﻿ 基于改进BOMP算法的水声信道估计
 舰船科学技术  2017, Vol. 39 Issue (8): 156-159 PDF

Channel estimation of UWA based on improved BOMP algorithm
ZHU Qin, WANG Biao
School of Electronic and Information, Jiangsu University of Science and Technology, Zhenjiang 212003, China
Abstract: Recently, the underwater acoustic channel estimation is mainly based on the sparse model. Underwater acoustic medium inhomogeneity etc make voice spread in the form of cluster, which result the underwater acoustic channel show as block structure sparse features. In OFDM communication system, based on the block structure sparse characteristics. This article proposed to use the improved BOMP algorithm to estimate the underwater acoustic channel. At a time, the BOMP algorithm filtrate a maximum relative block, but the improved algorithm can select t non-zero block, which reduce the algorithm reconstruction time t Times. The simulation results show that the improved BOMP algorithm ber and reconstruction time are superior to the traditional LS, the OMP algorithm based on compression perception; without reducing BOMP algorithm reconstruction precision, reduce reconstruction time t Times.
Key words: block structure sparse     OFDM     BOMP     channel estimation
0 引　言

 图 1 稀疏信道模型 Fig. 1 Sparse channel model

 图 2 块稀疏信道模型 Fig. 2 Block sparse channel model
1 压缩感知和块稀疏信号 1.1 压缩感知理论

CS理论通过测量矩阵将高维信号映射到低维空间，通过求解优化问题，高概率的重构出稀疏信号[8]。CS基本原理：设x是一维实值离散信号，长度为N，稀疏度为Kx中最多有K个非零值），且K<<N。对信号x进行线性测量，得到测量值yy是通过计算x和一组采样向量 $\{ {\phi _i}\} _{i = 1}^M$ 的内积得到：

 $y = {\varPhi} x,$ (1)

 $(1 - \varepsilon )\left\| x \right\|_2^2 \leqslant \left\| {{\varPhi} x} \right\|_2^2 \leqslant (1 + \varepsilon )\left\| x \right\|_2^3\text{。}$ (2)

1.2 块稀疏信号

 $\begin{array}{l}x = {[{x_1},...{x_m},{x_{m + 1}},...{x_{2m}},...{x_{N - m + 1}},...{x_N}]^{\rm{T}}},\\x[1] = {[x1,...,{x_m}]^{\rm{T}}},\\x[2] = {[{x_{m + 1}},...,{x_{2m}}]^{\rm{T}}},\\\;\; \vdots \\x[n] = {[{x_{N - m + 1}},...,{x_N}]^{\rm{T}}},\end{array}$ (3)

 ${\left\| x \right\|_{2,0}} = \sum\limits_{l = 1}^k {I({{\left\| {x[l]} \right\|}_{2,0}} > 0)},$ (4)

 $I({\left\| {x[l]} \right\|_2} > 0) = \left\{ \begin{array}{l}1, \;\;\;\;\;\;{\left\| {x[l]} \right\|_2} > 0,\\0, \;\;\;\;\;\;{\left\| {x[l]} \right\|_2} = 0\text{。}\end{array} \right.$ (5)

${\left\| x \right\|_{2,0}} \leqslant K$ ，则x称为K稀疏信号。当m=1时，块稀疏信号就退化为标准意义上的稀疏信号[12]。与式（3）类似，将测量矩阵Φ表示为n×m列块Φ[l]的连接：

 $\begin{array}{l}{\varPhi} = [{\phi _1},...{\phi _m},{\phi _{m + 1}},...{\phi _{2m}},...{\phi _{N - m + 1}},...{\phi _N}],\\\varPhi [1] = [{\phi _1},...{\phi _m}],\\\varPhi [2] = [{\phi _{m + 1}},...{\phi _{2m}}],\\\;\;\; \vdots \\\varPhi [n] = [{\phi _{N - m + 1}},...{\phi _N}]\text{。}\end{array}$ (6)
2 OFDM系统模型

 $h(t,\tau ) = \sum\limits_{n = 0}^{N - 1} {{h_n}(t)\delta (\tau - {\tau _n}(t))},$ (7)

 $h(n) = \sum\limits_{l = 0}^L {{h_l}\delta (n - l)},$ (8)

 $h = {[\underbrace {{h_1},...{h_d}}_{{h^T}[1]},\underbrace {{h_{d + 1}},...{h_{2d}}}_{{h^T}[2]},...,\underbrace {{h_{L - d + 1}},...{h_L}}_{{h^T}[c]}]^{\rm{T}}}\text{。}$ (9)

 $y = {X}H + n = X{W}h + n,$ (10)

 ${W} = \frac{1}{{\sqrt N }}\left[ \begin{array}{l}\;\;\;\;{w^{00}}\;\;\;\; \cdots \;\;\;\;{w^{(L - 1)0}}\;\\\;\;\;\;\; \vdots \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; \vdots \;\\{w^{0(N - 1)}}\;\;\; \cdots \;\;\;{w^{(L - 1(N - 1)}}\end{array} \right]\text{。}$ (11)

SP×N的选择矩阵，利用SN个子载波中选出P个导频所在的位置，从N×N单位矩阵中选择与导频位置对应的P行生成S矩阵。则接收的导频信号为：

 ${y_P} = {{X}_P}{{W}_P}h + {{n}_p}\text{。}$ (12)

3 改进BOMP算法的信道估计实现

1）首先，进行初始化：设置迭代次数l=0，残差r0=y，索引集 ${\varOmega _0} = \varphi$

2）迭代步骤，第l次迭代（ $l = 1,2,......$ ）；

①选择出t个与迭代余量最匹配的块的索引： ${i_l} = \mathop {\max (}\limits_i {\left\| {{{\varPhi} ^*}[i]{r_{l - 1}}} \right\|_2},t)$

②增大索引集 ${\varOmega _l} = {\varOmega _{l - 1}} \cup \{ {i_l}\};$

③利用LS算法得到近似的估计 ${\hat h_l}$ ，在索引集之内的 ${\hat h_{l|{\varOmega _l}}} = \arg \;\mathop {\min }\limits_{{h_l} \in C} \left\| {y - {{\varPhi} _p}h} \right\| = {\varPhi} _p^ + y$ ，在索引集Ωl之外的都是零，其中Φ+Φ的伪逆矩阵；

④计算观测向量的新近似 ${y_l} = {\varPhi} {\hat h_l} = {{\varPhi} _p}{\hat h_{l|{\varOmega _l}}}$ ，更新残差 ${r_l} = y - {y_l}$

⑤判断是否满足迭代停止条件，满足则停止，r=rl，输出 $\hat h$

4 仿真和结果分析

 $MSE = \frac{{E\left[ {\sum\limits_k {{{\left| {H(k) - \hat H(k)} \right|}^2}} } \right]}}{{E\left[ {\sum\limits_k {{{\left| {H(k)} \right|}^2}} } \right]}}\text{。}$ (16)

 图 3 LS，OMP，BOMP，改进BOMP算法BER比较 Fig. 3 LS，OMP，BOMP，improved BOMP algorithms BER comparison

 图 4 LS，OMP，BOMP，改进BOMP算法运算时间比较 Fig. 4 LS，OMP，BOMP，improved BOMP algorithms operate time comparison

5 结　语

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