TIME REVERSAL MIMO RADAR: IMPROVED CRB AND ANGULAR RESOLUTION LIMIT

Transcription

1 TIME EVESAL MIMO ADA: IMPOVED CB AND ANGULA ESOLUTION LIMIT Foroohar Foroozan, Amir Asif,andémy Boyer Computer Science and Engineering, York University, Toronto, ON, Canada MJ P Laboratoire des Signaux et Système, Université Paris-Sud XI,France ABSACT The paper derives closed form (nonmatrix) expression for the deterministic Cramér-ao bound (CB) for the direction-of-arrival associated with a target embedded in a noisy, multipath channel using the time reversal () MIMO system. By incorporating the built-in adaptive waveform processing feature to reshape the MIMO probing signals, we prove that the CBs for the direction of arrival can be improved in ways not foreseen with the conventional MIMO radars. Asecondcontributionofthepaperistheanalyticalderivation of the Angular esolution Limit (AL) defined as the minimal separation between two targets to be separately resolved by the MIMO radar. At high signal-to-noise ratio, we show that the AL inherits the properties of the CB and is superior to its conventional counterpart by a factor proportional to the order of the channel multipath. Index Terms Time eversal, MIMO radar, CB and AL.. INODUCTION Time reversal () utilizes the reciprocity property of wave propagation in a time-invariant medium. based methods time reverse, energy normalize, and retransmit (mathematically or physically) a previously recorded backscatter of a probing signal reflected from the dispersive medium. As the process is repeated, the signal becomes highly focused on the sources (targets) producing backscatters. This phenomenon is referred to as super-resolution focusing, which among other factors depends on the physical aperture of the transmitter-receiver array and the medium. A complex medium creates the so-called multipath effect and significantly increases the effective aperture of the transmitter-receiver array. Indeed, harnesses multipathing to enhance focusing resolution beyond the classical diffraction limit. This paper applies to multiple-input/multiple-output (MIMO) radars for improving the accuracy of the estimation algorithms associated with an aerospace target. Indeed, several researchers have reported that taking advantage of the super-resolution focusing property of improves the detection accuracy 4, 5 in radars. Previously, we have applied for radar target localization using both single-input/multiple-output (SIMO) 6, 7 and MIMO radar systems 8 0. MIMO radar has the ability to transmit multiple and potentially different probing signals. The resulting waveform diversity in MIMO radars enables superior performance including improved target identifiability, clutter/interference rejection that includes fading mitigation, and interference suppression 4. Thead- vantages of a MIMO radar system with both colocated and widely separated antenna elements are further investigated in 5 tooffer improved resolution and higher sensitivity towards detecting moving targets. In this work, we focus on the case of colocated antennas in a MIMO setup, where the transmit waveforms are adjusted according to the channel characteristics using. The first section of the paper derives and analyzes closed form (nonmatrix) expressions ofthedeterministic Cramer-ao bounds (CB) for the direction of arrival associated with a target embedded in a medium that is contaminated by multipath clutter and background noise using the /MIMO framework. To assess the accuracy of the estimation of the target parameters, the CB 6 is a universally accepted tool and provides the optimal accuracy achievable by any unbiased estimator of the target parameters. What distinguishes the CB derived in this paper from the great majority of existing derivations in literature, e.g., 7, is that the based MIMO CBs are formulated in a multipath environment. By incorporating the built-in adaptive waveform processing feature to design the probing signals, our intuition is that the CB can be further improved in ways which were not foreseen with the non-adaptive waveforms (referred to as conventional) MIMO radars. The second section of the paper focuses on the analytical derivation and analysis of the Angular esolution Limit (AL), which is the minimal separation between two closely located targets at which they can be resolved separately. Deriving the AL of a radar system is a fundamental problem and has previously been formulated in a clutter free environment. This paper considers multipath channels. For the simplified case of two targets and a single interference source, we show that the AL is lower than the conventional AL by a factor that depends on the strength of the multipath at equal Signal-to-Interference-Noise ratio (SIN). To the best of our knowledge and among all existing literature on the CB and AL surveyed by us, no such contribution has been made in the context of the AL for the MIMO radars. The paper is organized as follows. Section defines the notation and derives the mathematical formulation for the conventional MIMO system followed by the /MIMO system using a noisy - way parallel factor analysis (PAAFAC) model 9, 4. Section derives the CBs for the MIMO radar, while Section 4 formulates the AL. esults from numerical examples are discussed in Section 5. Finally, Section 6 concludes the paper.. POBLEM SETUP Standard MIMO: We consider a colocated MIMO radar system with an equal number P of transmit and receive antenna elements monitoring L moving targets modeled as narrowband point-sources in the far field. The received signal 5 resulting from the k th, (0 k (K )) transmittedpulse,withk being the number of pulses in one coherent pulse interval (CPI), is k = α l e jπflk a (ζ l )a T T (ζ l )F + N k, () where α l and f l denote, respectively, the reflection coefficient and the normalized Doppler frequency associated with the l th target. The target velocity from path l is denoted by v l and the radar pulse //$.00 0 IEEE 45 ICASSP 0

2 period by T p and the wavelength of the propagating wave by λ,then f l =T pv l /λ. LetQ be the number of samples in each CPI, then the known P Q probing signal matrix F is defined by F =f f P T where f i =F i(0) F i(q ) T,whereastheP Q noise matrix for the k th pulse is denoted by N k. The transmit and receive steering vectors are denoted by a T (.) and a (.) with the i th elements of these vectors given by a T (ζ l ) i =exp(j(π/λ)ζ l d T i ) and a (ζ l ) i =exp(j(π/λ)ζ l d i ), respectively,whereζ l =sin(θ l ) considering that the target direction of arrival (DOA) is denoted by θ l. Notations d T i and d i are the distances between the reference sensor and the i th transmission and reception arrays, respectively. We assume that the transmitted signals are orthogonal, then relations FF H = F F T = QI P hold true. After matched filtering, Eq. () is given as Y k = Q k F H = { }}{ Qαl e jπflk a (ζ l )a T T (ζ l )+V k, () α l where V k =/ Q N k F H denotes the noise matrix after matched filtering. Following the CanDecomp/Parafac model 4 y =vec(y 0) T vec(y K ) T T = α l g l + v, () where g l = a(f l ) a T (ζ l ) a (ζ l ),thedopplerfrequencyvector a(f l )=e jπfl e jπfl(k ) T,and denotes the Kronecker product. Notation v =vec(v 0) T vec(v K ) T T vectorizes the noise matrices of K pulses. Let the observation noise before matched filtering be complex circular Gaussian independent and identically distributed samples with zero mean and covariance of σvi. Due to the orthogonality of the probing signals, one can show that E(vv H )=σvi KP with E{.} being the expected operator. Without loss of generality, we assume that the targets of interest are the first and the second indices of l (i.e., with l =, ) and the rest of L signals are clutter. Then, Eq. () is given as y i y t y = α g + α g + α l g l +v, (4) where the signal of interest y t from the two targets is deterministic. In general, the clutter amplitudes α l are assumed to be zero-mean complex circular Gaussian independent random variables. Due to the orthogonality of the transmitted waveforms and the central limit theorem, it can be shown that observations y have circular complex Gaussian distribution with mean y t and covariance Σ. MIMO: In, the backscatter of the signals are reversed in time, energy normalized, conjugated, and re-transmitted a second time into the medium. The transmitted signal k = c k { k} P end:-: where c k is the energy normalization factor for the k th pulse. Here, the Matlab notation {end:-:} is used for showing the time reversal operation. Similar to (), the k th received signal in the stage is X k = l = l= α l e jπf l k a T (ζ l )a T (ζ l ) k + M k = c k l = α l α l e jπ(f l+f l )k a T (ζ l )a T (ζ l )a (ζ l )a H T (ζ l )F + W k (5) where F is the same as F except for time reversal. The accumulated observation noise W k takes both M k and N k into account. Due to the focusing property, Eq. (5) is approximated as,6. X k c k P α l e jπflk a T (ζ l )a H T (ζ l )F + W k. (6) Following the procedure used in conventional matched filtering, we apply the matched filter to the observation given in Eq. (6) by multiplying with F H. Using the orthogonality of F, itcanbe shown that F F H = QI p.usingthisequalityandaftermatched filtering of the observations (Eq. (6)), we have Z k = Q X k F H = P Q α l c k e jπflk a T (ζ l )a H T (ζ l )+U k. α l Stacking the returns of all k, 0 k (K ) pulses in vector z =vec(z 0) T vec(z K ) T T,themodelisrepresentedas z = α l g l + u, (7) in which gl = a (f l ) a T (ζ l ) a T (ζ l ) and a (f l ) = c 0 c e jπfl c K e jπfl(k ) T. Notation u vectorizes all the noise matrices of K pulses. Following the discussion for the standard MIMO observations, one can show that observations z is Gaussian with a covariance of Σ = Π,whereΠ is a positive definite matrix and is the noise variance.. DETEMINISTIC CB FO THE MIMO ADA In this section, we derive the CB expressions assuming that one target to be present. est of the sources are considered as interference. Based on Eq. (7), the observations become z = z t α g + l= α l g l + u. (8) The vector of unknown parameters is Φ =Φ T t Φ T i T,whereΦ t = θ,v, α T are the deterministic parameters of interest. The nuisance parameters are Φ i =θ,, θ L, α,, α L T,wherethe interferences are assumed stationary. We assume that the normalization factor is the same for all K pulses (i.e. c k = c for k), which is taken out from a (v l ) and considered in the attenuation factors... Deterministic CB for Gaussian Interference The deterministic CB for a Gaussian interference 7 is { ( J = ) H ( ) } zt zt Σ, (9) Φ t Φ t where Σ = σ uπ is the covariance of the clutter plus noise in the phase. Using the three auxiliary vectors e Π / g, e θ Π / g θ, e v Π / g v, where Π / and gv is the inverse of the square root of Π and gθ with respect to θ and v, are the partial derivative of g respectively. The closed form expression of the CB for the direction parameter is given as (Eq. () of 9) CB (θ e v )= sin (Θ), with (0) C ρ ( ) ρ = e θ e v sin (Θ)sin {e H (Ω) Ae } e 46

3 where C = c α and A =(e H θ e v )I e θ e H v.also,ineq. (0), we have introduced two distances as in 9: dist(e θ, E) = sin (Ω) = ( e H θ e / e θ e ) and dist(e v, E) = sin (Θ) = ( e H v e / e v e ) with Ω and Θ, respectively, being the largest canonical angles in 0, π/ between onedimensional linear subspaces E θ = {e θ }, E v = {e v },andthe linear subspace E = {e }... Special Case of a Nonmoving Target with Interference For a detailed analysis of the CB expression, this section assumes a simplified model consisting of one non-moving target and one unknown deterministic interference source as follows z = α g + α g + u. () Since the second target is denoted by index,werepresenttheinterference as the third source. In order to get a closed form expression of the CB, we assume that α and α are known. Directions θ and θ are unknown. The CB of the target s direction θ is ( ) CB (θ )= jθ θ j θ θ, with () jθ θ j θ i θ i =4C (P 4π cos (θ i) κ T j λ { θ θ = α α σ u g θ + 4π cos (θ i)κ T ) i =,, } λ H g, θ with κ T = P i= dt i and κ T = P i= (dt i ).Ifthetargetandinterference directions {ζ, ζ } are decoupled, the closed form CB of ζ is CB (ζ )= with σ 4C P π σt T = 4 ( ) κ P λ T κ T () P being the sample variance of the sensor positions in the transmit array and is equal to 4 κ T /(P λ ) for centro-symmetric arrays. esult. At equal SIN, the CB of θ based on the observations (Eq. ())islessthanthecbofθ based on the conventional MIMO radar in the presence of one interference source. Proof of esult. The CB of θ based on conventional MIMO, takes exactly the form of Eq. () except for a difference in the coefficient factors. Then, the gain defined as the ratio of the conventional CB(θ ) to the CB (θ ) is given as GI (θ CB(θ) ) CB (θ = C ) C =+P c α ( α α ). (4) Eq. (4) is based on comparing the ratio of C C for both the conventional and observations. under same SIN The second factor in Eq. (4) is directly proportional to the normalization factor c and the number of antenna elements P of the array while holding an inverse relationship with the strength (variance) of the observation noise. What is however more interesting to observe is the dependence of the gain in the second factor onthe attenuations associated with the target and the interference source. Term α ( α α maximizes when α =0.5 α with avalueapproaching0 when α =0and α = α. Itisobvious that the multipath factor should disappear in such a case. When α = α, boththetargetandtheinterferencehaveequalattenuation factors. In such a case, it is not possible to differentiate the target and interference responses. 4. AL FO THE MIMO ADA In this section, we derive and analyze the AL for both clutterfree and a single interference source models. In terms of the earlier notation, the AL is defined as ζ ζ. 4.. AL with No Interference With no interference, Eq. (7) can be represented as z t z = α g + α g +u. (5) Then, the AL is the solution to the Smith s equation, i.e., η CB t (δ) (the probabilities of detection and false alarm are selected such that η = ), where CB t (δ) is the CB associated with model (5). The closed from solution of the AL involves the analytic inversion of the FIM which is difficult to derive. Instead, we use a partially known radar model 8 to derive the CB. In this model, it is assumed that both velocity v and direction ζ of the second target are known. Then, the vector of unknown parameter Φ t =ζ v α α T.Consequently,theCBof Φ t is { ( CB t (ζ )= J where J = ) H ( ) } zt zt Φ t Φ. t As ζ is known and the CB is the lower bound of an unbiased estimator, then CB t (δ) =CB t (ζ ). Sincetheobservations (Eq. (5)) follow the -way noisy PAAFAC model 4, we use the results presented in Eq. (45) of 9 to derive CB t (ζ ) as CB t (ζ ) CB (ζ )+ σ u P K h (δ), (6) where CB (ζ ) is the lower bound when there is one target and no interference and h (δ) takes care of all the elements of the CB which are a function of the AL. For a nonmoving radar, centrosymmetric arrays, and high SN, h (δ) is given as 9 4 h (δ) = 4 α (CB (ζ )) g H ζ g (7) where gζ H g = p (δ){ p T (δ)p T (δ)}, (8) and the following notations are used based on the first-order Taylor series expansion in a neighborhood of 0: p T (δ)=a T T (ζ )a T (ζ ) P j(π/λ)κ T δ, (9) ( ) H p at (ζ ) T (δ)= a T (ζ ) (4π /λ ) κ T δ (jπ/λ)κ T ζ p (δ)=a H (v )a (v )= K k=0 c ke j8πtp/λ(v v )k. Assuming centro-symmetric arrays (κ T =0)andaftersimplifying, the magnitude square of the modulus term in (7) is expressed as gζ H g =4c P 4 π 4 s σt 4 δ, (0) where p (δ) =c s. UsingtheSmith scriterion CB t (ζ ) and using Eq. (6), the AL expression is given as CB (ζ ) (8P π 4 s σt 4 /K)(C)(CB (ζ )). () Substituting the lower bound in the absence of clutter in Eq. () and assuming high SN, the AL will be simplified to CB (ζ ). Consequently, the AL has the same properties as CB. 47

4 G I (θ ) db σ u =.4709 σ u =0.497 σ u =.586 σ u = β AL AL SIN (db) Fig.. CBgainasfunctionofinterferenceandnoise. Fig.. ALsforconventionalandMIMOarrays. 4.. AL of Nonmoving Targets with Interference Assume a model based on two targets and one interference source. The unknown parameters are ζ, ζ and α.estoftheparameters are assumed known. The FIM for the unknown parameters is J = σ u J ζ B(δ) B T (δ) P, () where J ζ is the FIM derived in Section. for one target and one interference; unknown parameters ζ =sin(θ ) and ζ =sin(θ ); and B(δ) = j ζ α j ζ α = { α ( g / ζ ) H g } { α ( g / ζ ) H g }. () Using the Schur complement inverse of J,theCBofζ is where CB (ζ ) = CB (ζ )+ σ u h (δ) = J ζ P h B(δ)B T (δ)j ζ (δ), (4). (5) In Eq. (4), the CB of ζ based on one target and one interference is denoted by CB (ζ ) and that of the two targets and one interference source is denoted by CB (ζ ). Defining notations C = c α / and C = c α /,Eq.(5)simplifiesto h (δ) = (6) ( C C CB (ζ ) gζ H σ g + CB (ζ, ζ ) gζ H u σ g ). u Note that gζ H g =(π σt P )δ. earrangetermsin(4)as (C 4π σt 4 P CB (ζ ) )δ + (7) (4 C C π σ T γ)δ + CB (ζ )+ C P CB (ζ, ζ )γ =0, where γ = gζ H g. TheALδ is the solution of the above polynomial. In order to compare the AL with the conventional one, we assume that the target direction ζ and the interference direction ζ are decoupled. Therefore, the second term in h reduces to zero. esult. At equal SIN and with one interference, the AL is less than the AL based on the conventional MIMO radar. Proof of esult. Using Eq. (4) and based on the fact that h = (C /)(4π 4 σt 4 P 4 )CB (ζ )δ,thealisgivenby CB (ζ ) (4P π 4 σt 4 )(C)(CB (ζ )). (8) Substituting the expression of CB (ζ ) from (), the AL in presence of one interference source is given as π σt (4CP ). (9) At high SN, C P and the AL simplifies to CB (ζ ). Based on esult, the ratio of the conventional AL to the AL is GI (ζ) which is greater than one. Therefore, the AL is always less than the conventional AL at equal SIN. 5. NUMEICAL ILLUSATION This section demonstrates the potential performance gain of the /MIMO framework by numerical examples. One colocated Uniform Linear Arrays (ULA) with 0 antenna elements and halfwavelength inter-element spacing is used as transmit and receive arrays. The targets are located at ζ = sin(π/6) = 0.5 and at ζ = 0.5 with respect to the broadside of the array. The direction of arrival associated with the interference source is set to ζ = sin(π/4). The attenuation factors for the two targets and one interference source are set to 0.8, 0.8, 0.4 T,respectively.Attenuations are assumed to be real to simplify the matrix inversion operation needed to derive the CBs. In Fig., we plot the gain for the direction of arrival θ as given in esult for a fixed direct path attenuation of α =0.8and different values of α = β α for (0 < β ). Thevarianceofnoiseinstage(i.e.)isvar- ied and the gain is plotted for different noise variances. As shown in this figure, the gain is maximum for β = or α = α, which is in agreement with esult. Fig. plots the AL obtained for the conventional and /MIMO systems at the same SINs set for β =0.5. As shown in this figure, the /MIMO offers lower AL in a one interference source model. For a fair comparison, we keep the same SINs for both the conventional and frameworks. 6. SUMMAY The paper derives and analyzes closed form expressions for the /MIMO CBs of the direction of arrival associated with a target embedded in a multipath environment corrupted with noise. Our derivations in a -path environment reveals that using, the CBs for direction can be improved over the conventional MIMO systems. Finally, the AL is derived for both environments with and without clutter. At high SN and equal SIN, the AL is shown to be lower than that for the convectional AL. 48

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