## 4 Proposed Method: XQIFFT
### Sensitivity to Noise  
We analyze how the parameter estimation is affected by the presence of noise in the signal. To do so we corrupt the signal \(\mathcal{A}\)\(\varepsilon^{\left|2\pi\mathcal{F}nT+\phi\right\rangle}\) with different realizations of an additive white Gaussian noise random process (with zero-mean and standard deviation \(\sigma\)) at different levels of signal-to-noise ratio (SNR). The SNR is computed as the ratio of the amplitude of the sinusoid by the standard deviation of the noise:  
\[\mathrm{SNR}=\mathcal{A}/\sigma \tag{41}\]  
Figure 4: Correction curves. (**a**) MQIFFT; (**b**) LQIFFT; (**c**) XQIFFT, \(p=0.1\); (**d**) MQIFFT; (**e**) LQIFFT; (**f**) XQIFFT, \(p=0.1\). In this analysis, we quantify the effect of added noise by looking at the distribution of the noisy estimates \(\hat{\mathcal{K}}_{G}\) around the noiseless estimates \(\hat{\mathcal{K}}\). As such, we decorrelate the influence of the systematic bias associated with the estimator from the influence of the noise by reporting the bias and variance of \(\hat{\mathcal{K}}_{G}-\hat{\mathcal{K}}\) for various bin index offsets. In the following experiments, we consider a length-64 Hann window. We sample uniformly the bin index \(\Delta_{\mathcal{K}}\) in \([-0.5,0]\). For each bin index, we generated \(10^{9}\) noise trials to get reliable estimates of the bin index error bias and variance. Figure 5 shows the bias of the bin index estimate (25) due to the noise corruption alone at various signal-to-noise ratios (SNR). As expected, for each estimator the bias decreases as the SNR increases. Across all \(\Delta_{\mathcal{K}}\) and tested SNRs, the XQIFFT (\(p=0.2776\)) has a noise bias between that of the MQIFFT and the LQIFFT. Considering that the XQIFFT tends towards the LQIFFT as \(p\) approaches zero (as shown in Section 5.3) and is identical to the MQIFFT when \(p=1\), the XQIFFT with a value of \(p=0.2776\) can be considered an intermediate between the MQIFFT and LQIFFT. So, it is unsurprising that its noise bias also falls between the two. Interestingly, the bias is nonzero for most offsets \(\Delta_{\mathcal{K}}\). This means that the presence of noise actually shifts the mean error of our estimator away from the systematic bias already present in the noiseless estimation. Figure 6 shows the bin index error variance around the error mean. Again, the variance of the XQIFFT (\(p=0.2776\)) is between the MQIFFT and the LQIFFT, and we can notice that its variance has less variation across the different bin index offsets \(\Delta_{\mathcal{K}}\). Figure 5: Bin index error bias for the MQIFFT (\(\circ\)), the LQIFFT (\(\times\)), and the XQIFFT (\(\cdot\), \(p=0.2276\)) at various SNR: 0, 10, 20, and 30 dB. Figure 6: Bin index error variance for the MQIFFT (\(\circ\)), the LQIFFT (\(\times\)), and the XQIFFT (\(\cdot\), \(p=0.2276\)) at various SNR: 0, 10, 20, and 30 dB. Figure 7 shows the bin index error variance of each estimator plotted against SNR, compared against the Cramer-Rao bound for unbiased estimators. Normalizing to bin width, the Cramer-Rao bound for a discrete-time signal corrupted by additive white Gaussian noise, with unknown phase and unknown amplitude is [44]  
\[\text{Var}(\hat{\mathcal{K}})=\frac{12\sigma^{2}N}{4\pi^{2}\mathcal{A}^{2}(N^{2} -1)}\,, \tag{42}\]  
where \(N=64\) is the signal length, \(\sigma\) is the noise standard deviation, and \(\mathcal{A}\) the sinusoid amplitude. To compare against the Cramer-Rao bound for unbiased estimation, we recompute the variance of the bin index estimation error under the assumption of a zero bias. Though we showed in Figure 5 that the estimation is not actually unbiased, the bias is commonly neglected for the noise robustness analysis of sinusoidal parameter analysis. Notice that all the QIFFT-based methods have a similar robustness against noise independently of the chosen magnitude spectrum scaling function. This result highly contrasts against the systematic estimation bias, which is highly sensitive to the choice of scaling function.