Hann Window

As noted in the previous subsection, we may think of applying the time domain Hann window as creating a weighted frequency domain sum. In the current situation, this again can become problematic. Since we are using the small limits approximation to obtain $Y(0)$, the Hann window requires us to consider $Y^{Rect}_a\left(-\ensuremath{\frac{K}{N}}\right)$ and $Y^{Rect}_a\left(\ensuremath{\frac{K}{N}}\right)$ as well. Mathematically, this presents a difficulty, since the small limits approximation is not valid for $Y^{Rect}_a\left(\pm\ensuremath{\frac{K}{N}}\right)$. We can still use the large limits approximation to consider those values. This, however, does not lead to an elegant or invertible expression for $\alpha$. Fortunately, it can be shown that $Y_a\left(\pm\ensuremath{\frac{K}{N}}\right)\approx 0$. Intuitively, this makes sense, because as $\alpha$ becomes very small, the ``chirp'' signal becomes a quasistationary sinusoid, and the FFT converges to the window transform of the signal which is identically zero at $\pm\ensuremath{\frac{K}{N}}$. Applying this idea, we have that

$$\fbox{$\displaystyle \alpha \approx 2\cdot\ensuremath{\frac{\Im{Y^{Hann}_a(0)}\cdot 4\pi}{N^3}}.$}$$ (131)