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Plancharel and Parseval's Theorems

Module by: Justin Romberg

Summary: This module contains the definition of the Plancharel theorem and Parseval's theorem along with proofs and examples.

Plancharel Theorem

theorem 1: Plancharel Theorem

The inner product of two vectors/signals is the same as the ℓ2 ℓ 2 inner product of their expansion coefficients.

Let b i b i be an orthonormal basis for a Hilbert Space H H. x∈H x H , y∈H y H x=∑ α i b i x i α i b i y=∑ β i b i y i β i b i then <x,y> H =∑ α i β i ¯ x y H i α i β i

Example

Applying the Fourier Series, we can go from ft f t to c n c n and gt g t to d n d n ∫0Tftgt¯dt=∑n=-∞∞ c n d n ¯ t 0 T f t g t n c n d n inner product in time-domain = inner product of Fourier coefficients.

Proof

x=∑ α i b i x i α i b i y=∑ β j b j y j β j b j <x,y> H =<∑ α i b i ,∑ β j b j >=∑ α i < b i ,∑ β j b j >=∑ α i ∑ β j ¯< b i , b j >=∑ α i β i ¯ 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

note:
< b i , b j >=0 b i b j 0 when i≠j i j and < b i , b j >=1 b i b j 1 when i=j i j

If Hilbert space H has a ONB, then inner products are equivalent to inner products in ℓ2 ℓ 2 .

All H with ONB are somehow equivalent to ℓ2 ℓ 2 .

point of interest:
square-summable sequences are important

Parseval's Theorem

theorem 2: Parseval's Theorem

Energy of a signal = sum of squares of it's expansion coefficients

Let x∈H x H , b i b i ONB

x=∑ α i b i x i α i b i Then ∥x∥H2=∑| α i |2 H x 2 i α i 2

Proof

Directly from Plancharel ∥x∥H2= <x,x> H =∑ α i α i ¯=∑| α i |2 H x 2 x x H i α i α i i α i 2

Example 1

Fourier Series 1Tⅇⅈ w 0 nt 1 T w 0 n t ft=1T∑ c n 1Tⅇⅈ w 0 nt f t 1 T n c n 1 T w 0 n t ∫0T|ft|2dt=∑n=-∞∞| c n |2 t 0 T f t 2 n c n 2

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