DS-CDMA Synchronization in Time-VaryingFading Channels�Erik G. Str�om1 Stefan Parkvall1 Scott L. Miller2 Bj�orn E. Ottersten1April 26, 19951Royal Institute of TechnologyDept. of Signals, Sensors and SystemsSignal ProcessingS-100 44 Stockholm, SwedenPhone: +46 8 790 8430FAX: +46 8 790 7329email: strom@elixir.e.kth.se2University of FloridaDept. of Electrical Engineering405 CSE Bldg. 42Gainesville, Florida 32611, USAPhone: +1(904) 392-2665FAX: +1(904) 392-0044email: miller@synapse.ee.ufl.eduAbstractHerein, the problem of estimating propagation delays of the transmittedsignals in a direct-sequence code-division multiple access (DS-CDMA) systemoperating over fading channels is considered. Even though this study is limitedto the case when the propagation delays are �xed during the observation inter-val, the channel gain and phase are allowed to vary in time. Special attention isgiven to the near-far problem which is catastrophic for the standard acquisitionalgorithm. A near-far robust estimator based on subspace identi�cation tech-niques is proposed, and the Cram�er-Rao bound, which serves as an optimalitycriterion, is derived.1 IntroductionDue to the nature of the code waveforms typically used in DS-CDMA systems, ac-curate propagation delay estimation is important. See, for instance, [11], in which�This paper was presented in part at the IEEE Communication Theory Mini-Conference held inconjunction with Globecom '94. The authors are grateful for the �nancial support from MotorolaInc., Plantation, FL, and the Swedish National Board for Industrial and Technical Development.1

we evaluated the e�ect of imperfect propagation delay estimation on the populardecorrelating receiver. The classical method for achieving synchronization, the so-called sliding correlator, is described in [5]. Other methods are: rapid acquisition bysequential estimation (RASE) proposed in [15], and the multiple user extension ofRASE developed in [4]. These methods assume that the user, whose propagation de-lay is to be estimated, transmits a known data sequence (e.g., all ones). This impliesthat these methods cannot be used directly for tracking changes in the propagationdelays during data transmission. Furthermore, these algorithms are known to be sen-sitive to the near-far problem (i.e., when the received powers from the users are verydissimilar) [12, 10].Madhow and Pursley suggest in [3] that the capacity of a DS-CDMA systemis limited by the acquisition problem. Furthermore, the overwhelming majority ofthe proposed near-far resistant receivers assume perfect synchronization. This of-fers strong motivation for studying more accurate algorithms for propagation delayestimation.In this paper we extend the subspace-based propagation delay estimator, whichwe proposed in [12, 10], to a fading environment. The algorithm is experimentallyshown to be near-far resistant. Furthermore, since the estimator does not assume thatthe data sequences are known, the technique can be used for tracking as well as foracquisition. The Cram�er-Rao bound (CRB) is derived and shown to be independentof the near-far problem. This result indicates that there is no fundamental reason forsynchronization to be near-far limited.Bensley and Aazhang have also proposed a subspace-based propagation delay es-timator [1]. Our approaches di�er in that the algorithm in [1] is capable of estimatingthe channel gain and phase as well as propagation delays, whereas the algorithm inthis paper only estimates the propagation delays. However, the Bensley and Aazhang2

algorithm is limited to channels with time-invariant gain and phase shifts, while theestimator proposed here is designed for time-varying channels.2 System ModelThe system under consideration is an asynchronous K-user DS-CDMA system oper-ating in a fading environment. The modulation scheme is BPSK with bit durationT and chip duration Tc = T=N , where N is an integer. The code waveforms haveunit amplitude and are assumed to be rectangular and periodic with period T . Asa general rule, a subscript k implies that the quantity is due to the kth user. Forinstance, a period of the kth user's code waveform is denoted by bk(t), where bk(t) = 0for t =2 [0; T ).The baseband signal, sk(t), is formed by pulse amplitude modulating the datastream, dk(m) 2 f+1;�1g, with a period of the code waveform, i.e.,sk(t) = 1Xm=�1 dk(m)bk(t�mT ) (1)The transmitted signal is formed by multiplying sk(t) with the carrier p2Pk cos(!ct+�0k), where Pk is the transmitted power and �0k is the random carrier phase uniformlydistributed in [0; 2�). We assume, without loss of generality, that P1 = 1 and Tc = 1.The channel model proposed by Turin [14] is adopted. The model is intuitivelypleasing and has been shown to provide a good �t to measured channels [13, 2]. Thechannel for the kth user is modeled as a time-varying �lter with impulse responsehk(�; t), hk(�; t) = Rk(t)Xr=0 �k;r(t)�(� � �k;r(t)) (2)where �(t) is the Dirac delta function, Rk(t) is the number of paths (or rays), �k;r(t) is3

the path gain (or fading process), and �k;r(t) is the propagation delay. We assume thatthe channel is slowly varying compared to the observation time for the received signal.In particular, the number of paths and the propagation delays are considered to be�xed, i.e., Rk(t) = Rk and �k;r(t) = �k;r 2 [0; T ) for t 2 [0;MT ). Furthermore, thefading processes �k;r(t) are assumed to be wide-sense stationary and to vary slowlyin time compared to the symbol time, i.e., �k;r(t) � �k;r(mT ) for t 2 [mT; (m+1)T ).The received signal can be written asr(t) = n(t) + KXk=1 RkXr=1�k;r(t)sk(t� �k;r)q2Pk cos(!ct + �k;r) (3)where �k;r = �0k � !c�k;r and n(t) is a white Gaussian noise waveform with two-sidedpower spectral density N0=2.The receiver front-end is a standard IQ-stage followed by an integrate-and-dumpsection, as shown in Figure 1. The integration time is Ti = Tc=Q, where the integerQ is called the oversampling factor. Ignoring double frequency terms, the equivalentcomplex received sequence, r(l) = rI(l) + jrQ(l), may be written asr(l) = n(l) + KXk=1 RkXr=1�k;r(l mod QN) 1Ti Z lTi(l�1)Ti sk(t� �k;r) dt (4)where �k;r(m) = �k;r(mT )pPk exp(j�k;r) and n(l) is a white Gaussian sequence withvariance �2 = E[jn(l)j2] = N0QN=Eb;1, where Eb;1 is the transmitted energy per bit forthe �rst user. The received vector, r(m) 2 CQN , and the noise vector, n(m) 2 CQN ,are de�ned asr(m) = [ r(mQN +QN) r(mQN +QN � 1) � � � r(mQN + 1) ]T (5)n(m) = [n(mQN +QN) n(mQN +QN � 1) � � � n(mQN + 1) ]T (6)4

It is easy to show that E[n(m)] = 0 and that3E[n(m)n�(p)] = �2IQN�K(m� p); E[n(m)nT (p)] = 0 (7)After some straight-forward calculations we can formulate the contribution from thekth user to r(m) as rk(m) = Aksk(m), whereAk = [ ak;1 ak;2 � � � ak;2Rk ] 2 IRQN�2Rk (8)sk(m) = [ sk;1(m) sk;2(m) � � � sk;2Rk(m) ]T 2 C2Rk (9)The elements of sk(m) are sk;2r�1(m) = �k;r(m)z2k�1(m) and sk;2r(m) = �k;r(m)z2k(m),where z2k�1(m) = [dk(m) + dk(m+ 1)]=2 and z2k(m) = [dk(m)� dk(m + 1)]=2.The columns of Ak are the contributions from the kth user's Rk rays to r(m).For instance, if dk(m) = dk(m� 1), the kth user's rth ray will contribute the vector�k;r(m)dk(m)ak;2r�1. On the other hand, if dk(m) = �dk(m � 1), the contributionwill be �k;r(m)dk(m)ak;2r. The form of ak;2r and ak;2r�1 are de�ned by �k;r and bk(t)as ak;2r�1 = "�k;rTi D(pk;r+1)1 + 1� �k;rTi !D(pk;r)1 # ck (10)ak;2r = "�k;rTi D(pk;r+1)�1 + 1� �k;rTi !D(pk;r)�1 # ck (11)where �k;r = pk;rTi+ �k;r, such that pk;r is an integer and �k;r 2 [0; Ti), and ck 2 IRQN3We denote the r� r identity matrix by Ir and the Kronecker delta function with �K(p). Trans-pose, conjugate transpose and the 2-norm of a vector x are denoted by xT , x� and kxk = px�x,respectively.5

is de�ned asck = [ ck(QN) ck(QN � 1) � � � ck(1) ]T ; ck(l) = 1Ti Z lTi(l�1)Ti bk(t) dt (12)D(p)s 2 IRQN�QN is de�ned in block form asD(p)s = 24 0 IQN�psIp 0 35 (13)We adopt the conventions that D(0)s = IQN and D(QN)s = sIQN .The received vector, r(m), may be expressed asr(m) = As(m) + n(m) 2 CQN (14)where A = [A1 A2 � � � AK ] 2 IRQN�2R (15)s(m) = [ sT1 (m) sT2 (m) � � � sTK(m) ]T 2 C2R (16)and where R is the total number of rays, R = PKk=1Rk.3 Cram�er-Rao BoundThe derivation of the Cram�er-Rao bound (CRB) is somewhat lengthy and the deriva-tion is therefore deferred to Appendix A.Our main objective in this work is to estimate the propagation delays. It istherefore of interest to establish a bound on the accuracy with which the delayscan be estimated. If we restrict our attention to unbiased estimators, the natural6

performance measure is the error variance, and the CRB is a bound on the smallestcovariance matrix that can be achieved by an unbiased estimator. In particular,CRB(� ) � E[(� � � )(� � � )T ] (17)where � is any unbiased estimator of � ,� = [ � T1 � T2 � � � � TK ]T 2 IRR; � k = [ �k;1 �k;2 � � � �k;Rk ]T 2 IRRk (18)In order to compute the CRB, we condition on the data and treat the fadingprocesses as unknown deterministic parameters. The CRB is therefore dependent onthe transmitted bits and should be considered as being conditioned on the fading.The resulting bound is still valid for the case when data and fading are random,although the bound will not be tight. In Appendix A it is shown thatCRB�1(� ) = 2�2 MXm=1Ref��(m)�P?AZ(m)�(m)g (19)where P?AZ(m) is the projection matrix onto [range(AZ(m))]?,P?AZ(m) = IQN �AZ(m)[Z�(m)A�AZ(m)]�1Z�(m)A� (20)The matrix 2 IRQN�2R is de�ned as = [1 2 � � � K ] (21)k = [ k;1 k;2 � � � k;2Rk ] 2 IRQN�Rk (22) k;2r�1 = @ak;2r�1@�k;r = 1Ti hD(pk;r+1)1 �D(pk;r)1 i ck (23)7

k;2r = @ak;2r@�k;r = 1Ti hD(pk;r+1)�1 �D(pk;r)�1 i ck (24)and Zk(m) 2 f�1; 0; 1g2Rk�Rk , Z(m) 2 f�1; 0; 1g2R�R, �k(m) 2 C2Rk�Rk , and�(m) 2 C2R�R are de�ned asZ(m) = diag(Z1(m); : : : ;ZK(m)) (25)(Zk(m))(i; j) = 8>>>><>>>>: z2k�1(m) if i = 2j � 1z2k(m) if i = 2j0 otherwise (26)�(m) = diag(�1(m); : : : ;�K(m)) (27)(�k(m))(i; j) = 8<: sk;i(m) if i 2 f2j � 1; 2jg0 otherwise (28)We will now show that the CRB is independent of the near-far problem, i.e., theCRB for an estimator of the propagation delays for kth user's rays is independentof the received power for the other K � 1 users. To show this, let the square-rootof the mean power of the kth user's rth ray be qk;r = qE[j�k;r(m)j2], and considerQ 2 IRR�R which is de�ned as Q = diag(Q1;Q2; : : : ;QK) whereQk = diag(qk;1; qk;2; : : : ; qk;Rk) 2 IRRk�Rk (29)Moreover, let �norm(m)Q = �(m). Now, if all rays arrive with unit mean power, theCRB is CRB�1norm(� ) = 2�2 MXm=1Ref��norm(m)�P?AZ(m)�norm(m)g (30)8

From (19) and (30) we see thatCRB(� ) = Q�1CRBnorm(� )Q�1 (31)and from this equation we conclude that the CRB on an estimator of a particularray's propagation delay|given by the appropriate diagonal element of CRB(� )|is independent of the mean power of the other rays. In other words, the CRB isindependent of the near-far problem. This result is important since it tells us thatthere may exist near-far resistant estimators of � .4 Propagation Delay EstimationThe maximum-likelihood (ML) estimate of � is found by minimizing the negative log-likelihood function (49). However, the ML estimator is very computational expensiveand thus of limited practical use. We will therefore study the performance of asuboptimum, but more practical, estimator. In [12, 10] we proposed a propagationdelay estimator based on the well-known MUSIC algorithm [7]. We will here extendthat estimator from the additive white Gaussian noise channel to the more generalchannel model outlined in Section 2.The correlation matrix, R, for r(m) isR = E[r(m)r�(m)] = ASA� + �2IQN (32)where S = E[s(m)s�(m)]. We assume that A and S have full rank, and we de�ne thesignal subspace as the column space of A, i.e., range(A), and the noise subspace asthe orthogonal complement to range(A) = range(ASA�). Furthermore, since A hasrank 2R, the signal subspace will have dimensionality 2R.9

SinceR is hermitian and positive de�nite, there exists an eigenvalue decompositionof R as R = Es�sE�s + �2EnE�n (33)where Es 2 CQN�2R and En 2 CQN�(QN�2R) are such that [Es En ] is unitary, and�s 2 IR2R�2R is de�ned as�s = diag(�1 + �2; �2 + �2; : : : ; �2R + �2) (34)where f�1; : : : ; �2Rg are the nonzero eigenvalues of ASA�. Since ASA� is hermitianand positive semi-de�nite, these eigenvalues will be real and positive.Note that range(A) = range(Es) and, consequently, the columns of A (which arefunctions of � ) are orthogonal to the columns of En. Consider the vectors bk;1(�)and bk;2(�): bk;1(�) = " �TiD(p+1)1 + 1� �Ti!D(p)1 # ck (35)bk;2(�) = " �TiD(p+1)�1 + 1� �Ti!D(p)�1# ck (36)where � = pTi + � such that p is an integer and � = [0; Ti). We observe thatak;2r�1 = bk;1(�k;r) and ak;2r = bk;2(�k;r). Thus, given knowledge of R we can�nd f�k;1; : : : ; �k;Rkg as the solutions to kE�nbk;1(�)k2 = 0, or as the solutions tokE�nbk;2(�)k2 = 0. In practice, the correlation matrix is unknown and is thereforeestimated by the sample correlation matrix, RM , and a consistent estimate of En isfound in the eigenvalue decomposition of RMRM = 1M MXm=1 r(m)r�(m) = Es�sE�s + En�nE�n (37)10

where the columns of En are the eigenvectors corresponding to the QN �2R smallesteigenvalues of RM . Note, however, that the columns of A will now be only approxi-mately orthogonal to the columns of En.To form En, we need knowledge the dimension of the signal subspace, 2R. TheMUSIC algorithm is quite robust against overestimating R, and several methods forestimating R exist [16, 7]. Therefore, we will ignore the problem of determining Rsince this does not seem to be a critical issue.The original MUSIC algorithm is formulated for the case where each column inA is parameterized by a distinct parameter. However, this is not the case here andwe will therefore formulate one possible modi�cation of the MUSIC algorithm. Letthe MUSIC cost function for the kth user be de�ned asJk(�) = kE�nbk;1(�)k2kbk;1(�)k2 + kE�nbk;2(�)k2kbk;2(�)k2 (38)As seen from (35) and (36), bk;1(�) and bk;2(�) are piecewise linear in � . In particular,for � 2 [pTi; (p+ 1)Ti) where p is an integer,bk;1(�) = �Tibk;1(t1) + 1� �Ti!bk;1(t0) (39)bk;2(�) = �Tibk;2(t1) + 1� �Ti!bk;2(t0) (40)where t0 = pTi, t1 = (p+1)Ti and � = � � t0. By substituting (39) and (40) into (38),we obtain Jk(�) = s2K11 + (1� s)2K12 + 2s(1� s)K13(s2 + (1� s)2)QN + 2s(1� s)K14+ s2K21 + (1� s)2K22 + 2s(1� s)K23(s2 + (1� s)2)QN + 2s(1� s)K24 (41)11

where we have de�ned s = �=Ti, K11 = kE�nbk;1(t1)k2, K12 = kE�nbk;1(t0)k2, K13 =Refb�k;1(t1)EnE�nbk;1(t0)g, K14 = b�k;1(t1)bk;1(t0) and where K21, K22, K23 and K24are de�ned in an analogous manner by changing bk;1 to bk;2.From (41) we see that Jk(�) can be written as a rational function of two polyno-mials of degree four and that Jk(�) is di�erentiable for � 2 [0; T ) except when � = pTifor an integer p. This suggests an e�cient search procedure of the cost function. Weform the set of candidate estimates, T , of the propagation delays f�k;1; : : : ; �k;Rkg asthe union of f0; Ti; 2Ti; : : : ; (QN � 1)Tig and the solution set to the equationdJk(�)d� = 0 (42)The �nal estimates, f�k;1; : : : ; �k;Rkg, are found as the members of T correspondingto the Rk smallest values of the cost function.5 Numerical ResultsThe simulated system is a K = 5-user system with N = 31 chips per bit Goldsequences, SNR1 = 2Eb;1=N0 = 15 dB, and Rk = 2 rays per user. The rays from thedesired user were assumed to have the same mean (unit) power. The fading processeswere standard Rayleigh fading processes with a Doppler frequency of 2:60� 10�3=T(carrier frequency of 900 MHz, data rate of 32 kbit/s, and a transmitter-receiverrelative speed of 100 km/h). Independent realizations of fdk(m); �k;r(m);n(m)g wereused for each Monte-Carlo run. The propagation delays were (randomly) set to�Tc = [ 8:25 15:31 2:81 29:38 2:29 15:52 11:91 8:60 28:33 16:42 ]T (43)12

The received vector was observed for M = f100; 200; 300; 400; 500; 600g symbols and�1;1 was estimated by the MUSIC estimator. The variance of the MUSIC algorithmwas estimated as the sample variance found from 103 independent Monte-Carlo trials.Since the Cram�er-Rao bound is dependent on the realizations of dk(m) and �k;r(m),the CRB plotted in Figures 2 and 3 is the average CRB. The near-far ratio, Pk=P1,for k = 2; : : : ; 5 was varied from 0 dB to 30 dB in order to investigate the near-farresistance of the MUSIC estimator.The results of the experiment are shown in Figures 2 and 3 for Q = 1 and Q = 2,respectively. The average CRB is the bottommost dash-dotted line and the estimatedMUSIC variance is plotted for near-far ratios P2=P1 = 0 dB: (o), 10 dB: (x), 20 dB:(+), 30 dB: (*). Recall that the CRB is invariant to the near-far ratio.The plots indicates that the MUSIC estimator is near-far robust. The CRB is notattained by the MUSIC algorithm. However, recall that the CRB will not be a tightbound since the number of parameter increases with M , i.e., no e�cient estimatormay be found for �nite Q. On the other hand, note from Figure 3 that oversamplingincreases the accuracy of the algorithm and decreases the distance to the CRB. Ofcourse, the price paid for this is increased complexity.The strategy outlined in Section 4 calls for searching allQN intervals � 2 [pTi; (p+1)Ti) of the cost function. The estimates, �1;1 and �1;2, are picked as the � 2 Tcorresponding to the two smallest values of the cost function. The procedure isillustrated by Figure 4, where a typical cost function is plotted. The circles indicatethe members of T . In this case, the MUSIC algorithm will pick the right bins, i.e.,the estimates will be close to 8:25Tc and 15:31Tc. However, if the cost function hadbeen more perturbed, we might have chosen the wrong bin for one or both of �1;1and �1;2. This type of error, referred to as an outlier error, can be quite large. Onthe other hand, if we choose the correct bin, or restrict the search to be inside the13

correct bin (as the case will be in tracking), the error is much smaller. To make anmeaningful comparison with the CRB, we therefore exclude the outlier errors beforecomputing the variance of �1;1. The relative number of outlier errors can be foundin Table 1. Note that these number are not very reliable since only 103 Monte-Carloruns were made. However, the table indicates that for large M , there will be only afew outliers. The reason why outlier errors are frequent for small M is that the ray,whose delay is to be estimated, might be in a deep fade during the entire observationinterval which leads to a very noisy estimate.M = 100 200 300 400 500 600P2=P1 = 0 [dB] 0.36 0.20 0.12 0.04 0.02 0.0210 0.37 0.22 0.19 0.13 0.08 0.0620 0.40 0.27 0.22 0.17 0.10 0.0830 0.40 0.32 0.22 0.17 0.11 0.07Table 1: Relative number of outlier errors for Q = 1.6 ConclusionsIn this paper we have investigated the propagation delay estimation problem in a DS-CDMA system operating over fading channels. The CRB was presented and shown tobe independent of the near-far ratio. This is encouraging since this result implies thatthere may exist near-far resistant estimators. The standard sliding correlator is knownto have poor performance in a near-far environment [12, 10]. The maximum-likelihoodestimator would perform better, however, the overwhelming complexity associatedwith the maximum-likelihood estimator limits its practical use. We therefore proposeda MUSIC based estimator and devised an e�cient method for searching the costfunction. The MUSIC estimator was experimentally shown to be robust against the14

near-far problem. However, the MUSIC estimator does not attain the CRB.A Cram�er-Rao Bound DerivationThe derivation closely follows the procedure devised by Stoica and Nehorai [8]. Dueto limited space, the following discussion is rather succinct; a more verbose derivationcan be found in [9].We assume that �2 and �k;r are unknown and deterministic parameters that are�xed during the observation interval t 2 [0;MT ) and that �k;r 6= pTi for any integer p,which guarantees that A is di�erentiable with respect to �k;r. Furthermore, we con-sider f�k;r(m)g to be unknown and deterministic form = 1; 2; : : : ;M , k = 1; 2; : : : ; K,and r = 1; 2; : : : ; Rk.We denote the real part of a complex quantity x by �x and the imaginary part by~x. The parameter vector � 2 IR1+2RM+R is de�ned as� = [ �2 �T � T ]T (44)� = [ ��T (1) ~�T (1) ��T (2) ~�T (2) � � � ��T (M) ~�T (M) ]T 2 IR2RM (45)�(m) = [�T1 (m) �T2 (m) � � � �TK(m) ]T 2 IRR (46)�k(m) = [ �k;1(m) �k;2(m) � � � �k;Rk(m) ]T 2 IRRk (47)The Fisher information matrix, J 2 IR(1+2RM+R)�(1+2RM+R), is found asJ = E 24 @ lnL(r)@� ! @ lnL(r)@� !T35 (48)where lnL(r) is the log-likelihood function (conditioned on z = [ zT (1) � � �zT (M) ]T )of r = [ rT (1) � � � rT (M) ]T with respect to �. Thus, the resulting CRB should be inter-15

preted as being conditioned on the fading processes and dependent on the transmittedbits.It is easy to show that the log-likelihood function conditioned on z (ignoringconstants independent of �) islnL(r) = �MQN ln�2 � 1�2 MXm=1 kr(m)�As(m)k2 (49)After some algebra, we conclude that@ lnL(r)@�2 = �MQN�2 + 1�4 MXm=1 kn(m)k2 (50)@ lnL(r)@ ��(m) = 2�2 Re fZ�(m)A�n(m)g (51)@ lnL(r)@ ~�(m) = 2�2 Im fZ�(m)A�n(m)g (52)@ lnL(r)@� = 2�2 MXm=1Re f��(m)�n(m)g (53)where 2 IRQN�2R, Zk(m) 2 f�1; 0; 1g2Rk�Rk , Z(m) 2 f�1; 0; 1g2R�R, �k(m) 2C2Rk�Rk , and �(m) 2 C2R�R are de�ned in (21){(28).The Fisher information matrix, J, can now be formed by taking the expected valueof all possible outer products of the gradients (50){(53). After some manipulation weconclude that J = 266664 MQN�4 0 00 Jss Js�0 JTs� J�� 377775 (54)where Jss = diag(Jss(1); : : : ;Jss(M)) (55)16

Jss(m) = 2�2 24 RefZ�(m)A�AZ(m)g � ImfZ�(m)A�AZ(m)g� ImTfZ�(m)A�AZ(m)g RefZ�(m)A�AZ(m)g 35 (56)and Js� = [JTs� (1) � � � JTs� (M) ]T (57)Js� (m) = 2�2 24RefZ�(m)A��(m)gImfZ�(m)A��(m)g 35 (58)J�� = 2�2 MXm=1Ref��(m)��(m)g (59)The CRB matrix is found as the inverse of the Fisher information matrix. Weare mostly interested in bounding the performance of a propagation delay estimator.The bound on the covariance matrix of such an estimator is found as the lower righthand side R � R block of the CRB matrix. In other words, we can write the CRBmatrix as J�1 = 266664CRB(�2) 0 00 CRB(s) �0 � CRB(� ) 377775 (60)where � denotes matrices that are of no particular interest to us and where CRB(� ) �E[(� � � )(� � � )T ] for any unbiased estimator � of the propagation delay vector � .Invoking a standard result on the inverse of a block matrix (see [6], Lemma 1,page 53), we see from (54) thatCRB�1(� ) = J�� � JTs�J�1ss Js� (61)After some algebraic manipulations we obtain (19).17

References[1] S. E. Bensley and B. Aazhang. \Subspace-based estimation of multipath chan-nel parameters for cdma communication systems." In Proceedings IEEE GlobalTelecommunications Conference, Communication Theory Mini-Conference Vol-ume, pp. 154{157, 1994.[2] H. Hashemi. \Simulation of the urban propagation channel." IEEE Transactionson Vehicular Technology , vol. 28, no. 3, pp. 213{225, August 1979.[3] U. Madhow and M. B. Pursley. \Acquisition-based capacity of direct-sequencespread-spectrum communication networks." In Proceedings of the Conference onInformation Sciences and Systems, 1991.[4] T. K. Moon, R. T. Short, and C. K. Rushforth. \A RASE approach to acquisitionin SSMA systems." In Proceedings IEEE Military Communications Conference,pp. 1037{1041, 1991.[5] R. L. Pickholtz, D. L. Schilling, and L. B. Milstein. \Theory of spread-spectrumcommunications|A tutorial." IEEE Transactions on Communications, vol. 30,no. 5, pp. 855{884, May 1982.[6] L. L. Scharf. Statistical Signal Processing: Detection, Estimation and TimeSeries Analysis. Addison-Wesley, Reading, Massachusetts, 1991.[7] R. O. Schmidt. A Signal Subspace Approach to Multiple Emitter Location andSpectral Estimation. Ph.D. thesis, Stanford University, Stanford, California,November 1981.18

[8] P. Stoica and A. Nehorai. \MUSIC, maximum likelihood, and Cram�er-Raobound." IEEE Transactions on Acoustics, Speech, and Signal Processing , vol. 37,no. 5, pp. 720{741, May 1989.[9] E. G. Str�om. Direct-Sequence Code-Division Multiple Access Systems: Near-Far Resistant Parameter Estimation and Detection. Ph.D. thesis, University ofFlorida, Dept. of Electrical Engineering, Gainesville, FL, December 1994.[10] E. G. Str�om, S. Parkvall, S. L. Miller, and B. E. Ottersten. \Propagation delayestimation in asynchronous direct-sequence code-division multiple access sys-tems.", August 1994. Submitted to IEEE Transactions on Communications.[11] E. G. Str�om, S. Parkvall, S. L. Miller, and B. E. Ottersten. \Sensitivity analysisof near-far resistant DS-CDMA receivers to propagation delay estimation errors."In Proceedings IEEE Vehicular Technology Conference, pp. 757{761, 1994.[12] E. G. Str�om, S. Parkvall, and B. E. Ottersten. \Near-far resistant propagationdelay estimators for asynchronous direct-sequence code division multiple accesssystems." In C. G. G�unther, editor, Lecture Notes in Computer Science: Mo-bile Communications, Proceedings 1994 International Zurich Seminar on DigitalCommunications, pp. 251{260. Springer-Verlag, New York, New York, March1994.[13] H. Suzuki. \A statistical model for urban radio propagation." IEEE Transactionson Communications, vol. 25, no. 7, pp. 673{680, July 1977.[14] G. L. Turin. \Introduction to spread-spectrum antimultipath techniques andtheir application to urban digital radio." Proceedings of the IEEE , vol. 68, ,pp. 328{352, 1980. 19

[15] R. B. Ward and K. P. Yiu. \Acquisition of pseudonoise signals by sequentialestimation." IEEE Transactions on Communications Technology , vol. 13, no. 4,pp. 475{483, December 1965.[16] M. Wax and T. Kailath. \Detection of signals by information theoretic criteria."IEEE Transactions on Acoustics, Speech, and Signal Processing , vol. 33, no. 2,pp. 387{392, April 1985.

20

r(t) p2 cos(!ct) 1Ti Z tt�Ti( ) dt1Ti Z tt�Ti( ) dt�p2 sin(!ct) t = lTi rI(l)

rQ(l)t = lTi

Figure 1: Receiver front-end.

100 200 300 400 500 60010

−5

10−4

10−3

10−2

Observation interval (M)[T2 c]

Variance of �1;1

Figure 2: Plot of measured variance of the MUSIC estimate of �1;1 with Q = 1 andfor near-far ratios P2=P1 = 0 dB: (o), 10 dB: (x), 20 dB: (+), 30 dB: (*), and averageCRB: dash-dotted line. 21

100 200 300 400 500 60010

−5

10−4

10−3

10−2

Observation interval (M)[T2 c]

Variance of �1;1

Figure 3: Plot of measured variance of the MUSIC estimate of �1;1 with Q = 2 andfor near-far ratios P2=P1 = 0 dB: (o), 10 dB: (x), 20 dB: (+), 30 dB: (*), and averageCRB: dash-dotted line.

22

0 5 10 15 20 25 300

0.2

0.4

0.6

0.8

1

Delay � [Tc]

J1(�)

Figure 4: Typical MUSIC cost function for M = 600 and P2=P1 =30 dB. The circlesindicate the members of T .

23