A periodic signal
x(t)x(t) size 12{x \( t \) } {}can be expressed by an exponential Fourier series as follows:
x(t)=∑n=−∞∞cnej2πntTx(t)=∑n=−∞∞cnej2πntT 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=1T∫Tx(t)e−j2πntTdtcn=1T∫Tx(t)e−j2πntTdt 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
∣c−n∣=∣cn∣∣c−n∣=∣cn∣ size 12{ lline c rSub { size 8{ - n} } rline = lline c rSub { size 8{n} } rline } {}
(3)∠c−n=−∠cn∠c−n=−∠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=1∞ancos(2πntT)+bnsin(2πntT)x(t)=a0+∑n=1∞ancos(2πntT)+bnsin(2πntT) 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=1T∫Tx(t)dta0=1T∫Tx(t)dt size 12{a rSub { size 8{0} } = { {1} over {T} } Int cSub { size 8{T} } {x \( t \) ital "dt"} } {}
(6)an=2T∫Tx(t)cos(2πntT)dtan=2T∫Tx(t)cos(2πntT)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=2T∫Tx(t)sin(2πntT)dtbn=2T∫Tx(t)sin(2πntT)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
1T∫T∣x(t)∣2dt=∑n=−∞∞∣cn∣21T∫T∣x(t)∣2dt=∑n=−∞∞∣cn∣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} } } } } {}
(13)More theoretical details of Fourier series are available in signals and systems textbooks (Reference) - (Reference) .
