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Lecture 7:Continuous Time Fourier Series For Periodic Signals

Module by: Thanh Dinh Vu, Truc Pham-Dinh, Anh Tuan Hoang, Tam Huynh-Ngoc

Summary: Representation of continuous time, periodic signals in the frequency domain. Periodic signals occur frequently — motion of planets and their satellites, vibration of oscillators, electric power distribution, beating of the heart, vibration of vocal chords, etc.

Lecture #7:
CONTINUOUS TIME FOURIER SERIES FOR PERIODIC SIGNALS
Motivation:
  • Representation of continuous time, periodic signals in the frequency domain
  • Periodic signals occur frequently — motion of planets and their satellites, vibration of oscillators, electric power distribution, beating of the heart, vibration of vocal chords, etc.
Outline:
  • Fourier series of periodic functions
  • Examples of Fourier series — periodic impulse train
  • Fourier transforms of periodic functions — relation to Fourier series
  • Conclusions
I. FOURIER SERIES OF A PERIODIC FUNCTION
1/ Periodic time function
x(t) is a periodic time function with period T.
Figure 1
Such a periodic function can be expanded in an infinite series of exponential time functions called the Fourier series,
x ( t ) = ∑ n = − ∞ ∞ X [ n ] e j2π nt / T x ( t ) = ∑ n = − ∞ ∞ X [ n ] e j2π nt / T size 12{x \( t \) = Sum cSub { size 8{n= - infinity } } cSup { size 8{ infinity } } {X \[ n \] e rSup { size 8{j2π ital "nt"/T} } } } {}
2/ Fourier series coefficients
The coefficients of the Fourier series can be found as follows.
1 T ∫ − T / 2 T / 2 x ( t ) e − j2π nt / T dt = 1 T ∫ − T / 2 T / 2 ( ∑ k = − ∞ ∞ X [ k ] e j2π kt / T ) e − j2π nt / T dt 1 T ∫ − T / 2 T / 2 x ( t ) e − j2π nt / T dt = 1 T ∫ − T / 2 T / 2 ( ∑ k = − ∞ ∞ X [ k ] e j2π kt / T ) e − j2π nt / T dt size 12{ { {1} over {T} } Int rSub { size 8{ - T/2} } rSup { size 8{T/2} } {x \( t \) e rSup { size 8{ - j2π ital "nt"/T} } } ital "dt"= { {1} over {T} } Int rSub { size 8{ - T/2} } rSup { size 8{T/2} } {} \( Sum cSub { size 8{k= - infinity } } cSup { size 8{ infinity } } {X \[ k \] e rSup { size 8{j2π ital "kt"/T} } } \) e rSup { size 8{ - j2π ital "nt"/T} } ital "dt"} {}
1 T ∫ − T / 2 T / 2 x ( t ) e − j2π nt / T dt = ∑ k = − ∞ ∞ X [ k ] 1 T ∫ − T / 2 T / 2 e j2π ( k − n ) t / T dt 1 T ∫ − T / 2 T / 2 x ( t ) e − j2π nt / T dt = ∑ k = − ∞ ∞ X [ k ] 1 T ∫ − T / 2 T / 2 e j2π ( k − n ) t / T dt size 12{ { {1} over {T} } Int rSub { size 8{ - T/2} } rSup { size 8{T/2} } {x \( t \) e rSup { size 8{ - j2π ital "nt"/T} } } ital "dt"= Sum cSub { size 8{k= - infinity } } cSup { size 8{ infinity } } {X \[ k \] } { {1} over {T} } Int rSub { size 8{ - T/2} } rSup { size 8{T/2} } {e rSup { size 8{j2π \( k - n \) t/T} } } ital "dt"} {}
The integral can be evaluated as follows.
1 ifk = n 0 ifk ≠ n 1 T ∫ − T / 2 T / 2 e j2π ( k − n ) t / T dt = { 1 ifk = n 0 ifk ≠ n 1 T ∫ − T / 2 T / 2 e j2π ( k − n ) t / T dt = { size 12{ { {1} over {T} } Int rSub { size 8{ - T/2} } rSup { size 8{T/2} } {e rSup { size 8{j2π \( k - n \) t/T} } } ital "dt"=alignl { stack { left lbrace 1 ital "ifk"=n {} # right none left lbrace 0 ital "ifk" <> n {} # right no } } lbrace } {}
The set of exponential time functions are said to be an orthonormal basis.
The coefficients are
X [ n ) = 1 T ∫ − T / 2 T / 2 x ( t ) e − j2π nt / T dt X [ n ) = 1 T ∫ − T / 2 T / 2 x ( t ) e − j2π nt / T dt size 12{X \[ n \) = { {1} over {T} } Int rSub { size 8{ - T/2} } rSup { size 8{T/2} } {x \( t \) } e rSup { size 8{ - j2π ital "nt"/T} } ital "dt"} {}
3/ Definition of line spectra, harmonics
The fundamental frequency fo = 1/T . The Fourier series coefficients plotted as a function of n or nfo is called a Fourier spectrum.
Figure 2
II. EXAMPLES OF FOURIER SERIES OF PERIODIC TIME FUNCTIONS
1/ Periodic impulse train
The periodic impulse train is an important periodic time function and we derive its Fourier series coefficients.
Figure 3
s ( t ) = ∑ n = − ∞ ∞ δ ( t − nT ) s ( t ) = ∑ n = − ∞ ∞ δ ( t − nT ) size 12{s \( t \) = Sum cSub { size 8{n= - infinity } } cSup { size 8{ infinity } } {δ \( t - ital "nT" \) } } {}
The Fourier series coefficients are found as follows
S [ n ] = 1 T ∫ − T / 2 T / 2 s ( t ) e − j2π nt / T dt , 1 T ∫ − T / 2 T / 2 ∑ n = − ∞ ∞ δ ( t − nT ) e − j2π nt / T dt , 1 T ∫ − T / 2 T / 2 δ ( t ) e − j2π nt / T dt = 1 T S [ n ] = 1 T ∫ − T / 2 T / 2 s ( t ) e − j2π nt / T dt , 1 T ∫ − T / 2 T / 2 ∑ n = − ∞ ∞ δ ( t − nT ) e − j2π nt / T dt , 1 T ∫ − T / 2 T / 2 δ ( t ) e − j2π nt / T dt = 1 T alignl { stack { size 12{S \[ n \] = { {1} over {T} } Int rSub { size 8{ - T/2} } rSup { size 8{T/2} } {s \( t \) e rSup { size 8{ - j2π ital "nt"/T} } } ital "dt",} {} # = { {1} over {T} } Int rSub { size 8{ - T/2} } rSup { size 8{T/2} } { Sum cSub { size 8{n= - infinity } } cSup { size 8{ infinity } } {δ} } \( t - ital "nT" \) e rSup { size 8{ - j2π ital "nt"/T} } ital "dt", {} # = { {1} over {T} } Int rSub { size 8{ - T/2} } rSup { size 8{T/2} } {δ \( t \) e rSup { size 8{ - j2π ital "nt"/T} } } ital "dt"= { {1} over {T} } {} } } {}
The Fourier series coefficients are
S [ n ] = 1 T S [ n ] = 1 T size 12{S \[ n \] = { {1} over {T} } } {}
The time function and spectrum are shown below.
Figure 4
To summarize, the periodic impulse train can be represented by its Fourier series,
s ( t ) = ∑ n = − ∞ ∞ δ ( t − nT ) = 1 T ∑ n = − ∞ ∞ e j2π nt / T s ( t ) = ∑ n = − ∞ ∞ δ ( t − nT ) = 1 T ∑ n = − ∞ ∞ e j2π nt / T size 12{s \( t \) = Sum cSub { size 8{n= - infinity } } cSup { size 8{ infinity } } {δ \( t - ital "nT" \) = { {1} over {T} } } Sum cSub { size 8{n= - infinity } } cSup { size 8{ infinity } } {e rSup { size 8{j2π ital "nt"/T} } } } {}
The Fourier series of the periodic impulse train is
s ( t ) = ∑ n = − ∞ ∞ δ ( t − nT ) = 1 T ∑ n = − ∞ ∞ e j2π nt / T s ( t ) = ∑ n = − ∞ ∞ δ ( t − nT ) = 1 T ∑ n = − ∞ ∞ e j2π nt / T size 12{s \( t \) = Sum cSub { size 8{n= - infinity } } cSup { size 8{ infinity } } {δ \( t - ital "nT" \) = { {1} over {T} } } Sum cSub { size 8{n= - infinity } } cSup { size 8{ infinity } } {e rSup { size 8{j2π ital "nt"/T} } } } {}
It is not obvious that the two expressions are equal. To investigate this, we define the partial sum of the Fourier series, sN(t),
s N ( t ) = 1 T ∑ n = − N N e j2π nt / T s N ( t ) = 1 T ∑ n = − N N e j2π nt / T size 12{s rSub { size 8{N} } \( t \) = { {1} over {T} } Sum cSub { size 8{n= - N} } cSup { size 8{N} } {e rSup { size 8{j2π ital "nt"/T} } } } {}
and investigate its behavior as N →∞.
The partial sum of the Fourier series is
S N ( t ) = 1 T ∑ n = − N N e j2π nt / T = 1 T ∑ n = − N N ( e j2πt / T ) n S N ( t ) = 1 T ∑ n = − N N e j2π nt / T = 1 T ∑ n = − N N ( e j2πt / T ) n size 12{S rSub { size 8{N} } \( t \) = { {1} over {T} } Sum cSub { size 8{n= - N} } cSup { size 8{N} } {e rSup { size 8{j2π ital "nt"/T} } } = { {1} over {T} } Sum cSub { size 8{n= - N} } cSup { size 8{N} } { \( e rSup { size 8{j2πt/T} } } \) rSup { size 8{n} } } {}
Figure 5
We can use the summation formula for a finite geometric series (Lecture 10) to sum this series,
s N ( t ) = 1 T (