nd Intenational Wokshop on ateials Engineeing and Copute Sciences (IWECS 05) An Adaptive Diagonal oading Covaiance atix Estiato in Spatially eteogeneous Sea Clutte Yanling Shi, Xiaoyan Xie College of elecounications & Infoation Engineeing Nanjing Univesity of Posts and elecounications Nanjing 0003, China Eail:ylshi@njupt.edu.cn 8655678@63.co eywods: diagonal loading; covaiance atix estiato; heteogeneous sea clutte; ada taget detection Abstact. he ost typical ethod to estiate the covaiance atix of sea clutte by seconday saples is the saple covaiance atix (SC), iplying seconday saples with the sae statistical popety. he hypothesis established o not depends on whethe o not sea clutte is spatially hoogeneous. In ode to get id of the hypothesis on seconday saples, a novel covaiance atix estiation algoith based on adaptive diagonal loading technique is pesented. he loading coefficient is a facto to easue the statistical consistency of seconday saples. Expeiental esults show that, to detect the distibuted tagets in eal sea clutte by genealized likelihood atio detecto (GR), the poposed estiato is obust to jaing and the pefoance ipoves in diffeent ange esolution sea clutte, especially as high as 5dB in 3 ange esolution. Intoduction It is well known that saple covaiance atix (SC) estiato is the axiu likelihood estiate of covaiance atix in Gaussian white noise []. It is siple in opeation and sall in coputation. Fo taget detection in sea clutte, SC estiato woks well in the low ange esolution whee sea clutte can be odelled as coplex Gaussian ando pocess. Seconday saples ae with the sae statistical popeties, such as shae the sae covaiance atix. oweve, with the ipoveent of ada ange esolution, sea clutte in the ange cell is the vecto supeposition of only a few scattees. Sea clutte is spatially heteogeneous [], which is ainly eflected in the inconsistency of covaiance atix of sea clutte. Fo the high ange esolution, due to the statistical inconsistency of seconday saples, SC estiato has a lage eo in estiating covaiance atix, which will lead to a pefoance loss of detecto. In this lette, an adaptive covaiance atix estiato based on the diagonal loading (D) is poposed. It cobines SC with the unit atix adaptively, whee the coefficient of cobination changes with the statistical popety of seconday saples adaptively. Expeiental esults show that, to detect the distibuted taget in eal sea clutte by genealized likelihood atio test (GR) detecto [3], the poposed estiato is obust to jaing, and its pefoance has a significant ipoveent in high ange ada esolution sea clutte, copaed with SC. Poble Desciption et N -diensional coplex vecto z, c, p be the eceived echoes, sea clutte and Dopple steeing vecto, =,,,, whee is the nube of ange cells that taget pesences. α, =,,,, ae unknown constants elevant to taget ada coss section. z, =,,, ae seconday saples. aget detection in sea clutte is a conventional binay hypothesis testing, defined as 05. he authos - Published by Atlantis Pess 470
0 : z = c, =,,,, +,, z = αp+ c, =,,, () : z = c, =,, Fo the distibuted taget, the GR detecto [3] is ˆ p z t = N ln () ( ˆ )( ˆ = p p z z) whee ˆ is the estiated tepoal covaiance atix of the clutte vecto c. In the lette, the supescipt denotes conjugate tanspose, and the supescipt denotes tanspose. Poposed Estiato Accoding to the statistical inconsistency of sea clutte in the high ange esolution ada, a new estiato of covaiance atix of the clutte vecto c is pesented ˆ = ( β) + βi (3) whee is SC, given by + = = + zz (4) and I is an unit atix. he weighted coefficient β is a facto to easue the statistical consistency of seconday saples, calculated as follows β = zz = Sybol A = t( AA ). Foula (3) involves Ο ( + ) N floating-point opeations. In addition, it is easy to pove 0 β. he facto β eflects the statistical consistency of covaiance atix of seconday saples z, =,,, the nueato of facto β is equivalent to the vaiance of exteio poduct of z, eflecting the deviation of exteio poduct fo the ean of SC. If the degee of deviation is lage, the exteio poduct atix is instable, and thee is a bad statistical consistency of covaiance atix of z s in diffeent ange cells, which is coesponding to heteogeneous sea clutte backgound. hus, SC will obviously bing in estiation eos. ence, it is necessay to use β to odify the estiation esults. And, vice vesa. (5) Fig. he change of β with the inceasing In addition, because β eflects the degee of deviation between two atixes, consideed the change of β with the nube of seconday saples, the gaph of β vesus is shown in Fig by 47
using the siulation data, whee z ~ CN( 0I, ), N = 8, = 0. he siulation was conducted with 50 onte-calo uns. Fo Fig, the cuve shows the fall in β with inceasing. he lage, the salle β, and tend to 0 eventually, naely li β = 0. When the clutte is spatially hoogeneous fo a lage, coesponding to an abundant of seconday saples available, ˆ tends to. hus, the covaiance atix can alost be estiated by SC accuately. So, it akes no sense to calculate β any oe. When the nube of the seconday saple is sall, a deviation will be bought in by SC. heefoe, it is indispensable to intoduce the paaete β to evise the estiation esults. he conclusion can be eached that the poposed estiato is suitable fo the spatially heteogeneous and/o a sall nube of seconday saples available. et the eigendecoposition of be = UΛU (6) whee U is the clutte subspace atix, Λ is diagonal atix coposed of the eigenvalues of clutte, Λ = diag( λ, λ,, λ N ) λ > λ > > λn (7) Substituting (6) into (), it can be shown that ˆ β = ( β ) U[ Λ + IU ] (8) β he eigenvalues of ˆ ae ( βλ+ β, n=,,, N. he invese atix of ˆ is, ) n ˆ β = ( β ) U Λ + I U (9) β Consideing the influence of clutte powe, we noalize the eigenvalue by λ. hus ˆ β = ( β) U Λ / λ + I β U (0) λ λβ = U + β Λ I β U Substituting (0) into (), the GR detecto is elated with the loading coefficient λβ /( β). Fo foula (8), ˆ has a siila fo to the D algoith in space-tie adaptive pocessing (SAP) [4] which copesses eigenvalues of noise, weakens the adaptive capacity to sall jaings, with the ai of deceasing the syste sensitivity. he loading coefficient in SAP is geneally set by expeience, which ay deviate away fo an actual envionent and bing in an eo. As to guaantee the invetibility of the covaiance atix with sall nube of seconday saples, the D algoith is used to estiate the covaiance atix of clutte, whee the loading coefficient is -3 ties of the noise powe [5]. So the loading coefficient is also an epiical value. In the lette, β is adaptive with the statistical consistency of seconday saples. he loading coefficient is set with an actual condition, no longe an expeience value. Fo spatially heteogeneous sea clutte in high ada ange esolution, the covaiance atix of sea clutte vaies with the spatial seconday saples [6]. Seconday saples do not necessaily shae the sae covaiance atix. oe pecisely, we assue that the seconday saples can be clusteed into goups, each one of the containing snapshots and shaing the sae covaiance atix. he paticula case = coesponds to the scenaio whee sea clutte is spatially hoogeneous. he extee case = fo =,, would coespond to a situation whee all seconday saples would have a diffeent covaiance atix. et = [,,,, ], = =. Regoup the seconday saples as Z = [ z,, z,, z ] = [ Z,, Z,, Z ], whee C N N z and Z C. he SC based on Z is denoted as = Z Z () 47
Siilaly, based on (3), the estiation of covaiance atix ˆ is ˆ = ( β) + βi () = U[( β) Λ + βiu ] whee U, Λ and β ae the eigenvecto atix, eigenvalue atix and weighted coefficient of, espectively. Noalize the eigenvalue by λ,, ˆ β λ,β = U Λ + I U (3) λ, β whee λ, is the axiu eigenvalue of Λ. So, the covaiance atix estiato of Z is given by ˆ = ˆ = (4) ( β ) λ,β = U Λ + I U = λ, β Substituting (4) into (), the GR detecto is elated not only with the loading coefficient λ, β /( β) but also with ( β) / λ,. Expeiental Wok We now illustate the pefoance of the vaious estiation schees intoduced in the pevious sections. he following expeient uses the eal sea clutte data collected by IPIX ada (http://soa.caste.ca/ipix.php). he file is 99803-70435 with the ange esolution 5. he synthetic taget etuns ae expessed as: S( n) = aexp(i( pnfd + ϕ)), n=,,, N, =,, (5) whee fd is the taget Dopple fequency, is the coheent pocessing inteval, the initial phase ϕ is a unifo distibuted ando nube in [ ππ, ] in tials, and a is a constant elevant to ada coss section of taget used to adjust signal-to-clutte atio (SCR), defined by SCR = 0log a / P (6) 0 ( c) whee P c is the aveage powe of sea clutte tie seies. a = a = = a. he false ala pobability is taken as 0.00. he paaetes ae set as =, N = 4, = 8, f d =-00z. We vay the nube of goups. oe specifically, we conside fou cases with = (coespond to foula (8)), = and = 4, espectively. Fo = 4, we conside two cases: = = = = and diffeent in each goup. his eans that thee exist, o 4 diffeent covaiance atices aong the seconday saples. he fou cases ae defined by case : = 8 ;case : = [4,4] ;case 3: = [7,7,7,7] ;case 4: = [6,8,8,6]. Fig gives the detection pefoance of SC and poposed estiatos in fou cases. Fo detecting the distibuted taget in eal sea clutte by GR, the poposed estiato obviously outpefos SC. In ode to easue the obust of the poposed estiato, we add a jaing at the fist ange cell. he jaing is I( n) = Bexp(i( pnfi + ϕi)) (7) whee f i = 400z, ϕ i is a unifo distibuted ando nube in [ ππ, ] in tials, and B is adjust as intefeence-to-clutte atio (ICR), defined by ICR = 0log B P (8) 0 ( / c ) 473
Fig 3 gives the detection pefoance of SC and poposed estiatos in fou cases with ICR = 0dB. Copaed fig. with fig. 3, the esults pesented pove gouping is of advantage in ipoving anti-jaing capability. Fig. Pd vesus SCR of GR detecto with diffeent covaiance atix estiatos without intefeence. Fig. 3 Pd vesus SCR of GR detecto with diffeent covaiance atix estiatos with ICR = 0dB. (a) (b) (c) Fig. 4 Pd vesus SCR of GR detecto with poposed estiato and SC with the ange esolution (a) 5, (b) 9, (c) 3. 474
In following, we analyze the pefoance of spatially heteogeneous sea clutte in diffeent ange esolution. Suppose that the length of taget is 30 and N = 8, =. he paaetes ae = 8and = fo 99803-70435( the ange esolution is 5), = 6 and = 4 fo 99803-73950( the ange esolution is 9), = 0 and = 0 fo 99803-740( the ange esolution is 3), espectively. Fig. 4 gives the detection pefoance of SC and poposed estiato. he poposed estiato pefos even bette than SC with the highe ange esolution. Conclusion In ode to ovecoe the statistical inconsistency of seconday saples in the high ange esolution, a obust covaiance atix estiato based on the adaptive D is poposed. It is suitable fo spatially heteogeneous and/o a sall nube of seconday saples available. he expeiental esults show that, to detect the distibuted taget in eal sea clutte by GR, the poposed estiato is obust to jaing and the pefoance ipoves in diffeent ange esolution sea clutte, especially as high as 5dB in 3 ange esolution. Acknowledgent his wok was suppoted by National Natual Science Fund (6035) and Jiangsu Govenent Scholaship fo Oveseas Studies. Refeences []. Rangasway, Statistical analysis of the nonhoogeneity detecto fo non-gaussian intefeence backgounds, IEEE ans. Signal Pocessing, vol. 53, no.6, pp. 0-, 005. []. Geco, P. Stinco, F. Gini, Ipact of sea clutte nonstationaity on distubance covaiance atix estiation and CFAR detecto pefoance, IEEE ans. Aeosp. Electon. Syst, vol. 46, no.3, pp. 50-53, Septebe, 00. [3]X. F. Shuai,. J, ong and J. Y. Yang, Pefoance analysis of GR-based adaptive detecto fo distibuted tagets in copound-gaussian clutte, Signal pocessing, Vol. 90, pp. 6-3, 00. [4]G. Ginolhac, P. Foste, J. P. Ovalez, et al. Spatio-tepoal adaptive detecto in nonhoogeneous and low-ank clutte, IEEE Intenational Confeence on Acoustics, Speech and Signal Pocessing, pp. 045-048, 009. [5]I. A.Yui, A. J. Ben,. S. Nicholas, Saple-Deficient adaptive detection: adaptive scala thesholding vesus CFAR detecto pefoance, IEEE ans. on Aeospace and Electonic Systes, vol. 46, no., pp. 3-46, Januay, 00. [6]B. Olivie, B. Stéphanie,. Jean-Yves, Covaiance atix estiation with heteogeneous saples, IEEE ans. Signal Pocessing, vol.56, no. 3, pp. 909-90, 008. 475