# Analysis of SNR for High-Orbit Target Detected by Ground-Based Photoelectric System
## 2 Signal-to-Noise Model of Satellite  
The HOT works at a distance farther than 8000 km above the Earth, where the environment around the target is at absolute zero temperature; therefore, the target can be regarded as a point source black-body from the observer, the schematic of target model is shown in Figure 1.  
According to the definition of [PERSON]'s black-body radiation spectrum formula, the total spectral radiance of the target relative to the observer can be obtained by  
\[L_{t}(\lambda,T)=\frac{2h\epsilon(\lambda,T)c^{2}A_{t}}{\lambda^{5}}\Big{[}e^ {\frac{hc}{kT}}-1\Big{]}^{-1} \tag{1}\]  
where \(T\) is the temperature of the radiation cavity of the black-body, \(A_{t}\) is the area seen from the observer, \(\lambda\) is the detecting band, \(\epsilon(\lambda,T)\) is the effective emissivity, \(h\) is the Planck constant, and \(c\) is the speed of light.  
The power of the target through the atmospheric incidence on the telescope pupil entrance can be expressed as  
\[E_{t}(\lambda)=\tau_{a}(\lambda)L_{t}(\lambda,T)\frac{A_{o}}{R^{2}} \tag{2}\]  
where \(\tau_{a}\) is the atmospheric transmission, \(A_{o}\) is the input pupil area of the telescope, and \(R\) is the distance from target to observer.  
Figure 1: The schematic of infrared telescope detecting the high-orbit target (HOT).  
The energy received by the telescope was reflected repeatedly by an optical system, then it reached an infrared focal plane array. The integration time of the detector varied with different circumstances, so the energy that eventually reached the focal plane can be expressed as follows [9]:  
\[E_{d}(\lambda,t)=\int\limits_{t_{i}}^{t_{f}}dt\int_{\lambda_{i}}^{\lambda_{f}}d \lambda\tau_{a}\tau_{o}(\lambda)L_{t}(\lambda,T)\frac{A_{o}}{R^{2}} \tag{3}\]  
where \(\tau_{o}\) is the transmission of the optics and \(t\) is exposure time of detector.  
When telescopes track targets, the sky background covers the entire view of the telescope. The individual field of view for one pixel can be defined as the source area, and the power incident on one pixel is expressed by Reference [10]:  
\[L_{B}(\lambda,t)=\ \tau_{o}E_{B}(\lambda,t)A_{d}\frac{\pi}{4F^{2}} \tag{4}\]  
where \(E_{B}\) is the spectral radiation of the sky, \(F\) is the f-number of the system, and \(A_{d}\) is the area of the detector pixel.  
Infrared detector noises included readout noise, dark current noise, and photon noise [11]. Therefore, the energy of the target and sky background was converted into electronics to calculate the SNR using the formula \(\left(\frac{\lambda}{hc}\right)\) to convert joules to photons. In the range of the infrared shortwave, photons and electrons are converted equally; however, the availability electronic number depends on the quantum efficiency \(q(\lambda)\) of the detector. So, the electronic of the target and sky background is  
\[S_{t}(\lambda,t)=\int\limits_{t_{i}}^{t_{f}}dt\int_{\lambda_{i}}^{\lambda_{f}} d\lambda\tau_{a}\tau_{o}q(\lambda)L_{t}(\lambda,T)\frac{A_{o}\lambda}{R^{2}hc} \tag{5}\]  
\[S_{B}(\lambda,t)=\ \tau_{o}E_{B}(\lambda,t)A_{d}q(\lambda)\frac{\pi}{4F^{2}} \left(\frac{\lambda}{hc}\right). \tag{6}\]  
As HOTs were too far away from the telescope, the target was projected on the focal plane in point form. Assuming that the target occupied only one pixel, the SNR can be expressed as  
\[\text{SNR}=\frac{S_{t}(\lambda,t)}{\sqrt{S_{t}(\lambda,t)+S_{B}(\lambda,t)+{ \sigma_{n}}^{2}+tD_{e}+\sigma_{l}}} \tag{7}\]  
where \(\sigma_{n}\) is the number of read noise electrons per pixel, \(D_{e}\) is the dark current noise expressed in electrons, and \(\sigma_{l}\) is the noise of stray light.