Rodrigo de Lamare - Multi-Branch MMSE Decision Feedback Detection Algorithms with Error Propagation Mitigation for MIMO Systems

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April 01, 2010

Rodrigo de Lamare - Multi-Branch MMSE Decision Feedback Detection Algorithms with Error Propagation Mitigation for MIMO Systems

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SCEE Team

April 01, 2010
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  1. Multi-Branch MMSE Decision Feedback Detection Algorithms with Error Propagation Mitigation

    for MIMO Systems Rodrigo C. de Lamare † Communications Research Group, University of York, UK in collaboration with Didier Le Ruyet, Conservatoire National des Arts et M´ etiers (CNAM), Paris, France. Emails : rcdl500@ohm.york.ac.uk, leruyet@cnam.fr 1
  2. Outline – Introduction – MIMO System and Data Model –

    Proposed Multi-Branch MMSE Decision Feedback Detection – MMSE Design of Filters – Design of Cancellation Patterns – Ordering Algorithm – Multistage Detection – Complexity – Simulations – Conclusions 2
  3. Introduction – MIMO systems offer high performance and capacity but

    also present many design challenges [1]. – Interference in MIMO reduces capacity and performance. – Mitigation of interference and exploitation of diversity → MIMO detectors – MIMO detectors with different trade-offs between performance and complexity : ML, Sphere Decoder [3], lattice reduction, VBLAST with ordered successive interference cancellation (SIC) [2], Linear and Decision Feedback (DF) [7]. – Challenge → design of detectors with near ML performance and low complexity. – Contributions : MMSE DF detector with multiple cancellation branches, MMSE design of the filters and ordering strategies, and multistage scheme with the proposed detector . 3
  4. MIMO System Model – Consider a MIMO system with NT

    transmit antennas and NR receive antennas in a spatial multiplexing configuration. – The signals are transmitted over single-path channels. – We assume that the channel is constant during each packet transmission (block fading) and the receiver is perfectly synchronized. 4
  5. MIMO Data Model – The received signal is applied to

    a matched filter, sampled and collected into a NR × 1 vector r[i] given by r[i] = Hs[i] + n[i], – The NR × 1 vector n[i] is a zero mean complex circular symmetric Gaussian noise vector with E n[i]nH[i] = σ2 nI, where σ2 n is the noise variance. – The symbol vector s[i] has mean zero and a covariance matrix E s[i]sH[i] = σ2 s I, where σ2 s is the signal power. – The elements hnR,nT of the NR × NT channel matrix H correspond to the complex channel response from the nT th transmit antenna to the nRth receive antenna. 5
  6. Proposed Multi-Branch Decision Feedback Detection – The proposed multi-branch detector

    considers the following combination of weights : zj,l[i] = wH j,lr[i] − fH j,l [i]ˆ so[i], for l = 1, . . . , L, m = 1, . . . NT – wj,l is the NR × NT feedforward filter matrix, the vector of initial decisions ˆ so[i] is fed back through the NT × 1 feedback filter fj,l[i]. 6
  7. Proposed Multi-Branch Decision Feedback Detection (cont.) – The proposed MB-MMSE-DF

    detector selects the best branch according to lopt = arg min 1≤l≤L MMSE(sj[i], wj,l, fj,l), j = 1, . . . , NT where MMSE(sj[i], wj,l, fj,l) corresponds to the MMSE produced by the pair of filters wj,l and fj,l. – The final detected symbol of the MB-MMSE-DF detector is obtained by : ˆ sj[i] = Q zj,lopt [i] = Q wH j,lopt r[i] − fH j,lopt ˆ so j,lopt [i] , j = 1, . . . , NT where Q(·) is a slicing function that makes the decisions about the symbols, which is drawn from an M-PSK or a QAM constellation. 7
  8. MMSE Filter Design – The design of the MMSE filters

    of the proposed MB-MMSE-DF detector must solve the following optimization problem min MSE(sj[i], wj,l, fj,l) = E |sj[i] − wH j,l r[i] + fH j,l ˆ so j,l [i]|2 subject to Sj,lfj,l = vj,l and ||fj,l||2 = γj,l||fc j,l ||2 for j = 1, . . . , NT and l = 1, . . . , L, where -the NT × NT shape constraint matrix is Sj,l, -vj,l is the resulting NT × 1 constraint vector and -fc j,l is a feedback filter without constraints on the magnitude of its squared norm. 8
  9. MMSE Filter Design (cont.) – Expressions for wj,l and fj,l

    obtained after solving the optimization problem : wj,l = R−1(pj + Qfj,l), fj,l = βj,lΠj,l(QHwj,l − tj) + (I − Πj,l)vj,l, where Πj,l = I − SH j,l (SH j,l Sj,l)−1Sj,l is a projection matrix that ensures the shape constraint Sj,l and βj,l = (1 − αj,l)−1 is a factor that adjusts the magnitude of the feedback, 0 ≤ βj,l ≤ 1 and αj,l is the Lagrange multiplier. – The NR × NR covariance matrix of the input data vector is R = E[r[i]rH[i]], pj = E[r[i]s∗ j [i]], Q = E r[i]ˆ so, H j,l [i] , and tj = E[ˆ so j,l [i]s∗ j [i]] is the NT ×1 vector of correlations between ˆ so j,l [i] and s∗ j [i]. 9
  10. MMSE Filter Design (cont.) – Simplification of the filter expressions

    (using the fact that the quantity tj = 0 for interference cancellation, vj,l = 0, and assuming perfect feedback (s = ˆ s)) : wj,l = HHH + σ2 n/σ2 s I −1 H(δj + fj,l) fj,l = βj,lΠj,l σ2 s HHwj,l , where δj = [0 . . . 0 j−1 1 0 . . . 0 NT −j−2 ]T is a NT × 1 vector with a one in the jth element and zeros elsewhere. – The proposed MB-MMSE-DF detector expressions above require the channel matrix H (in practice an estimate of it) and the noise variance σ2 n at the receiver. 10
  11. MMSE Filter Design (cont.) – In terms of complexity, it

    requires for each branch l the inversion of an NR ×NR matrix and other operations with complexity O(N3 R ). However, the matrix inver- sion is identical for all branches and the MB-MMSE-DFE only requires further additions and multiplications of the matrices. – Moreover, we can verify that the filters wj,l and fj,l are dependent on one ano- ther, which means the designer has to iterate them before applying the detector. – The MMSE associated with the pair of filters wj,l and fj,l and the statistics of data symbols sj[i] is given by MMSE(sj[i], wj,l, fj,l) = σ2 s − wH j,l Rwj,l + fH j,l fj,l where σ2 s = E[|sj[i]|2] is the variance of the desired symbol. 11
  12. Design of Cancellation Patterns – Design of the shape constraint

    matrices Sj,l and vectors vj,l : pre-stored pat- terns at the receiver for the NT data streams and for the L branches. – Basic idea : to shape the filters fj,l for the NT data streams and the L branches with the matrices Sj,l such that resulting constraint vectors vj,l are null vectors. – For the first branch of detection (l = 1), we can use the SIC approach and Sj,lfj,l = 0, l = 1 Sj,l = 0j−1 0j−1,NT −j+1 0NT −j+1,j−1 INT −j+1 , j = 1, . . . , NT , where 0m,n denotes an m×n-dimensional matrix full of zeros, and Im denotes an m-dimensional identity matrix. 12
  13. Design of Cancellation Patterns (cont.) – For the remaining branches,

    we adopt an approach based on permutations of the structure of the matrices Sj,l, which is given by Sj,lfj,l = 0, l = 2, . . . , L Sj,l = φl 0j−1 0j−1,NT −j+1 0NT −j+1,j−1 INT −j+1 , j = 1, . . . , NT , where the operator φl[·] permutes the columns of the argument matrix such that one can exploit different orderings via SIC. – These permutations are straightforward to implement and allow the increase of the diversity order of the proposed MB-MMSE-DF detector. – An alternative approach for shaping Sj,l for one of the L branches is to use a PIC approach and design the matrices as follows Sj,lfj,l = 0, l Sj,l = diag (δj), j = 1, . . . , NT , 13
  14. Ordering Algorithm – The proposed ordering algorithm for l =

    1, . . . , L is given by {o1,l, . . . , oNT ,l} = arg min o1,l,...,oNT ,l L l=1 NT j=1 MMSE(sj[i], wj,l, fj,l) – The ordering for the proposed MB-MMSE-DF detector is based on determining the optimal ordering for the first branch, which employs a SIC-based DFE, and then uses phase shifts for increasing the diversity for the remaining branches. – The algorithm finds the optimal ordering for each branch. For a single branch detector this corresponds to the optimal ordering of the V-BLAST detector. – The idea with the multiple branches and their orderings is to attempt to benefit a given data stream or group for each decoding branch. 14
  15. Multistage Detection – Main ideas : to combat error propagation

    by refining the decision vectors with multiple stages, and to equalize the performance over the data streams. – The MB-MMSE-DF detector with M stages can be described by z(m+1) j,l (i) = ˜ wH j,l r[i] − ˜ fH j,lˆ so,(m) j,l [i], m = 0, 1, . . . , M, where ˆ so,(m) j,l [i] is the vector of tentative decisions from the preceding itera- tion that is described by ˆ so,(1) k,j,l [i] = Q wH j,l r[i] , k = 1, . . . , NT , and ˆ so,(m) k,j,l [i] = Q z(m) j,l [i] , m = 2, . . . , M,. – In order to equalize the performance over the data streams population, we consider an M-stage structure with output given by z(m+1) j,l [i] = [T wj,l]Hr[i] − [T fj,l]H]ˆ s0,(m) j,l [i] (1) where z(m+1) j,l [i] is the output of jth data stream and T is a square permutation matrix with ones along the reverse diagonal and zeros elsewhere. 15
  16. Computational Complexity 2 4 6 8 10 12 100 102

    104 106 108 1010 1012 Number of Antenna Elements (N R =N T ) Number of Multiplications QPSK, SNR=12 dB MMSE−Linear MMSE−SIC MB−MMSE−DF(L=8) LR−MMSE−SIC ML(SD) 16
  17. Simulation Results – We assess the bit error rate (BER)

    performance of the proposed and analyzed MIMO detection schemes. – Compared schemes : the sphere decoder (SD) [3], the linear [6], the VBLAST [2], the S-DF [7], the lattice-reduction versions of the linear and the VBLAST detectors [5], the P-DF [8] and the proposed MB-MMSE-DF detector. – The channels’s coefficients are taken from complex Gaussian random variables with zero mean and unit variance. – We employ QPSK modulation and use packets with Q = 200 symbols. – We average the experiments over 10000 runs . – The signal-to-noise ratio is defined as SNR = 10 log10 NT σ2 s σ2 , where σ2 s is the variance of the symbols and σ2 is the noise variance. 17
  18. – BER Performance of the detectors with perfect channel estimation

    for multiple branches L . 0 5 10 15 20 10−4 10−3 10−2 10−1 100 SNR BER N T = N R = 4 antennas MMSE−Linear MMSE−SIC MB−MMSE−DF(L=1,γ j,l =0) MB−MMSE−DF(L=2,γ j,l =0) MB−MMSE−DF(L=8,γ j,l =0) MB−MMSE−DF(L=N T !=24) ML 18
  19. – BER Performance with perfect channel estimates. 0 5 10

    15 20 10−4 10−3 10−2 10−1 100 SNR BER N T = N R = 8 antennas MMSE−Linear LR−MMSE−Linear MMSE−SIC LR−MMSE−SIC MB−MMSE−DF (L=4,γ j,l =0) MB−MMSE−DF (L=4,optimized γ j,l ) MB−MMSE−DF w/ M stages (M=2,L=4,optimized γ j,l ) ML 19
  20. Conclusions – A novel MMSE multi-branch decision feedback detector was

    proposed. – The key strategy lies in the use of multiple branches for interference cancella- tion which allows further exploitation of the diversity at the receiver. – We have derived MMSE expressions for the filters of the proposed multi-branch decision feedback detectors with shape and magnitude constraints on the feed- back filters. – The proposed multi-branch detector has a performance superior to the linear, VBLAST and existing DF detectors and very close to the ML detector, while it is simpler than the SD detector. – Future work will investigate iterative detection techniques with the aid of chan- nel codes. 20
  21. References 1. G. J. Foschini and M. J. Gans, “On

    limits of wireless communications in a fading environment when using multiple antennas”, Wireless Pers. Commun., vol. 6, pp. 311-335, Mar. 1998. 2. G. D. Golden, C. J. Foschini, R. A. Valenzuela and P. W. Wolniansky, “Detection algorithm and initial laboratory results using V-BLAST space-time communication architecture”, Electronics Letters, vol. 35, No.1, January 1999. 3. E. Viterbo and J. Boutros, “A universal lattice code decoder for fading channels”, IEEE Trans. on Inf. Theory, vol. 45, no. 5, pp.1639-1642, July 1999. 4. B. Hassibi and H. Vikalo, “On the sphere decoding algorithm : Part I, the expected com- plexity”, IEEE Trans. on Sig. Proc., vol 53, no. 8, pp. 2806-2818, Aug 2005. 5. D. Wuebben, R. Boehnke, V. Kuehn, and K. D. Kammeyer, “Near maximum- likelihood detec- tion of MIMO systems using MMSE-based lattice reduction,” in Proc. IEEE Int. Conf. Com- mun., Paris, France, June 2004, pp. 798-802. 6. A. Duel-Hallen, “Equalizers for Multiple Input Multiple Output Channels and PAM Systems with Cyclostationary Input Sequences,” IEEE J. Select. Areas Commun., vol. 10, pp. 630- 639, April, 1992. 7. N. Al-Dhahir and A. H. Sayed, ”The finite-length multi-input multi-output MMSE-DFE,” IEEE Trans. on Sig. Proc., vol. 48, no. 10, pp. 2921-2936, Oct., 2000. 8. G. Woodward, R. Ratasuk, M. L. Honig and P. Rapajic, “Minimum Mean-Squared Error Mul- tiuser Decision-Feedback Detectors for DS-CDMA,” IEEE Trans. on Commun., vol. 50, no. 12, December, 2002. 21