## 4 Proposed Method: XQIFFT
### Choice of the Power-Scaling Factor \(p\)
XQIFFT cases which heuristically minimize worst case and mean error bin and magnitude errors are shown. Numbers are given to five significant figures and minimized values are shaded. Figure 3: Mean and worst-case for absolute value of errors for the QIFFT method on a length-4096 Hann window, including bin error and detail on bin error, magnitude error and detail on magnitude error. Horizontal lines indicate the worst case and mean errors for the nearest bin, MQIFFT, and LQIFFT methods for comparison. Shaded regions indicate the range of the detail in corresponding figures to the right. The thick lines indicate the XQIFFT. (**a**) Bin error; (**b**) Bin error (zoom); (**c**) Magnitude error (zoom); (**d**) Magnitude error (zoom). asymptotic behavior of the XQIFFT estimator as \(p\) approaches \(\infty\). Because we have \(0\leq\alpha/\beta<1\) and \(0\leq\gamma/\beta<1\), \(\lim_{p\to\infty}{(\alpha/\beta)^{p}}=0\) and \(\lim_{p\to\infty}{(\gamma/\beta)^{p}}=0\) and the XQIFFT estimate (13), (14), (17) and (18) reduces to the nearest neighor method:  
\[\hat{\mathcal{K}} =\lim_{p\to\infty}\left[k_{m}+\frac{1}{2}\frac{\alpha^{p}-\gamma^ {p}}{\alpha^{p}-2\beta^{p}+\gamma^{p}}\right]=\lim_{p\to\infty}\left[k_{m}+ \frac{1}{2}\frac{(\alpha/\beta)^{p}-(\gamma/\beta)^{p}}{(\alpha/\beta)^{p}-2+ (\gamma/\beta)^{p}}\right]=k_{m} \tag{33}\] \[\hat{\mathcal{X}} =\lim_{p\to\infty}\left[\left(\beta^{p}-\frac{1}{8}\frac{(\alpha ^{p}-\gamma^{p})^{2}}{\alpha^{p}-2\beta^{p}+\gamma^{p}}\right)^{1/p}\right]= \lim_{p\to\infty}\left[\beta\left(1-\frac{1}{8}\frac{\left[(\alpha/\beta)^{p} -(\gamma/\beta)^{p}\right]^{2}}{(\alpha/\beta)^{p}-2+(\gamma/\beta)^{p}} \right)^{1/p}\right]=\beta\,. \tag{34}\]  
Additionally, we can show that, as suggested by Figure 3, the XQIFFT converges towards the LQIFFT as \(p\) approaches \(0\). Indeed, from the Taylor expansion of the bin estimate (13) using the XQIFFT scaling (17) and (18), we have that \(\alpha^{p}=\exp{(p\log\alpha)}\sim 1+p\log\alpha+\mathcal{O}(p^{2})\), \(\beta^{p}\sim 1+p\log\beta+\mathcal{O}(p^{2})\) and \(\gamma^{p}\sim 1+p\log\gamma+\mathcal{O}(p^{2})\). Hence, we get:  
\[\hat{\mathcal{K}}=\lim_{p\to 0}\left[k_{m}+\frac{1}{2}\frac{ \alpha^{p}-\gamma^{p}}{\alpha^{p}-2\beta^{p}+\gamma^{p}}\right] \sim\lim_{p\to 0}\left[k_{m}+\frac{1}{2}\frac{p\log\alpha-p\log \gamma+\mathcal{O}(p^{2})}{p\log\alpha-2p\log\beta+p\log\gamma+\mathcal{O}(p^ {2})}\right]\] \[=k_{m}+\frac{1}{2}\frac{\log\alpha-\log\gamma}{\log\alpha-2\log \beta+\log\gamma} \tag{35}\]  
This matches (13) for a logarithmic scaling, verifying that the XQIFFT's bin estimate (17) approaches the LQIFFT as \(p\to 0\). For the amplitude estimate (14) using the XQIFFT scaling (17) and (18) we have  
\[\beta^{p}-\frac{1}{8}\frac{(\alpha^{p}-\gamma^{p})^{2}}{\alpha^{p}-2\beta^{p }+\gamma^{p}}\sim 1+p\left(\log\beta+\mathcal{O}(p)-\frac{1}{8}\frac{(\log \alpha-\log\gamma)^{2}+\mathcal{O}(p)}{\log\alpha-2\log\beta+\log\gamma+ \mathcal{O}(p)}\right) \tag{36}\]  
Similarly, \(\log(1+x)\sim x+\mathcal{O}(x^{2})\), such that:  
\[\log\left[\beta^{p}-\frac{1}{8}\frac{(\alpha^{p}-\gamma^{p})^{2}}{\alpha^{p}- 2\beta^{p}+\gamma^{p}}\right]\sim p\left(\log\beta+\mathcal{O}(p)-\frac{1}{8} \frac{(\log\alpha-\log\gamma)^{2}+\mathcal{O}(p)}{\log\alpha-2\log\beta+\log \gamma+\mathcal{O}(p)}\right)+\mathcal{O}(p^{2}) \tag{37}\]  
Furthermore, since \((\cdot)^{1/p}=\exp{\left[\frac{1}{p}\log(\cdot)\right]}\), we have:  
\[\left[\beta^{p}-\frac{1}{8}\frac{(\alpha^{p}-\gamma^{p})^{2}}{ \alpha^{p}-2\beta^{p}+\gamma^{p}}\right]^{1/p} =\exp{\left[\frac{1}{p}\log\left[\beta^{p}-\frac{1}{8}\frac{(