OpenStax-CNX module: m32881 1

Parseval's Theorem for the Fourier

Series∗

Carlos E. Davila

This work is produced by OpenStax-CNX and licensed under the

Creative Commons Attribution License 3.0†

Recall that in Chapter 1, we dened the power of a periodic signal as

px =1T

∫ t0+T

t0

x2 (t) dt (1)

where T is the period. Using the complex form of the Fourier series, we can write

x(t)2 =

( ∞∑n=−∞

cnejnΩ0t

)( ∞∑m=−∞

cmejmΩ0t

)∗(2)

where we have used the fact that x(t)2 = x (t) x(t)∗, i.e. since x (t) is real x (t) = x(t)∗. Applying here1 andhere2 gives

x(t)2 =(∑∞

n=−∞ cnejnΩ0t) (∑∞

m=−∞ c∗me−jmΩ0t)

=∑∞

n=−∞∑∞

m=−∞ cnc∗mej(n−m)Ω0t

=∑∞

n=−∞ |cn|2 +∑

n 6=m cnc∗mej(n−m)Ω0t

(3)

Substituting this quantity into (1) gives

px = 1T

∫ t0+T

t0

[∑∞n=−∞ |cn|2 +

∑n 6=m cnc∗mej(n−m)Ω0t

]dt

=∑∞

n=−∞ |cn|2 + 1T

∫ t0+T

t0

∑n 6=m cnc∗mej(n−m)Ω0tdt

(4)

It is straight-forward to show that

1T

∫ t0+T

t0

∑n 6=m

cnc∗mej(n−m)Ω0tdt = 0 (5)

This leads to Parseval's Theorem for the Fourier series:

px =∞∑

n=−∞|cn|2 (6)

∗Version 1.3: Nov 19, 2009 12:37 pm -0600†http://creativecommons.org/licenses/by/3.0/1"Complex Numbers and Complex Arithmetic", (9) <http://cnx.org/content/m32867/latest/#uid6>2"Complex Numbers and Complex Arithmetic", (10) <http://cnx.org/content/m32867/latest/#uid7>

http://cnx.org/content/m32881/1.3/

OpenStax-CNX module: m32881 2

which states that the power of a periodic signal is the sum of the magnitude of the complex Fourier seriescoecients.

http://cnx.org/content/m32881/1.3/