CFAR BI DETECTOR IN BINOMIAL DISTRIBUTION PULSE JAMMING 1. I. Garvanov. (Submitted by Academician Ivan Popchev on June 23, 2003)

Transcription

1 FA BI EEO I BIOMIAL ISIBUIO PULSE JAMMIG I. Gavanov (Submtted by Academcan Ivan Popchev on June 3, 3) Abtact: In many pactcal tuaton, howeve, the envonment peence of tong pule ammng (PJ) wth hgh ntenty; the pule ammng wth bnomal dtbuton. In th cae, the pule ammng can caue datc degadaton n the pefomance of ontant Fale Alam ate (FA) poceo. In th pape, A FA and A FA BI (bnay ntegaton) poceo n bnomal dtbuted pule ammng ae tuded. he mathematcal epeon fo calculatng of the pobablty of detecton, pobablty of fale alam and the aveage decon thehold (A) fo A FA detecto n bnomal dtbuted pule ammng ae offeed. he eult fo the A ae eceved analytcally and the pobablty functon (S). eeach pefomed n MALAB envonment. Keywod: ada etecto, FA Poceo, Pule Jammng, Aveage econ hehold (A), Pobablty of detecton, Pobablty of fale alam.. Intoducton. Bnay ntegaton poceo ae ued fo taget ada detecton (BIpoceo) and employ a two-tep thehold technque. In the ft tep a pelmnay decon made about each pule of the tan eflected fom the taget. he pule detecton declaed f the ft adaptve thehold eceeded n the cell unde tet. In conventonal mean level contant fale alam ate (A FA) detecto popoed by Fnn and Johnon n [], the detecton thehold detemned a a poduct of the noe level etmate n the efeence wndow and a cale facto to acheve the degn pobablty of fale alam. In the econd tep the numbe of decon whee the ft thehold eceeded ae counted and th numbe compaed wth the econd thehold. he taget ada detecton declaed f the econd thehold eceeded. he peence of pule ammng (PJ) n both, the tet eoluton cell and the efeence cell, can caue datc degadaton n the pefomance of a A FA poceo a hown n []. he law of dtbuton of tong PJ change fom Poon to bnomal wth the nceang of the pobablty fo the appeaance of pule ammng. he optmal algothm fo A FA poceo n tong pule ammng wth a poy known paamete and Poon dtbuton ae peented n pape [4,5,7-]. eeach caed out fo the whole ange of appeaance of pule ammng. he autho ue n pape [4, 5, 7-] the acheved analytcal epeon fo FA detecto opeatng n the peence of pule ammng wth Poon dtbuton, n cae of PJ wth a epetton fequency hghe than.. It obvou that n th cae the dtbuton law change fom Poon to bnomal. he autho conde that t poble to tudy the behavo of FA detecto fo hghe epetton fequency of PJ by ung the above-mentoned eult, a thee two dtbuton law ae mla. he eult acheved and peented by the autho n pape [9] and n the peent pape, eveal that the autho hypothe tue. he chaacte of the pobablty functon of the A and the cale facto of A FA poceo ae equvalent at hghe fequence of PJ fo both type of PJ dtbuton. Becaue of the change n the PJ dtbuton, the loe fo keepng of contant fale alam ato ae hghe than the A (S) loe. h wok uppoted by II 44, MPS Ltd. Gant, Bulgaan F S Gant ¹J I 9/99 and Mnty of efene Gant ¹

2 In th pape, contnued the eeach wok concenng the above-mentoned tude. he acheved eult ae optmal fo hgh epetton fequency of pule ammng. he optmal analytcal epeon fo the pobablty functon, A and the cale facto of A FA, FA BI and API FA detecto, n the peence of PJ wth epetton fequency up to., ae peented n pape [4, 5, 7-]. he epeon fo A FA and FA BI detecto n the peence of hgh epetton fequency PJ ae peented n th pape. In th wok the loe of A FA and A FA BI poceo n bnomal dtbuted pule ammng towad the homogeneou eceve noe ae tuded. he noe n the tet cell aylegh envelope dtbuted and the taget etun ae fluctuatng accodng to Swellng II model, a n [3,5]. he eult, acheved fo the A and the cale facto of A FA and A FA BI poceo, ae mla fo pule ammng wth bnomal dtbuton n the ente ange of the pobablty fo the appeaance of pule ammng. he epemental eult how that the A FA BI poceo wth the bnay ule M-out-of-L 6/6 ae mot effectve fo lowe value of the pobablty fo the appeaance of pule ammng to.5. Fo hghe value of the pobablty fo the appeaance of pule ammng above.5 ecommended bnay ule /6.. Sgnal model. onde a ada detecto fo whch the eceved gnal ampled n ange by the () eoluton cell, eultng n a vecto of () obevaton. he amplng ate uch that the ample ae tattcally ndependent. Afte fltaton the gnal appled to a quae-law detecto and then poceed n the A FA decon element. In condton of bnomal dtbuton of pule ammng, the backgound envonment nclude the ntefeence-plu-noe tuaton, whch may appea at the output of the eceve wth the pobablty, two ntefeence-plu-noe tuaton wth the pobablty and the noe only tuaton wth the pobablty ( ), whee t c F, F the aveage epetton fequency of PJ and t c the length of pule tanmon [6]. he dtbuton bnomal when the pobablty of PJ above.. In hee tuaton the output of the efeence wndow ae obevaton fom tattcally ndependent eponental andom vaable. onequently, the pobablty denty functon (pdf) of the efeence wndow output may be defned by: () f λ ep λ λ ( ) λ ( ) λ ( ) λ ( ) ep ep, whee λ the aveage powe of the eceve noe, the aveage ntefeence-to-noe ato (I) of pule ammng and the numbe of obevaton n a efeence wndow. he ample fom the tet eoluton cell aume to be dtbuted accodng to Swelng II cae wth the pdf gven by: () f ep λ λ ( ) λ ( ) ep ep ( ) ( ) ( ) λ λ λ whee the pe pule aveage gnal-to-noe ato (S). he et of ample { } poceed, eultng n a noe level etmate. he etmate multpled by a pedetemned cale facto, eultng n a pule detecton thehold ( H ). he ample fom the tet eoluton cell compaed wth the thehold H. he taget gnal detecton declaed, f the ample eceed the thehold H :

3 (3) H H o : : o o H < H whee H the hypothe that the tet eoluton cell contan the echoe fom the taget and the hypothe that the tet eoluton cell contan the pule ammng only. H o 3. Analy of the A FA detecto. 3.. Pobablty of detecton and fale alam of a A FA detecto Accodng to the ule (3), the pobablty of pule detecton P may be found a: (4) () ( / ) P P d f H d whee the etmate of the noe level n the cell unde tet, P, and f ( H ) the pdf of the etmate / the condtonal pdf of the tet ample unde hypothe H. Ung ( and 4) the pobablty of pule detecton P evaluated a: (5) P ( ) M λ ( ) ( ) M λ M λ whee M (). the moment geneatng functon (mgf) of the noe level etmate. In a conventonal A FA detecto the noe level etmate output of the efeence wndow: fomed a a um of all the. In th cae the mgf of the etmate defned to be M ( U ) M ( U ), whee M ( U ) the mgf of the andom vaable. he mgf of the andom vaable found a a mathematcal epectaton: (6) M ( U ) ep( U ) f () d whee the pobablty denty functon f () defned n (). (7) ( ) ( ) M ( U ) Uλ Uλ ( ) Uλ ( ) hen the mgf of the etmate take the fom: (8) M ( U) ( ( ( ) Uλ Uλ ( ) ( Uλ ) he pobablty of taget gnal detecton computed by the epeon, peented n [], on bae epeon fo Poon dtbuton eceve n [7]. 3

4 4 (9) { } 3 P, whee 3 he pobablty of fale alam evaluated by (9), ettng. 3.. Aveage decon thehold of FA detecto he aveage decon thehold (A) defned a a nomalzed quantty []: () A E FA / λ whee the andom vaable the eult of the etmaton method ued n the FA ytem, the calng facto fo thehold adutment adapted to the etmaton method and equed P FA. E tand fo the epectaton. () () E d d M / / λ λ evatng fom the method uually decbed n ada lteatue, ae ue the aveage decon thehold (A) fo compaon of vaou FA poceo. h povde the advantage that the dffeence, etng between vaou FA ytem, then epeed by a nglevalued meaue. he dffeence between two FA ytem can be epeed by the ato between the two A meaued n db []. () [ ] db A A E E log fo P P P P fa fa 5,. etemnaton of the A fo a A FA detecto n bnomal dtbuted PJ: Ung (8, ) and ubttute U / λ, fo, the A epeon fom [] eceved: (3) A

5 whee computed by epeon (9), ettng. Fo o wthout pule ammng A / AFA a n [3], whee P fa 4. Analy of the A FA BI detecto. Let u aume that L pule ht the taget, whch modeled accodng to Swelng cae II. he eceved gnal ampled n ange by ung () eoluton cell eultng n a mat wth () ow and L column. Each column of the data mat cont of the value of the gnal obtaned fo L pule nteval n one ange eoluton cell. Let u alo aume that the ft (/) and the lat (/) ow of the data mat ae ued a a efeence wndow n ode to etmate the noe-plu-ntefeence level n the tet eoluton cell of the ada. In th cae the ample of the efeence cell eult n a mat X of the ze * L. he tet cell o the ada taget nclude the element of the (/) ow of the data mat and a vecto X of length L. he element of the efeence wndow ae ndependent andom vaable wth the compound eponental dtbuton law (). In the peence of a deed gnal fom a taget the element of the tet eoluton cell ae ndependent andom vaable wth pobablty denty functon gven by (). All ample fom the tet eoluton cell at the nput of the bnay ntegato l (l,..,l) ae compaed wth the thehold H accodng to the ule: (4) Φ ( ) l l, f l H, othewe he bnay ntegato pefom ummng of L decon Φ l. he taget ada detecton declaed f th um eceed the econd thehold M.. (5) H H : ( t ag et peent), :( not aget), f L Φ M l l othewe he pobablty of taget detecton fo A FA BI poceo n [] computed by the epeon: L L (7) l l l P P P l M whee P pobablty of detecton fom (9). he pobablty of fale alam evaluated by (7), ettng. 5. umecal eult. he new analytcal epeon obtaned n the pevou ecton make poble the etmaton of the qualty of A FA and A FA BI detecto n the peence of vey ntenve pule ammng wth bnomal dtbuted. eeach pefomed n MALAB envonment. he aveage decon thehold A and the cale facto fo A FA and A FA BI poceo n pule ammng wth bnomal dtbuted ae hown on fg. and. he epemental eult ae obtaned fo the followng paamete: aveage powe of the eceve noe λ ; aveage ntefeence-to noe ato (I) 3 [db]; pobablty fo the appeaance of pule ammng fom to ; numbe of efeence cell 6 and pobablty of L 5

6 fale alam P fa 6 and the bnay ule M-out-of-L (/6 and 6/6). he eult fo the A ae eceved ung the pobablty functon (S). Fg.. Aveage decon thehold A and cale facto fo A FA poceo Fg.. Aveage decon thehold A and cale facto fo A FA BI poceo In a A FA poceo, the noe level etmate n efeence wndow nceae wth nceae of the aveage ntefeence-to-noe ato and the pobablty fo the appeaance of pule ammng wth length n the ange cell. In ode to keep contant the fale alam pobablty n a A FA poceo, the cale facto deceae wth the nceae of the pobablty fo the appeaance of pule ammng. Fg. 3. A FA loe Fg. 4. A FA BI loe he aveage decon thehold (A) pmaly nceae wth the nceae of the pobablty fo the appeaance of pule ammng up to. -.3, and then deceae fat (fg.). A a eult of the deceang of the A, the loe of the A FA detecto alo deceae (fg. 3). he FA loe n bnomal dtbuton pule ammng towad the homogeneou eceve noe ae hown on fg.3 and 4. In fg.4 can be een that FA BI poceo wth the bnay ule M-out-of-L6/6 bette n all aea fo the appeaance of pule ammng towad the cae wth eceve noe. Fom fg. can be een that FA BI poceo wth the bnay ule M-out-of-L6/6 bette n cae of lowe value of the pobablty fo the appeaance of pule ammng to.5. Fo hghe value of the pobablty fo the appeaance of pule ammng above.5, the ung of the bnay ule M-out-of-L/6 eult n lowe loe. 6

7 6. oncluon. he change of the dtbuton law make mpoble the keepng of contant fale alam ate of the A FA poceo and lead to deceang of the pobablty of detecton. Unfotunately, the law of dtbuton of tong PJ change fom Poon to bnomal wth the nceang of the pobablty fo the appeaance of pule ammng. In th pape the mathematcal equaton fo etmaton of the pobablty of detecton, pobablty of fale alam and the aveage decon thehold (A) fo A FA detecto n the ente aea of the pobablty fo the appeaance of tong pule ammng ae offeed. he obtaned eult fo the A and fo the cale facto of A FA and A FA BI poceo ae mla fo pule ammng wth bnomal dtbuton n the ente ange (fom up to ) of the pobablty fo the appeaance of pule ammng. he epemental eult how that the A FA BI poceo wth the bnay ule 6/6 ae mot effectve fo lowe value of the pobablty fo the appeaance of pule ammng. Fo hghe value of the pobablty fo the appeaance of pule ammng above.5 ecommended bnay ule /6. he optmal analytcal epeon fo the pobablty functon, A and the cale facto of A FA, FA BI and API FA detecto, n the peence of PJ wth epetton fequency up to., ae peented n pape [4, 5, 7-]. he epeon fo A FA and FA BI detecto n the peence of hgh epetton fequency PJ ae peented n th pape. EFEEES [] Fnn H. M., P. S. Johnon. A eve, 9, 968, ¹e 3, [] ohlng H. IEEE an., AES-9, 983, ¹ 4, [3] Gandh P. P., S. A. Kaam IEEE an., AES-4, 988, ¹ 4, [4] Hmona S., M. Bakat, IEEE an., AES-8, 99, ¹, [5] Hmona S. IEEE an., AES-3, 994, ¹Ñ 3, [6] Akmov P., F. Evtatov, S. Zahaov, ado Sgnal etecton, 989, 95-3, (n uan). [7] Beha., ompt. end. Acad. Bulg. Sc., 49, 996, ¹ 7/8, [8] Gavanov I.,. Kabakchev, Poc. GS,, [9] Gavanov I.,. Beha, h. Kabakchev, LS 54, pp. 9-98, 3. [] Gavanov I., h. Kabakchev, I,, ¹^, [] Gavanov I. II/WP-45,, -. Inttute of Infomaton echnologe Bulgaan Academy of Scence Akad. G. Bonchev St., bl., 3 Sofa, Bulgaa E-mal: gavanov@t.ba.bg 7