## 4 Proposed Method: XQIFFT
### Choice of the Power-Scaling Factor \(p\)  
To demonstrate the XQIFFT and the process of choosing \(p\), we study one particular representative window in detail: a length-4096 (\(N=4096\)) Hann window (p. 98, [42]) [43], defined as  
\[w_{\text{Hann}}(n)=\left[\frac{1}{2}+\frac{1}{2}\text{cos}\left(\frac{2\pi n }{N}\right)\right]\;. \tag{32}\]  
When computing the worst-case and mean errors for various values of power-scaling factor \(p\) (Figure 3), we can observe that the XQIFFT strongly outperforms the other QIFFT-derived methods if \(p\) is picked appropriately. In particular, we see that there seems to exist a single value \(p\) for which each statistic is minimized. This optimal value of \(p\) is typically different for each statistic, as well as for the chosen smoothing window, DFT length, and zero-padding factor. As the function exhibits a single minimum (i.e., it is unimodal), we propose here to find an optimal \(p\) value numerically finding that minimum. Finding the minimum of a unimodal function can be done numerically by running a Fibonacci search on the desired error statistic. Such a heuristic yields a choice of \(p\) minimizing the desired error statistic up to an arbitrary precision order, while minimizing the number of power-scaling factors \(p\) for which we need to evaluate the statistic (using the proposed approach in Section 5.2), and still without requiring the complex derivation of a closed form solution for the different statistics as a function of \(p\). Other estimators which allow for arbitrary windows also favor a computational rather than analytic approach [14; 15]. Table 1 shows the different error statistics from Figure 3 for the four different optimal cases. From the results, we can see that (1) for each error statistic, its associated optimal \(p\) yields an error much smaller than the MQIFFT and LQIFFT; and (2) a \(p\) value optimized for a given error statistic still yields small errors for the other ones, in particular, \(p\) values that yield small fractional bin index estimation error also yield small amplitude estimation error. Notice that in Figure 3, the mean and worst-case bin and magnitude errors of the XQIFFT approach the performance of the nearest neighbor method for large values of \(p\). This can be explained by the \begin{table}
\begin{tabular}{c c c c c c} \hline
**Case** & \(\mathbf{e_{\mathcal{K}}^{\text{N}\text{C}.}}\) & \(\mathbf{e_{\mathcal{K}}^{\text{N}\text{C}.}}\) & E[\(\left|\mathbf{e_{\mathcal{K}}}\left(\mathbf{\mathcal{K}}\right)\right|\)] & E[\(\left|\mathbf{e_{\mathcal{K}}}\left(\mathbf{\mathcal{K}}\right)\right|\)] & **Minimizes** \\ \hline nearest & \(5.0000\times 10^{-1}\) & \(1.5110\times 10^{-1}\) & \(2.5000\times 10^{-1}\) & \(5.1688\times 10^{-2}\) & n/a \\ MQIFFT & \(5.2764\times 10^{-2}\) & \(6.6237\times 10^{-2}\) & \(3.4221\times 10^{-2}\) & \(2.5601\times 10^{-2}\) & n/a \\ LQIFFT & \(1.5997\times 10^{-2}\) & \(3.7932\times 10^{-1}\) & \(1.0392\times 10^{-2}\) & \(1.3121\times 10^{-2}\) & n/a \\ XQIFFT, \(p=0.23086\) & \(2.4484\times 10^{-4}\) & \(9.5196\times 10^{-4}\) & \(1.5693\times 10^{-4}\) & \(2.0239\times 10^{-4}\) & \(e_{\mathcal{K}}^{\text{N}\text{C}.}\) \\ XQIFFT, \(p=0.23437\) & \(4.4380\times 10^{-4}\) & \(4.7735\times 10^{-4}\) & \(2.3462\times 10^{-4}\) & \(2.5251\times 10^{-4}\) & \(e_{\mathcal{K}}^{\text{N}\text{C}.}\) \\ XQIFFT, \(p=0.22917\) & \(3.1861\times 10^{-4}\) & \(1.1803\times 10^{-3}\) & \(1.4645\times 10^{-4}\) & \(2.0637\times 10^{-4}\) & E[\(\left|\mathbf{e_{\mathcal{K}}}\left(\mathcal{K}\right)\right|\)] \\ XQIFFT, \(p=0.23039\) & \(2.6445\times 10^{-4}\) & \(1.0149\times 10^{-3}\) & \(1.5203\times 10^{-4}\) & \(2.0170\times 10^{-4}\) & E[\(\left|\mathbf{e_{\mathcal{K}}}\left(\mathcal{K}\right)\right|\)] \\ \hline \end{tabular}
\end{table} 
Table 1: Comparing \(\mathbf{e_{\mathcal{K}}}\left(\mathcal{K}\right)\) and \(\mathbf{e_{\mathcal{X}}}\left(\mathcal{K}\right)\) for various estimation approaches on a length-4096 Hann window.