## Appendix A  
In this appendix, _SNR_ improvement at the peak of cross-correlation \(i(t)\) is derived. First, \(i_{S}(\[PERSON])\) and \(i_{R}(\[PERSON])\) are given respectively as:  
\[i_{S}(\[PERSON]_{P})=\exp\{-\left[\frac{\[PERSON]_{P}-\[PERSON]_{S}}{\pi\!\!L_{C}/\lambda} \right]^{2}\}\cos(\[PERSON]_{P}-\[PERSON]_{S})\] (A-1)  
\[i_{R}(\[PERSON]_{P})=\exp\{-\left[\frac{\[PERSON]_{P}-\[PERSON]_{R}}{\pi\!\!L_{C}/\lambda} \right]^{2}\}\cos(\[PERSON]_{P}-\[PERSON]_{R})\] (A-2)  
where \(\[PERSON]_{P}=\frac{2\pi\!\!L_{P}}{\lambda}\), \(\Phi_{S}=\frac{2\pi\!\!\!L_{S}}{\lambda}=\Phi_{R}+\Phi_{d}\), \(\Phi_{R}=\frac{2\pi\!\!\!mL_{R}}{\lambda}\)  
Here \(L_{P}\), \(L_{S}\), \(L_{R}\) are the path length difference of the Mach-Zehnder processing interferometer, sensing FFPI (twice the Fabry-Perot cavity length), reference FFPI and \(n\) is the effective refractive index of core of the optical fiber (not to be confused with sample number \(n\) in \(i(n)\)). Notation \(\otimes,*\) and \(\Leftrightarrow\) are reserved to denote the cross-correlation, convolution and Fourier transform relationship.  
When \(L_{P}\) is scanned \(L_{P}\) is a function of time \(t\) and also \(\Phi_{P}\) is a function of time \(t\). Then:  
\[\Phi_{P}=2\pi\!f\!t=w_{0}t\] (A-3)  
where \(f\) (or \(w_{0}\)) is a modulation frequency (or angular velocity) of fringe scan. Following the same token, \(\Phi_{S_{s}}\), \(\Phi_{R}\) are given by:  
\[\Phi_{S}=2\pi\!f\!t_{S}=w_{0}t_{S}\] (A-4)  
\[\Phi_{R}=2\pi\!f\!t_{R}=w_{0}t_{R}\] (A-5)  
and the relation between \(t_{S}\) and \(t_{R}\) is given by:  
\[t_{S}=t_{R}+t_{d}\] (A-6)  
where \(t_{d}\) is the time delay between sensing FFPI fringe scan and reference FFPI fringe scan. Using the following substitutions:  
\[\Phi_{P}-\Phi_{S}=w_{0}(t-t_{S})=w_{0}(t-t_{R}-t_{d})\] (A-7)  
\[\Phi_{P}-\Phi_{R}=w_{0}(t-t_{R})\] (A-8)  
\[L_{C}=N_{f}\lambda\] (A-9)  
and considering the noise in \(i_{S}(\[PERSON]_{P})\) and \(i_{R}(\Phi_{P})\), time domain representation of the Equation A-1 and Equation A-2 can be rewritten as:  
\[i_{S}(t)=e^{-a(t-t_{d})^{2}}\cos(w_{0}(t-t_{d}))+w_{S}(t)\] (A-10)  
\[i_{R}(\[PERSON]_{P})=\exp\{-\left[\frac{\[PERSON]_{P}-\[PERSON]_{R}}{\pi\!\!L_{C}/\lambda} \right]^{2}\}\cos(\[PERSON]_{P}-\[PERSON]_{R})\]0 (A-11)  
In Equation A-10 and A-11 we used a substitution \(a=(w_{0}/\pi N)^{2}\) and it is assumed that \(\Phi_{R}\)=0 without loss of generality and that \(w_{S}(t)\), \(w_{R}(t)\) are white noise with zero mean and variance \(\sigma_{{}_{N}}^{2}\) of \(i_{S}(t)\), \(i_{R}(t)\) respectively and independent each other. Also, \(w_{0}\) was set to \"1\" for normalization in order to make \(i_{S}(t)=e^{-1}\) at \(t=(N_{f}/2)\) x \(2\pi\). The first part of Appendix A shows that the increase of coherence length from \(L_{C}\) to \(\sqrt{2}L_{C}\) due to cross-correlation. We know that cross-correlation and convolution are identical operations for the case where either \(i_{S}(t)\) or \(i_{R}(t)\) is an even symmetric function. Then, the cross-correlation \(i(t)\) of \(i_{S}(t)\) and \(i_{R}(t)\) is given as follows:  
\[ \begin{array}{l}i_{S}(t)\otimes i_{R}(t)=i_{S}(t)*i_{R}(t)=(e^{-a(t-t_{d})^ {2}}\cos(t-t_{d})+w_{S}(t))*(e^{-at^{2}}\cos t\ +w_{R}(t))\\ =\int\limits_{-\infty}^{+\infty}(e^{-a\lambda^{2}}\cos\lambda\ +w_{R}( \lambda))\ (e^{-a(t-\lambda-t_{d})^{2}}\cos(t-\lambda-t_{d})+w_{S}(t-\lambda))\,d \lambda\\ =\int\limits_{-\infty}^{+\infty}((e^{-a\lambda^{2}}\cos\lambda\ )\ (e^{-a(t-\lambda-t_{d})^{2}}\,\cos(t-\lambda-t_{d}))\,d \lambda\ +\int\limits_{-\infty}^{+\infty}(e^{-a\lambda^{2}}\cos\lambda\ )\ (w_{S}(t-\lambda))\,d\lambda\\ +\int\limits_{-\infty}^{+\infty}(w_{R}(\lambda))\ (e^{-a(t-\lambda-t_{d})^{2}}\cos(t- \lambda-t_{d}))\,d\lambda\ +\int\limits_{-\infty}^{+\infty}w_{R}(\lambda\ )\ w_{S}(t-\lambda))\,d\lambda\end{array}