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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?

The Dude
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2 Answers2

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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.

Matt L.
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  • I see. That is what I was looking for. Thanks! – The Dude Aug 18 '20 at 15:42
  • Matt,

    I have to disagree with you here (assuming I am understanding your terminology correctly). The eigenvector window when applied to a pure complex tone, simply slides the profile up the spectrum, shape and size intact. Thus $|W(f)|=|W(0)|$ for all $f$. As the church lady would say: "Isn't that special?"

     – Cedron Dawg Aug 18 '20 at 16:21
  • @CedronDawg: You mean $w(t)=e^{j\omega t}$? – Matt L. Aug 18 '20 at 16:26
  • No, I think I misunderstood what you are saying. The profile of the eigenvector has a peak in the center, and it is indeed bigger than any of the side bins in magnitude. If you apply the window to a pure complex tone $x[n]=e^{i\omega n}$ then take the DFT, the profile is preserved. – Cedron Dawg Aug 18 '20 at 16:31
  • @CedronDawg: Yes, but how can $|W(f)|$ be constant then? – Matt L. Aug 18 '20 at 16:31
  • @CedronDawg: If $w(t)\ge 0$ is satisfied, it follows that $|W(f)|<W(0)$ for $f>0$. – Matt L. Aug 18 '20 at 16:32
  • No, you're right, the center is taller than the sides, I did misunderstand what you were saying, sorry. – Cedron Dawg Aug 18 '20 at 16:48
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From the seminal paper on the subject by harris:

enter image description here

"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.

Cedron Dawg
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