The window function

Starting with a reasonable window function $\varphi:\ensuremath{\mathbb{R}}\rightarrow\ensuremath{\mathbb{R}}$ , one assumes that its 1-periodic version $\tilde \varphi$ , i.e.

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

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

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

with the Fourier coefficients

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

Figure 2.1: From left to right: Gaussian window function $\varphi$ , its 1-periodic version $\tilde \varphi$ , and the integral Fourier-transform $\hat \varphi$ (with pass, transition, and stop band) for $N=24,\,\sigma=\frac{4}{3},\,n=32$ .