## 3 Proposed Signal Processing Algorithm
### Fine Tuning  
Figure 8 shows the vicinity of zero order fringe peak of cross-correlation \(i(n)\). From Figure 8 it is clear that the discrete sample zero order fringe peak position \(p_{d}\) of the cross-correlation \(i(n)\) identified by the hypothesis test is not necessarily same as the \"true zero order fringe peak position\", \(p_{t}\) and the goal of fine tuning algorithm is to find the distance between \(p_{d}\) and \(p_{t}\). Figure 7: Distribution \(d(j,i)\) of noise-free cross-correlation. First, fine tuning algorithm assumes that the cross-correlation \(i\left(n\right)\) is represented as:  
\[i(n)=\exp\Biggl{\{}-\Biggl{[}\frac{2\pi\bigl{(}n-p_{{}_{t}}\bigr{)}/\,f_{{}_{S}}}{ \pi\hat{L}_{{}_{C,\mathit{eff}}}\,/\,\lambda}\Biggr{]}^{2}\Biggr{\}}\cos\Biggl{\{} \frac{2\pi\bigl{(}n-p_{{}_{t}}\bigr{)}}{f_{{}_{S}}}\Biggr{\}} \tag{20}\]  
Note that \(p_{t}\) in Equation 20 is not necessarily a integer sample number. Also, fine tuning algorithm generates test cross-correlation, \(i_{test}(n)\) defined by:  
\[i_{test}(n) =\exp\Biggl{\{}-\Biggl{[}\frac{2\pi\bigl{(}n-p_{{}_{d}}\bigr{)}/ \,\hat{f}_{{}_{S}}}{\pi\hat{L}_{{}_{C,\mathit{eff}}}\,/\,\lambda}\Biggr{]}^{2 }\Biggr{\}}\cos\Biggl{\{}\frac{2\pi\bigl{(}n-p_{{}_{d}}\bigr{)}}{\hat{f}_{{}_{ S}}}\Biggr{\}} \tag{21}\] \[=\exp\Biggl{\{}-\Biggl{[}\frac{2\pi\bigl{(}n-n_{{}_{d}}\bigr{)}/ \,\hat{f}_{{}_{S}}}{\pi\hat{L}_{{}_{C,\mathit{eff}}}\,/\,\lambda}\Biggr{]}^{2 }\Biggr{\}}\cos\Biggl{\{}\frac{2\pi\bigl{(}n-n_{{}_{d}}\bigr{)}}{\hat{f}_{{}_ {S}}}\Biggr{\}}\]  
In Equation 21 it is assumed that the discrete sample zero order fringe peak position \(p_{d}\) happens ideally at \(n\!=n_{d}\) and also that the approximate value of the coherence length of the light source, \(\hat{L}_{{}_{C}}\) is known priori to us. This is possible by counting the number of fringes within full width at which the intensity of interferogram using a particular light source decreases down to \(e^{-1}\) of its maximum value. And value of \(\hat{f}_{{}_{S}}\) is given from \(\hat{f}_{{}_{S}}=\bigl{[}2b\bigr{]}\) as shown in Section 3.2. Then, the distance between \(p_{d}\) and \(p_{t}\) can be found by calculating _J(M)_ for varying integer number  
Figure 8: Illustration for true zero order fringe peak and discrete sample zero order fringe peak of \(i(n)\). \[J(M)=\sum_{n=-\infty}^{\infty}i(n)i_{test}\left(n-M\Delta n\right) \tag{22}\]  
where \(i_{test}\left(n\text{-}M\Delta n\right)\) is a computer-generated test cross-correlation with its zero order fringe peak at \(p_{d}\)\(+M\Delta n\) and \(\Delta n\) is the desired fine tuning resolution in sample. \[\Delta n=\frac{\hat{f}_{S}}{N_{sub}}\left[\text{sample}\right] \tag{23}\]  
In Equation 23\(N_{sub}\) is the number of sub-divisions in one fringe (the desired fine tuning resolution in fringe). Then \(J(M)\) becomes:  
\[\exp\left\{-\left[\frac{2\pi\left(n-n_{d}-M\Delta n\right)/\hat{f}_{S}}{\pi \hat{L}_{C,eff}/\lambda}\right]^{2}\right\}\cos\left\{\frac{2\pi\left(n-n_{d}- M\Delta n\right)}{\hat{f}_{S}}\right\} \tag{24}\]  
Interpretation of Equation 24 is that fine tuning algorithm generates its own test cross-correlation, \(i_{test}\left(n\right)\) with the true zero order fringe peak positioned at \(n_{d}\)\(+M\Delta n\) and calculate \(J(M)\) for varying M. Then \(J(M)\) has a peak value at the certain \(M_{f}\) which makes \(M_{f}\Delta n\approx p_{t}\)- \(n_{d}\) because the best similarity between \(i\left(n\right)\) and \(i_{test}\left(n\right)\) is attained when \(n_{d}\)\(+M_{f}\Delta n\approx p_{t}\). Then, fine tuning algorithm announces (\(n_{d}\)\(+M_{f}\Delta n\)) as the estimated true zero order fringe peak location \(\hat{p}_{t}\). Calculating \(J(M)\) for the range of - \(\infty\leq\text{M}\leq+\infty\) takes many calculations and is time-consuming. The true zero order fringe peak \(p_{t}\) normally is within several samples of the discrete sample zero order fringe peak \(p_{d}\).