Frequency Support of Window Functions

Question

All window functions, which go from zero amplitude, up, then back to zero amplitude, are all base-band signals. Why is this? Are all functions that have this behavior always band-band aka low-pass?

Answer 1

A window function $w(t)$ satisfies $w(t)\ge 0$. The value of a window's frequency response $W(f)$ at DC ($f=0$) equals its integral

$$W(0)=\int_{-\infty}^{\infty}w(t)dt>0\tag{1}$$

which is clearly greater than zero because $w(t)\ge 0$. For all other frequencies we obtain the following bound:

$$|W(f)|=\left|\int_{-\infty}^{\infty}w(t)e^{-j2\pi ft}dt\right|\le\int_{-\infty}^{\infty}\left|w(t)e^{-j2\pi ft}\right|dt=\int_{-\infty}^{\infty}w(t)dt=W(0)\tag{2}$$

Eq. $(2)$ means that $|W(f)|$ cannot be greater than its DC value. Actually, it can be shown that $(2)$ is a strict inequality for any $f\neq 0$: $|W(f)|<W(0)$ for any $f>0$. This shows that $W(f)$ has a lowpass characteristic.

To give a more intuitive explanation, a rectangular window's frequency response is a sinc function, which has a lowpass characteristic. All other windows are just smoother versions of a rectangular window, which means that their spectra decay faster than a sinc. I.e., non-rectangular windows have an even more pronounced low pass characteristic than a rectangular window.

Answer 2

From the seminal paper on the subject by harris:

"On the Use of Windows for Harmonic Analysis with the Discrete Fourier Transform. FREDRIC J. HARRIS"

I've recently discovered a new family of windows. They are very special.

The Zeroing Sine Family of Window Functions

You can think of them as the discrete version of the Raised Sine Windows (Equation (25b) in harris' paper).

These windows literally zero out the sample values to achieve this effect.