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2 where b [n] contains the previous, current and next transmitted symbols for all M users, H [n] contains the associated time-varying composite channel vectors, and n [n] is complex AWGN of power σ 2. For this case, the standard MMSE receiver is ( 1 W [n] = H [n] H [n] H H [n] + σ I) 2. (2) Note that the weight matrix is time-varying because of the rapidly fluctuating composite channel vectors. To avoid evaluating Equation 2 every symbol period, W [n] can be considered to be constant for the channel coherence time. However, when the radio channels are fast fading the standard MMSE receiver becomes expensive to implement. In [3] it was shown that the composite channel vector for the i th user can be decomposed as h i [n] = C i s i [n], where C i is the code matrix for the i th user and s i [n] contains the channel coefficients at different delays. Considering there to be D temporally resolvable paths for each user then a related decomposition of the channel vector is h i [n] = V i u i [n]. Here u i [n] consists of the stacked non-zero elements of s i [n] and V i contains the corresponding columns of C i at the D time delays. Hence, x [n] can be written as x [n] = Vu [n] + n [n], (3) where the substitution u [n] = U [n] b [n] has been applied and the following definitions are used [( V = I ( J T ) ) L V, V, ( I J L) ] V, (4) V = [V 1, V 2,..., V M ], U [n] = diag {[ U [n 1], U [n], U [n + 1] ]}, U [n] = diag {[u 1 [n], u 2 [n],..., u M [n]]}. The diag{ } notation means form a block diagonal matrix and is the Kronecker product. Furthermore, J is a 2L 2L shift matrix, where L is the number of temporal samples per symbol. J is defined to be a 2L 1 column identity matrix with an additional top row and right hand column containing zero elements. A modified MMSE receiver can now be formulated as W = V ( V H V + σ 2 R 1 uu ) 1. (5) This weight matrix is invariant during fading and is valid whilst the time of arrivals (TOAs) of the temporally resolvable paths remain approximately constant relative to the sampling period. If the output of the modified MMSE receiver is z [n] = W H x [n], then decision variables for all symbols are generated by diversity combining the signals from different antennas, paths and subcarriers so that d [n] = U [n] H z [n]. 2

3 Blind implementation Initially an estimate of each users composite channel vector is found by recursive application of the following equation for the i th user (based on the theorem of alternating projection) ĥ i [n] = P i P s [n] ĥi [n 1], (6) where the initial value of ĥi [n] is any non-zero vector. The projector P i = ( ) C i C H i C i C H i, where ( ) denotes the pseudoinverse, restricts the solution to the subspace spanned by the code matrix, and the time-varying projector P s [n] = Ês [n] Ês [n] H restricts the solution to the signal subspace. Ê s [n] is an orthonormal basis of the signal subspace provided by a subspace tracking algorithm such as RO-FST [5]. To calculate matrix V, the TOAs of the D temporally resolvable paths for each user must be estimated. Correlating ĥi [n] with the temporal vector for the k th subcarrier at all possible delays will produce an interference free spectrum where peaks correspond to estimated TOAs. We define the temporal vector for paths arriving with an integer delay of l sample periods as a ik [l] = [ a ik [0, l], a ik [1, l],..., a ik [L 1, l], 0 T L] T. (7) The elements of a ik [l] are given by [ ] m a ik [m, l] = α i exp (j2πf k (m l) T s ), (8) qn sc where α i is the PN-code, q is the oversampling factor, N sc is the number of subcarriers, F k is the k th subcarrier frequency, T s is the sampling period and means round down to integer. To provide diversity, the TOA correlation spectrum is averaged over all subcarriers and antennas. Even greater accuracy is achieved by building up a histogram of the peak positions over a block of B symbols during which TOAs are assumed constant. In practise the block period can be significantly greater than the channel coherence time. Knowledge of the TOAs and C i i means that V can be generated. The i th component vector of matrix U [n] is estimated from û i [n] = V i ĥi [n]. (9) All M component vectors of U [n] are estimated and then matrix U [n] is generated by applying Equation 4. With estimates of V and U [n] available, Equation 5 can be formulated and the decision variables calculated. A block diagram of the proposed MMSE receiver is shown in Fig. 1. Simulation results In the simulations a carrier frequency of 2 GHz was used and the data rate was set to 1 Mbit/s. Differential QPSK modulation and Gold codes of length seven 3

5 [2] J. Namgoong and J. S. Lehnert, Performance of multicarrier DS/SSMA systems in frequency selective fading channels, IEEE Trans. Wireless Commun., vol. 1, no. 2, pp , Apr [3] D. J. Sadler and A. Manikas, Blind reception of multicarrier DS-CDMA using antenna arrays, accepted for publication in IEEE Trans. Wireless Commun. [4] M. Latva-aho and M. J. Juntti, LMMSE detection for DS-CDMA systems in fading channels, IEEE Trans. Commun., vol. 48, no. 2, pp , Feb [5] D. J. Rabideau, Fast, rank adaptive subspace tracking and applications, IEEE Trans. Signal Processing, vol. 44, no. 9, pp , Sept [6] R. H. Clarke, A statistical theory of mobile radio reception, Bell Systems Tech. J., vol. 47, no. 6, pp , July Aug

6 A m m BÒ 8 Ó DÒ 8 Ó.Ò 8 Ó S u b s p a c e tr a c k in g s Ò 8 Ó T O es tim a tio n Weight a tr ix c o n s tr u c tio n s C a l c u l a te c o v a r ia n c e C ha n n el es tim a tio n s Ò 8 Ó D iv er s ity a tr ix c o n s tr u c tio n Ò s 8 Ó Figure 1: Proposed blind MMSE receiver for fading channels. BER 3 D R A K E B l o c k M M S E P r o p o s e d M M S E E / N [ d B ] Figure 2: Performance when there are five 120 km/h users. 6

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