#### 5.1.1. Optical Lens Analysis

Theoretically, starlight distribution before the optical lens can be regarded as an energetic impulse function

Eδ(

x−

x_{0},

y−

y_{0}). When the lenses are modulated, the impulse point blurs into a spot. Assuming that the lenses are equivalent to an ideal convex lens, and the PSF of the lens

h_{0}(

x,

y) will be the same Gaussian function everywhere, as follows:

where 3σ is the radius of the spot. The MTF of the lenses is as follows:

Then, the star spot behind the lenses can be written as follows:

The ideal optical lens is translation invariant; hence, it has no effect on centroid localization according to Equation (10), as follows:

That is, centroid localization will remain error-free after passing through the lenses.

#### 5.1.2. Systematic Centroid Error of an FOFP

According to the previous analysis, the effect of an FOFP on the input image can be divided into three separate processes: integration, sampling and reimaging. The reimaging process is a translation-invariant cylinder function (as shown in Equation (16)). According to the analysis in

Section 3.2, no systematic error will be evolved by the reimaging process. That is, the systematic centroid error caused by an FOFP is formed during integration and sampling.

In the integration and sampling processes, the input image initially convolves with the cylinder function circ(

r), and then, the result is sampled by a hexagonal comb function. The transform of the spatial hexagonal comb function in the frequency domain is still a hexagonal comb function. Any image pattern that goes through the plate will have its frequency spectrum replicated and overlapped. The original frequency spectrum will be replicated to areas centered on each frequency point shown in the shaded area in

Figure 6. These frequency points are assigned to a hexagonal grid structure that is perpendicular to one of the sample points in the spatial domain, with a distance of

$2/\sqrt{3}a$ (approximately 1.15/

a) between two neighboring points. These points are called overlapping frequency points.

**Figure 6.**
Overlapping frequency points in the frequency domain.

**Figure 6.**
Overlapping frequency points in the frequency domain.

The overlapping of the frequency spectrum can cause frequency aliasing. The effect of frequency aliasing on centroid localization is analyzed in this section. Assuming that starlight spot input in the FOFP obeys Gaussian distribution, the exact spot in the origin spot can be denoted as

f_{0}(

x,

y), and then, the distribution

f(

x,

y) of a starlight spot in the arbitrary position

$\left(\overline{x},\overline{y}\right)$ can be expressed as follows:

After integration, the PSF is convolved with the cylinder function as follows:

After being transformed in the frequency domain, the Gaussian spectrum is multiplied by the first-order Bessel curved surface as follows:

where

G_{0}(

u,

v) represents the after-integration frequency spectrum of the star spot at the origin point. It would still be rotationally symmetric to the origin point in the frequency domain. After being transformed into frequency polar coordinates, spectrum

G_{0}(

u,

v) and its partial derivative can be expressed as follows:

As shown in

Figure 6, assuming that only the first-order overlapping spectra will interfere with the origin spectrum, then the aliasing spectrum

$\tilde{G}\left(u,v\right)$ after the sampling process can be written as follows:

where

D represents the intervals between two overlapping frequency points (

i.e.,

$2/\sqrt{3}a$. According to Equations (28)–(29), we obtain

Compared with the first item in

$\tilde{G}\left(0,0\right)$, the second one is significantly smaller. From Equation (12), the estimated centroid location under the aliasing influence of the first-order overlapping spectra can be approximately expressed as follows:

The magnitude in frequency

D is considerably less than that in the origin, and thus, the two preceding equations can be approximated as follows:

The actual center of the input starlight spot is known as

$\left(\overline{x},\overline{y}\right)$; hence, the systematic centroid error caused by the FOFP is as follows:

According to the result, the overall magnitude of the systematic error is determined by the ratio of the derivative value in the first-order overlapping points

${G}_{0}^{\prime}\left(D\right)$ to the spectrum value in the origin point

G_{0}(0). The repetition period of the systematic error is determined by the distance

D between two neighboring overlapping frequency points. As shown in

Figure 7, the systematic error magnitude distribution with starlight spots enters the FOFP from different places. If the spots are located at the center of a fiber or at the border of two neighboring fibers, then they would be affected by a minor systematic error. When the spots move from the border area and towards the fiber center, the error magnitude will initially increase and then decrease.

**Figure 7.**
Magnitude of the systematic centroid error with respect to the entrance position of the starlight spot.

**Figure 7.**
Magnitude of the systematic centroid error with respect to the entrance position of the starlight spot.

#### 5.1.3. Error Reduction

The integration process causes the Gaussian spot to become convoluted with the cylinder function. The Fourier transform of the cylinder function is the first-order Bessel function. Nearly all of the energy of this function is generally acknowledged to focus within the zero-order central circular area with a diameter of 1.22/

r_{0}. The sampling process will cause the original frequency spectrum to shift and overlap, with the interval of two neighboring overlapping frequency points being 1.15/

a. Disregarding the thickness of the shield wrap (

r_{0} =

a), the diameter of the spectrum after integration becomes:

If the thickness of the shield wrap is considered, then the spectrum diameter will be larger. In the analysis of the frequency domain, the accurate recovery of an image requires no aliasing in the spectrum after sampling. This condition is the essence of Shannon’s sampling theorem. However, the only requirement of a star tracker is that no systematic centroid error will result when an image passes through the component. According to the conclusion in

Section 3.2, this condition indicates that the value at the origin point of the spectrum and its derivative value should not be affected by the component. As shown in

Figure 6, the circle marks the minimum border of the zero-order central circle of the first-order Bessel function, within which lies six first-order overlapping shift points. Therefore, if no constraint is placed upon the star spot that is entering the input FOFP, then systematic error will be inevitable. The constraint to the spectrum of the entering image

F(

u,

v) is as follows:

The spectrum of the starlight spot from the lenses in the frequency domain still obeys Gaussian distribution. The star spot described by Equation (23) will have its frequency spectrum distribution within a circle that is 6

σ in diameter. Thus, the upper frequency limit is as follows:

By combining Equations (40) and (41), we obtain

According to Equation (42), the circle of confusion (6σ) should not be less than 2.49 times that of the fiber interval to free the star tracker from systematic errors caused by the input FOFP.