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DEMO_GABFIR - Working with FIR windows

Description

This demo demonstrates how to work with FIR windows in Gabor systems.

FIR windows are the windows traditionally used in signal processing. They are short, much shorter than the signal, and this is used to make effecient algorithms. They are also the only choice for applications involving streaming data.

It is very easy to compute a spectrogram or Gabor coefficients using a FIR window. The hard part is reconstruction, because both the window and the dual window used for reconstruction must be FIR, and this is hard to obtain, if the window is longer than the number of channels.

This demo demonstrates two methods:

  1. Using a Gabor frame with a simple structure, for which dual/tight FIR windows are easy to construct. This is a very common technique in traditional signal processing, but it limits the choice of windows and lattice parameters.
  2. Cutting a canonical dual/tight window. We compute the canonical dual window of the analysis window, and cut away the parts that are close to zero. This will work for any analysis window and any lattice constant, but the reconstruction obtained is not perfect.
demo_gabfir_1.png

Hanning FIR window

This figure shows the a Hanning window in the time domain and its magnitude response.
demo_gabfir_2.png

Kaiser-Bessel FIR window

This figure shows a Kaiser Bessel window and its magnitude response, and the same two plots for the canonical dual of the window.
demo_gabfir_3.png

Gaussian FIR window for low redundancy

This figure shows a truncated Gaussian window and its magnitude response. The same two plots are show for the truncated canonical dual window.
demo_gabfir_4.png

Almost tight Gaussian FIR window

This figure shows a tight Gaussian window that has been truncated and its magnitude response.