Target Detection Performance of Spectrum Sharing MIMO Radars

Transcription

1 Target Detecton Performance of Spectrum Sharng MIMO Radars Awas Khawar, Ahmed Abdelhad, and T. Charles Clancy arxv:48.54v [cs.it] 4 Aug 4 Abstract Future wreless communcaton systems are envsoned to share rado frequency (RF) spectrum, wth other servces such as radars, n order to meet the growng spectrum demands. In ths paper, we consder co-channel spectrum sharng between cellular systems and radars. We address the problem of target detecton by radars that are subject to shape ts waveform n a way that t doesnt cause nterference to cellular systems. We consder a multple-nput multple-output (MIMO) radar and a MIMO cellular communcaton system wth K base statons (BS). We propose a spectrum sharng algorthm whch steers radar nulls, by projectng radar waveform onto the null space of nterference channel, towards a selected BS, thus, protectng t from radar nterference. Ths BS s selected, among K BSs, on the bass of guaranteeng mnmum waveform degradaton. We study target detecton capabltes of ths null-space projected (NSP) waveform and compare t wth the orthogonal waveform. We derve the generalzed lkelhood rato test (GLRT) for target detecton and derve detector statstc for NSP and orthogonal waveform. The target detecton performance for NSP and orthogonal waveform s studed theoretcally and va Monte Carlo smulatons. Index Terms MIMO Radar, Null Space Projecton, Cellular System Coexstence, Target Detecton, GLRT, ML Estmaton. I. INTRODUCTION Spectrum sharng between wreless communcaton systems and radars s an emergng area of research. In the past, spectrum has been shared prmarly between wreless communcaton systems usng opportunstc approaches by users equpped wth cogntve rados []. Ths type of spectrum sharng has been made possble wth the use of spectrum sensng [], or geolocaton databases [3], or a combnaton of both n the form of rado envronment maps (REM) [4]. Some recent efforts have explored co-channel sharng approaches among secondary network enttes, please see [5] and reference theren. However, n contrast, co-channel spectrum sharng between wreless systems and radars has receved lttle attenton thus far because of regulatory concerns. Spectrum polcy regulators, n the past, have not allowed commercal wreless servces n the radar bands, except n Awas Khawar (awas@vt.edu) s wth Vrgna Polytechnc Insttute and State Unversty, Arlngton, VA, 3. Ahmed Abdelhad (aabdelhad@vt.edu) s wth Vrgna Polytechnc Insttute and State Unversty, Arlngton, VA, 3. T. Charles Clancy (tcc@vt.edu) s wth Vrgna Polytechnc Insttute and State Unversty, Arlngton, VA, 3. Ths work was supported by DARPA under the SSPARC program. Contract Award Number: HR-4-C-7. The vews expressed are those of the author and do not reflect the offcal polcy or poston of the Department of Defense or the U.S. Government. Dstrbuton Statement A: Approved for publc release; dstrbuton s unlmted. few cases, due to the fear of harmful nterference from these servces to radar systems. Recently, n the Unted States (U.S.), the Federal Communcatons Commsson (FCC), has proposed to use the MHz band for commercal broadband use [6]. The ncumbents n ths band are radar and satellte systems. The Commsson has proposed that ncumbents share ths band wth commercal communcaton systems. The Commsson s spectrum sharng ntatve s motvated by many factors ncludng the Presdent s Natonal Broadband plan, whch called to free up to 5 MHz of federal-held spectrum by [7]; surge n consumers demand for access to moble broadband, whch operators can t meet wth current spectrum allocaton; the report on effcent spectrum utlzaton by Presdent s Councl of Advsers on Scence and Technology (PCAST), whch emphaszed to share MHz of government-held spectrum [8], and the low utlzaton of MHz band by federal ncumbents [9]. In the future, when rado frequency (RF) spectrum wll be shared among many dfferent systems, e.g., radars and cellular systems, t s mportant to access the nterference scenaro. Of course, radars wll cause nterference to communcaton systems and vce versa f proper nterference mtgaton methods and novel sharng algorthms are not employed. In a study conducted by Natonal Telecommuncatons and Informaton Admnstraton (NTIA), t was observed that n order to protect commercal cellular communcaton systems, from hgh power radar sgnal, large excluson zones are requred [9]. These excluson zones cover a large porton of the U.S. majorty of the populaton lves and, thus, does not make a busness case for commercal deployment n radar bands. In order to share radar bands for commercal operaton we have to address the nterference mtgaton technques at both the systems. In ths work, we focus on nterference caused by radar systems to communcaton system and propose methods to mtgate ths nterference. The federal-commercal spectrum sharng s not a new practce. In fact, n the past, commercal wreless systems have shared government bands on a low transmt power bass, n order to protect ncumbents from nterference. An example of such a scenaro s wreless local area network (WLAN) n the MHz and MHz radar bands []. So the Commsson s latest ntatve to share 3.5 GHz radar band wth small cells,.e. wreless base statons operatng on a low power, s n harmony wth prevous practces [6]. A. Related Work On gong research efforts have shown that there are numerous ways to share spectrum between radars and commu-

2 ncaton systems. Cooperatve sensng based spectrum sharng approaches can be utlzed a radar s allocated bandwdth s shared wth communcaton systems [] [3]. A jont communcaton-radar platform can be envsoned n whch a spectrally-agle radar performs an addtonal task of spectrum sensng and upon fndng of unused frequences t can change ts operatng frequency. In addton of spectrum sharng such a settng can enable co-located radar and communcaton system platforms for ntegrated communcatons and radar applcatons [4] [7]. Radar waveforms can be shaped n a way that they don t cause nterference to communcaton systems [8] [3]. Moreover, database-aded sensng at communcaton systems [4] and beamformng approaches at MIMO radars can be realzed for spectrum sharng [5]. B. Our Contrbutons The problem of target estmaton, detecton, and trackng les at the heart of radar sgnal processng. Ths problem becomes crtcally mportant when we talk about sharng radar spectrum wth other systems, say cellular systems. The regulatory work gong on n the 3.5 GHz band to share radar spectrum wth commercal systems s the motvaton of ths work. The focus of ths work s to study target detecton performance of a radar that s subject to share ts spectrum wth a cellular system. We consder spectrum sharng between a MIMO radar and a cellular system wth many base statons. In our prevous work, we have addressed the problem of radar waveform projecton onto the null space of nterference channel, n order to mtgate radar nterference to communcaton system, and the problem of selecton of nterference channel for projecton when we have a cellular system wth K base statons [9]. In ths work, we consder the same sharng scenaro but study the target detecton performance of radar for the null-space projected (NSP) waveform and compare t wth that of the orthogonal radar waveform. We use the generalzed lkelhood rato test (GLRT) for target detecton and derve detector statstc for NSP and the orthogonal waveform. C. Notatons Matrces are denoted by bold upper case letters, e.g. A, and vectors are denoted by bold lower case letters, e.g. a. Transpose, conjugate, and Hermtan operators are denoted by ( ) T,( ), and ( ) H, respectvely. Moreover, notatons used throughout the paper are provded n Table I along wth descrptons for quck reference. D. Organzaton Ths paper s organzed as follows. Secton II dscuss MIMO radar, target channel, orthogonal waveforms, nterference channel, and our cellular system model. Moreover, t also dscusses modelng and statstcal assumptons. Secton III dscusses spectrum sharng between MIMO radar and cellular system and ntroduces sharng archtecture and projecton algorthms. Secton IV presents the generalzed lkelhood rato test (GLRT) for target detecton and derves detector statstc for NSP and orthogonal waveform. Secton V dscusses numercal results and compares performance of NSP and orthogonal waveform. Secton VI concludes the paper. Notaton x(t) a(θ) y(t) R x s UE j (t) L UE K M N BS H r (t) P Descrpton TABLE I TABLE OF NOTATIONS Transmtted radar (orthogonal) waveform Steerng vector to steer sgnal to target angle θ Receved radar waveform Correlaton matrx of orthogonal waveforms Sgnal transmtted by the jth UE n the th cell Total number of user equpments (UEs) n the th cell Total number of BSs Radar transmt/receve antennas BS transmt/receve antennas th nterference channel Receved sgnal at the th BS Projecton matrx for the th channel II. SYSTEM MODEL In ths secton, we ntroduce prelmnares of MIMO radar, pont target n far-feld, orthogonal waveforms, nterference channel, and cellular system model. Moreover, we also dscuss modelng and statstcal assumptons along wth RF envronment assumptons used throughout the paper. A. Radar Model The radar we consder n ths paper s a colocated MIMO radar wth M transmt and receve antennas and s mounted on a shp. The colocated MIMO radar has antennas that have spacng on the order of half the wavelength. Another class of MIMO radar s wdely-spaced MIMO radar elements are wdely-spaced whch results n enhanced spatal dversty [6]. The colocated radar gves better spatal resoluton and target parameter dentfcaton as compared to the wdelyspaced radar [7]. B. Target Model/Channel In ths paper, we consder a pont target model whch s defned for targets havng a scatterer wth nfntesmal spatal extent. Ths model s a good assumpton and s wdely used n radar theory for the case when radar elements are colocated and ther exsts a large dstance between the radar array and the target as compared to nter-element dstance [8]. The sgnal reflected from a pont target wth unt radar crosssecton (RCS) s mathematcally represented by the Drac delta functon. C. Sgnal Model Let x(t) be the sgnal transmtted from the M-element MIMO radar array, defned as [ ] T x(t) = x (t)e jωct x (t)e jωct x M (t)e jωct () x k (t)e jωct s the baseband sgnal from the k th transmt element, ω c s the carrer angular frequency, t [,T o ], wth

3 3 T o beng the observaton tme. We defne the transmt steerng vector as [ ] T a T (θ) e jωcτt (θ) e jωcτt (θ) e jωcτt M (θ). () Then, the transmt-receve steerng matrx can be wrtten as A(θ) a R (θ)a T T(θ). (3) Snce, we are consderng M transmt and receve elements, we defne a(θ) a T (θ) a R (θ). The sgnal receved from a sngle pont target, n far-feld wth constant radal velocty v r, at an angle θ can be wrtten as y(t) = αe jωdt A(θ)x(t τ(t))+n(t) (4) τ(t) = τ Tk (t)+τ Rl (t), denotng the sum of propagaton delays between the target and the k th transmt element and the l th receve element, respectvely; ω D s the Doppler frequency shft, and α represents the complex path loss ncludng the propagaton loss and the coeffcent of reflecton. D. Modelng Assumptons In order to keep the analyss tractable we have made the followng assumptons about our sgnal model: The path loss α s assumed to be dentcal for all transmt and receve elements, due to the far-feld assumpton [9]. The angle θ s the azmuth angle of the target. After compensatng the range-doppler parameters, we can smplfy equaton (4) as E. Statstcal Assumptons y(t) = αa(θ)x(t)+n(t). (5) We make the followng assumptons for our receved sgnal model n equaton (5): θ and α are determnstc unknown parameters representng the target s drecton of arrval and the complex ampltude of the target, respectvely. The nose vectorn(t) s ndependent, zero-mean complex Gaussan wth known covarance matrxr n = σ ni M,.e. n(t) N c ( M,σ ni M ), N c denotes the complex Gaussan dstrbuton. Wth the above assumptons, the receved sgnal model n equaton (5) has an ndependent complex Gaussan dstrbuton,.e., y(t) N c (αa(θ)x(t),σ n I M). F. Orthogonal Waveforms In ths paper, we consder orthogonal waveforms transmtted by MIMO radars,.e., R x = x(t)x H (t)dt = I M. (6) T o The transmsson of orthogonal sgnals gves MIMO radar advantages n terms of dgtal beamformng at the transmtter n addton to recever, mproved angular resoluton, extended array aperture n the form of vrtual arrays, ncreased number of resolvable targets, lower sdelobes, and lower probablty of ntercept as compared to coherent waveforms [9]. G. Communcaton System In ths paper, we consder a MIMO cellular system, wth K base statons, each equpped wth N BS transmt and receve antennas, wth th BS supportng L UE user equpments (UEs). The UEs are also mult-antenna systems wth N UE transmt and receve antennas. If s UE j (t) s the sgnals transmtted by the j th UE n the th cell, then the receved sgnal at the th BS recever can be wrtten as r (t) = j H N BS N UE j s UE j (t)+w(t) (7) for K and j L UE w(t) s the addtve whte Gaussan nose. H. Interference Channel In ths secton, we characterze the nterference channel that exsts between a MIMO cellular base staton and a MIMO radar. In our paper, we are consderng K cellular BSs that s why our model has H, =,,...,K, nterference channels, the entres of H are denoted by H = h (,).. h (N BS,) h (,M)..... h (N BS,M) (N BS M) (8) h (l,k) denotes the channel coeffcent from the k th antenna element at the MIMO radar to the l th antenna element at the th BS. We assume that elements of H are ndependent, dentcally dstrbuted (..d.) and crcularly symmetrc complex Gaussan random varables wth zero-mean and untvarance, thus, havng a..d. Raylegh dstrbuton. I. Cooperatve RF Envronment In the wreless communcatons lterature, t s usually assumed that the transmtter (mostly BSs) has channel state nformaton ether by feedback from the recever (mostly UEs), n FDD systems [3], or transmtters can recprocate the channel, n TDD systems [3]. The feedback and recprocty are vald and practcal as long as the feedback has a reasonable overhead and coherence tme of the RF channel s larger than the two-way communcaton tme, respectvely. In the case of radars sharng ther spectrum wth communcatons systems one way to get CSI from communcaton systems s through feedback. Snce, a radar sgnal s treated as nterference at a communcaton system, we can characterze the channel as nterference channel and refer to nformaton about t as nterference-channel state nformaton (ICSI). Spectrum sharng between radars and communcatons systems can be envsoned n two domans: mltary radars sharng spectrum wth mltary communcaton systems, we call t MlMl sharng; another possblty s mltary radars sharng spectrum wth commercal communcaton systems, we call t MlCom sharng. In MlMl sharng, ICSI can be acqured by radars farly easly as both systems belong to mltary. In MlCom sharng, ICSI can be acqured by gvng ncentves

4 4 p mn(n BS,M) and σ, > σ, > > σ,q > σ,q+ = σ,q+ = = σ,p = are the sngular values of H. Next, we defne Σ dag( σ,, σ,,..., σ,m ) () Fg.. Spectrum Sharng Scenaro: A seaborne MIMO radar detectng a pont target whle smultaneously sharng spectrum wth a MIMO cellular system wthout causng nterference to the cellular system. to commercal communcaton system. The bggest ncentve n ths scenaro s null-steerng and protecton from radar nterference. Thus, regardless of the sharng scenaro, MlMl or MlCom, we have ICSI for the sake of mtgatng radar nterference at communcaton systems. III. RADAR-CELLULAR SYSTEM SPECTRUM SHARING After ntroducng our radar and cellular system models we can now dscuss the spectrum sharng scenaro between radar and cellular system. In our sharng archtecture MIMO radar and cellular systems are the co-prmary users of the MHz band under consderaton. In the followng sectons, we wll dscuss the archtecture of spectrum sharng problem whch s followed by our spectrum sharng algorthm. A. Archtecture We llustrate our coexstence scenaro n Fgure the martme MIMO radar s sharng K nterference channels wth the cellular system. Consderng ths scenaro, the receved sgnal at the th BS recever can be wrtten as r (t) = H N BS M x(t)+ j H N BS N UE j s UE j (t)+w(t). (9) The goal of the MIMO radar s to map x(t) onto the nullspace of H n order to avod nterference to the th BS,.e., H x(t) =, so that r (t) has equaton (7) nstead of equaton (9). B. Projecton Matrx In ths secton, we defne the projecton algorthm whch projects radar sgnal onto the null space of nterference channel H. Assumng, the MIMO radar has channel state nformaton of all H nterference channels, through feedback, n MlMl or MlCom scenaro, we can perform sngular value decomposton (SVD) to fnd the null space and then construct a projector matrx. We proceed by frst fndng SVD of H,.e., Now, let us defne H = U Σ V H. () Σ dag( σ,, σ,,..., σ,p ) () σ,u {, for u q,, for u > q. (3) Usng above defntons we can now defne our projecton matrx,.e., P V Σ VH. (4) In order to show that P s a vald projecton matrx we prove two results on projecton matrces below. Property. P C M M s a projecton matrx f and only f P = P H = P. Proof: Let s start by showng the only f part. Frst, we show P = P H. Takng Harmton of equaton (4) we have P H = (V Σ VH )H = P. (5) Now, squarng equaton (4) we have P = V Σ V H V Σ V H = P (6) above equaton follows from V H V = I (snce they are orthonormal matrces) and ( Σ ) = Σ (by constructon). From equatons (5) and (6) t follows that P = P H = P. Next, we show P s a projector by showng that f v range (P ), then P v = v,.e., for some w,v = P w, then P v = P (P w) = P w = P w = v. (7) Moreover, P v v null(p ),.e., P (P v v) = P v P v = P v P v =. (8) Ths concludes our proof. Property. P C M M s an orthogonal projecton matrx onto the null space of H C N BS M. Proof: Snce P = P H, we can wrte H P H = U Σ V H V Σ VH =. (9) The above results follows from notng that Σ Σ = by constructon. In ths paper, we are dealng wth K nterference channels. Therefore, we need to select the nterference channel whch results n least degradaton of radar waveform n a mnmum norm sense,.e., mn argmnp x(t) x(t) () K P P mn. () Once we have selected our projecton matrx t s straght forward to project radar sgnal onto the null space of nterference channel va x(t) = Px(t). ()

5 5 The correlaton matrx of our NSP waveform s gven as R x = T o x(t) x H (t)dt (3) whch s no longer dentty and ts rank depends upon the rank of the projecton matrx. C. Spectrum Sharng and Projecton Algorthms The process of spectrum sharng by formng projecton matrces and selectng nterference channels s executed wth the help of Algorthms () and (). Frst, at each pulse repetton nterval (PRI), the radar obtans ICSI of all K nterference channels. Ths nformaton s sent to Algorthm () for the calculaton of null spaces and formaton of projecton matrces. Algorthm () process K projecton matrces, receved from Algorthm (), to fnd the projecton matrx whch results n least degradaton of radar waveform n a mnmum norm sense. Ths step s followed by the projecton of radar waveform onto the null space of the selected BS,.e, the BS to the correspondng selected projecton matrx, and waveform transmsson. Algorthm Spectrum Sharng Algorthm loop for = : K do Get CSI of H through feedback from the th BS. Send H to Algorthm () for the formaton of projecton matrx P. Receve the th projecton matrx P from Algorthm (). end for Fnd mn = argmn K Set P = P mn P x(t) x(t). as the desred projector. Perform null space projecton,.e., x(t) = Px(t). end loop Algorthm Projecton Algorthm f H receved from Algorthm () then Perform SVD on H (.e. H = U Σ V H ) Construct Σ = dag( σ,, σ,,..., σ,p ) Construct Σ = dag( σ,, σ,,..., σ,m ) Setup projecton matrx P = V Σ VH. Send P to Algorthm (). end f IV. STATISTICAL DECISION TEST FOR TARGET DETECTION In ths secton, we develop a statstcal decson test for target llumnated wth the orthogonal radar waveforms and the NSP projected radar waveforms. The goal s to compare the performance of the two waveforms by lookng at the test decson on whether the target s present or not n the range- Doppler cell of nterest. We present a system-level archtecture x.... x..... x.... x Detecton and Estmaton Recever y MIMO Radar Array Projector Channel State Informaton Projecton Algorthm Waveform Generaton Transmtter Fg.. Block dagram of spectrum sharng radar. The transmtter s modfed to perform the functons of ICSI collecton, projecton matrx formaton, nterference channel selecton, and radar waveform projecton on to the selected nterference channel for spectrum sharng. On the other hand, the recever s a tradtonal radar recever performng functons of parameter detecton and estmaton on radar returns. of the spectrum sharng radar n Fgure. In our archtecture, the transmtter performs the functons of waveform generaton, channel selecton, and projecton; and the recever performs the functons of sgnal detecton and estmaton. For target detecton and estmaton, we proceed by constructng a hypothess test we seek to choose between two hypothess: the null hypothess H whch represents the case when the target s absent or the alternate hypothess H whch represents the case when the target s present. The hypothess for a sngle target model n equaton (5) can be wrtten as { H : αa(θ)x(t)+n(t), t T o, y(t) = (4) H : n(t), t T o. Snce, θ and α are unknown, but determnstc, we use the generalze lkelhood rato test (GLRT). The advantage of usng GLRT s that we can replace the unknown parameters wth ther maxmum lkelhood (ML) estmates. The ML estmate of α and θ are found for varous sgnal models, targets, and nterference sources n [9], [3] when usng orthogonal sgnals. In ths paper, we consder a smpler model wth one target and no nterference sources n order to study the mpact of NSP on target detecton n a tractable manner. Therefore, we present a smpler dervaton of ML estmaton and GLRT. The receved sgnal model n equaton (5) can be wrtten as y(t) = Q(t,θ)α+n(t) (5) Q(t,θ) = A(θ)x(t). (6) We use Karhunen-Loève expanson for dervaton of the log-lkelhood functon for estmatng θ and α. Let Ω denote the space of the elements of {y(t)},{q(t,θ)}, and {n(t)}. SVD

6 6 Moreover, let ψ z, z =,,..., be an orthonormal bass functon of Ω satsfyng < ψ z (t),ψ z (t) >= ψ z (t),ψz (t) = δ zz (7) T δ zz s the Krönecker delta functon. Then, the followng seres can be used to expand the processes, {y(t)},{q(t, θ)}, and {n(t)}, as y(t) = Q(t,θ) = n(t) = y z ψ z (t) (8) z= Q z (θ)ψ z (t) (9) z= n z ψ z (t) (3) z= y z,q z, and n z are the coeffcents n the Karhunen- Loève expanson of the consdered processes obtaned by takng the correspondng nner product wth bass functon φ z (t). Thus, an equvalent dscrete model of equaton (5) can be obtaned as y z = Q z (θ)α+n z, z =,,... (3) For whte crcular complex Gaussan processes,.e, E[n(t)n(t τ(t))] = σni M δ(τ(t)), the sequence {n z } s..d. and n z N c ( M,σn I M). Thus, we can express the log-lkelhood functon as ( L y (θ,α) = M log(πσn ) y z Q z (θ)α ). z= Maxmzng wth respect to α yelds σ n L y (θ, ˆα) = Γ ( ) σn E yy e H QyE QQ e Qy (3) (33) Γ M log(πσn ) (34) E yy y z (35) e Qy E z= Q H z y z (36) z= QQ Q H z Q z. (37) z= Note that, n equaton (33), apart from the constant Γ, the remanng summaton goes to nfnty. However, due to the non-contrbuton of hgher order terms n the estmaton of θ and α the summaton can be fnte. Usng the dentty T o v (t)v H (t)dt = v z vz H (38) z= for v (t) = z= v zψ z (t), =,, equatons (35)-(37) can be wrtten as E yy y(t) dt (39) T o e Qy Q H (t,θ)y(t)dt (4) T o E QQ Q H (t,θ)q(t,θ)dt. (4) T o Usng the defnton of Q(t,θ) n equaton (6), we can wrte the f th element of e Qy as [e Qy ] f = a H (θ f )E T a(θ f ) (4) E = y(t)x H (t)dt. (43) T o Smlarly, we can wrte the fg th element of E QQ as [E QQ ] fg = a H (θ f )a(θ g )a H (θ f )R T x a(θ g). (44) Snce, e Qy and E QQ are ndependent of the receved sgnal, the suffcent statstc to calculate θ andαs gven bye. Usng equaton (4)-(44) we can wrte the ML estmate n matrxvector form as L y (ˆθ ML ) = argmax θ a H (ˆθ ML )Ea (ˆθ ML ) Ma H (ˆθ ML )R T xa(ˆθ ML ) (45) Then, the GLRT for our hypothess testng model n equaton (4) s gven as L y = max {logf y(y,θ,α;h )} logf(y;h ) H δ (46) θ,α H f y (y,θ,α;h ) and f(y;h ) are the probablty densty functons of the receved sgnal under hypothess H and H, respectvely. Hence, the GLRT can be expressed as L y (ˆθ ML ) = argmax θ a H (ˆθ ML )Ea (ˆθ ML ) Ma H (ˆθ ML )R T xa(ˆθ ML ) H H δ. (47) The asymptotc statstcs of L(ˆθ ML ) for both the hypothess s gven by [3] { H : χ L(ˆθ ML ) (ρ), H : χ, (48) χ (ρ) s the noncentral ch-squared dstrbutons wth two degrees of freedom, χ s the central ch-squared dstrbutons wth two degrees of freedom, and ρ s the noncentralty parameter, whch s gven by ρ = α σn a H (θ)r T x a(θ). (49) For the general sgnal model, we set δ accordng to a desred probablty of false alarm P FA,.e., P FA = P(L(y) > δ H ) (5) δ = F ( P χ FA ) (5)

7 7 F s the nverse central ch-squared dstrbuton χ functon wth two degrees of freedom. The probablty of detecton s gven by = P(L(y) > δ H ) (5) ( ) = F χ (ρ) ( P FA ) (53) F χ F χ (ρ) s the noncentral ch-squared dstrbuton functon wth two degrees of freedom and noncentralty parameter ρ. A. For orthogonal waveforms R T x = I M, therefore, the GLRT can be expressed as a H (ˆθ ML )Ea (ˆθ ML ) H L Orthog (ˆθ ML ) = δ Orthog (54) Ma H (ˆθ ML )a(ˆθ ML ) H and the statstcs of L(ˆθ ML ) for ths case s { H : χ L Orthog (ˆθ ML ) (ρ Orthog ), H : χ, (55) ρ Orthog = M α σn (56) We set δ Orthog accordng to a desred probablty of false alarm P PF-Orthog,.e., δ Orthog = F ( P χ PF-Orthog ) (57) and then the probablty of detecton for orthogonal waveforms s gven by -Orthog = F χ (ρ Orthog) B. ( F χ ) ( P PF-Orthog ). (58) For spectrum sharng waveforms R T x = RT x, therefore, the GLRT can be expressed as a H (ˆθ ML )Ea (ˆθ ML ) H L NSP (ˆθ ML ) = Ma H (ˆθ ML )R T x a(ˆθ δ NSP (59) ML ) H and the statstcs of L(ˆθ ML ) for ths case s { H : χ L NSP (ˆθ ML ) (ρ NSP ), H : χ, (6) ρ NSP = α σn a H (θ)r T x a(θ). (6) We set δ NSP accordng to a desred probablty of false alarm P PF-NSP,.e., δ NSP = F ( P χ PF-NSP ) (6) and then the probablty of detecton for orthogonal waveforms s gven by ( ) -NSP = F χ (ρ NSP) ( P PF-NSP ). (63) F χ V. NUMERICAL RESULTS In order to study the detecton performance of spectrum sharng MIMO radars, we carry out Monte Carlo smulaton usng the radar parameters mentoned n Table II. At each run of Monte Carlo smulaton we generate K Raylegh nterference channels each wth dmensons N BS M, calculate ther null spaces and construct correspondng projecton matrces usng Algorthm (), determne the best channel to perform projecton of radar sgnal usng Algorthm (), transmt NSP sgnal, estmate parameters θ and α from the receved sgnal, and calculate the probablty of detecton for orthogonal and NSP waveforms. TABLE II MIMO RADAR SYSTEM PARAMETERS Parameters Notatons Values Radar/Communcaton System RF band MHz Radar antennas M 8, 4 Communcaton System Antennas N BS Carrer frequency f c 3.55 GHz Wavelength λ 8.5 cm Inter-element antenna spacng 3λ/4 6.4 cm Radal velocty v r m/s Speed of lght c 3 8 m/s Target dstance from the radar r 5 Km Target angle θ ˆθ Doppler angular frequency ω D ω cv r/c Two way propagaton delay τ r r /c Path loss α ˆα A. Performance of Algorthms () and () In Fgure 3, we demonstrate the use of Algorthms () and () n mprovng target detecton performance when multple BSs are present n detecton space of radar and the radar has to relably detect target whle not nterferng wth communcaton system of nterest. As an example, we consder a scenaro wth fve BSs and the radar has to select a projecton channel whch mnmzes degradaton n ts waveform, thus, maxmzng ts probablty of detecton of the target. In Fgure 3(a), we consder the case when N BS < M. We show detecton results for fve dfferent NSP sgnals,.e, radar waveform projected onto fve dfferent BSs. Note that, n order to acheve a detecton probablty of 9%, we need 6 db to 3 db more gan n as compared to the orthogonal waveform, dependng upon whch channel we select. Usng Algorthms () and () we can select nterference channel that results n mnmum degradaton of radar waveform and results n enhanced target detecton performance wth the mnmum addtonal gan n requred. For example, Algorthms () and () would select BS#5 because n ths case NSP waveform requres least gan n to acheve a detecton probablty of 9% as compared to other BSs. In Fgure 3(b), we consder the case when N BS M. Smlar to Fgure 3(a) we show detecton results for fve

8 8 dfferent NSP sgnals but now MIMO radar has a larger array of antennas as compared to the prevous case. In ths case, n order to acheve a detecton probablty of 9%, we need 3 db to 5 db more gan n as compared to the orthogonal waveform. As n the prevous case, usng Algorthms () and () we can select nterference channel that results n mnmum degradaton of radar waveform and results n enhanced target detecton performance wth the mnmum addtonal gan n requred. For example, Algorthms () and () would select BS# because n ths case NSP waveform requres least gan n to acheve a detecton probablty of 9% as compared to the other BSs. The above two examples demonstrates the mportance of Algorthms () and () n selectng nterference channel for radar sgnal projecton to maxmze detecton probablty and mnmze gan n requred as a result of NSP of radar waveforms for spectrum sharng. B. Case : dmn[h ] = In Fgure 4, we plot the varatons of probablty of detecton as a functon of sgnal-to-nose rato () for varous values of probablty of false alarm P FA. Each subplot represents the for a fxed P FA. We choose to evaluate aganst P FA values of, 3, 5 and 7 when the nterference channel H has dmensons 4,.e., the radar has M = 4 antennas and the communcaton system has N BS = antennas, thus, we have a null space dmenson of dmn[h ] =. When we compare the detecton performance of two waveforms we note that n order to get a desred for a fxed P FA we need more for NSP than orthogonal waveforms. For example, say we desre =, then accordng to Fgure 4 we need 6 db more gan n for NSP waveform to get the same result produced by the orthogonal waveform. C. Case : dmn[h ] = 6 In Fgure 5, smlar to Fgure 4, we do an analyss of aganst the same values of P FA but for nterference channel H havng dmensons 8,.e., now the radar has M = 8 antennas and the communcaton system has N = antennas, thus, we have a null space dmenson of dmn[h ] = 6. Smlar to Case, when we compare the detecton performance of two waveforms we note that n order to get a desred for a fxed P FA we need more for NSP than the orthogonal waveforms. For example, say we desre =, then accordng to Fgure 5 we need 3.5 to 4.5 db more gan n for the NSP waveform to get the same result produced by the orthogonal waveform. D. Comparson of Case and Case As expected, when ncreases detecton performance ncreases for both waveforms. However, when we compare the two waveforms at a fxed value of, the orthogonal waveforms perform much better than the NSP waveform n detectng target. Ths s because our transmtted waveforms are no longer orthogonal and we lose the advantages promsed 6dB = 3dB to BS to BS to BS 3 to BS 4 to BS (a) Probablty of detecton when N BS < M. As an example we use 4 confguraton. Note that 6 db to 3 db of addtonal gan n s requred to detect target wth 9% probablty, dependng upon the NSP waveform transmtted.. 3dB = 5 5dB to BS to BS to BS 3 to BS 4 to BS (b) Probablty of detecton when N BS M. As an example we use 8 confguraton. Note that 3 db to 5 db of addtonal gan n s requred to detect target wth 9% probablty, dependng upon the NSP waveform transmtted. Fg. 3. Performance of Algorthms () and (): Usng our spectrum sharng and projecton algorthms, we can select nterference channel for radar sgnal projecton to maxmze detecton probablty and mnmze gan n requred as a result of NSP of radar waveforms. For example, Algorthms () and () select BS#5 and BS# for N BS < M and N BS < M cases, respectvely, as they requre mnmum addtonal gan n. by orthogonal waveforms when used n MIMO radars as dscussed n Secton II-F, but, we ensure zero nterference to the BS of nterest, thus, sharng radar spectrum at an ncreased cost of target detecton n terms of. In Case, n order to acheve a desred for a fxed P FA we need more for NSP as compared to Case. Ths s because we are usng more radar antennas, whle the antennas at the BS reman fxed, n Case whch ncreases the dmenson of the null space of the nterference channel. Ths yelds better detecton performance even for NSP waveform. So, n order to mtgate the effect of NSP on radar performance one way s to employ a larger array at the radar transmtter.

9 9 = = 3 6dB 6dB = 5 6dB = 7 6dB Fg. 4. Case : dmn[h ] = : as a functon of for varous values of probablty of false alarm P FA,.e., P FA =, 3, 5 and 7. The nterference channel H has dmensons 4,.e., the radar has M = 4 antennas and the communcaton system has N BS = antennas, thus, we have a null space dmenson of dmn[h ] =. Note that we need 9 to db more gan n for the NSP waveform to get the same result produced by the orthogonal waveform. = = 3 3.5dB 4dB = 5 4.5dB P for P = 7 D FA 4.5dB Fg. 5. Case : dmn[h ] = 6 : as a functon of for varous values of probablty of false alarm P FA,.e., P FA =, 3, 5 and 7. The nterference channel H has dmensons 8,.e., the radar has M = 8 antennas and the communcaton system has N = antennas, thus, we have a null space dmenson of dmn[h ] = 6. Note that we need 3.5 to 4.5 db more gan n for the NSP waveform to get the same result produced by the orthogonal waveform.

10 VI. CONCLUSION In the future, radar RF spectrum wll be shared wth wreless communcaton systems to meet the growng bandwdth demands and mtgate the effects of spectrum congeston for commercal wreless servces. In ths paper, we analyzed a smlar spectrum sharng scenaro between radars and cellular systems. We proposed a spectrum sharng scenaro n whch a MIMO radar s sharng spectrum wth a cellular system. We proposed algorthms for nterference channel selecton and projecton of radar waveform onto the selected nterference channel n order to mtgate nterference to the selected BS. We evaluated the detecton performance of spectrum sharng MIMO radars. We formulated the statstcal detecton problem for target detecton and used generalzed lkelhood rato test to decde about the presence of target when usng orthogonal waveforms and null-space projected (NSP) waveforms. 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