FFT of the Given Signal

We assume for now that $w(n)$ is a rectangular windowing function from $-\ensuremath{\frac{N-1}{2}}$ to $\ensuremath{\frac{N-1}{2}}$. (We will consider other windowing functions in a section below.) The DFT of our rectangular windowed $y(n)$ may be written

$$Y(k) = \sum_{n=-\ensuremath{\frac{N-1}{2}}}^{\ensuremath{\frac{N-1}{2}}}e^{i \alpha n^2} e^{-i2\pi k n/K }$$ (18)

$$= \sum_{n=-\ensuremath{\frac{N-1}{2}}}^{\ensuremath{\frac{N-1}{2}}}\exp(i( \alpha n^2 -2\pi k n/ K)).$$ (19)

where $K$ is the length of the optionally zero padded transform. (If no zero padding is used, $K=N$.) Using Euler's identity ( $e^{i\theta}=\cos(\theta)+i\sin(\theta)$), it is easy to see that the real and imaginary parts of $Y(k)$ are, respectively,

$$\Re{Y(k)} = \sum_{n=-(N-1)/2}^{(N-1)/2}\cos( \alpha n^2 -2\pi k n/ K)$$ (20)

$$\Im{Y(k)} = \sum_{n=-(N-1)/2}^{(N-1)/2}\sin( \alpha n^2 -2\pi k n/ K).$$ (21)