Efficient Markov Chain Monte Carlo Algorithms For MIMO and ISI channels

Transcription

1 Efficient Markov Chain Monte Carlo Algorithms For MIMO and ISI channels Rong-Hui Peng Department of Electrical and Computer Engineering University of Utah /7/6

2 Summary of PhD work Efficient MCMC algorithms for communication Application to MIMO channel ([Chen-Peng-Ashikhmin-Farhang], To appear IEEE Trans. Comm.) Application to noncoherent channel ([Chen-Peng], to appear IEEE Trans. Comm. 9) Application to ISI channel and extend to underwater channel ([Peng- Chen-Farhang], To appear IEEE Trans. Signal Process.) Achieve near optimal performance with reduced complexity Solve the slow convergence problem MIMO-HARQ schemes and combining algorithm ([Peng-Chen], to be submitted to IEEE Trans. Comm.) Propose new retransmission scheme Propose new combining algorithm Increase the throughput significantly Applicable to Wimax and LTE /7/6

3 Summary of PhD work Nonbinary LDPC codes ([Peng-Chen], IEEE Trans. Wireless Comm. 8) Nonbinary LDPC coded MIMO system, achieve good performance with reduced complexity Code design based on EXIT chart Hardware-friendly construction Low encoding complexity Parallel architecture Low error floor Intern at MERL Low complexity MIMO detection algorithm for MIMO-OFDMA Wimax link-level simulation paper, patent 3 /7/6

4 Outline Background and motivation Introduction to MCMC technology MCMC MIMO detection Review of MIMO detection QRD-M and MCMC detector Hybrid MIMO detector Complexity analysis MCMC ISI equalization Bit-wise MCMC equalizer Group-wise MCMC equalizer Conclusion 4 /7/6

5 Background and motivation uk h b d k y b k k ˆk k ENC MOD CHANNEL DEM/DET DEC MAP detection: y Hd n bˆ k max( P b k,) y H P( b k,)( y H,)() p Y H d P d b k MAP detector has optimal performance but the complexity is exponential with the dimension of b k IDD uˆk Our objective: reduce the complexity but still obtain near optimal performance 5 /7/6

6 Markov Chain Monte Carlo MCMC is a class of algorithms for sampling from probability distributions based on constructing a Markov chain that has the desired distribution as its stationary distribution. The state of the chain after a large number of steps is then used as a sample from the desired distribution. MCMC is suitable for addressing problems involving highdimensional summations or integrals f(x) X 6 /7/6

7 Gibbs sampling One kind of the MCMC methods. The point of Gibbs sampling is that given a multivariate distribution it is simpler to sample from a conditional distribution rather than integrating over a joint distribution. Generate an initial sample as start state Using conditional distribution as state transition probability Each state jumping allows only one variable change Return best state seen across all iterations (may not be the last one) Stop after a fixed number of iterations Solution is sensitive to the starting state, so we typically run the algorithm several times from different starting points 7 /7/6

8 State transition in Gibbs sampling d y Hd n d d d 3 Sampling d according to p( d using conditional PMF p( d i y, H) y, H, d i ) State d 3 d d S S ` S S S S S S /7/6

9 Gibbs sampler Generate initial d for n= to I () randomly generate d from distribution p( d b d, d, L, d,) ()( n )( )( ) n n n N generate d from distribution p( d b d, d, L, d,) M ()()( n )( ) n n n N ()()()() n n n n generate d from distribution ( p d b d, d, L, d,) end for N N N y y y 9 /7/6

10 Why Gibbs sampling works Retains elements of the greedy approach weighing by conditional PDF makes likely to move towards locally better solutions Allows for locally bad moves with a small probability, to escape local maxima (with limitations) /7/6

11 ISI channel Multipath fading channel L y()()()(), m h m d m l n m l l for m,, L, N L d (m ) D D h ( ) h () m h () m () h m m L + n (m ) y (m ) /7/6

12 ISI channel In matrix form y Hd n where d C y C N H C n C : transmitting vector N L N L N N L : received vector : channel matrix : dditive noise vector N : Equalizer block length L : Channel memory H h L L h h L M O O M hl L h L M O O M L hl hl L h L O M L hl hl h L L L /7/6

13 Detection for ISI channel MAP detection is optimal Efficient implementation of MAP detection is Forward backward algorithm (BCJR) Still exponential complexity with channel memory Low complexity algorithm MMSE Decision-feedback 3 /7/6

14 Bit-wise MCMC detector Transition probability a P( b a b,)( y )() p y b P b a k k k a p( y )() d P b a N L j p( y )() d P b a a j jl: j k i N L il a a a { ( j )( d jl: )( j )() j d jl: j } j d jl: j k j jil ji il Cg p( y )() d P b a b d ji a ()() n n k a a j jl: j k { b, L, b, a, b k p y p y p y P b a ( n)( ) n k M N denote the symbol vector corresponding to bk is mapped to d i, L, b }, b b a 4 /7/6

15 Bit-wise MCMC detector Computing the a posteriori LLR Accurate posteriori LLR involve computations over the whole block (may be very cumbersome since N can be very large) Truncated window to approximate I k, i : i I i : i i: il i : i l k, i : i l i e b I i : i e, k ln i, k i : i b p( y )() b p( y )() b i: il i : i l li the set which truncate sequences in I with bits { b, i k i } k i P b P b 5 /7/6

16 Severe ISI channel For channels with severe ISI, bit-wise MCMC suffers from slow convergence problem We found slow convergence problem for ISI channel is mainly caused by high posterior correlation PSD, db h [ n].7 [ n].46 [ n ].688 [ n ].46 [ n 3].7 [ n 4] h [ n]( 4) [ i ](5 n 8) [ i ] n [ ] n -5 C C Normalized Frequency ( 3) i[ n3](8 6) [ 4] n 6 /7/6

17 Severe ISI channel Our solution Grouping (blocking) the highly correlation variables in Gibbs sampler st iteration: nd iteration: 3 rd iteration: 4 th iteration: Shift grouping different adjacent symbols over iterations to speed up the mixing rate of Gibbs sampler 7 /7/6

18 Transition probability Grouping multiple variables allow a large sample space M containing b G r values y h L L d n h h M L M M y i M O O M d i ni yi hl L h L di ni M M O O M M M yip L hl hl L h dig ni P y ip L O M d ig ni P M L hl hl M M y N L h L d L L N nn L P G L 8 /7/6

19 Transition probability y% y H d H d il: i ig: ip i: ip i: ip i: ip il: i i: ip ig: ip i: ip H d n ig: ip i: ip i: ig i: ip n Can be thought as received signals of a MIMO channel with G transmit antenna and P+ receive antenna Can apply QRD-M to find r = r samples with large APPs Gibbs sampler randomly chooses one sample from samples r 9 /7/6

20 Simulation results Performance of the channel with strong ISI /7/6

21 Summary of results MCMC equalizer for ISI channel is studied Near optimal performance can be obtained Slow convergence is mainly caused by high posterior correlation QRD-M algorithm is applied to reduce the complexity of group-wise MCMC equalizer Parallel implementation is proposed /7/6

22 Conclusions Efficient MCMC algorithms for MIMO and ISI channel are studied For MIMO, MCMC works well at low SNR region For ISI, MCMC works well for moderate ISI Solutions for slow convergence are proposed For MIMO, use QRD-M as good start point For ISI, Future work Use group-wise Gibbs sampler to group high correlated variables Use QRD-M to reduce the complexity of group-wise Gibbs sampler Study the sensitivity of proposed MIMO detector with more practical channel model Study the time-varing ISI channel and adaptive grouping scheme for group-wise MCMC Hardware implementation of MCMC /7/6

23 Motivation Optimal binary code has been designed to approach channel capacity. Long codes Irregular Nonbinary LDPC code design has been studied for AWGN and shows better performance than binary codes. Shorter codes More regular 3 /7/6

24 Contribution Apply nonbinary LDPC codes to fading channels and MIMO channels and provide comparison with optimal binary LDPC coded systems Propose modified nonbinary LDPC decoding algorithm. Extend EXIT chart to nonbinary code design Propose parallel sparse encodable nonbinary code with low encoding and decoding complexity 4 /7/6

25 /7/6 5 Introduction of binary LDPC codes A subclass of linear block codes Specified by a parity check matrix (n-k) n n: code length k: length of information sequence Variable Nodes check Nodes x x x 3 x 4 x 5 x 6 x 7 c c c Hx H T c c c x x x : : : x x x x c x x x x c x x x x c

26 /7/6 6 Definition of nonbinary LDPC codes For nonbinary codes, the ones in parity check matrix are replaced by nonzero elements in GF(q) H

27 Application to fading channels Channel model X M ρ HS V Assume each entry of channel matrix is independent, follows Rayleigh fading, and is known by receiver 7 /7/6

28 System block diagram Separate detection and decoding: the detection is performed only once. 8 /7/6

29 Performance comparison Performance: GF56>GF6> binary>8.6e GF56 SDD Complexity: GF6<GF56 8.6e<binary GF6 JDD Binary JDD 8.6e Performance comparison of GF(56) SDD, GF(6) JDD and binary JDD system 9 /7/6

30 Construction of Parallel sparse encodable codes Motivation Low encoding complexity Allow parallel implementation Parallel sparse encodable (PSE) codes = Qausi cyclic (QC) cycle codes + tree codes QC codes: parallel implementation Cycle code: Low encoding complexity 3 /7/6

31 Quasi-cyclic construction Quasi-cyclic structure A, A, A, n A, A, A, n H A i,j is a circulant: each row is a right cycle-shift of the row above it and the first Am, Am, Am, n row is the right cycle-shift of the last row The advantage of QC structure Allow linear-time encoding using shift register Allow partially parallel decoding Save memory 3 /7/6

32 /7/6 3 A new QC structure for GF(q),,,,, ' q j i j i j i j i j i A A i,j is a nonbinary multiplied circulant permutation matrix ) GF( of primitive element is a },,, randomly choosen from { is ') GF( of primitive element is a ); GF( ') GF( ) ' /( (, q q q q q q j i

33 Cycle codes With degree variable nodes only Can be represented by normal graph, every vertex imposes one linear constraint Columns => edges; rows => vertices b c c b b b 5 H m c c 3 b 4 b 5 b 3 b Parity check matrix 33 c 3 b c /7/6 Normal graph representation

34 H based encoding of cycle codes b c c b 4 b 5 b 3 b c 3 b c Find the spanning tree: b, b 3, b 4 Information bits => Edges outside SP b =, b =, b 5 = 3 Compute coded bits b 3 =b +b 5 = b 4 =b +b = b =b +b 5 = For binary code: check c is always satisfied. b +b 3 +b 4 = Not work for nonbinary cycle codes! Why? 34 /7/6

35 PSE codes b 9 c 5 b 6 c 3 b 3 b c c b 4 b b c b 5 c 3 b 3 b c c b 7 b 4 b b c b 5 c 3 b 8 b 3 b c c b 4 b b c b 5 PSE codes = QC cycle codes + tree codes 35 /7/6

36 Encoding of PSE codes Parity check matrix based encoding Parallel encoding for QC cycle subcode Much lower encoding complexity than normal LDPC codes using generator matrix based encoding because the generator matrix is usually dense 36 /7/6

37 Simulation results Performance comparison of PSE codes over GF(56) 37 with QPP (QC) codes and PEG (non-qc) codes /7/6

38 Thank you! 38 /7/6