Summaries of Other Implemented Algorithms for Future Publication

In this appendix, we present summaries of additional techniques for the chirp parameter estimation problem. Each of these techniques has been implemented by the author, and examples as well as in-depth discussion of each are forthcoming in future reports.

These approaches are still relevant to Fresnel analysis, however, as each relies on Fresnel integrals, but simply uses different approximations of them than those used in the paper above. Specifically, the issue becomes our use of equations 66 and 67 as ``valid approximations'' of the Fresnel integrals. As mentioned in the writing above, those approximations are not always accurate, depending on $\alpha$ and the frequency bins $k$ considered. Below, we reconsider the approximations used in those equations for

$$\int_{\pm l_1}^{\pm l_2} \cos\left(\frac{\pi}{2}u^2\right) du$$ (114)

and

$$\int_{\pm l_1}^{\pm l_2} \sin\left(\frac{\pi}{2}u^2\right) du,$$ (115)

where $l_1 =\sqrt{\frac{2\alpha}{\pi}}\left(-\frac{N}{2}-\frac{\pi k}{\alpha K}\right)$ and $l_2 =\sqrt{\frac{2\alpha}{\pi}}\left(\frac{N}{2}-\frac{\pi k}{\alpha K}\right)$.

We show that when $\alpha$ is smaller than permitted above and we choose frequency bins carefully, we may again obtain useful expressions for the real and imaginary parts of $Y_a(k)$, by using what we term the ``same sign large limits approximation'' and the ``small limits approximation.''

To compare the performance of these new models to those discussed in the body of the paper (and to Liu's model in appendix B) we conclude this section with comparative plots.