# An Observation Scheduling System for Radio Telescope Array
## 3 The Overall Architecture of the System
### uv Coverage Module  
_uv_ coverage is one of the basic functions of the radio astronomical observation scheduling system. Radio astronomical observation schedules are generally created through the optimization of uv coverage. The SCHED software, for instance, can plot the uv coverage in a schedule and supports moving stations around to explore _uv_ coverages in array configuration design projects. Therefore, we also include this basic function in the system. This function in our system has two main goals: one is to make the astronomical schedule plans for the current telescope array; the other is to determine the optimal layout of the upgraded telescope array. The theory of uv coverage is introduced in detail below. Synthetic aperture is an important radio astronomy observation technique that provides high angular resolution images of radio sources by measuring the phase differences  
Figure 5: Flowchart of the calibrator prediction module. between radio telescopes, a method that has been widely used in the field of radio astronomy [12]. During synthetic aperture measurements, radio telescopes interfere in pairs (i.e., interferometers), leveraging Earth's rotation to form multiple distinct interference baselines. These baselines sample the area of the sky where the radio source is located [13]. Following the standard processing steps in radio astronomy, the sampling data of radio sources yield sky images of the radio sources. To facilitate the transformation from the sampling domain to the sky image domain, the foremost task is to establish the sampling coordinate system and the imaging coordinate system [14]. The sampling coordinate system is referred to as the (\(u,v,w\)) coordinate system. Generally, the projection of the radio telescopes' baselines (the distance between two telescopes) in the (\(u,v,w\)) coordinate system is referred to as \(uv\) coverage. \(uv\) coverage reflects the degree of sampling of the target source by the telescope array and indirectly indicates the quality of the telescope array layout. The following introduces the (\(u,v,w\)) coordinate system, which is derived from the transformation of the coordinate system of radio telescope antennas. As shown in Figure 6, this diagram illustrates the transformation relationship between the telescope antenna coordinate system and the sampling coordinate system. In the diagram, the (X, Y, Z) coordinates represent the equatorial coordinate system, where the X and Y planes are parallel to the celestial equator. The X-axis is aligned along the meridian of the reference station of the radio telescope array, the Y-axis points East, and the Z-axis points towards the North Pole. In the (\(u,v,w\)) coordinate system, the \(w\)-axis is directed towards the target. The angle between the \(w\)-axis and the XOY plane denotes the declination (\(\delta\)) of the target S, while the angle between the meridian plane of the \(w\)-axis and the local meridian plane signifies the hour angle (\(H\)). The \(u\)-axis lies in the XOY plane, and the angle between the \(u\)-axis and the Y-axis represents the hour angle (\(H\)). The zenith angle of the meridian plane containing the \(v\)-axis represents the declination (\(\delta\)) of the radio source. Given the components of a baseline vector (\(D_{\lambda}\)) in the (X, Y, Z) coordinate system: \(X_{\lambda}\), \(Y_{\lambda}\), \(Z_{\lambda}\), and in accordance with the coordinate transformation relationships illustrated in Figure 6, the components in the (\(u,v,w\)) coordinate system are calculated as follows [12]:  
Figure 6: Diagram illustrating the transformation between antenna coordinate system (X, Y, Z) and sampling coordinate system (\(u,v,w\)). Modified from [12]. \[ \begin{bmatrix}u\\ v\\ w\end{bmatrix} = \begin{bmatrix}\sin(H)&\cos(H)&0\\ -\sin(\delta)\cos(H)&\sin(\delta)\sin(H)&\cos(\delta)\\ \cos(\delta)\cos(H)&-\cos(\delta)\sin(H)&\sin(\delta)\end{bmatrix}\begin{bmatrix} X_{\lambda}\\ Y_{\lambda}\\ Z_{\lambda}\end{bmatrix}