﻿ 基于分数阶傅里叶变换的声呐探测信号鉴别
 舰船科学技术  2017, Vol. 39 Issue (9): 142-147 PDF

Identification of Sonar detection signal based on fractional Fourier transform
DING Zhi-hui, WANG Biao
School of Telecommunications, Jiangsu University of Science and Technology, Zhenjiang 212000, China
Abstract: In view of underwater acoustic emission source is more. To identify the enemy emission source sonar accurately. Using the digital watermarking technology, combined with the good time-frequency characteristics of fractional Fourier transform (FRFT), Sonar watermarking method is proposed based on fractional Fourier transform, digital watermark embedding in fractional Fourier transform domain of sonar signals, combined with the coefficients features in the fractional fourier transform domain of sonar signal, Selecting the appropriate watermark position. According to the characteristic of the signal before embedding and after embedding, the watermark detection threshold is adaptive. To achieve the identification of sonar signals. The feasibility of this method is verified by the simulation analysis, This method is shown in the simulation results with higher resolution and larger watermark capacity, better robustness, At the same time the performance of the detection accuracy can be further improved.
Key words: fractional fourier transform     watermark     sonar     robustness
0 引　言

1 基于FRFT的声呐数字水印嵌入原理 1.1 理论基础

FRFT作为傅里叶变换的一种广义形式，它可以解释为信号在时频平面内坐标轴绕原点逆时针旋转任意角度后构成的分数阶傅里叶变换域上的表示方法。如果信号的傅里叶变换可看成将其在时间轴上逆时针旋转π/2到频率轴上的表示，则FRFT可以看成将信号在时间轴上逆时针旋转角度αμ轴上的表示（μ轴被称为分数阶Fourier域），信号xt）的分数阶Fourier变换（FRFT）定义为：

 ${X_\alpha }(u) = \{ {F^u}[X(t)]\} (u) = \int_{ - \infty }^{ + \infty } {X(t){K_\alpha }} (t,u){\rm{d}}t\text{，}$ (1)

 ${K_\alpha }(t,u) = \left\{ \begin{array}{l}\sqrt {\displaystyle\frac{{1 - j\cot (\alpha )}}{{2\pi }}} \exp (j \displaystyle\frac{{{t^2} + {u^2}}}{2} \times \\\cot (\alpha ) - tu\csc (\alpha ))\;\;\;\;\;\;\;\;\;\;\;\alpha \ne n\pi \\\delta (t - u)\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\alpha = 2n\pi \\\delta (t + u)\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\alpha = (2n + 1)\pi \end{array} \right\}\text{，}$ (2)

 $x(t) = \int_{ - \infty }^{ + \infty } {{X_\alpha }(u){K_\alpha }} (t,u){\rm{d}}t\text{。}$ (3)

p=1（α=π/2）时，FRFT就退化为传统的傅里叶变换，当变换阶数接近于1时反应的是频域特性，当变换阶数接近于0时候反应的是时域特性。

1.1.1 分解法离散分数傅里叶变换

 $\Delta x = \sqrt {\Delta t\Delta f}\text{，} \;\;x = t/s\text{，}\;\;\;v = fxs\text{，}$ (4)

 $\begin{split}& {X_p}(u) = {A_\alpha }\exp [j\pi {u^2}\cot \alpha ] \times \\& \int_{ - \infty }^{ + \infty } {x(t)\exp [j\pi {t^2}\cot \alpha ]} \exp [ - j2\pi ut\csc \alpha ]{\rm{d}}t\text{，}\end{split}$ (5)

 $\begin{array}{l}0 < \left| \alpha \right| < \pi \\{A^\alpha } = \exp [j(P - 1)\pi /4]/\sqrt {\left| {\sin \alpha } \right|} ,\alpha = P\pi /2\text{。}\end{array}$

1）信号与线性调频函数的相乘，

2）傅里叶变换（变元乘以尺度系数csc α），

3）再与线性调频函数相乘，

4）乘以一复数因子。

 \setcounter{equation}{6}\begin{aligned}& {X_P}(\displaystyle\frac{m}{{2\Delta x}}) = \displaystyle\frac{{{A_\alpha }}}{{2\Delta x}}\sum\limits_{n = - N}^N {\exp [\displaystyle\frac{{j\pi (\cot \alpha ){m^2}}}{{{{(2\Delta x)}^2}}} - \displaystyle\frac{{j2\pi (\csc \alpha )mn}}{{{{(2\Delta x)}^2}}} + } \\& \displaystyle\frac{{j\pi (\cot \alpha ){n^2}}}{{{{(2\Delta x)}^2}}}]x(\displaystyle\frac{n}{{2\Delta x}}) =\\& \displaystyle\frac{{{A_\alpha }}}{{2\Delta x}}[\exp \frac{{ - j\pi \tan \frac{\alpha }{2}{m^2}}}{{{{(2\Delta x)}^2}}}]X\sum\limits_{n = - N}^{n = N} {\exp [\displaystyle\frac{{j\pi (\csc \alpha ){{(m - n)}^2}}}{{{{(2\Delta x)}^2}}}]} \times \\& [\exp \displaystyle\frac{{ - j\pi \tan \frac{\alpha }{2}{m^2}}}{{{{(2\Delta x)}^2}}}]x(\displaystyle\frac{n}{{2\Delta x}})\text{。}\\& {A^\alpha } = \exp [j(P - 1)\pi /4]/\sqrt {\left| {\sin \alpha } \right|}\text{，} \\& \alpha = P\pi /2\text{，}\;\;N = {(\Delta x)^2}\text{，}\;\;\;0.5 \leqslant p \leqslant 1.5\text{。}\end{aligned} (6)

1.2 声呐数字水印嵌入系统

1.2.1 水印生成

1.2.2 水印嵌入算法

 $X = \{ x(n),n = 1,2...,N\} \text{，}$ (7)

 $S_i^w = {S_i} + {\rm{a}}W = {\rm{a}}(R_1 + jR_2),i = L + 1,...L + M\text{。}$ (8)

1.2.3 水印嵌入准则

 ${E_{xw}} = \int {{x^2}(t){\rm{d}}t\;} + {k^2}\int {{w^2}} (t){\rm{d}}t + k\int {x(t)w(t){\rm{d}}t} \text{。}$ (9)

1.2.4 水印检测方法

 $\begin{split}d \!= &\!\!\! \sum\limits_{i \!=\! L + 1}^{i = L + M} \!\!\!{[R_{1i} \!-\! jR_{2i}]} S_i^{(a)} \!=\!\!\! \sum\limits_{i = L + 1}^{i = L + M} \!\!\! {[R_{1i} \!-\! jR_{2i}][{S_i} \!\!+\!\! a(R_{1i} \!\!+\!\! R_{2i})]} \!= \\[7pt] & a \!\! \sum\limits_{i = L + 1}^{i = L + M} \!\! {[R_{1i}^2 + R_{2i}^2]} + \!\! \sum\limits_{i = L + 1}^{i = L + M} {[{S_i}R_{1i} - j{S_i}R_{2i}]} \text{，} \end{split}$ (10)

 $E(d) = aM(\sigma _1^2 + \sigma _2^2)\text{，}$ (11)

 $E(d) = 0\text{。}$ (12)

2 仿真分析

 $SWR = 10\lg \frac{{{\text{载体信号的功率}}}}{{{\text{嵌入水印的功率}}}}\text{。}$

2.1 信号时频图
 图 1 时频图 Fig. 1 Time frequency diagram

2.2 水印容量

 图 2 不同水印序列和阶数的检测 Fig. 2 Detection of different watermark sequence and order

 图 3 不同SWR的检测器响应 Fig. 3 Detector response of different SWR

 图 4 不同水印长度检测器响应 Fig. 4 Detector response of different watermark length

2.3 P阶数的选择和水印鲁棒性分析
 图 5 不同变换阶数对应的信噪比检测器响应 Fig. 5 Response of signal to noise ratio detectors with different transform orders

3 结　语

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