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Iterative Detection in Coded Linear SystemsBased on Orthogonal AMP

Columbia University,City University of Hong Kong,

University of Electronic Science and Technology of China.

Junjie Ma, Lei Liu, Xiaojun Yuan, and Li Ping

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Contents

§ Background

§ Orthogonal AMP

§ Details of OAMP

§ Numerical results

§ Conclusions

References:J. Ma and L. Ping, “Orthogonal AMP,” IEEE Access, 2017.

X. Wang and H. V. Poor, “Iterative (turbo) soft interference cancellation and decoding forcoded CDMA,” IEEE Trans. Commun., vol. 47, no. 7, pp. 1046–1061, Jul 1999.

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Contents

§ Background

§ Orthogonal AMP

§ Details of OAMP

§ Applications and numerical results

§ Conclusions

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Problem formulation

Applications: compress sensing, MIMO, equalization, multi-user system, …

A coded/modulated linear system can be characterized as

,= +y Ax n~ , .jx S j",CÎx

The solution is well-known if xj ~ PX(x) is Gaussian. For other discrete distributions, the optimal solution is generally NP hard.

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A Encoder

LMMSE Decoder

transmitter/channel

receiver

noise

Turbo algorithm

x

y

d

d

W&P (Turbo) Receiver:

Turbo detector:

where .

( )h=s r( )LMMSE+ -=r s W y As

extrinsic information

( ) 1LMMSE 2 T 2 T 2v v s-

= +W A AA I

extrinsic informationLinear detector :

Non-linear decoder:

X. Wang and H. V. Poor, “Iterative (turbo) soft interference cancellation and decoding forcoded CDMA,” IEEE Trans. Commun., vol. 47, no. 7, pp. 1046–1061, Jul 1999.

r

sEstimates of x

Intuitions:• The use of APP messages achieves locally minimum MSE. However, it may cause

correlation during iterative detection.• The use of extrinsic messages avoids correlation by imposing independency

between input and output errors.

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Turbo algorithm

X. Wang and H. V. Poor, “Iterative (turbo) soft interference cancellation and decoding for codedCDMA,” IEEE Trans. Commun., vol. 47, no. 7, pp. 1046–1061, Jul. 1999.

Xiaojun Yuan, Qinghua Guo, Xiaodong Wang, and Li Ping, “Evolution analysis of low-costiterative equalization in coded linear systems with cyclic prefixes,” IEEE JSAC, Feb. 2008.

The use of extrinsic information is key in Turbo.

LMMSE Decodery dextrinsic extrinsic

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H. A. Loeliger, J. Dauwels, J. Hu, S. Korl, L. Ping, and F. R. Kschischang, “The factor graph approach to model-based signal processing,” Proc. IEEE, vol. 95, no. 6, pp. 1295–1322, June 2007.R. Visoz, A. O. Berthet, and M. Lalam, “Semi-analytical performance prediction methods for iterative mmse-icmultiuser mimo joint decoding,” IEEE Trans. on Commun., vol. 58, no. 9, pp. 2576–2589, September 2010.L. Liu, C. Yuen, Y. L. Guan, Y. Li, and C. Huang, “Gaussian message passing for overloaded massive MIMO-NOMA systems with massive access,” in 2018 IEEE TWC, Dec 2018.X. Yuan, Q. Guo, X. Wang, and L. Ping, “Evolution analysis of low-cost iterative equalization in coded linearsystems with cyclic prefixes,” IEEE JSAC, Feb. 2008.C. Huang, L. Liu, C. Yuen, and S. Sun, "Iterative Channel Estimation Using LSE and Sparse Message Passing forMmWave MIMO Systems ," IEEE TSP, 2018.X. Yuan, L. Ping, C. Xu and A. Kavcic, “Achievable rates of MIMO systems with linear precoding and iterativeLMMSE detector,” IEEE Trans. Inf. Theory, vol. 60, no.11, pp. 7073-7089, Oct. 2014.L. Liu, C. Yuen, Y. L. Guan, and Y. Li, “Capacity-achieving iterative LMMSE detector for MIMO-NOMAsystems,” 2016 IEEE ICC, Malaysia, May 2016.

• Due to its excellent performance, Turbo has been widely used for various applications:MIMO, multiple access, channel estimation, equalization, joint decoding…

• Turbo achieves the sum capacity of the linear system, if Gaussian signaling is used. However, for discrete signaling, Turbo may not be capacity optimal any more.

Turbo algorithm

Can we do better than Turbo?

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Motivation

A

LMMSE demodulator

transmitter/channel

receiver

noise

no extrinsic information

For un-coded system, the extrinsic information from a soft-output demodulator is zero.

Turbo does not work in un-coded discrete systems, where OAMP works well.

modulator (un-coded)

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Motivation

Those motivate us to study OAMP for the discrete coded linear system.

Decoded MSEs of Turbo and OAMP:

Thus, OAMP potentially performs better than Turbo for discrete coded systems.

(3, 6) LDPC code

OAMPTurbo OAMP

Turbo

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APP vs Turbo vs OAMP

uncorrelated

APP( )h=s r

( )+ -=r s W y Aslocally optimal

locally optimal

= +y Ax n~ , .jx S j",CÎx

System model:

Linear:Non-linear:

Turbo( )h=s r

( )+ -=r s W y AsLinear:Non-linear:

OAMP( )h=s r

( )+ -=r s W y AsLinear:Non-linear:

extrinsic

extrinsic

uncorrelated

Intuitions:

• APP messages achieve locally optimal MSE, but cause correlation during iteration.

• Extrinsic messages avoid correlation by imposing independency between input-output errors, but it increases MSE.

• Uncorrelated messages impose orthogonality on input-output errors.

• It achieves lower MSE than ‘extrinsic’, and avoids the correlation of ‘APP’.

Linear constraint:Non-linear constraint :

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OAMP and Turbo

Theorem 1: OAMP achieves lower MSE than Turbo.

OAMP is less restrictive than Turbo, so OAMP can achieve lower MSE (and so better performance) than Turbo.

Intuitive explanation:

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Contents

§ Background

§ Orthogonal AMP

§ Details of OAMP

§ Numerical results

§ Conclusions

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Orthogonal AMP (OAMP)

J. Ma and L. Ping, “Orthogonal AMP,” IEEE Access, 2017.

uncorrelated

= +y Ax n~ , .jx S j",CÎx

System model:

OAMP:

( )h=s r( )+ -=r s W y AsLinear detector:

Non-linear decoder:uncorrelated

Linear constraint:Non-linear constraint :

Details:• How to choseW to construct an uncorrelated linear detector?• How to design to construct an uncorrelated non-linear decoder?!

Lemma 1: If , the input and output errors of a lineardetector are statistically orthogonal.

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Orthogonality for a linear detector

Assume that zin consists of IID entries with zero mean and second moment E(| zin |2). Then

Thus, the errors in zin and zout are statistically orthogonal to each other.

( )out in-=z I WA z

( )tr 0=I -WA

( ) ( )( )H H Hin out in in in in- tr - .= I Iz z z WA z = WA z z

( ) ( )2Hin out intr - E 0.z =Iz z = WA

Proof: ( )+ -=r s W y AsFrom , we have . Then

For an arbitrary , we can meet using:ˆ

.ˆtr( ) / NWW =WA

( )tr 0=I -WAW

Construction:

For an arbitrary , we can meet using:

If holds, then

The above is an direct application of Stein’s Lemma. Thus the errors at the input and output of the decoder are statically orthogonal to each other.

Lemma 2: If the input and output errors of a non-linear decoderare statistically orthogonal.

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Orthogonality for a non-linear decoder

Proof:

( ){ }E ' 0,Rh =

( ) ( ){ }E 0.R X R Xhé ù- × - =ë û

( ){ }E ' 0Rh =

( ) ( ) ( ){ }{ }ˆ ˆE ' .R C R R Rh h h= × - ×

( )ˆ Rh ( ){ }E ' 0Rh =

Construction:

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Contents

§ Background

§ Orthogonal AMP

§ Intuitions

§ Numerical results

§ Conclusions

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Comparison for large-scale MIMO

Ø M = N =105.

Ø (23, 25) convolutional code.

Ø QPSK modulation.

Ø Codeword length = 105.

Ø Condition number of A = 50.

OAMP

OAMP

q OAMP has a better BER than Turbo and APP.

q Turbo has a better BER than APP.

Turbo

= +y Ax n~ , .jx S j",CÎx

System model:

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Comparison for medium sized MIMO

Ø M = N = 60.

Ø Regular (3, 6) LDPC code.

Ø 16QAM modulation.

Ø Codeword length = 4096.

Ø Left: = 0 (IIDG).

Ø Right: = 0.3 (Non-IIDG).

q AMP works well in IIDG MIMO, but fails in non-IIDG MIMO.

q OAMP has the best BER both in IIDG and non-IIDG MIMO.

q APP has a better BER than Turbo.

APP

Turbo Turbo

APP

= +y Ax n~ , .jx S j",CÎx

System model:

AMP

OAMP

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Contents

§ Background

§ Orthogonal AMP

§ Details of OAMP

§ Numerical results

§ Conclusions

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Conclusions

• OAMP outperforms turbo in discrete coded linear systems.

• The advantage of OAMP is achieved by using an “orthogonal” requirement to replace the “extrinsic” requirement.

Thanks!

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ISTC 2018, Hong Kong 06 December, 2018