文章快速检索 高级检索
 浙江大学学报(工学版)  2018, Vol. 52 Issue (9): 1747-1752  DOI:10.3785/j.issn.1008-973X.2018.09.015 0

### 引用本文 [复制中英文]

dx.doi.org/10.3785/j.issn.1008-973X.2018.09.015
[复制中文]
MA Heng-da, YUAN Wei-na, FU Wei. Group sparse channel estimation method based on pilot placement optimization[J]. Journal of Zhejiang University(Engineering Science), 2018, 52(9): 1747-1752.
dx.doi.org/10.3785/j.issn.1008-973X.2018.09.015
[复制英文]

### 作者简介

orcid.org/0000-0001-9333-7953.
E-mail: m499581@hotmail.com.

### 通信联系人

E-mail: wnyuan_ice@163.com
.

### 文章历史

Group sparse channel estimation method based on pilot placement optimization
MA Heng-da , YUAN Wei-na , FU Wei
School of information Science and Engineering, East China University of Science and Technology, Shanghai 200237, China
Abstract: Channel estimation using group sparsity methods were studied for sparsity of OFDM system. These methods considered both the time-selective fading and frequency-selective fading of the channel. The concept of group sparsity was introduced by the sparse representation of the channel coefficients. The non-zero components of sparse signals tended to cluster in a region. This characteristic could be used to improve the quality of the reconstruction. At the same time, the pilot pattern played an important role in channel estimation. Estimation of distribution algorithm (EDA) was used to optimize pilot pattern in sparse channel estimation. The algorithm is more robust than other methods and unlikely to trap into local minima. Both theoretical analysis and simulation results show that the scheme is more effective than the traditional estimation method. The simulation results indicate the applicability of the scheme by using different reconstruction methods and group sizes.
Key words: compressive sensing    sparse signal    channel estimation    pilot pattern optimization    orthogonal frequency division multiplexing (OFDM)

1 系统模型

 $h(t,\tau ) = \sum\limits_{p = 1}^{P(t)}\, {\left[ {{\eta _p}(t)\delta (\tau - {\tau _p}(t))\exp \left( {{\rm j}2{\rm{\pi }}{v_p}(t)t} \right)} \right]} .$ (1)

 $r(t) = \int_{ - \infty }^\infty {h(t,\tau )s(t - \tau ){\rm{d}}\tau } + z\left( t \right).$ (2)

 $r[n] = \sum\limits_{m = 0}^{M - 1} {\Big\{ {h[n,m]s[n - m]} \Big\}} + z[n].$ (3)

 $\begin{array}{l}\!\!s[n]\!\! =\! \!\displaystyle\frac{1}{{\sqrt K }}\displaystyle\sum\limits_{l = 0}^{L - 1}{\left\{ {\displaystyle\sum\limits_{k = 0}^{k - 1}{\left[ {{a_{l,k}}\exp \left( {{\rm j}2{\rm{\pi }}kn/K} \right)} \right]g[n - lN]} } \right\}} .\end{array}$ (4)

 $\begin{array}{l}{r_{l,k}} = \displaystyle\frac{1}{{\sqrt k }}\displaystyle\sum\limits_{n = - \infty }^\infty {\Big\{ {r[n]\exp \left( { - {\rm j}2{\rm{\pi }}k(n - lN)/K} \right)\gamma [n - lN]} \Big\}} .\end{array}$ (5)

 ${r_{l,k}} = {H_{l,k}}\,{a_{l,k}} + {z_{l,k}}.$ (6)

 ${H_{l,k}} = \sum\limits_{m = 0}^{K - 1}\; {\sum\limits_{i = - L/2}^{L/2 - 1} \,{\left[ {{F_{m,i}}\exp \left( { - {\rm j}2\pi \left(\frac{{km}}{K} - \frac{{li}}{L}\right)} \right)} \right]} } ,$ (7)
 ${F_{m,i}} = \sum\limits_{q = 0}^{N - 1}\, {\left\{ {{S_h}[m,i + qL]A_{\gamma ,g}^*\left(m,\frac{{i + qL}}{{{N_0}}}\right)} \right\}} .$ (8)

2 基于组稀疏压缩感知的信道估计

${\cal{J}}{\rm{ = }}\left\{ {{I_b}} \right\}_{b = 0}^{B - 1}$ 为集合 $\{ 0, 1,\cdots ,N - 1\}$ 的一种分组方式，即有

 $\cup _{b = 0}^{B - 1}{I_b} = \{ 0, 1,\cdots ,N - 1\},$
 $\displaystyle\sum\nolimits_{b = 0}^{B - 1} {|{I_b}|} = N.$

 图 1 组稀疏压缩感知(GSCS)信号模型 Fig. 1 Signal model for group sparse compressed sensing (GSCS) method

 ${{y}} = {{\varPhi x}} + {{z}}.$ (9)

 $\mathop {\arg \min }\limits_{{x}} {\left\| {{x}} \right\|_{2|\cal{J}}},\;{\rm s}.{\rm t}.\,{\left\| {{{y}} - {{\varPhi x}}} \right\|_2} \leqslant \varepsilon.$

 ${\cal{G}}: = \left[ (\lambda \Delta L,\,\kappa \Delta K)\,|\,\lambda = 0,1, \cdots ,J - 1;\kappa = 0,1, \cdots, D - 1\right] ,$

 $\begin{split}&{H_{\lambda \Delta L,\kappa \Delta K}} =\\ & \displaystyle\sum\limits_{m = 0}^{K - 1} \;{\displaystyle\sum\limits_{i = - L/2}^{L/2 - 1} \,{\left[ {{F_{m,i}}\exp \,\left( { - {\rm j}2\pi \left(\displaystyle\frac{{\kappa \Delta Km}}{D} - \displaystyle\frac{{\lambda \Delta Li}}{L}\right)} \right)} \right]} } = \\ & \displaystyle\sum\limits_{m = 0}^{D - 1} \;{\displaystyle\sum\limits_{i = - J/2}^{J/2 - 1}\, {\left[ {{F_{m,i}}\exp \,\left( { - {\rm j}2\pi \left(\displaystyle\frac{{\kappa m}}{D} - \displaystyle\frac{{\lambda i}}{J}\right)} \right)} \right]} } .\end{split}$ (12)

 ${\cal{B}_b} = \left\{ {{k_b}\Delta m, \cdots ,\left( {{k_b} + 1} \right)\Delta m} \right\} \times \left\{ {{l_b}\Delta i, \cdots ,\left( {{l_b} + 1} \right)\Delta i} \right\}.$

 $\begin{split}&{k_b} \in \{ 0, \cdots ,(K/\Delta m) - 1\} ,\\ &{l_b} \in \{ - L/(2\Delta i), \cdots ,(({N_0} - L/2)/\Delta i) - 1\}. \end{split}$

 $\cal{J}{\rm{ = }}\left\{ {{I_b}} \right\}_{b = 0}^{{B_D}{B_J} - 1},$

${I_b}$ ${\cal{B}_b}$ 一一对应.

 ${H_{\lambda \Delta L,\kappa \Delta K}} = \sum\limits_{m = 0}^{D - 1} \;{\sum\limits_{i = - J/2}^{J/2 - 1} \left({{G_{m,i}}\,{u_{m,i}}[\lambda ,\kappa ]} \right)} .$ (13)

 ${u_{m,i}}\left[ {\lambda ,\kappa } \right] = \left( {1/JD} \right)\exp \left( { - {\rm j}2{\rm{\pi }}\left( {\displaystyle\frac{{\kappa m}}{D} - \displaystyle\frac{{ i}}{J}} \right)} \right)$

 \begin{aligned}&{\left[ {{{{h}}_\Delta }} \right]_{\kappa J + \lambda ,(i + J/2)D + m}} = {H_{\lambda \Delta L,\kappa \Delta K}},\\ & {\left[ {{g}} \right]_{(i + J/2)D + m}} = {G_{m,i}},\\ & {\left[ {{{{u}}_{m,i}}} \right]_{\kappa J + \lambda }} = {u_{m,i}},\end{aligned}

 ${{{h}}_\Delta } = \sum\limits_{m = 0}^{D - 1} \;{\sum\limits_{i = - J/2}^{J/2 - 1} {\left( {{G_{m,i}}{{{u}}_{m,i}}} \right) = } } {{Ug}}.$ (18)

 ${{{h}}_{(p)}} = {{\varPhi x}} + {{{z}}_{(p)}}.$ (19)

1）通过 ${r_{l,k}}/{p_{l,k}}$ 求得导频位置处的 $H$ 估计值 ${\hat H_{l,k}}$ ，将其写成向量形式 ${{{h}}_{(p)}}$ .

2）利用现有的GSCS算法重构出 ${{x}}$ 的估计值，利用这个估计值得到系数向量g的估计值.

3）通过向量g的估计值和矩阵 ${{U}}$ 计算得到完整的信道系数值.

$t \leqslant S$ ${\left\| {{{{r}}_t}} \right\|_2} \geqslant \rho {\left\| {{Y}} \right\|_2}$

1. $\varLambda _{t - 1}^c = \cal{B}\backslash {\Lambda _{t - 1}}$ ;

2. ${\lambda _t} = \arg \mathop {\rm{max}}\limits_{\lambda \in \varLambda _{t - 1}^c} {\left\| {{\varPhi }} \right\|_{2,\lambda }}$ ;

3. ${\varLambda _t} = {\varLambda _{t - 1}} \cup {\lambda _t}$ ;

4. ${{{x}}_t} = \arg \mathop {\rm{min}}\limits_{{u}} {\left\| {{{Y}} - {{\varPhi x}}} \right\|_2},{\rm{ }}{\rm s}.{\rm t}.\,{\rm{ supp}}\left( {{x}} \right) \subset {\varLambda _t}$ ;

5. ${{{r}}_t} = {{Y}} - {{\varPhi }}{{{x}}_t}$ ;

6. $t = t + 1$ .

 ${{f}} = {\left[ {f\left( 0 \right), f(1) ,f\left( {N - 1} \right)} \right]^{\rm T}}$

 ${Q_1}:{\rm{ }}\mathop {\rm{min}}\limits_{{{f}} \in \cal{S}}\, \mu \left( {{{A}}\left( {{f}} \right)} \right),\;{\rm{ s}}{\rm{.t}}{\rm{. }}\sum\limits_{n = 0}^{N - 1} \,\,{f\left( n \right)} = P.$ (22)

 $\mu \left( {{A}} \right) = \mathop {\rm{max}}\limits_{d \in \cal{D}} \frac{1}{P}\left| {\sum\limits_{i = 1}^P\, {{\omega ^{{k_i}d}}} } \right|.$ (23)

1. 对当前总体 ${{\cal{F}}^{\left( i \right)}}$ ，根据式（15）计算 $\mu \left( {{{f}}_1^{\left( i \right)}} \right), \mu \left( {{{f}}_2^{\left( i \right)}} \right),\cdots ,\mu \left( {{{f}}_M^{\left( i \right)}} \right)$ 并升序排列;

2. 按1中排序选取总体中前T个个体 $\left\{ {{{\hat f}}_1^{\left( i \right)},\, {{\hat f}}_2^{\left( i \right)},\cdots\, , {{\hat f}}_T^{\left( i \right)}} \right\}$ ,令 ${{\hat f}}_1^{\left( i \right)} = {\widetilde{ f}}_1^{\left( i \right)}$ ,

$t = 2, 3,\cdots ,T-1 ,T$ ，计算 $\widehat {{f}}_t^{\left( i \right)}$ $\widetilde {{f}}_1^{\left( i \right)}$ 的循环卷积 ${{g}}_t^{\left( i \right)}$ ${{g}}_{t,{\rm{flip}}}^{\left( i \right)}$ ,得到最大匹配指数：

 ${N_0} = \arg \mathop {\rm{max}}\limits_{{{n}} = 0,1, \cdots ,N - 1} \left\{ {{g_t}\left( n \right),{g_{t,{\rm{flip}}}}\left( n \right)} \right\}.$

${{\hat f}}_{t,{\rm{flip}}}^{\left( i \right)}$ ${{\hat f}}_t^{\left( i \right)}$ 循环移位N0位得到 $\widetilde {{f}}_{\rm{t}}^{\left( i \right)}$ ;

3. 得到新总体 ${\tilde \cal{F}^{\left( i \right)}} = \left[ {{{\tilde f}}_1^{\left( i \right)}, {{\tilde f}}_2^{\left( i \right)}, \cdots ,{{f}}_T^{\left( i \right)}} \right]$ ;

4. 计算导频概率向量 ${{{p}}^{\left( i \right)}}$ :

 ${{ p}^{\left( i \right)}}\left( n \right) = \frac{1}{T}\sum\limits_{t = 1}^T\, {{{{\tilde f}}^{\left( i \right)}}\left( n \right)}; \; \; {n = 0,1, \cdots ,N - 1} .$

5. 保留 $\widetilde {{f}}_1^{\left( i \right)}$ ${{f}}_1^{\left( {i + 1} \right)}$ ，从分布 ${{P}^{\left( {{i}} \right)}}$ 中抽取新的个体组成总体 ${{\cal{F}}^{\left( {i + 1} \right)}} = \left[ {{{f}}_1^{\left( {i + 1} \right)}, {{f}}_2^{\left( {i + 1} \right)},\cdots ,{{f}}_M^{\left( {i + 1} \right)}} \right]$ ;

3 仿真结果与分析

 图 2 组稀疏与传统稀疏信道估计方法对比（采用G-OMP和G-BPDN重构算法） Fig. 2 Compressive channel estimation utilizing group sparsity compared to conventional compressive channel estimation (using G-OMP and G-BPDN reconstruction technique)
 图 3 基于不同优化方案的组稀疏信道估计方法对比（均采用G-OMP重构算法） Fig. 3 Comparison of group sparsity channel estimation based on different optimization schemes (using G-OMP reconstruction technique)
 图 4 基于不同优化方案的组稀疏信道估计方法对比（均采用G-BPDN重构算法） Fig. 4 Comparison of group sparsity channel estimation based on different optimization schemes (using G-BPDN reconstruction technique)
 图 5 采用不同块大小的组稀疏信道估计性能对比 Fig. 5 Performance of compressive channel estimator utilizing group sparsity for various block sizes

1）组稀疏算法复杂度取决于相应的CS算法，GSCS算法复杂度记为 $O({\rm{GSCS}})$ ，根据不同的重构算法记作不同的具体值，缩放运算复杂度为 $O(JD)$

2）计算式(11)、(12)中 ${H_{\lambda \Delta L,\;\kappa \Delta K}}$ 的复杂度，JD×JD的矩阵 ${{U}}$ 需要做(JD)2次乘法；

3）式(7)做快速傅里叶变换(fast Fourier transformation, FFT)，复杂度为 $O(LK\,{\rm{log}}\,(LK))$

4）总复杂度为

$O({\rm{GSCS}}) + O({(JD)^2}) + O(LK\,{\rm{log}}\,(LK))$ .

 $O({\rm{GSCS}}){\rm{ = }}O(M{(n{'_{{\rm{G{\text{-}}OMP}}}})^2} + {n_{{\rm{G{\text{-}}OMP}}}}{{\varPhi }}).$

4 结　语

 [1] ENGELS M. Wireless OFDM systems: How to make them work? [M]. Norwell: Springer Science & Business Media, 2002: 33–36. [2] BERTHOLD U, JONDRAL F K, BRANDES S, et al. OFDM-based overlay systems: a promising approach for enhancing spectral efficiency [Topics in radio communications][J]. IEEE Communications Magazine, 2007, 45(12): 52-58. DOI:10.1109/MCOM.2007.4395365 [3] DONOHO D L. Compressed sensing[J]. IEEE Transactions on Information Theory, 2006, 52(4): 1289-1306. DOI:10.1109/TIT.2006.871582 [4] BAJWA W U, HAUPT J, RAZ G, et al. Compressed channel sensing [C] // 2008 42nd Annual Conference on Information Sciences and Systems. New Jersey: IEEE, 2008: 5–10. [5] BAJWA W U, HAUPT J, SAYEED A M, et al. Compressed channel sensing: a new approach to estimating sparse multipath channels[J]. Proceedings of the IEEE, 2010, 98(6): 1058-1076. DOI:10.1109/JPROC.2010.2042415 [6] TAUBOCK G, HLAWATSCH F, EIWEN D, et al. Compressive estimation of doubly selective channels in multicarrier systems: leakage effects and sparsity-enhancing processing[J]. IEEE Journal of Selected Topics in Signal Processing, 2010, 4(2): 255-271. DOI:10.1109/JSTSP.2010.2042410 [7] EIWEN D, TAUBÖCK G, HLAWATSCH F, et al. Group sparsity methods for compressive channel estimation in doubly dispersive multicarrier systems [C] // 2010 IEEE 11th International Workshop on Signal Processing Advances in Wireless Communications (SPAWC). Marrakech: IEEE, 2010: 1–5. [8] TROPP J A. Greed is good: algorithmic results for sparse approximation[J]. IEEE Transactions on Information Theory, 2004, 50(10): 2231-2242. DOI:10.1109/TIT.2004.834793 [9] HE X, SONG R. Pilot pattern optimization for compressed sensing based sparse channel estimation in OFDM systems [C] // 2010 International Conference on Wireless Communications & Signal Processing (WCSP). Suzhou: IEEE, 2010: 1–5. [10] QI C, WU L. Optimized pilot placement for sparse channel estimation in OFDM systems[J]. IEEE Signal Processing Letters, 2011, 18(12): 749-752. DOI:10.1109/LSP.2011.2170834 [11] QI C, WU L. A study of deterministic pilot allocation for sparse channel estimation in OFDM systems[J]. IEEE Communications Letters, 2012, 16(5): 742-744. DOI:10.1109/LCOMM.2012.032612.112553 [12] QI C, YUE G, WU L, et al. Pilot design schemes for sparse channel estimation in OFDM systems[J]. IEEE Transactions on Vehicular Technology, 2015, 64(4): 1493-1505. DOI:10.1109/TVT.2014.2331085 [13] QI C, YUE G, WU L, et al. Pilot design for sparse channel estimation in OFDM-based cognitive radio systems[J]. IEEE Transactions on Vehicular Technology, 2014, 63(2): 982-987. DOI:10.1109/TVT.2013.2280655 [14] NAJJAR L. Pilot allocation by Genetic Algorithms for sparse channel estimation in OFDM systems [C] // 21st European Signal Processing Conference (EUSIPCO 2013). Marrakech: IEEE, 2013: 1–5. [15] HE X, SONG R, ZHU W P. Pilot allocation for sparse channel estimation in MIMO-OFDM systems[J]. IEEE Transactions on Circuits and Systems II: Express Briefs, 2013, 60(9): 612-616. DOI:10.1109/TCSII.2013.2268433 [16] LARRAÑAGA P., LOZANO J. A. Estimation of distribution algorithms: a new tool for evolutionary computation [M]. New York: Springer Science & Business Media, 2001: 58–59 [17] FLANDRIN P. Time-frequency/time-scale analysis [M]. Orlando: Academic Press, 1998: 237–240. [18] FORNASIER M. Theoretical foundations and numerical methods for sparse recovery [M]. New York: Walter de Gruyter, 2010: 108–111.