## 3 Comparison among Different AC Signal Analysis Methods
### Influence of Spectral Leakage on the Accuracy of the Amplitude and Phase
#### 3.3.1 Amplitude AC Signal Analysis Accuracy  
Theoretically, all the AC signal analysis methods project the signals into an orthogonal Fourier basis formed by trigonometric reference signals, hence, no signal interference is expected, if the signal is a multisine. In practice, the signals are finite, hence, truncated. In the point-of-view of Fourier analysis, this truncation is none other than multiplication with a rectangular window in the time-domain and a convolution by sinc in the frequency domain. For incomplete signals, not only the desired frequency but also nearby frequencies are influenced. For example, for DTFT, the extra coefficients (10) and (11) become remarkable and are not nullified. The same problem is also spotted in DFT; Considering a distance to the closest frequency bin \(\delta m\), its amplitude deviation is, therefore, \(A(1-sinc(\delta m))\). The use of window function, such as Hanning window function [32], for example, minimizes, but does not eliminate, the deviation to \(A(1-\frac{sinc(\delta m)}{1-\delta m^{2}})\)[28]. Similarly for DTFT, the deviation formula is demonstrated in Appendix A and depicted in (26) where \(R=\frac{sin(2\pi fNdt)}{sin(2\pi ft)}\).  
\[\overline{\|X_{DTFT}\|}=A\left(1-\sqrt{1+\left(\frac{R}{N}\right)^{2}-2\frac{ R}{N}cos(2\varphi+2\pi f(N-1)\delta t)}\right) \tag{26}\]  
Figure 3: (**a**) Memory usage of DTFT/Goertzel vs. FFT as a function of \(N\) and \(N_{f}\). (**b**) Memory usage of Sine-fit (linear) vs. FFT as a function of \(N\) and \(N_{f}\)  
However, this expression is not predictable when the initial phase is unknown. Therefore, assuming the worst case, it can be simplified to (27)  
\[\left\|X_{DTFT\_sim p_{\text{simp}}}\right\|=A\frac{sin(2\pi fN\delta t)}{Nsin(2 \pi fdt)}\approx A\frac{sin(2\pi\delta m)}{Nsin(2\pi fdt)} \tag{27}\]  
Since the FFT correction based on Section 2.1.3 generates a new frequency that emulates the DTFT, its behavior could be modeled as in (26), but with a different \(R\)-expression, since it inherits the behavior of the rectangular window from the original FFT. \(R\) then becomes \(R=\frac{sinc(\delta m)sin(2\pi fNdt)}{sin(2\pi fdt)}\). The expression of the simplified corrected FFT for the worst case is then \(A\frac{sinc(\delta m)sin(2\pi fdt)}{Nsin(2\pi fdt)}\).  
Cross-correlation loses its accuracy if the amplitudes from the AC signal analysis are very different due to the use of \(L_{\infty}\) norm for amplitude and phase determination. In addition, it is also affected by the signal incompleteness, as shown in (12), as well as the resolution of the \(\varphi\) phase. To obtain accurate results, two conditions must be met: For the frequency in question, an integer number of periods; and an adequate phase resolution are required to satisfy the existence of an integer time \(l\) that satisfies \(l=-\varphi\frac{F_{l}}{2\pi f}\). This means that \(\frac{F_{l}}{f}\) should be as low as possible. The maximum amplitude deviation expression, given a good phase resolution, could be simplified to the same expression as for DTFT (27), albeit with a negative sign.  
Finally, sine-fitting methods show little to no sensitivity to spectral leakage due to both of them being solutions to inverse algebra problems.  
For the exemplary signal and as shown in Figure 4b and summarized in Table 4, the sine-fitting methods show no uncertainty. The second best is DFT with correction with a maximum deviation of 0.08%, then cross-correlation with a maximum deviation of 0.27%, then DTFT with a maximum deviation of 0.32%. Although the windowing allowed a lower maximum deviation of 15%, it is still considered high. The strongest influence is seen with the DFT algorithm without correction, with a maximum deviation of 36%.