Improved MUSIC Algorithm for Estimation of Time Delays in Asynchronous DS-CDMA Systems Thomas Ostman, Stefan Parkvall and Bjorn Ottersten Department of Signals, Sensors and Systems, Royal Institute of Technology (KTH), S-100 44 Stockholm, Sweden E-mail: ostman @ s3.kth.se Abstract An algorithm for time delay estimation in an asynchronous direct-sequence code-division multiple access (DS-CDMA) system which exploits the structure of the system better than previously known algorithms [I], is presented. The different algorithms are compared through simulation and asymptotic analysis (large number of vector samples). For a typical scenario it is shown that the proposed algorithm decreases the standard deviation of the time delay errors with a factor of approximately 2. 1 Introduction The aim of this work is to present an algorithm for time delay estimation in an asynchronous direct sequence code division multiple access (DS-CDMA) system which exploits the structure of the system better than [I], and hence has better performance. The basic observation is that in [ 11 only the correlation matrix of lag 0 is considered. However, since the algorithm uses vector samples which are asynchronous to the sent data, two symbols will contribute to each vector sample and the sequence will be correlated in time. We propose an algorithm which uses a sequence of vector samples, where each vector sample consists of two or more of the original vector samples, and thus exploiting more than the zeroth lag of the covariance matrix. 2 Vector Model The DS-CDMA system under consideration is a K-user system having N chips per symbol. The front end of the receiver consists of an integrate and dump filter, which is sampled Q times per chip, thereafter the samples are stacked in a vector sample r(m). The vector samples are asynchronous to the sent symbols', r(m) = H(r)Bd(m) + n(m) where H(r) E is a function of the spreading codes and the time delays r E IRK, B E C22Kx2K is a Matrices and vectors are set in bold face, AT is the transpose of A, and A* is the conjugated transpose of A. diagonal matrix with the complex amplitudes, and the data vector d(m) is defined as d(m) = [dt(m)... d$(m)it dk(m) = [dl,(m) d,,(m-1)it, where quantities with index k are due to the kfh user. The matrix H(r) is defined as H(T) = [Hi... HK] HI, = p2rc-1 h2k]. For more details of the vector model see [2]. Furthermore, the model is closely related to the one used in [ 11 as shown in [3]. 3 Algorithm First, we note that since the noise and the data are independent the covariance of the received vector samples is, Rrr(Z) =E {r(m)r*(m + I )} =HBRdd(Z)B*H* + Rnn(Z), (1) where Rdd(Z) and Rnn(I) is the covariance matrix of the data and the noise, respectively. In (1) the explicit T dependence on H is dropped. This is done whenever it does not lead to confusion. The data is assumed to be white, hence Rdd(1) is zero for 2 2. The asynchronous behavior of the system (two symbols for each user in d(m) ) yields that Rdd(1) # 0. The noise is also assumed to be white, thus Rnn(Z) is zero for III 2 1 and R,,(Z) is zero for IZI 2 2. Hence, we want to derive an algorithm which take this fact into account. First, we define new vector sequences which span longer intervals. And hence exploit the asynchronous behavior of the system better than the original sequence. Alternative 1: (no overlap) Form a new sequence as, ~(m) = [rt(mc- (c- 1))... r~(m~)]~ m=1,2,...,u (2) 1058-6393/98 $10.00 0 1998 IEEE 838
where M = Luj, 1x1 is the integer part of x, M is the number of original vector samples, and 5 is the number of symbol intervals in one vector. Alternative 2: (overlap) Form a new sequence as, ~(m) = [rt(m).-. rt(c - 1 + m)lt - m =1,2,...,M, (3) where M = M - Fig. 1. + 1. The sequences are illustrated in 1 symbol 0 I symbol 1 I symbol 2 1 symbol 3 1 Original r(1):. 42).. 43).. rf4)- Alt. 1 f(l) f(2) e * i;o r(2) c -., e Alt. 2 * iq3) ~ Figure 1: Illustration of the different vector samples. where Pk and ek is the power and the phase of k user, respectively, The Kronecker product (8) used in (5) is defined in [4]. The two alternatives differ in dk(m). The definitions are for alternative 1, dk (m) = [dk (mc)... dk (m5-5)] E (C(C+l), and for sequence 2, ;I&) = [dk(m + c - 1),&+l).... dk (m) dk (m - I)] Note that H k E RQNCX(C+l) is an extension of Hk and given by H k = One important difference between the alternatives is that the first has an uncorrelated noise sequence which yields a simple analysis and compact expressions of the asymptotic performance. Alternative 2 has correlated noise samples which complicates the analysis. On the other hand, since the sequence has more samples the convergence of the covariance matrix is probable faster. One might argue that since the covariance matrix is zero for 1 2 2 increasing C beyond 2 would not be necessary. However, since the new sequences still are correlated it might be useful to increase 5 beyond 2. Both sequences can be written as. f(m) =HBd(m) +n(m), (4) The received vectors F(m) E CQNC are, in absence of noise, confined to the K(( + 1)-dimensional signal subspace spanned by the columns of a matrix E, E RQNCxK(C+l). The QNC-K(C+l)-dimensional orthogonal complement, called the noise subspace, is spanned by E,. This splitting of the total space into signal and noise subspaces is valid if QNC > K(< + 1). If a basis spanning the noise subspace were known, the delays could be find as the value of r making H(r) orthogonal to E,. However, E, is unknown, but an estimate can be found by estimating the covariance as..m 1 RW(O) =- i-(m)r*(m>, M m=l where F(m) is given by either (2) or (3), and performing an eigenvalue decomposition of Rff (0) as RFf(0) =E,A,,E: + E,AnE;, where A, is a diagonal matrix with the K(< + 1) largest eigenvalues and E, are the corresponding eigenvectors. Estimate of the users delays are found one at a time and the estimate of the ICth user s delay is given by the minimizing argument of, J~,MU(T) = { uk(t)h;(t)enefh/c(t)} (6) 1, is an identity matrix of size p x p and Opxl is a matrix of zeros of size p x 1. where UI, E C(c+l)x(c+l) is the krh user s Hermitian weighting matrix, [3]. We refer to this technique as the improved modified MUSIC algorithm. 839
The MUSIC algorithm was originally proposed in [5] and modified for timing estimation in DS-CDMA systems in [I]. Note that the modified MUSIC estimator proposed in [I] is identical to the special case C = 1 and a particular weighting. This is shown in Appendix A. 4 Analysis The asymptotic analysis is derived from a Taylor expansion of the cost function around the true value and extensions of the analysis in [6] is used. The derivation is carried out with the assumption of a full rank H in [7] and the result is where 5 A Numerical Example To compare the performance of the algorithms the following scenario is considered. BPSK modulation is used, &(m) E { -1,l). No oversampling is used (Q = 1), K = 5, Gold codes are used with N = 15, the SNR is defined as 2Eb,l/No = 2PlTS/No. We consider the performance for user #1 (k = 1). The NFR (near-far ratio) is defined as NFR = P2/Pl = P3IPl =... - PK/P1. The time delays are r = (6.2 5.9698 0.1343 7.787 1.8551 1' which is identical to [3]. The CRB (Cram&-Rao Bound) in the figures is derived from a Gaussian assumption of the input signals, see [8]. The weighting Uk = I/Tr{H;Hk} is used throughout the simulations. As is discussed in [7] the weighting Uk = I yields the same performance for high SNR and large number of vector samples, but not for small samples. In Section 4 we concluded that the algorithm is near-far resistant. This is illustrated in Fig. 2 for the sequence with overlap. The number of vector samples, M, is 500 and the SNR is 15 db. It is seen in the figure that the old algorithm (C = 1) has approximately a factor 2 worse performance in this scenario. Furthermore, the near-far resistance of the algorithm is clearly seen. Simulation Theoretical and For alternative 1 the following expression is obtained, I 5 25 30 lo NFR5[dB] 2o Figure 2: The near-far resistance for the sequence with overlap. where n2 = (QNO)/Tc is the noise variance, T, = Ts/N is the chip duration, T, is the symbol time, and NO is the spectral density of the white noise on the channel. The important conclusion from the analysis is that the performance is independent of the other users' power. The algorithm is near-fur resistant. The performance as a function of the SNR is illustrated in Fig. 3. It is seen that there is a small bias, the rms-values are above the standard deviation, for small SNR, but not for large SNR. Moreover, the gain going from C = 1 to C = 2 is approximately 3 db. The performance as a function of < is illustrated in Fig. 4 both for a sequence with overlap and for the one without. In the simulation M = 1000, SNR= 15 db, and 840
Appendix A: Similarities to Previous Proposed Algorithm for 5 = 1 In [I] linear combinations of the h-vectors are used, { a2k a2k-1 = h2k-1 + h2k = h2k--1 -h2k. The cost function proposed in [ 11 can be written as, - a;k-1e,eka2k-1 + a;ke:ne;a21c = E. J= afk- 1 a2 k - 1 a;ka2k Y (7) 10 15 SNR [db] Figure 3: The performance as a function of the SNR for the sequence with overlap. the near-far ratio is 0 db. It is seen in the figure that the analytical and the simulated results for alternative 2 are very close for ( = 1,...,7. Furthermore, it is also seen in Fig. 4 that the asymptotic analysis for alternative 1 is not a good approximation for large 5 in this scenario. This is since the number of vector samples will be very few when ( is increased. Simulation (No overlap) Next we show that j M ~J~,Mu(T) when Uk = I/ Tr{H;Hk}, thus the weighting used in this work is approximately the same as in [I]. The basic observation we use to show this is the fact Ih;k-lh2k( 5 1, i.e, h2k--1 and h2k stem from two different symbols which overlap at most for one chip (the sampling is chip-asynchronous). Then by using (7) these preliminary results are obtained, aak-1a2k-1 =h;k-1h2k--l + hikh2k + 2 Re(hak-ih2k) a;ka2k =h;k-lh2k-l + h a 2 k - 2Re(h;k-lh~k) A A aak-le,e:a2k-l =h;k-1e:neth2k-1 + h;,e:,e:hzk Thus, a;ke:neza2k =h;k-lene:h2k--l + 2 Re(hak-lEnEth2k) + h&e,ezhzk *..,... - 2 Re(h;,-,EnEkh2k). Figure 4: The performance as a function of c. 6 Conclusion By increasing the number of considered lags in the covariance matrix of the received vector samples the performance can be significantly improved. 84 1
eferences [I] E. G. Strom, S. Parkvall, S. L. Miller, and B. E. Ottersten, Propagation Delay Estimation in Asynchronous Direct-Sequence Code-Division Multiple Access Systems IEEE Transactions on Communications, vol. 44, pp. 84-93, January 1996. [2] T. Ostman and B. Ottersten, Low Complexity Asynchronous DS-CDMA Detectors in Proceedings IEEE Vehicular Technology Conference, pp. 559-563, IEEE, April 1996. [3] S. Parkvall, Direct-Sequence Code-Division Multiple Access Systems: Near-far Resistant Parameter Estimation and Data Detection. PhD thesis, Kungliga Tekniska Hogskolan, Stockholm, Sweden, October 1996. [4] A. Graham, Kronecker Products and Matrix Calculus with Applications. Ellis Horwood Ltd, 198 1. [5] R. 0. Schmidt, A Signal Subspace Approach to Multiple Emitter Location and Spectral Estimation. PhD thesis, Stanford University, Stanford, CA, November 1981. [6] P. Stoica and T. Soderstrom, Statistical Analysis of MUSIC and Subspace Rotation Estimates of Sinusoidal Frequencies IEEE Transactions on Signal Processing, vol. 39, pp. 1836-1847, August 1991. [7] T. Ostman, S. Parkvall, and B. Ottersten, Analysis of an Improved MUSIC Algorithm for Estimation of Time Delays in Asynchronous DS-CDMA Systems Submitted to IEEE Transactions on Communications, 1997. [8] T. Ostman, On the Bounds of Performance in Communications Systems Internal report (IR-S3-SB- 9725), Signal Processing, Royal Institute of Technology, Sweden, Available by WWW, document URL: http://www2.e.kth.se/s3/signal /INDEX.html or by anonymous ftp to: ftp.e.kth.se directory /pub/signal/reports., September 1997. 842