H.M. AI kussayer Y.J.Guo S.K.Barton Indexing terms: Multiuser detector,, Asynchronous CDMA, Detector configuration, Near-fur environment I I Abstract: A novel multiuser detector for asynchronous code division multiple access (CDMA) system is presented. Using a new type of spreading sequences and shortened correlators with linear combining, an unbiased one-shot suboptimum detection is achieved without increasing the receiver complexity. Two decorrelating algorithms, the direct and the recursive methods, are given for the proposed detector configuration. Theoretical and simulation results show that the receiver performs well in severe near-far environments. 1 Introduction To overcome the near-far problem in CDMA systems, intensive research on multiuser detection techniques has been carried out recently. Although the maximum likelihood detector proposed by Verdu offers optimum performance [l], its exponential complexity in the number of users inherently prohibits its application in practice, and attention has been focused on various suboptimum detectors, a promising one of which is the linear decorrelating detector (LDD) [2, 31. For asyn- chronous CDMA system, the LDD is a block detector in its original form, and detection cannot be started until the whole data sequence is received, thus leading to unacceptable detection delay. To realise real time bit-by-bit detection, a one-shot solution was proposed in [4, 51, in which the correlation window is chosen to coincide with one interested user and each of the other users is treated as two independent interferers. There- fore, to obtain an unbiased detection, a bank of linear decorrelators, each devoted to one user, must be used at the central station, which will result in a complicated receiver configuration. Another approach to simplify- ing the original LDD is to use the isolation bit inser- tion (IBI) technique [6]. The essence of the IBI technique is to periodically embed isolation bits into the information bearing sequences at the transmitters, so the correlating matrix can be broken into a series of by-block basis. The disadvantage of this approach is sinaller blocks and detection is performed on a block- 0 IEE, 1997 IEE Proceedings online no. 19971246 Paper first received 25th March and in revised form 2nd December 1996 H.M. Alkussayer and S.K. Barton are with the Department of Electronic and Electrical Engineering, University of Bradford, Bradford BD7 lDP, UK Y.J. Guo was with the University of Bradford and is now with Fujitsu Europe Telecom R&D Centre Ltd. (FTRC), 2 Longwalk Rd., Stockley Park, Uxbridge, Middlesex UBll lAB, UK that it may significantly reduce the bandwidth utilisa- tion efficiency if large numbers of isolation bits are used. In this paper, a novel one-shot multiuser detector for asynchronous CDMA is presented. At each transmit- ter, a special kind of spreading sequence whose second half is the replica of the first is employed. At the cen- tral station, correlation is performed in each half bit period and the two received signals in every bit period are combined to provide an estimate for the transmit- ted data bit. Two decorrelation algorithms, the direct and the recursive methods, are proposed for the new detector. For an asynchronous CDMA system with K active users, only one 2K by 2K decorrelation matrix for the direct method, or one K by K matrix for the recursive method, is required by the proposed detector, whereas Lupas' one-shot LDD involves K decorrela- tion matrices of 2K - 1 by 2K - 1 dimension. Com- pared with the IBI technique when isolation bits are inserted in every other information bearing data bit to realise bit-by-bit detection, the two schemes lead to matrices of similar dimension, but the data transmis- sion rate of the new scheme is twice as fast as that of the IBI. SI (t - h'T/2) bl (n) I bKM user K I T/3 decorrelator Fig. 1 Asynchronous BPSK CDMA system using proposed detector 2 New CDMA system model Consider an asynchronous K-user BPSK CDMA sys- tem in a AWGN channel as shown in Fig. 1. Assume the spread spectrum processing gain allocated to the IEE Proc.-Commun.. Vol. 144, No. 5, October 1997 336
I I Abstract: A novel multiuser detector for asynchronous code division multiple access (CDMA) system is presented. Using a new type of spreading sequences and shortened correlators with linear combining, an unbiased one-shot suboptimum detection is achieved without increasing the receiver complexity. Two decorrelating algorithms, the direct and the recursive methods, are given for the proposed detector configuration. Theoretical and simulation results show that the receiver performs well in severe near-far environments.
1 Introduction
To overcome the near-far problem in CDMA systems, intensive research on multiuser detection techniques has been carried out recently. Although the maximum likelihood detector proposed by Verdu offers optimum performance [l], its exponential complexity in the number of users inherently prohibits its application in practice, and attention has been focused on various suboptimum detectors, a promising one of which is the linear decorrelating detector (LDD) [2, 31. For asyn- chronous CDMA system, the LDD is a block detector in its original form, and detection cannot be started until the whole data sequence is received, thus leading to unacceptable detection delay. To realise real time bit-by-bit detection, a one-shot solution was proposed in [4, 51, in which the correlation window is chosen to coincide with one interested user and each of the other users is treated as two independent interferers. There- fore, to obtain an unbiased detection, a bank of linear decorrelators, each devoted to one user, must be used at the central station, which will result in a complicated receiver configuration. Another approach to simplify- ing the original LDD is to use the isolation bit inser- tion (IBI) technique [6]. The essence of the IBI technique is to periodically embed isolation bits into the information bearing sequences at the transmitters, so the correlating matrix can be broken into a series of
by-block basis. The disadvantage of this approach is sinaller blocks and detection is performed on a block-
0 IEE, 1997 IEE Proceedings online no. 19971246 Paper first received 25th March and in revised form 2nd December 1996 H.M. Alkussayer and S.K. Barton are with the Department of Electronic and Electrical Engineering, University of Bradford, Bradford BD7 lDP, UK Y.J. Guo was with the University of Bradford and is now with Fujitsu Europe Telecom R&D Centre Ltd. (FTRC), 2 Longwalk Rd., Stockley Park, Uxbridge, Middlesex UBll lAB, UK
that it may significantly reduce the bandwidth utilisa- tion efficiency if large numbers of isolation bits are used.
In this paper, a novel one-shot multiuser detector for asynchronous CDMA is presented. At each transmit- ter, a special kind of spreading sequence whose second half is the replica of the first is employed. At the cen- tral station, correlation is performed in each half bit period and the two received signals in every bit period are combined to provide an estimate for the transmit- ted data bit. Two decorrelation algorithms, the direct and the recursive methods, are proposed for the new detector. For an asynchronous CDMA system with K active users, only one 2K by 2K decorrelation matrix for the direct method, or one K by K matrix for the recursive method, is required by the proposed detector, whereas Lupas' one-shot LDD involves K decorrela- tion matrices of 2K - 1 by 2K - 1 dimension. Com- pared with the IBI technique when isolation bits are inserted in every other information bearing data bit to realise bit-by-bit detection, the two schemes lead to matrices of similar dimension, but the data transmis- sion rate of the new scheme is twice as fast as that of the IBI.
SI ( t - h'T/2)
b l (n)
I
b K M user K
I T/3 decorrelator
Fig. 1 Asynchronous BPSK CDMA system using proposed detector
2 New CDMA system model
Consider an asynchronous K-user BPSK CDMA sys- tem in a AWGN channel as shown in Fig. 1. Assume the spread spectrum processing gain allocated to the
IEE Proc.-Commun.. Vol. 144, No. 5, October 1997 336
system is Gp and the source data transmitted by the kth user are given by:
where N is the length of the data block and K is the number of active users. For each transmitter, a spread- ing sequence is so chosen that the second half is the replica of the first, which is equivalent to using spread spectrum sequences (signature waveforms) of TI2 period. At the receiver, a bank of correlators with TI2 correlation period is used. This generates a new stream of data dk(h) at the kth sampler:
{dk(h)ldk(h) = hi.; h = I , . . . ,2N; k = I,. . . , K }
As shown in Fig. 2, the duration of each new bit is Tl2, and its relation with bk (n) is given by:
(2) Although only half ol' the spread spectrum processing gain is used in the new data stream dk(h), it will be shown later that this can be compensated by using lin- ear combining in each bit period.
{bk(n)Jbk(n) = f l ; n = l , . . . , N ; k = 1 ) . " , K }
m l t K [ - . -. - . - . - . - . - . - . -. - - . - . - . - pT-1
b Fig.2 central station
Representation of (u,l source &tu of user I ; (b) received datu ut
R=
The received signal at the central station can be expressed as:
where v(t) is zero-mean additive white Gaussian noise with variance 02, and S(t, d) is given by:
r ( t ) := S( t , d) + w ( t ) ( 3 )
h=2N k=K
S ( t ,d ) = C d k ( h ) ~ ( h ) S k [ t - - ( h - l ) T / 2 - . r k l h = l k = l
(4) with sk(t) as the normalised signature waveform of user k, which is zero outside the interval [0, 2721, and rk as the time delay of the kth user. Without loss of general- ity, it is assumed that the users are numbered so that their delays satisfy 0 s *rl s z2 ... z, s Tl2. The received signal is passed through a bank of correlators, each matched to one user's known signature waveform. The sampled output of the correlator for the hth bit of the kth user is given by:
rhT/2+rh
-R(O) R(-1) 0 . . . 0 0
R(l) R(0) R(-l) R(1) R(O) R(-l)
0 0 0 . * . R(l) R(0)
Y k @ ) = J T ( t ) S k [ t - ( h - 1)T/2 - 7k]d t (h-I)T/2+7r,
(5) The output of the correlators can be written in the fol- lowing linear system equation:
IEE Proc.-Commun., Vol. 144, No. 5, October 1997
where
R is a block-tridiagonal, block-Toeplitz crosscorrela- tion matrix with block elements R(m) = [rij(m)lkxk rep- resenting the normalised correlation matrix of the delayed replicas of the signature waveforms [3-6, 81, and w(h) represents the power of the received signals and the entries of R(m) are given by:
00
rz3 (m) = s%(t - 7 % ) ~ ~ (t + mT/2 - rj )dt (8)
The vectors y, d, and v represent the correlator output, the data to be detected and the noise, respectively. Each of them is a 2NK elements vector having the fol- lowing structure:
L XT = [ X ( l ) , . . . , x ( h ) , . . . , x (2N)]
x ( h ) = [XI (h ) , . . . , Z k ( h ) , . . . , 5 K ( h ) l (9) The noise v is a vector with an auto-correlation matrix E[v v 7 = 0% which means that all the noise elements vk(h) have zero mean and equal variance. Based on the special relations between any two adjacent vectors given by eqn. 2, eqn. 6 can be simplified into a series of independent equations of much smaller size, so deci- sions based on linear combining can be made bit-by- bit.
3 Decorrelation algorithms
3. I Direct method Any two adjacent rows of eqn. 6 can be written as: y (h ) = R(l)d"(h - 1) + R(O)d"(h)
+R(-l)d"(h + 1) + ~ ( h ) (104
( l o b )
y ( h + 1) = R(l)d"(h) + R(O)d"(h + 1) +R(-l)dW(h + 2) + ~ ( h + 1)
When h is even, eqn. 2 gives: d"(2n - I) = d"(2n) d"(2n + 1) = d"(2n + 2)
so eqn. 10 can be written as the following equation, which is related with only the nth and the (n + 1)th transmitted bits and independent of the other bits:
where the correlation matrix of vD is given by:
E [ v ~ v ~ ] = a 2 R ~ (12) with E, as a 2K by 2K matrix given by:
331
Following [7], the best linear estimate of dw(h) is given by:
It should be noted that eqn. 14 provides an estimate of all the transmitted bits except the first and the last, which can be obtained by:
For every two received bit periods TI2 (or every single transmitted bit period T), eqn. 14 gives the estimates of both the present and the next transmitted symbols. To increase the detection reliability of the transmitted bits, various combining methods can be used before thresh- olding. Using equal-gain combining, the decision rule is given by:
i k ( n ) = sgn(dr(2n) + d ~ ( 2 n - 1))
= sgn{a&h(n) +VIZ> (16) The noise term v12 in eqn. 16 is a Gaussian random variable with zero mean, whose variance is given by d [ a f ' ] k , k + d [ R D - l ] K + k , K + k and the bit error ratio (BER) related with this decision can be written in the form:
(17) Pk(n) = P(b = i(n) = l l b k ( n ) = -1)
r 1
where e(.) is the Gaussian Q-function, w k is the power of user k arriving at the receiver, and [ R f ' ] k , k and [ R f l ] K + k , K+k denote, respectively, the (k, k ) and (K + k, K + k) diagonal elements of RD-l. It is seen from eqn. 18 that the BER is independent of the powers of the interfering users.
3.2 Recursive method The dimension of the matrix used in the direct method is 2K by 2K. Using a recursive method, it is shown in the following that the dimension of the matrix involved in decision making can be reduced to K by K.
Eqn. 19 gives two equations with three independent unknown vectors (since dw(2n - 1) = dw(2n)), and there- fore has no solution. However, it shows that the odd- numbered biis are dependent on the present bits and the previous even-numbered bits, and the even-num- bered bits are dependent on the present bits and the following odd-numbered bits, so either recursive or pre- dictive estimation method can be used. Since matrix R(1) is not well conditioned, however, a direct predic- tive estimate of the following bits based on the present bits should be avoided and a stable recursive algorithm for solving eqn. 19 is given as follows.
Replacing dw(2n - 2) with dw(2n - 3) and following [7], the best linear estimates of d"(2n - 1) and dw(2n)
(20b) First, since dW(- 1) = 0, an estimate of dW(l) can be obtained directly from eqn. 20a. Second, using dw(l), an estimate of dW(3) can also be obtained from eqn. 20a and substituting dW(3) in eqn. 20b gives the estimate of dW(2). Following this procedure, eqn. 20 can be solved recursively to obtain dw(2n + 1) and dw(2n), and deci- sions are made based on the combination of dw(2n - 1) and dw(2n).
The recursive method requires the inversion of a K by K matrix whereas the direct method requires the inversion of a 2K by 2K matrix. Since the computa- tional complexity for inverting an M by M matrix lies between O(&) and O(M3), the recursive method can be between four and eight times more efficient than the direct method. The disadvantage of the recursive method is that since estimates are made recursively, instead of independently as in the case of the direct method, an error in estimation for one bit might prop- agate to the estimation of the following bits.
4 Analytical and simulation results
Using the two decorrelation algorithms described in Section 3, simulations were carried out to study the performance of the proposed detector (PD). To illus- trate the effect of near-far resistance, the Kth user is allocated the lowest power in all the examples. For comparison, the signal of the Kth user is also detected using a conventional matched filter receiver (MF), and a single-user receiver (SU) which is a matched filter in a single link where there are no interfering users. Unless specified otherwise, all simulation results refer to those obtained using the direct method. Simulation results are shown with dashed lines, and analytical results are shown with solid lines.
l . O r
::I -0.8 -1.01 4
0 1 2 3 L 5 6 I Z,ChlP
Two 7-chip Gold codes used in example 1 and their cross-correla- Fig. 3 tion functions 0- - -0 r12(1) n--n r12(0)
Example 1 is a two-user case where the spreading sequences are chosen as 7-chip Gold codes replicated in each bit period. The two Gold codes and their cross- correlation function against the delay difference between the two users are shown in Fig. 3. Since the
IEE Proc-Commun., Vol. 144, No. 5, October 1997
cross-correlation between any two sequences is a func- tion of the delay difference between them, so is the BER performance of the detector. In the simulation, the delay difference was chosen to correspond to the highest cross correlation. Using the direct method, Fig. 4 shows the BER of the second user with its signal strength lOdB weakei than that of the first. It is seen that simulation result 3 agree with the theoretical ones in all cases, and the proposed detector significantly out- performs the conventional detector. For the same sce- nario, Fig. 5 shows the performance of the proposed detector using the recursive method. It is observed that the recursive method produces BERs similar to those obtained using the direct method.
BER ofsecond user in example I with w2/wI = - lOdB
10-61 I t , I I
0 1 2 3 4 5 6 7 8 9 1 0 SNR2,dB
Fig.5 x- - - -x MF 0---0 PD v--v sv
BER of second user in example I using recursive algorithm
Example 2 is a four-user case where the spreading sequences are chosen to be of 21-chip length generated by combining two nonniaximal length sequences in the same manner as Gold code, and the sequences are rep- licated in each bit-period to obtain a processing gain of 42. The signal strength of the fourth user is l l dB weaker than that of the other three at the receiver. Using the direct method, Fig. 6 shows the BER of the fourth user in this system. To illustrate how well the proposed detector performs, the delay differences were chosen so that the minimum cross-correlations were obtained, a scenario which favours the conventional
IEE Proc -Commun, Vol 144, No 5, October 1997
receiver. It is seen from Fig. 6 that the proposed detec- tor significantly outperforms the conventional receiver in the near-far environment and is almost as good as the single user receiver. Fig. 7 shows the average bit error ratio of the fourth user against the power of the interfering users for constant SNR values, to illustrate the near-far resistant capabilities of the proposed detector in comparison with that of the conventional receiver. It is shown that the performance of the pro- posed detector does not change with the power of the interfering users, whereas the performance of the con- ventional receiver deteriorates rapidly when the multi- ple access interference increases.
loo[
10-61 I
0 1 2 3 4 5 6 7 8 9 10 SNR4
BER of fourth usu in example 2 with wdw4 = l l d B Fig.6 k = 1, ..., 3 x- - - -x MF 0-0 PD v-v sv
_-,,---
0.0001 1 8 12 16 20
1010g,o(wK/w4)
Fig.7 BER of ourth user in example 2 against interfering power of three users for d d e n t SNR
MF 0 PD _ _ ~ _ SNR = 7 dB --_ ___--- SNR = 5 dB
Example 3 is a ten-user case, where the spreading sequences are 31-chip length Gold codes replicated in each bit period. The delay differences between users are arbitrarily chosen, and nine users are assumed to be of the same power level, each being 8dB stronger than the 10th user. The average bit error ratio of the 10th user is shown in Fig. 8. Again, it is seen that the proposed detector performs well in this near-far environment.
Two major obstacles to the application of multiuser detectors in practical wireless systems are processing
339
complexity and possible processing delay [8]. For linear decorrelating detection schemes, such as the one pro- posed in this paper, the main computational complex- ity is due to matrix inversion, but it only needs to be done when a user joins or leaves the network, or at the rate of timing drift. As inverting an M by A4 matrix is basically an 0(M3) process, or an 0(M2) process for some special matrices [6] the use of parallel algorithms seems to be an appropriate approach to handling large systems in real time. It should also be mentioned that the theoretical limit on the number of users which the proposed one-shot scheme can accommodate is the length of the half-bit spreading codes and the practical limit is set by the enhanced noise level in the system [8].
100,
10-61 I I I I
0 1 2 3 L 5 6 7 8 9 10 SNR 10
Fig.8 k = 1, ..., 9 x- - - -x MF 0-0 PD v--0 sv
BER of 10th user in example 3 with wdwl0 = 8dB
5 Conclusions
A novel simple near-far resistant multiuser detector for asynchronous CDMA in AWGN channels is proposed. Using replicated spreading sequences at the transmit-
ters and halved correlation period plus linear combin- ing at the central station, a one-shot decorrelating detection is achieved. Two decorrelating algorithms, the direct method and the recursive method, are pre- sented. The former is more reliable and involves the inversion of a 2K by 2K matrix, with K as the number of users in the system, whereas the performance of the latter is slightly worse than the former but involves only the inversion of a K by K matrix. Theoretical and simulation results showed that the new scheme signifi- cantly outperforms the conventional detector in the near far environment and its performance approaches that of the single user detector.
6 Acknowledgment
This work is partly supported by the EPSRC under grant GR K02961.
References
VERDU, S.: ‘Mimmum probability of error for asynchronous Gaussian multiple-access channels’, ZEEE Trans., 1986, IT-32,
LUPAS, R., and VERDU, S.: ‘Linear multi-user detectors for synchronous code-division multiple-access channel’, ZEEE Trans.,
LUPAS. R.. and VERDU, S.: ‘Near-far resistance of multi-user
(l), pp. 85-86
1989, IT-35, (I), pp. 123-136
detectors in’ asynchronous ’channels’, ZEEE Trans., 1990, COM- 38, (4), pp. 496-508 LUPAS. R.: ‘Near-far resistant linear multi-user detection’. PhD dissertation, Department of electrical engineering, Princeton Uni- versity, 1989 VERDU, S.: ‘Recent progress in multi-user detection’ in ‘Advances in communications and control systems, lecture notes in control and information science series’ (Springler-Verlag, NY, 1988) ZHENG, F.C., and BARTON, S.K.: ‘Near-Far resistant detec- tion of CDMA signals via isolation bit insertion’, ZEEE Trans., 1995, COM-43, (2), pp. 1313-1317 PROAKIS, J.: ‘Digital communications’ (McGraw Hill, New York, 1995, 3rd edn.) DUEL-HAALEN, A., HOLTZMAN, J., and ZVONAR, Z.: ‘Multiuser detection for CDMA system’. IEEE personal commu- nication, 1995, pp. 46-57
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