Spectrum Analysis Windows

In spectrum analysis of naturally occurring audio signals, we nearly always analyze a short segment of a signal, rather than the whole signal. This is the case for a variety of reasons:

• Perhaps most fundamentally, the ear similarly Fourier analyzes only a short segment of audio signals at a time (on the order of 10-20 ms worth). Therefore, to match our spectrum analysis to human hearing, we desire to limit the time window of the analysis.
• Audio signals typically have spectra which change over time. It is therefore usually most meaningful to restrict analysis to a time window over which the spectrum stays rather constant.
• It can be extremely time consuming to compute the Fourier transform of an audio signal of typical length, and it will rarely fit in computer memory all at once.

We will see that the proper way to extract a ``short time segment'' of length from a longer signal is to multiply it by a window function such as the Hann window:

We will see that the main benefit of choosing a good Fourier analysis window function is minimization of side lobes, which cause ``cross-talk'' in the estimated spectrum from one frequency to another.

The study of spectrum-analysis windows serves other purposes as well. Most immediately, it provides an array of useful window types which are best for different situations. Second, by studying windows and their Fourier transforms, we build up our knowledge of Fourier dualities in general. Finally, the defining criteria of different window types often involve interesting and useful analytical techniques.

In this chapter, we begin with a summary of the rectangular window, followed by a variety of additional window types, including the generalized Hamming and Blackman-Harris families (sums of cosines), Bartlett (triangular), Poisson (exponential), Kaiser (Bessel), Dolph-Chebyshev, Gaussian, and other window types.

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Subsections

• Rectangular Window
  • Definition ( odd):
  • Transform:
  • Properties:
  
  
• Generalized Hamming Window Family
  • Hann or Hanning or Raised Cosine
  • Matlab for the Hann Window
  • Summary of Hann window properties:
  • Hamming Window
  • Matlab for the Hamming Window
  • Summary of Generalized Hamming Windows
  • Definition:
  • Transform:
  • Common Properties
  • Rectangular window properties:
  • Hann window properties:
  • Hamming window properties:
  • The MLT Sine Window
    • Properties:
  
  
• Blackman-Harris Window Family
  • Window Definition:
  • Window Transform:
  • Blackman Window Family
  • Classic Blackman
  • Matlab for the Classic Blackman Window
  • Three-Term Blackman-Harris Window
  • Frequency-Domain Blackman-Harris
  • Power-of-Cosine Window Family
    • Definition:
    • Properties:
    • Special Cases:
  
  
• Bartlett (``Triangular'') Window
  • Definition:
  • Transform:
  • Properties:
  • Matlab for the Bartlett Window:
  
  
• Poisson Window
  • Definition:
  
  
• Hann-Poisson Window
  • Definition:
  • Matlab for the Hann-Poisson Window
  
  
• Slepian or DPSS Window
  • Matlab for the DPSS Window
  
  
• Kaiser Window
  • Definition:
  • Window transform:
  • Kaiser Window Beta Parameter
  • Kaiser Windows and Transforms
  • Minimum Frequency Separation vs. Window Length
  • Kaiser and DPSS Windows Compared
  
  
• Dolph-Chebyshev Window
  • Matlab for the Dolph-Chebyshev Window
  • Example Chebyshev Windows and Transforms
  • Dolph-Chebyshev and Hamming Windows Compared
  • Dolph-Chebyshev Window Theory
    • Chebyshev Polynomials
    • Dolph-Chebyshev Window Definition
    • Dolph-Chebyshev Window Main-Lobe Width
    • Dolph-Chebyshev Window Length Computation
  
  
• Gaussian Window and Transform
  • Matlab for the Gaussian Window
  • Gaussian Window and Transform
  • Exact Discrete Gaussian Window
  
  
• Optimized Windows
  • Optimal Windows for Audio Coding
  • Conclusion
  
  
• Exercises
• Lab Assignments

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Order a Hardcopy of Spectral Audio Signal Processing

Previous: Relation of Smoothness to Roll-Off Rate
Next: Rectangular Window

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written by Julius Orion Smith III
Julius Smith's background is in electrical engineering (BS Rice 1975, PhD Stanford 1983). He is presently Professor of Music and Associate Professor (by courtesy) of Electrical Engineering at Stanford's Center for Computer Research in Music and Acoustics (CCRMA), teaching courses and pursuing research related to signal processing applied to music and audio systems. See http://ccrma.stanford.edu/~jos/ for details.