﻿ 改进间接滤波噪声抵消算法在VLF中的应用
 舰船科学技术  2021, Vol. 43 Issue (7): 148-152    DOI: 10.3404/j.issn.1672-7649.2021.07.030 PDF

Application of improved indirect filter noise cancellation
JIA Shu-yang, JIANG Yu-zhong, NIU Zheng, LIU Gang
College of Electronic Engineering, Naval University of Engineering, Wuhan 430033, China
Abstract: Generally speaking, adaptive filtering algorithm is widely used in signal communication to reduce noise. Its main function is to recover useful signals directly from noisy signals, so as to improve the SNR gain. However, for the signal with very low SNR, the filter coefficient deviation is often large because the expected signal is weak, and this method is often unable to recover the ideal noise reduction signal. In order to solve the above problems, the original indirect filtering algorithm is adopted and improved, and the Hermite quadratic form is used to reduce the calculation amount. A new indirect filtering algorithm is proposed based on the wavelet transform algorithm. A vertical dual channel antenna receiving model is built, and the performance of the algorithm is evaluated with the signal to noise ratio gain as the main index.
Key words: improved adaptive filter noise reduction algorithm     Hermite quadratic form     wavelet transform     vertical dual channel antenna structure
0 引　言

1 信号与天线模型 1.1 天线模型

 图 1 数据传输硬件系统结构图 Fig. 1 Structure diagram of data transmission hardware system
1.2 信号建模

 $x(i) = e(i) + n(i){\text。}$ (1)

2 干扰抵消算法 2.1 间接滤波算法

 图 2 间接自适应滤波器原理图 Fig. 2 Schematic diagram of indirect adaptive filter

 ${{y}}(i) = {{x}}(i) + {{n}}(i){\text{。}}$ (2)

 $\hat n(i)= {{{w}}^{\rm H}}{{y}}(i) = {{{w}}^{\rm H}}{{x}}(i) + {{{w}}^{\rm H}}{{n}}(i){\text，}$ (3)

 $\begin{split}\hat x(i) =& y(i) - \hat n(i) = y(i) - {{{w}}^{\rm H}}{{y}}(i) =\\& \left({{{I}}_1} - {{{w}}^{\rm H}}\right){{x}}(i) + \left({{{I}}_1} - {{{w}}^{\rm H}}\right){{n}}(i)\text{。}\end{split}$ (4)

 $\begin{split}n(i) = &x(i) - \hat x(i) =\\ &{{{I}}_1}{{x}}(i) - \left({{{I}}_1} - {{{w}}^{\rm H}}\right){{x}}(i) - \left({{{I}}_1} - {{{w}}^{\rm H}}\right){{n}}(i) = \\ &{{{w}}^{\rm H}}{{x}}(i) - \left({{{I}}_1} - {{{w}}^{\rm H}}\right){{n}}(i)\text{。}\end{split}$ (5)

$w(n)$ 采用最陡下降法更新，公式为 $w(i + 1) = w(i) -$ $\mu \nabla J$ ，其中μ是一个影响因数，一般取 $0 < \mu < \frac{1}{{N{P_y}}}$ [6] ${P_y}$ 为输入信号的功率；根据Widrow-Hoff[7]提出的最小均方算法（LMS），为使估计期望信号中残留噪声最小，代价函数 $J$ 为噪声的最小均方误差 $J = \min \left\{ E\left[{n^2}(n)\right]\right\}$

 $\nabla J = \frac{{\partial J}}{{\partial {{w}}}}{\rm{ = }}\frac{{\partial [{n^2}(i)]}}{{\partial {{w}}}}{\rm{ = 2}}\left( {{{{w}}^{\rm H}}{{{R}}_{{{xx}}}} - {{{R}}_{{{nn}}}}{{{I}}_1} + {{{w}}^{\rm H}}{{{R}}_{{{nn}}}}} \right){\rm{ = }}0$ (6)

 ${{w}} = ({{{R}}_{{{xx}}}} + {{{R}}_{{{nn}}}}{)^{ - 1}}{{{R}}_{{{nn}}}}{{{I}}_1} = {{R}}_{{{yy}}}^{ - 1}{{{R}}_{{{nn}}}}{{{I}}_1}\text{。}$ (7)
2.2 改进的间接滤波算法

 图 3 改进的间接滤波算法原理图 Fig. 3 Schematic diagram of improved indirect filtering algorithm
2.2.1 厄米特二次型分解的间接滤波算法

 $\begin{split} {{w = U}}{{{w}}_{{0}}} =& [{u_1},{u_2} \cdots \cdots {u_{{r_x}}}] \cdot [{w_1},{w_2} \cdots \cdots {w_{{r_x}}}] +\\ &[{u_{{r_x} + 1}},{u_{{r_x} + 2}} \ldots \cdots {u_{rn}}] \cdot [{w_{{r_x} + 1}},{w_{{r_x} + 2}} \cdots \cdots {w_{{r_n}}}]= \\ &{{{U}}_{{r_x}}}{{{w}}_{0{r_x}}}{{ + }}{{{U}}_{{r_n} - {r_x}}}{{{w}}_{0{r_n} - {r_x}}}\text{。} \end{split}$ (8)

 $\begin{split} J =& \min \left\{ E[n_{{r_n} - {r_x}}^2(n)]\right\}= \\ & {\rm min}\Big\{ ({{I}}_{{1}}^{} - {{U}}_{{r_n} - {r_x}}^{}{{w}}_{0{r_n} - {r_x}}^{}{)^{\rm H}}{{{R}}_{{{vv}}}}({{{I}}_{{1}}} - {{{U}}_{{r_n} - {r_x}}}{{{w}}_{0{r_n} - {r_x}}}{\rm{\Big\} }}=\\ &{\rm min}\Big\{ {{I}}_{{1}}^{\rm H}{{{R}}_{{{vv}}}}{{{I}}_{{1}}} \!-\! {{I}}_{{1}}^{\rm H}{{{R}}_{{{vv}}}}{{{U}}_{{r_n} \!-\! {r_x}}}{{{w}}_{0{r_n} \!-\! {r_x}}} \!-\! {{w}}_{0{r_n} \!-\! {r_x}}^{\rm H}{{U}}_{{r_n} \!-\! {r_x}}^{\rm H}{{{R}}_{{{vv}}}}{{{I}}_{{1}}}\!+ \\ & {{w}}_{0{r_n} - {r_x}}^{\rm H}{{U}}_{{r_n} - {r_x}}^{\rm H}{{{R}}_{{{vv}}}}{{{U}}_{{r_n} - {r_x}}}{{{w}}_{0{r_n} - {r_x}}}{\rm{\Big\} }}=\\ &{\rm min}\Bigg\{ {{I}}_{{1}}^{\rm H}{{{R}}_{{{vv}}}}{{{I}}_{{1}}} - {{I}}_{{1}}^{}{{{R}}_{{{vv}}}}\left( {{{{U}}_{{r_n} - {r_x}}}{{{w}}_{0{r_n} - {r_x}}}+ {{w}}_{0{r_n} - {r_x}}^{\rm H}{{U}}_{{r_n} - {r_x}}^{\rm H}} \right)+\\ & {{w}}_{0{r_n} - {r_x}}^{\rm H}{{U}}_{{r_n} - {r_x}}^{\rm H}{{{R}}_{{{vv}}}}{{{U}}_{{r_n} - {r_x}}}{{{w}}_{0{r_n} - {r_x}}}{\rm{\Bigg\} }}\text{。} \end{split}$ (9)

 $\frac{{\partial J}}{{\partial {{w}}}} \!=\! - \!\left( {{{U}}_{{r_n} \!-\! {r_x}}^{} \!+\! {{U}}_{{r_n} \!-\! {r_x}}^{\rm H}} \right){{{R}}_{{{vv}}}}{{{I}}_{{1}}}\!{\rm{ + }}\!2{{U}}_{{r_n} \!-\! {r_x}}^{\rm H}{{{R}}_{{{vv}}}}{{{U}}_{{r_n} \!-\! {r_x}}}{{{w}}_{0{r_n} \!-\! {r_x}}}{\rm{ = }}0,$ (10)

 ${{{w}}_{0{r_n} - {r_x}}}= \frac{1}{2}{\left( {{{U}}_{{r_n} - {r_x}}^{\rm H}{{{R}}_{{{vv}}}}{{{U}}_{{r_n} - {r_x}}}} \right)^{ - 1}}\left( {{{U}}_{{r_n} - {r_x}}^{} + {{U}}_{{r_n} - {r_x}}^{\rm H}} \right){{{R}}_{{{vv}}}}{{{I}}_1},$ (11)

 $\begin{split} {{{w}}_{{r_n} - {r_x}}} =& {{{I}}_1} - {{{U}}_{{r_n} - {r_x}}}{{{w}}_{0{r_n} - {r_x}}}=\\ & {{{I}}_1} - {{{U}}_{{r_n} - {r_x}}} \cdot \frac{1}{2}{\left( {{{U}}_{{r_n} - {r_x}}^{\rm H}{{{R}}_{{{vv}}}}{{{U}}_{{r_n} - {r_x}}}} \right)^{ - 1}}\cdot\\ &\left( {{{U}}_{{r_n} - {r_x}}^{} + {{U}}_{{r_n} - {r_x}}^{\rm H}} \right){{{R}}_{{{vv}}}}{{{I}}_1}=\\ & {{{I}}_1} \!-\! \frac{1}{2}{{{U}}_{{r_n} \!-\! {r_x}}}{\rm{D}}_{2,{r_n} \!-\! {r_x}}^{ - 1}\left( {{{U}}_{{r_n} \!-\! {r_x}}^{} \!+\! {{U}}_{{r_n} \!-\! {r_x}}^{\rm H}} \right){{{R}}_{{{vv}}}}{{{I}}_1}\text{。} \end{split}$ (12)

 ${{D}}_2^{}{{ = D}}_{2,{r_x}}^{} + {{D}}_{2,{r_n} - {r_x}}^{}\text{。}$ (13)
2.2.2 小波变换分解

 图 4 Db10小波基函数与频谱 Fig. 4 Db10 wavelet basis function and spectrum
 ${\varphi _{a,b}}\left( i \right) = \frac{1}{{\sqrt {\left| a \right|} }}\varphi \left( {\frac{{i - b}}{a}} \right),\begin{array}{*{20}{c}} {}&{a,b \in R;a \ne 0} {\text，} \end{array}$ (14)

1）期望信号 ${{\hat x(i)}}$ n层小波分解；

2）噪声信号 ${{n}}(i)$ n层小波分解；

3）找出不同层分解信号并重构期望信号 ${{\hat x}}'{{(i)}}$

 $\begin{split} SNR({{W}})=&\frac{E\left\{{\left|{x}_{r}\left(i\right)\right|}^{2}\right\}}{E\left\{{\left|{n}_{r}\left(i\right)\right|}^{2}\right\}}=\frac{{{{W}}}^{\rm H}{{{R}}}_{xx}{{W}}}{{{{W}}}^{\rm H}{{{R}}}_{vv}{{W}}}=\\ &\frac{{{{w}}}_{0,{r}_{n}-{r}_{x}}^{\rm H}{{{U}}}_{{r}_{n}-{r}_{x}}^{\rm H}{{{R}}}_{xx}{{{U}}}_{{r}_{n}-{r}_{x}}{{{W}}}_{0,{r}_{n}-{r}_{x}}}{{{{W}}}_{0,{r}_{n}-{r}_{x}}^{\rm H}{{{U}}}_{{r}_{n}-{r}_{x}}^{\rm H}{{{R}}}_{vv}{{{U}}}_{{r}_{n}-{r}_{x}}{{{W}}}_{0,{r}_{n}-{r}_{x}}}=\\ &\frac{{\displaystyle \sum _{i=1}^{{r}_{n}-{r}_{x}}{\left|{w}_{r+i}\right|}^{2}{\lambda }_{r+i}}}{{\displaystyle \sum _{i=1}^{{r}_{n}-{r}_{x}}{\left|{w}_{r+i}\right|}^{2}}}{\text。}\end{split}$ (15)
3 仿真及实测结果验证

3.1 仿真

 图 5 仿真信号功率谱 Fig. 5 Power spectrum of simulation signa

3.2 实验

 图 6 实验信号功率谱 Fig. 6 Power spectrum of experimental signal

4 结　语

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