Phase Based Estimator

Our phase-based (equation 8) estimator ensures neutrality to phase shift in the input by analyzing the curvature of the phase via evaluation of a second order difference (again with sufficiently large $K$). Doing so and solving for $\alpha$ yields

$$\alpha \approx \frac{-2\pi^2}{K^2} \left(\frac{\Delta^2\angle(Y(k))}{\Delta k^2}\right)^{-1}.$$ (17)

where we estimate the second order phase difference by averaging the second order difference values obtained when considering some $k \in k_{hh}$.

The exact portion of $k$ examined by our estimator may be chosen as a function of the desired Fresnel model accuracy.

• Choosing a greater range of $k$ increases the amount of data included in the average, but strains the Fresnel approximations by requiring limits to be smaller (recall the discussion in section 2.1).
• It may be shown [1] that to ensure the limits in the Fresnel integrals are at least $c$, the range of $k$ used must be
  

  $$\vert k\vert \leq \left(2-\ensuremath{\frac{2c}{M}}\sqrt{\ensuremath{\frac{2 \pi}{\hat{\alpha}}}}\right)\cdot k_{hh},$$ (18)

  
  
  where $\hat{\alpha}$ in this case is the estimate given by the magnitude model estimator above, and preliminary experiments show that the ideal choice of the Fresnel limits parameter is around $c=2.5$.
• This limits our estimator to cases where $\vert\alpha\vert N^2 \geq 50\pi$.
• For $\vert\hat{\alpha}\vert$ smaller than this, equation 18 will lead to a range of $k$ entirely within the $k_{hh}$ for a stationary sinusoid (the window transform), or to no $k$ at all. In those cases, our estimator will use only $k\in \{-1,0,1\}$, generating estimates that are not accurate but that provide useful information for our neural net.
• The neural net is used in the transition region between our large $\alpha$ Fresnel model, and small $\alpha$ Taylor series model.