## 3 Comparison among Different AC Signal Analysis Methods
### Influence of Spectral Leakage on the Accuracy of the Amplitude and Phase
#### 3.3.2 Phase AC Signal Analysis  
Spectral leakage affects the signal's shape in the frequency domain, which eventually affects the phases for DFT and DTFT-based methods unless a correction formula is present for Fourier-based methods. On the other hand, the phase in the cross-correlation is mainly affected by the \(L_{\infty}\) norm in addition to the phase resolution, which is \(\frac{2\pi f}{F_{n}}\). Among all the solutions, only sine fitting solutions have the least influence by the spectral leakage during phase determination. \begin{table}
\begin{tabular}{c c c c} \hline \hline \multirow{2}{*}{**Method**} & **Sensitivity to** & **Sensitivity to** & **Approx. Max. amp.** & **Max Deviation** \\  & **Interference** & **Spectral Leakage** & **Deviation Formula.** & **(Application/Figure 4a)** \\ \hline DFT-based & Low & Very High & \(A(1-sinc(\delta m))\) & 36\% \\ \hline DFT-based and & Low & High & \(A(1-\frac{sinc(\delta m)}{1-\delta m^{2}})\) & 15\% \\ windowed (Hann) & & & \\ \hline DFT-based and & Low & Very Low & \(\frac{Asinc(\delta m)}{N}\frac{sin(2\pi\delta m)}{sin(2\pi fdt)}\) & 0.08\% \\ corrected & & & \\ \hline DTFT & Low & Low & \(\frac{A}{N}\frac{sin(2\pi\delta m)}{sin(2\pi fdt)}\) & 0.32\% \\ \hline X-corr & High & Low & \(-\frac{A}{N}\frac{sin(2\pi\delta m)}{sin(2\pi fdt)}\) & 0.27\% \\ \hline Sine-fitting & Low & Insensitive & - & 0\% \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Comparison between the sensitivity of the amplitude AC signal analysis methods to interference and to spectral leakage. Analytically, the deviation expression can be intuitive, such as in the case of DFT and sine-fitting, but can also be convoluted in the other methods due to the non-linear behavior of the arc tangent of imaginary and real parts used in the phase calculation. In these cases, it is necessary to calculate the deviation in Cartesian coordinates beforehand. In DFT, with or without windowing, the phase deviation is always \(\delta m\pi\). The phase analysis using cross-correlation depends mainly on the phase resolution. Sine-fitting methods show lower sensitivity to phase uncertainty. However, for DTFT and DFT with correction, the Cartesian calculation is necessary to derive the phase uncertainty. For this purpose, numerical simulations and test scenario studies are preferably used. Table 5 summarizes these deviations. Figure 4: (**a**) Full Comparison of the Different Signals. (**b**) A Zoom into the most Precise Methods. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Method** & **Sensitivity to** & **Sensitivity to** & **Max Deviation** \\  & **Interference** & **Spectral Leakage** & **(Application/Figure 4a)** \\ \hline DFT-based & Low & Very High & \(\pi/2\) rad \\ DFT-based and windowed (Hann) & Low & Very High & \(\pi/2\) rad \\ DFT-based and corrected & Low & Low & 0.0078 rad \\ DTFT & Low & Low & 0.0029 rad \\ X-corr & High & Low & 0.0293 rad \\ Sine-fitting & Low & Insensitive & \(1.44\times 10^{-15}\) rad \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Comparison between the sensitivity of the phase AC signal analysis methods to interference and to spectral leakage. When applied to the example signal for phase spectroscopy, it can be seen that sine fitting methods give the most accurate results. The maximum deviation is in the range of \(10^{-15}\) resp. \(10^{-14}\). DTFT has the third-lowest maximum deviation of 0.0293 rad, followed by DFT with correction, with 0.0078 rad. While the cross-correlation is not affected by the spectral leakage, as shown in Figure 5a, the phase resolution defines its precision, and therefore the constant deviation, independent of the spectral leakage, is 0.0293 rad. Finally, both the DFT with and without windowing yielded a maximum deviation of \(\frac{\pi}{2}\), as shown in Figure 5a. A summary of the methods is shown in Table 5. The values of the phases are plotted against spectral leakage in Figure 5a, with a closer look in Figure 5b. Figure 5: (**a**) Full Comparison of the Different Signals. (**b**) A Zoom into the most Precise Methods.