º [ ] isthe modified Bessel function of order º a is a positive constant. These distributions are the K, Gamma, Rayleigh distributions, respectively. Inasmuch as sea clutter, particularly when viewed at low grazing angles, has often been reported as K-distributed, we describe a model for a correlated K-distribution which is appropriate to suggested by scattering from the sea. Let Z be the amplitude of the sea clutter return. The compound K-distribution clutter model [1, 25, 33] represents the power from a sea clutter return, Z 2, as the product of two independent RVs, X s Y. Z 2 i = X si Y i, i =1,2,3::: : (14) Two methods are available to compute the pdf for Z, f Z (z). We could compute it directly by stard methods if the pdfs of both X s Y were known. An equivalent but more elegant approach employs a Bayesian pdf formulation. The second approach is more appealing because the resulting model accounts for the shape of the waves the sea surface thereby providing an intuitive physical justification for the result. Therefore, we adopt the Bayesian approach; however, it is easily shown that, for the pdfs under consideration, either method provides exactly the same result for f Z (z). X s represents the speckle attributed to capillary waves which is modeled by an exponential pdf, the square of a Rayleigh. The second rom variable, Y, models 1/2 of the local mean of the speckle power with a Gamma pdf. Y represents the swell of the waves which is generally correlated over distances of several meters. In [25], X s was assumed to change rapidly its samples were modeled to be uncorrelated. Thus, the correlation features of Z in (14) were controlled solely by Y, Z would never achieve perfect correlation because of the presence of X s in the product. In contrast to this, we assume that the X s may be highly correlated a necessary condition to model time correlation or spatial correlation between overlapping clutter patches. Using a Bayesian formulation to compute the pdf of the K-distributed romz variable z i,weform: 1 f Z (z i )=K º [z i ;a,º]= f ZjY (z i j y)f Y (y)dy: (15) We set f ZjY (z i j y)=r[z i ;y] (16) f Y (y)=g[y;2a 2,º]: (17) Equations (15) (17) account for the fact that, given the mean value of the speckle component, the amplitude return z i is Rayleigh distributed in accordance with experimental evidence. A straightforward evaluation of the integral in (15) averages the speckle power over all mean values results in the desired K-distribution. This model is appropriate to the sea but might not seem suitable for l clutter in which there is no clear 572 IEEE TRANSACTIONS ON AEROSPACE AND ELECTRONIC SYSTEMS VOL. 31, NO. 2 APRIL 1995

counterpart to speckle wave swells. Nevertheless, measurements [13] show that l clutter does possess a K-distribution under some conditions so it should be modeled accordingly. The realization of K-distributed vectors to be used for simulation purposes will employ an approach expressed by the product in (14), viz., a Rayleigh RV times the square root of a Gamma RV resulting in a K-distributed RV. Observe that because Rayleigh Gamma RVs are functions of normal RVs, it is easy to generate correlated vectors for them by the same linear filtering methods used for log normal Weibull vectors. We next compute the joint pdf for two Rayleigh-distributed correlated RVs. The approach is to compute the pdf for a correlated exponential RV with a result from Raghavan [25] then compute the pdf of the square root of the exponential which is Rayleigh. This is later combined with the joint pdf of Y to compute a joint pdf for Z in (15). Before proceeding, we set forth some of the conditions which govern the joint pdf of Z i Z j, which we hope to satisfy. As described in [24], the joint pdf f Z (z i,z j ) should meet several conditions including: Z Z Z f Z (z i,z j )dz i = f Z (z j ) f Z (z i,z j )dz j = f Z (z i ) z i z j f Z (z i,z j )dz i dz j = R z where R z is the autocorrelation of Z. Also, f Z (z i,z j )=f Z (z i )f Z (z j ) (18a) (18b) (18c) (18d) when the samples i,j are separated greatly in time or space, i.e., decorrelated, f Z (z i,z j )=f Z (z i )±(z i z j ) (18e) when the samples i,j are the same in time or space, i.e., perfectly correlated ±( ) isthediracdelta function. In general, there is no unique probability distribution function f Z (z i,z j ) which meets the above conditions. Complete knowledge of all the moments, E[zi lzk j ], l,k =,1,2,:::, is required to find a unique solution for f Z (z i,z j ). This is especially true for the f Z under consideration here because it will later be seen that f Z contains two variable parameters. These parameters are correlation coefficients whose values can be adjusted together to achieve any desired autocorrelation for z an infinite number of acceptable combinations exist. Nevertheless, the resultant probability f Z is valuable because, as is shown, it can be used to generate K-distributed RVs, with the proper correlation. We next compute the pdf of two Rayleighdistributed RVs, f Z (z i,z j j y i,y j ). The details are in Appendix A we repeat the result here: f Z (z i,z j j y i,y j )=4z i z j where 1 X n= p n,s G(z 2 i ;d i,n)g(z2 j ;d j,n) (19) d i =2y i (1 t ), (2) d j =2y j (1 t ), (21) t = E[x s i x sj ] E 2 [x s ] ¾x 2 s (22) p n,s =(1 t )t n : (23) The joint pdf of the correlated Gamma distributed RVs, y i y j, each with marginal pdf (17), is given by [25], f Y (y i,y j )= where p m G(y i ;b,m + º)G(y j ;b,m + º) (24) m= b =2a 2 (1 ½ ) (25) º+1 (º + m +1) ½ m p m =(1 ½ ) (º +1) m!, m =,1,2,::: (26) ½ = E[y i y j ] E2 [y] ¾y 2 : (27) To compute the desired joint pdf f Z (z i,z j )weextend (15) (17) to two RVs, f Z (z i,z j )= Z 1 Z 1 f Z [z i,z j j y i,y j ]f y (y i,y j )dy i dy j : (28) Inserting (19) (26) into (28) (the details are in Appendix B), we find where f Z (z i,z j )= 4p n,s p m m+º+n+1 i m+º+n+1 j = [n! (m + º +1)ac 1=2 ] 2 K m+º n (2 i)k m+º n (2 j) (29) i = z i =(2ac 1=2 ) (3a) j = z j =(2ac 1=2 ) (3b) c =(1 ½ )(1 t ): (3c) This expression is the joint pdf for the two correlated, K-distributed RVs Z i Z j.asacheck,setting t =, reduces f Z to the same expression as in [25]. MARIER: CORRELATED K-DISTRIBUTED CLUTTER GENERATION FOR RADAR DETECTION AND TRACK 573

Fig. 6. Generation of K-distributed clutter. The moments of the RVs z i z j are computed in Appendix C are reproduced here. The l,kth moment is given by E[z l (º + l=2+1) (º + k=2+1) (l=2+1) (k=2+1) i zk j ]= 2 (º +1) (2a) l+k 2 F 1 ( l=2, k=2;º +1;½ ) 2 F 1 ( l=2; k=2;1;t ) (31) where 2 F 1 (, ; ; ) is the Gauss hypergeometric function. Appendix D shows that the conditions (18a) (18e) on the joint pdf for Z are met. VI. CORRELATED K-DISTRIBUTION VECTORS In Fig. 6, we show the processing necessary to generate correlated K-distributed vectors. The RVs v 1,i,:::,v μ+2,i, are independent, uncorrelated identically distributed. Hence, in a manner similar to that described previously for the Weibull case, terms of the same sequences fw 1,i g fw μ+2,i g are correlated but terms drawn from any two different sequences are not. Expressions for all correlation coefficients are given below. A total of μ + 2 RVs are processed; the first μ comprise the RV Y, the remaining two comprise the exponentially distributed RV, X s.itiseasytoshow that the K-distributed pdf of z, as formed from the product of the square roots of X s Y, is identical to that formed using the Bayesian approach of (28). It is now necessary to define the two filters H 1 (!) H 2 (!) sothatz i is properly correlated after being formed by the nonlinear processing. We begin by computing s, (1). We note that for a K-distributed RV Z, with the distribution K(z;a,º) (º = μ=2 1), 2a (º +3=2) (3=2) E(z)=, (32) (º +1) E(z 2 )=4a 2 (º + 1), (33) D 2 (z)=4a 2 º +1 2 (º +3=2) 2 (3=2) 2 : (34) (º +1) Equations (32) (33) follow immediately from (31) upon substitution of l =1,k =l =2, k =, respectively. In these cases, the Gaussian hypergeometric function equals 1. We define the correlation coefficients r q,as r = E[w k,i w k,j ] E[w k,i ]E[w k,j q ], k =1,2,:::μ D 2 [w k,i ]D 2 [w k,j ] q is the same expression for k = μ +1 μ +2. To compute ½ (27), we find E[y i y j ]=E[(w 2 1,i + w2 μ,i )(w2 1,j + w2 μ,j )] = μ 2 a 4 +2a 4 μr 2 : Substituting this into (27), it is easily shown that, similarly, ½ = r 2, t = q 2 : (35a) (35b) Substituting this result into (31) then substituting (31) (with l = k = 1), (32), (34) into (1), we obtain, s = 2 [ 2 F 1 ( 1 2, 1 2 ;º +1;r2 ) 2 F 1 ( 1 2, 1 2 ;1;q2 ) 1] º +1 2 (36) 574 IEEE TRANSACTIONS ON AEROSPACE AND ELECTRONIC SYSTEMS VOL. 31, NO. 2 APRIL 1995

Fig. 7. s(i,j) versusr(i,j) K-distributed clutter constant º. q(i,j)=r(i,j). Fig. 8. s(i,j) versusr(i,j) K-distributed clutter constant º. q(i,j)=r(i,j) 1. with = (º +3=2) (3=2)= (º +1): (37) As described in Section V, the probabilities, moments, correlation coefficients of the joint pdf for Z are functions of two parameters: r q. Clearly, in (36), a value of s does not allow us to define uniquely either parameter. An unlimited number of combinations will satisfy the equation. Two combinations were selected for evaluation of s.first, all correlation coefficients were set equal, i.e., r = q. Fig. 7 shows the resulting graph for s. Second, the two RVs comprising the Rayleigh distribution were set to be far less correlated than the μ RVs comprising the Gamma: q = r 1,Fig.8. In the former case, the curves are only weakly dependent on º. In the latter, the curves separate to show a dependence on º. As before, the transfer functions of Fig. 6 are defined as jg 1 (!)j 2 =F(r) jg 2 (!)j 2 =F(q) where F is the Fourier transform operator, Á Z 1 1 jh 1 (!)j 2 = jg 1 (!)j 2 jg 2¼ 1 (!)j 2 d! 1 Á Z 1 1 jh 2 (!)j 2 = jg 2 (!)j 2 jg 2¼ 2 (!)j 2 d!: 1 MARIER: CORRELATED K-DISTRIBUTED CLUTTER GENERATION FOR RADAR DETECTION AND TRACK 575

The graphs in Figs. 7 8 together with the processing of Fig. 6, completely define an approach for the generation of the K-distributed clutter vectors of any correlation. The model employs μ RVs (w 1,i,w 2,i,:::,w μ,i ) to form the Gamma distributed RV Y i, which is combined with the exponentially distributed RV X s to find the K-distributed Z i of order º = μ=2 1. Because μ =1,2,:::, the parameter º assumes half integer integer values beginning with 1=2. While the expressions for the pdf moments of Z computed in Section V are correct for any º, the generation of correlated, K-distributed RVs is not defined for general º. We suspect that the model set forth in Fig. 6 bounds the problem of clutter representation for general º. However,[34]reports on a method to extend the results to general º for K-distributed, spatially correlated clutter. VII. CONCLUSIONS In this paper, we have provided a complete methodology for the generation of noncoherent vectors of arbitrary correlation from the three most common non-gaussian clutter distributions: log normal, Weibull, K. We presented processing diagrams showing details of each RVs realizations corresponding correlation coefficients. We extended previous results for K-distributed clutter to provide for the generation of samples of any correlation. This permits the useful representation of highly correlated spatial temporal clutter for radar detection track simulations. Joint pdfs moments for the K-distribution were derived shown to meet prescribed conditions for correct, jointly correlated RVs. APPENDIX A. DERIVATION OF CORRELATED RAYLEIGH-DISTRIBUTED MARGINAL PDF To compute f Z (z i,z j j y i,y j )wherez i z j are correlated Rayleigh-distributed RVs, recall that p x si, the square root of an exponential RV, is Rayleigh distributed, from (14) from (16), Defining H = Z 2, Z i = p x si p yi f Z (z j y)=r(z;y): f H ( j y)= 1 2y exp( =2y) = G( ;2y,): (38) The next step is to compute the joint pdf, f H ( i, j j y i,y j ), for the two correlated Gamma-distributed RVs, i j, with a result from Raghavan, [25, eq. (7)]. But a problem arises because Raghavan s result requires that the two s be identically distributed RVs.Thisisnotthecaseherebecausethesecond arguments of (38), 2y i 2y j for the distributions of i j, respectively, differ. Hence the distributions are not identical. To overcome this problem, we define anewvariable i,where i = iy j =y i : Then Because f H ( i j y i )= 1 2y j exp( i =2y j ) = G( i ;2y j,): f H ( j j y j )=G( j;2y j,) the distributions for i j are identical. Their joint pdf can now be computed from [25, eq. (7)]. Then 1 f H ( i, j j y i,y j )= X p n,s G( i ;d j,n)g( j;d j,n) where n= (39) d j =2y j (1 t ) (4) p n,s =(1 t )t n (41) t is the correlation coefficient between i j given by Ef[( i t = E( i j y i )) j y i ][( j E( j j y j )) j y j ]g ¾ i jy i ¾ jjy j where ¾ 2 i jy = Ef[ i E( i j y i i )]2 j y i g ¾ 2 jjy j is similarly defined. The statistics of i j are identical we write t more compactly as t = E[ i j j y i,y j ] E2 [ j j y j ] ¾ 2 jjy : j Because j = x sj y j i = i y j =y i = x s i y j t = E[x s i x sj ] E 2 [x s ] ¾ 2 x s (42) (39) (42) provide the joint pdf of i j.tofindthe desired joint pdf for z i z j,wetransformthe s to zs: s yi z i = q i y j Then, z j = p j: f Z (z i,z j j y i,y j )= f H ( i, j j y i,y j ) jj( i, j)j 576 IEEE TRANSACTIONS ON AEROSPACE AND ELECTRONIC SYSTEMS VOL. 31, NO. 2 APRIL 1995

where J is the Jacobian of the transformation of to z, J =1=(4z i z j y j =y i ): We obtain But where f Z (z i,z j j y i,y j ) Finally, =4z i z j y j y i n= p n,s G(z 2 i y j =y i ;d j,n)g(z2 j ;d j,n): (43) y j y i G(z 2 i y j =y i ;d j,n)=g(z2 i ;d i,n) f Z (z i,z j j y i,y j )=4z i z j d i =2y i (1 t ): (44) 1 X n= p n,s G(z 2 i ;d i,n)g(z2 j ;d j,n): APPENDIX B. DERIVATION OF CORRELATED K-DISTRIBUTED PDF The correlated, K-distributed pdf, f Z (z i,z j )is obtained from (28): f Z (z i,z j )= Z 1 Z 1 f Z [z i,z j j y i,y j ]f y (y i,y j )dy i dy j : Substituting (19) (24) into (28), we obtain f Z (z i,z j )= Z 1 Z 1 4z i z j X 1 p n,s n= (45) G[zi 2 ;2y i (1 t ),n]g[z2 j ;2y j (1 t ),n] p m G[y i ;b,m + º] m= G[y j ;b,m + º]dy i dy j : (46) Consider one term, I nm, of this double sum: I nm = Z 1 G[z 2 i ;2y i (1 t ),n]g[y i ;b,m + º]dy i : (47) Substituting (13b) into (47) making the following substitutions, we find, I nm = n!b n+1 i = y i =b ¹ 2 i =4z 2 i =[2b (1 t )] z 2n i (m + º +1) Z 1 m+º n 1 i exp( ¹ 2 i =4 i + i )d i [2(1 t )] n+1 : (48) From [35, pg. 34, 3:471:12] 2zi m+º+n K m+º n (z i =ac 1=2 ) I nm = n!(4a 2 c ) n+1 (m + º + 1)(4a 2 c ) (m+º n)=2 (49) where c =(1 ½ )(1 t )K º ( ) is the modified Bessel function of order º. Defining J nm similarly for z j y j,wehave, f Z (z i z j )=4z i z j 1 X n= m= p n,s p m I nm J nm : (5) After rearranging collecting terms, we find, where f Z (z i,z j )= 4p n,s p m m+º+n+1 i m+º+n+1 j = [n! (m + º +1)ac 1=2 ] 2 K m+º n (2 i)k m+º n (2 j) (51) i = z i =(2ac 1=2 ) j = z j =(2ac 1=2 ): (52) The form of f Z is quite complicated. Because of the K term (itself an infinite series), f Z cannot be factored into the product of two infinite series. For t =, the Rayleigh variates become uncorrelated the series simplify considerably. In this case, p n,s =1 n = = n 6= the sum runs only over m. Then f Z (z i,z j )= p m K[z i ;a(1 ½ ) 1=2,m + º] K[z j ;a(1 ½ ) 1=2,m + º] (t =) which matches the result for the same conditions in [25]. APPENDIX C. DERIVATION OF MOMENTS FOR CORRELATED K-DISTRIBUTION To compute the l,k moments (l,k =,1,2,:::)ofthe correlated K-distribution we form E[z l i zk j ]= ZZ z l i zk j f Z (z i,z j )dz i dz j (53) where f Z is given by (51). To minimize the computations, we set Z I l = z i m+º+n+1 l i K (2 i)dz m+º n i : (54) MARIER: CORRELATED K-DISTRIBUTED CLUTTER GENERATION FOR RADAR DETECTION AND TRACK 577

Integrating over i, as defined in (52), we have Z 1 I l =(2ac 1=2 ) l+1 m+º+n+1+l i K m+º n (2 i)d i: (55) From [35, pg. 684, 6:561:16], I l = (m + º +1+l=2) (n +1+l=2)(2ac 1=2 ) l+1 =4: (56) Note that the last integral has produced an important simplification: the modified Bessel function has been removed I l now has separable m n portions. I k is defined similarly to I l but applies to j. The final expression for all orders of the moments can be written as the simple product of two series the computer evaluation of the moments becomes far less difficult. Denoting the l,kth moment by M l,k, i.e., we have, M l,k = m= n= E[z l i zk j ]=M l,k I l I k 4p n,s p m =[n! (m + º +1)ac 1=2 ] 2 (57) = X X (m + º +1+l=2) m n (m + º +1+k=2)½ m =[m! (m + º +1)] (n +1+l=2) (n +1+k=2)t n =(n!) 2 (2ac 1=2 ) l+k (1 t)(1 ½) º+1 = (º + 1) (58) where we have suppressed the i j subscripts on ½, t, c. M l,k now consists of the products of the two infinite series over m n. We first consider the sum over m. Then m= (m + º +1+l=2) (m + º +1+k=2)½ m =[m! (m + º +1)] = (º + l=2+1) (º + k=2+1) (º +1) 2 F 1 (º + l=2+1,º + k=2+1;º +1;½) (59) where from [31], the Gaussian hypergeometric function 2 F 1,isgivenby 2 F (, ;,z)=1+ ( +1) ( +1)z2 1 z + + : ( +1)2! (6) Considering next the sum over n, (n +1+l=2) (n +1+k=2)t n =(n!) 2 m= = (l=2+1) (k=2+1) 2 F 1 (l=2+1,k=2+1;l,t) (61) which is similar to (59) but with º =. The remaining terms can be put in the form: (2ac 1=2 ) l+k (1 t)(1 ½) º+1 = (º +1) =(2a) l+k (1 ½) l=2+k=2+º+1 Using the identity [36, pg. 15], (1 t) l=2+k=2+1 = (º +1): (62) (1 z) 2 F 1 (, ; ;z)= 2 F 1 (, ; ;z) then the product of (59), (61), (62) becomes M l,k = (º + l=2+1) (º + k=2+1) (l=2+1) (k=2+1)= 2 (º +1) (2a) l+k 2 F 1 ( l=2, k=2;º +1;½) 2 F 1 ( l=2, k=2;1;t) (63) which is (31). It is not difficult to check that (63) gives the anticipated values for various moments. For example, M, =1 M 1,1 =4a 2 2 (º +3=2) 2 (3=2)= 2 (º +1) = E 2 (z), (½ = t =) in agreement with (32) for z given by the distribution (13a), M 1,1 =4a 2 (º +1)=E(z 2 ) (½ = t =1) in agreement with (33) where we have made use of the identity [36, pg. 14], 2 F 1 (, ; ;1)= ( ) ( + + )=[ ( ) ( )]: Similarly, μ M l,k =(2a) l+k º +1+ l + k 2 μ 1+ l + k Á (º +1) 2 = E[z l+k ] for ½ = t = 1 (64) i.e., when z i z j are perfectly correlated, the moment correctly depends only on the sum, l + k. APPENDIX D. VERIFICATION THAT CONDITIONS (18A) (18E) ARE MET 1) Marginal Distribution f Z (z): We show that the marginal distribution of (29) is given exactly by the K-distribution, (13a). We begin by computing the 578 IEEE TRANSACTIONS ON AEROSPACE AND ELECTRONIC SYSTEMS VOL. 31, NO. 2 APRIL 1995

marginal distribution of f Y (y i,y j ) from (24). Then, f Y (y j )= = Z 1 Z 1 f y (y i,y j )dy i p m G(y i ;b,m + º) m= G(y j ;b,m + º)dy i (65) = p m G(y j ;b,m + º): (66) m= Substituting from (26) for p m from (13b) for G, we find that terms containing m are a power series expansion for expf½y=[2a 2 (1 ½)]g f Y (y j )=G[y j ;2a 2,º]: (67) Thus the marginal pdf of (24) is a Gamma distribution, as expected. We make use of this fact shortly. We now compute the marginal distribution of f Z (z i,z j ) begin with the form of (28). We first integrate f Z over z i, by reversing the order of integration: 5 f Z (z j )= = = Z 1 f Z (z i,z j )dz i Z 1 Z 1 Z 1 f Z [z i,z j j y i,y j ] f Y (y i,y j )dz i dy i dy j (68) Z 1 Z 1 f Z [z j j y j ]f y (y i,y j )dy i dy j : (69) Note from (19) that following integration over z i in (68), y i which is associated only with z i, no longer appears in f Z [z j j y j ]. Then, integrating over y i, f Z (z j )= Note that (from (16)) Z 1 f Z (z j j y j )f y (y j )dy j : (7) f Z (z j j y j )=R[z j ;y j ] f Y (y j ) is given by (67). With these substitutions, a straightforward evaluation of (7) shows that f Z (z j )=K[z i ;a,º] as expected, (18a) (18b) are confirmed. 2) Autocorrelation; Decorrelated Correlated z: Note that condition (18c) is automatically met because the correlation coefficients r q were chosen accordingly in (36). 5 This is permited because the appropriate integral is uniformly convergent. To confirm (18d), we set ½ i,j = t i,j =in (29). Figs. 7 8 show that then s i,j becomes, decorrelating the samples z i z j. Then, p n,s p m become zero except for n = m = 1, (29) reduces to the product of two K-distributed pdfs for z i z j, each of the form (13a). To confirm (18e), the pdf of two perfectly correlated RVs, z i z j, refer to (64) for M l,k,the l,kth moment of z i z j.observethat,first,for perfectly correlated RVs, all moments are a function of the sum l + k. Second, the expression for M l,k can easily be shown to be identical to the l + kth moment of the K-distribution, (13a). Therefore, the pdf of two perfectly correlated RVs, z i z j,hasthesame characteristic function as the K-distribution, their pdfs must be equal (18e) is confirmed. REFERENCES [1] Trunk, G. V., George, S. F. (197) Detection of targets in non-gaussian sea clutter. IEEE Transactions on Aerospace Electronic Systems, AES-6, 5 (Sept. 197), 62 628. [2] Daley,J.C.,Ransone,J.T.,Burkett,J.A.,Duncan,J.R. (1968) Sea clutter measurements on four frequencies. Report 686, Naval Research Laboratory, Washington, DC, Nov. 1968. [3] Valenzuela, G. R., Laing, M. B. (1972) Point-scatterer formulation of terrain clutter statistics. Report 7459, Naval Research Laboratory, Sept. 1972. [4] Schleher, D. C. (1976) Radar detection in Weibull clutter. IEEE Transactions on Aerospace Electronic Systems, AES-12, 6 (Nov. 1976), 736 743. [5] Sekine, M., Mao, Y. (199) Weibull Radar Clutter. London: Peregrinus, 199. [6] Boothe, R. R. (1969) The Weibull distribution applied to the ground clutter backscatter coefficient. Report RE-TR-69-15, U.S. Army Missile Comm, 1969. [7] Sekine, M., Ohtani, S., Musha, T., Irabu, T., Kuichi, E., Hagisawa, T., Tomita, Y. (1981) Weibull distributed ground clutter. IEEE Transactions on Aerospace Electronic Systems, AES-17, 4 (July 1981), 596 598. [8] Schleher, D. C. (1978) MTI Radar. Dedham, MA: Artech House, 1978. [9] Oliver, C. J. (1988) Representation of radar sea clutter. IEE Proceedings, 135, Pt. F, 6 (Dec. 1988), 497 5. [1] Ward, K. D. (1981) Compound representation of high resolution sea clutter. Electronics Letters, 17, 16 (Aug. 6, 1981), 561 563. [11] Watts, S., Ward, K. D. (1987) Spatial correlation in K-distributed sea clutter. IEE Proceedings, 134, Pt.F, 6 (Oct. 1987), 526 532. [12] Jakeman, E., Pusey, P. N. (1976) A model for non-rayleigh sea echo. IEEE Transactions on Antennas Propagation, AP-24, 6 (Nov. 1976), 86 814. MARIER: CORRELATED K-DISTRIBUTED CLUTTER GENERATION FOR RADAR DETECTION AND TRACK 579

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