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笔者所在课题组前期工作使用共横向空间模式耦合从湍流光无线信道中的信号衰落生成共享密钥的理论分析[7]并搭建了试验系统对大气光信道互易性进行了验证[18]。为了克服湍流本身对于光信道测量样本数量的限制,提出了一种方法,在大气湍流光信号衰落自相关时间内对信道的发射光信号强度进行随机调制,使得发射光信号在大气光信道功率传输系数自相关时间内能够快速随机变化,则在用比其自相关时间长度更小的采样时间间隔条件下,对连续的接收光信号测量样本之间实现去相关[17]。收发端机原理示意图如图1所示。
测量两端分别为Alice和Bob两个合法收发端。发射端光信号LSA、LSB分别从激光器A和激光器B发射出来,经过电光调制器后,调制信号源${x_{\rm{A}}}\left( t \right)$、${x_{\rm{B}}}\left( t \right)$分别调制光信号LSA、LSB,${x_A}\left( t \right)$、${x_B}\left( t \right)$会随着发射时间变化且相互独立,在不同t时刻的取值不同且$0 \leqslant {x_{\rm{A}}}\left( t \right) \leqslant 1,0 \leqslant {x_{\rm{B}}}\left( t \right) \leqslant 1$,同时调制信号源变化的自相关远小于大气湍流光信号衰落的自相关时间并进行存储。通过收发光学系统进行发射进入到互易的大气光信道中,从而被Alice和Bob两个合法接收端接收。
Alice和Bob在t时刻接收到的光信号分别为[17]:
$$ \begin{gathered} {y_{\rm{A}}}\left( t \right) = {s_{\rm{B}}}\left( t \right){x_{\rm{B}}}\left( t \right){h_{\rm{M}}}\left( t \right) \\ {y_{\rm{B}}}\left( t \right) = {s_{\rm{A}}}\left( t \right){x_{\rm{A}}}\left( t \right){h_{\rm{M}}}\left( t \right) \\ \end{gathered} $$ (1) 式中:${y_{\rm{A}}}\left( t \right)$和${y_{\rm{B}}}\left( t \right)$分别表示收发光学系统A和收发光学系统B在t时刻收到的光信号功率;${s_{\rm{A}}}\left( t \right)$和${s_{\rm{B}}}\left( t \right)$分别表示激光器A和激光器B发射的光信号功率;${h_{\rm{M}}}\left( t \right)$表示大气光信道功率传输系数,$0 \leqslant {h_{\rm{M}}}\left( t \right) \leqslant 1$。由于大气光信道具有互易性,在合适的条件下,理论上t时刻从Bob到Alice和从Alice到Bob两传输链路终端的大气光信道功率传输系数是相等的。实际实验中,采取一些合理的系统设计和具有优势的算法优化,相关系数可以保持近似为1[18]。Alice 和 Bob 构建共享的公共随机源变为[17]:
$$ \begin{gathered} {\phi _{\rm{A}}}\left( t \right) = {y_{\rm{A}}}\left( t \right){x_{\rm{A}}}\left( t \right) \\ {\phi _{\rm{B}}}\left( t \right) = {y_{\rm{B}}}\left( t \right){x_{\rm{B}}}\left( t \right) \\ \end{gathered} $$ (2) 调制信号源${x_{\rm{A}}}\left( t \right)$、${x_{\rm{B}}}\left( t \right)$在生成的过程中,在本地服务器已经进行了存储,因此完成这一操作是可行的,同时降低了在网络信道传输过程中泄露的风险。于是,${\phi _{\rm{A}}}\left( t \right)$和${\phi _{\rm{B}}}\left( t \right)$可看作是加入随机调制后的接收光信号。因为大气光信道具有互易性,理论上,在t时刻,${\phi _{\rm{A}}}\left( t \right)$和${\phi _{\rm{B}}}\left( t \right)$的互相关系数等于1。这是针对合法双方两端进行都进行随机调制的情形(之后分析过程中用双调制表示)。
当进行合法双方只有一端进行调制时(之后分析过程中用单调制表示),Alice和Bob在t时刻接收到的光信号分别为:
$$ \begin{gathered} {{\tilde y}_{\rm{A}}}\left( t \right) = {s_{\rm{B}}}\left( t \right){x_{\rm{B}}}\left( t \right){h_{\rm{M}}}\left( t \right) \\ {{\tilde y}_{\rm{B}}}\left( t \right) = {s_{\rm{A}}}\left( t \right){h_{\rm{M}}}\left( t \right) \\ \end{gathered} $$ (3) 此时,Alice 和 Bob 构建共享的公共随机源变为:
$$ \begin{gathered} {{\tilde \phi }_{\rm{A}}}\left( t \right) = {{\tilde y}_{\rm{A}}}\left( t \right) \\ {{\tilde \phi }_{\rm{B}}}\left( t \right) = {{\tilde y}_{\rm{B}}}\left( t \right){x_{\rm{B}}}\left( t \right) \\ \end{gathered} $$ (4) 同理,单调制情形下,${\tilde \phi _{\rm{A}}}\left( t \right)$和${\tilde \phi _{\rm{B}}}\left( t \right)$可看作是加入随机调制后的接收光信号。因为大气光信道具有互易性,理论上,在t时刻 ${\tilde \phi _{\rm{A}}}\left( t \right)$和${\tilde \phi _{\rm{B}}}\left( t \right)$的互相关系数等于1。
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该节通过时间序列自协方差函数分析调制信号源${x_{\rm{A}}}\left( t \right),{x_{\rm{B}}}\left( t \right)$和大气光信道功率传输系数${h_{\rm{M}}}\left( t \right)$,作用是分析在任意两个不同时刻的取值之间的二阶混合中心矩,描述在两个时刻取值的起伏变化的相关程度。${\phi _{\rm{A}}}\left( t \right),{\phi _{\rm{B}}}\left( t \right)$两次连续观测值的相关系数分别为[17]:
$$ \begin{gathered} {b_{\phi ,{\rm{A}}}}\left( \tau \right) \simeq \frac{{{b_h}\left( \tau \right)}}{{\zeta \left( {{\gamma _{\rm{A}}},\sigma _{x,{\rm{A}}}^2,\sigma _{x,{\rm{B}}}^2} \right)}} \\ {b_{\phi ,{\rm{B}}}}\left( \tau \right) \simeq \frac{{{b_h}\left( \tau \right)}}{{\zeta \left( {{\gamma _{\rm{B}}},\sigma _{x,{\rm{B}}}^2,\sigma _{x,{\rm{A}}}^2} \right)}} \\ \end{gathered} $$ (5) 式中:$\sigma _{x,{\rm{A}}}^2,$$\sigma _{x,{\rm{B}}}^2,$$ \sigma _h^2 $分别代表${x_{\rm{A}}}\left( t \right),$ ${x_{\rm{B}}}\left( t \right),$ ${h_{\rm{M}}}\left( t \right)$的归一化方差反映了信号的波动性的变化;${\gamma }_{{\rm{A}}}和{\gamma }_{{\rm{B}}}$分别是 Alice 和 Bob 光电探测器输出端的平均信噪比 (SNR);${b_h}\left( \tau \right) = \exp \left( { - {\tau ^2}/\tau _{h,0}^2} \right)$,${\tau _{h,0}}$是大气光信道传输系数的相干时间,理论分析过程中令${b_h}\left( \tau \right) = {{\mathrm{e}}^{ - 1}}$。${b_h}\left( \tau \right)$是大气光信道传输系数${h_{\rm{M}}}\left( t \right)$归一化时间自协方差函数。
$$ \begin{gathered} \zeta \left( {{\gamma _{\rm{A}}},\sigma _{x,{\rm{A}}}^2,\sigma _{x,B}^2} \right) = \left( {\sigma _{x,{\rm{A}}}^2 + 1} \right)\left( {\sigma _{x,{\rm{B}}}^2 + 1} \right) + \left[ {\left( {\sigma _{x,{\rm{A}}}^2 + 1} \right)\left( {\sigma _{x,{\rm{B}}}^2 + 1} \right) + \gamma _{\rm{A}}^{ - 1}\left( {\sigma _{x,{\rm{A}}}^2 + 1} \right) - 1} \right]\sigma _h^{ - 2} \\ \zeta \left( {{\gamma _{\rm{B}}},\sigma _{x,{\rm{B}}}^2,\sigma _{x,{\rm{A}}}^2} \right) = \left( {\sigma _{x,{\rm{B}}}^2 + 1} \right)\left( {\sigma _{x,{\rm{A}}}^2 + 1} \right) + \left[ {\left( {\sigma _{x,{\rm{B}}}^2 + 1} \right)\left( {\sigma _{x,{\rm{A}}}^2 + 1} \right) + \gamma _{\rm{B}}^{ - 1}\left( {\sigma _{x,{\rm{B}}}^2 + 1} \right) - 1} \right]\sigma _h^{ - 2} \\ \end{gathered} $$ (6) $\zeta \left( {{\gamma _{\rm{A}}},\sigma _{x,{\rm{A}}}^2,\sigma _{x,{\rm{B}}}^2} \right)$和$\zeta \left( {{\gamma _{\rm{B}}},\sigma _{x,{\rm{B}}}^2,\sigma _{x,{\rm{A}}}^2} \right)$是由${\gamma _{\rm{A}}},$${\gamma _{\rm{B}}},$$\sigma _{x,{\rm{A}}}^2,$$\sigma _{x,{\rm{B}}}^2,$$ \sigma _h^2 $针对$ {b_h}\left( \tau \right) $构建的参数比例因子。调制信号源${x_{\rm{A}}}\left( t \right),{x_{\rm{B}}}\left( t \right)$的归一化方差$\sigma _{x,{\rm{A}}}^2,$$\sigma _{x,{\rm{B}}}^2$分别对于 ${\tilde \phi _{\rm{A}}}\left( t \right)$,${\phi _{\rm{A}}}\left( t \right)$的两次连续观测值的相关系数的影响如图2所示。分析过程中限制${\gamma _{\rm{A}}} \equiv \infty$${\gamma _{\rm{B}}} = \infty$,同时接收信号样本采样间隔时间远小于大气光信道功率传输系数自相关时间。
图 2 (a)单调制情况下对${\tilde \phi _{{{\rm{A}}}}}\left( t \right)$两次连续观测值的相关系数分析图,$\sigma _{x,{{{\rm{A}}}}}^2 > 0,\sigma _{x,{{{\rm{B}}}}}^2 \equiv 0$;(b)双调制情况下对${\phi _{\rm{A}}}\left( t \right)$两次连续观测值的相关系数分析图,$\sigma _{x,{{{\rm{A}}}}}^2 \equiv \sigma _{x,{{{\rm{B}}}}}^2 > 0$
Figure 2. (a) Correlation coefficient analysis of two consecutive observations of ${\tilde \phi _{{{\rm{A}}}}}\left( t \right)$ in the case of signal modulation $\sigma _{x,{{{\rm{A}}}}}^2 > 0,\sigma _{x,{{{\rm{B}}}}}^2 \equiv 0$;(b) Correlation coefficient analysis of two consecutive observations of ${\phi _{{{\rm{A}}}}}\left( t \right)$ in the case of double modulation$\sigma _{x,{{{\rm{A}}}}}^2 \equiv \sigma _{x,{{{\rm{B}}}}}^2 > 0$
图2(a)为单调制情况下对${\tilde \phi _{\rm{A}}}(t)$分析图, ${x_{\rm{A}}}(t)$随时间随机变化,$\sigma _{x,{\rm{A}}}^2 > 0$,${x_{\rm{B}}}(t)$不随时间随机变化,$\sigma _{x,{\rm{B}}}^2 = 0$。横坐标为$\sigma _{x,{\rm{A}}}^2$,代表调制信号的波动性变化,纵坐标为${b_{\tilde \phi ,{\rm{A}}}}\left( \tau \right)$,代表单调制情况两次连续观测值的相关系数,反映相邻测量样本的自相关情况。$ \sigma _h^2 $反映大气光信道功率传输系数的波动性变化。图2(b)为双调制情况下对${\phi _{\rm{A}}}(t)$的分析图,${x_{\rm{A}}}(t)$,${x_{\rm{B}}}(t)$都随时间发生变化,$\sigma _{x,{\rm{A}}}^2 = \sigma _{x,{\rm{B}}}^2 > 0$;纵坐标为${b_{\phi ,{\rm{A}}}}\left( \tau \right)$,代表单调制情况两次连续观测值的相关系数,反映相邻测量样本的自相关情况。下面从调制情况、大气光信道功率传输系数的波动性变化情况和调制信号的波动性变化情况三个角度,分析对于相邻测量样本自相关影响。
对于不同的调制情况,对比图2(a)和图2(b)中$\sigma _h^2 = 0.3$曲线可以发现,两种情况下,随着自变量$\sigma _{x,{\rm{A}}}^2$增加,两次连续观测值的相关系数不断降低,双调制情况下相关系数随着$\sigma _{x,{\rm{A}}}^2$增加,下降的更快。同时单调制情况下$\sigma _{x,{\rm{A}}}^2 \equiv 1$时,两次连续观测值的相关系数在0.2附近,而双调制情况下两次连续观测值的相关系数小于0.1,变化幅度更明显。
对于大气光信道功率传输系数的波动性变化情况,通过对比图2(a)中$\sigma _h^2 = 0.3$,$\sigma _h^2 = 0.2$,$\sigma _h^2 = 0.1$,发现随着$\sigma _h^2$减小,大气光信道功率传输系数的单位时间内波动性变化减弱,两次连续观测值的相关系数的下降幅度变得更大。具体来说,当$\sigma _{x,{\rm{A}}}^2 = 1$时,在此条件下,$\sigma _h^2 = 0.3$时的相关系数小于0.2,而$\sigma _h^2 = 0.1$时,相关系数小于0.1,变化幅度更大。分析得出大气光信道功率传输系数的波动性变化对于测量样本的自相关性具有重要影响。
对于调制信号的波动性变化情况,观察图2(a)和图2(b)可以发现,随着 $\sigma _{x,{\rm{A}}}^2$的增加,调制信号单位时间内波动性变化增强,两次连续观测值的相关系数减小。
Random modulation-based realization of optical channel measurement sample de-correlation analysis
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摘要: 从大气光信道中可以提取两个合法通信方用来加密其传输的机密信息的公共密钥。首先,为了打破湍流引起的光学波动的相关时间对每秒提取的不相关光信道测量样本的数量限制,提出了为合法通信双方配备随机调制的方法,并针对其实现测量样本去相关性进行了理论分析;其次,利用OptiSystem软件仿真分析了大气光信道水平路径传输距离为1 km,中湍流,调制方式为幅度调制条件下,合法通信方在无调制、单调制、双调制三种不同调制的情况下对光信道测量样本自相关性的影响。针对单调制的情形,分析了不同大气光信道传输系数相干时间、伪随机码生成速率、传输距离和信噪比对光信道测量样本自相关性的影响。结果表明,对比无调制情况,在相同采样间隔内,观测到的光信道测量样本的测量数据变化次数增多;同时施加随机调制的发射光信号,在小于湍流引起的光学波动的相关时间的采样间隔下,观测到的光信道测量样本,在滞后100个样本的延迟时间后的自相关性,从无调制情况下的0.676降低到单调制情况下的0.083和双调制情况下的0.035。Abstract:
Objective Based on channel-based key extraction technology, the reciprocal random channel is treated as a public random source from which shared keys are generated. During the optical channel transmission, the received optical signals may exhibit correlation due to unstable factors such as atmospheric turbulence, indicating a degree of correlation between adjacent signal samples. In order to extract highly random key sources from the optical channel, it is necessary to reduce the measurement rate of the channel state based on the correlation time of the channel, ensuring the lack of correlation between consecutive channel measurements. To some extent, the correlation time of optical fluctuations caused by turbulence limits the number of uncorrelated optical channel measurement samples that can be obtained per second. Therefore, it is necessary to perform decorrelation on the continuous observed optical channel measurement samples obtained by the legitimate party at a sampling interval shorter than the correlation time of optical fluctuations caused by turbulence. In this paper, a simulation experiment based on random modulation is constructed to achieve decorrelation of measurement samples, and the impact of random modulation on the autocorrelation of measurement samples is analyzed. Methods Based on existing relevant theories, an experimental system for measuring sample decorrelation based on random modulation has been designed. The schematic diagram of the transmitting and receiving terminal principles based on random modulation is shown (Fig.1), and theoretical analysis has been conducted (Fig.2). Through analysis, the impact of the normalized variance of the modulation signal source on the correlation coefficient of consecutive measurements is explained. Utilizing OptiSystem software, a simulation experiment of the sample measurement system based on random modulation in optical channels was constructed (Fig.3). Results and Discussions The power samples of received signals over time were analyzed under three conditions of no modulation, single modulation, and double modulation (Fig.4). Moreover, the autocorrelation of measurement samples as a function of lag time was examined under different modulation conditions (Fig.5). In the absence of random modulation, it was observed that the autocorrelation after a lag of 100 samples is 0.676. Contrarily, under single modulation and double modulation conditions, the autocorrelation after a lag of 100 samples is 0.083 and 0.035, respectively. The utilization of random modulation effectively decreased the autocorrelation of optical channel measurement samples. Additionally, concerning the single modulation case, a separate analysis was conducted to assess the impact of varying the coherence time of the atmospheric optical channel transmission coefficient (Fig.6) and the effect of different pseudo-random code generation rates on the autocorrelation of optical channel measurement samples (Fig.7). When the sampling rate is 6.4 × 106 Hz, and the generation rate is 6 × 106 bit/s, the autocorrelation of adjacent measurement samples is 0.112. This indicates that the use of random modulation enables obtaining nearly uncorrelated consecutive observations of optical channel measurement samples at a smaller sampling interval than the correlated time caused by turbulence-induced optical fluctuations. Finally, the impact of transmission distance (Fig.8) and signal-to-noise ratio (Fig.9) on the autocorrelation of measurement samples was analyzed. Conclusions The impact of three scenarios, namely no modulation, single modulation, and dual modulation, on the autocorrelation of the observed optical channel measurement samples was studied. The research results show that compared to the case without modulation, the number of measurement data variations in the observed optical channel measurement samples increases within the same time interval. Moreover, applying random amplitude modulation to the transmitted optical signal reduces the autocorrelation of the observed optical channel measurement samples at a sampling interval shorter than the correlation time of optical fluctuations caused by turbulence. When other parameters remain unchanged, as the coherence time of the atmospheric optical channel transmission coefficients decreases and the delay time increases, the autocorrelation of the measurement samples decays faster and the variation becomes more pronounced. Similarly, when other parameters remain unchanged, as the generation rate of pseudo-random codes increases and the delay time increases, the autocorrelation of the measurement samples decays faster. Additionally, the impact of increasing generation rate of pseudo-random codes on the autocorrelation of adjacent optical channel measurement samples was analyzed. -
Key words:
- laser communication /
- autocorrelation /
- random modulation /
- atmospheric turbulence
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图 2 (a)单调制情况下对${\tilde \phi _{{{\rm{A}}}}}\left( t \right)$两次连续观测值的相关系数分析图,$\sigma _{x,{{{\rm{A}}}}}^2 > 0,\sigma _{x,{{{\rm{B}}}}}^2 \equiv 0$;(b)双调制情况下对${\phi _{\rm{A}}}\left( t \right)$两次连续观测值的相关系数分析图,$\sigma _{x,{{{\rm{A}}}}}^2 \equiv \sigma _{x,{{{\rm{B}}}}}^2 > 0$
Figure 2. (a) Correlation coefficient analysis of two consecutive observations of ${\tilde \phi _{{{\rm{A}}}}}\left( t \right)$ in the case of signal modulation $\sigma _{x,{{{\rm{A}}}}}^2 > 0,\sigma _{x,{{{\rm{B}}}}}^2 \equiv 0$;(b) Correlation coefficient analysis of two consecutive observations of ${\phi _{{{\rm{A}}}}}\left( t \right)$ in the case of double modulation$\sigma _{x,{{{\rm{A}}}}}^2 \equiv \sigma _{x,{{{\rm{B}}}}}^2 > 0$
图 4 (a)无调制情况下实时测量数据图;(b)单调制情况下实时测量数据图;(c)双调制情况下实时测量数据图;(d)、(e)和(f)分别为不同调制情况下 0 ~ 0.119 × 10−3 s测量数据图
Figure 4. (a) Real-time measurement data plot in the case of no modulation; (b) Real-time measurement data plot in the case of single modulation; (c) Real-time measurement data plot in the case of double modulation; (d), (e), and (f) are the 0 -0.119 × 10−3 s measurement data plots in different modulation cases, respectively
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