## 2 Previous works  
Two major classes of signal processing algorithms for WLI are the hardware approach and the software approach. Both approaches have a more or less \"tracking zero order fringe peak\" feature. [PERSON] proposed a hardware approach which locks the zero order fringe position of interferometer by a feedback loop [4]. An improvement of the sensitivity up to 1/240 fringe was claimed. To the author's best knowledge this method, while still dependent on the incremental characteristic of laser interferometry and not fully taking advantage of WLI's potential to identify the interference fringe, demonstrated the feasibility of locking and tracking the fringe peak for absolute measurement for the first time. There are many software algorithms to estimate phase delay \(\Phi^{d}\)[3, 5]. Among many detection methods, the cross-correlation method dominates the field of phase delay estimation in practice due to its easier implementation [6]. Many other TDE methods are based on this algorithm. The cross-correlation method cross-correlates the two fringe signals \(i_{S}(n)\) and \(i_{R}(n)\), which are sampled versions of \(I_{S}(\Phi_{P})\) and \(I_{R}(\Phi_{P})\) respectively, into \(i\left(n\right)\) and considers the sample number argument \(n\)=\(n_{d}\) that corresponds to the maximum peak in cross-correlation \(i\left(n\right)\) as the estimated time delay [7]. While WLI has the potential to identify the interference fringe order from the output pattern of an interferometer [8], it is difficult to distinguish the zero order fringe peak from its adjacent first order fringe peaks when noise is present in the interferometer output, as shown in Figure 3. From Equations 1 and 2, the amplitude difference \(\Delta I\) between the zero order fringe peak and the first order fringe peaks is represented as:  
\[\Delta I=\frac{1}{2}\Bigg{(}1-\exp\Bigg{[}-\Bigg{(}\frac{2}{L_{C}\,/\,\lambda} \Bigg{)}^{2}\Bigg{]}\Bigg{)}\] (3a) Clearly, if a system has a noise level which is equal or greater than \[\Delta I\]  the zero order fringe peak cannot be identified directly, simply through inspection of its amplitude. If the normalized zero order fringe peak value is defined as unit signal, a minimum signal-to-noise ratio \[SNR_{\min}\] required to identify the zero order fringe peak [9] is given by \[SNR_{\min}=-20\log\Bigg{[}\frac{\Delta I}{\left(1/2\right)}\Bigg{]}=-20\log \Bigg{\{}1-\exp\Bigg{[}-\Bigg{(}\frac{2}{L_{C}\,/\,\lambda}\Bigg{)}^{2} \Bigg{]}\Bigg{\}} \tag{3b}\]  
Representative values of the coherence length of different light sources like white light lamp, light emitting diode (LED), superluminescent diode (SLD), are about 10, 20, and 40 in terms of optical fringes. The \(SNR_{\min}\) required to identify the zero order fringe peak by amplitude difference \(\Delta I\) is given from Equation 3 as 28 dB, 40 dB, and 52 dB respectively [9]. One disadvantage related to cross-correlation is the broadening of envelope (or peak) (hereafter termed as envelope) from \(L_{C}\) to \(\sqrt{2}\,L_{C}\) due to the cross-correlation of two almost identical Gaussian envelope signals [17]. This results in higher \(SNR_{\min}\) in Equation 3 requiring 6 dB more than before cross correlation. Then, with this broadening, the attainable resolution is often not better than one fringe and a precise peak location may be somewhat questionable. This difficulty has inhibited the application of fiber optic sensors using WLI, for example, absolute OPD determination [9]. Additionally, once the zero order fringe peak is identified, then for a more accurate sub-sample resolution time delay estimation we will have to use interpolation which is possible by either quadratic interpolation in time domain or frequency domain zero-padding [10, 11]. But quadratic interpolation uses three cross-correlation values centered at the estimated peak of cross-correlation. This method has a shortcoming of bias caused by time domain sample rate [12, 13] and the difference between true peak shape and the fitted quadratic polynomial.