Low Complexity, Pairwise Layered Tabu Search for Large Scale MIMO Detection

Abstract

This paper presents a low complexity pairwise layered tabu search based detection algorithm for a large-scale multiple-input multiple-output system. The proposed algorithm can compute two layers simultaneously and reduce the effective number of tabu searches. An efficient Gram matrix and matched filtered output update strategy is developed to reuse the computations from past visited layers. Also, a precomputation technique is adapted to reduce the redundancy in computation within tabu search iterations. Complexity analysis shows that the upper bound of initialization complexity in the proposed algorithm reduces from $O(N_t^4)$ to $O(N_t^3)$. The detection performance of the proposed detector is almost the same as the conventional complex version of LTS for 64QAM and 16QAM modulations. However, the proposed detector outperforms the conventional system for 4QAM modulation, especially in $16 \times 16$ and $8 \times 8$ MIMO. Simulation results show that the percent of complexity reduction in the proposed method is approximately 75% for $64 \times 64$, 64QAM and 85% for $64 \times 64$ 16QAM systems to achieve a BER of $10^{-3}$. Moreover, we have proposed three layer-wise iteration allocation strategies that can further reduce the upper bound of complexity with minor degradation in detection performance.

Appendix: Proof of property 2 and 3 of W

Proof of property 2: $W_{2i,2i-1} = W_{2i-1,2i} = 0$

Using (30), $(2i,2i-1)$th element of $\varvec{W}$ can be expressed as

$$\begin{aligned} \begin{aligned} W_{2i,2i-1}&= \sum _{m=1}^{2N_t} R_{m,2i} R_{m,2i-1} \\&= \sum _{n = 1}^{N_t} \{ R_{2n,2i} R_{2n,2i-1} + R_{2n-1,2i} R_{2n-1,2i-1}\} \quad \text {Where},\, m=2n \end{aligned} \end{aligned}$$

(51)

From (5) and (6), we can write, $R_{2n,2i} = R_{2n-1,2i-1}$ and $R_{2n,2i-1} = - R_{2n-1,2i}$. Substituting $R_{2n,2i}$ and $R_{2n,2i-1}$ in (51) we can obtain $W_{2i,2i-1} = 0$. Also, $W_{2i-1,2i}= 0$ as $W_{i,j} = W_{j,i}$ from property 1 of $\varvec{W}$.

Proof of property 3: $W_{2i-1,2j-1} = W_{2i,2j}$ and $W_{2i-1,2j} = -W_{2i,2j-1}$

From (30), $W_{2i-1,2j-1}$ can be computed as

$$\begin{aligned} \begin{aligned} W_{2i-1,2j-1}&= \sum _{m=1}^{2N_t}R_{m,2i-1}R_{m,2j-1} \\&= \sum _{n=1}^{N_t}\{R_{2n,2i-1}R_{2n,2j-1}+R_{2n-1,2i-1}R_{2n-1,2j-1}\} \quad \text {Where},\, m=2n \end{aligned} \end{aligned}$$

(52)

Using (5) and (6), we can substitute $R_{2n,2i-1}$, $R_{2n,2j-1}$, $R_{2n-1,2i-1}$ and $R_{2n-1,2j-1}$ in (52) by $-R_{2n-1,2i}$, $-R_{2n-1,2j}$, $R_{2n,2i}$ and $R_{2n,2j}$ respectively. Therefore, (52) can be expressed as

$$\begin{aligned} \begin{aligned} W_{2i-1,2j-1}&= \sum _{n=1}^{N_t}\{R_{2n,2i-1}R_{2n,2j-1}+R_{2n-1,2i-1}R_{2n-1,2j-1}\}\\&= W_{2i,2j} \end{aligned} \end{aligned}$$

(53)

In a similar way we can prove $W_{2i-1,2j} = -W_{2i,2j-1}$. The element $W_{2i-1,2j}$ can be expressed as

$$\begin{aligned} \begin{aligned} W_{2i-1,2j}&= \sum _{m=1}^{2N_t}R_{m,2i-1}R_{m,2j} \\&= \sum _{n=1}^{N_t}\{R_{2n,2i-1}R_{2n,2j}+R_{2n-1,2i-1}R_{2t-1,2j}\} \quad \text {Where},\, m=2n \end{aligned} \end{aligned}$$

(54)

Using (5) and (6), we can substitute $R_{2n,2i-1}$, $R_{2n,2j-1}$, $R_{2n-1,2i-1}$ and $R_{2n-1,2j-1}$ in (54) by $-R_{2n-1,2i}$, $-R_{2n-1,2j}$,$R_{2n,2i}$ and $R_{2n,2j}$ respectively. Therefore, (54) can be expressed as

$$\begin{aligned} \begin{aligned} W_{2i-1,2j}&= -\sum _{n=1}^{N_t}\{R_{2n,2i}R_{2n,2j-1}+R_{2n-1,2i}R_{2n-1,2j-1}\}\\&= -W_{2i,2j-1} \end{aligned} \end{aligned}$$

(55)