Low-Complexity Belief Propagation Detection for Correlated Large-Scale MIMO Systems

Abstract

In this paper, belief propagation (BP) detection in real domain for large-scale multiple-in multiple-out (MIMO) systems is proposed. The mathematical analysis of message updating rules for independent identically distributed (i.i.d.) and correlated fading MIMO channels are given in detail. The damped BP with damping on the a priori probability vector is employed to improve the performance for the uplink large-scale MIMO systems with correlation among transmitting antennas or loading factor ρ = 1. Based on the convergence analysis, the method of selecting message damping factor δ is presented also. In addition, the adaptive message updating for BP detection is first proposed to provide a good trade-off between performance and complexity. Simulation results have shown that, for 16 × 16 MIMO with quadrature phase shift keying (QPSK) modulation, this approach can show 1 dB performance improvement at BER of 10⁻², compared to complex domain single-edge based BP (SE-BP). For 8 × 32 MIMO with correlation among transmitting and receiving antennas, where 16-Quadrature Amplitude Modulation (16-QAM) is employed, simulation results have shown that the proposed adaptive BP detection achieves a complexity reduction of 50% compared to the general BP detection with negligible performance loss. For i.i.d. and correlated fading channels with various antennas configurations, advantages of the proposed approach over existing BP detections as well as MMSE approach have been demonstrated by thorough simulations. Hence, the proposed BP detection is suitable for large-scale MIMO systems, especially for those with high-order modulations. Furthermore, the adaptive BP detection together with message damping is expected to be a good choice for low complexity detection.

Appendices

Appendix A: Proof of Theorem 1

Assume r and $\mathbf {H}=(\mathbf {h}_{1}^{T},\ldots ,\mathbf {h}_{2N}^{T})^{T}$ are given. Define

$$\left\{ \begin{array}{l} {{\mathbf{x}}^{2M\backslash i}} = ({x_{1}},\ldots,{x_{i - 1}},{x_{i + 1}},\ldots,{x_{2M}}),\\ {\mathbf{x}}_{0}^{2M\backslash i} = \{{{x}_{1}} = {s_{0}},\ldots,{x_{i - 1}} = {s_{0}},{x_{i + 1}} = {s_{0}},\ldots,{x_{2M}}={s_{0}}\}. \end{array} \right. $$

For the l-th iteration in BP detection, $\beta _{j,i}^{(l)}({s_{k}})$ can be calculated as follows:

$$\begin{array}{@{}rcl@{}} \beta_{j,i}^{(l)}({s_{k}})&=& \ln \frac{{{p^{(l)}}({x_{i}} = {s_{k}}|{r_{j}},\mathbf{H})}}{{{p^{(l)}}({x_{i}} = {s_{0}}|{r_{j}},\mathbf{H})}}\\ &=&\ln \frac{{{\sum}_{{\textbf{x}}:{x_{i}} = {s_{k}}} {{p^{(l)}}({r_{j}}|\mathbf{x},\mathbf{H}){p^{(l - 1)}}({{\textbf{x}}^{2M\backslash i}})} }}{{{\sum}_{{\textbf{x}}:{x_{i}} = {s_{0}}} {{p^{(l)}}({r_{j}}|\mathbf{x},\mathbf{H}){p^{(l - 1)}}({{\textbf{x}}^{2M\backslash i}})}}}\\ &=& \ln\frac{\sum\limits_{{\textbf{x}}:{x_{i}} = {s_{k}}}{p^{(l)}}\left( {r_{j}}|\mathbf{x},\mathbf{H}\right) \exp\underset{t\neq i}{\sum\limits_{\mathbb{T}:x_{t}=s_{k}}} \alpha_{t,j}^{(l-1)} (s_{k})}{\sum\limits_{{\textbf{x}}:{x_{i}} = {s_{0}}}{p^{(l)}}\left( {r_{j}}|\mathbf{x}, \mathbf{H}\right) \exp\underset{t\neq i}{\sum\limits_{\mathbb{T}:x_{t} = s_{0}}}\alpha_{t,j}^{(l-1)}(s_{k})}. \end{array} $$

Combining Eq. 19 with the iterative process, we have

$$p^{(l)}({r_{j}}|\mathbf{x},\mathbf{H})=\frac{1}{{\sqrt {2\pi } {\sigma_{{z_{j,i}}}^{(l-1)}}}}\exp \left\{\frac{{({r_{j}} - {\mu_{{z_{j,i}}}^{(l-1)}} - {h_{j,i}}{x_{i}})^{2}}}{{2(\sigma_{{z_{j,i}}}^{2})^{(l-1)}}}\right\}. $$

Then $\beta _{j,i}^{(l)}({s_{k}})$ could be rewritten as follows:

$$\begin{array}{@{}rcl@{}} \beta_{j,i}^{(l)}({s_{k}})= \ln \left\{ \frac{\sum\limits_{{\textbf{x}}:{x_{i}} = {s_{k}}}\exp\left( -\frac{\left( r_{j}- \mu_{z_{j,i}}^{(l-1)}-h_{j,i}x_{i}\right)^{2}}{2(\sigma_{z_{j,i}}^{2})^{(l-1)}}\right)}{\sum\limits_{{\textbf{x}}: {x_{i}} ={s_{0}}}\exp\left( -\frac{\left( r_{j}-\mu_{z_{j,i}}^{(l-1)}-h_{j,i}x_{i}\right)^{2}}{2(\sigma_{z_{j, i}}^{2})^{(l-1)}}\right)}\cdot \right.\\ \left.\frac{\exp\sum\limits_{\mathbb{T}:x_{t}=s_{k} t\neq i} \alpha_{t,j}^{(l-1)} (s_{k})}{\exp\sum\limits_{\mathbb{T}:x_{t}=s_{0} t\neq i}\alpha_{t,j}^{(l-1)}(s_{k})}\right\} \end{array} $$

Employing Jacobi logarithm, we have

$$\begin{array}{@{}rcl@{}} \beta_{j,i}^{(l)}(\mathbf{s}_{k}) & \approx &\underset{{\textbf{x}}:{x_{i}} = {s_{k}}}{\max}\left( \underset{t\neq i}{\sum\limits_{\mathbb{T}:x_{t}=s_{k}}}\alpha_{t,j}^{(l - 1)}(s_{k}) - \frac{\left( r_{j} - \mu_{z_{j,i}}^{(l - 1)} - h_{j,i}x_{i}\right)^{2}}{2(\sigma_{z_{j,i}}^{2})^{(l - 1)}}\right)\\ &&- \underset{{\textbf{x}}:{x_{i}} = {s_{0}}} {\max}\left( \underset{t\neq i}{\sum\limits_{\mathbb{T}:x_{t} = s_{0}}}\alpha_{t,j}^{(l - 1)}(s_{k}) - \frac{\left( r_{j} - \mu_{z_{j,i}}^{(l - 1)} - h_{j,i}x_{i}\right)^{2}}{2(\sigma_{z_{j,i}}^{2})^{(l - 1)}}\right). \end{array} $$

Appendix B: Proof of Lemma 1

Giving that

$$\left\{ \begin{aligned} \mathbf{H}&=(\mathbf{H}_{1},\ldots,\mathbf{H}_{2M})= (\mathbf{h}_{1}^{T},\ldots,\mathbf{h}_{2N}^{T})^{T},\\ \mathbf{H}_{w}&=(\mathbf{H}_{w,1},\ldots,\mathbf{H}_{w,2M})= (\mathbf{h}_{w,1}^{T},\ldots,\mathbf{h}_{w,2N}^{T})^{T}, \end{aligned} \right. $$

together with the i.i.d. Gaussian matrix H _w . According to Eq. 6, we have:

$$\mathbf{H}=\mathbf{R}_{r}^{1/2}\mathbf{H}_{w}\Sigma_{t}^{1/2} \Rightarrow\mathbf{H}_{k}=\sigma_{t,k}^{1/2} (\mathbf{R}_{r}^{1/2})\mathbf{H}_{w,k}. $$

Since H _k is the linear combination of the entries from i.i.d. column vector H _w,k , the column vectors {H _k , 1 ≤ k ≤ 2M} are independent and Gaussian. According to Eq. 7, we have:

$$\mathbf{H}^{T}=(\mathbf{R}_{t}^{1/2})^{T}\mathbf{H}_{w}^{T}(\Sigma_{r}^{1/2})^{T}\Rightarrow\mathbf{h}_{m}= \sigma_{r,m}^{1/2}\mathbf{h}_{w,m}(\mathbf{R}_{t}^{1/2}). $$

Similarly, the row vectors {h _m , 1 ≤ m ≤ 2N} are also independent. However the entries {h _m,k , 1 ≤ m ≤ 2N} of H _k are dependent. Therefore, column vectors of H are not independent now. Similarly, column vectors of H with correlation among both Tx and Rx antennas are dependent also.

Appendix C: Proof of Theorem 2

According to Eq. 2, we have

$$\mathbf{r}=\sum\limits_{k=1}^{2M}\mathbf{H}_{k}x_{k}+\mathbf{n}=\mathbf{H}_{i}x_{i}+\sum\limits_{k=1,k\neq i}^{2M}\mathbf{H}_{k}x_{k}+\mathbf{n}. $$

The column vectors {H _k , 1 ≤ k ≤ 2M} are independent and Gaussian by Theorem 2. Together with the independent transmitted symbols s _k and AWGN vector n, the received vector r are independent and Gaussian.

Given H, x _i = s, and $\mathbf {r}\sim \mathcal {N}\left (\mathbf {m}_{\mathbf {r}}, \Sigma _{\mathbf {r}}\right )$, where

$$\left\{ \begin{aligned} \mathbf{m}_{\mathbf{r}}&=(m_{1}, m_{2},\ldots,m_{2N}),\\ \Sigma_{\mathbf{r}}&=\mathbb{E}\{(\mathbf{r}-\mathbf{m}_{\mathbf{r}})(\mathbf{r}-\mathbf{m}_{\mathbf{r}})^{T}\} =\text{diag}(\sigma_{1},\ldots,\sigma_{2N}), \end{aligned} \right. $$

we can further acquire that for 1 ≤ j ≤ 2N:

$$\left\{ \begin{aligned} m_{j}&=\mathbb{E}\{r_{j}\}=h_{j,i}s+\sum\limits_{k=1,k\neq i}h_{j,k}\mathbb{E}\{s_{k}\},\\ {\sigma_{j}^{2}}&=\mathbb{E}\{(r_{j}-m_{j})^{2}\} =\sum\limits_{k=1,k\neq i}^{2N}h_{j,k}^{2}\text{Var}\{s_{k}\}+\sigma^{2}. \end{aligned} \right. $$

The likelihood probability can be derived as follows:

$$\begin{array}{@{}rcl@{}} p(\mathbf{r}|x_{i})& = &\prod\limits_{j=1}^{2N}\frac{1}{\sqrt{2\pi}\sigma_{j}}\exp\left\{ - \sum\limits_{j=1}^{2N}\frac{(r_{j}-m_{j})^{2}}{{\sigma_{j}^{2}}}\right\}\\ & = &\prod\limits_{j=1}^{2N}\frac{1}{\sqrt{2\pi}\sigma_{z_{j,i}}}\exp\left\{ - \sum\limits_{j=1}^{2N}\frac{(r_{j}-\mu_{z_{j,i}}-h_{j,i}s)^{2}}{\sigma_{z_{j,i}}^{2}}\right\}, \end{array} $$

where $\mu _{z_{j,i}}$ and $\sigma _{z_{j,i}}$ are given by Eq. 18. Compared with Eq. 19, we have the following formula

$$p(\mathbf{r}|x_{i} = s,\mathbf{H}) = \prod\limits_{j=1}^{2N}p(r_{j}|\mathbf{H},x_{i} = s)\propto\prod\limits_{j = 1}^{2N}p(x_{i} = s|r_{j},\mathbf{H}). $$