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由于电磁波在水下传播时的衰减率随着频率的增大而显著增加,岸基与水下潜器通信系统一般采用频段范围为3~30 kHz[1]的甚低频。为了兼顾能量效率和频谱效率,甚低频岸基至水下潜器通信通常采用MSK信号[1]。为了叙述简洁,本文将其称为水下甚低频MSK信号。
甚低频MSK信号传播到水下的过程中会受海水衰减和天电噪声的共同影响。一方面,甚低频电磁波在海水中传播时具有较大的衰减率(约为1.9~6dBm)[2],同时,甚低频岸基与水下潜器通信系统通常的接收深度为10 m左右,而海面的起伏严重时可达几米,因此必须考虑海浪起伏对信号的影响[1]。另一方面,水下甚低频MSK信号还会受到天电噪声的干扰,天电噪声是典型的脉冲噪声[3]。由于海水对甚低频信号具有显著的低通滤波特性,脉冲噪声经过海水传播后在时域上会被展宽[4],不利于对脉冲噪声的处理,必须进行准确的海洋补偿滤波。由于上述两方面因素的影响,需要精确同步的相干解调算法,实现起来比较复杂,所以水下甚低频MSK信号通常采用非相干的差分解调算法[5]。
现有的水下甚低频MSK信号差分解调一般采用简单的两符号次优解调方法,性能不佳[1]。为此,本文综合考虑脉冲噪声、白噪声和海洋补偿滤波器对水下甚低频MSK信号的影响,基于最大似然原理,推导多符号水下甚低频MSK信号差分解调算法。
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电磁波在传播过程中,满足Maxwell方程组及其辅助方程。假设所研究的电磁波为简谐波,波的传播方向竖直向下,则其对时间的关系为ejωt,ω为信号角频率。令导电介质中的电荷体密度为零,即ρ=0,将其代入辅助方程,并将辅助方程代入到Maxwell方程组,得海水中Maxwell方程组为[6]:
$$ \left\{ \begin{gathered} \nabla \times {\mathbf{H}} = \sigma {\mathbf{E}} + j\omega {\mathbf{D}} \hfill \\ \nabla \times {\mathbf{E}} = - j\omega {\mathbf{B}} \hfill \\ \nabla {\mathbf{D}} = 0 \hfill \\ \nabla {\mathbf{B}} = 0 \hfill \\ \end{gathered} \right. $$ (1) 求解式可得海水中电磁波的衰减规律为[1]:
$$ \left\{ {\begin{array}{*{20}{l}} {H(z) \approx H(0)2{e^{ - \alpha z}}2{e^{ - j\beta z}}} \\ {E(z) \approx E(0)2{e^{ - \alpha z}}2{e^{ - j\beta z}}} \end{array}} \right.{\text{ }} $$ (2) 式中,E(z)和H(z)分别表示水下离海面深度为z的接收点的水平电场和磁场;E(0)和H(0)分别表示海面上的水平电场和磁场;α和β分别为衰减常数和相移常数,满足$\alpha = \beta = \sqrt {\omega \mu \sigma /2} $。本文中,取海水的导电率σ=4 S/m,磁导率$\mu = 4{\text{π }} \times {10^{ - 7}}{\text{ }}H/m $[7]。
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接收深度的变化对信号的幅度、相位和脉冲噪声的补偿存在影响[1, 7]。海浪对接收深度的影响如图 1所示。显然,海面的起伏会使水下接收天线的深度发生变化。
本文使用常用的线性海浪模型[8]:
$$ \zeta (t) = \sum\limits_{i = 1}^\infty {{a_i}\cos ({\omega _i}t + {\varepsilon _i})} $$ (3) 式中,ai为振幅;ωi为角频率;εi为初始相位,服从0~2π的均匀分布,即:εi~U(0, 2π)。线性海浪模型认为海浪是平稳正态过程,且具有各态历经性,由无限多个不同波幅、不同频率的组成波叠加而成,并且这些组成波是随机且互相独立的,叠加的结果也具有各态历经性的平稳正态过程。本文借助ITTC(international towing tank conference)[9]参数谱海浪谱(海浪内部结构)来对海浪进行描述,由于海浪谱与叠加海浪模型的振幅之间满足:
$$ {S_\zeta }({\omega _i})\Delta \omega = \frac{1}{2}a_i^2 $$ (4) 则简化的海浪模型波面高度为:
$$ \zeta (t) = \sum\limits_{i = 1}^\infty {\sqrt {2{S_\zeta }({\omega _i})\Delta \omega } } \cos ({\omega _i}t + {\xi _i}) $$ (5) -
海水信道可近似为一个低通滤波,在可以忽略移位电流的情况下,其复传递函数与接收点深度z处的水平电场和水平面水平磁场有关。海水的复传递函数为[7]:
$$ {W_O}(f,d) = {\left( {\frac{{j2\pi f\mu }}{\sigma }} \right)^{1/2}}\exp \left[ { - z{{(j2\pi f\mu \sigma )}^{1/2}}} \right] $$ (6) 式中,f是载波频率。
海水信道会使脉冲噪声的波形被展宽,尖锐程度锐减。这使得后续在对脉冲噪声进行处理后,信号中残留的噪声功率会大幅增加,严重影响系统性能。为此,需要对海水信道进行补偿。海洋补偿滤波器对海水低通特性补偿平衡,其幅频特性与海水的低通特性完全相反[7],脉冲噪声通过合适的海洋补偿滤波器后可恢复其脉冲特性。
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对脉冲噪声的抑制常用限幅器(clipper)[10]:
$$ {x_{clipper}} = \left\{ \begin{gathered} - C{\text{ }}x < - C \hfill \\ x{\text{ }}\left| x \right| \leqslant C \hfill \\ C{\text{ }}x > C \hfill \\ \end{gathered} \right. $$ (7) 式中,x为限幅器输入信号;xclipper为其输出信号。限幅器的门限c与脉冲噪声的强度、白噪声强度和信号幅度密切相关,最优门限可通过仿真实验得到。脉冲噪声通过限幅器后为准高斯噪声[11]。
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经过海水衰减、叠加脉冲和高斯白噪声的水下甚低频信号经过海洋补偿滤波和脉冲噪声预处理后,解调器输入端信号为:
$$ r(t) = s(t){e^{{\text{j}}\theta }}{{\text{e}}^{ - \alpha z - {\text{j}}\beta z}} + n(t) $$ (8) 式中,s(t)是发送信号;z是实际接收深度,服从N(l, (H/3)2)的正态分布,l是平均接收深度,H是波高;接收相偏θ~U(0, 2π);n(t)是高斯白噪声。
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任意二进制全响应CPM信号可以表示为[12]:
$$ s(t) = \sqrt {\frac{{2{E_b}}}{T}} \sum\limits_i {{c_i}} g(t - iT) $$ (9) 式中,Eb是每比特信号的发送能量;1/T是比特速率;g(t)是持续时间为2T的有限时间冲击响应;{ci}是数据序列,数据ci与信息相位ϕi之间满足关系:
$$ {c_i} = {c_{i - 1}}{e^{j\pi h{\varphi _i}}} $$ (10) 对于MSK信号,h=1/2,g(t)可表示为:
$$ g(t) = \sin \left( {\frac{{\pi t}}{{2T}}} \right){u_{2T}}(t) $$ (11) 且ϕi=±1,式变为:
$$ {c_i} = {\text{j}}{c_{i - 1}}{\varphi _i} $$ (12) 水下甚低频信号接收模型为:
$$ r(t) = s(t){{\text{e}}^{{\text{j}}\theta }}{{\text{e}}^{ - \alpha z - {\text{j}}\beta z}} + n(t) $$ (13) -
传统MSK差分解调利用前后两个码元的信息。从式~式可以发现:ϕk的变化范围与冲击响应$ {c_{k - 1}}g(t - kT + T) $(从(k-1)T持续到k+1)T)和${c_k}g(t - kT) $(从kT持续到k+2)T))有关。为了对ϕk作出判决,可至少选择3 bit的接收信号区间 $\left[{(k-1)T, (k + 2)T} \right] $。更精确一点,可以利用(2N+1)比特区间$ \left[{(k-N)T, (k + N + 1)T} \right] $对中间比特的信息相位ϕk作出判决。
基于最大似然原理,接收信号在相偏$\theta = \tilde \theta $、接收深度$ z = \tilde z $、数据向量${\tilde c_k} = \left[{{{\tilde c}_{k-N-1}}, {{\tilde c}_{k-N}}, \cdots, {{\tilde c}_{k + N}}} \right] $和接收信号模型式的条件下,似然函数为[2]:
$$ \begin{gathered} L(r(t)|{{\tilde c}_k}, \tilde \theta, \tilde z) = \\ \exp \left\{ { - \frac{1}{{2{N_0}}}\int_{{\text{ }}(k - N)T}^{{\text{ }}(k + N + 1)T} {\left| {r(t)} \right.} - {{\left. {s(t, {{\tilde c}_k}){{\text{e}}^{{\text{j}}\tilde \theta }}{{\text{e}}^{ - {\text{j}}\beta \tilde z - \alpha \tilde z}}} \right|}^2}{\text{d}}t} \right\} \\ \end{gathered} $$ (14) 将式(9)代入到式(14)中,得到:
$$ \begin{gathered} L(r(t)|{{\tilde c}_k}, \tilde \theta, \tilde z) = \exp \left\{ {\frac{1}{{{N_0}}}\sqrt {\frac{{2{E_b}}}{T}} \operatorname{Re} \left[{{{\text{e}}^{-{\text{j}}\beta \tilde z-\alpha \tilde z}}} \right.} \right. \times \\ {\text{ }}{{\text{e}}^{-{\text{j}}\tilde \theta }}\left. {\left. {\left( {\sum\limits_{i = k - N - 1}^{k + N} {\tilde c_i^ * \int_{(k - N)T}^{(k + N + 1)T} {r(t)} } g(t - iT){\text{d}}t} \right)} \right]} \right\} \\ \end{gathered} $$ (15) 记:
$$ \begin{gathered} F(r(t), {{\tilde c}_k}) = \sum\limits_{i = k - N - 1}^{k + N} {\tilde c_i^ * \int_{{\text{ }}(k - N)T}^{{\text{ }}(k + N + 1)T} {r(t)} } g(t - iT){\text{d}}t \\ \left\{ \begin{gathered} g(t) = {g_1}(t) + {g_2}(t - T) \hfill \\ {g_1}(t) = \sin \left( {\frac{{{\text{\pi }}t}}{{2T}}} \right){u_T}(t) \hfill \\ {g_2}(t) = \cos \left( {\frac{{{\text{\pi }}t}}{{2T}}} \right){u_T}(t) \hfill \\ \end{gathered} \right. \\ {x_n}(k) = \int_{{\text{ }}kT}^{{\text{ }}(k{\text{ + 1}})T} {r(t)} {g_n}(t - kT){\text{d}}t{\text{ }}n = 0, 1 \\ \end{gathered} $$ (16) 可得:
$$ \begin{gathered} F(r(t), {{\tilde c}_k}) = \tilde c_{k - N - 1}^ * {x_2}(k - N) + \\ \sum\limits_{i = k - N}^{k + N - 1} {\tilde c_i^ * \left[{{x_1}(i) + {x_2}(i + 1)} \right] + } \tilde c_{k + N}^ * {x_1}(k + N) \\ \end{gathered} $$ (17) 将$ L(r(t){\text{|}}{\tilde c_k}, \tilde \theta, \tilde z) $对θ积分得到:
$$ \begin{gathered} {L_1}(r(t)|{{\tilde c}_k}, \tilde z) = \\ {I_0}\left( {\frac{1}{{{N_0}}}\sqrt {\frac{{2{E_b}}}{T}} \left| {F(r(t), {{\tilde c}_k})} \right|{{\text{e}}^{ - {\text{j}}\beta \tilde z - \alpha \tilde z}}} \right) \\ \end{gathered} $$ (18) 式中,I0 (x)是零阶修正贝塞尔函数。再对z积分,于是求$ {L_2}(r(t){\text{|}}{\tilde c_k})$的最大值就转变为求$\left| {F(r(t), {{\tilde c}_k})} \right| $的最大值。定义:
$$ \begin{gathered} X = \left[{{x_1}(k-N), {x_2}(k-N), {x_1}(k-N + 1), } \right. \\ \left. {{x_2}(k - N + 1), \cdots, {x_1}(k + N), {x_2}(k + N)} \right] \\ \end{gathered} $$ (19) 相邻的2N个信息相位组成向量:
$$ \left\{ \begin{gathered} \tilde \varphi _{k - 1}^{( \leftarrow )} = ({{\tilde \varphi }_{k - N}}, {{\tilde \varphi }_{k - N + 1}}, \cdots, {{\tilde \varphi }_{k - 1}}) \hfill \\ \tilde \varphi _{k + 1}^{( \to )} = ({{\tilde \varphi }_{k + 1}}, {{\tilde \varphi }_{k + 2}}, \cdots, {{\tilde \varphi }_{k + N}}) \hfill \\ \end{gathered} \right. $$ (20) 结合式和式,并借助式和式的定义可得:
$$ \begin{gathered} F(X(k);\tilde \varphi _{k - 1}^{( \leftarrow )};{{\tilde \varphi }_k};\tilde \varphi _{k + 1}^{( \to )}) = {x_2}(k - N) + \\ \sum\limits_{i = k - N}^{k + N - 1} {\left[{{x_1}(i) + {x_2}(i + 1)} \right]} \exp \left( { - {\text{j}}\frac{{\text{\pi }}}{2}\sum\limits_{l = k - N}^i {{{\tilde \varphi }_l}} } \right){\text{ + }} \\ {x_1}(k + N)\exp \left( { - {\text{j}}\frac{{\text{\pi }}}{2}\sum\limits_{l = k - N}^{k + N} {{{\tilde \varphi }_l}} } \right) \\ \end{gathered} $$ (21) 又有:
$$ \begin{gathered} F(r(t), {{\tilde c}_k}) = \tilde c_{k - N - 1}^ * F(X(k);\tilde \varphi _{k - 1}^{( \leftarrow )};{{\tilde \varphi }_k};\tilde \varphi _{k + 1}^{( \to )}) \\ {\text{ }}\left| {\tilde c_{k - N - 1}^ * } \right| = 1 \\ \end{gathered} $$ (22) 利用式和判决反馈,水下MSK信号多符号差分解调算法的最终检测统计量为:
$$ {\hat \varphi _k} = \arg \mathop {\max }\limits_{{{\tilde \varphi }_k}, \tilde \varphi _{k + 1}^{( \to )}} \left\{ {\left| {F(X(k);\tilde \varphi _{k - 1}^{( \leftarrow )};{{\tilde \varphi }_k};\tilde \varphi _{k + 1}^{( \to )})} \right|} \right\} $$ (23)
Maximum Likelihood Multiple Symbol Differential Demodulation Algorithms for Underwater VLF MSK Signal
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摘要: 水下甚低频(VLF)最小频移键控(MSK)信号受脉冲噪声、海水衰减和海面起伏的共同影响。为了降低对相位同步的要求,提高解调算法的适应性,该文研究了水下甚低频MSK信号的差分解调算法,建立了甚低频MSK调制信号在海水中的传播模型。考虑脉冲噪声处理算法的影响,基于最大似然(ML)原理推导了水下甚低频MSK信号多符号差分解调算法。仿真结果表明:水下甚低频MSK信号最大似然多符号差分解调算法在相位同步下可获得逼近相干解调的性能,在相位不同步下可获得优于相干解调的性能,具有显著的优势。Abstract: Underwater very low frequency (VLF) minimum shift keying (MSK) signals are distorted by the impulsive noise, the propagation attenuation of sea water and the fluctuation of sea wave. In order to reduce the requirement for carrier phase synchronization and improve the robustness of demodulation algorithm, the differential demodulation algorithm of the underwater VLF MSK signal is investigated in this paper. The propagation attenuation of sea water is first modeled. By considering the impact of suppressing impulsive noise, the multiple-symbols differential demodulation of the underwater VLF MSK signal is then derived based on the principle of maximum likelihood (ML) detection. Extensive simulation results show that the performance of the ML multiple-symbols differential demodulation algorithm can closely approach that of the coherent demodulation algorithm if the carrier phase is recovered and a relative large amount of symbols are utilized. On the other hand, the differential demodulation algorithm is superior to the coherent demodulation algorithm if the carrier phase is not recovered.
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Key words:
- differential demodulation /
- impulsive noise /
- MSK /
- sea wave /
- VLF
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