Selected Continuous-Time Fourier Theorems

This appendix presents Fourier theorems which are nice to know, but which do not, strictly speaking, pertain to the DFT. The differentiation theorem for Fourier Transforms comes up quite often, and its dual pertains as well to the DTFT (Appendix B). The scaling theorem provides an important basic insight into time-frequency duality. Finally, the very fundamental uncertainty principle is related to the scaling theorem.

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Subsections

• Differentiation Theorem
• Scaling Theorem
• The Uncertainty Principle
  • Second Moments
  • Time-Limited Signals
  • Time-Bandwidth Products are Unbounded Above

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Order a Hardcopy of Mathematics of the DFT

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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.