Non-Binary Belief Propagation in Large-Dimension Communication - - PowerPoint PPT Presentation

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

nt: # of transmit antennas, nr: # receive antennas

# Antennas Error Probability (Pe) Capacity (C), bps/Hz SISO nt = nr = 1 Pe ∝ SNR−1 C = log(SNR) SIMO nt = 1, nr > 1 Pe ∝ SNR−nr C = log(SNR) MIMO nt > 1, nr > 1 Pe ∝ SNR−ntnr C = min(nt, nr) log(SNR) Increased spectral efficiency (bps/Hz) with increased nt, nr

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 xc = [x1, x2, · · · , xK]T: transmitted vector xk ∈ B: transmitted symbol from user k B: modulation alphabet (e.g., M-QAM) Hc ∈ CN×K: channel gain matrix hjk ∼ CN(0, σ2

k): gain from kth user to jth rx ant at BS

nc: noise vector with i.i.d. CN(0, σ2) entries Complex system model: yc = Hcxc + nc 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: xML = arg min

x∈A2Ky − Hx2

Maximum a posteriori (MAP) decision rule: xMAP = arg max

x∈A2K 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: xMF = HTy Zero-Forcing (ZF) solution: xZF = (HTH)−1HTy MMSE solution: xMMSE = (HTH + σ2I)−1HTy

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, xk ∈ {±1} Each entry of y is treated as a function (observation) node Each symbol, xk ∈ {±1}, is treated as a variable node Scalar Gaussian approximation of interference (GAI)

yi = hikxk +

interference

• 2K
• j=1,j=k

hijxj + ni,

• △

= zik

i = 1, · · · , 2N

zik modeled as CN(µzik , σ2

zik ) with

µzik =

2K

• j=1,j=k

hijE(xj), σ2

zik = 2K

• j=1,j=k

h2

ij Var(xj) + σ2

hij is the (i, j)th element in H

BP Detection - Binary modulation LLR of xk at observation node i is Λk

i

= log p(yi|H, xk = 1) p(yi|H, xk = −1) = 2 σ2

zik

hik(yi − µzik) LLRs computed at observation nodes are passed to variable nodes Using these LLRs, variable nodes compute probabilities pk+

i △

= pi(xk = +1|y) = exp(2N

l=1,l=i Λk l )

1 + exp(2N

l=1,l=i Λk l )

and pass them back to observation nodes This message passing done for a certain no. of iterations At the end, xk is detected as

• xk

= sgn 2N

• i=1

Λk

i

BP Detection - Binary modulation

x1 x1 x2 x2 p2+

i

p1+

i

Λ2

i

Λ1

i = f({pj+ i }, j = 1)

yi pnt+

i

xnt xnt Λnt

i

y2 y1 y1 ynr Λk

nr

Λk

2

Λk

1

pk+

2

ynr y2 pk+

nr

pk+

1

= g({Λk

l }, l = 1)

xk

Figure : Message passing between variable and observation nodes

Complexity: O(KN) MMSE Complexity: O(KN2)

BP Detection - Binary modulation

2 4 6 8 10 12 10

• 5

10

• 4

10

• 3

10

• 2

10

• 1

10

Average received SNR (dB) Bit Error Rate

N=K=8 N=K=16 N=K=24 N=K=32 N=K=64 SISO AWGN BER improves with increasing N=K 4-QAM # BP iterations = 20 Message damping factor = 0.4

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 = log2 √ M bits xi =

q−1

• j=0

2j b(j)

i ,

i = 0, 1, · · · , 2K − 1

c

△

= [20 21 · · · 2q−1], b

△

=

• b(0)

· · · b(q−1) · · · b(0)

2K−1 · · · b(q−1) 2K−1

T

x = (I2K ⊗ c)b System model in bit vector form: y =

△

= H′

• H(I2K ⊗ c) b + n

Run binary message passing on this system model

BP Detection - Higher-order modulation Another approach

Vector messages

b b b b b b b

y1 yN y2 x1 x2 xK aij vji

BP Detection - Higher-order modulation yi = hijxj + zij, i = 1, · · · , 2N, j = 1, · · · , 2K zij

△

=

2K

• l=1,l=j

hilxl + wi approximate the scalar term zij as Gaussian with mean and variance µij =

2K

• l=1,l=j

hilE(xl), σ2

ij = 2K

• l=1,l=j

h2

il Var(xl) + σ2

aij: √ M-length vector message passed from ith

• bservation node to jth variable node (likelihood)

vji: √ M-length vector message passed from jth variable node to ith observation node (posterior probabilities)

BP Detection - Higher-order modulation Likelihood and posterior probabilities are approximated as

Pr(yi|H, xj = s) ≈ 1 σij √ 2π exp −(yi − µij − hijs)2 2σ2

ij

• ,

s ∈ A Pr(xj = s|y, H) ∝

2N

• i=1

Pr(yi|H, xj = s) ≈

2N

• i=1

exp

• −(yi−µij−hijs)2

2σ2

ij

• σij

Messages

aij(s) = 1 σij √ 2π exp −(yi − µij − hijs)2 2σ2

ij

• vji(s)

=

2N

• l=1,l=i

alj(s)

BP Detection - Higher-order modulation Mean & variance at ith observation node are computed as

µij =

2K

• l=1,l=j

hilsTvli σ2

ij

=

2K

• l=1,l=j

h2

il

• (s ⊙ s)Tvli − (sTvli)2

+ σ2

s: vector of all elements in A (for M = 16, s = [−3 − 1 + 1 + 3]T) Symbol probabilities at the end (after iterations)

Pxj(s)

△

= Pr(xj = s) ∝

2N

• l=1

alj(s)

Bit decisions are made on probability values computed as

Pr(bp

j = 1) =

• ∀s∈A: pth bit in s is 1

Pxj(s)

bp

j : pth bit in the jth user’s symbol

BP Detection - Higher-order modulation 16-QAM

8 10 12 14 16 18 20 22 24 26 28 30 10

• 5

10

• 4

10

• 3

10

• 2

10

• 1

10 Average SNR in dB Uncoded BER

MF, N=32, 64, 128, 256 MMSE, N=32 MMSE, N=64 MMSE, N=128 MMSE, N=256 B-BP in [11], N=32 B-BP in [11], N=64 B-BP in [11], N=128 B-BP in [11], N=256

• Prop. NB-BP, N=32
• Prop. NB-BP, N=64
• Prop. NB-BP, N=128
• Prop. NB-BP, N=256

SISO AWGN

NB-BP (Proposed) B-BP (in [11]) MMSE MF

BP Detection - Higher-order modulation Complexity: O(KN √ M), MMSE complexity: O(KN2)

0.2 0.4 0.6 0.8 1 10

• 5

10

• 4

10

• 3

10

• 2

10

• 1

10

Loading factor, K/N Uncoded BER

Average SNR = 17dB N=128, 16-QAM MF ZF MMSE NB-BP

(a) Performance

0.2 0.4 0.6 0.8 1 10

4

10

5

10

6

10

7

10

8

Loading factor, K/N Number of computations required

N=128, 16-QAM

MF ZF MMSE NB-BP

(b) Complexity

Channel estimation Transmission frame format

1 Pilot Block

User 1

Frame 1 Frame 2 L Data Blocks DB-i DB-L

User 1

DB-2 PB DB-1

K pilot symbols User K K pilot symbols User K

K information symbols K information symbols

1 Data Block

Obtain MMSE channel estimate in the pilot phase

Channel estimation 16-QAM, MMSE channel estimate

10 12 14 16 18 20 22 24 26 28 30 10

• 5

10

• 4

10

• 3

10

• 2

10

• 1

10 Average SNR in dB Uncoded BER MMSE with perfect CSI B-BP with perfect CSI

• Prop. NB-BP with perfect CSI

MMSE with estimated CSI B-BP with estimated CSI

• Prop. NB-BP estimated CSI

SISO AWGN N=K=128

Channel estimation 16-QAM, MMSE channel estimate

10 12 14 16 18 20 22 24 10

• 5

10

• 4

10

• 3

10

• 2

10

• 1

10 Average SNR in dB Uncoded BER MMSE with perfect CSI

• Prop. NB-BP with perfect CSI

MMSE with estimated CSI

• Prop. NB-BP with estimated CSI

N=128, K=64

Optimized LDPC code design Design optimized q-ary LDPC codes

by matching EXIT charts of the proposed detector and the LDPC decoder

6 7 8 9 10 11 12 10

• 5

10

• 4

10

• 3

10

• 2

10

• 1

10 Average SNR in dB Coded BER

• Min. SNR at capacity (6.8 dB)
• Reg. 16-ary LDPC
• Irreg. 16-ary LDPC in [25]
• Opt. binary LDPC in [23]
• Prop. opt. 16-ary LDPC

(c)

0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

• 10
• 5

5 10 15 Loading factor, K/N Average SNR required to achieve a coded BER of 10

• 5 , in dB
• Prop. 16-ary LDPC with NB-BP
• Irreg. LDPC in [25] with NB-BP
• Irreg. LDPC in [25] with MMSE det.

(d)

Concluding remarks Large MIMO systems with tens to hundreds of antennas

fast becoming a reality enable multi-Gigabit wireless transmissions potential candidate technology for 5G

Tools/algorithms from machine learning

key enablers for large-dimension communication/signal processing much more needs to be explored and remains to be exploited

Thank you