Optimum range of angle tracking radars: a theoretical computing

Received Mar 7, 2018 Revised Dec 22, 2018 Accepted Jan 3, 2019 In this paper, we determine an optimal range for angle tracking radars (ATRs) based on evaluating the standard deviation of all kinds of errors in a tracking system. In the past, this optimal range has often been computed by the simulation of the total error components; however, we are going to introduce a closed form for this computation which allows us to obtain the optimal range directly. Thus, for this purpose, we firstly solve an optimization problem to achieve the closed form of the optimal range (Ropt.) and then, we compute it by doing a simple simulation. The results show that both theoretical and simulation-based computations are similar to each other.


INTRODUCTION
There are different kinds of radar systems for different applications in industries, airports, military organs such as navy and so on. A type of radars is the tracking radars which are applied in fire control and etc [1]. Generally, two classes of tracking radars can be considered as angle tracking and range tracking radars. Our aim in this study attends to the first class. Figure 1 shows a real sample of tracking radar which has been placed on a ground platform for the fire control usages. Tracking radar for finding angle can use several tracking modes (scanning techniques) including sequential lobing, conical scan and monopulse (e.g. in terms of amplitude) [1]. These three techniques are different in terms of the error status. So, the mode used in each angle tracking radar must be considered in computing the error and then the optimal range of radar. Figure 2 shows a schematic which explains the mechanism of angle tracking radars. From step 1 to step 6, a target is seen in PPI display which tracked during these steps. Figure 3 and Figure 4 show the block diagram of two types of angle tracking radars based on monopulse and conical scan techniques, respectively [2], [3]. In [1]- [3], it is explained that in terms of angle error, two techniques of sequential lobing and conical scan are similar to each other. In the next parts, we specify the differences between the impacts of monopulse and conical scan/sequential lobing (non-monopulse) techniques. Computing the optimal range of each angle tracking radar is an optimization problem which depicts one of the main parameters of the radar. In [1], this problem has been represented, but it is only solved by simulation and numerical computation. As a completing approach for [1], we wish to solve this problem theoretically. In addition, we obtain the simulation result to compare to the proposed approach.
In the appendix of this paper, we give the readers a sample MATLAB code in order to the simulation-based computing of the optimal range. This paper is organized as follows. In the second part, we have an overview on the resources of the angle error. In the third part, we theoretically solve the problem. In the forth part, we solve a typical example based on both simulation-based and theoretical approaches which allows us to have a comparison between them. The latest parts are allocated to the conclusions of the work and the appendix.

RESOURCES OF ERROR IN ATRS
According to [1], the general standard deviation of the angle error (  t ) in ATRs is mainly depended on five components (five different standard deviations) including angle fluctuation error related to angle noise and glint phonomena (  g ), mechanical error (  mech. ), control loop error (  cont. ), scintillation error related to amplitude fluctuation and target fading (  sc. ). And finally the error created by thermal noise of the radar system (  n ). As follows, we introduce the detailed information about  t . (1) shows the general formula of  t .
In the equations, L is the largest size of the target and R is the distance between radar and target.

SNR
is signal-to-noise ratio of radar while receiving an echo from the target. SNR is based on the radar equation; we will discuss more about it later. PRF shows the pulse repetition frequency which is equal to the inverse of pulse repetition interval (shown by T p ) in pulse radar-based tracking systems. About SNR and radar equation, we can say that SNR is desired at the input side of the receiver and is therefore computable by the radar equation as (7); for simplicity, we omit some details of radar equation which are reachable in [1], [4].
Where t P the maximum of transmission power of the radar is,  is the wavelength used in the radar. a G is antenna gain of the radar (assume that we use a monostatic pulse radar for angle tracking) and described by (8) where 32000  G , and L el and L az are the size of the radar's antenna for elevation and azimuth, respectively, see more details in [1].
RCS shows the radar cross section of target and is not often fixed, especially in fast maneuvering and high rate fluctuation, however it is usually replaced with an average value which is at most equal to the geometrical cross section of the target (target size). So in here, we suppose that it is a constant. Finally, noise P is power of the thermal noise of radar's receiver which is computed by (9); in this equation B is the noise bandwidth of the radar's receiver in which In addition, there is another form for representing the noise power noise P , refer to [1]. It is observable that in a specific example, all values of

PROPOSED METHOD
In this part, the proposed method is represented based on solving a mathematical optimization problem. In the previous part, we reviewed all details of angle error. It is noticeable that in an example with the specified parameters,  t is only in terms of R (distance to the target). Therefore, if we minimize the general standard deviation for a sample range such as .  opt RR , then the total angle error becomes the minimum (because the mean of error variable is assumed to be zero, so a minimum variance/standard deviation shows the lowest error). In this case, . opt R is the optimal range. The equation of  t is rewritten as (10). If the SNR equation is to be rewritten as (11), we accordingly have  t as (12).  In (12), all K 1 to K 4 coefficients are fixed, so we find the optimal range which can minimize the term for  t , as (13). After solving the optimization problem, we see that K 1 has no impact on the The roots of (13) include all roots of two new equations of  The achieved relation for the optimal range is a theoretical solution. This recent equation is a generally closed form for all scanning techniques due to the consideration of the coefficient 2 q (embedded in K 2 ) and the standard deviation of .  sc in terms of different aspects related to the scanning techniques. In order to have a clear approach, we integrate all discussed items through the Consequence 1. Consequence 1. The optimal range of an angle tracking radar for all scanning techniques is as (15)-(18). 6 1 2 3 All angles are in degree(s) and other parameters are in linear space (not in dB) and some of them which have physical dimension are in S.I. units. In the next part, we solve an example based on the proposed approach and then compare it to the simulation result.

NUMERICAL EXAMPLE AND SIMULATION
In this part, we firstly use the MATLAB code in the appendix to solve the problem by simulation (classically). Its result is shown in Figure 5. It is clear that the optimal range when the simulation parameters are selected as Table 1 is between 5 and 10 meters.
Note that in theoretical computation of the optimal range, some of the parameters like v k max ,  and so on are not generally used (against the simulation-based solution, The MATLAB code for solving the problem based on simulation is in Table 2.).

CONCLUSIONS
In this work, our aim is to introduce a computational method [5]- [10] in which we theoretically computed the optimal range of ATRs. Our approach's result was matched with the simulation result in which they are sufficiently similar to each other. In fact, main aim of the work is to create an extension for the approaches represented in [1]. The proposed approach is suitable for educational aspects too.