Approximation of the FFT Sums with Integrals

To apply the Fresnel analysis suggested above, we will need to convert the sums in equations 20 and 21 above to integrals, and will need to manipulate the resulting integrals into Fresnel integrals.

We begin this process by considering the above sums as midpoint approximations of an integral.⁵ Stated formally, these approximations are:

$$\Re{Y_a(k)} = \int_{-N/2}^{N/2}\cos( \alpha n^2 -2\pi k n/ K) dn \approx \sum_{n=-(N-1)/2}^{(N-1)/2}\cos( \alpha n^2 -2\pi k n/ K)$$ (24)

$$\Im{Y_a(k)} = \int_{-N/2}^{N/2}\sin( \alpha n^2 -2\pi k n/ K) dn \approx \sum_{n=-(N-1)/2}^{(N-1)/2}\sin( \alpha n^2 -2\pi k n/ K).$$ (25)

where the subscript $a$ reflects that the given expression is an inverse midpoint integral approximation. Note that we have omitted any reference to sampling frequency, since we are treating our integral as an approximation of the sum of the given samples, not as the area under the discrete curve.