Fresnel Analysis Based Model

It may be shown [1] that for sufficiently large $\alpha$ and $N$,

$$Y(k) \approx Y(\omega) = \int_{-T}^T \exp(j( \alpha t^2- \omega t)) dt,$$ (3)

where we have applied the midpoint approximation to the definite integral in the Fourier transform of the analogous continuous time chirp, $y(t) = e^{j\alpha t^2}.$ Applying this approximation, it may be shown [1] that the real and imaginary parts of $Y(k)$ are given by:

$$\Re{Y(k)} \approx \pm\sqrt{\frac{\pi}{2\alpha}} \Big( \cos\left(\phi\right) \int_{l... ...sin\left(\phi\right) \int_{l_1}^{l_2}\sin\left(\frac{\pi}{2}u^2\right) du \Big)$$ (4)

$$\Im{Y(k)} \approx \pm\sqrt{\frac{\pi}{2\alpha}} \Big( \cos\left(\phi \right) \int_{... ...n\left(\phi\right) \int_{l_1}^{l_2} \cos\left(\frac{\pi}{2}u^2\right) du \Big),$$ (5)

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

We recognize the integrals in the above expressions as Fresnel integrals.

When $\alpha$, $N$, $k$,and $K$ are such that $l_1\ll-1$ and $l_2\gg1$, we may apply what we call the large limits approximations,

$$\int_{0}^{l_2}\sin\left(\frac{\pi}{2}u^2\right) du \approx \frac{1}{2} - \frac{1}{\pi l_2} \cos\left(\frac{\pi}{2} l_2^2\right)$$ (6)

$$\int_{0}^{l_2} \cos\left(\frac{\pi}{2}u^2\right) du \approx \frac{1}{2} + \frac{1}{\pi l_2} \sin\left(\frac{\pi}{2} l_2^2\right)$$ (7)

and where the odd symmetry of the Fresnel integrals leads to analogous negative results when the limits are $[l_1,0]$ rather than $[0,l_2]$ as above.

Figure 1: C(u) and S(u) (solid) with large limits approximations (dashed)

These approximations will allow us to create an invertible model of the signal's DFT, and incur a modeling error of less than 1% when $\{-l_1,l_2\}\geq 3$. (When $\{-l_1,l_2\}\geq 2$ or $\{-l_1,l_2\}\geq 1$ the error bounds are 2.3% or 14% respectively.)