## 2 Error Analysis of the Center of Mass Method
### Discrete Approximation Error  
The star image sampling process includes three steps for star sensors, as illustrated in Figure 1. Firstly, the ideal star signal intensity function \(I(x)\) projected on the surface of the Charge-coupled Device (CCD) is convoluted with the pixel response function \(p(x)\) to generate the continuing pixel signal function \(f(x)\). Then, after multiplying the pixel sampling function \(s(x),f(x)\) is discretized and the corresponding sampled pixel signal function \(f_{s}(x)\) is obtained. Finally, \(f_{s}(x)\) is multiplied by the window response function \(r(x)\) to give the windowed pixel signal function \(g(x)\) utilized in star centroid estimation by the CoM method. The above process can be written as  
\[g(x)=f(x)\times s(x)\times r(x)\]where  
\[f(x)=I(x,x_{0})\otimes p(x) \tag{5}\]  
where \(I(x,x_{0})\) denotes the ideal star signal intensity function with ideal centroid position \(x_{0}\) in the \(x\) direction of the image plane. We assume that the fill factor is 100%, pixel sensitivity is uniform, and each pixel in the image plane has the same photon response. Then, \(p(x)\) and \(r(x)\) are equal to the rectangle function \(rect(x)\), \(s(x)\) is equal to a comb function \(comb(x)\) with \(1/T\) sampling frequency (\(T\) is the length of a pixel), and Equation (4) can be written as  
\[g(x)=I(x,x_{0})\otimes rect(\frac{x}{T})\times comb(\frac{x}{T})\times rect( \frac{x}{n_{w}T}). \tag{6}\]  
If we represent all distances in unit \(T\), Equation (6) can be rewritten as  
\[g(x)=I(x,x_{0})\otimes rect(x)\times comb(x)\times rect(\frac{x}{n_{w}}). \tag{7}\]  
In this section, we discuss the systematic error under the case where almost the complete sampled pixel signal \(f_{s}(x)\) is included in the window, so \(f_{s}(x)\) can be considered equivalent to \(g(x)\); then  
\[g(x)=I(x,x_{0})\otimes rect(x)\times comb(x). \tag{8}\]  
According to the properties of Fourier transform, the estimated centroid \(\hat{x}_{g}\) is represented as follows [15]:  
\[\hat{x}_{g}=\frac{\int_{-\infty}^{\infty}xg(x)dx}{\int_{-\infty}^{\infty}g(x) dx}=-\frac{G^{\prime}(0)}{2\pi iG(0)}. \tag{9}\]  
where \(G(s)\) is the Fourier transform of \(g(x)\). We set a pixel center as the coordinate system origin \(O\); then the function \(f(x)\) of the star spot located at \(d\) can be expressed as the function \(f_{e}(x)\) shifted by offset \(d\) from the origin \(O\), i.e.,  
\[f(x)=f_{e}(x-d). \tag{10}\]  
From Equations (5) and (10), we notice that \(d=x_{0}\). Thus, the Fourier transform of the function \(f(x)\) is  
\[F(s)=\exp(-2\pi ids)F_{e}(s). \tag{11}\]  
where \(F_{e}(s)\) is the Fourier transform of \(f_{e}(x)\). From Equations (8) and (11), \(G(s)\) is given as  
\[G(s)=F(s)\otimes S(s)=\exp(-2\pi ids)F_{e}(s)\otimes S(s). \tag{12}\]  
\[g(x)=f(x)\times s(x)\times r(x)\]0 Decomposed graph of the star image sampling process. where \(S(s)\) is the Fourier transform of \(s(x)\). Then, we have the derivative of \(G(s)\):  
\[ \begin{array}{ll}G^{\prime}(s)&=F^{\prime}(s)\otimes S(s)\\ &=[-2\pi id\exp(-2\pi ids)F_{e}(s)+\exp(-2\pi ids)F_{e^{\prime}}(s)]\otimes S(s).\end{array}  \tag{13}\]  
Substituting Equations (12) and (13) into Equation (9) yields  
\[\hat{x}_{g}=-\frac{G^{\prime}(0)}{2\pi iG(0)}=d-\frac{\exp(-2\pi ids)F_{e^{ \prime}}(s)\otimes S(s)}{2\pi i\exp(-2\pi ids)F_{e}(s)\otimes S(s)}\bigg{|}_{s =0}. \tag{14}\]  
Therefore, the systematic error is obtained as  
\[\sigma_{\hat{x}_{g}}=\hat{x}_{g}-x_{0}=-\frac{1}{2\pi i}\frac{\sum_{n=-\infty} ^{\infty}\exp(2\pi idn)F_{e^{\prime}}(-n)}{\sum_{n=-\infty}^{\infty}\exp(2\pi idn )F_{e}(-n)}. \tag{15}\]  
Equation (15) is the general expression of the systematic error in the CoM method considering only the sampling frequency limitation. Then, we can derive a more explicit relationship by substituting the specific function \(f(x)\) of the star spot in the image sensor into the above equation.