﻿ 基于分数阶傅里叶变换的非同步水下测距影响因素研究
 舰船科学技术  2019, Vol. 41 Issue (3): 131-136 PDF

1. 大连测控技术研究所，辽宁 大连 116000;
2. 大连海洋大学，辽宁 大连 116023

Research on influence factor of non-synchronization ranging based on fractional fourier transform
CAO Qing-gang, SHI Mi-na
1. Dalian Scientific Test and Control Technology Institute, Dalian 116000, China;
2. Dalian Ocean Universit, Dalian 116023, China
Abstract: Improving ranging accuracy in the water is very important and difficult. This article chooses the pulse with linear frequency-modulation (LFM) for transmit signal. For improved ranging accuracy, using fractional Fourier transform (FrFt) analyzed signals. When the bed deep was knew, using distance differences of interface echo and direct sound, research on influence factor of non-synchronization measuring distance exactly of source in the water. The experimentation was performed, and results are as follows: When LFM was used and using FrFt analysis, to compare with tradition method ranging accuracy was improved. The ranging accuracy depend on the bed deep and bandwidth of signal.
Key words: LFM     fractional fourier transform     non-synchronization     measuring distance exactly     influence factor
0 引　言

1 分数阶傅里叶变换理论

 ${X_p}(u) = \{ {F^p}[x(t)]\} (u) = \int_{ - \infty }^\infty x(t){K_p}(t,u){{\rm {d}}t} \text{，}$ (1)
 $x(t) = \{ {F^{ - p}}[x(u)]\} (t) = \int_{ - \infty }^\infty {{X_p}(u){K_{ - p}}(t,u){\rm d}u}\text{。}$ (2)

 ${K_p}\left( {t,u} \right) = \left\{ {\begin{array}{*{20}{l}} {\sqrt {\frac{{1 - i\cot \alpha }}{{2\pi }}} {e^{\left( {i\frac{{{t^2} + {u^2}}}{2}\left( {\cot \alpha - itu\csc \alpha } \right)} \right)}},}&{\alpha \ne n\pi }\text{；}\\ {\delta \left( {t - u} \right),}&{\alpha = 2n\pi }\text{；}\\ {\delta \left( {t - u} \right),}&{\alpha = \left( {2n + 1} \right)\pi } \text{。} \end{array}}\right.$ (3)

xt）中处理的主要信号为LFM信号 ${x_1}(t)$ 时FrFt的变换核在u域上表现为一个冲击函数，即FrFt某个阶次的分数阶域对给定的LFM信号具有很好的聚能性。这种聚能性可以很好地提高LFM信号的检测及两信号的分离，提高信号的时间分辨率。

${x_1}(t) = \left\{ \begin{array}{l}A{e^{i\pi (2{f_0}t + k{t^2})}},t \in [ - T/2,T/2]\\0,\text{其他}\end{array} \right.$ ，其中kF/TF为调频宽度，T为信号脉宽。

 $p_0 = \frac{2}{\pi }\arctan (\kappa ) - 1\text{，}$ (4)

FrFt可以认为是一种广义的傅里叶变换。傅里叶变换是线性变换，是FrFt的一个特例。即若将FrFt的变换算子逆时针旋转时间轴到u轴的任意角度 $\alpha = \pi /2$ 时就是傅里叶变换。当处理的信号为LFM信号时，其原理如图1所示。

 图 1 FrFt变换原理示意图 Fig. 1 Schematic diagram of the principle of FrFt transformation
2 声传播理论模型

 图 2 海洋近程声线传播图 Fig. 2 2 Marine short-range ray travel map

 $x = \frac{{4zh - {s^2}}}{{2 s}}\text{，}$ (5)
 $x = \frac{{4(H - z)(H - h) - {s^2}}}{{2 s}}\text{。}$ (6)
3 计算机仿真与可行性分析

 图 3 时间差值为0.000 05 s时的混跌信号 Fig. 3 Mixed drop signal with a time difference of 0.000 05 s

 图 4 时间差值为0.000 05 s时分数阶傅里叶变换后的信号 Fig. 4 The signal after fractional Fourier transform when the time difference is 0.000 05s.

 图 5 时间差值为0.000 5 s时的混跌信号 Fig. 5 Mixed drop signal with a time difference of 0.00 05 s

 图 6 时间差值为0.000 5 s时分数阶傅里叶变换后的信号 Fig. 6 The signal after fractional Fourier transform when the time difference is 0.0005 s

 图 7 直达声程随水平距离和发射深度变化规律图 Fig. 7 The change rule of direct distance of sound with horizontal distance and emission depth

 图 8 经海面反射的距离差随水平距离和发射深度变化规律图 Fig. 8 Variation rule of distance difference with horizontal distance and emission depth reflected by sea surface

 图 9 经海底反射的距离差随水平距离和发射深度变化规律图 Fig. 9 Variation rule of distance difference with horizontal distance and launching depth reflected by sea bottom

 图 10 经海面反射的时间差随水平距离和发射深度变化规律图 Fig. 10 The time difference of horizontal reflection and the depth of horizontal transmission

 图 11 经海底反射的时间差随水平距离和发射深度变化规律图 Fig. 11 The time difference of seafloor reflection with horizontal distance and emission depth

 图 12 经海面海底反射的时间差随水平距离和发射深度变化规律 Fig. 12 The time difference of sea bottom reflection with horizontal distance and emission depth

 图 13 海深变化对反演结果的影响 Fig. 13 Changes of deep impact on the inversion results of the sea
4 误差分析验证与仿真预测

 $\Delta = \sqrt {{{\left( {\frac{{\partial x}}{{\partial H}}} \right)}^2} + {{\left( {\frac{{\partial x}}{{\partial h}}} \right)}^2} + {{\left( {\frac{{\partial x}}{{\partial z}}} \right)}^2} + {{\left( {\frac{{\partial x}}{{\partial s}}} \right)}^2}} \text{，}$ (7)

 $s = \sqrt {m{h^2} + {{(2 H - h - z)}^2}} - \sqrt {m{h^2} + {{(h - z)}^2}} \text{，}$ (8)
 $\frac{{\partial x}}{{\partial H}} = \frac{{4 H - 2(z + h)}}{s}{\rm{ d}}H\text{，}$ (9)
 $\frac{{\partial x}}{{\partial z}} = \frac{{2(h - H)}}{s}{\rm d}z\text{，}$ (10)
 $\frac{{\partial x}}{{\partial h}} = \frac{{2(z - H)}}{s}{\rm d}h\text{，}$ (11)
 $\frac{{\partial x}}{{\partial s}} = (\frac{{ - 4(H - z)(H - h)}}{{{s^2}}} - 0.5){\rm d}s\text{。}$ (12)

 图 14 通过海底海面反演误差随发射深度和水平距离变化图 Fig. 14 The variation of error with depth of emission and horizontal distance is retrieved from the bottom and sea surface

 $s = \sqrt {{{(z + h)}^2} + m{h^2}} - \sqrt {{{(z - h)}^2} + m{h^2}}\text{，}$ (13)
 $\Delta = \sqrt {{{(\frac{{2h}}{s}dz)}^2} + {{(\frac{{2z}}{s}dh)}^2} + {{( - \frac{{2zh}}{{{s^2}}}ds)}^2} + {{( - \frac{{ds}}{2})}^2}} \text{。}$ (14)
5 试验验证

 图 15 海上实验实际获得信号的时域图和经FrFt变换图 Fig. 15 Real time signal obtained from marine experiment and FrFt transformation map

 图 16 反演时所用海深简图（精度0.1 m） Fig. 16 Sea depth map used for inversion (precision 0.1m)

 $x = \frac{{4zh - {{(ct)}^2}}}{{2ct}}$ (15)

 图 17 通过海底发射反演值与真实值对比及误差 Fig. 17 Comparison and error between inversion value and real value of submarine launching

 图 18 通过海面发射反演值与真实值对比及误差 Fig. 18 Comparison and error between the inversion value and the real value of the sea surface launch
6 结　语

1）使用海底反射信号与直达声信号差值反演直达距离时，受到海深的影响较大，当获得海深的精度较高时，其定位精度较高。且随发射深度增加，误差变大，距离变远，误差增大。

2）使用海面反射信号与直达声信号差值反演直达距离时，随发射深度减小，误差变大；距离变远，误差增大。由于海面起伏较大，所以其发射距离海面深度变化大，只有当海面较平稳时使用此种定位较好。

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