Optimal Waveform Design with Constant Modulus Constraint for Rank-One Target Detection

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Senor & Tranduer Vol. 63 Iue January 4 pp. 39-43 Senor & Tranduer 4 by IFSA Publihing S. L. p://www.enorporal.

1 Senor & Tranduer Vol. 63 Iue January 4 pp Senor & Tranduer 4 by IFSA Publihing S. L. p:// Opimal Waveform Deign wih Conan odulu Conrain for Rank-One Targe Deeion Fengming XIN Bin WANG Jinkuan WANG Xin SONG Shool of Informaion Siene and Engineering Norheaern Univeriy Shenyang 4 China Engineering Opimizaion and Smar Anenna Iniue Norheaern Univeriy a Qinhuangdao 664 China wjk@mail.neuq.edu. xfm_neu@6.om Reeived: 4 Oober 3 /Aeped: 9 January 4 /Publihed: 3 January 4 Abra: In radar yem many reearher foued on he waveform deign o improve he performane. And many algorihm ju only onider he energy of he ranmied waveform. owever in praie only he energy onrain i inuffiien o guaranee ha he ignal will aify ommon envelope requiremen uh a onan modulu ha i imporan o radar ranmier. In hi paper we propoe a onan modulu waveform deign mehod for rank-one arge deeion. Firly he opimal waveform under he energy onrain i obained. Then he opimal waveform own he onan modulu propery in ime domain from i Fourier ranform magniude i obained finally. Finally imulaion reul how he effeivene of he propoed algorihm. Copyrig 4 IFSA Publihing S. L. Keyword: Waveform deign Cogniive radar Relaive enropy Conan modulu onrain Targe deeion.. Inroduion Cogniive radar i propoed a a new generaion radar yem by aykin [ ] whih i no only dependen on he adapive ignal proeing in he radar reeiver bu alo hrough an adapive radar ranmier in order o improve he performane. Thu he opimal waveform deign problem i he key ehnology of he ogniive radar. Lo of reearh effor have been foued on he radar waveform opimizaion. Bell propoed o maximize he muual informaion (I beween he reeived ignal and he arge impule repone (TIR o opimize he ranmied waveform [3]. In [4] he auhor exended he I mehod o he muliple-inpu muliple-oupu (IO radar and propoed anoher mehod ha adoped he mean-quare error (SE in eimaing he TIR a he o funion and minimized SE o opimize he waveform for arge idenifiaion and laifiaion. The propoed wo mehod have he ame oluion. Pillai e al. developed an eigenoluion for opimal ignal/filer pair for arge deeion when he arge and luer an be een a ime invarian random proee [5 6] whih ued he ignal o luer-plu-noie raio (SCNR a he o funion and maximized he SCNR o opimize he waveform. Goodman e al. publihed a erie of paper abou he waveform deign baed on he equenial hypohei eing ha onrol when hard deiion may be made wih adequae onfidene o deign waveform. The deailed work an be found in [7 8]. In [9] Kay derived he opimal NP deeor firly whih how ha he NP deeion performane doe no immediaely lead o an obviou ignal deign rierion o ha a divergene rierion i propoed for Arile number P_798 39

2 Senor & Tranduer Vol. 63 Iue January 4 pp ignal deign alo baed on he relaive enropy in ignal inpu muliple oupu (SIO radar enario. owever he Kay mehod ju only onider he energy onrain in he ranmier ide. owever he onan modulu i alo imporan. Sine he radar yem ommonly require onan modulu waveform in order o uilize nonlinear power amplifier oherwie he arge deeion performane will be degraded. In hi paper we olve he opimal waveform deign wih onan modulu problem for arge deeion in rank-one arge model. In nex eion he ignal model and problem formulaion i given. The opimal waveform deign onidering he onan modulu onrain i olved in eion 3. Simulaion reul are hown in eion 4. The onluion i ummarized in eion 5.. Signal odel and Problem Formulaion When he ranmied ignal band B i uffiienly large o ha B i omparable o /d where i he peed of he lig and d i he range pan of he arge he arge i no wihin a ignal range ell. In hi ae i hould be modeled a exended arge whoe impule repone funion an be expreed a []: L h ( a ( ( l l l where al l L are he omplex refleion oeffiien of he arge. The luer impule repone funion an be expreed a h ( a ( ( m m m where am m are he omplex refleion oeffiien of he luer. So he reeived ignal in addiive Gauian noie ondiion an be expreed a x( ( h ( ( h ( n( (3 where ( i he ranmied ignal and n ( i he omplex addiive Gauian noie. denoe onvoluion operaor. Aording o [] formula (3 an be expreed in frequeny domain ha i X S S N diag( diag( N (4 where SF SF ( ( T S diag( N/ N/ exp( j FN/ exp( j FN/L exp( j F exp( j F (5 a N/ N/ L T a exp( j FN/ exp( j FN/ exp( j F exp( j F (6 a N/ N/ T a i a zero mean random Gauian veor wih ovariane R. When Rank( R he ampliude veor of he arge aer funion a i fixed he j phae of he arge aer funion e may fluuae randomly due o hange in he diane beween he arge and he radar ha i he aer of a rigid arge exi no relaive Doppler. Some paper named hi kind of arge afer rank-one arge. The ovariane R an be deompoed R VV by he low rank deompoiion he V i defined by he arge ignaure veor. The luer i alo omplex random Gauian veor wih zero mean and ovariane R h. And he R h i alo an be deompoed a Rh VhV h. When a arge exi in radar environmen he deeion proedure i given by deiion on one of he wo poible hypohee a : X S N : X S S N (7 Our aim i o deign he opimal waveform o maximize he probabiliy of deeion. In nex eion we will elaborae how o opimize he waveform deign. Thi doumen i prepared aording o our journal' manurip inruion. You an ue i a a emplae. 3. Waveform Deign wih Conan odulu Conrain Aording o model (5 he probabiliy deniy funion (PDF under an be wrien a p( X exp ( N de( R X R X (8 The PDF under an be wrien a 4

3 Senor & Tranduer Vol. 63 Iue January 4 pp p( X N de( R R exp ( X R R X (9 where R i he ovariane marix of S N R i he ovariane marix of S and R VV. Thu he likelihood radio e an be expreed a l X p( X ( X R R R X ( ln p ( X ( de ( R R where ln. The fir iem in de R he rig of equal ign in ( i a onan. Aording o he Woodbury ideniy ( an be expreed a RV V X R V l( X V R V V R V ( The relaive enropy i defined a he exponen in probabiliy of error in a hypohei e beween diribuion p and p []. There exi he probabiliy of error in eah hypohei an be expreed a DD( p p D( p p p( X p X dx ( ln p ( X p( X p X dx ( ln p ( X p( X p( X Eln Eln p( X p( X ( where D( pi pj denoe he relaive enropy in favor of i again j. In radar yem he arge deeion problem ha wo probabiliie of error fale alarm probabiliy P fa and mi probabiliy P m exp NDp ( p and exp NDp ( p are equal o P fa and P m repeively. I an be een ha maximizing he relaive enropy ould reul in minimizing he P fa and P m. Auming V R V [9] Subiue ( ino ( we have DP P V R V fa m N / V( Fn P( F P( F nn/ n n n N / SF ( n V( Fn nn/ SFn Ph Fn Pn Fn ( ( ( (3 Noe ha Pm Pd where P d i he arge deeion probabiliy. We hope he arge deeion probabiliy P d a large a poible while he fale alarm probabiliy P fa a mall a poible. Thu he opimizaion rierion an be deigned a max D max D( p p D( p p (4 Conidering he oal energy limi he opimizaion model i max D B/ B/ B/ B/.. S( F df E SF ( VF ( df SF ( Ph ( F Pn ( F (5 Afer uing Lagrange muliplier he opimal waveform i VF ( PF n( / PF n( SF ( max Ph( F (6 The parameer i found by olving B/ V ( F Pn ( F/ Pn ( F max df E B/ Ph( F (7 The opimal waveform i he ame a he reul of []. owever he opimal waveform of equaion (6 ju only onider he energy onrain. In modern radar yem he onan modulu waveform are ommonly required in praie. Sine he nonlinear power amplifier in radar yem require he inpu ignal o be onan modulu. Thu he opimal waveform wih he onan modulu onrain ould be onidered. The mehod o he onan modulu i baed on he ehnique in [3].he onan modulu ignal i generaed in ime domain afer given i Fourier ranform magniude. Le C denoe he e of funion g( ha have he ame Fourier ranform magniude over he frequeny inerval. A magniude projeion operaor P defined an make f proje o a neare he arbirary funion neighbor P f ha belong o C. For arbirary funion f i Fourier ranform an be j repreenaion by F F e. The magniude projeion of P f F x an be defined a j e ' (8 4

4 Senor & Tranduer Vol. 63 Iue January 4 pp Equaion (8 guaranee ha radar yem mainain he ranmi ignal o own he opimal waveform in frequeny domain. Coninue he onan modulu elemen. Le C A denoe he e of funion g whih have onan envelope level A There exi a ampliude projeion operaor PA o aign an arbirary funion f a" neare neighbor Pf A on C A o ha here i no elemen g C A o aify x g x PA x. Given an j arbirary funion f a e he projeion proedure i Pf j Ae T (9 a e oherwie A j he ranmied energy among mo of he peral peak of he arge ignaure and he wo waveform are imilar whih demonrae he opimal waveform onidering he onan modulu onrain ill guaranee he deeion performane. The magniude of he onan modulu waveform are maller han he one of he non-onan modulu waveform. owever he opimal waveform wih onan modulu alloae he ranmied energy ino addiional frequeny band. Fig. 4 i he omplex onellaion of onan modulu onrained waveform. I an be een ha he opimal waveform ha he onan envelope in ime domain afer onidering he onan modulu onrain whih enure he modulu of he ranmied ignal ould no exeed he maximum inpu value of DAC in praie. The magniude and ampliude projeion are ombined aording o f ( k PAP fk ( The equaion ( ha he error reduion propery [3]: Power Targe peral ignaure Cluer PSD d f PP f f PP f d ( k k A k k A k k where fk i he h k projeion ieraed funion. When he opimal waveform wihou onidering he onan modulu ( ha own he magniude S f i obained he opimal waveform g wih he onan modulu ould be obained by he equaion (8 (9 and (. The equaion ( guaranee he error redue when he k. Afer a number of ieraion he ignal ha he onan modulu envelope wih he preribed Fourier ranform magniude. 4. Simulaion Reul In hi eion we will give he imulaion o demonrae he effeivene of he propoed mehod. Suppoe he frequeny of he ignal i normalized o be. The lengh of he arge impule repone i 3. The ampling frequeny i. Fig. how he arge peral ignaure and he luer power peral deniy (PSD. The energy i onrained o be 5 (energy uni. The noie PSD i.5. Fig. how he opimal waveform wihou onan modulu onrain and he Fig. 3 how he opimal waveform wih onan modulu onrain. The wo figure how ha he opimal waveform wih/wihou onan modulu onrain alloae Energy peral deniy Energy peral deniy Normalied Frequeny Fig.. The arge peral ignaure and he luer PSD Opimal waveform wihou onan modulu onrain Normalized frequeny Fig.. The opimal waveform wihou onan modulu onrain. Opimal waveform wih onan modulu onrain Normalied Frequeny Fig. 3. The opimal waveform wih onan modulu onrain. 4

5 Senor & Tranduer Vol. 63 Iue January 4 pp x -3 Referene Im(fk Re(fk Fig. 4. Complex Conellaion of opimal waveform wih onan waveform. 5. Conluion Thi paper foue on he waveform deign for arge deeion onidering he onan modulu onrain. aximizing he relaive enropy a he opimizaion rierion deign he opimal waveform wih he ranmied energy onrain for he rankone arge. In addiion onidering he onan modulu in praie he onan modulu onrain i applied o deign he opimal waveform. The energy peral deniy of he opimal waveform wihou onan modulu i regarded a he preribed Fourier ranform magniude. Then he onan modulu waveform ould be obained in ime domain. I i imilar o he opimal waveform ha only onider he oal energy onrain. The imulaion reul preen he propoed mehod validaion. Aknowledgmen Thi work wa uppored by he Naional Naural Siene Foundaion of China (No. 645 and No. 645 he Naural Siene Foundaion of Liaoning provine under Gran (No. 57 he Naural Siene Foundaion of ebei Provine (No. F544 and No. F3575 and he Dooral Sienifi Reearh Foundaion of Liaoning Provine (No. 33. x -3 []. S. aykin Cogniive radar: a way of he fuure IEEE Signal Proeing agazine Vol. 3 Iue January 6 pp []. S. aykin Cogniion i he key o he nex generaion of radar yem in Proeeding of he IEEE Signal Proeing Eduaion Workhop 9 pp [3].. R. Bell Informaion heory and radar waveform deign IEEE Tranaion on Informaion Theory Vol. 39 Iue 5 Sepember 993 pp [4]. Y. Yang and R. S. Blum IO radar waveform deign baed on muual informaion and minimum mean-quare error eimaion IEEE Tranaion on he Aeropae and Eleroni Syem Vol. 43 Iue January 7 pp [5]. S. U. Pillai D. C. Youla. S. Oh and J. R. Gueri Opimum ranmi-reeiver deign in he preene of ignal-dependen inerferene and hannel noie in Proeeding of he 33 rd Ailomar Conferene on Signal Syem and Compuer Paifi Grove California USA 999 Vol. pp [6]. S. U. Pillai. S. Oh D. C. Youla and J. R. Gueri Opimum ranmi-reeiver deign in he preene of ignal-dependen inerferene and hannel noie IEEE Tranaion on Informaion Theory Vol. 46 Iue pp [7]. N. A. Goodman Adapive waveform deign and equenial hypohei eing for arge reogniion wih aive enor IEEE Journal of Seleed Topi in Signal Proeing Vol. Iue June 7 pp [8]. R. A. Romero J. Bae and N. A. Goodman Theory and appliaion of SNR and muual informaion mahed illuminaion waveform IEEE Tranaion on Aeropae and Eleroni Syem Vol. 47 Iue pp [9]. S. Kay Waveform deign for muliai radar deeion IEEE Tranaion on Aeropae and Eleroni Syem Vol. 45 Iue 3 9 pp []. A. Lehem O. Naparek A. Nehorai Informaion heorei adapive radar waveform deign for muliple exended arge IEEE Journal of Seleed Topi in Signal Proeing Vol. Iue 7 pp []. X. Deng C. Qiu Z. Cao. orelande and B. oran Waveform deign for enhane deeion of exended arge in ignal-dependen inerferene IET Radar Sonar and Navigaion Vol. 6 Iue pp []. T.. Cover and J. A. Toma Elemen of informaion heory nd ediion Wiley- Ineriene 6. [3]. S. U. Pillai K. Y. Li and. Beyer Conruion of onan envelope ignal wih given Fourier ranform magniude in Proeeding of he IEEE Inernaional Radar Conferene 9 pp Copyrig Inernaional Frequeny Senor Aoiaion (IFSA Publihing S. L. All rig reerved. (p:// 43

Let. x y. denote a bivariate time series with zero mean.

Let. x y. denote a bivariate time series with zero mean. Linear Filer Le x y : T denoe a bivariae ime erie wih zero mean. Suppoe ha he ime erie {y : T} i conruced a follow: y a x The ime erie {y : T} i aid o be conruced from {x : T} by mean of a Linear Filer.