Fresnel Model Estimators

We present two estimators based on Fresnel analysis, one each from equations 8 and 9. Our magnitude-based (equation 9) estimator will rely on the fact that the cosine function in the chirp magnitude model has an argument valued at $\ensuremath{\frac{\pi}{2}}$ at the $k$ value where the magnitude is at half the maximum height on a linear scale. Thus, we may solve for $\alpha$ via

$$\vert\alpha\vert \approx \ensuremath{\frac{4\pi k_{hh}}{KN}},$$ (16)

where $k_{hh}$ corresponds to the $k$ value at half the peak's height in linear amplitude. To get very accurate $k_{hh}$ estimates, we linearly interpolate between the appropriate nearest $k$ values, recalling that a point of inflection exists as a cosine argument passes through $\ensuremath{\frac{\pi}{2}}$. A rough phase curvature estimate (below) may be used to determine the sign of $\alpha$. Other similar magnitude estimators relying on a pre-specified or smaller range of $k$ (even as small as $k\in \{-1,0,1\}$) are indeed possible, but are less accurate both with and without noise. In applications where interference at higher (analogous) magnitude $k$ values is likely, such estimators may prove useful, or may at least highlight such interference. This is left as an area for future exploration. 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 [6] 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.