Magnitude-based estimator

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.