## 6 Results
### Bin Estimate MSE as a Function of SNR (Twelve Windows)  
In this third study, the performance of the XQIFFT and other QIFFT-derived methods is compared to the state-of-the-art direct methods of Duda [14] and Candan [15] for _twelve_ common length-4096 windows. These windows include the [PERSON] window, the Bartlett-Hann window [45], the [PERSON] window [4], the Hamming window [4], the [PERSON] window [4], the Blackman-Harris window [43], the Gaussian window [43], the Digital Prolate Spheroidal Sequence (DPSS/Slepian) window (with time-halfbandwidth product \(NW=3\)) [4], the Kaiser-Bessel window (\(\beta=0.5\)) [46], the Nuttall window [47], the Dolph-Chebyshev window (sidelobes at \(-100\) dB) [43], and the Tukey window (with a 50% taper) [48]. For each of the twelve windows, \(p\) is chosen to minimize \(\mathrm{E}[|e_{\mathcal{K}}\left(\mathcal{K}\right)|]\) using the procedure explained earlier. The values of \(p\) used for each window are given in Table 2.  
Again, the complex sinusoids are corrupted by additive white Gaussian noise. For each SNR from \(-40\) to \(120\), in intervals of \(5\) dB, \(20,000\) sinusoids are tested with bin offsets of \(-0.5\) to \(+0.5\). The MSE of the bin estimate is plotted against SNR for each method and the Cramer-Rao bound (CRB) in Figure 10.  
Figure 9: MSE of bin error as a function of singal-to-noise ratio. (**a**) Full SNR range; (**b**) Zoom.  
\begin{table}
\begin{tabular}{c c c c|c} \hline
**Window** & **Length-512** & **Length-1024** & **Length-2048** & **Length-4096** \\ \hline [PERSON] & 0.22903 & 0.22911 & 0.22915 & 0.22917 \\ [PERSON]–[PERSON] & 0.21635 & 0.21642 & 0.21645 & 0.21647 \\ [PERSON] & 0.22530 & 0.22535 & 0.22538 & 0.22539 \\ [PERSON] & 0.18505 & 0.18575 & 0.18611 & 0.18628 \\ [PERSON] & 0.13056 & 0.13057 & 0.13058 & 0.13058 \\ [PERSON]–[PERSON] & 0.08552 & 0.08553 & 0.08553 & 0.08554 \\ [PERSON] & 0.12024 & 0.12074 & 0.12099 & 0.12112 \\ DPSS/[PERSON] (\(NW=3\)) & 0.11144 & 0.11144 & 0.11144 & 0.11144 \\ [PERSON]–[PERSON] (\(\beta=0.5\)) & 0.28214 & 0.28270 & 0.28298 & 0.28312 \\ [PERSON] & 0.08153 & 0.08155 & 0.08157 & 0.08157 \\ Dolph–Chebyshev (sidelobes at \(-100\)dB) & 0.08403 & 0.08403 & 0.08404 & 0.08404 \\ Tukey window (50\% taper) & 0.50592 & 0.50609 & 0.50618 & 0.50622 \\ \hline \end{tabular}
\end{table}
Table 2: The values of \(p\) that minimize \(\mathrm{E}[|e_{\mathcal{K}}(\mathcal{K})|]\) for the twelve length–4096 windows (highlighted column) used for tests in Figure 10, as well as length-512, -1024, and -2048 versions of the same windows. \(p\) is rounded to five decimal points.  
Figure 10: _Cont._Figure 10: _Cont._  
It can be seen in Figure 10 that the XQIFFT (with a properly-chosen \(p\) according to Table 2) has the lowest MSE error across the entire range of SNRs for every tested window except the Hann window and the Tukey window (50% taper). In some cases the XQIFFT outperforms its closest competitor by several orders of magnitude. Figure 10 demonstrates that the XQIFFT is applicable to a wide range of common windows and that it even outperforms the state of the art for most tested windows. In a broader sense, this demonstrates that the best choice of method is highly dependent on the window type which is used--the performance ordering of different methods is highly variable, although for most windows tested the XQIFFT has the best performance. It is also interesting to note that of all the windows tested, the Nuttall window using the XQIFFT has the lowest MSE of bin estimate at high SNR. This implies that Nuttall using XQIFFT achieves the lowest systematic bias of the windows and estimators tested.