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本文中选用常用的非直视(non-line of sight, NLOS)紫外光单次散射通信模型[7-10]来进行建模,光子从发射到接收的整个传输过程中只受到了一次大气散射作用[11]。由于非直视紫外光通信属于短距离通信,满足单次散射模型。通过计算分析接收端靠单次散射接收到的能量和时间的函数关系,得到信道的脉冲响应h(t)[12]为:
$ h\left( t \right) = \frac{{{k_{\rm{s}}}{\theta _{\rm{r}}}\theta _{\rm{t}}^2\sin \left( {{\beta _{\rm{r}}} + {\beta _{\rm{t}}}} \right)\exp \left( {-{k_{\rm{e}}}ct} \right)}}{{4{\pi ^3}r\sin {\beta _{\rm{t}}}\left( {1-\cos {\theta _{\rm{t}}}} \right)}} $
(1) 式中, c为光速,t为时间,θr为接收端的半视场角,βr为接收端顶角,θt为发射端光束的发散角,βt为发射端顶角,ks为大气散射系数,ke为大气衰减系数,r为收发间距。
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紫外光通信的码间干扰(inter symbol interference, ISI)信道可以看作是有限脉冲响应(finite impulse response, FIR)滤波器,相应的信道模型图如图 1所示。图中,z-1表示延迟。
紫外光通信的信道长度p是有限长的,由图 1可看出,信道输出符号yk依赖当前数据输入符号xk和已传输的后(p-1)个数据符号{xk-1,xk-2,…,xk-p+1}。所以接收到的信号可写作:
$ {y_k} = \sum\limits_{p = 0}^{p-1} {{h_p}{x_{k-p}} + {n_k}} $
(2) 式中,nk是背景噪声,如果在发射端并没有采用光学放大器件,nk可以忽略不计,对于以下的讨论中,都以二进制开关键控(on-off keying, OOK)调制为例。
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紫外光发射机发射符号序列{X}在通过图 1中的码间干扰信道模型后, 通过MLE算法进行最佳符号序列的判决,假设发射机发射的符号序列{X}已知,yk间彼此相互独立,从而得到整个接收序列{Y}的条件概率, 则整个接收序列{Y}的似然函数[13]为:
$ \begin{array}{l} f\left( {Y|X} \right) = \prod\limits_{k = 0}^{k-1} {f\left( {{y_k}|{x_k}, {x_{k-1}}, \cdots, {x_{k-p + 1}}} \right)} = \\ \;\;\;\;\;\;\;c\exp \left( { - \frac{1}{{{n_0}}}\sum\limits_{k = 1}^{k - 1} {{{\left| {{y_k} - \sum\limits_{p = 0}^{p - 1} {{h_p}{x_{k - p}}} } \right|}^2}} } \right) \end{array} $
(3) 最大似然函数可以等价于发射序列{X}的最小价值函数:
$ M\left( X \right) = \sum\limits_{k = 1}^{k-1} {{{\left| {{y_k}-\sum\limits_{p = 0}^{p-1} {{h_p}{x_{k - p}}} } \right|}^2}} $
(4) 在MLE准则下的最佳符号序列为M(X)最小时, 对应的序列为:
$ {\left\{ X \right\}_{{\rm{MLE}}}} = \arg \mathop {\min }\limits_{\left\{ X \right\}} M\left( X \right) $
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在启用自适应算法之前,首先要在发射端发射一个短的训练序列对紫外光非直视通信信道做一个最初的估计,对信道估计器的抽头系数做初始调整。LMS-MLE结构框图见图 2。k-1时信道状态为:
$ s_{k-1}^{\left( n \right)} = \left\{ {x_{k-1}^{\left( n \right)}, x_{k-2}^{\left( n \right)}, \cdots, x_{k - p + 1}^{\left( n \right)}} \right\} $
(6) 以下为算法执行过程。
(1) 对所有状态点的累积度量值进行比较,获取最优幸存序列。计算k-1时从状态sk-1(n)出发,当且仅在k时刻到达m,状态k时刻转移到状态sk(m)的度量计算为:
$ M_k^{\left( {n \to m} \right)} = {\left| {{y_k}- {{\left[{\left( {{h_{k-1}}\left( n \right)} \right.} \right]}^{\rm{T}}}s_{k -1}^{\left( n \right)}} \right|^2} $
(7) 对于所有的2p-1个传输可能所对应的幸存路径的相应状态sk-1(n0)为:
$ s_{k-1}^{\left( {{n_0}} \right)} = {\left( {M_k^{\left( {n \to m} \right)}} \right)_{{\rm{MLE}}}} = \arg \mathop {\min }\limits_{s_{k-1}^{\left( n \right)}} \left( {M_k^{\left( {n \to m} \right)}} \right) $
(8) 即可得到最终的判决信息为:
$ {\hat y_k} = {h_{k-1}} \cdot s_{k-1}^{\left( {{n_0}} \right)} $
(9) 式中,n0表示最优序列。
(2) 使用判决延迟和LMS算法对信道参量进行更新:若延迟量为D,对于当前延迟后的信道参量hk-D并不代表当前的信道条件,其误差信号可以表示为:
$ {e_{k-D}} = {y_{k-D}}-{h_{k - D}}{\hat x_{k - d}} $
(10) 式中, ${\hat x_{k-d}}$表示判决后数据的延迟输出, d表示D的重复迭代过程。那么最终得到的信道估计输出为:
$ \hat h = {h_{k-d}} + \mu {e_{k-D}}{\left( {{{\hat x}_{k-d}}} \right)^{\rm{T}}} $
(11) 式中,μ为步长因子, T表示矩阵的转置。在判决过程中,随着延迟量的增大,判决的可靠性增强,系统的跟踪性能也降低,但是信道估计随着延迟增加不能及时地进行信道跟踪,所以要在保证可靠性的同时,合理地选择延迟量,这里的判决器更新系数为某个码元周期后的输入信号。
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基于信道估计的判决延迟算法与常规的LMS算法相比,延迟算法使用的并不是当前的误差信号和接收信号,而是经过延迟器延迟后的误差信号和接收信号来进行信道预测和权系数更新,通过对延迟算法的性能分析,延迟量D的引入对算法的稳态影响不大,但是对步长因子μ的选取相比传统LMS算法的步长因子选取(0 < μ < 2/λmax)更为苛刻,它的收敛条件为[14]:
$ 0 < \mu < \frac{2}{{{\lambda _{\max }}}}\sin \left[{\frac{\pi }{{2\left( {2D + 1} \right)}}} \right] $
(12) 式中,λmax表示输入信号自相关函数的最大特征值,是一个常数。随着延迟量的增大,为保证算法收敛,μ的取值为保证系统收敛的上限值。如图 3所示, 随着延迟量的增加,μ逐渐减小,各种近似的误差也逐渐增大。所以在算法运行中,要根据不同的延迟深度选择合适的步长值。
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下面分析不同步长时算法的收敛特性,采用判决延迟的MLE算法性能主要依赖信道估计的差错,跟LMS算法的输入向量的自相关矩和收敛速度相关[15]。如图 4所示,步长因子μ越大,收敛速度越快,但是越小,信道估计的差错越小,所以μ的选取最好是在满足收敛要求的前提下尽可能取最小值。
适用于紫外光通信的延迟判决均衡算法
Equalization algorithm with delayed decision suitable for UV communication
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摘要: 为了消除紫外光通信过程中强烈散射所引起的码间干扰,采用一种带信道估计的最小均方误差-最大似然估计(LMS-MLE)延迟判决均衡算法进行了理论分析和仿真验证。通过选取合适的判决延迟深度来调整LMS自适应滤波器抽头系数进行信道跟踪,获取新的信道估计向量,最后利用MLE均衡算法得到最优序列输出。结果表明, 该算法可以明显提升紫外光通信系统的性能,在没有提高复杂度的情况下,性能接近最优MLE均衡算法,并且可以实现信道跟踪, 紫外光通信中算法的最佳延迟量取值为20。这一结果对紫外光通信性能提升以及MLE均衡器的工程实现是有帮助的。Abstract: In order to eliminate inter symbol interference caused by strong scattering in ultraviolet communication, one delay decision equalization algorithm based on least mean square-maximum likelihood estimation (LMS-MLE) with channel estimation was adopted. The theoretical analysis and simulation verification were carried out. The appropriate decision delay depth was selected to adjust the LMS adaptive filter tap coefficient for channel tracking, and to obtain a new channel estimation vector. Finally, the optimal sequence output was obtained by using the MLE equalization algorithm. The results show that, the algorithm can obviously improve the performance of ultraviolet communication system. Without increasing the complexity, the performance is close to the optimal MLE equilibrium algorithm. And the channel tracking can be achieved. The optimal delay of the algorithm in ultraviolet (UV) communication is 20. This result is helpful for improving the performance of UV communication and engineering implementation of MLE equalizer.
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