On the Lower Performance Bounds for DOA Estimators from Linearly-Modulated Signals

Transcription

1 On the Lower Performance Bounds for DOA stimators from Lineary-oduated Signas Faouzi Beii, Sofiène Affes, and Aex Stéphenne INRS-T, 800, de a Gauchetière Ouest, Bureau 6900, ontrea, Qc, H5A 1K6, Canada mais: beii@emt.inrs.ca, affes@emt.inrs.ca, stephenne@ieee.org Abstract In this paper, the probem of direction of arriva (DOA) estimation from ineary-moduated signas over AWGN channes is considered. We derive cosed-form expressions for the inphase/quadrature Cramér-Rao ower bounds of the dataaided (DA) DOA estimates from any ineary-moduated signa corrupted by additive white circuar compex Gaussian noise (AWCCGN). We consider the case of singe-source signas impinging on mutipe receiving antenna eements, commony known as singe input mutipe output (SIO) configurations. An anaytica approach is conducted to compare the achievabe performance in coherent estimation against noncoherent estimation over uniform inear array (ULA) and uniform circuar array (UCA) configurations. It wi be shown that the CRLBs that can be achieved over a ULA are ower than those that can be achievabe over a UCA up to a given anguar aperture whose expression is aso derived in this paper. It wi be shown aso that ULAs exhibit ower CRLBs in coherent estimation than in noncoherent estimation and that the CRLBs hod, however, the same for UCAs in both estimation schemes. I. INTRODUCTION Direction of arriva (DOA) estimation of mutipe pane waves impinging on an arbitrary array of antennas is an important task in array signa processing [1, 2]. Therfore, DOA estimation has attracted much interest during the ast few decades since the knowedge of this parameter has been crucia for many appications. For exampe, in modern communication systems, estimating the DOAs of the desired users and those of the interference signas aows their extraction and canceation, respectivey, by beamforming technoogies [3, 4]. Roughy speaking, DOA estimators can be mainy categorized into two categories: data-aided (DA) and non data-aided (NDA) estimators. DA DOA techniques assume perfect aprioriknowedge of the transmitted symbos. NDA DOA techniques assume the atters to be competey unknown. DA estimators imit the throughput of the sytem. However, perfecty known sequences are usuay transmitted for synchronization purposes. In this case, these known symbos can be used to design DA DOA estimation techniques without any addititiona penaty. In this context, a number of high-resoution DOA estimation agorithms have been deveoped, incuding Utipe SIgna Cassification (USIC) [5] and stimation of Signa parameters via Rotationa Invariance Technique (SPRIT) [6]. oreover, in the ast few decades, there has been interest in deveoping other agorithms to improve the DOA estimation capabiity, especiay the maximum ikeihood (L) approaches [7], the decouped maximum ikeihood (DL) ange estimator [8] and the modified ikeihood function approach [9]. In both DA and NDA cases, the variance of any DOA estimastor is usuay compared to the Cramér-Rao ower bound (CRLB) which is a we-known fundamenta performance imit that serves as a usefu benchmark for practica estimators [10]. In the NDA mode, the CRLB is often numericay or empiricay computed. But even when a cosed-form expression can be obtained, it is usuay compex and requires tedious agebraic manipuations. However, in the DA scenario, the CRLB is reativey easy to derive in cosed-from expression. In this paper, we imit our anaysis to the DA case for which we derive the cosed-form expression for the CRLB of DA DOA unbiased estimates from any ineary-moduated signa. Actuay, the DA CRLB was recenty derived by Deams et a. in [11], but ony for BPSK and QPSK transmissions. In this paper, however, we extend these resuts to any one-/twodimensiona consteation. Then, we focus our anaysis on the comparison of the achievabe performance between coherent and noncoherent estimations over the most used antenna configurations, namey the ULA and the UCA. We mention, however, that the anaytica comparison carried out in this paper between ULA and UCA configurations, in the DA mode, is aso vaid in the NDA mode since this comparison is based on a geometrica factor which is the same for both DA and NDA scenarios. This can be seen in [11] for the specia cases of BPSK and QPSK transmissions and for higher-order square QA consteations in [12]. The rest of this paper is organized as foows. In section II, we introduce the system mode that wi be used throughout the artice. In section III, expicit expressions for the DA Fisher information matrix (FI) and the DA CRLB of the DOA estimates wi be derived for any ineary-moduated signa. In section IV, the anaytica comparison of the CRLB in coherent and noncoherent estimation schemes wi be conducted over ULA and UCA configurations. Graphica representations and discussions wi be given in section V and some concuding remarks wi be drawn out in section VI. Throughout this paper, matrices and vectors are represented by bod upper case and bod ower case etters, respectivey. II. SYST ODL Consider a ineary-moduated (i.e., PA, PSK, QA) signa impinging on an arbitrary array of antennas from a singe source situated in the far-fied region. We suppose that the received signa is corrupted by an additive white compex circuar Gaussian noise (AWCCGN), spatiay /10/$ I 381

2 uncorreated between antenna eements, with unknown noise power σ 2 over each antenna branch. Assuming a receiver with idea time and frequency synchronization, the received signa at the output of the array matched fiter can be modeed as a compex signa as foows: y(n) =Se jφ a x(n)+w(n), n =1, 2,...,N, (1) where, at time indices n N n=1, x(n) N n=1 are the N transmitted independent and identicay-distributed (iid) symbos and y(n) N n=1 are the correspending vectors of received sampes on the antenna array. In this paper, we consider DA estimation scenarios and, therefore, the sequence x(n) N n=1 is considered as perfecty known to the receiver. a is the steering vector parametrized by the scaar DOA parameter θ. The unknown parameters S and φ stand for the constant channe gain and phase distortion, respectivey. For any panar configuration of the receiving antenna array, the steering vector can be written as: [ a = e j2f0(θ),e j2f1(θ),...,e j2f 1(θ)] T, (2) where the superscript T stands for the transpose operator and f i (θ) i=1,2,...,( 1) are transformations of the scaar DOA parameter θ, which vary from one configuration to another. The steering vector verifies the property a 2 =, where. returns the second norm of any vector. oreover, we assume that the transmitted symbos x(n) n=1,2,...,n are independent from the noise components w(n) n=1,2,...,n which are modeed by iid -variate zero-mean compex circuar Gaussian random vectors with independent rea and imaginary parts and w(n)w(n) H = σ 2 I, where I refers to the ( ) identity matrix. Furthermore, in order to derive standard CRLBs, the consteation energy is supposed to be normaized to one, i.e., x(n) 2 =1, where. and. return the expectation of any random variabe and the moduus of any compex number, respectivey. The unknown parameters σ, S, φ and θ are more convenienty gathered in the foowing parameter vector: α =[σ S φ θ] T, (3) The true SNR of the system is defined as foows: ρ = S2 P σ 2, where P is the consteation power given for the DA and NDA N n=1 modes, respectivey, as P = x(n) 2 N 1 for N 1 and P = x(n) 2 =1. III. DRIVATION OF TH I/Q DATA-AIDD FI FOR DOA STIATS In this section, we consider a genera antenna array configuration. Since the transmitted symbos are assumed apriori known and the circuar noise components are Gaussian distributed, then the received signa at the output of the array matched fiter is circuar Gaussian distributed. In fact, it can be shown that P DA [y(n); α] =P [y(n) x(n); α], the probabiity of the received vector y(n) parameterized by the unknown parameter vector α, is given by: P DA [y(n); α] = y(n) Se jφ 1 x(n)a σ 2 e ( ) H (y(n) Se jφ x(n)a) σ 2. (4) After some agebraic manipuations, the og-ikeihood function of the received sampes reduces to: n(p DA [y(n); α])= n ( σ 2) 1 σ 2 ( y(n) 2 S 2 x(n) 2 with +SRx(n)U(n) SIx(n)V (n)), (5) U(n) = 2 Re jφ y(n) H a, (6) V (n) = 2 Ie jφ y(n) H a. (7) In the above equations, R. and I. refer to the rea and imaginary parts of any compex number, respectivey. In the ony cases of BPSK and QPSK considered in [11], note that the DA CRLB for the DOA estimates is derived using the Sepian-Bangs formua [13, (B.3.25)]. In this paper, for any ineary-moduated signa, we wi use another technique which is based on the evauation of the expectation of the second partia derivatives of (5). Therefore, we need to evauate beforehand the partia derivatives of the og-ikeihood function and then derive the DA FI associated with the four considered parameters (σ, S, φ, θ). First, we partition the parameter vector into two parameter vectors α (1) =[S, σ 2 ] and α (2) =[φ, θ] and it wi be ater verified that: =0 i, =1, 2, (8) α (1) i α (2) where α (m) i is the i th eement of α (m) for m =1, 2. Therefore, α (1) and α (2) are two decouped parameter vectors and we are actuay deaing with two seperate probems regarding the estimation of S and σ 2, on the one hand, and the estimation of θ and φ on the other hand. This means that the DA FI is bock-diagona structured as foows: I DA (α) = IDA 1 (σ, S) 0, (9) 0 I2 DA (φ, θ) where 0 is a (2 2) zero matrix. I1 DA (σ, S) is the DA FI pertaining to the estimation of the channe ampitude and the noise power. I2 DA (φ, θ) is the DA FI associated with the unknown channe phase and signa DOA. At this stage, as inferred from the bock-diagona structure of the goba FI, I DA (α), we mention that whether the channe ampitude and the noise power are aprioriknown or not does not affect the utimate accuracy on the DAO DA estimation. This is because, as I DA (α) is bock diagona, the DOA CRLB (which corresponds to the fourth diagona eement of the inverse of I DA (α) [15]) does not depend on the eements I DA 1 (σ, S). 382

3 oreover, since y(n) N n=1 are i.i.d random vectors, the entries of the matrices I1 DA (σ, S) and I2 DA (φ, θ) are given by: N [I1 DA 2 n(p (σ, S)] i, = DA [y(n); α]),i, =1, 2, (10) n=1 α (1) i α (1) N [I2 DA 2 n(p (φ, θ)] i, = DA [y(n); α]),i, =1, 2.(11) n=1 α (2) i α (2) In (10) and (11), the expectation. is taken with respect to y(n). We show subsequenty that for i,, m =1, 2, wehave: = C α (m) i α (m) i,,m 1 + Ci,,m 2 U(n) + Ci,,m 3 V (n), (12) where Ci,,m 1, C2 i,,m and C3 i,,m are deterministic coefficients which depend on the unknown parameters α (m) i 2 i,m=1. To that end, we wi ony detai the derivation of = and the other 2 n(p DA[y(n);α]) α (2) 2 α(2) 1 2 n(p DA[y(n);α]) terms can then be easiy derived in the same way. Indeed, we have: = SRx(n) V (n) σ 2 SIx(n) U(n) σ 2, where U(n) = U(n), θ V (n) V (n) =. θ Hence, we deduce: 2 n (P DA [y(n); α]) = SRx(n) σ 2 SIx(n) σ 2 V (n) U(n), and it can be shown that: U(n) = jȧh a V (n), V (n) = jȧh a U(n), with ȧ = a θ. Consequenty, we obtain: 2 n (P DA [y(n); α]) = jȧh as Rx(n) σ 2 U(n) + jȧ H as Ix(n) σ 2 V (n).(13) We note from (13) that has the same 2 n(p DA[y(n);α]) form given in (12) with C 1 2,1,2 =0. In order to find an expicit expression for (13), we define the scaar random variabe ϑ(n) = 2 e jφ y(n) H a = U(n) + jv (n) whose probabiity, in the DA case, is P DA [ϑ(n); α] = P [ϑ(n) x(n); α] = P DA [(U(n),V(n)); α]. oreover, taking into account the hypothesis of the independence of Rx(n) and Ix(n), we show that U(n) and V (n) are independant and, therefore, we have: P DA [(U(n),V(n)); α] =P DA [U(n); α]p DA [V (n); α], where it can be shown that: exp 2(2σ P DA [U(n); α] = 2 ) 2 (2σ2 ) P DA [V (n); α] = exp (U(n) 2S Rx(n))2 (V (n)+2s Ix(n))2 2(2σ 2 ) 2 (2σ2 ), (14). (15) We note from (14) and (15) that U(n) and V (n) are rea Gaussian random variabes with the same variance 2σ 2 and with means U(n) = 2S Rx(n) and V (n) = 2S Ix(n), respectivey. Therefore, from (13), we obtain: 2 n (P DA [y(n); α]) = 2(jȧH a)s 2 σ 2 x(n) 2. (16) We note that in the specia case of normaized-energy BPSK and QPSK consteations, we have x(n) = 1, for n = 1, 2,...,N and hence 2 n(p DA[y(n);α]) = 2(jȧH a)s 2 σ. 2 Therefore, we have [ I DA (α) ] = 2N ( jȧ H a ) ρ, which is 3,4 the same resut previousy obtained for these two specia consteations in [11]. For any higher-order consteation, this resut hods aso as S2 N σ 2 n=1 x(n) 2 = Nρ. In fact, from (11), we have: [ I DA (α) ] 3,4 = 2 ( jȧ H a ) S 2 σ 2 = 2N ( jȧ H a ) ρ. N x(n) 2, n=1 oreover, we note that the diagona structure of the DA FI is verified because 2 n(p DA[y(n);α]) α i α invoves the i=1,2; =3,4 sum of two opposite terms. In fact, it can be verified that for i, =1, 2: α (1) i α (2) =2SC i, (Rx(n)Ix(n) Rx(n)Ix(n)), =0. Deriving the other terms of the DA FI, in the same way, we obtain the foowing resut: 4 σ I DA σ 0 0 (α) =N 2. (17) 0 0 2ρ 2ρ(jȧ H a) 0 0 2ρ(jȧ H a) 2ρ ȧ 2 383

4 Now that we have derived the expression of the FI, in the DA mode, the considerd CRLBs can be easiy deduced and, in the next section, we wi compare these ower bounds between ULA and UCA configurations in both coherent and noncoherent estimations. IV. COPARISON OF TH DA DOA CRLBS BTWN ULA AND UCA CONFIGURATIONS For a normaized consteation, inverting I DA (α), we deduce the expression for the DA CRLB in the presence of an unknown phase offset (i.e., noncoherent estimation) 1, CRLB NCO DA,ofany ineary-moduated signa and for any panar antenna-array configuration as foows: CRLB NCO DA (θ) = 1 Nγ ρ, (18) where γ is a purey geometrica factor defined as γ = 2ȧ H Π a ȧ with Π a = I aa H /. It can be verified that the expression given by (18) generaizes the expression recenty derived in [11] from BPSK and QPSK signas to any higherorder one-/two-dimensiona normaized consteation. Note aso that this geometrica factor γ appears as we in the expressions of the NDA DOA CRLBs of BPSK and QPSK signas [11] and higher-order square QA signas [12]. Therefore, the anaytica comparison between ULAs and UCAs that wi be from now on conducted, using the expression of γ in each configuration, appies aso in the NDA mode. To begin with, using (2), we show that γ can be written as: γ = 2 ( ȧ 2 ah ȧ 2 ), (19) 1 = 8 2 f k 2 (θ) ( 1 f k (θ) ) 2. (20) We first show that, as it shoud be 2, γ is stricy superior to zero. In fact, using the Cauchy-Schwarz inequaity which states that for any ineary independant ( 1)-dimensiona rea vectors x =[x 1,x 2,...,x ] and y [y 1,y 2,...,y ],wehave: ( 1 ) 2 x k y k < x 2 y 2. Repacing x k by f k and y k by 1, we obtain: ( 1 f k (θ)) 2 < 1 f k (θ) 2. Then, from (20), it foows that γ is stricy superior to zero. In the absence of the phase offset or assuming that it was perfecty recovered (i.e., the parameter φ is supposed to 1 From now on, the superscripts CO and NCO wi refer to coherent and noncoherent estimation scenarios, repectivey. 2 This is because the CRLB is aways positive and γ soud not be equa to zero as it appears in the denomeraor. be known, coherent estimation), the data-aided DOA CRLB, DA, can be obtained by simpy inverting the second diagona eement of I2 DA to yied: 1 DA(θ) = 2 Nρ ȧ 2. (21) We notice in (18) and (21) that the DA DOA CRLBs are inversey propotiona to N. Therefore, they can be easiy deduced for any observation interva size N 2 if they were aready computed for a given window size N 1, by simpy scaing with the factor N1 N 2. A. Uniform inear array (ULA) configuration Considering a uniform inear array (ULA), the steering vector is given by: [ a = 1,e j sin(θ),e 2j sin(θ),...,e j( 1) sin(θ)] T. Consequenty, from (19), γ ULA can be written as: ( 1 k 1 γ ULA = 2 2 cos 2 (θ) k 2 k=1 k=1 ) 2, = 2 ( 2 1) cos 2 (θ). (22) 6 Therefore, from (18), the expression of CRLB NCO DA for a ULA configuration is: CRLB NCO DA (θ) = 6 N( 2 1) 2 cos 2 (θ)ρ. (23) On the other hand, for a ULA configuration, ȧ 2 is given by: 1 ȧ 2 = 2 cos 2 (θ) k 2, k=1 2 ( 1)(2 1) = cos 2 (θ), 6 = γ ULA. Consequenty, from (18) and (21), we deduce that for a ULA configuration: DA(θ) = 3 ( 1)(2 1) 2 Nρcos 2 (θ), (24) (25) Then, it can be easiy deduced that: DA(θ) = +1 2(2 1) CRLBNCO DA (θ). (26) We show that for any > 1, +1 < 2(2 1). Therefore, from (26), DA(θ) is ower than CRLB NCO DA (θ) for any antenna-array size > 1. This is hardy surprising since in the absence of any phase offset, we have one ess nuisance parameter which utimatey resuts in better achievabe performance. It can aso be seen from (22) that γ ULA is an even function of 384

5 θ and maxima for θ =0since θ [ 2, 2 ]. Therefore, from (23) and (24), the CRLB is an even function of θ and minima for this vaue of θ, thereby refecting the symmetry of the ULA around the broadside axis. B. Uniform circuar array (UCA) configuration For a uniform circuar array (UCA), the steering vector is given by [14]: ] j cos(θ) j cos(θ 2/) j cos(θ 2( 1)/) T a = [e 2sin(/),e 2sin(/),...,e 2sin(/). Hence, γ UCA can be given by: 1 2 ( sin 2 θ 2k γ UCA = ) Using the identities sin ( θ 2k cos ( ) θ 2k = R e that γ UCA reduces simpy to: j(θ 2k ( 1 k=1 ( sin θ 2k ) ) 2 2sin 2 ( ). ) ) and 1 2k j(θ = I e ), e 2jk =0,weshow 2 γ UCA = 4sin 2 ( ), (27) which means that the CRLB for a UCA is the same for any θ, thereby refecting the circuar symmetry of the UCA. Consequenty, from (18), the expression of CRLB NCO DA for a UCA configuration is given by: ( ) CRLB NCO DA (θ) = 4sin2 N 2 ρ. (28) On the other hand, for a UCA configuration, ȧ 2 is given by: ȧ = 4sin 2 ( ( ) sin 2 θ 2k ), 2 = 8sin 2 ( ), = γ UCA 2. Consequenty, from (18) and (21), we deduce that, for a UCA configuration, the DA CRLB is the same for both coherent and noncoherent estimations and we have: ( ) DA(θ) = CRLB NCO DA (θ) = 4sin2 Nρ 2. (29) We concude from (29) that, for a DA scenario, the knowedge of the phase offset for a UCA configuration does not bring any additiona CRLB performance gain, contrariy to the ULA configuration. V. GRAPHICAL RPRSNTATIONS In this section, to iustrate the behavior of the derived ower bounds, we provide a graphica representation of the DA CRLBs of both coherent and noncoherent DOA estimations as a function of the SNR for a fixed DOA parameter θ =0. CRLB UCA, DA NCO;DA CO ULA, DA NCO ULA, DA CO SNR [db] Fig. 1. DA CRLBs for the DOA estimates with ULA and UCA configurations, p =2, N =1, =8. In Fig. 1, we compare the DOA CRLBs for both the ULA and UCA antenna-array configurations for =8receiving antenna eements. We see from this figure that, as shown anayticay, the achievabe performance on the DOA estimates depends on the geometrica configuration of the receiving antenna array. In fact, for the considered vaue of θ =0,a ULA offers better estimation performance than UCA. Actuay, for a fixed number of antennas, we prove that the CRLBs obtained for a ULA configuration are ower than the CRLBs obtained with a UCA for any vaue of θ [ θ c,θ c ] with θ c = arccos ( 3 2sin 2 ( )( 1)(+1) ).UCAsare,however, more used in practice. Indeed, the major avantage of UCAs is their 360 degrees azimutha coverage which is necessary in many appications such as wireess communications, radar and sonar and their amost invariant directiona pattern. This is in strong contrast with the widey studied ULA that ony covers 180 degrees. VI. CONCLUSION In this paper, we deveoped anaytica expressions for the CRLBs of the DA DOA CRLBs for any ineary-moduated signa as a function of the true SNR vaues. The received sampes are assumed to be corrupted by additive white circuar compex Gaussian noise. We proved that the achievabe performance on the DA DOA estimates depends on the geometrica configuration of the antenna array. In fact, with a ULA, we showed that the CRLBs are even functions of the DOA that reach their minima at broadside direction due to the ULA s axia symmetry 385

6 around it. oreover, the ULA CRLBs in coherent estimation are ower than those obtained in noncoherent estimation. With a UCA, we showed however that the CRLBs no onger depend on the DOA due to its circuar symmetry and that the CRLB in both coherent and noncoherent estimations are equa. We aso showed that, for a given antenna-array size, ULAs exhibit ower CRLBs than UCAs up to an ange aperture that depends on. RFRNCS [1] D. H. Johnson, and D.. Dudgeon, Array Signa Processing Concepts and Techniques, Prentice Ha, ngewood Ciffs, NJ, [2]. Wax, Detection and stimation of Superimposed Signas, Ph.D. Dissertation, Stanford University, [3] T. S. Rappaport, Smart antennas: adaptive arrays, agorithms, and wireess position ocation, I Press, [4] S. D. Bostein and H. Leib, utipe antenna systems: roe and impact in future wireess access, I Commun. ag., vo. 41, no. 7, pp , Ju [5] R. O. Schmidt, utipe emitter ocation and signa parameter estimation, Proc. RADC, Spectra stimation Workshop, Rome, NY, 1979, pp [6] R. Roy, A. Pauraj, and T. Kaiath, SPRIT - A subspace rotation approach to estimation of parameters of cisoids in noise, I Trans. Acoust., Speech, Signa Process., vo. ASSP-34, pp , Oct [7] P. Stoica and K. C. Sharman, aximum ikeihood methods for direction-of-arriva estimation, I Trans. Acoust., Speech, Signa Process., vo. 38, pp , Juy [8] J. Li, B. Hader, P. Stoica, and. Viberg, Computationay efficient ange estimation for signas with known waveforms, I Trans. Signa Process., vo. 43, pp , Sept [9]. Agrawa and S. Prasad, A odified ikeihood function approach to DOA estimation in the presence of unknown spatiay correated gaussian noise using a uniform inear array, I Trans. Signa. Process., vo. 48, no. 10, pp , Oct [10] P. Stoica, A. G. Larsson, and A. B. Gershman, The stochastic CRB for array processing: A textbook derivation, I Signa Process. Lett., vo. 8, pp , ay [11] J. P. Demas and H. Abeida, Cramér-Rao bounds of DOA estimates for BPSK and QPSK moduated signas, I Trans. Signa Process., vo. 54, no. 1, pp , Jan [12] F. Beii, S. Ben Hassen, S. Affes, and A. Stéphenne, Cramér- Rao bound for NDA DOA estimates of square QA-moduated signas, accepted for pubication in GLOBCO 09. [13] W. J. Bangs, Array Processing with Generaized Beamformers, Ph.D. Dissertation, Yae Univ., New Haven, CT, [14] J.-J. Fuchs, On the appication of the goba matched fiter to DOA estimation with uniform circuar arrays, I Trans. on Signa Process., vo. 49, no. 4, pp , Apri [15] S.. Kay, Fundamentas of Statistica Signa Processing, stimation Theory, Upper Sadde River, NJ : Prentice Ha,