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# Fourier Series

## Background

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

x(t)=n=cnejntTx(t)=n=cnejntT 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} } } } } } {}
(1)

where T indicates the period of the signal and cncn 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

cn=1TTx(t)ejntTdtcn=1TTx(t)ejntTdt 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"} } {}
(2)

which possesses the following symmetry properties

cn=cncn=cn size 12{ lline c rSub { size 8{ - n} } rline = lline c rSub { size 8{n} } rline } {}
(3)
cn=cncn=cn size 12{∠c rSub { size 8{ - n} } = - ∠c rSub { size 8{n} } } {}
(4)

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

x(t)=a0+n=1ancos(ntT)+bnsin(ntT)x(t)=a0+n=1ancos(ntT)+bnsin(ntT) 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" $${ {2π ital "nt"} over {T} }$$ } } {}
(5)

where

a0=1TTx(t)dta0=1TTx(t)dt size 12{a rSub { size 8{0} } = { {1} over {T} } Int cSub { size 8{T} } {x $$t$$ ital "dt"} } {}
(6)
an=2TTx(t)cos(ntT)dtan=2TTx(t)cos(ntT)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"} } {}
(7)
bn=2TTx(t)sin(ntT)dtbn=2TTx(t)sin(ntT)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"} } {}
(8)

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

a0=c0a0=c0 size 12{a rSub { size 8{0} } =c rSub { size 8{0} } } {}
(9)
a n = 2 Re { c n } a n = 2 Re { c n } size 12{a rSub { size 8{n} } =2"Re" lbrace c rSub { size 8{n} } rbrace } {}
(10)
b n = 2 Im { c n } b n = 2 Im { c n } size 12{b rSub { size 8{n} } = - 2"Im" lbrace c rSub { size 8{n} } rbrace } {}
(11)
c n = 1 2 ( a n jb n ) c n = 1 2 ( a n jb n ) size 12{c rSub { size 8{n} } = { {1} over {2} } $$a rSub { size 8{n} } - ital "jb" rSub { size 8{n} }$$ } {}
(12)

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

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

1TTx(t)2dt=n=cn21TTx(t)2dt=n=cn2 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} } } } } {}
(13)

More theoretical details of Fourier series are available in signals and systems textbooks (Reference) - (Reference) .

## Fourier Series Numerical Computation

Fourier series coefficients are often computed numerically – in particular, when an analytic expression for x(t)x(t) size 12{x $$t$$ } {} is not available or the integration in Equation 6 - Equation 8 is difficult to perform. By approximating the integrals in Equation 6 - Equation 8 with a summation of rectangular strips, each of width ΔtΔt size 12{Δt} {}, one can write

a0=1Mm=1Mx(mΔt)a0=1Mm=1Mx(mΔt) size 12{a rSub { size 8{0} } = { {1} over {M} } Sum cSub { size 8{m=1} } cSup { size 8{M} } {x $$mΔt$$ } } {}
(14)
an=2Mm=1Mx(mΔt)cos(mnM)an=2Mm=1Mx(mΔt)cos(mnM) size 12{a rSub { size 8{n} } = { {2} over {M} } Sum cSub { size 8{m=1} } cSup { size 8{M} } {x $$mΔt$$ "cos" $${ {2π ital "mn"} over {M} }$$ } } {}
(15)
bn=2Mm=1Mx(mΔt)sin(mnM)bn=2Mm=1Mx(mΔt)sin(mnM) size 12{b rSub { size 8{n} } = { {2} over {M} } Sum cSub { size 8{m=1} } cSup { size 8{M} } {x $$mΔt$$ "sin" $${ {2π ital "mn"} over {M} }$$ } } {}
(16)

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

Similarly, by approximating the integrals in Equation 2 with a summation of rectangular strips, each of width ΔtΔt size 12{Δt} {}, one can write

cn=1Mm=MMx(mΔt)exp(j2πmnM)cn=1Mm=MMx(mΔt)exp(j2πmnM) size 12{c rSub { size 8{n} } = { {1} over {M} } Sum cSub { size 8{m=M} } cSup { size 8{M} } {x $$mΔt$$ "exp" $${ {j2π ital "mn"} over {M} }$$ } } {}
(17)

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