# A Phase Correction Model for Fourier Transform Spectroscopy
## 3 Phase Correlation Method  
The proposed PCM-APF model first requires the correction of linear phase errors, primarily caused by the ZPD offset. In some scholars' research, to determine the linear phase component, they often perform linear fitting of the phase curve. In [19], polynomial fitting is used for phase correction. However, it must be considered that the phase error curve calculated by the inverse tangent function may jump if there is nonlinear phase error [20]. It is obvious that we have to unwrap the phase error curve before we can use the linear fit in this case, which increases the uncertainty and potential computation in practice; so we use the phase correlation method. The phase correlation method is one of the most commonly used techniques for signal time-delay estimation. By calculating the cross-correlation function of two signals and identifying the location of the maximum value of the function, we can find the time delay which is equal to the distance in the \(x\)-axis between the maximum value point and the zero point [21]. In the PCM method proposed in [11], a Hanning window is applied to the interferogram initially. Subsequently, data points of equal length on both sides of the sampled point with maximum interference intensity are taken as two signals. The phase correlation method is then employed to calculate the time delay between these two signals. Differing from the PCM method, the PCM-APF model in this paper modifies the phase correlation method used. It is essential to understand that the application of the Hanning window in the PCM method serves a purpose distinct from apodization in FTS data processing. Its function is more akin to the general phase correlation method [21; 22], aiming to enhance the SNR of the selected signals. Moreover, due to the inherent asymmetry of the interferogram caused by the error phase, applying a Hanning window may lead to increased errors in time-delay estimation. In the PCM-APF model proposed in this paper, we employ a new approach to selecting two signals. Let us assume that the collected interferogram only has ZPD-point drift, meaning it contains only linear phase error. Just like the Mertz method, we pre-define the number of data points for two small double-sided interferograms. Subsequently, we extract two small double-sided interferograms near the point of maximum interference intensity (sampled ZPD point). As illustrated in Figure 2, A and B represent the two endpoints of the larger double-sided interferogram, and similarly, C and D represent the two endpoints of the smaller double-sided interferogram. Then, we select the interference signal from point A to D as Signal1, and similarly, choose the interference signal from point B to C as Signal2. Additionally, since the selected signals are all in regions of the original interferogram with higher SNR, there is no need to apply a Hanning window to the selected signals. Suppose the distance of the real ZPD position of the interferogram from the sampled ZPD position is \(\varepsilon\). By geometric considerations, it can be easily deduced that the theoretical time delay between Signal1 and Signal2 should be \(2\varepsilon\). In other words, by following the method described above to obtain the two interferometric signals, then using the previously mentioned phase correlation method to calculate the time delays of the two signals, and finally, dividing the result by 2, we can determine the linear ZPD-point drift of the interferogram. Since the method described above can only provide ZPD-point drift values in integer multiples of the sampling interval, so further sub-pixel peak estimation of the obtained cross-correlation function is needed. [PERSON] [11] presented a fitting method for peak detection at peak positions [11; 23]. In our PCM-APF model in this paper, we utilize the Hilbert difference method for sharpening the correlation peaks to achieve peak position determination. Figure 3a shows the basic process of ZPD offset calculation in proposed PCM-APF model.