Fresnel Model Result

Finally, to obtain the Fresnel model:

1. Substitute the approximations above.
2. Apply a time domain Hann window via a frequency domain convolution [1].

This leads to the following expressions for the magnitude and phase of $Y(k)$, valid for $k$ as noted above:

$$\angle Y(k) \approx \pm \frac{\pi}{4} - \frac{\pi^2 k^2}{K^2\alpha}$$ (8)

$$\vert Y(k)\vert \approx \sqrt{\frac{\pi}{\vert\alpha}\vert} \left(\ensuremath{\frac{1}{2}... ...math{\frac{1}{2}}\cos\left(\ensuremath{\frac{2\pi^2 k}{KN\alpha}}\right)\right)$$ (9)

where the $\ensuremath{\frac{\pi}{4}}$ term in the phase expression is positive for $\alpha>0$ and negative for $\alpha<0$.

As we will see in the next section, these models are invertible.