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作为分析的前提条件,系统中存在以下基本情况:
(1)回波时刻点的界定
进行100 Hz的激光测距时,主波与回波呈现的一一对应关系相对明显。回波可能会由于传播发生脉冲展宽,或由于目标的复杂反射面产生多重回波脉冲的情况[9]。短时间内较集中的回波脉冲群组都可以认为是由单一一次主波发射造成的一组回波。在文中的分析中,回波时刻以单个测距周期内探测器响应输出的第一个脉冲信号为准。
(2)采用TOF理论值的模拟与实际的差别
光子的飞行时间(Time of Flight,TOF)是指测距时激光光子从测站发射、被目标反射、最后返回测站所需的时间,剔除系统延迟后,可用于计算测站与目标之间的距离。目标与测站间的理论回波时刻来自于测距目标的轨道预报,理论回波时刻与实际测量的回波时刻间的偏差称为残差。以1 km级的预报精度为例(实际预报精度通常优于此值),根据真空中光速计算,其对应的残差范围达到约3微秒(1 000 m/(3×108 m/s)),而转镜的穿孔时段通常为几百微秒甚至达到毫秒量级,前者远小于后者,且该影响随着测距精度的提升而更加微小,故利用TOF理论值进行计算是有效可行的。
(3)转镜与信号时序
测距过程中转镜的位置以及主回波重叠的情况见图1,测距时的信号时序见图2。以穿孔时段表示转镜上的孔经过发射光轴路径位置所需的时间。
图 1 转镜位置与主回波重叠。(a)正常的发射;(b)正常的接收;(c)主回波重叠
Figure 1. Rotating mirror’s position and transmitting/echo signal overlapping. (a) Normal transmitting; (b) normal receiving; (c) transmitting/echo signal overlapping
在图2的时序中,在n时刻(n为整数)时,转镜产生了同步信号时刻R(n),对应地产生了激光脉冲的主波时刻T(n),主波时刻总是位于穿孔时段内。穿孔时段(Hole-pass)是转镜上的开孔经过发射光轴所需时间,起始于R´(n),持续时间为tw。回波时刻E(n)表示回波返回到探测器的时刻,E(n)与T(n)间满足:
$$ E\left( n \right) - T\left( n \right) = {\rm{ }}TOF\left( n \right) $$ (1) 重叠在时序上的表现为,回波时刻E(n)存在于穿孔时段中间。造成重叠的通常是第n时刻产生的主波,经过长时间传播得到的E(n),落在了后续某个周期的穿孔时段内,产生了主回波的重叠。
图3显示了重叠发生时系统的信号时序。一方面,作为转速稳定的同步信号R(n),以t代表转镜周期,转镜上开孔数量为2,在转镜转速r=3 000 r/min情况下,t为0.01 s(100 Hz频率)。R(n)与n存在关系:
$$ R\left( n \right) = {\rm{ }}n \times t $$ (2) 以R´(n)代表主波开始穿孔的时刻,tr代表该时刻与转镜同步信号间延时,则:
$$ R'\left( n \right) = {\rm{ }}R\left( n \right) + {\rm{ }}{t_r} $$ (3) 另一方面,一次测距实验中,在时间序列上一定存在一个与测距目标当前时刻的TOF值所对应的m(n)值,m(n)为整数,使得:
$$m\left( n \right) \times t < TOF\left( n \right) \leqslant \left( {m\left( n \right) + 1} \right) \times t $$ (4) 由于系统中是由转镜同步信号R(n)触发激光器产生主波T(n),二者间延时以ts表示,则:
$$ T\left( n \right) = {\rm{ }}R\left( n \right) + {\rm{ }}{t_s} $$ (5) 根据公式(2)和公式(5),作为按照固定频率产生的激光发射主波时刻序列,T(n)满足:
$$ T\left( {n + m\left( n \right)} \right) - T\left( n \right) = {\rm{ }}m\left( n \right) \times t $$ (6) 由此可得重叠发生时的回波时刻条件,tw作为穿孔时段时长。对于在n时刻的激光脉冲产生的回波E(n),则当:
$$ R'\left( {n + m\left( n \right)} \right) < E\left( n \right) \leqslant R'\left( {n + m\left( n \right)} \right) + {t_w} $$ (7) 时,发生主回波重叠,如图3所示。
结合前文分析,将公式(1)~(6)代入公式(7),逐步推导,可得:
$$ R\left( {n + m\left( n \right)} \right) + {t_r} < E\left( n \right) \leqslant R\left( {n + m\left( n \right)} \right) + {t_r} + {t_{w}} $$ (8) $$ \begin{split} & {n \times t + m\left( n \right) \times t + {t_r} < T\left( n \right) + TOF\left( n \right) \cdots }\\ & \quad { \cdots \leqslant n \times t + m\left( n \right) \times t + {t_r} + {t_w}} \end{split} $$ (9) $$ \begin{split} & {n \times t + m\left( n \right) \times t < R\left( n \right) + {t_s} + TOF\left( n \right) - {t_r} \cdots }\\ & \quad { \cdots \leqslant n \times t + m\left( n \right) \times t + {t_w}} \end{split} $$ (10) 将式中延迟量统一为综合延迟 td,令:
$$ {t_d} = {t_s} - {t_r} $$ (11) 则公式(10)可化简为:
$$m\left( n \right) \times t - {t_d} < TOF\left( n \right) \leqslant {t_w} - {t_d} + m\left( n \right) \times t $$ (12) 若是将整个过程理想化,假设综合延迟项td=0,则条件可进一步化简为:
$$ m\left( n \right) \times t < TOF\left( n \right) \leqslant {t_w} + m\left( n \right) \times t $$ (13) 它表示了目标所对应TOF的变化,也即目标与测站间距离的变化是产生重叠现象的主要原因。在任一时刻n,当TOF相对于测距整周期时段的差值小于转镜穿孔时段时,会发生主回波重叠。根据以上推导,参考目标预报,可模拟出百赫兹激光测距系统对目标进行测距时主回波重叠现象的情况。
Study on transmitting/echo signal overlapping in common-optical-path 100 Hz-rate LLR system
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摘要: 高重频激光测月技术是月球激光测距的一个重要发展趋势,对获得更多的激光测月数据具有重要意义。在之前的10 Hz测月频率基础上,中国科学院云南天文台开发了共光路百赫兹激光测月系统。然而在共光路系统中,存在发射激光光子的主波时刻与接收激光光子的回波时刻相重叠的现象,影响了对回波光子的探测。文中从信号时序角度对主回波重叠现象进行了分析,研究了光子飞行时间与重叠现象间的关联性。利用某天的月面反射器的轨道预报对系统的重叠发生比率进行了模拟,得出了当天观测时段内整体的重叠发生率约为8%。同时开展了基于每个发射时刻的分段模拟,结果与前者吻合。对微调转镜转速以避开主回波重叠的方法进行了分析,开展了对应的转镜转速调整试验,结果表明当重叠发生时,对转速进行适量调整可以避开主回波重叠。Abstract: In Lunar Laser Ranging (LLR), higher repetition rate is an important trend for development, which is also of great significance for acquiring more LLR data. Based on the previous 10 Hz ranging rate system, a 100 Hz lunar ranging system with common-optical-path structure was developed by Yunnan Observatories, CAS. However, overlapping of the transmitting/echo signals exists in such a common-optical-path system, and seriously disturbs echo-photon detection. In the paper, the overlapping phenomenon was analyzed from the perspective of time sequence, and the correlation between the overlapping and the Time of Flight (TOF) was studied. Overlapping in the system was simulated with the orbital prediction of a lunar retro-reflector of a certain day, and the general overlapping ratio was approximately 8% for the observational duration. Sectional simulation based on each transmitting epoch was also performed, its result was consistent with the previous one. The method of avoiding overlapping by slightly adjusting rotating-mirror's speed was analyzed, and corresponding experiment was carried out, as proper speed adjustment turned out to be a solution when the overlapping happened.
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Key words:
- LLR /
- common-optical-path /
- high repetition rate /
- transmitting/echo signal overlapping
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