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 perform a spectrum analysis having time- and frequency-resolution comparable to human hearing, we must limit the time-window accordingly. 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
• Generalized Hamming Window Family
  • Hann or Hanning or Raised Cosine
  • Matlab for the Hann Window
  • Hamming Window
  • Matlab for the Hamming Window
  • Summary of Generalized Hamming Windows
  • The MLT Sine Window
  
  
• 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
  
  
• Spectrum Analysis of an Oboe Tone
  • Rectangular-Windowed Oboe Recording
  • Hamming-Windowed Oboe Recording
  • Blackman-Windowed Oboe Recording
  • Conclusions
  
  
• Bartlett (``Triangular'') Window
  • Matlab for the Bartlett Window:
  
  
• Poisson Window
• Hann-Poisson Window
  • Matlab for the Hann-Poisson Window
  
  
• Slepian or DPSS Window
  • Matlab for the DPSS Window
  
  
• Kaiser Window
  • 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
  • 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
  • General Rule
  
  
• Optimal Window Design by linprog
  • Linear Programming (LP)
  • LP Formulation of Chebyshev Window Design
  • Symmetric Window Constraint
  • Positive Window Sample Constraint
  • DC Constraint
  • Sidelobe Specification
  • LP Standard Form
    • Normal Chebyshev Window
  • Remez Exchange Algorithm
    • Convergence of Remez Exchange
  • Monotonicity Constraint
    • Monotonic Chebyshev Window
  • L-Infinity Norm of Derivative Objective
  • L-One Norm of Derivative Objective

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Previous: Matlab/Octave fftshift utility
Next: Rectangular Window

Order a Hardcopy of Spectral Audio Signal Processing

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About the Author: 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.