Lecture 4 - EE 104: Introduction to Communications - Winter 2003
Fourier Series and its Properties.
Lecture Outline
Basis Functions for Periodic Signals
Fourier Series Transform Pair
LTI Filtering of Exponential Functions
Fourier Series Examples
Signal Construction: Demo
Fourier Series with Sinusoidal Bases
Properties of Fourier Series
1.
Basis Functions for Periodic Signals
2.
Basic idea of Fourier series is to project periodic signals onto a set of basis functions
The basis functions are orthogonal and span the space of periodic functions
Any periodic signal can be written as a weighted sum of these basis functions.
Fourier series uses exponential functions fej 2nt=T0 g as basis functions to represent
periodic signals with fundamental period T0 .
Fourier Series Transform Pair:
For a periodic signal xp (t) with a fundamental period T0 = 1=f0 , the Fourier series
representation, if it exists, is
xp (t) =
3.
n=
1
cn ej 2nf0t :
The Fourier series coeÆcients fcn g are given by
cn =
1
X
Z
1
x (t)e j 2nf0t dt;
T0 T0 p
where the integral is taken over any period of the signal.
The Fourier series coeÆcients are the frequency components of the periodic signal.
Linear Time-Invariant Filters
P
P
A lter is linear if it obeys the properties of superposition: If input xn (t) leads to
output yn (t) then input n cn xn (t) leads to output n cn yn (t)
A lter is time-invariant if for any lter input x(t) with lter output y (t), the input
x(t ) leads to output y (t ).
4.
LTI Filtering of Exponential Functions
5.
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Fourier series coeÆcients are cn = Af0 T sinc(nf0 t), where sinc(x) = sin(x)=(x):
CoeÆcients are symmetric (cn = c n ) and indicate innite frequency content of the
signal.
Constructing signal from nite number of coeÆcients can only approximate sharp
corners.
Real Signals
7.
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Periodic Square Wave Example:
6.
The exponential basis functions in Fourier series are eigenfunctions of LTI lters.
Filter eigenfunctions have the property that if the they are input into the lter, the
lter output is a constant times the original eigenfunction.
For exponential eigenfunctions, if x(t) = ej 2ft is input into the lter, the lter output
is y (t) = H (f )ej 2ft. The complex constant H (f ) is called the frequency response of
the lter at frequency f .
The Fourier series representation and the eigenfunction property of exponentials allows
us to nd the lter output for any periodic input.
j 2nt=T0 is an LTI lter input, the lter output is y (t) = 1
j 2nt=T0
If xp (t) = 1
n= 1 cn e
n= 1 cn H (n=T0 )e
CoeÆcients always symmetric (cn = c n ).
Simplied representation: xp (t) = c0 + 2 1
n=1 jcn j cos(2nf0 t + 6 cn )
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Sinusoidal Basis Functions:
Sinusoids fcos(2ff0t); sin(2ff0t); n = 1; 2; : : :g are basis functions for periodic signals with period T0 .
Sinusoids are also eigenfunctions of LTI lters.
Sinusoidal Fourier Series Representation:
xp (t) = a0 + 2
where
8.
an =
Z
1
X
[a
n=1
n cos 2nf0 t + bn sin 2nf0 t];
1
x (t) cos 2nf0 t;
T0 T0 p
bn =
Exponential to sinusoidal conversion: an = :5(cn + c n ), bn = :5j (c n cn ).
Sinusoidal to exponential conversion: c0 = a0 , cn = an jbn ; n > 0, and cn = an +
jbn ; n < 0
Properties of Fourier Series:
assume xp (t) , fcn g and vp (t) , fdn g.
Z
1
x (t) sin 2nf0 t:
T0 T0 p
Linearity: Axp (t) + Bvp (t) , fAcn + Bdn g.
Multiplication: xp (t)vp (t) , f 1
k= 1 ck dn k g.
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Time Shifting: xp (t ) , fe j 2nf0 cn g:
Time Reversal: xp ( t) , fc n g:
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j 2nf0 (at)
Time Scaling: xp (at) = 1
n= 1 cn e
Conjugation: xp (t) , fc n g.
Parseval's Relation:
1
1
j
xp (t)j2 dt =
jcnj2 :
T0 T0
n= 1
Z
X
Main Points:
Fourier series represents periodic signals as a weighted sum of exponential functions.
Square wave has innite frequency content with FS following a sinc function.
Exponentials are eigenfunctions of LTI lters, so using FS can obtain lter output to any
periodic signal.
Real signals have symmetric FS coeÆcients.
Sinusoids are an alternate FS representation.
Fourier series properties used in analysis of periodic signals.