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 应用科技  2018, Vol. 45 Issue (2): 38-42,48  DOI: 10.11991/yykj.201610013 0

### 引用本文

YAO Yougen, HE Zhongqiu. Time domain channel estimation technology for FBMC[J]. Applied Science and Technology, 2018, 45(2), 38-42,48. DOI: 10.11991/yykj.201610013.

### 文章历史

FBMC时域信道估计技术

Time domain channel estimation technology for FBMC
YAO Yougen, HE Zhongqiu
College of Information and Communication Engineering, Harbin Engineering University, Harbin 150001, China
Abstract: Aiming at the problems existing in filter bank multicarrier (FBMC) system that the time domain estimation method fails to sufficiently utilize pilot frequency information and the spectrum effectiveness is not high, the original estimation expression was extended and two cases including non-sparse pilot frequency and sparse pilot frequency were investigated. In the case that all pilot frequencies are not zero, a group of optimized pilot frequencies were given. The influence of different shapes of auxiliary points on estimated performance were explored when the pilot frequency was sparse. In addition, a joint estimation method was proposed by utilizing the correlation among the noises in the neighboring symbol subcarriers. Both theoretic analysis and simulation results show that, at the same pilot overhead, the extended algorithm is superior to the original algorithm and the combined multi-symbol estimation can further reduce the estimated Cramer-Rao bound.
Key words: filter bank multicarrier    channel estimation    weighted least square estimate    pilot frequency structure    sparse    joint estimation method    pilot overhead    Cramer-Rao bound

1 系统模型

FBMC的基带输出信号s[k]的表达式为

 $s\left[ k \right] = \sum\limits_{m = 0}^{M - 1} {\sum\limits_{n \in Z} {{a_{m,n}}} } g\left[ {k - n\frac{M}{2}} \right]{{{e}}^{{{j2\pi }}mk/M}}{{{e}}^{{{j\pi }}\left( {m + n} \right)/2}}$

 $\begin{split}{{\hat a}_{m,n}} & = \sum\limits_{l = 0}^{L - 1} {\sum\limits_{k = - \infty }^\infty {\sum\limits_{p = 0}^{M - 1} {\sum\limits_{q \in Z} {{a_{p,q}}g\left[ {k - l - q\frac{M}{2}} \right]} } } } g\left[ {k - n\frac{M}{2}} \right]\times \\& {{{e}}^{{{j2\pi }}\left( {p - m} \right)k/M}} \cdot {{{e}}^{{{j\pi }}\left( {p + q - m - n} \right)/2}} \cdot {{{e}}^{ - {{j2\pi }}pl/M}}h\left[ l \right] + {\eta _{m,n}}\end{split}$

 $\begin{split}{{\hat a}_{m,0}} & = \sum\limits_{l = 0}^{L - 1} {\sum\limits_{k = - \infty }^\infty {\sum\limits_{p = 0}^{M - 1} {{a_{p,0}}} } } g\left[ {k - l} \right]g\left[ k \right]{{{e}}^{{{j2\pi }}\left( {p - m} \right)k/M}}\times \\& {{{e}}^{{{j\pi }}\left( {p - m} \right)/2}}{{{e}}^{ - {{j2\pi }}pl/M}}h\left[ l \right] + {\eta _{m,0}}\end{split}$ (1)

 ${{{r}}_{{0}}} = {{\varGamma h}} + {{{\eta }}_0}$

 $\begin{split}{\varGamma _{m,l}} & = \sum\limits_{k = - \infty }^\infty {\sum\limits_{p = 0}^{M - 1} {{a_{p,0}}} } g\left[ {k - l} \right]g\left[ k \right]{{{e}}^{{{j2\pi }}\left( {p - m} \right){k / M}}} \times \\& {{{e}}^{{{j2\pi }}{{\left( {p - m} \right)} / 2}}}{{\mathop{ e}\nolimits} ^{ - {{j2\pi }}p{l / M}}}\end{split}$

 ${{\hat{ h}}_{{{WLS}}}} = {\left( {{{{\varGamma }}^{{H}}}{{{C}}^{ - 1}}{{\varGamma }}} \right)^{ - 1}}{{{\Gamma }}^{{H}}}{{{C}}^{ - 1}}{{{r}}_0}$

 ${{C}} = \left[ {\begin{array}{*{20}{c}} {{\sigma ^2}} & {{\sigma ^2}\zeta _{0,0}^{1,0}} & \cdots & {{\sigma ^2}\zeta _{0,0}^{M - 2,0}} & {{\sigma ^2}\zeta _{0,0}^{M - 1,0}} \\ [6pt] {{\sigma ^2}\zeta _{1,0}^{0,0}} & {{\sigma ^2}} & \cdots & {{\sigma ^2}\zeta _{1,0}^{M - 2,0}} & {{\sigma ^2}\zeta _{1,0}^{M - 1,0}} \\ [6pt] \vdots & \cdots & \cdots & \cdots & \vdots \\ [6pt] {{\sigma ^2}\zeta _{M - 2,0}^{0,0}} & {{\sigma ^2}\zeta _{M - 2,0}^{1,0}} & \cdots & {{\sigma ^2}} & {{\sigma ^2}\zeta _{M - 2,0}^{M - 1,0}} \\ [6pt] {{\sigma ^2}\zeta _{M - 1,0}^{0,0}} & {{\sigma ^2}\zeta _{M - 1,0}^{1,0}} & \cdots & {{\sigma ^2}\zeta _{M - 1,0}^{M - 2,0}} & {{\sigma ^2}} \end{array}} \right]$ (2)

2 算法扩展

 $\begin{split}{{\hat a}_{m,0}} & = \sum\limits_{l = 0}^{L - 1} {\sum\limits_{k = - \infty }^\infty {\sum\limits_{p = 0}^{M - 1} {\sum\limits_{q \in \Psi } {{a_{p,q}}g\left[ {k - l - q\frac{M}{2}} \right]} g\left[ k \right]} } } \times \\& {{{e}}^{{{j2\pi }}\left( {p - m} \right)k/M}}{{{e}}^{{{j\pi }}\left( {p + q - m} \right)/2}}{{{e}}^{ - {{j2\pi }}pl/M}}h\left[ l \right] + {\eta _{m,0}}\end{split}$ (3)

 ${{{r}}_{{0}}} = \left( {{{S}} + {{U}}} \right){{h}} + {{{\eta }}_{{0}}}$ (4)

 ${{{S}}_{m,l}} = \sum\limits_{k = - \infty }^\infty {{a_{m,0}}g\left[ {k - l} \right]g\left[ k \right]{{ e}^{ - j2\pi m{l / M}}}} ,$
 $\begin{split}{{{U}}_{m,l}}& = \sum\limits_{k = - \infty }^\infty {\sum\limits_{p = 0,p \ne m}^{M - 1} {\sum\limits_{q \in \psi } {{a_{p,q}}} } } g\left[ {k - l - q\frac{M}{2}} \right]g\left[ k \right] \times \\[8pt]& {{{e}}^{j2\pi \left( {p - m} \right)k/M}}{{{e}}^{{{j\pi }}\left( {p + q - m} \right)/2}}{{{e}}^{ - {{j2\pi }}pl/M}}\end{split}$

 $\hat{ h}={\left( {{{\left( {{{S}} + {{U}}} \right)}^{{H}}}{{{C}}^{ - 1}}\left( {{{S}} + {{U}}} \right)} \right)^{ - 1}}{\left( {{{S}} + {{U}}} \right)^{{H}}}{{{C}}^{ - 1}}{{{r}}_0}$

 ${{tr}}\left\{ {{{\left[ {{{\left( {{{S}} + {{U}}} \right)}^{{H}}}{{{C}}^{ - 1}}\left( {{{S}} + {{U}}} \right)} \right]}^{ - 1}}} \right\}$

 Download: 图 1 两种导频结构示意
2.1 全导频情况

 $a = {1_{\frac{M}{4}}} \otimes b$

 ${{b}} = {\left[ {\begin{array}{*{20}{c}} 1 & { - j} & { - 1} & j \\ { - j} & { - 1} & j & 1 \\ { - 1} & j & 1 & { - j} \end{array}} \right]^{{T}}}$
2.2 稀疏导频情况

 $\left\{ {\left. p \right|p = {p_0} + i \cdot u + 1,i = 0,1, \cdots ,\left\lfloor {M/u} \right\rfloor } \right\},\;u = \left\lfloor {M/L} \right\rfloor$

3 联合估计

 $\begin{split}{{\hat a}_{m,1}} & = \sum\limits_{l = 0}^{L - 1} {\sum\limits_{k = - \infty }^\infty {\sum\limits_{p = 0}^{M - 1} {\sum\limits_{q = 0}^2 {{a_{p,q}}} } } } g\left[ {k - l - q\frac{M}{2}} \right]g\left[ {k - \frac{M}{2}} \right] \times \\& {{{e}}^{{{j2\pi }}\left( {p - m} \right)k/M}}{{\mathop{ e}\nolimits} ^{{{j\pi }}\left( {p + q - m - 1} \right)/2}}{{{e}}^{ - {{j2\pi }}pl/M}}h\left[ l \right] + {\eta _{m,1}}\end{split}$ (5)

 ${{{r}}_0} = \left( {{{{S}}_0} + {{{U}}_0}} \right){{\hat{ h}}_0} + {{{\eta }}_0}$
 ${{{r}}_1} = \left( {{{{S}}_1} + {{{U}}_1}} \right){{\hat{ h}}_1} + {{{\eta }}_1}$

 ${\hat{ h}} = {{\left( {{{{\hat{ h}}}_0} + {{{\hat{ h}}}_1}} \right)} / 2}$

${{r}} = \left[ {\begin{array}{*{20}{c}} {{{{r}}_0}} \\ {{{{r}}_1}} \end{array}} \right]$ ${{A}} = \left[ {\begin{array}{*{20}{c}} {{{{A}}_0}} \\ {{{{A}}_1}} \end{array}} \right] = \left[ {\begin{array}{*{20}{c}} {{{{S}}_0} + {{{U}}_0}} \\ {{{{S}}_1} + {{{U}}_1}} \end{array}} \right]$ ${{\eta}} = \left[ {\begin{array}{*{20}{c}} {{{{\eta }}_0}} \\ {{{{\eta }}_1}} \end{array}} \right]$ ，建立新方程组

 ${{r}} = {{Ah}} + {{\eta }}$

 \begin{aligned}& {{Cov}}\left[ {{\eta _{{m_1},1}},{\eta _{{m_2},1}}} \right] = {{E}}\left[ {{\eta _{{m_1},1}}\eta _{{m_2},1}^ * } \right] - {{E}}\left[ {{\eta _{{m_1},1}}} \right]{{E}}\left[ {\eta _{{m_2},1}^ * } \right]{{ = }}\\& \quad { E}\left[ {\left( {\left( { - {{j}}} \right)\sum\limits_{k = - \infty }^{ + \infty } {\eta \left[ k \right]g\left[ {k - \frac{M}{2}} \right]} {{{e}}^{ - {{j2\pi }}{m_1}k/M}}{{{e}}^{ - {{j}}{m_1}{{\pi }}/2}}} \right)} \right. \times \\& \quad \left. {\left( {{{j}}\sum\limits_{k = - \infty }^{ + \infty } {{\eta ^ * }\left[ k \right]g\left[ {k - \frac{M}{2}} \right]{{{e}}^{{{j2\pi }}{m_2}k/M}}{{{e}}^{{{j\pi }}{m_2}/2}}} } \right)} \right] = \\& \quad {\sigma ^2}\sum\limits_{k = - \infty }^{ + \infty } {g\left[ {k - \frac{M}{2}} \right]} g\left[ {k - \frac{M}{2}} \right]{{{e}}^{{{j2\pi }}\left( {{m_2} - {m_1}} \right)k/M}}{{{e}}^{{{j}}\left( {{m_2} - {m_1}} \right){{\pi }}/2}} = \\& \quad {\sigma ^2}\zeta _{{m_{1,}}1}^{{m_2},1}\;{{ e}^{{{j\pi }}\left( {{m_2} - {m_1}} \right)}}{\sigma ^2}\zeta _{{m_1},0}^{{m_2},0}\end{aligned}

 $\begin{split}& {{Cov}}\left[ {{\eta _{{m_1},1}},{\eta _{{m_2},0}}} \right] = - {{j}}{\sigma ^2}\sum\limits_{k = - \infty }^\infty {g\left[ k \right]} g\left[ {k - \frac{M}{2}} \right] \times\\& \quad \quad {{{e}}^{{{j2\pi }}\left( {{m_2} - {m_1}} \right)k/M}}{{{e}}^{{{j}}\left( {{m_2} - {m_1}} \right){{\pi }}/2}} = {\sigma ^2}\zeta _{{m_1},1}^{{m_2},0}\end{split}$
 $\begin{split}& {{Cov}}\left[ {{\eta _{{m_1},0}},{\eta _{{m_2},1}}} \right] = { j}{\sigma ^2}\sum\limits_{k = - \infty }^{ + \infty } {g\left[ k \right]} g\left[ {k - \frac{M}{2}} \right] \times \\& \quad \quad {{{e}}^{{{j2\pi }}\left( {{m_2} - {m_1}} \right)k/M}}{{{e}}^{{{j}}\left( {{m_2} - {m_1}} \right){{\pi /2}}}} = {\sigma ^2}\zeta _{{m_1},0}^{{m_2},1}\end{split}$

 ${ V}=\left[ {\begin{array}{*{20}{c}}{{\sigma ^2}} & {{\sigma ^2}\zeta _{0,0}^{1,0}} & \cdots &{{\sigma ^2}\zeta _{0,0}^{M - 2,0}}& {{\sigma ^2}\zeta _{0,0}^{M - 1,0}}& {{\sigma ^2}\zeta _{0,0}^{0,1}} & {{\sigma ^2}\zeta _{0,0}^{1,1}} & \cdots & {{\sigma ^2}\zeta _{0,0}^{M - 2,1}}& {{\sigma ^2}\zeta _{0,0}^{M - 1,1}}\\[5pt]{{\sigma ^2}\zeta _{1,0}^{0,0}}& {{\sigma ^2}}& \cdots & {{\sigma ^2}\zeta _{1,0}^{M - 2,0}}& {{\sigma ^2}\zeta _{1,0}^{M - 1,0}}& {{\sigma ^2}\zeta _{1,0}^{0,1}}& {{\sigma ^2}\zeta _{1,0}^{1,1}}& \cdots & {{\sigma ^2}\zeta _{1,0}^{M - 2,1}}& {{\sigma ^2}\zeta _{1,0}^{M - 1,1}}\\[5pt] \vdots & \cdots & \cdots & \cdots & \cdots & \cdots & \cdots & \cdots & \cdots & \vdots \\[5pt] {{\sigma ^2}\zeta _{M - 2,0}^{0,0}}&{{\sigma ^2}\zeta _{M - 2,0}^{1,0}}& \cdots & {{\sigma ^2}}& {{\sigma ^2}\zeta _{M - 2,0}^{M - 1,0}}& {{\sigma ^2}\zeta _{M - 2,0}^{0,1}}& {{\sigma ^2}\zeta _{M - 2,0}^{1,1}}& \cdots & {{\sigma ^2}\zeta _{M - 2,0}^{M - 2,1}}& {{\sigma ^2}\zeta _{M - 2,0}^{M - 1,1}}\\[5pt]{{\sigma ^2}\zeta _{M - 1,0}^{0,0}}&{{\sigma ^2}\zeta _{M - 1,0}^{1,0}}& \cdots & {{\sigma ^2}\zeta _{M - 1,0}^{M - 2,0}}& {{\sigma ^2}}& {{\sigma ^2}\zeta _{M - 1,0}^{0,1}}& {{\sigma ^2}\zeta _{M - 1,0}^{1,1}}& \cdots & {{\sigma ^2}\zeta _{M - 1,0}^{M - 2,1}}& {{\sigma ^2}\zeta _{M - 1,0}^{M - 1,1}}\\[5pt]{{\sigma ^2}\zeta _{0,1}^{0,0}}& {{\sigma ^2}\zeta _{0,1}^{1,0}} & \cdots & {{\sigma ^2}\zeta _{0,1}^{M - 2,0}}& {{\sigma ^2}_{0,1}^{M - 1,0}}& {{\sigma ^2}} & {{\sigma ^2}\zeta _{0,1}^{1,1}}& \cdots & {{\sigma ^2}\zeta _{0,1}^{M - 2,1}}& {{\sigma ^2}\zeta _{0,1}^{M - 1,1}}\\[5pt]{{\sigma ^2}\zeta _{1,1}^{0,0}}& {{\sigma ^2}\zeta _{1,1}^{1,0}} & \cdots & {{\sigma ^2}\zeta _{1,1}^{M - 2,0}}& {{\sigma ^2}\zeta _{1,1}^{M - 1,0}}& {{\sigma ^2}\zeta _{1,1}^{0,1}}& {{\sigma ^2}}& \cdots & {{\sigma ^2}\zeta _{1,1}^{M - 2,1}}& {{\sigma ^2}\zeta _{1,1}^{M - 1,1}}\\[5pt] \vdots & \cdots & \cdots & \cdots & \cdots & \cdots & \cdots & \cdots & \cdots & \vdots \\[5pt] {{\sigma ^2}\zeta _{M - 2,1}^{0,0}}& {{\sigma ^2}\zeta _{M - 2,1}^{1,0}}& \cdots & {{\sigma ^2}\zeta _{M - 2,1}^{M - 2,0}}& {{\sigma ^2}\zeta _{M - 2,1}^{M - 1,0}}& {{\sigma ^2}\zeta _{M - 2,1}^{0,1}}& {{\sigma ^2}\zeta _{M - 2,1}^{1,1}}& \cdots &{{\sigma ^2}}& {{\sigma ^2}\zeta _{M - 2,1}^{M - 1,1}}\\[5pt]{{\sigma ^2}\zeta _{M - 1,1}^{0,0}} & {{\sigma ^2}\zeta _{M - 1,1}^{1,0}}& \cdots & {{\sigma ^2}\zeta _{M - 1,1}^{M - 2,0}}& {{\sigma ^2}\zeta _{M - 1,1}^{M - 1,0}}& {{\sigma ^2}\zeta _{M - 1,1}^{0,1}}& {{\sigma ^2}\zeta _{M - 1,1}^{1,1}}& \cdots & {{\sigma ^2}\zeta _{M - 1,1}^{M - 2,1}}& {{\sigma ^2}}\end{array}} \right]$ (6)

 ${{V}} = \left[ {\begin{array}{*{20}{c}} {{{{V}}_1}} & {{{{V}}_2}} \\ {{{{V}}_3}} & {{{{V}}_4}} \end{array}} \right]$ (7)

 ${{{V}}_4} = {{{V}}_1}^{{T}}, \; {{{V}}_3} = {{{V}}_2}^{{H}}$

${{V}}$ 矩阵可简化为

 ${{V}} = \left[ {\begin{array}{*{20}{c}} {{{{V}}_1}} & {{{{V}}_2}} \\ {{{{V}}_2}^{{H}}} & {{{{V}}_1}^{{T}}} \end{array}} \right]$

 ${\hat{ h = }}{\left( {{{{A}}^{{H}}}{{{V}}^{{{ - 1}}}}{{A}}} \right)^{{{ - 1}}}}{{{A}}^{{H}}}{{{V}}^{{{ - 1}}}}{{r}}$

 ${{CRB}} = {{tr}}\left[ {{{\left( {{{{A}}^{{H}}}{{{V}}^{ - 1}}{{A}}} \right)}^{ - 1}}} \right]$ (8)

4 仿真结果

4.1 扩展后时域估计

 Download: 图 2 非稀疏结构不同方法估计效果

 Download: 图 3 仿真中用到的导频形状

 Download: 图 4 稀疏结构不同方法估计效果
4.2 联合估计

 Download: 图 5 导频非稀疏情况各算法性能
 Download: 图 6 导频稀疏情况下各算法性能

 ${{{V}}_2} = {\sigma ^2}\zeta _{0,0}^{0,1}{{I}} , \; {{{V}}_3} = - {{{V}}_2}$

 ${{V}} = \left[ {\begin{array}{*{20}{c}} {{\sigma ^2}{{I}}} & {{\sigma ^2}\zeta _{0,0}^{0,1}{{I}}} \\ [8pt] { - {\sigma ^2}\zeta _{0,0}^{0,1}{{I}}} & {{\sigma ^2}{{I}}} \end{array}} \right]$

 ${{{V}}^{{{ - }}1}} = \frac{1}{{{\sigma ^2}}} \cdot \left[ {\begin{array}{*{20}{c}} {\displaystyle\frac{1}{{1 - {\alpha ^2}}}{{I}}} & {\displaystyle\frac{{ - j\alpha }}{{1 - {\alpha ^2}}}{{I}}} \\ [8pt] {\displaystyle\frac{{j\alpha }}{{1 - {\alpha ^2}}}{{I}}} & {\displaystyle\frac{1}{{1 - {\alpha ^2}}}{{I}}} \end{array}} \right]$

 $\begin{split}& {\sigma ^2} \cdot {{tr}}\left[ {\left( {\frac{1}{{1 - {\alpha ^2}}} \cdot \left( {{{A}}_0^{{H}}{{{A}}_0}{{ + A}}_1^{{H}}{{{A}}_1}} \right)} \right.} \right. + \\& \frac{{{{j}}\alpha }}{{1 - {\alpha ^2}}}\left. {{{\left. {\left( {{{A}}_1^{{H}}{{{A}}_0} - {{A}}_0^{{H}}{{{A}}_1}} \right)} \right)}^{ - 1}}} \right]\end{split}$

5 结论

1) 本文分析了原有时域估计方式的缺陷，为充分利用导频数据，对估计方程进行了扩展，并针对导频稀疏和非稀疏两种情况进行了研究。

2) 通过限制导频数值范围和导频结构，分别求解出稀疏和非稀疏情况下的一组估计误差最小的导频。仿真结果表明，使用本文导频构造的观测矩阵进行信道估计，效果优于原有算法。

3) 针对能同时获得相邻两列估计结果的情况进行了研究，将两个估计方程联立，推导出加权矩阵的表达式并给出了联合估计的克拉美罗界。

4) 对联合估计的理论进行仿真验证，结果表明联合估计效果优于单独一列的估计结果，也优于两列结果求平均。此外，性能提升结果与导频相关，部分情况下可以用求平均来代替联合估计。联合估计方式的缺点是频谱效率降低且计算复杂度增大。

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