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Fourier series

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Background

A periodic signal x ( t ) size 12{x \( t \) } {} can be expressed by an exponential Fourier series as follows:

x ( t ) = ∑ n = − ∞ ∞ c n e j 2π nt T size 12{x \( t \) = Sum cSub { size 8{n= - infinity } } cSup { size 8{ infinity } } {c rSub { size 8{n} } e rSup { size 8{j { {2π ital "nt"} over {T} } } } } } {}

where T indicates the period of the signal and c n size 12{c rSub { size 8{n} } } {} ’s are called Fourier series coefficients, which, in general, are complex. Obtain these coefficients by performing the following integration

c n = 1 T ∫ T x ( t ) e − j 2π nt T dt size 12{c rSub { size 8{n} } = { {1} over {T} } Int cSub { size 8{T} } {x \( t \) e rSup { size 8{ - j { {2π ital "nt"} over {T} } } } ital "dt"} } {}

which possesses the following symmetry properties

∣ c − n ∣ = ∣ c n ∣ size 12{ lline c rSub { size 8{ - n} } rline = lline c rSub { size 8{n} } rline } {}
∠ c − n = − ∠ c n size 12{∠c rSub { size 8{ - n} } = - ∠c rSub { size 8{n} } } {}

where the symbol ∣ . ∣ size 12{ lline "." rline } {} denotes magnitude and ∠ size 12{∠} {} phase. Magnitudes of the coefficients possess even symmetry and their phases odd symmetry.

A periodic signal x ( t ) size 12{x \( t \) } {} can also be represented by a trigonometric Fourier series as follows:

x ( t ) = a 0 + ∑ n = 1 ∞ a n cos ( 2π nt T ) + b n sin ( 2π nt T ) size 12{x \( t \) =a rSub { size 8{0} } + Sum cSub { size 8{n=1} } cSup { size 8{ infinity } } {a rSub { size 8{n} } "cos" \( { {2π ital "nt"} over {T} } \) +b rSub { size 8{n} } "sin" \( { {2π ital "nt"} over {T} } \) } } {}

where

a 0 = 1 T ∫ T x ( t ) dt size 12{a rSub { size 8{0} } = { {1} over {T} } Int cSub { size 8{T} } {x \( t \) ital "dt"} } {}
a n = 2 T ∫ T x ( t ) cos ( 2π nt T ) dt size 12{a rSub { size 8{n} } = { {2} over {T} } Int cSub { size 8{T} } {x \( t \) "cos" \( { {2π ital "nt"} over {T} } \) ital "dt"} } {}
b n = 2 T ∫ T x ( t ) sin ( 2π nt T ) dt size 12{b rSub { size 8{n} } = { {2} over {T} } Int cSub { size 8{T} } {x \( t \) "sin" \( { {2π ital "nt"} over {T} } \) ital "dt"} } {}

The relationships between the trigonometric series and the exponential series coefficients are given by

a 0 = c 0 size 12{a rSub { size 8{0} } =c rSub { size 8{0} } } {}
a n = 2 Re { c n } size 12{a rSub { size 8{n} } =2"Re" lbrace c rSub { size 8{n} } rbrace } {}
b n = − 2 Im { c n } size 12{b rSub { size 8{n} } = - 2"Im" lbrace c rSub { size 8{n} } rbrace } {}
c n = 1 2 ( a n − jb n ) size 12{c rSub { size 8{n} } = { {1} over {2} } \( a rSub { size 8{n} } - ital "jb" rSub { size 8{n} } \) } {}

where Re size 12{"Re"} {} and Im size 12{"Im"} {} denote the real and imaginary parts, respectively.

According to the Parseval’s theorem, the average power in the signal x ( t ) size 12{x \( t \) } {} is related to the Fourier series coefficients c n size 12{c rSub { size 8{n} } } {} ’s, as indicated below

1 T ∫ T ∣ x ( t ) ∣ 2 dt = ∑ n = − ∞ ∞ ∣ c n ∣ 2 size 12{ { {1} over {T} } Int cSub { size 8{T} } { lline x \( t \) rline rSup { size 8{2} } ital "dt"= Sum cSub { size 8{n= - infinity } } cSup { size 8{ infinity } } { lline c rSub { size 8{n} } rline rSup { size 8{2} } } } } {}

More theoretical details of Fourier series are available in signals and systems textbooks [link] - [link] .

Fourier series numerical computation

Fourier series coefficients are often computed numerically – in particular, when an analytic expression for x ( t ) size 12{x \( t \) } {} is not available or the integration in [link] - [link] is difficult to perform. By approximating the integrals in [link] - [link] with a summation of rectangular strips, each of width Δt size 12{Δt} {} , one can write

a 0 = 1 M ∑ m = 1 M x ( mΔt ) size 12{a rSub { size 8{0} } = { {1} over {M} } Sum cSub { size 8{m=1} } cSup { size 8{M} } {x \( mΔt \) } } {}
a n = 2 M ∑ m = 1 M x ( mΔt ) cos ( 2π mn M ) size 12{a rSub { size 8{n} } = { {2} over {M} } Sum cSub { size 8{m=1} } cSup { size 8{M} } {x \( mΔt \) "cos" \( { {2π ital "mn"} over {M} } \) } } {}
b n = 2 M ∑ m = 1 M x ( mΔt ) sin ( 2π mn M ) size 12{b rSub { size 8{n} } = { {2} over {M} } Sum cSub { size 8{m=1} } cSup { size 8{M} } {x \( mΔt \) "sin" \( { {2π ital "mn"} over {M} } \) } } {}

where x ( mΔt ) size 12{x \( mΔt \) } {} are M size 12{M} {} equally spaced data points representing x ( t ) size 12{x \( t \) } {} over a single period T size 12{T} {} , and Δt size 12{Δt} {} denotes the interval between data points such that Δt = T M size 12{Δt= { {T} over {M} } } {}

Similarly, by approximating the integrals in [link] with a summation of rectangular strips, each of width Δt size 12{Δt} {} , one can write

c n = 1 M ∑ m = M M x ( mΔt ) exp ( j2π mn M ) size 12{c rSub { size 8{n} } = { {1} over {M} } Sum cSub { size 8{m=M} } cSup { size 8{M} } {x \( mΔt \) "exp" \( { {j2π ital "mn"} over {M} } \) } } {}
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Source:  OpenStax, An interactive approach to signals and systems laboratory. OpenStax CNX. Sep 06, 2012 Download for free at http://cnx.org/content/col10667/1.14
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