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SCIENCE CHINA Information Sciences, Volume 62, Issue 4: 042302(2019) https://doi.org/10.1007/s11432-018-9637-5

Lattice reduction aided belief propagation for massive MIMO detection

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  • ReceivedSep 4, 2018
  • AcceptedNov 12, 2018
  • PublishedDec 29, 2018

Abstract

Efficient massive MIMO detection for practical deployment, which is with spatially correlated channel and high-order modulation, is a challenging topic for the fifth generation mobile communication (5G).In this paper, lattice reduction aided belief propagation (LRA-BP) is proposed for massive MIMO detection.LRA-BP applies the message updating rules of Markov random field based belief propagation (MRF-BP) in lattice reduced MIMO system.With the lattice reduced, well-conditioned MIMO channel, LRA-BP obtains better message updating and detection performance in spatially correlated channel than MRF-BP.Log-domain arithmetic is used in LRA-BP for computational complexity reduction.Simulation result shows that LRA-BP outperforms MRF-BP with 3–10 dB in terms of required SNR for 1% packet error rate in spatially correlated channel for 256-QAM.We also show that LRA-BP requires much lower complexity compared with MRF-BP.


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  • Figure 1

    Real-domain constellation for 256-QAM.

  • Figure 2

    Pairwise graph for MIMO detection.

  • Figure 3

    Comparison of receiver performance: $12\times~12$, Rayleigh channel.

  • Figure 4

    Comparison of iteration number: $32\times~4$, 3GPP 3D channel, SU.

  • Figure 5

    Comparison of receiver performance: $32\times~4$, 3GPP 3D channel, SU-MIMO. (a) $N_r=32~~(M=16,N=1,P=2),~N_t=4$; (b) $N_r=32~(M=8,N=2,P=2),~N_t=4$.

  • Figure 6

    Comparison of receiver performance: $32\times~4$, 3GPP 3D channel, MU-MIMO (two uplink 2-antenna UEs). protectłinebreak(a) $N_r=32~~(M=16,N=1,P=2),~N_t=4$; (b) $N_r=32~~(M=8,N=2,P=2),~N_t=4$.

  • Table 1   General computational complexity (in terms of equivalent additions)
    Log-domain MRF-BP LRA-BP
    Iteration (38) $2TN_t(2N_t-1)^2~\big|\boldsymbol{\Omega}\big|^2$
    Translation function (20) $80N_r^2N_t$
    LLL reduction 0 $24N_r^2N_t$
    In total $2TN_t(2N_t-1)^2~\big|\boldsymbol{\Omega}\big|^2~+~80N_r^2N_t$ $2TN_t(2N_t-1)^2~\big|\boldsymbol{\Omega}\big|^2~+~104N_r^2N_t$
  •   

    Algorithm 1 LRA-BP

    Initialization: $m_{i\to~j}^{(0)}(z_j)~=~\mathrm{log}(|\boldsymbol{\Omega_z}^{(j)}|^{-1})$ for $j=1,2,\ldots,2N_t$. Pre-processing: $\boldsymbol{K}_{\{j,i\}}=\frac12\sigma^2\boldsymbol{I}+\sum_{k\ne~j,i}\boldsymbol{\tilde~h}_k\boldsymbol{t}_k\boldsymbol{t}_k^{\rm~T}\boldsymbol{\tilde~h}_k^{\rm~T}$; $y^\prime_{j|i}=\boldsymbol{\tilde~h}_j^{\rm~T}\boldsymbol{K}^{-1}_{\{j,i\}}\boldsymbol{y}$; $a_{j|i,i}=\boldsymbol{\tilde~h}_j^{\rm~T}\boldsymbol{K}^{-1}_{\{j,i\}}\boldsymbol{\tilde~h}_i$; $\sigma^2_{j|i}=\boldsymbol{\tilde~h}_j^{\rm~T}\boldsymbol{K}^{-1}_{\{j,i\}}\boldsymbol{\tilde~h}_j$. $ p\big(z_j~\big|~z_i,~\boldsymbol{y}~\big) = \sqrt{ \frac {\beta} {2\pi} } {\rm~e}^{d_{i\to~j}(z_j)} $, $\beta=\sigma^2_{j|i}+\sigma_j^{-2}$, $d_{i\to~j}(z_j)=-(\beta~z_j-y^\prime_{j|i}+a_{j|i,i}z_i)^2/(2\beta)$.

    Iteration: For all possible $(i,j)$ pairs repeat message updating as $t=1,2,\ldots,T$ and $\forall~z_j\in~\boldsymbol{\tilde~\Omega_z}^{(j)}$ $m_{i\to~j}^{(t)}(z_j)= \mathop{\mathrm{argmax}}_{z_i\in~\boldsymbol{\tilde~\Omega_z}^{(i)}} \left\{ d_{i\to~j}(z_j)~+~\sum_{k\in~V(i)\setminus~j}{m_{k\to~i}^{(t-1)}(z_i)} \right\}$

    Output: $M(z_j)=\sum_{k\in~V(j)}m_{k\to~j}^{(T)}(z_j),~\forall~z_j\in~\boldsymbol{\tilde~\Omega_z}^{(j)}$ $P(z_j)=\alpha~{\rm~e}^{M(z_j)}$

  • Table 2   Computational complexity with specific parameters (in terms of equivalent additions)
    Rayleigh channel 3GPP 3D channel
    Log-domain MRF-BP LRA-BP Log-domain MRF-BP LRA-BP
    $N_r$ 12 32
    $N_t$ 12 4
    $T$ 5 5 2
    $\big|\boldsymbol{\Omega}\big|$ 16 9 16 9
    Complexity 16389120 5321592 829440 489488

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