﻿ 基于FRFT的频域自适应匹配滤波器检测方法
 舰船科学技术  2024, Vol. 46 Issue (5): 69-73    DOI: 10.3404/j.issn.1672-7649.2024.05.013 PDF

Frequency domain adaptive matching filter detection method based on FRFT
LIANG Qin, HU Peng-peng, CHEN Zhou, LI Jie
Sichuan Jiuzhou Electric Group Co., Ltd., Chengdu 610000, China
Abstract: To solve the problem of the degradation of the detection performance of the frequency domain adaptive matching filter in shallow sea reverberation background, a frequency domain adaptive matching filter detection method based on fractional Fourier transform (FRFT) is proposed. The method uses template matching technology and sliding window to perform the optimal fractional Fourier transform on the received signal. Then, the FRFT domain map obtained in this process is matched with the FRFT domain map of the reference signal with the optimal order Fourier transform, and the sum of the squares of deviation is used as the index to evaluate the similarity, that is, the position of the minimum sum of the squares of deviation is filtered, and the optimal order fractional inverse Fourier transform is carried out on the filtered signal, so as to realize the target detection under the reverberation background. Simulation results show that the proposed algorithm can significantly improve the detection performance of the frequency-domain adaptive matching filter when the signal-mixing ratio is -15dB.
Key words: reverberation suppression     fractional Fourier transform     frequency domain adaptive matching filter     signal detection
0 引　言

1 分数阶傅里叶变换

$p$阶傅里叶变换可以看做信号在时频域上坐标轴绕原点逆时针旋转${{p{\text{π}} } \mathord{\left/ {\vphantom {{p{\text{π}}} 2}} \right. } 2}$角度后构成的分数阶傅里叶域上的表示，如图1所示。

 图 1 分数阶傅里叶变换示意图 Fig. 1 Schematic diagram of the fractional Fourier transform

 ${X_p}(u) = {F_p}[x(t)] = \int_{ - \infty }^\infty {x(t){K_p}} (t,u){\mathrm{d}}t。$ (1)

 $\begin{split}& {K_p}(t,u) =\\ &\left\{ \begin{array}{l} {\sqrt {(1 - j\cot \alpha )} {e^{j\text{π} [({t^2} + {u^2})\cot \alpha - 2tu\csc \alpha ]}}}，{\alpha \ne n\text{π} } ，\\ {\delta (t - u)}，{\alpha = 2n\text{π} }，\\ {\delta (t + u)}，{\alpha = (2n \pm 1)\text{π} }。\end{array} \right. \end{split}$ (2)

$x(t)$的FRFT变换为：

 $\begin{split} &{X_p}(u) =\\ & \left\{ {\begin{array}{l} {\int_{ - \infty }^\infty {x(t)\sqrt {(1 - j\cot \alpha )} {e^{j\text{π}[({t^2} + {u^2})\cot \alpha - 2tu\csc \alpha ]}}} }，{\alpha \ne n\text{π} }，\\ {x(t)}，{\alpha = 2n\pi }，\\ {x( - t)}，{\alpha = (2n \pm 1)\text{π}} 。\end{array}} \right. \\[-1pt]\end{split}$ (3)

FRFT是一种广义的傅里叶变换，可以解释为将信号的时频平面旋转任意角度$\alpha$到分数阶傅里叶$u$域的线性变换，当$\alpha = \text{π} /2$时，FRFT就变成了传统傅里叶变换。因此，傅里叶变换是FRFT的一个特例，而FRFT是傅里叶变换的推广[13]

FRFT的逆变换定义为：

 $x(t) = \int_{ - \infty }^\infty {{X_p}(u){K_{ - p}}} (t,u){\rm{d}}u 。$ (4)

 图 2 原始信号分数阶傅里叶变换图 Fig. 2 Fractional Fourier transform diagram of the original signal
2 匹配滤波器检测

 $x(t) = s(t) + n(t) 。$ (5)

 $X(\omega ) = S(\omega ) + N(\omega )。$ (6)

 $Y(\omega)=H(\omega)X(\omega)=H(\omega)S(\omega)+H(\omega)N(\omega)。$ (7)

 $H(\omega)=kS^*(\omega)e^{-j\omega t_0}。$ (8)

 $\begin{split} Y(\omega)=& kS(\omega)S^*(\omega)e^{-j \omega t_0}+kN(\omega)S^*(\omega)e^{-j \omega t_0}=\\& k|S(\omega)|^2 e^{-j \omega t_0}+kN(\omega)S^*(\omega)e^{-j \omega t_0} 。\end{split}$ (9)

 图 3 频域自适应匹配滤波器原理框图 Fig. 3 Block diagram of frequency domain adaptive matching filter

 图 4 SNR=−10 dB时匹配滤波归一化结果 Fig. 4 When SNR=−10 dB, the result of filtering normalization is matched
3 基于FRFT的频域自适应匹配滤波技术

 $s(t)=\left\{\begin{array}{lc}A\exp(j(2{\text{π}} f_0t+k{\text{π}} t^2))，t\in[0,T]，\\ 0，其他。\qquad\qquad\qquad\qquad\qquad\qquad\qquad \end{array} \right.$ (10)

 $y(t)=a(t)+ \omega (t) +n(t) 。$ (11)

 图 5 算法流程图 Fig. 5 Algorithm flow chart

 ${Y'_i}(u)={Y_i}(u) M(u)。$ (12)

4 计算机仿真

 图 6 LFM信号的最优阶次FRFT域图 Fig. 6 The optimal order FRFT domain graph of LFM signal

 图 7 混响信号时域图和频域图 Fig. 7 Time domain and frequency domain of reverberation signal

 图 8 总接收信号时域图和频域图 Fig. 8 Time domain diagram and frequency domain diagram of total received signal

 图 9 滑动过程中时频图误差值 Fig. 9 Time frequency graph error value during sliding

 图 10 混响背景下信号最优阶次时频图 Fig. 10 Optimal order time-frequency diagram of signal in reverb background

 图 11 带通滤波器 Fig. 11 Bandpass filter

 图 12 混响抑制后LFM信号 Fig. 12 LFM signal after reverb suppression

 图 13 混响抑制前后FDAMF图 Fig. 13 FDAMF before and after reverberation suppression
5 结　语

 [1] 郝程鹏, 施博, 闫晟, 等. 主动声呐混响抑制技术与目标检测技术[J]. 科技导报, 2017, 35(20): 102-108. [2] 兰同宇, 周胜增. 一种级联型自适应滤波器的混响抑制技术[J]. 舰船科学技术, 2021, 43(3): 130-133. [3] COX H, LAI H. Geometric comb waveforms for reverberation suppression[C]//Proceeding of IEEE, 1995, 154(6): 347-352. [4] 姚东明, 蔡志明. 主动声呐梳状谱信号研究[J]. 信号处理, 2006(2): 256-259. [5] KAY S, SALISBURY J. Improved active sonar detection using autoregressive prewhiteners[J]. Journal of the Acoustical Society of America. 1900, 87(4): 1603−1611. [6] 舒象兰, 孙荣光, 马鑫. 强混响背景下LFM信号回波检测[J]. 电讯技术, 2016, 56(1): 82-87. DOI:10.3969/j.issn.1001-893x.2016.01.015 [7] 张宗堂, 杨锡铅, 戴卫国. 混响背景下回波信号起始位置提取[J]. 舰船电子工程, 2014, 11: 73-76. [8] 石亚莉. 多基地混响模拟及抑制技术研究[D]. 哈尔滨: 哈尔滨工程大学, 2018. [9] 赵智姗. 基于匹配滤波器的频域自适应线谱增强技术研究[D]. 哈尔滨: 哈尔滨工程大学, 2018. [10] HUANG Xiang, ZHANG Linrang, ZHANG Juan, et al. Efficient angular chirp-Fourier transform and its application to high-speed target detection[J]. Signal Processing, 2019, 164: 234-248. DOI:10.1016/j.sigpro.2019.06.011 [11] MIAO Hongxia, ZHANG Feng, TAO Ran. Fractional Fourier analysis using the Möbius inversion formula[J]. IEEE Transactions on Signal Processing, 2019, 67(12): 3181-3196. DOI:10.1109/TSP.2019.2912878 [12] 徐伟, 许传, 马艳秋, 等. 一种基于分数阶傅里叶变换的主动声自导回波检测方法[J]. 舰船科学技术, 2022, 44(2): 161-165. [13] 刘利民, 李豪欣, 李琦, 等. 基于分数阶傅里叶变换的低信噪比线性调频信号参数快速估计算法[J]. 电子与信息学, 2021, 43(10): 2798-2804. [14] 董仲臣, 李亚安, 金彦丰. 鱼雷浅海海底混响建模及仿真[J]. 鱼雷技术, 2013, 21(2): 100-104.