Power Theorem

Theorem: For all ,

Proof:

As mentioned in §5.8, physical power is energy per unit time.^7.19 For example, when a force produces a motion, the power delivered is given by the force times the velocity of the motion. Therefore, if and are in physical units of force and velocity (or any analogous quantities such as voltage and current, etc.), then their product is proportional to the power per sample at time , and becomes proportional to the total energy supplied (or absorbed) by the driving force. By the power theorem, can be interpreted as the energy per bin in the DFT, or spectral power, i.e., the energy associated with a spectral band of width .^7.20

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Subsections

• Normalized DFT Power Theorem

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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.