Window Function (Part Two)
I introduced the rectangular window in the previous session but it has a problem. The window is very simple but the rapid transition in both ends makes a negative effect. To avoid this problem, we can use the following window function.
The window is called Hanning window. You can see the effect of the Hanning window in the following applet, changing the phase and frequency by clicking [<] and [>] of Phase and [Down] and [Up] of Frequency, and monitoring the spectrum in the right half.
Did you sense some changes from the rectangular window? You can see that the shape of spectrum does not change depending on the frequency while it significantly changes when using the rectangular window. Especially, the peak level change is reasonably suppressed in the Hanning window while in the rectangular window the maximum peak level, where the energy is converged, is about twice of the minimum, when the energy is dispersed.
See the left side of the applet, where the gray lines show the original signal, the red line shows the windowed signal, and the blue line shows the window function itself. As the window function is represented in zero-to-one scale vertically, I think you can intuitively understand the shape of windowed signal.
Windows functions I provided here other than the rectangular window or Hanning window are the following.
Hamming Window: 
Blackman Window: 
Kaiser Window: 
See the difference of the spectrum characteristics among windows. In the Kaiser window, the spectrum characteristic depends on the value of Alpha. (Alpha can be adjusted by [Alpha] field in the applet.)
Using windows other than the rectangular window reduces the variance of spectrum according to the frequency change. Using the Blackman or Kaiser (Alpha is more than ten) window can reasonably suppress the variance of the peak level while adjacent frequencies cannot be separated even if the width of the window includes an integral multiple of the fundamental wave. (Using the rectangular window, when a window includes just an integral multiple of the fundamental wave, each harmonic component is converged to a line.)
As you see in the applet, introducing windows other than rectangular window provides a solution to the problem that the shape of spectrum varies depending on the frequency but it degrades the frequency resolution. If the number of points of FFT is larger, the interval of frequencies is reduced, resulting that the same number of lines (for example four lines) represents a narrower frequency band. However, a larger number of points means a longer window in FFT, it is not suitable to monitor a part with a certain length, in which the wave form is fairly stable. Therefore, if you want a higher resolution of time, you have to sacrifice the frequency resolution, and vice versa.
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