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 Métiers (CNAM), Paris, France. Emails : rcdl500@ohm.york.ac.uk, leruyet@cnam.fr 1
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
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
MIMO System Model Consider a MIMO system withn T transmit antennas andn R 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
MIMO Data Model The received signal is applied to a matched filter, sampled and collected into a N R 1 vector r[i] given by r[i] = Hs[i]+n[i], The N R 1 vector n[i] is a zero mean complex circular symmetric Gaussian noise vector with E [ n[i]n H [i] ] = σ 2 ni, where σ 2 n is the noise variance. The symbol vector s[i] has mean zero and a covariance matrix E [ s[i]s H [i] ] = σsi, 2 where σs 2 is the signal power. The elements h nr,n T of the N R N T channel matrix H correspond to the complex channel response from the n T th transmit antenna to the n R th receive antenna. 5
Proposed Multi-Branch Decision Feedback Detection The proposed multi-branch detector considers the following combination of weights : z j,l [i] = w H j,l r[i] fh j,l [i]ŝ o[i], for l = 1,...,L, m = 1,...N T w j,l is the N R N T feedforward filter matrix, the vector of initial decisionsŝ o [i] is fed back through the N T 1 feedback filter f j,l [i]. 6
Proposed Multi-Branch Decision Feedback Detection (cont.) The proposed MB-MMSE-DF detector selects the best branch according to l opt = arg min 1 l L MMSE(s j[i],w j,l,f j,l ), j = 1,..., N T wheremmse(s j [i],w j,l,f j,l ) corresponds to the MMSE produced by the pair of filters w j,l and f j,l. The final detected symbol of the MB-MMSE-DF detector is obtained by : ŝ j [i] = Q [ z j,lopt [i] ] = Q [ w H j,l opt r[i] f H j,l opt ŝ o j,l opt [i] ], j = 1,..., N T 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
MMSE Filter Design The design of the MMSE filters of the proposed MB-MMSE-DF detector must solve the following optimization problem min MSE(s j [i],w j,l,f j,l ) = E [ s j [i] w H j,l r[i]+fhj,lŝo j,l [i] 2] subject to S j,l f j,l = v j,l and f j,l 2 = γ j,l f c j,l 2 for j = 1,...,N T and l = 1,...,L, where -the N T N T shape constraint matrix is S j,l, -v j,l is the resulting N T 1 constraint vector and -f c j,l norm. is a feedback filter without constraints on the magnitude of its squared 8
MMSE Filter Design (cont.) Expressions for w j,l and f j,l obtained after solving the optimization problem : where w j,l = R 1 (p j +Qf j,l ), f j,l = β j,l Π j,l (Q H w j,l t j )+(I Π j,l )v j,l, Π j,l = I S H j,l (SH j,l S j,l) 1 S j,l is a projection matrix that ensures the shape constraint S j,l and β j,l = (1 α j,l ) 1 is a factor that adjusts the magnitude of the feedback, 0 β j,l 1 and α j,l is the Lagrange multiplier. The N R N R covariance matrix of the input data vector is R = E[r[i]r H [i]], p j = E[r[i]s j [i]],q = E[ r[i]ŝ o, H j,l [i] ], andt j = E[ŝ o j,l [i]s j [i]] is the N T 1 vector of correlations between ŝ o j,l [i] and s j [i]. 9
MMSE Filter Design (cont.) Simplification of the filter expressions (using the fact that the quantityt j = 0 for interference cancellation, v j,l = 0, and assuming perfect feedback (s = ŝ)) : where δ j = [0...0 }{{} w j,l = ( HH H +σ 2 n /σ2 si ) 1 H(δj +f j,l ) j 1 f j,l = β j,l Π j,l ( σ 2 s H H w j,l ), 1 0...0 }{{} N T j 2 element and zeros elsewhere. ] T is a N T 1 vector with a one in the jth The proposed MB-MMSE-DF detector expressions above require the channel matrixh (in practice an estimate of it) and the noise varianceσ 2 n at the receiver. 10
MMSE Filter Design (cont.) In terms of complexity, it requires for each branchlthe inversion of ann R N R matrix and other operations with complexityo(nr 3 ). However, the matrix inversion 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 filtersw j,l andf j,l are dependent on one another, which means the designer has to iterate them before applying the detector. The MMSE associated with the pair of filters w j,l and f j,l and the statistics of data symbols s j [i] is given by MMSE(s j [i],w j,l,f j,l ) = σ 2 s wh j,l Rw j,l +f H j,l f j,l where σ 2 s = E[ s j[i] 2 ] is the variance of the desired symbol. 11
Design of Cancellation Patterns Design of the shape constraint matrices S j,l and vectors v j,l : pre-stored patterns at the receiver for the N T data streams and for the L branches. Basic idea : to shape the filtersf j,l for then T data streams and thelbranches with the matricess j,l such that resulting constraint vectorsv j,l are null vectors. For the first branch of detection (l = 1), we can use the SIC approach and S j,l f j,l = 0, l = 1 S j,l = [ 0 j 1 0 j 1,NT j+1 0 NT j+1,j 1 I NT j+1 ], j = 1,...,N T, where0 m,n denotes anm n-dimensional matrix full of zeros, andi m denotes an m-dimensional identity matrix. 12
Design of Cancellation Patterns (cont.) For the remaining branches, we adopt an approach based on permutations of the structure of the matrices S j,l, which is given by S j,l f j,l = 0, l = 2,...,L S j,l = φ l [ 0 j 1 0 j 1,NT j+1 0 NT j+1,j 1 I NT j+1 ], j = 1,...,N T, where the operator φ l [ ] 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 S j,l for one of the L branches is to use a PIC approach and design the matrices as follows S j,l f j,l = 0, l S j,l = diag (δ j ), j = 1,...,N T, 13
Ordering Algorithm The proposed ordering algorithm for l = 1,..., L is given by {o 1,l,...,o NT,l} = arg min o 1,l,...,o NT,l L N T l=1j=1 MMSE(s j [i],w j,l,f j,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
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) = w H j,l r[i] f H j,lŝo,(m) [i], m = 0, 1,..., M, where ŝ o,(m) j,l [i] is the vector of tentative decisions from the preceding iteration that is described by ŝ o,(1) k,j,l [i] = Q( w H j,l r[i]), k = 1,..., N T, and ŝ o,(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] = [Tw j,l ] H r[i] [Tf j,l ] H ]ŝ 0,(m) j,l [i] (1) wherez (m+1) j,l [i] is the output ofjth data stream andt is a square permutation matrix with ones along the reverse diagonal and zeros elsewhere. 15
Computational Complexity Number of Multiplications 10 12 10 10 10 8 10 6 10 4 QPSK, SNR=12 db MMSE Linear MMSE SIC MB MMSE DF(L=8) LR MMSE SIC ML(SD) 10 2 10 0 2 4 6 8 10 12 Number of Antenna Elements (N =N ) R T 16
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. N The signal-to-noise ratio is defined as SNR = 10log T σs 2 10 σ 2, where σ2 s is the variance of the symbols and σ 2 is the noise variance. 17
BER Performance of the detectors with perfect channel estimation for multiple branches L. N T = N R = 4 antennas 10 0 10 1 10 2 BER 10 3 10 4 MMSE Linear MMSE SIC MB MMSE DF(L=1,γ =0) j,l MB MMSE DF(L=2,γ =0) j,l MB MMSE DF(L=8,γ =0) j,l MB MMSE DF(L=N!=24) T ML 0 5 10 15 20 SNR 18
BER Performance with perfect channel estimates. 10 0 10 1 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) BER 10 2 MB MMSE DF (L=4,optimized γ j,l ) MB MMSE DF w/ M stages (M=2,L=4,optimized γ j,l ) ML 10 3 10 4 0 5 10 15 20 SNR 19
Conclusions A novel MMSE multi-branch decision feedback detector was proposed. The key strategy lies in the use of multiple branches for interference cancellation 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 feedback 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 channel codes. 20
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