Group signature sequence optimisation for highly loaded MIMO MC-CDMA
mean and covariance matrix s2INMr , where INMr is the NMr NMr identity matrix.
H.-Y. Lu, W.-H. Fang and H.-P. Hsu
Proposed recursive approach: To construct an adequate set of signature sequences, fsðgÞ gg¼1 . . . G, in this Letter we develop a recursive scheme to form an orthonormal set of effective signature vectors fqðgÞ i gi¼1... Mt , g¼1...G . For brevity, assume the effective signature vectors fqðlÞ i gi¼1... Mt ;l¼1...g1 are mutually orthogonal. Further, to combat the MAI between the ith transmitted symbol of the gth group and the previous groups, we minimise the projection of the effective signature onto the subspace constructed by the effective signature vector q (g) i matrix fQðlÞ g; 1 l g 1 and then the constrained optimisation problem can be posed as
A simple, yet effective, method is proposed to form the group signature sequences for multiple input multiple output (MIMO) multicarrier code-division multiple access (MC-CDMA) systems. By minimising the projection onto a properly constructed subspace, the signature sequences are recursively determined group by group. The simulation results show the effectiveness of the proposed method.
Introduction: MIMO MC-CDMA, a promising candidate for the next generation wireless communication systems, possesses merits such as spectral efficiency and robustness against inter-symbol interference. Hence, a flurry of studies has been inspired to develop feasible systems and schemes [1]. In particular, to reduce the resource management overhead, a group-based MIMO MC-CDMA system has been proposed in [2, 3], where only a unique signature sequence is assigned for spreading the transmitted symbols of each group. However, the performance of CDMA-like systems is multiple access interference (MAI) limited [1, 4]. To facilitate MAI mitigation or avoidance, several approaches, including multiuser detection (MUD) [4] and signature sequence optimisation [5, 6], have received much attention. However, the performance of MUD significantly degrades in highly loaded scenarios. As for signature sequence optimisation, the previous developments, in general, are iterative schemes, which call for more computational complexity. Also, the previous work cannot be directly applied to group-based MIMO MC-CDMA systems owing to the transmitted symbols of a group sharing a unique signature sequence. In this Letter, to facilitate MAI avoidance in the aforementioned group-based MIMO MC-CDMA, a simple, yet effective, scheme of signature sequence optimisation is presented, which first recursively finds the effective signature vectors of a new group by minimising the projection onto the subspace constructed by the effective signature vectors of the previous groups. After that, matrix inversion of the frequency response of the channel matrix of the corresponding group is carried out to attain the signature sequence. Thanks to the orthogonality among the effective signature vectors of the groups, the disturbance of MAI, therefore, can be completely avoided to enhance the performance even in highly loaded scenarios. In addition, owing to the channels in MC-CDMA being commonly assumed to be static [1], the proposed scheme can be carried out in only one pass in a data frame duration. Furthermore, in contrast to previous work [5, 6], the proposed scheme is independent of the detector and, therefore, is naturally easy for implementation. Also, since the ordinary MIMO MC-CDMAs can be seen as a special case of [2, 3], the proposed scheme can be intuitively extended to the ordinary ones. Finally, conducted simulations show that the proposed scheme can indeed enhance the performance, especially in highly loaded scenarios.
ðgÞ ðgÞ min kP H g1 qi k subject to kqi k ¼ 1; 1 i Mt ; qðgÞ i
1 g G ð2Þ
where P g1 ¼ ½Qð1Þ Qð2Þ . . . Qðg1Þ is the NMr Mt (g 2 1) effective signature matrix of the previous groups. Thereafter, employing the Lagrange multipliers method and conducting some manipulations yields ðgÞ ðgÞ P g1 P H g1 qi ¼ li qi 1 i Mt ; 1 g G
ð3Þ
where li is the Lagrange multiplier. Obviously, the solution of q(g) i in (3) is equivalent to the (normalised) eigenvector corresponding to the minimum eigenvalue of Pg-1 PH g-1. Note that since all the effective signature vectors, fq(g) i gi¼1. . .Mt , have to meet (3), therefore, we simulthe Mt minimum taneously choose Mt eigenvectors corresponding to T ðgÞT ðgÞT ðgÞT T ðgÞ ¼ ½qðgÞ eigenvalues of Pg-1 PH g-1 , and form q 1 q2 . . . qMt qMt . Then, multiplying the inverse matrix of the frequency response of the Rayleigh fading channel of the gth group, the corresponding signature vector s(g) can be found. Furthermore, the new effective signature sequence fqi(g)gi¼1...Mt. can be used to construct the matrix Q(g) and Pg , which can be used for the (g þ 1)th group. To sum up, the proposed scheme consists of the following steps: Step 1: (Initialisation) Arbitrarily choose a set of normalised vectors, (1) , and Q (1), and set k ¼ 2. fqð1Þ i gi¼1Mt . After that, construct q ðgÞ Step 2: Recursively determine fqi gi¼1Mt , and form them as fq (g) and Q (g), where fqðgÞ i gi¼1Mt amount to the (normalised) eigenvectors corresponding to the Mt minimum eigenvalues of P g1 P H g1 ; g ¼ 2; 3; . . . ; K: Step 3: The signature sequence of the gth group, s(g) , 1 g G, can T T y ðgÞT T H ðgÞ then be determined by H ðgÞ , q (g) where H ðgÞ ¼ ½H ðgÞ Mt 1 H2 is the frequency response of the fading channel matrix of the gth group and (†) is the matrix pseudo-inverse operation. 100
10–1
Data model: Consider a synchronous uplink group-based MIMO MCCDMA system, where there are K transmit antennas at the transmitter side, which are divided into G groups so that each group has Mt transmit antennas, i.e. K ¼ G Mt. Also, assume that each group is assigned a ðgÞ ðgÞ T unique, length N signature sequence, sðgÞ ¼ ½sðgÞ 1 ; s2 ; . . . ; sN ; 1 g G; where (.)T denotes matrix transposition. For simplicity, assume that the number of subcarriers employed is also N. Then, after removing the cyclic prefix, performing N-point DFT, and staking the discrete signal of the receive antennas, the total received signal in a symbol duration can be described as [2, 3] r¼
G X
QðgÞ bðgÞ þ n
ð1Þ
g¼1
where r is the NMr 1 total received signal vector. QðgÞ ¼ ðgÞ ðgÞ ½qðgÞ 1 q2 . . . qMt ; 1 g G, is the NMr Mt effective signature (g) (g) ; 1 i Mt , is the matrix of the gth group, in which q (g) i ¼ Hi s NMr 1 effective signature vector for the ith, 1 i Mt , transmitted symbol of the gth group. H (g) i , 1 i Mt , is the NMr N frequency response of the Rayleigh fading channel for the gth group and s(g) is ðgÞ ðgÞ T the corresponding signature sequence. bðgÞ ¼ ½bðgÞ 1 ; b2 ; . . . ; bMt ; 1 g G; is the BPSK modulated transmission symbol vector of the gth group. n is the additive white Gaussian noise (AWGN) with zero
average BER
10–2
10–3
10–4
10–5 ZF LAST MF with proposed scheme
–6
10
2
3
4
5
6 SNR
7
8
9
10
Fig. 1 BER comparisons against SNR with G ¼ 8
Experimental results and discussion: Assume that the transmitted symbols of each user are BPSK modulated and the channels are quasistatic [2, 3], i.e. the channel coefficients remain constant during a data frame. Also, assume that the number of multipaths, L, is 3. Consider a full loaded scenario, i.e. N Mr ¼ G Mt , in which the number of subcarriers N, the number of the receive antennas Mr , and the number of the transmit antenna of each group Mt are 12, 2, and 3, respectively. Three schemes: the zero forcing (ZF) [4], the layered space-time (LAST) [2], and the matched filter (MF) with the proposed signature
ELECTRONICS LETTERS 1st January 2009 Vol. 45 No. 1
optimisation method, are carried out for comparison in terms of the average bit error rate (BER) performance. We consider the number of groups, G ¼ 8, and show the results in Fig. 1. We can observe from Fig. 1 that, since the proposed method makes the effective signature vectors mutually orthogonal, the MF with the proposed signature optimisation scheme indeed outperforms the other two. Further, as the detection of MIMO systems, the performance of the ZF and the LAST inevitably degrades significantly owing to Mr , Mt , whereas the MF with the proposed scheme still provides substantial performance gain even in the full loaded scenario. Conclusion: The proposed method can not only adapt the group signature sequences easily but also attain significant performance gain and thus provide an appealing scheme to overcome the MAI, especially in highly loaded scenarios.
References 1 Hanzo, L., Mu¨nster, M., Choi, B.J., and Keller, T.: ‘OFDM and MCCDMA’ (John Wiley and IEEE Press, 2003) 2 Sfar, S., Murch, R.D., and Letaief, K.B.: ‘Layered space-time multiuser detection over wireless uplink systems’, IEEE Trans. Wirel. Commun., 2003, 2, (4), pp. 653–668 3 Lu, H.-Y., and Fang, W.-H.: ‘Heterogeneous information aided semiblind group MUD for MIMO MC-CDMA systems’. Proc. IEEE Vehicular Technology Conf., October 2007, pp. 730– 734 4 Verdu´, S.: ‘Multiuser detection’ (Cambridge University Press, UK, 1998) 5 Rajappan, G.S., and Honig, M.L.: ‘Signature sequence adaptation for DS-CDMA with multipath’, IEEE J. Select. Areas Commun., 2002, 20, (2), pp. 384– 395 6 Ulukus, S., and Yates, R.: ‘Iterative construction of optimum signature sequence sets in synchronous CDMA systems’, IEEE Trans. Inf. Theory, 2001, 47, (5), pp. 1989– 1998
# The Institution of Engineering and Technology 2009 28 August 2008 Electronics Letters online no: 20092474 doi: 10.1049/el:20092474 H.-Y. Lu (Department of Electronic Engineering, Huafan University, No. 1 Huafan Road, Shihding Township, Taipei County, Taiwan, Republic of China) E-mail:
[email protected] W.-H. Fang (Department of Electronic Engineering, National Taiwan University of Science and Technology, No. 43, Sec. 4, Keeling Road, Taipei, Taiwan, Republic of China) H.-P. Hsu (Department of Information Communications, Kainan University, No. 1, Kainan Road, Luchu, Taoyuan County, Taiwan, Republic of China)
ELECTRONICS LETTERS 1st January 2009 Vol. 45 No. 1
