Non-Binary Belief Propagation in Large-Dimension Communication Receivers A. Chockalingam Department of Electrical Communication Engineering Indian Institute of Science Bangalore CEFIPRA Workshop 2014, IISc, Bangalore 14 January 2014
Outline Large MIMO systems Motivation and challenges Signal detection BP based near-optimal detection Binary BP Non-binary BP Concluding remarks
Large MIMO Exploitation of large spatial dimensions Potential to practically realize the theoretically predicted benefits of MIMO very high spectral efficiencies / sum rates tens to hundreds of bps/Hz increased reliability Transmit / receive diversity power efficiency Green communications Use of large number of antennas getting recognized as a good approach to fulfill high throughput requirements in future wireless systems
MIMO capacity n t : # of transmit antennas, n r : # receive antennas # Antennas Error Probability ( P e ) Capacity ( C ) , bps/Hz SISO P e ∝ SNR − 1 n t = n r = 1 C = log ( SNR ) SIMO P e ∝ SNR − n r n t = 1 , n r > 1 C = log ( SNR ) MIMO P e ∝ SNR − n t n r n t > 1 , n r > 1 C = min ( n t , n r ) log ( SNR ) Increased spectral efficiency (bps/Hz) with increased n t , n r
Large MIMO systems (a) Point-to-point MIMO (b) Multiuser MIMO Multiuser MIMO with hundreds of antennas at the BS: ‘Massive MIMO’ System loading factor α = K / N ≤ 1
Large MIMO systems Release by January/February 2014
Large MIMO systems
Challenges Placement of large # of antennas in communication terminals Feasible in moderately sized communication terminals use high carrier frequencies for small carrier wavelengths (e.g., 5 GHz, 60 GHz) RF technologies Multiple IF/RF transmit and receive chains Signal detection Need low-complexity detectors Channel estimation Estimation and feedback of large # of channel coefficients
Large multiuser MIMO systems Tens to hundreds of antennas at the base station Same or less number of users Users can have one or more antennas Uplink synchronization channel estimation, detection, decoding multi-cell operation Downlink precoding CSI for precoding pilot contamination in multi-cell operation
System model x c = [ x 1 , x 2 , · · · , x K ] T : transmitted vector x k ∈ B : transmitted symbol from user k B : modulation alphabet (e.g., M -QAM) H c ∈ C N × K : channel gain matrix h jk ∼ C N ( 0 , σ 2 k ) : gain from k th user to j th rx ant at BS n c : noise vector with i.i.d. C N ( 0 , σ 2 ) entries y c = H c x c + n c Complex system model: Work with real-valued system model: y = Hx + n For M -QAM alphabet B , elements of x come from √ underlying M -PAM alphabet A
Optimum detection Maximum likelihood (ML) decision rule: x ∈ A 2 K � y − Hx � 2 x ML = arg min Maximum a posteriori (MAP) decision rule: x MAP = arg max x ∈ A 2 K Pr ( x | y , H ) Complexity: exponential in K
Detection algorithms Low-complexity, near-optimal detection a challenge in large dimensions Traditional sub-optimum solutions Matched filter (MF) solution: x MF = H T y Zero-Forcing (ZF) solution: x ZF = ( H T H ) − 1 H T y MMSE solution: x MMSE = ( H T H + σ 2 I ) − 1 H T y Problem: Poor performance in large dimensions Sphere decoder and variants achieves ML / near-ML performance Problem: does’nt scale well beyond 32 dimensions Encouraging progress in large-MIMO detection using algorithms rooted in AI and machine learning
Detection algorithms Algorithms of demonstrated near-optimal performance and low complexity for large MIMO detection 1. Local neighborhood search based Likelihood Ascent Search (LAS) and variants Reactive Tabu Search (RTS) and variants 2. Belief propagation (BP) based Message passing on graphical models Scalar Gaussian approximation of interference 3. Probabilistic Data Association (PDA) based Vector Gaussian approximation of interference 4. Markov Chain Monte Carlo (MCMC) based Sampling from mixture distribution
BP Detection - Binary modulation Consider 4-QAM, x k ∈ {± 1 } Each entry of y is treated as a function (observation) node Each symbol, x k ∈ {± 1 } , is treated as a variable node Scalar Gaussian approximation of interference (GAI) interference � �� � 2 K � = h ik x k + h ij x j + n i , i = 1 , · · · , 2 N y i j = 1 , j � = k � �� � △ = z ik z ik modeled as C N ( µ z ik , σ 2 z ik ) with 2 K 2 K � � σ 2 h 2 ij Var ( x j ) + σ 2 µ z ik = h ij E ( x j ) , z ik = j = 1 , j � = k j = 1 , j � = k h ij is the ( i , j ) th element in H
BP Detection - Binary modulation LLR of x k at observation node i is = log p ( y i | H , x k = 1 ) 2 Λ k p ( y i | H , x k = − 1 ) = h ik ( y i − µ z ik ) i σ 2 z ik LLRs computed at observation nodes are passed to variable nodes Using these LLRs, variable nodes compute probabilities exp ( � 2 N l = 1 , l � = i Λ k l ) △ p k + p i ( x k = + 1 | y ) = = 1 + exp ( � 2 N i l = 1 , l � = i Λ k l ) and pass them back to observation nodes This message passing done for a certain no. of iterations At the end, x k is detected as � 2 N � � Λ k � x k = sgn i i = 1
BP Detection - Binary modulation y 1 x 1 x 1 p 1+ i = f ( { p j + Λ 1 i } , j � = 1) y 1 Λ k p k + = g ( { Λ k l } , l � = 1) i 1 1 p 2+ Λ 2 p k + i i Λ k y i x 2 x 2 2 x k 2 y 2 y 2 p n t + Λ n t p k + i i Λ k n r n r x n t x n t y n r y n r Figure : Message passing between variable and observation nodes Complexity: O ( KN ) MMSE Complexity: O ( KN 2 )
BP Detection - Binary modulation 0 10 4-QAM # BP iterations = 20 Message damping factor = 0.4 BER improves -1 10 with increasing N=K Bit Error Rate -2 10 N=K=8 N=K=16 -3 10 N=K=24 N=K=32 -4 10 N=K=64 SISO AWGN -5 10 0 2 4 6 8 10 12 Average received SNR (dB)
BP Detection - Higher-order modulation Consider square M -QAM √ Each element of x belongs to underlying M -PAM One approach √ Write each M -QAM symbol in the form of linear √ combination of q = log 2 M bits q − 1 � 2 j b ( j ) i = 0 , 1 , · · · , 2 K − 1 x i = i , j = 0 � T = [ 2 0 2 1 · · · 2 q − 1 ] , △ △ � b ( 0 ) · · · b ( q − 1 ) · · · b ( 0 ) 2 K − 1 · · · b ( q − 1 ) c b = 0 0 2 K − 1 x = ( I 2 K ⊗ c ) b △ = H ′ � �� � H ( I 2 K ⊗ c ) b + n System model in bit vector form: y = Run binary message passing on this system model
b b b b b b b BP Detection - Higher-order modulation Another approach Vector messages x 1 x 2 x K a ij v ji y 1 y 2 y N
BP Detection - Higher-order modulation y i = h ij x j + z ij , i = 1 , · · · , 2 N , j = 1 , · · · , 2 K � 2 K △ z ij = h il x l + w i l = 1 , l � = j approximate the scalar term z ij as Gaussian with mean and variance 2 K 2 K � � σ 2 h 2 il Var ( x l ) + σ 2 µ ij = h il E ( x l ) , ij = l = 1 , l � = j l = 1 , l � = j √ a ij : M -length vector message passed from i th observation node to j th variable node (likelihood) √ v ji : M -length vector message passed from j th variable node to i th observation node (posterior probabilities)
BP Detection - Higher-order modulation Likelihood and posterior probabilities are approximated as � − ( y i − µ ij − h ij s ) 2 � 1 Pr ( y i | H , x j = s ) ≈ √ exp s ∈ A , 2 σ 2 2 π σ ij ij � � − ( y i − µ ij − h ij s ) 2 exp 2 N 2 N � � 2 σ 2 ij Pr ( x j = s | y , H ) ∝ Pr ( y i | H , x j = s ) ≈ σ ij i = 1 i = 1 Messages � − ( y i − µ ij − h ij s ) 2 � 1 a ij ( s ) = √ exp 2 σ 2 2 π σ ij ij � 2 N v ji ( s ) = a lj ( s ) l = 1 , l � = i
BP Detection - Higher-order modulation Mean & variance at i th observation node are computed as 2 K � h il s T v li = µ ij l = 1 , l � = j � ( s ⊙ s ) T v li − ( s T v li ) 2 � 2 K � σ 2 h 2 + σ 2 = ij il l = 1 , l � = j s : vector of all elements in A ( for M = 16, s = [ − 3 − 1 + 1 + 3 ] T ) Symbol probabilities at the end (after iterations) 2 N � △ P x j ( s ) = Pr ( x j = s ) ∝ a lj ( s ) l = 1 Bit decisions are made on probability values computed as � Pr ( b p j = 1 ) = P x j ( s ) ∀ s ∈ A : p th bit in s is 1 b p j : p th bit in the j th user’s symbol
BP Detection - Higher-order modulation 16-QAM 0 10 MF -1 10 MMSE NB-BP Uncoded BER (Proposed) -2 10 MF, N=32, 64, 128, 256 B-BP (in [11]) MMSE, N=32 MMSE, N=64 MMSE, N=128 -3 MMSE, N=256 10 B-BP in [11], N=32 B-BP in [11], N=64 B-BP in [11], N=128 B-BP in [11], N=256 -4 Prop. NB-BP, N=32 10 Prop. NB-BP, N=64 Prop. NB-BP, N=128 Prop. NB-BP, N=256 SISO AWGN -5 10 8 10 12 14 16 18 20 22 24 26 28 30 Average SNR in dB
BP Detection - Higher-order modulation √ MMSE complexity: O ( KN 2 ) Complexity: O ( KN M ) , 8 0 10 10 MF ZF Number of computations required MMSE -1 10 MF 7 10 NB-BP ZF MMSE N=128, 16-QAM Uncoded BER NB-BP -2 10 Average SNR = 17dB 6 10 N=128, 16-QAM -3 10 5 10 -4 10 -5 4 10 10 0.2 0.4 0.6 0.8 1 0.2 0.4 0.6 0.8 1 Loading factor, K/N Loading factor, K/N (a) Performance (b) Complexity
Channel estimation Transmission frame format Frame 1 Frame 2 1 Pilot L Data Blocks Block DB- i PB DB-1 DB-2 DB- L K pilot symbols K information symbols User 1 User 1 User K K pilot symbols K information symbols User K 1 Data Block Obtain MMSE channel estimate in the pilot phase
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