The window function

Starting with a window function $\varphi \in L_2\left(\mathbb{R}\right)$, one assumes that its 1-periodic version $\tilde \varphi$, i.e.,

$$\tilde \varphi\left(x\right):=\sum_{r \in \mathbb{Z}} \varphi\left(x+r\right)$$

has an uniformly convergent Fourier series and is well localised in the time/spatial domain $\mathbb{T}$ and in the frequency domain $\mathbb{Z}$. The periodic window function $\tilde \varphi$ may be represented by its Fourier series

$$\tilde \varphi\left(x\right)= \sum_{k\in \mathbb{Z}} c_k\left(\tilde\varphi\right) {\rm e}^{-2\pi{\mbox{\scriptsize {i}}} k x}$$

with the Fourier coefficients

$$c_k\left( \tilde \varphi \right) :=\int\limits_{\mathbb{T}} \til... ...e {i}}} k x} \, {\rm d} x = \hat \varphi \left(k\right),\quad k\in\mathbb{Z}.$$

Figure: From left to right: Gaussian window funtion $\varphi$, its 1-periodic version $\tilde \varphi$, and the integral Fourier-transform $\hat \varphi$ (with pass, transition, and stop band) for $N=24,\,\sigma=\frac{4}{3},\,n=32$.