Selected Continuous Fourier Theorems

This section presents continuous-time Fourier theorems that go beyond obvious analogs of the DTFT theorems proved in §2.3 above. The differentiation theorem comes up quite often, and its dual pertains as well to the DTFT. The scaling theorem provides an important basic insight into time-frequency duality. The Poisson Summation Formula (PSF) in continuous time extends the discrete-time version presented in §8.3.1. Finally, the extremely fundamental uncertainty principle is derived from the scaling theorem.

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Subsections

• Radians versus Cycles
• Differentiation Theorem
• Differentiation Theorem Dual
• Scaling Theorem
• Shift Theorem
• Convolution Theorem
• Flip Theorems
• Power Theorem
• The Continuous-Time Impulse
• Gaussian Pulse
• Rectangular Pulse
• Sinc Impulse
• Impulse Trains
• Poisson Summation Formula
• Sampling Theory
• The Uncertainty Principle
  • Second Moments
  • Time-Limited Signals
  • Time-Bandwidth Products are Unbounded Above

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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.