Gaussian Function Properties

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Subsections

• Gaussian Window and Transform
• Integral of a Complex Gaussian
  • Area Under a Real Gaussian
  
  
• Gaussian Integral with Complex Offset
• Fourier Transform of Complex Gaussian
  • Alternate Proof
  
  
• Gaussian Windowed Chirps
  • Modulated Gaussian Windowed Chirp
  
  
• Identifying Chirp Rate
  • Notes
  
  
• Simulation Results
• Gaussians Closed under Multiplication
  • Product of Two Gaussian PDFs
  • Gaussians are Closed under Convolution
  
  
• Infinite Flatness at Infinity
• Gaussian Probability Density Function
• Maximum Entropy Property
  • Entropy of a Probability Distribution
  • Example: Random Bit String
  • Maximum Entropy Distributions
    • Uniform Distribution
    • Exponential Distribution
    • Gaussian Distribution
  
  
• Uncertainty Principle
• Gaussian Moments
  • Gaussian Mean
  • Gaussian Variance
  • Higher Order Moments Revisited
  • Moment Theorem
  • Gaussian Characteristic Function
  • Gaussian Central Moments
  
  
• Sums of Gaussian Random Variables
• Why Gaussian?
  • Central Limit Theorem
  • Iterated Convolutions
  • Binomial Distribution
  • Boltzman Energy Distribution for a Potential Well
  
  
• History of the Gaussian Function
• References

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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.