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

Parseval's theorem

Continuous Time Fourier Series preserves signal energy

i.e.:

0 T | f ( t ) | 2 d t = T n = - | C n | 2 with unnormalized basis e j 2 π T n t 0 T | f ( t ) | 2 d t = n = - | C n ' | 2 with unnormalized basis e j 2 π T n t T | | f | | 2 2 L 2 [ 0 , T ) e n e r g y = | | C n ' | | 2 2 l 2 ( Z ) e n e r g y

Prove: plancherel theorem

Given f ( t ) C T F S c n g ( t ) C T F S d n Then 0 T f ( t ) g * ( t ) d t = T n = - c n d n * with unnormalized basis e j 2 π T n t 0 T f ( t ) g * ( t ) d t = n = - c n ' ( d n ' ) * with normalized basis e j 2 π T n t T f , g L 2 ( 0 , T ] = c , d l 2 ( Z )

Periodic signals power

Energy = | | f | | 2 = - | f ( t ) | 2 d t = Power = lim T Energy in [ 0 , T ) T = lim T T n | c n | 2 T = n Z | c n | 2 (unnormalized FS)

Fourier series of square pulse iii -- compute the energy

f ( t ) = n = - c n e j 2 π T n t FS c n = 1 2 sin π 2 n π 2 n energy in time domain: | | f | | 2 2 = 0 T | f ( t ) | 2 d t = T 2 apply Parseval's Theorem: T n | c n | 2 = T 4 n sin π 2 n π 2 n 2 = T 4 4 π 2 n sin π 2 n 2 n 2 = T π 2 π 2 4 + n odd 1 n 2 π 2 4 = T 2

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Plancharel theorem

Plancharel theorem

The inner product of two vectors/signals is the same as the 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.

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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 2 .

All H with ONB are somehow equivalent to 2 .

square-summable sequences are important

Plancharels theorem demonstration

PlancharelsTheoremDemo
Interact (when online) with a Mathematica CDF demonstrating Plancharels Theorem visually. To Download, right-click and save target as .cdf.

Parseval's theorem: a different approach

Parseval's theorem

Energy of a signal = sum of squares of its 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

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Source:  OpenStax, Signals and systems. OpenStax CNX. Aug 14, 2014 Download for free at http://legacy.cnx.org/content/col10064/1.15
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