A comprehensive windows tutorial

This article is a tutorial attempt to provide an easier analysis of how windows work. I begin by looking at individual spectrum bins as affected by off-bin-centered tones with six different windows. I define and use the convolution theorem to explain why the windows do what they do. I review the sophisticated misunderstood fat-center-peaked, picket-fence-looking FFTs of various windows that come from adding zeros to increase resolution. I then use these drawings in a step-by-step graphical convolution to show how this or that window will affect the FFT of your data. This is a procedure to demonstrate how any oddly shaped, even graphically defined, window affects data. These techniques are tools to allow you to invent a graphical window to affect data in new ways. To demonstrate how to use graphically defined windows, I use digitized drawings of a Hawaiian mountain and the popular cartoon beagle as windows. To use these windows, I demonstrate FFT-IFFT interpolation and resampling, which in itself is interesting. Finally, I compare how eight windows affect the spectrum of typical vibration data.