Parameter-Optimized Extended MMSE-DFE based ...

Parameter-Optimized Extended MMSE-DFE based Detection for V-BLAST Van-Su Pham, Minh-Tuan Le, Linh Mai and Giwan Yoon Communication Electronics Laboratory Information and Communications University (ICU) Yuseong-gu, Daejeon 305-732 Email: {vansu pham,letuan,mailinh,gwyoon}@icu.ac.kr Abstract— In this work, the optimal value of parameter for complexity reduction of an Extended MMSE-DFE based Detection for V-BLAST system is considered. The detection, which is based on the fast sphere decoder for Phase-Shift Keying (PSK) signals in uncoded V-BLAST systems, namely PSD, generalizes the parameter β in the extended channel matrix. It is shown that for M-ary PSK (M-PSK) signal, the PSD is always an optimal decoder irrespective of β. In addition, with appropriately set the value of β the PSD can operate at optimal low, i.e. minimum, computational load. In this work, from computer simulation, the trend of impact of β on the complexity of the detection is presented.

I. I NTRODUCTION Recently, multiple input multiple output (MIMO) systems, which use multiple transmit and/or receive antennas, have been an active research topic because MIMO systems are theoretically shown to significantly improve spectral efficiencies [1]. To achieve high spectral efficiencies, a MIMO technique, known as the Vertical Bell Laboratories Layered Space Time (V-BLAST) architecture [2], has been reported. In order to achieve optimal system performance, maximumlikelihood (ML) decoder is required for the V-BLAST. Unfortunately, brute-force maximum likelihood (BFML) detection is an impractical approach due to its complexity, which is exponential with the number of transmit antennas and level of modulation scheme. In order to achieve ML performance at reduced complexity, a class of detection algorithms, collectively referred to as sphere decoders (SDs), has been developed [3]- [10]. Both computer simulation and theoretical analysis showed that the average complexity of SDs was remarkably smaller than that of the BFML decoder in many practical scenarios [6]- [7]. Most of the sphere decoders [3]- [8] were originally designed based on integer lattice theory, hereafter referred to as the real SDs. In order for the real SDs to be employed, a complex MIMO system needs to be decoupled into its real and imaginary parts so as to form an equivalent real-valued system. Therefore, real SDs are most appropriate for lattice-based modulation schemes such as QAM or PAM. For other complex constellations, such as PSK, the real SDs are inefficient due to the existence of invalid candidates. The problem of invalid candidates facing the real SDs can be solved by using a complex SD introduced in [9] because this complex SD does not require the decoupling of the complex

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system. A reduced complexity version of the complex SD in [9], referred to as the Pham SD, was presented in [10]. The key idea behind the Pham SD is the utilization of the depth-first branch and bound algorithm [11]. In the Pham SD, potential candidates at each layer are determined based on the original idea in [9]. Then, an ordering procedure is applied to those candidates so as to maximize the probability that the first feasible solution is the optimal one. In addition, instead of restarting the search from the root node, the search resumes with the path, which is closest to completion. As a consequence, the Pham SD offers a noticeable reduction in the computational load as compared to the original SD in [9]. Nonetheless, one disadvantage of both the original complex SD in [9] and the Pham SD is that they have to deal with the computationally inefficient cos−1 operation, thereby slowing down the decoding process. In order to eliminate the computationally inefficient cos−1 operation in V-BLAST systems employing M-PSK signals, a fast sphere decoder detection , namely PSD, exploiting the inspiration of [15], has been proposed [16]. The PSD is composed of the preprocessing and search stages. The preprocessing is based on the enhanced MMSE-DFE (E-MMSEDFE) technique along with the sorted QR decomposition [14]. It can be seen that the MMSE-DFE filter can be established via the definition of the augmented channel matrix [12] or the extended channel matrix [13]. The E-MMSE-DFE technique herein means that the extended channel matrix is given in a general form with generalized parameter β. It is shown that there exists an optimal value of β that the PSD can operate at minimum computational load. Moreover, the PSD is always an optimal decoder for V-BLAST system employing M-PSK signal irrespective of β. In this work, from computer simulation, we explore the trend of impact of β on the complexity of the PSD. The rest of the paper is organized as follows. In Section II, the system model is introduced. Section III investigates the impact of the generalized parameter β on the complexity of the PSD. Simulation results and discussion are presented in Section IV. And the final section will be our conclusions. II. S YSTEM MODEL We consider an uncoded V-BLAST configuration with nT transmit and nR receive antennas, denoted as (nT , nR ) system.

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At the transmitter, the input data sequence is partitioned into nT sub-streams (layers), each of which is then modulated by an M -PSK modulation scheme, M = 2n for some positive integer n, and transmitted from a different transmit antenna. For the sake of simplicity, we investigate one-time-slot complex baseband signal model, where at each symbol period a nT × 1 transmit signal vector s consisting of nT symbols, si , i = 1, . . . , nT , is sent through nT transmit antennas. Under the assumptions that the signals are narrow-band and the channel is quasi-static, i.e., it remains constant during some block of arbitrary length and changes from one block to another, the relationship between transmitted and received signals can be expressed in the following form: r = Hs + w

(1)

where r = [r1 , . . . , rnR ]T is the nR × 1 received signal vector, T denotes the transpose of a matrix, w = [w1 , . . . , wnR ]T represents the noise samples at nR receive antennas, which are modelled as independent samples of a zero-mean complex Gaussian random variable with noise variance σ 2 , H is the nR × nT channel matrix, whose entries are the path gains between transmit and receive antennas modelled as the samples of a zero-mean complex Gaussian random variable with equal variance of 0.5 per real dimension. Besides, we assume that the signals transmitted from individual antenna have equal powers of P/nT , i.e., E{ssH } = nPT InT , where H denotes the Hermitian transpose of a matrix, InT indicates the nT × nT identity matrix, and E{.} denotes the expectation operator. Under the assumption that H is perfectly known at the receiver, the transmitted vector is recovered by using ML decoder according to: ˆ s = arg min kr − Hsk2 s∈Ω

where β is an arbitrary and real number. By applying the sorted QR decomposition [14] to H we obtain (4) H = QR where Q is a (nT +nR )×nT matrix with orthonormal columns and R is a nT × nT upper triangular matrix. Rather than detecting the transmitted vector s based on (2), the proposed PSD decoder will detect s using s∈Ω

= arg min s∈Ω

nT X

|ξk − Rk,k sk |2

(5) (6)

k=1

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III. I MPACT OF β ON C OMPLEXITY OF THE PSD In this subsection, we qualitatively investigate the role of β in the PSD’s complexity. For simplicity, let us consider a VBLAST system with 8-PSK modulation. For this system, the PSD tries to test only those s ∈ Ω that satisfy: 2

kv − Rsk =

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nT X

|ξk − Rk,k sk |2 ≤ C

(7)

k=1

Assuming at layer k = m + 1, the PSD has successfully selected the tentative decisions sˆnT , . . . , sˆm+1 respectively for symbols snT , . . . , sm+1 such that Tm+1 = PnTtransmitted ˆk − Rk,k sˆk |2 < C. Then, at layer k = m, a | ξ k=m+1 constellation symbol xi , i = 1, . . . , 8, is to be selected as the candidate for sk must satisfy:

(2)

where Ω is the transmission constellation, kAk denotes the 2 Euclidean norm of matrix A defined by kAk = tr(AH A), tr(A) denotes the trace of A. Based on the concept of extended channel matrix and extended received signal vector in [13], we define a (nT + nR ) × nT extended channel matrix H and a (nT + nR ) × 1 extended received signal vector r as follows: · ¸ · ¸ H r and r = (3) H= βInT 0nT ×1

ˆ s = arg min kv − Rsk2

where v = QH r is the nT × 1 received signal vector after being PnT filtered by the feed-forward matrix Q, ξk = vk − i=k+1 Rk,i si . From (3) one can see that if β = 0, Q and R are the Zero-forcing DFEpforward and backward filters. Whereas, if β = βMMSE = nT /γ, where γ = P/σ 2 is the SNR at each receive antenna (see the Appendix), Q and R become the MMSE-DFE forward and backward filters. β can also be set equal to any nonzero value. Therefore, the exploited preprocessing technique is referred to as the enhanced MMSEDFE preprocessing. It is worth noting that by selecting β 6= 0, the augmented channel matrices H in (3) is always of fullcolumn rank for any nR , thus resulting in the upper triangular matrix R with all positive diagonal entries. Therefore, the EMMSE-DFE preprocessing enables the PSD to work with any number of receive antennas.

Tm+1 + |ξˆk − Rk,k xi |2 < C

(8)

|ξˆk − Rk,k xi |2 < C − Tm+1

(9)

Or, equivalently,

This equation implies that, in a complex plane, if Rk,k xi lies inside a circle p whose center and radius are respectively given by ξˆk and C − Tm+1 , then pxi can be a candidate for sk . Therefore, for a fixed radius C − Tm+1 and a certain noise variance, Rk,k plays an important role in the detection complexity. As illustrated in Fig.1, a large value of Rk,k make the distances among the constellation points widen, thereby reducing the number of signal points to be examined. In contrast, a small value of Rk,k causes the distances among the signal points to reduce, leading to more signal points to be visited, and thus higher complexity. Because, for a given channel realization, Rk,k is proportional to β, a large value should be assigned to β in order to have low detection complexity. The relationship of β and the average of Rk,k is illustrated in Fig.2. On the other hand, if we partition the (nT +nR )×nT matrix Q into the nR × nT matrix Q1 and the nT × nT matrix Q2 , then the signal vector v in (5) can be rewritten as: v

=

H Rs + QH 1 w − βQ2 s

Feb. 17-20, 2008 ICACT 2008

(10)

IV. C OMPUTER SIMULATION RESULTS This section examines the optimal value of β for achieving minimal complexity of the PSD by applying them to different V-BLAST architectures with different PSK modulation schemes. In the simulations, signals are transmitted in a burst by burst basis with burst length of 100 symbol durations. In addition, the channel matrix H is assumed to remain fixed within one burst and changes randomly from this burst to the next. Thus, the preprocessing is carried out once per burst. The initial square sphere radius C is set equal to 100. If no signal point exists inside the sphere, the radius will be increased by a step of 0.2C until a point is found. The algorithms are all implemented in floating-point C, then converted into mex files and used in MATLAB 6. We use the number of floating point operations (flops), i.e., addition, subtraction, multiplication, division, and cos−1 operation, as a measure for complexity. We only count the flops of the searching stage without taking into account the complexity of the preprocessing stage. Fig. 1. The effect of Rk,k on the number of constellation points to be visited at layer k, and thus on the detection complexity; 8-PSK modulation.

From (10) one can see that the power of the data-dependent 2 noise term, βQH 2 s, is in proportion to β . Therefore, if β is large, the power of this noise term is large, thereby making the transmitted symbols become more uncertain, and thus leading to high detection complexity. 3.5

3

Average Rii

2.5

Uncoded V-BLAST System (4, 4) (8, 8) (6, 6) (6, 3)

Optimal value of β 0.501 0.621 0.552 0.474

The Table.1 presents the achieving optimal value of β for minimal complexity of PSD. The values here obtained by tuning β from 0.0 to 1.0 and the optimal value of β leading minimal computational load is recorded. As we can see, for the the optimal β for system employing equal number of transmit and receive antennas increases when the number of antennas increases. V. C ONCLUSION

2

1.5

4x4 6x6 8x8 6x3

1

0.5

TABLE I O PTIMAL VALUE OF β FOR ACHIEVING MINIMAL COMPUTATIONAL LOAD OF PSD IN (4, 4) AND (8, 8) EMPLOYING 8-PSK AND (6, 6) AND (6, 3) EMPLOYING QPSK AT Eb/N 0 = 15dB.

0

0.1

Fig. 2.

0.2

0.3

0.4

0.5 β

0.6

0.7

0.8

0.9

1

The average of Rk,k as a function of β.

The impact of generalized parameter β in PSD for uncoded V-BLAST systems using M-PSK signal is examined. By exploiting a selectable parameter β in the extension channel matrix, we has been shown that β plays a key role in determining the PSD’s complexity. Computer simulation verifies that there exist the optimal β which allows the PSD have the minimal computational load. In addition, for system using equal number of transmit and receive antennas, the optimal β increases as number of antennas increases. ACKNOWLEDGMENT

From the above analyses we come to a conclusion that there exists an optimal value of β, which results in the PSD with the lowest detection complexity. It can be seen that, in order for the PSD to achieve reasonably low complexity β should not be too large or too small. Motivated by that, we further examine the trend of impact of β on the complexity of the PSD by computer simulation.

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This work was supported by the ERC program of MOST/KOSEF (Intelligent Radio Engineering Center). R EFERENCES [1] G. J. Foschini and M. J. Gans, “On limits of wireless communications in a fading environment when using multiple antennas,” Wireless Personal Communications, vol. 6, pp. 311-335, 1998.

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[2] G. D. Golden, G. J. Foschini, R. A. Valenzuela, P. W. Wolniansky, “Detection algorithm and initial laboratory results using the V-BLAST space-time communication architecture,” Electronics Letters, vol. 35, no. 1, pp. 14-15, Jan. 1999. [3] E. Viterbo and J. Boutros, “A universal lattice code decoder for fading channels,” IEEE Trans. Inform. Theory, vol. 45, pp. 1639-1642, July 1999. [4] M.O. Damen, A. Chkeif, and J. Belfiore, ”Lattice code decoder for spacetime codes,” IEEE Commun. Lett., vol. 4, pp. 161-163, May 2000. [5] E. Agrell, T. Eriksson, A. Vardy, and K. Zeger, “Closest point search in lattices,” IEEE Trans. Inform. Theory, vol. 48, no. 8, pp. 2201-2214, Aug. 2002. [6] M.O. Damen, H.E. Gamal, and G. Caire, “On maximum-likelihood detection and the search for the closest lattice point,” IEEE Trans. Inform. Theory, vol. 49, no. 10, Oct. 2003. [7] B. Hassibi and H. Vikalo, “On the sphere-decoding algorithm I. Expected complexity,” IEEE Trans. Signal Processing, vol. 53, no. 8, pp. 28062818, Aug. 2005. [8] A. M. Chan and I. Lee, “A new reduced-complexity sphere decoder for multiple antenna systems,” in Proc. IEEE Int. Conf. Commun., vol. 1, pp. 460-464, May 2002. [9] B.M. Hochwald and S.T. Brink, “Achieving near-capacity on a multipleantenna channel,” IEEE Trans. Commun., vol. 51, no. 3, Mar. 2003. [10] D. Pham, K. R. Pattipati, P. K. Willett, and J. Luo, “An improved complex sphere decoder for V-BLAST systems,” IEEE Signal Process. Letter, vol. 11, no. 9, pp. 748-751, Sep. 2004. [11] J. Luo, K. Pattipati, P. Willett, and L. Brunel, “Branch-and-bound-based fast optimal algorithm for multiuser detection in synchronous CDMA,” in Proc. IEEE Int. Conf. Commun., vol. 5, pp. 3336-3340, May 2003. [12] A. D. Murugan, H. E. Gamal, M. O. Damen, and G. Caire, “A unified framework for tree search decoding: rediscovering the sequential decoder,” IEEE Trans. Inform. Theory, vol. 52, no. 3, Mar. 2006, [13] D. W¨ubben, R. B¨ohnke, V. K¨uhn, and K. D. Kammeyer, “MMSE Extension of V-BLAST based on sorted QR decomposition,” in IEEE Proc. VTC 2003-Fall, vol. 1, pp. 508 - 512, Oct. 2003. [14] D. W¨ubben, R. B¨ohnke, J. Rinas, V. K¨uhn, and K. D. Kammeyer, “Efficient algorithm for decoding layered space-time codes,” IEE Electronic Letters, vol. 37, no. 22, pp. 1348-1350, Oct. 2001. [15] Minh-Tuan Le, Van-Su Pham, Linh Mai, and Giwan Yoon, “Rate-One Full-Diversity Quasi-Orthogonal STBCs with Low Decoding Complexity,” IEICE Trans. Commun., vol. E89-B, no. 12, pp. 3376-3385, Dec 2006. [16] Minh-Tuan Le, Van-Su Pham, Linh Mai, and Giwan Yoon, “A Low Complexity Branch-and-Bound-Based Decoder for V-BLAST Systems with PSK Signals,” submitted to IEICE Trans. Commun. Dec 2007.

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Feb. 17-20, 2008 ICACT 2008