## Abstract

In order to ensure the communication link stability in mobile FSO system, a new omni-directional optical antenna is designed. Being aimed at discontinuous tracking, a novel beam control method based on the error correction Kalman prediction algorithm (EC-KPA) is proposed. The comparison of EC-KPA and the conventional Kalman prediction algorithm (KPA) is given. Numerical simulations about beam control method are carried out. The results show that the prediction accuracy of EC-KPA is improved about 77% than that of KPA in Gaussian noise situation, and that the increase is up to 12.92 times in strong noise situation. Therefore, the beam control method is feasible, and this optical antenna can meet the demands of fast mobile FSO.

©2013 Optical Society of America

## 1. Introduction

Recently, free space optical communication (FSO) has received increasing attention because of its advantages such as potential wide bandwidth, the flexibility of building network, no need to apply for frequency license, immunity to electromagnetic interference, high security, low cost and so on [1,2]. It is widely used in civil and military communications, such as broadband access, emergency or provisional communication, optical fiber alternative, satellite communication, marine communication and so on [3,4].

Because of the narrow beam for the line-of-sight (LOS) system and the remarkable atmospheric effects [5,6], the acquisition, pointing and tracking (APT) is the key technology for FSO system. For the fixed FSO, APT could guarantee the link stability on the whole. However, for mobile FSO, it often requires machine servo and precise transmission system to drive optical antenna, which could give rise to the problems such as low tracking accuracy, slow processing speed and large system volume, and would eventually cause the link frequent interruption and instability [7–11]. Many scholars had put forward ideas to solve these problems. M. Bilgi and M. Yuksel et al. proposed an optical antenna which has greatly simplified the complexity of the APT [12,13]. This design used multi-transceivers composed of multi-LEDs (light emitting diodes) to achieve angle diversity and spatial reuse, but it may be unsuitable for long-distance outdoor communications. K.H. Heng et al. utilized several spherical lens and corresponding fiber array to receive multiple beams from large-scale angle in space [14]. This kind of antenna could improve the real-time performance of communications. However, the numerical aperture is limited for the restricted size of the optical device and the distinct distribution rule of the fiber array corresponding to different terminal positions, so the system is appropriate to the short-distance communication. Y. Zhu et al. presented an optical antenna based on phased array [15]. Through changing the phase of the light beam, the optical antenna is able to cover 360 degrees space. But it needed thousands of optical transmission units, which made it almost impossible for a small antenna. Literature [16] proposed an omni-directional regular icosahedron-shaped optical antenna. The spatial coverage area of the transmission unit’s beam divergence angle regularly covers the entire space. According to the terminal’s movement, the available transmission units can be determined to open through the feedback information. When the communication distance increases, the beam divergence angle will become small, it also needs lots of transmission units.

In order to ensure the communication link stability in terrestrial mobile FSO, a novel optical antenna and its beam control method are proposed. Primarily, the array element’s distribution (the multiple transmission/receiving units’ distribution) is designed. Considering the factors like the communication distance and the antenna’s size, the divergence angle of laser transmission unit cannot be too big and the number of the transmission units cannot be too large, which lead to discontinuous coverage area of the two adjacent transmission units. In order to achieve 360 degrees spatial coverage, the communication antennas’ self-rotation is chosen. Even so, the two adjacent communication processes still have a very small interval time ($\Delta t$). The accurate calculation of$\Delta t$is the key technology to open the appropriate transmission unit. Under this condition that the communication process is not continuous, namely, the transmitting antenna cannot track the receiving antenna in real time, the conventional tracking algorithm, Kalman prediction algorithm (KPA) cannot meet the requirements of the antenna. In order to solve this problem, for the first time, an error correction Kalman prediction algorithm (EC-KPA) is proposed. Because the predicted interval time of the two adjacent communication processes (between the moment *k* and the moment *k-1* in EC-KPA) is very short, it can be considered that the relative velocity remains unchanged in this interval time. In our proposed EC-KPA, the uniform motion model is built. Based on EC-KPA, through the status information which has been received by the transmitting antenna in the last communication process (The communication is full-duplex. For example, we take one antenna as the transmitting antenna and the other as the receiving antenna), the receiving antenna’s location can be predicted, and then the value of$\Delta t$is accurately calculated. According to the$\Delta t$, the appropriate transmission unit is timely opened in the next communication process. That is, the calculated$\Delta t$is fed back to the optical control system to control the “on-off” state of the transmission units, so that the APT function is realized in fast mobile FSO. However, it is unavoidable that the prediction error of EC-KPA will cause the miscalculation of$\Delta t$, so the influence of the EC-KPA’s prediction error on communication performance is also evaluated. Although our proposed antenna model leads to the discontinuous communication (the interval time_{$\Delta t$}is very small), because the laser communication has wide bandwidth, the practical transmission rate can achieve a higher level.

Finally, the prediction accuracy of the KPA and EC-KPA are compared through the numerical simulation. Here, two kinds of noise, weak noise and strong noise are considered. The weak noise signifies the Gaussian noise, and the strong noise includes the effect of atmospheric scintillation and atmospheric turbulence when the laser beam propagates in complex environment. Moreover, the bandwidth utilization and the prediction error of EC-KPA on communication performance are also simulated. The results show that whether weak noise or strong noise conditions the EC-KPA is more suitable for our antenna, meanwhile the bandwidth utilization and the communication link stability can meet the practical requirements.

The rest of the paper is organized as follows: In section 2, the antenna model is presented. In section 2.1, the array elements’ distribution is introduced; in section 2.2, the antenna self-rotation is designed, and the application conditions and the bandwidth utilization are analyzed; in section 2.3, the EC-KPA is proposed, which can partly eliminate the residual error to increase the anti-noise ability. Through the predicted trajectory by our proposed algorithm, the accurate calculation of_{$\Delta t$}and the beam control method are discussed in detail. Meantime, the influence of the EC-KPA’s prediction error on communication performance is evaluated. In section 3, the numerical simulations about EC-KPA and the bandwidth utilization are carried out respectively, and the comparison with KPA is also made. The analysis and discussion are given. Section 4 concludes this paper.

## 2. Antenna model

The design of the communication antenna includes transmitting part, receiving part, antenna tracking and control module, optical switch control module and mechanical rotating device. As shown in Fig. 1, the received data containing the status information of the opposite antenna (coordinates and velocity vector) is first sent to the antenna tracking control module, which is processed through EC-KPA, and then the status information of the opposite antenna could be predicted. When the communication is continuous, the state switch of the transmission unit is not required, and the current working transmission unit will keep open. But when the communication is interrupted, the state switch is necessary. Namely, the interval time of beam switch could be calculated through the predicted status information of the opposite antenna, and then the current working unit is closed and the appropriate one will be opened. In order to enlarge the spatial coverage, adequate transmission units (such as spherical lens) will be needed. However, the volume of the communication antenna cannot be too large, so the beam splitter and its control device are needed. The laser control module selects the suitable lasers to open. The appropriate transmission unit is selected through the optical switch control module.

#### 2.1. **Array elements’ distribution**

The antenna is a cylinder-shaped structure with radius *R*, and height *H*. Transmission/receiving units are evenly distributed on the cylinder surface. The radius of the transmission unit and the receiving unit is _{${r}_{1}$}and _{${r}_{2}$}respectively, and the beam divergence angles of the transmission units are all_{$\theta $}. The distribution of the array elements is shown in Fig. 2. In a typical FSO system, the field-of-view (FOV) of the receiver is important. In our antenna structure, the FOV of each receiving unit can reach several degrees to dozens of degrees with different numerical aperture lens. At the same time, a large number of the receiving units are distributed in the same line. Through optimization design, the antenna can work as a receiver to receive signals in all-directions. For example, if the FOV of each receiving unit is 5 degrees and the number of the receiving units at the same line is 80, the equivalent FOV of the antenna can reach 360 degrees. In practical applications, the distribution of the array elements, the transceiver number and the optical properties (the beam divergence angle and FOV) can be adjusted according to different requirements. For examples, when the antenna is used for short-distance communication, it is suitable to enlarge the beam divergence angle and/or decrease the transceiver number; But for long-distance it is better to reduce the beam divergence angle and/or increase the transceiver number (accordingly decrease the FOV of the receiving unit).

Each transmission unit is numbered, so in the process of communication it can be switched through the corresponding driver circuit in control system.

In order to ensure the spatial coverage large enough and continuous in the height direction, the coverage area of arbitrary adjacent transmission unit lines should have overlap in excess of a certain distance.

#### 2.2. Antenna self-rotation design and bandwidth analysis

Considering the factors of communication distance and antenna size, the beam divergence angle of each transmission unit cannot be too big and the number of the transmission units cannot be too large, which lead to discontinuous coverage area of the transmission units in horizontal directions. In order to achieve 360 degrees spatial coverage, the beam control by the antenna’s rotation is chosen.

Without loss of generality, taking any one of the communication antennas as a transmitter, and the other as a receiver, Fig. 3 shows the schematic of the communication process between the two communication antennas. The situation 1 shows that the receiver has just entered into the transmitter’s coverage area; the situation 2 shows that the receiver is within the coverage area; the situation 3 shows that the receiver will leave the coverage area. In practical application, the two communication antennas work in full-duplex communication mode. The communication antennas are all transmitting/receiving antenna, and the communication process is similar to the above process.For terrestrial mobile FSO system, considering the situation that the antennas mainly move in the 2-D plane, the communication angle of a transmission unit is shown in Fig. 4(a). $\gamma $is the angle determined by the size of the receiving antenna. The communication distance is far greater than the radius of the antenna, so the transmitting antenna can be regarded as a point in the communication process. From Fig. 4(a), it can be found that the communication rotating angle of a transmission unit is _{$\phi \_com\text{=}\gamma +\text{\theta}$} whether $\gamma <\theta $ or$\gamma \ge \text{\theta}$. Set the distance between the communication antennas as $L(t)$, so_{$\gamma \approx 2\mathrm{arctan}(\text{R}/L(t))$}, and the communication rotating angle of each transmission unit during its working time is:

_{$$\phi \_com=2\mathrm{arctan}(\frac{R}{L(t)}\text{)}+\text{\theta}\approx \frac{2R}{L(t)}+\text{\theta}\text{.}$$}

The interrupted rotating angle of two adjacent transmission units can be expressed as follows, where *N* is the number of the transmission units in the same line:

_{$$\phi \_nocom=\{\begin{array}{l}\frac{2\pi}{N}-(\frac{2R}{L(t)}+\text{\theta ),}(2R/L(t)+\text{\theta )}N2\pi \\ \text{0,}(2R/L(t)+\text{\theta )}N\ge 2\pi \end{array}.$$}

_{$$\{\begin{array}{c}T{(t)}_{com\_once}=\ufffd\theta \text{+2}R\text{/}L\text{(t)}\ufffd/\omega \text{'}(t)\\ T{(t)}_{nocom\_once}\text{={}\frac{2\pi}{N}-(\frac{2R}{L(t)}+\text{\theta )}}/\omega \text{'}(t)\\ \omega \text{'}(t)=\omega +\nu (t)\mathrm{cos}(\beta )/L(t)\end{array}.$$}

When the distance between the antennas is very short or the relative velocity is very high, link break will occur. Therefore,$\omega \text{'}(t)$ must greater than or equal to 0, so:

when_{$\omega \text{'}(t)$}=0, $T{(t)}_{com\_once}$or $T{(t)}_{nocom\_once}$is infinity, which represents both of communication antennas keeping a relatively static status. It can result in continuous communication or link break, which is determined by the current communication status. For example, when $L(t)$changes from 30m to 2000m and $\nu (t)$from 15m/s to 50m/s, the calculation results are shown in Fig. 5. In order to show the applicable situation, we set $\beta $as zero to make the inequality stricter.

The first four lines which be clearly marked is for 30 meter, 40 meter, 50 meter, and 60 meter distance respectively. The rest lines are stand for the longer distance. The distance difference between any adjacent lines is 10 meters. Figure 5 depicts the change of _{$\omega $}while$\beta $=0, where X-axis expresses the relative velocity between the two antennas, and Y-axis expresses$\nu (t)/L(t)$, which equates with $\omega $. It is shown that all the conditions under the dotted line can support normal communication, i.e. link break doesn’t occur. In condition that _{$\omega $}is 0.78rad/s, when the distance is 30m, the relative velocity must be less than 23m/s (point a), otherwise the communication link will break; when $L(t)$is 40m, $\nu (t)$must be less than 31m/s (point b), otherwise the communication link will also break. Similarly, when$\omega $is 0.6rad/s and$L(t)$is 40m, the communication link will break if$\nu (t)$is greater than 24m/s (point c). It can be seen that at the same communication distance the relative velocity of the antennas reduces when the rotation rate decreases. In order to the increase the relative velocity or to enlarge the communication distance, the antenna rotation rate should have an appropriate increase. However, the high rotation rate will make the system unstable, so the rotation rate should be chosen eclectically according to the practical circumstances.

Because$L(t)$and$\nu (t)$are changing all the time, the interval time is a variable. In order to switch the transmission unit precisely, it is necessary to acquire$T{(t)}_{nocom\_once}$in real time. For most terrestrial FSO systems, the range of relative velocity is about 0m/s to 50m/s (equals to 0km/h to 180km/h). Since the antenna size is limited by its platform, the antenna radius should be not too large and within several decimeters. Meantime, the communication distance should be long enough, so the beam divergence angle should be small, which is within 0.8mrad-30mrad. From the Eq. (3), considering the ranges of the parameters mentioned above ($\nu (t)$, _{$R$}and$\text{\theta}$), when_{$L(t)$}is short, there exists a small variation range of$T{(t)}_{nocom\_once}$and the value of$T{(t)}_{nocom\_once}$is the order of 1/10 second. When_{$L(t)$}is long, $T{(t)}_{nocom\_once}$changes very little and approximates a constant, and the constant is within the above range. For example, set *N*=80, _{$\omega $}=0.78 rad/s, $\nu (t)$=20m/s, $\text{\theta}$=0.8mrad and *R*=0.3m. When$L(t)$is 50m, the range of$T{(t)}_{nocom\_once}$is from 0.055s to 0.171s; when_{$L(t)$}becomes longer than 200m, it is close to 0.1s. Therefore, the value of$T{(t)}_{nocom\_once}$is small, the terrestrial FSO system has a good real-time performance.

From Eq. (3), the bandwidth utilization rate$\eta $can be obtained as:

_{$$\{\begin{array}{c}\eta \text{=(}{T}_{com}/{T}_{all})=\frac{N(\theta +2R/L(t))/\omega \text{'}(t)}{2\pi /\omega \text{'}(t)}=N(\theta +\text{2}R/L(t))/\text{2}\pi \\ 2{r}_{1}N\le 2\pi R\end{array}.$$}

In Eq. (5), it can be seen that, compared with the communication distance and the antenna radius, the beam divergence angle and the number of transmission units have a great impact on the bandwidth utilization.

#### 2.3. Antenna tracking and control module

**2.3.1. The design of **antenna** tracking and control module**

In order to realize the beam control between the mobile communication antennas, the tracking and control module based on the EC-KPA is a core to the antenna. The main role of this module is trajectory prediction and beam switch. Through the predicted trajectory, the transmitting antenna can estimate the status of the receiving antenna very well, such as coordinate and velocity. When the communication process is interrupted, in order to precisely switch the transmission units, it is very important to accurately calculate the interval time of the two adjacent communication processes. That is, the predicted status information through the EC-KPA is used to calculate the interval time, and then according to the interval time, the antenna tracking and control module can control the beam switch. So the appropriate transmission unit will be opened to ensure the communication link stability.

In the acquisition stage, each antenna confirm its location information through GPS, at the same time all the transmission units are opened to capture each other. After a reliable communication link has been created, only the current working transmission unit is opened and the GPS will be closed. During the communication process, the real status information of the receiving antenna is fed back to the transmitting antenna, so the trajectory prediction is not necessary. But when the communication process is broken, using the feedback information gotten in the last time, the transmitting antenna predicts the trajectory of the receiving antenna through antenna tracking and control module, and then the interval time between the two adjacent communication processes can be calculated. After this interval time, the appropriate transmission unit will be opened to communicate with the receiving antenna. Because the communication processes is full-duplex, the opening or switching of the transmission units can be realized through the real-time feedback, i.e. the handshaking procedure. Of course, when the communication link is lost, the antennas will re-enter the acquisition stage. There is a special case: Owing to the rotation of the antenna, when both communication antennas keep a relatively static status, the communication process will continue or break. If broken, the non-communication time lasts, the acquisition process will begin until the communication link rebuilds.

### 2.3.2. Error correction Kalman prediction algorithm

The principle of KPA is to use the dynamic information of the antenna to acquire a precise predicted location [17,18]. At present, KPA is widely used in radar detection, image processing and many other fields [19,20]. KPA is described as follows [21]:

_{$$\{\begin{array}{c}{\hat{X}}_{k,k-1}={\Phi}_{k,k-1}{\hat{X}}_{k-1}+{W}_{k-1}\\ {\hat{Z}}_{k,k-1}={H}_{k}{\hat{X}}_{k,k-1}+{V}_{k}\\ {\hat{X}}_{k}={\hat{X}}_{k,k-1}+{K}_{k}({Z}_{k}-{H}_{k}{\hat{X}}_{k,k-1})\\ \begin{array}{c}{K}_{k}={P}_{k,k-1}{H}_{k}^{T}{[{H}_{k}{P}_{k,k-1}{H}_{k}^{T}+{R}_{k}]}^{-1}\\ \begin{array}{c}{P}_{k,k-1}=E\left[\left({X}_{k}-{\hat{X}}_{k,k-1}\right){\left({X}_{k}-{\hat{X}}_{k,k-1}\right)}^{T}\right]={\Phi}_{k,k-1}{P}_{k-1}{\Phi}^{T}{}_{k,k-1}+{Q}_{k-1}\\ {P}_{k}=E[\stackrel{~}{{X}_{k}}\stackrel{~}{{X}^{T}{}_{k}}]=[I-{K}_{k}{H}_{k}]{P}_{k,k-1}\end{array}\end{array}\end{array}.$$}

*k-1*and

*k*denote the moment

*k-1*and the moment

*k*respectively. The subscript

*k, k-1*denotes a predicted value of the moment

*k*in the moment

*k-1*. The superscript denotes the transpose of a matrix. $\hat{X}$is the predicted status, $\hat{Z}$is the observed variable. ${K}_{k}$is the Kalam gain matrix, ${P}_{k,k-1}$is the one-step variance matrix of the prediction error, ${P}_{k}$is the one-step variance matrix of filtering error, both$W$and$V$are noise matrixes. If the predicted interval time between the moment

*k*and the moment

*k-1*is very short, it can be considered that the relative velocity remains unchanged in this interval time, so the uniform motion model is built. Then, the status transition matrix${\Phi}_{k,k-1}$and the observation matrix${H}_{k}$can be defined as:

_{$${\Phi}_{k,k-1}=\left[\begin{array}{cc}\begin{array}{cc}1& \Delta t\\ 0& 1\end{array}& \begin{array}{cc}0& 0\\ 0& 0\end{array}\\ \begin{array}{cc}0& 0\\ 0& 0\end{array}& \begin{array}{cc}1& \Delta t\\ 0& 1\end{array}\end{array}\right],{H}_{k}=\left[\begin{array}{c}\begin{array}{cccc}1& 0& 0& 0\end{array}\\ \begin{array}{cccc}0& 0& 1& 0\end{array}\end{array}\right].$$}

*k*and the moment

*k-1*.

For terrestrial mobile FSO system, because the antennas mainly move in the 2-D plane, along X axis and Y axis, let the status variable${X}_{k}$contain the coordinate value and velocity respectively, namely${X}_{k}={\left(\begin{array}{cccc}{x}_{k}& x{v}_{k}& {y}_{k}& y{v}_{k}\end{array}\right)}^{T}$. The observed variable${Z}_{k}$only contains the coordinate values, namely${Z}_{k}={\left(\begin{array}{cc}{x}_{k}& {y}_{k}\end{array}\right)}^{T}$. Using the initial information (${X}_{1}$and${Z}_{1}$), the predicted information${X}_{k}$can be obtained through *k-1* time’s iterative calculation.

Because the movement velocity is variable in the predicted interval time, a prediction deviation is produced in the uniform motion model. The prediction error of the KPA (from moment *k-1* to moment *k*) consists of two parts: residual error${\hat{\Delta X}}_{k-1}$and the new-added error${D}_{k}$. The residual error mainly results from the initial error and the accumulated error mainly results from the prediction process, and the new-added error is the prediction error introduced at the current moment. So, the prediction error is derived as:

*k*is approximately equal to the one (${\hat{\Delta X}}_{k-1}$) of the moment

*k-1*(reflected in coordinate like$\Delta {\widehat{x}}_{k}\approx \Delta {\widehat{x}}_{k-1}$,$\Delta {\widehat{y}}_{k}\approx \Delta {\widehat{y}}_{k-1}$). In KPA, the actual value of the moment

*k*equals to the sum of the predicted value and the prediction error: ${\hat{X}}_{k,k-1\_acture}={\hat{X}}_{k,k-1}+{\hat{\Delta X}}_{k,k-1}$. The predicted value${\hat{X}}_{k,k-1}$is the final result for predicting the status of the moment

*k*generally and the prediction error is${\hat{\Delta X}}_{k,k-1}$, but it is impossible to get${\hat{\Delta X}}_{k,k-1}$until the moment

*k*arrives. Because$\Delta t$is very small, and the acceleration in$\Delta t$,$a(t)$is not large,${D}_{k}$stays at a very small level.

Therefore, it can be concluded that${\hat{X}}_{k,k-1\_acture}={\hat{X}}_{k,k-1}+{\hat{\Delta X}}_{k,k-1}\approx {\hat{X}}_{k,k-1}+{\hat{\Delta X}}_{k-1}$. Now, the new prediction result,${\hat{X}}_{k,k-1}+{\hat{\Delta X}}_{k-1}$is closer to the actual result than${\hat{X}}_{k,k-1}$, so the new prediction error${\hat{\Delta X}}_{k,k-1\_new}={\hat{X}}_{k,k-1\_acture}-{\hat{X}}_{k,k-1}-{\hat{\Delta X}}_{k-1}={\hat{\Delta X}}_{k,k-1}-{\hat{\Delta X}}_{k-1}$.

This improved prediction algorithm is called the error correction Kalman prediction algorithm. The prediction error of KPA and the EC-KPA is expressed respectively:

_{$$\begin{array}{l}{\hat{\Delta X}}_{k,k-1}={\Phi}_{k,k-1}{\hat{\Delta X}}_{k-1}+{D}_{k}\\ =\left[\begin{array}{cc}\begin{array}{cc}1& \Delta t\\ 0& 1\end{array}& \begin{array}{cc}0& 0\\ 0& 0\end{array}\\ \begin{array}{cc}0& 0\\ 0& 0\end{array}& \begin{array}{cc}1& \Delta t\\ 0& 1\end{array}\end{array}\right]\cdot {\left[\begin{array}{cccc}\Delta {x}_{k-1}& \Delta x{v}_{k-1}& \Delta {y}_{k-1}& \Delta y{v}_{k-1}\end{array}\right]}^{\text{T}}+{D}_{k}\\ =\left[\begin{array}{c}\Delta {x}_{k-1}+\Delta x{v}_{k-1}\cdot \Delta t\\ \Delta x{v}_{k-1}\\ \Delta {y}_{k-1}+\Delta y{v}_{k-1}\cdot \Delta t\\ \Delta y{v}_{k-1}\end{array}\right]+{D}_{k}.\end{array}$$}

_{$$\begin{array}{l}{\hat{\Delta X}}_{k,k-1\_new}={\hat{\Delta X}}_{k,k-1}-{\hat{\Delta X}}_{k-1}\\ ={\Phi}_{k,k-1}{\hat{\Delta X}}_{k-1}+{D}_{k}-{\hat{\Delta X}}_{k-1}\\ =\left[\begin{array}{c}\Delta {x}_{k-1}+\Delta x{v}_{k-1}\cdot \Delta t\\ \Delta x{v}_{k-1}\\ \Delta {y}_{k-1}+\Delta y{v}_{k-1}\cdot \Delta t\\ \Delta y{v}_{k-1}\end{array}\right]+{D}_{k}-\left[\begin{array}{c}\Delta {x}_{k-1}\\ \Delta x{v}_{k-1}\\ \Delta {y}_{k-1}\\ \Delta y{v}_{k-1}\end{array}\right]\\ =\left[\begin{array}{c}\Delta x{v}_{k-1}\cdot \Delta t\\ 0\\ \Delta y{v}_{k-1}\cdot \Delta t\\ 0\end{array}\right]+{D}_{k}.\end{array}$$}

Through the comparison between the EC-KPA and the KPA, it is shown the EC-KPA has higher tracking accuracy and strong anti-noise ability. Because our antenna model has a small interval time ($\Delta t$, i.e. from *k-1* to *k* moment) between the two adjacent communication processes, and the beam switch is based on the accurate calculation of the$\Delta t$by using of the predicted trajectory, the conventional tracking algorithm KPA is not suitable for our antenna. The proposed EC-KPA is based on the discontinuous tracking and can accurately calculate the value of$\Delta t$ that is to say that the beam switch can realize effectively.

In our antenna system, the interrupted time $T{(t)}_{nocom\_once}$between adjacent two communications is very short, so $T{(t)}_{nocom\_once}$can be regarded as$\Delta t$. The coordinate components of${D}_{k}$can be expressed as:

_{$${D}^{c}{}_{k}\approx \pm \frac{\overline{a(t)}}{2}{\Delta}^{2}t\text{=}\pm \frac{\overline{a(t)}}{2}{\{[\frac{2\pi}{N}-(\frac{2R}{L{({t}_{k})}^{1}}+\text{\theta )]}/\omega \text{'}{({t}_{k})}^{1}\}}^{2}.$$}

*k.*

From Eq. (10) and Eq. (11), it can be seen that the residual error and the new-added error is proportional to$\Delta t$and${\Delta}^{2}t$respectively. Because$\Delta t$is very short in our antenna model, the revised predicted result will be more accurate, which can make the beam switch more precise.

The prediction error of the prediction algorithm can cause the miscalculation of$\Delta t$. When the error is large, the communication break would occur in the process of the transmission unit’s switch. It is because that the receiving antenna in fact has moved out of the coverage area of the transmission unit that will be opened after the interrupted time. Therefore, the influence of the EC-KPA’s prediction error on communication performance can be evaluated through the ratio of the prediction error and the width of the communication coverage. Because the acceleration has little change in the two adjacent interrupted times (such as in the interrupted time from *k-2* to *k-1*moment, and in the one from *k-1* to *k* moment), the following approximation can be made:$\Delta x{v}_{k-1}\cdot \Delta t\approx \overline{a(t)}\cdot {\Delta}^{2}t$. In the 2-D plane, under a certain communication distance, the width of the communication coverage (i.e. the light spot’s diameter of the coverage area) is$\theta L(t)$. Without loss of generality, considering the component of X-axis, the ratio of the prediction error and the width of the communication coverage,$\mu $is derived as:

_{$$\mu =\frac{{D}^{c}{}_{k}\text{+}\Delta x{v}_{k-1}\cdot \Delta t}{\theta L({t}_{k})}=\frac{3\overline{a(t)}{\Delta}^{2}t}{2\theta L({t}_{k})}=\frac{\text{3}\overline{a(t)}}{\text{2}\theta L({t}_{k})}{\{[\frac{\text{2}\pi}{N}-(\frac{\text{2}R}{L{({t}_{k})}^{1}}+\text{\theta )]}/\omega \text{'}{({t}_{k})}^{1}\}}^{2}.$$}

*k*.

## 3. Simulation results and discussion

Taking into account the attenuation of the transmitting system *A _{T}*, the attenuation of the receiving system

*A*, and the loss of light pulse in wireless laser communication which includes the geometric attenuation and the transmission attenuation, in order to ensure the detector correct reception of optical signals, it is required that:

_{R}_{$$\text{S}\le {P}_{T}-\text{20}\mathrm{lg}(\theta L(t)\text{/(2}{r}_{2}))-\alpha L(t)-{A}_{T}-{A}_{R}-M.$$}

*S*is the receiving sensitivity; and

*P*is the transmission power. In practical application the field distribution of laser beam is usually Gaussian. Due to the beam divergence angle is defined as the angle when the light power drops to 1/e

_{t}^{2}of the peak value, in order to accurately describe the Gaussian field distribution model at the receiver side, a power margin

*M*(M=3dB) is considered in our system.

Considering the normal weather condition and the inequality in Eq. (5), we set$\alpha $ as 1dB/km and other parameters are shown in Table 1. Combined Eq. (13) with inequality (4), the communication distance is within 0.05km and 1.4km.

Primarily, the performance of the EC-KPA is analyzed. Set the antennas’ velocity along X axis and Y axis varying from 15m/s to 25m/s, the acceleration being within 0m/s^{2}-10m/s^{2}, and $\theta $=0.8mrad. Other parameters are the same as Table 1. In Eq. (3), the interval time (the interrupted time between the moment *k* and the moment *k-1* in KPA) $\Delta t$is within 0.096s-1.23s, so the step of the simulation is set as 0.1s. The total simulation time is 100s.

Figure 6 shows the performance of tracking antenna through KPA and EC-KPA with weak noise respectively. Here the average noise power is set as 0dBm. Due to the communication distance varies from 50m to 1000m, without loss of generality, the communication distance is chosen from 840m to 850m here. It can be seen that the predicted trajectory almost completely overlap the actual trajectory in the right figure, and the prediction error of KPA is about 1m in the left figure, so it is verified that EC-KPA performs more efficiently than KPA in our proposed antenna model.

Figure 7 shows the performance of tracking antenna through KPA and EC-KPA with strong noise respectively. In strong noise situation, the atmosphere effects have a great influence on the propagation of the laser beam, which often leads to the beam scintillation. Here, the average noise power is increased by 30 times which equals to 14.77dBm [22]. The result shows that under a strong noise condition the EC-KPA preforms much better than KPA. The prediction deviation of KPA increases to 4m, but that of EC-KPA is quite small. Through the calculation, it is found that the EC-KPA’s prediction accuracy is increased by 12.92 times to KPA’s. Comparing the Eq. (9) and Eq. (10), the major difference is that EC-KPA reduces the residual error. As for the residual error, the noise has great influence on it, so the EC-KPA has a strong resistance to noise than KPA. That is, our proposed EC-KPA can improve the antenna efficiency under the complex environment.

The prediction error of KPA on X axis and Y axis is depicted In Fig. 8. It is shown that the prediction error is still less than 1m, when the velocity of the antenna reaches 20m/s. Through 100 seconds simulation, the average prediction error on X axis and Y axis is 0.3182m and 0.3109m respectively. Figure 9 shows the prediction error of the EC-KPA on X axis and Y axis, in which_{$\theta $}is within 0.8mrad to 30mrad; _{$\overline{a(t)}$} is from 0m/s^{2} to 10m/s^{2}; and the other parameters are identical to Table 1. Comparing with Fig. 8, the maximum prediction error decreases from 1.5m to 0.15m. From Eq. (10), the maximum of the acceleration is 10m/s^{2}, so the maximum of ${\hat{\Delta X}}_{k,k-1\_new}$is 0.15m, which includes 0.1m of the residual error ${\hat{\Delta X}}_{k-1}$and 0.05m of the new-add error${D}^{c}{}_{k}$. Therefore, the theoretical calculation results are consistent with our simulation results. Through 100 seconds simulation, the average prediction error on X axis and Y axis is 0.0729m and 0.0736m respectively. It can be known that the prediction accuracy of the EC-KPA is improved by an average of 77% than that of KPA. Based on the above analysis, it can be conclude that the EC-KPA predicts the trajectory and calculates $T{(t)}_{nocom\_once}$more accurately than KPA, and the switch of the transmission units is more effective.

Besides, the influence of the EC-KPA’s prediction error on communication performance is discussed. From Fig. 8 and Fig. 9, it can be seen that at 850m communication distance, the maximum prediction error of the EC-KPA is 0.15m, which just takes up 22.05% width of the communication coverage. Comparing with the ratio of 147.05% in KPA, the impact of the prediction error on the communication performance becomes very small. The 147.05% width of the laser spot means that the receiving antenna in fact has moved out of the coverage area of the transmission unit that will be opened after the interrupted time, and this situation leads to communication break.

In Fig. 10, it can be seen that, with the same $\theta $, the $\overline{a(t)}$impacts the$\mu $linearly. For example, when $\theta $=18.0mrad and $\overline{a(t)}$=2.107m/s^{2}, the$\mu $≈0.6846%; when$\theta $=13.6mrad and$\overline{a(t)}$=1.094m/s^{2}, the $\mu $≈0.4736%. It can be also seen that, with the same $\overline{a(t)}$, the smaller beam divergence angle has a larger effect on$\mu $. Set $\overline{a(t)}$as 2.047m/s^{2}, when$\theta $becomes from 20.7mrad to 14.5mrad, the $\mu $ increases 0.262%. In order to reduce$\mu $, we can increase the $\theta $, but will lead to short communication distance, so the other ways should be considered such as increasing the number of the transmission units or speeding up the rotation rate of the antenna. But a large number of transmission units will increase the cost and the difficulty of the antenna’s manufacture. In addition, the high rotation rate will make the antenna unstable. So, it is necessary to eclectically select the number of the transmission units, the rotation rate of the antenna and the beam divergence angle and so on in practical applications.

Finally, the relationships among the bandwidth utilization, the communication distance and the beam divergence angle are analyzed. The antenna parameters are also the same as Table 1, and the simulation results are shown in Fig. 11. It can be seen that when the beam divergence angle increases or the communication distance decreases, the bandwidth utilization will rise. For example, when $\theta $=0.8mrad and$L(t)$=890m, $\eta $is 1.87% (the actual bandwidth is 18.7MHz and the original bandwidth is 1GHz). But when_{$\theta $}=9.8mrad and_{$L(t)$}=90m, _{$\eta $}is 20.97%, i.e. the actual bandwidth is 209MHz. Therefore, in order to achieve better communication performance (long communication distance, high bandwidth utilization, and high tracking accuracy), it is important to adjust the design parameters eclectically in certain situation, such as beam divergence angle, the number of the transmission units, the rotation rate of the antenna and so on.

## 4. Conclusion

Due to the notable influence of atmospheric effects and the objects’ relative motion, APT is the key technologies to establish the communication link in FSO system. In this paper, being aiming at ensuring the communication link stability in terrestrial fast mobile FSO system, a new omni-directional optical antenna is designed, and a novel beam control method based on the EC-KPA is proposed. Considering the factors like the communication distance and the antenna’s size, the divergence angle of laser transmission unit cannot be too big and the number of the transmission units cannot be too large, which lead to discontinuous coverage area of the two adjacent transmission units. In order to cover 360 degrees space and to increase communication range, the special transmission/receiving units’ distribution is designed and the self-rotation antenna is adopted.

Even so, two adjacent communication processes still have a very small interval time ($\Delta t$). The accurate calculation of$\Delta t$is the key technology to open the appropriate transmission unit. In order to solve the problem of the discontinuous tracking in our antenna, the beam control method based on the EC-KPA is investigated theoretically and numerically. That is, the predicted status information through EC-KPA is used to calculate the** _{$\Delta t$}**, and then according to the interval time, the antenna tracking and control module can control the beam switch. So the appropriate transmission unit will be opened to ensure the communication link stability. Furthermore, the bandwidth utilization is analyzed and the prediction error of the EC-KPA on communication performance is evaluated. The simulation results show that the average prediction accuracy of the EC-KPA is improved 77% than that of the conventional KPA with weak noise (Gaussian noise), and that the prediction accuracy difference between EC-KPA and KPA can reach 12.92 times with strong noise (including the effect of atmospheric scintillation and atmospheric turbulence). It is also shown that the maximum prediction error of EC-KPA on communication performance just takes up 22.05% width of the communication coverage comparing with the ratio of 147.05% in KPA.

Based on the theoretical calculation and simulation analyses, there is reason to believe that through optimizing the parameters and the distribution of the transmission/receiving units, this kind of optical antenna and its beam control method can improve the efficiency of fast mobile FSO in certain application situations.

## Acknowledgment

This work was supported by the National Natural Science Foundation of China (Grant No.61172080)

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