1. Home
  2. Signals and systems
  3. Signals and systems
  4. Lecture 7:continuous time fourier

0.6 Lecture 7:continuous time fourier series for periodic signals

<< Chapter < Page
  Signals and systems     Page 1 / 3
Chapter >> Page >
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.

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 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 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 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 = { 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 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.

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.

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 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 size 12{S \[ n \] = { {1} over {T} } } {}

The time function and spectrum are shown below.

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 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 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 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 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} } } {}

We can use the summation formula for a finite geometric series (Lecture 10) to sum this series,

s N ( t ) = 1 T ( e j2πt / T ) − N − ( e j2πt / T ) N + 1 1 − e j2πt / T size 12{s rSub { size 8{N} } \( t \) = { {1} over {T} } { { \( e rSup { size 8{j2πt/T} } \) rSup { size 8{ - N} } - \( e rSup { size 8{j2πt/T} } \) rSup { size 8{N+1} } } over {1 - e rSup { size 8{j2πt/T} } } } } {}

s N ( t ) = 1 T e j ( 2N + 1 ) πt / T − e − j ( 2N + 1 ) πt / T e jπt / T − e − jπt / T size 12{s rSub { size 8{N} } \( t \) = { {1} over {T} } { {e rSup { size 8{j \( 2N+1 \) πt/T} } - e rSup { size 8{ - j \( 2N+1 \) πt/T} } } over {e rSup { size 8{jπt/T} } - e rSup { size 8{ - jπt/T} } } } } {}

s N ( t ) = 1 T sin ( 2N + 1 ) πt / T sin πt / T size 12{s rSub { size 8{N} } \( t \) = { {1} over {T} } { {"sin" \( 2N+1 \) πt/T} over {"sin"πt/T} } } {}

s N ( t ) = 1 T sin ( 2N + 1 ) πt / T sin πt / T size 12{s rSub { size 8{N} } \( t \) = { {1} over {T} } { {"sin" \( 2N+1 \) πt/T} over {"sin"πt/T} } } {}

Note that this function is periodic with period T, and

s N ( t ) = 2N + 1 T size 12{s rSub { size 8{N} } \( t \) = { {2N+1} over {T} } } {}

The first zero of sN(t) is at

t = T 2N + 1 size 12{t= { {T} over {2N+1} } } {}

Thus, as N → ∞, each lobe gets larger and narrower. To determine if each lobe acts as an impulse, we need to find its area.
<< Chapter < Page Page > Chapter >>

Read also:

Get Jobilize Job Search Mobile App in your pocket Now!

Get it on Google Play Download on the App Store Now




Source:  OpenStax, Signals and systems. OpenStax CNX. Jul 29, 2009 Download for free at http://cnx.org/content/col10803/1.1
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'Signals and systems' conversation and receive update notifications?

Ask