## 2 Principles
### Principles of MJI-HI
By treating \(B(v)T_{FPI}(v)\) as an integrated spectral term (schematically represented in Figure 3a), the FITS effectively performs an optical Fourier transform on this composite signal. Consequently, \(I_{2}\) is mathematically equivalent to the convolution of the Fourier transform of \(FT[T_{FPI}(v)]\) with the original interferogram \(I_{0}\). Mirroring the constraints of \(I_{1}(\Delta)\), the OPD range of \(I_{2}(\Delta)\) remains bounded by \(\Delta\in[-L,L]\), dictated by the FITS's maximum OPD. In Equation (1), \(T_{FPI}(v)\) can be expressed as comprising a DC component \(a_{0}\), a fundamental frequency component \(2a_{1}\)cos(\(2\pi v\Delta_{FPI}\)), and higher-order harmonic components \(2a_{m}\)cos(\(2\pi v\Delta_{FPI}\)). By combining Equation (1) with Equation (5), the derived relationship is as follows:  
\[ \begin{array}{ll}I_{2}(\Delta)&=I_{0}(\Delta)*\left\{a_{0}\delta(\Delta)+ \sum\limits_{m=1}^{\infty}a_{m}[\delta(\Delta+m\cdot\Delta_{FPI})+\delta( \Delta-m\cdot\Delta_{FPI})]\right\}\\ &=a_{0}I_{0}(\Delta)+\sum\limits_{m=1}^{\infty}a_{m}[I_{0}(\Delta+m\cdot \Delta_{FPI})+I_{0}(\Delta-m\cdot\Delta_{FPI})]\end{array} , \tag{6}\]  
where \(\delta\) represents the Dirac function. The presence of the DC component \(a_{0}\) ensures that \(I_{2}(\Delta)\) inherently contains the original interferogram \(I_{0}(\Delta)\), where \(a_{0}I_{0}(\Delta)\) is defined as the DC component of \(I_{2}(\Delta)\). Simultaneously, the existence of fundamental  
Figure 3: Schematic diagram of combined interference principle: (**a**) object spectrum \(B(v)\) and (**b**) its interferogram \(I_{0}(\Delta)\) and (**c**) FPI-modulated spectrum \(B(v)T_{FPI}(v)\) and (**d**) its interferogram \(I_{2}(\Delta)\). and higher-order harmonic components introduces frequency-shifted interferometric terms \(I_{0}(\Delta\pm m\Delta_{FPI})\) into \(I_{2}(\Delta)\), with \(a_{m}I_{0}(\Delta\pm m\Delta_{FPI})\) identified as the \(m\)-th order frequency-shifted component of \(I_{2}(\Delta)\), where \(m\Delta_{FPI}\) quantifies the frequency-shifted magnitude. Although the maximum OPD of \(I_{2}(\Delta)\) remains confined to \(L\), the incorporation of the FPI enables these frequency-shifted components to superimpose high-frequency information from \(I_{0}(\Delta)\) beyond \(L\) onto \(I_{2}(\Delta)\) via \(I_{0}(\Delta\pm m\Delta_{FPI})\), as schematically illustrated in Figure 3b. Consequently, the synergistic integration of \(I_{1}(\Delta)\) and \(I_{2}(\Delta)\) facilitates the reconstruction of \(I_{0}(\Delta)\)'s high-frequency components through \(I_{0}(\Delta\pm m\Delta_{FPI})\), effectively extending the maximum OPD of \(I_{0}(\Delta)\) and thereby enhancing spectral resolution by overcoming the Fourier uncertainty limit. In Equation (6), the Fourier transform of the ideal FPI transmittance function, \(FT[T_{FPI}(\
u)]\), can be rigorously expressed as a summation of \(\delta\) functions. It is precisely the presence of these delta functions that ensures a strict linear relationship between the frequency-shifted components and \(I_{0}(\Delta)\). These delta functions can be interpreted as the response functions of the \(I_{2}(\Delta)\) DC component and frequency-shifted components with respect to \(I_{0}(\Delta)\).