﻿ 大规模MIMO系统中基于模等式约束的降维去相干DOA估计

# 大规模MIMO系统中基于模等式约束的降维去相干DOA估计Modulus Equation Constraints Based Decoherent Dimension Reduction 2-D DOA Estimation for Massive MIMO Systems

Abstract: There are large numbers of coherent signals in massive MIMO system. Simple decoherence pro-cessing leads to the lower accuracy of the 2-D DOA (Two-Dimensional Direction of Arrival). This paper will propose a modulus equation constraints based dimension reduction MUSIC algorithm which can greatly improve the performance of two-dimensional DOA estimation of coherent signals. The algorithm transforms the two-dimensional DOA estimation problem into an optimization problem, and uses the modulus equation constraints to define the additional conditions and impose strong constraints on the direction vector so that the optimization equation is solved more close to the optimal solution. The results of theoretical analysis and simulation experiments show that the proposed DOA algorithm has high reliability and precision. Such algorithm is able to meet the requirements of DOA estimation performance in massive MIMO system, and can also provide high feasibility and practicability.

1. 引言

2. 信号模型与相关性分析

$r=\frac{\mathrm{cov}\left({s}_{1},{s}_{2}\right)}{\sqrt{D\left({s}_{1}\right)D\left({s}_{2}\right)}}=\frac{E\left\{\left[{s}_{1}-E\left\{{s}_{1}\right\}\right]\left[{s}_{2}-E\left\{{s}_{2}\right\}\right]\right\}}{\sqrt{E\left\{{\left[{s}_{1}-E\left\{{s}_{1}\right\}\right]}^{2}\right\}E\left\{{\left[{s}_{2}-E\left\{{s}_{2}\right\}\right]}^{2}\right\}}}$ (1)

$r=0$ 时，信号不相关；当 $0 时，信号相关；当 $r=1$ 时，信号相干。

$X\left(t\right)={A}_{x}S\left(t\right)+{N}_{x}\left(t\right)$ (2)

$Y\left(t\right)={A}_{y}S\left(t\right)+{N}_{y}\left(t\right)$ (3)

Figure 1. Schematic diagram of l-shaped array antenna

$\begin{array}{l}{a}_{x}\left({\theta }_{k}\right)={\left[{a}_{x,1}\left({\theta }_{k}\right),{a}_{x,2}\left({\theta }_{k}\right),\cdots ,{a}_{x,M}\left({\theta }_{k}\right)\right]}^{\text{T}}={\left[1,{\text{e}}^{-j2\text{π}d\mathrm{cos}{\phi }_{1}\mathrm{sin}{\theta }_{1}/\lambda },\cdots ,{\text{e}}^{-j2\text{π}\left(M-1\right)d\mathrm{cos}{\phi }_{k}\mathrm{sin}{\theta }_{k}/\lambda }\right]}^{\text{T}}\\ {a}_{y}\left({\phi }_{k}\right)={\left[{a}_{y,1}\left({\phi }_{k}\right),{a}_{y,2}\left({\phi }_{k}\right),\dots {a}_{y,M}\left({\phi }_{k}\right)\right]}^{\text{T}}={\left[1,{\text{e}}^{-j2\text{π}d\mathrm{sin}{\phi }_{1}\mathrm{sin}{\theta }_{1}/\lambda },\cdots ,{\text{e}}^{-j2\text{π}\left(M-1\right)d\mathrm{sin}{\phi }_{k}\mathrm{sin}{\theta }_{k}/\lambda }\right]}^{\text{T}}\end{array}$ (4)

$\left\{\begin{array}{l}\mathrm{cos}{\alpha }_{k}=\mathrm{cos}{\phi }_{k}\mathrm{sin}{\theta }_{k}\\ \mathrm{cos}{\beta }_{k}=\mathrm{sin}{\phi }_{k}\mathrm{sin}{\theta }_{k}\end{array}$ (5)

3. 去相干算法

${x}_{k}\left(t\right)=\left[{x}_{k}\text{\hspace{0.17em}}\text{\hspace{0.17em}}{x}_{k+1}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\cdots \text{\hspace{0.17em}}\text{\hspace{0.17em}}{x}_{k+m-1}\right]={A}_{xk}{D}^{\left(k-1\right)}s\left(t\right)+{n}_{k}\left(t\right)$ (6)

${R}_{xk}={A}_{xk}{\eta }^{\left(k-1\right)}{R}_{s}{\left({D}^{\left(k-1\right)}\right)}^{H}{A}_{xk}^{H}+{\sigma }^{2}I$ (7)

${R}_{x}^{f}=\frac{1}{p}{\sum }_{i=1}^{p}{R}_{ip}={A}_{xk}{R}_{xs}^{f}{A}_{xk}^{H}+{\sigma }^{2}I$ (8)

${R}_{x}^{b}=\frac{1}{p}{\sum }_{i=1}^{p}{R}_{xp}^{b}={A}_{xk}{R}_{xs}^{b}{A}_{xk}^{H}+{\sigma }^{2}I$ (9)

${R}_{x}^{fb}=\frac{1}{2}\left({R}_{x}^{f}+{R}_{x}^{b}\right)$ (10)

${R}_{y}^{fb}=\frac{1}{2}\left({R}_{y}^{f}+{R}_{y}^{b}\right)$ (11)

$Z={\left[{R}_{x}^{fb}\text{\hspace{0.17em}}\text{\hspace{0.17em}}{R}_{y}^{fb}\right]}^{\text{T}}$ (12)

4. 降维MUSIC算法

${P}_{2dmusic}=\frac{1}{{\left[{a}_{x}\left({\theta }_{k}\right)\otimes {a}_{y}\left({\phi }_{k}\right)\right]}^{H}{U}_{n}{U}_{n}^{H}\left[{a}_{x}\left({\theta }_{k}\right)\otimes {a}_{y}\left({\phi }_{k}\right)\right]}$ (13)

$Q\left({\theta }_{k},{\phi }_{k}\right)={\left[{a}_{x}\left({\theta }_{k}\right)\otimes {a}_{y}\left({\phi }_{k}\right)\right]}^{H}{U}_{n}{U}_{n}^{H}\left[{a}_{x}\left({\theta }_{k}\right)\otimes {a}_{y}\left({\phi }_{k}\right)\right]$ (14)

$Q\left({\theta }_{k},{\phi }_{k}\right)={a}_{y}^{H}\left({\phi }_{k}\right){\left[{a}_{x}\left({\theta }_{k}\right)\otimes {I}_{M}\right]}^{H}{U}_{n}{U}_{n}^{H}\left[{a}_{x}\left({\theta }_{k}\right)\otimes {I}_{M}\right]{a}_{y}\left({\phi }_{k}\right)$ (15)

$G\left({\theta }_{k}\right)={\left[{a}_{x}\left({\theta }_{k}\right)\otimes {I}_{M}\right]}^{H}{U}_{n}{U}_{n}^{H}\left[{a}_{x}\left({\theta }_{k}\right)\otimes {I}_{M}\right]$ (16)

$\mathrm{min}{a}_{y}^{H}\left({\phi }_{k}\right)G\left({\theta }_{k}\right){a}_{y}\left({\phi }_{k}\right)=\mathrm{min}Q\left({\theta }_{k},{\phi }_{k}\right)$ (17)

${a}_{y}\left(y\right)=\mathrm{exp}\left[-j2\text{π}d\left(m-1\right)\mathrm{cos}{\beta }_{k}/\lambda \right]={\left[\mathrm{exp}\left(-j2\text{π}d\mathrm{cos}{\beta }_{k}/\lambda \right)\right]}^{m-1}={a}_{y}^{y-1}=\left\{\begin{array}{l}1,\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}m=1\\ {a}_{y}\left(2\right),\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{ }\text{\hspace{0.17em}}m=2\\ {a}_{y}\left(2\right){a}_{y}\left(y-1\right),\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}m\ge 3\end{array}$ (18)

5. 改进的降维MUSIC (MRD-MUSIC)算法

$\left\{\begin{array}{l}{\mathrm{min}}_{w}{a}_{y}^{H}G\left({\theta }_{k}\right){a}_{y}\\ \text{s}\text{.t}\text{.}\text{\hspace{0.17em}}{‖{a}_{y}\left({\phi }_{k}\right)‖}^{2}=M\end{array}$ (19)

$L\left({a}_{y},{\theta }_{k}\right)={a}_{y}^{H}G\left({\theta }_{k}\right){a}_{y}-\lambda \left({a}_{y}^{H}{a}_{y}-M\right)$ (20)

$\frac{\partial }{\partial {a}_{y}}L\left({a}_{y},{\theta }_{k}\right)=2G\left({\theta }_{k}\right)\cdot {a}_{y}-2\lambda \cdot {a}_{y}=0$ (21)

$G\left({\theta }_{k}\right)\cdot {a}_{y}=\lambda \cdot {a}_{y}$ (22)

${a}_{y}^{H}\cdot G\left({\theta }_{k}\right)\cdot {a}_{y}=\lambda \cdot {a}_{y}^{H}\cdot {a}_{y}=\lambda \cdot M$ (23)

${a}_{y}={P}_{\mathrm{min}}\left[G\left({\theta }_{k}\right)\right]$ (24)

${\stackrel{^}{\theta }}_{k}=\mathrm{arg}\mathrm{min}{P}_{\mathrm{min}}^{H}\left[G\left({\theta }_{k}\right)\right]\cdot G\left({\theta }_{k}\right){P}_{\mathrm{min}}\cdot \left[G\left({\theta }_{k}\right)\right]=\mathrm{arg}\mathrm{max}\frac{1}{{P}_{\mathrm{min}}^{H}\left[G\left({\theta }_{k}\right)\right]\cdot G\left({\theta }_{k}\right){P}_{\mathrm{min}}\cdot \left[G\left({\theta }_{k}\right)\right]}$ (25)

6. 实验仿真与结果分析

$\text{RMSE}=\frac{1}{K}{\sum }_{k=1}^{K}\sqrt{\frac{1}{Num}{\sum }_{num=1}^{Num}{\left({\stackrel{^}{\theta }}_{k,num}-{\theta }_{k}\right)}^{2}+{\left({\stackrel{^}{\phi }}_{k,num}-{\phi }_{k}\right)}^{2}}$ (26)

DOA精度：令信噪比SNR范围从−10 dB到10 dB，快拍数为1000，进行500次蒙特卡罗仿真实验。将文献 [10] 去相干算法和本文算法的均方根误差曲线进行对比。对比如图5所示。

Figure 3. Estimation of pitch angle and azimuth angle

Figure 4. Comparison of success rate

Figure 5. The change curve of RMSE with SNR

Figure 6. The variation curve of RMSE with array elements

7. 结论

NOTES

*通讯作者。

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