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FFT Window Functions

A window function is a mathematical function that is normally symmetric around the middle of the interval, usually near a maximum in the middle, and usually tapering away from the middle.

Each window function has its own characteristics and suitability for different applications (some are more frequency accurate, others are more amplitude accurate). To choose a window function, you (the user) must select the most appropriate one. Some suggestions:

  • Sine wave or combination of sine waves: Hann
  • Sine wave (amplitude accuracy is important): Flat Top
  • Narrowband random signal (vibration data): Hann
  • Broadband random (white noise): Rectangular
  • Closely spaced sine waves: Uniform, Hamming
  • Excitation signals (hammer blow): Force
  • Response signals: Exponential
  • Unknown content: Hann
  • Two tones with frequencies close but amplitudes very different: Kaiser-Bessel
  • Two tones with frequencies close and almost equal amplitudes: Rectangular

Hann Window

Hann (also known as Hanning, raised cosine, or von Hann) window, is named after the Austrian meteorologist Julius von Hann. The math looks like:

w[n] = 1/2(1 - cos({2 pi n}/N))

Rectangular window

The rectangular window (also known as as boxcar, uniform or Dirichlet window) is the simplest window, equivalent to replacing all but N values of a data sequence by zeros, making it appear as though the waveform suddenly turns on and off.

w[n] = 1


Triangular windows are given by:

w[n] = 1 - delim{|}{{n - N/2}/{N/2}}{|}


The Welch window consists of a single parabolic section:

w[n] = 1 - ({n - {N/2}}/{N/2})^2


General purpose Blackman windows are defined as:

w[n] = a_{0} - a_{1}cos({2 pi n}/N) + a_{2}cos({4 pi n}/N)

The version that we implement is known as the Exact Blackman and uses:

  • a_{0} = 7938/18608
  • a_{1} = 9240/18608
  • a_{2} = 1430/18608


The cosine window is also known as the sine window, as it represents half a cycle of a sinusoidal function,

w[n] = cos({pi n}/{N} - pi/2) = sin({pi n}/{N})





Flat Top

3 Term Cosine

4 Term Cosine

5 Term Cosine

6 Term Cosine

7 Term Cosine

/srv/ · Last modified: 01 Jan 2022 07:36 by Robin Getz