Plancharel and Parseval's Theorems 2.5 2002/07/31 2006/08/02 14:32:40.801 GMT-5 Justin Romberg jrom@rice.edu Prashant Singh prash@ece.rice.edu Richard G. Baraniuk richb@rice.edu Mariyah Poonawala mariyah@rice.edu Matthew Hutchinson mhutch@rice.edu Parseval Plancharel This module contains the definition of the Plancharel theorem and Parseval's theorem along with proofs and examples.

Plancharel Theorem Plancharel Theorem The inner product of two vectors/signals is the same as the ℓ 2 inner product of their expansion coefficients. Let b i be an orthonormal basis for a Hilbert Space H . x H , y H x i α i b i y i β i b i then x y H i α i β i Applying the Fourier Series, we can go from f t to c n and g t to d n t 0 T f t g t n c n d n inner product in time-domain = inner product of Fourier coefficients. x i α i b i y j β j b j x y H i α i b i j β j b j i α i b i j β j b j i α i j β j b i b j i α i β i by using inner product rules b i b j 0 when i j and b i b j 1 when i j If Hilbert space H has a ONB, then inner products are equivalent to inner products in ℓ 2 . All H with ONB are somehow equivalent to ℓ 2 . square-summable sequences are important

Parseval's Theorem Parseval's Theorem Energy of a signal = sum of squares of it's expansion coefficients Let x H , b i ONB x i α i b i Then H x 2 i α i 2 Directly from Plancharel H x 2 x x H i α i α i i α i 2 Fourier Series 1 T w 0 n t f t 1 T n c n 1 T w 0 n t t 0 T f t 2 n c n 2