The Fourier Transform - Parseval's Theorem

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We've discussed how the Fourier Transform gives us a unique representation of the original underlying signal, g(t). That is, G(f) contains all the information about g(t), just viewed in another manner. To further cement the equivalence, on this page we present Parseval's Identity for Fourier Transforms.

Let g(t) have Fourier Transform G(f). Then the following equation is true:

parseval's theorem         [1]

The integral of the squared magnitude of a function is known as the energy of the function. For example, if g(t) represents the voltage across a resistor, then the energy dissipated in the resistor will be proportional to the integral of the square of g(t). Equation [1] states that the energy of g(t) is the same as the energy contained in G(f). This is a powerful result, and one that is central to understanding the equivalence of functions and their Fourier Transforms.

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