Expectation propagation for symbol detection in large-scale MIMO communications

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Expectation propagation for symbol detection in large-scale MIMO communications Pablo M. Olmos olmos@tsc.uc3m.

1 Expectation propagation for symbol detection in large-scale MIMO communications Pablo M. Olmos Joint work with Javier Céspedes (UC3M) Matilde Sánchez-Fernández (UC3M) and Fernando Pérez-Cruz (Bell Labs)

2 Today Probabilistic symbol detection in a MIMO system, combined with low-density parity-check (LDPC) channel coding. Approximate Inference. State of the art techniques: Soft MMSE, Gaussian tree approximations (GTA), message passing (BP)... Our proposal: approximate inference via Expectation Propagation (EP). Simulation results.

3 Index 1 Problem setup 2 Low-complexity probabilistic detection Soft MMSE GTA Message Passing 3 Approximate Inference with EP MIMO detection with an EP approximation 4 Simulation results

4 MIMO symbol detection Real-valued channel model. H is assumed known at the receiver. x i, i = 1,..., n are independent amplitude-modulated symbols x i A = {±1, ±2,..., ± M 2 } M symbols Probabilistic detection: Observed y, we are interested in computing p(x i y), i = 1,..., n.

5 MIMO symbol detection p(x i y) x 1 d N (y; Hx, σ ɛ I) p(x j ) M n operations!!! x 2 x i+1 x i 1 x d j=1 p(x j ) = { 1 xj A 0 otherwise

6 MIMO symbol detection p(x i y) x 1 d N (y; Hx, σ ɛ I) p(x j ) M n operations!!! x 2 x i+1 x i 1 x d j=1 p(x j ) = { 1 xj A 0 otherwise

7 Approximate Inference in MIMO detection Today approximate inference methods at O(n 3 ) complexity.

8 Approximate Inference in MIMO detection Today approximate inference methods at O(n 3 ) complexity. Large-scale! E.g., n = 128 and 64-QAM modulation.

9 Approximate Inference in MIMO detection Today approximate inference methods at O(n 3 ) complexity. Large-scale! E.g., n = 128 and 64-QAM modulation. Alternatives for small n, M: Sphere-decoding, MCMC methods,...

10 Index 1 Problem setup 2 Low-complexity probabilistic detection Soft MMSE GTA Message Passing 3 Approximate Inference with EP MIMO detection with an EP approximation 4 Simulation results

11 Index 1 Problem setup 2 Low-complexity probabilistic detection Soft MMSE GTA Message Passing 3 Approximate Inference with EP MIMO detection with an EP approximation 4 Simulation results

12 Soft MMSE d p(x y) N (y; Hx, σ ɛ I) p(x j ), p(x j ) = j=1 { 1 xj A 0 otherwise Replace the discrete priors by independent Gaussian priors with the same mean/variance: d q(x) N (y; Hx, σ ɛ I) q(x j ), q(x j ) = N (0, E s ) j=1

13 Soft MMSE d p(x y) N (y; Hx, σ ɛ I) p(x j ), p(x j ) = j=1 { 1 xj A 0 otherwise Replace the discrete priors by independent Gaussian priors with the same mean/variance: d q(x) N (y; Hx, σ ɛ I) q(x j ), q(x j ) = N (0, E s ) j=1

14 Soft MMSE d p(x y) N (y; Hx, σ ɛ I) p(x j ), p(x j ) = j=1 { 1 xj A 0 otherwise Replace the discrete priors by independent Gaussian priors with the same mean/variance: d q(x) N (y; Hx, σ ɛ I) N (x j : 0, E s ) j=1

15 Soft MMSE q(x) = N (x : µ MMSE, Σ MMSE ) ( Σ MMSE = H T H + σ ) 1 ɛ I E s µ MMSE = Σ MMSE H T y p(x i = a y) q(x i = a) q(x i = c), c A a A

16 Soft MMSE q(x) = N (x : µ MMSE, Σ MMSE ) ( Σ MMSE = H T H + σ ) 1 ɛ I E s µ MMSE = Σ MMSE H T y p(x i = a y) q(x i = a) q(x i = c), c A a A

17 Index 1 Problem setup 2 Low-complexity probabilistic detection Soft MMSE GTA Message Passing 3 Approximate Inference with EP MIMO detection with an EP approximation 4 Simulation results

18 The Gaussian tree approximation d p(x y) N (y; Hx, σ ɛ I) p(x j ), p(x j ) = j=1 { 1 xj A 0 otherwise Step 1: Ignore the discrete prior (or replace it by a constant term) q(x) N (y; Hx, σ ɛ I) = N (x; (H H) 1 H y, σ ɛ (H H) 1 )

19 The Gaussian tree approximation d p(x y) N (y; Hx, σ ɛ I) p(x j ), p(x j ) = j=1 { 1 xj A 0 otherwise Step 1: Ignore the discrete prior (or replace it by a constant term) q(x) N (y; Hx, σ ɛ I) = N (x; (H H) 1 H y, σ ɛ (H H) 1 )

20 The Gaussian tree approximation Step 2: Construct a Gaussian tree approximation to q(x) n q tree (x) = q tree (x j π(j)) q(x), j=1 Both q(x) and q tree (x) are Gaussian and have the same pairwise marginals.

21 The Gaussian tree approximation Step 3: Include the discrete priors and compute marginals using discrete message passing. q o tree(x) q tree (x) d p(x j ) j=1

22 Index 1 Problem setup 2 Low-complexity probabilistic detection Soft MMSE GTA Message Passing 3 Approximate Inference with EP MIMO detection with an EP approximation 4 Simulation results

23 Message Passing and Belief Propagation BP accurate marginal estimates in sparse factor graphs (such as LDPC code graphs). This is not the case in MIMO detection d { 1 xj A p(x y) N (y; Hx, σ ɛ I) p(x j ), p(x j ) = 0 otherwise j=1

24 Message Passing and Belief Propagation Approximate Message Passing (AMP): BP messages are approximated using Gaussian pdfs. Compressed Sensing applications: AMP achieves remarkable accuracy in a certain scenarios, despite the high-density in the factor graph.

25 AMP for MIMO detection Excellent performance/accuracy for QPSK modulations and large n. Larger QAM constellations: performance severely degraded. Reduce the loading factor: n/d < 1.

26 Index 1 Problem setup 2 Low-complexity probabilistic detection Soft MMSE GTA Message Passing 3 Approximate Inference with EP MIMO detection with an EP approximation 4 Simulation results

27 Expectation Propagation General purpose framework for approximating a probability distribution p(x) by a distribution q(x) that belongs to an exponential family F. Exponential Family: A family F of distributions with densities q(x θ) = h(x) exp(θ T φ(x) Φ(θ)), Φ(θ) = log exp(θ T φ(x))dh(x) θ Θ Family F of Gaussian distributions with diagonal covariance matrix: φ(x) = [ x 1, x 2,..., x n, x1 2, x2 2,..., xn 2 ] T The Moment Matching criterion: Find θ such that E q(x θ )[φ(x)] E p(x) [φ(x)]

28 EP Iterative algorithms

29 Index 1 Problem setup 2 Low-complexity probabilistic detection Soft MMSE GTA Message Passing 3 Approximate Inference with EP MIMO detection with an EP approximation 4 Simulation results

30 EP for MIMO symbol detection d p(x y) N (y; Hx, σ ɛ I) p(x j ), p(x j ) = j=1 { 1 xj A 0 otherwise

31 EP for MIMO symbol detection d p(x y) N (y; Hx, σ ɛ I) p(x j ), p(x j ) = j=1 { 1 xj A 0 otherwise A family of approximating distributions: n q(x γ, Λ) N (y; Hx, σ ɛ I) e γ j x j 1 2 Λ j x 2 j, where γ R n and Λ R n +. j=1

32 EP for MIMO symbol detection d p(x y) N (y; Hx, σ ɛ I) p(x j ), p(x j ) = j=1 { 1 xj A 0 otherwise A family of approximating distributions: n q(x γ, Λ) N (y; Hx, σ ɛ I) e γ j x j 1 2 Λ j x 2 j, where γ R n and Λ R n +. j=1 q(x γ, Λ) = N (x : µ, Σ) ( Σ = σɛ 1 H H + diag (Λ) ( ) µ = Σ H y + γ σ 1 ɛ ) 1

33 EP for MIMO symbol detection Note that q(x γ, Λ) N (y; Hx, σ ɛ I) ( Σ = ɛ ( µ = Σ σ 1 H H + diag (Λ) ) H y + γ σ 1 ɛ n e γ j x j 1 2 Λ j x 2 j = N (x : µ, Σ) j=1 ) 1 q(x m γ, Λ) = N (x m : µ m, Σ mm ) e γmxm 1 2 Λmx2 m N (y; Hx, σ ɛ I) n e γ j x j 1 2 Λ j x 2 j dx m j=1 j m

34 O(n 3 ) {}}{ Given the current value of γ, Λ µ, Σ q(x j γ, Λ), m = 1,..., n. Parallel update of all (γ m, Λ m ) pairs, m = 1,..., n: 1 Define ˆq(x m ) q(x m γ, Λ) p(x m ) e γmxm 1 2 Λmx2 m = { 0 xm A 0 otherwise 2 Compute 3 Find (γ m, Λ m) such that ˆµ m = Eˆq(xm)[x m ], ˆσ j = Eˆq(xm)[x 2 m] ˆµ 2 m. ˆq(x m ) q(x m γ, Λ) γ mx m 1 2 Λ mx 2 m e γmxm 1 2 Λmx2 m has mean and variance equal to ˆµ m, ˆσ m.

35 O(n 3 ) {}}{ Given the current value of γ, Λ µ, Σ q(x j γ, Λ), m = 1,..., n. Parallel update of all (γ m, Λ m ) pairs, m = 1,..., n: 1 Define ˆq(x m ) q(x m γ, Λ) p(x m ) e γmxm 1 2 Λmx2 m = { 0 xm A 0 otherwise 2 Compute 3 Find (γ m, Λ m) such that ˆµ m = Eˆq(xm)[x m ], ˆσ j = Eˆq(xm)[x 2 m] ˆµ 2 m. ˆq(x m ) q(x m γ, Λ) γ mx m 1 2 Λ mx 2 m e γmxm 1 2 Λmx2 m has mean and variance equal to ˆµ m, ˆσ m.

36 O(n 3 ) {}}{ Given the current value of γ, Λ µ, Σ q(x j γ, Λ), m = 1,..., n. Parallel update of all (γ m, Λ m ) pairs, m = 1,..., n: 1 Define ˆq(x m ) q(x m γ, Λ) p(x m ) e γmxm 1 2 Λmx2 m = { 0 xm A 0 otherwise 2 Compute 3 Find (γ m, Λ m) such that ˆµ m = Eˆq(xm)[x m ], ˆσ j = Eˆq(xm)[x 2 m] ˆµ 2 m. ˆq(x m ) q(x m γ, Λ) γ mx m 1 2 Λ mx 2 m e γmxm 1 2 Λmx2 m has mean and variance equal to ˆµ m, ˆσ m.

37 q(x m γ, Λ) e γmxm 1 2 Λmx2 m N (y; Hx, σ ɛ I) n e γ j x j 1 2 Λ j x 2 j dx m j=1 j m ˆq(x m ) = q(x m γ, Λ) p(x m ) e γmxm 1 2 Λmx2 m p(x m ) N (y; Hx, σ ɛ I) n e γ j x j 1 2 Λ j x 2 j dx m j=1 j m

38 O(n 3 ) {}}{ Given the current value of γ, Λ µ, Σ q(x j γ, Λ), m = 1,..., n. Parallel update of all (γ m, Λ m ) pairs, m = 1,..., n: 1 Define ˆq(x m ) q(x m γ, Λ) p(x m ) e γmxm 1 2 Λmx2 m = { 0 xm A 0 otherwise 2 Compute 3 Find (γ m, Λ m) such that ˆµ m = Eˆq(xm)[x m ], ˆσ m = Eˆq(xm)[x 2 m] ˆµ 2 m. ˆq(x m ) q(x m γ, Λ) γ mx m 1 2 Λ mx 2 m e γmxm 1 2 Λmx2 m has mean and variance equal to ˆµ m, ˆσ m.

39 O(n 3 ) {}}{ Given the current value of γ, Λ µ, Σ q(x j γ, Λ), m = 1,..., n. Parallel update of all (γ m, Λ m ) pairs, m = 1,..., n: 1 Define ˆq(x m ) q(x m γ, Λ) p(x m ) e γmxm 1 2 Λmx2 m = { 0 xm A 0 otherwise 2 Compute 3 Find (γ m, Λ m) such that ˆµ m = Eˆq(xm)[x m ], ˆσ m = Eˆq(xm)[x 2 m] ˆµ 2 m. q(x m γ, Λ) γ mx m 1 2 Λ mx 2 m e γmxm 1 2 Λmx2 m has mean and variance equal to ˆµ m, ˆσ m.

40 O(n 3 ) {}}{ Given the current value of γ, Λ µ, Σ q(x j γ, Λ), m = 1,..., n. Parallel update of all (γ m, Λ m ) pairs, m = 1,..., n: 1 Define 2 Compute ˆq(x m ) q(x m γ, Λ) p(x m ) e γmxm 1 2 Λmx2 m = { 0 xm A 0 otherwise 3 Find (γ m, Λ m) such that ˆµ m = Eˆq(xm)[x m ], ˆσ m = Eˆq(xm)[x 2 m] ˆµ 2 m. q(x m γ, Λ) γ mx m 1 2 Λ mx 2 m e γmxm 1 2 Λmx2 m has mean and variance equal to ˆµ m, ˆσ m.

41 The EP iterative algorithm MMSE initialization γ j = 0, Λ j = E 1 s, j = 1,..., n. Cost per iteration dominated by the matrix inversion: ( Σ = H H + diag (Λ) σ 1 ɛ ) 1 Smooth the parameter update to improve stability. For some β [0, 1] γ j βγ j + (1 β)γ j Λ j βλ j + (1 β)λ j Quick convergence to a stationary point. Number I of required iterations that does not scale neither with n nor M. Symbol marginals: p(x i = a y) q(x i = a γ, Λ) q(x i = c γ, Λ), c A a A

42 Index 1 Problem setup 2 Low-complexity probabilistic detection Soft MMSE GTA Message Passing 3 Approximate Inference with EP MIMO detection with an EP approximation 4 Simulation results

43 Binary LDPC channel code, block length N. Several channel uses are required to transmit a complete LDPC codeword. Channel coefficients are sampled from an independent complex zero-mean unit-variance Gaussian distribution. [ ] R(Hc ) I(H H = c ) I(H c ) R(H c ) SNR defined as: ( SNR(dB) = 10 log 10 n log 2 (M)R E ) b, σ ɛ

44 QPSK, 5 5 scenario, (3, 6)-regular LDPC code, N = 5120 bits BER SNR Optimal detector EP GTA APM Soft MMSE

45 QPSK, 5 5 scenario, (3, 6)-regular LDPC code, N = 5120 bits Calibration curves: q(x i ) vs. p(x i y) EP GTA pep(ui y) pgta(ui y) p(ui y) Soft MMSE p(ui y) AMP pmmse(ui y) pchemp(ui y) p(ui y) SNR = 13.5dB p(ui y)

46 scenario, (3, 6)-regular LDPC code, N = 5120 bits 10 0 QPSK 16-QAM 64-QAM BER SNR EP AMP GTA MMSE

47 32 32 scenario, 16-QAM, (3, 6)-regular LDPC code with N = 15k bits and N = 32k bits BER EP GTA MMSE SNR Dashed line N = 15k Solid line N = 32k

48 32 32 scenario, 16-QAM, capacity-achieving LDPC codes, N = 32k bits EP GTA MMSE BER SNR Dashed line Rate-1/2 Irregular LDPC code optimized for the BIAWNG threshold using density evolution. Solid line Rate-0.48 Convolutional LDPC code with L = 50 positions.

49 System performance is limited by the detector accuracy. Room for improvement. More involved EP approximating families can lead to further gains. Extension to partial CSI scenarios, where we only know the statistics of H. E.g., h i,j N (ĥi,j, σ h ) with known ĥi,j, σ h. Questions?

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