Signals and Systems
Fall 2003
Lecture #5
18 September 2003
1,Complex Exponentials as Eigenfunctions of LTI Systems
2,Fourier Series representation of CT periodic signals
3,How do we calculate the Fourier coefficients?
4,Convergence and Gibbs’ Phenomenon
Signals & Systems,2nd ed,Upper Saddle River,N.J.,Prentice Hall,1997,p,179.
Portrait of Jean Baptiste Joseph Fourier
Image removed due to copyright considerations.
Desirable Characteristics of a Set of,Basic” Signals
a,We can represent large and useful classes of signals
using these building blocks
b,The response of LTI systems to these basic signals is
particularly simple,useful,and insightful
Previous focus,Unit samples and impulses
Focus now,Eigenfunctions of all LTI systems
The eigenfunctions φ
k
(t) and their properties
(Focus on CT systems now,but results apply to DT systems as well.)
eigenvalue eigenfunction
Eigenfunction in → same function out with a,gain”
From the superposition property of LTI systems:
Now the task of finding response of LTI systems is to determine λ
k
,
Complex Exponentials as the Eigenfunctions of any LTI Systems
eigenvalue eigenfunction
eigenvalue eigenfunction
DT:
What kinds of signals can we represent as
“sums” of complex exponentials?
For Now,Focus on restricted sets of complex exponentials
CT & DT Fourier Series and Transforms
CT:
DT:
Magnitude 1
Periodic Signals
Fourier Series Representation of CT Periodic Signals
ω
o
=
2π
T
-smallest such T is the fundamental period
-i te fundamental frequency
- periodic with period T
-{a
k
} are the Fourier (series) coefficients
- k = 0 DC
- k = ±1 first harmonic
- k = ±2 second harmonic
Question #1,How do we find the Fourier coefficients?
First,for simple periodic signals consisting of a few sinusoidal terms
0 – no dc component
0
0
Euler's relation
(memorize!)
For real periodic signals,there are two other commonly used
forms for CT Fourier series,
Because of the eigenfunction property of e
jωt
,we will usually
use the complex exponential form in 6.003,
- A consequence of this is that we need to include terms for
both positive and negative frequencies:
Now,the complete answer to Question #1
Ex,Periodic Square Wave
DC component
is just the
average
Convergence of CT Fourier Series
How can the Fourier series for the square wave possibly make
sense?
The key is,What do we mean by
One useful notion for engineers,there is no energy in the
difference
(just need x(t) to have finite energy per period)
Under a different,but reasonable set of conditions
(the Dirichlet conditions)
Condition 1,x(t) is absolutely integrable over one period,i,e.
Condition 3,In a finite time interval,x(t) has only a finite
number of discontinuities.
Ex,An example that violates
Condition 3.
And
Condition 2,In a finite time interval,
x(t) has a finite number
of maxima and minima.
Ex,An example that violates
Condition 2.
And
Dirichlet conditions are met for the signals we will
encounter in the real world,Then
- The Fourier series = x(t) at points where x(t) is continuous
- The Fourier series =,midpoint” at points of discontinuity
-As N → ∞,x
N
(t) exhibits Gibbs’ phenomenon at
points of discontinuity
Demo,Fourier Series for CT square wave (Gibbs phenomenon).
Still,convergence has some interesting characteristics:
Fall 2003
Lecture #5
18 September 2003
1,Complex Exponentials as Eigenfunctions of LTI Systems
2,Fourier Series representation of CT periodic signals
3,How do we calculate the Fourier coefficients?
4,Convergence and Gibbs’ Phenomenon
Signals & Systems,2nd ed,Upper Saddle River,N.J.,Prentice Hall,1997,p,179.
Portrait of Jean Baptiste Joseph Fourier
Image removed due to copyright considerations.
Desirable Characteristics of a Set of,Basic” Signals
a,We can represent large and useful classes of signals
using these building blocks
b,The response of LTI systems to these basic signals is
particularly simple,useful,and insightful
Previous focus,Unit samples and impulses
Focus now,Eigenfunctions of all LTI systems
The eigenfunctions φ
k
(t) and their properties
(Focus on CT systems now,but results apply to DT systems as well.)
eigenvalue eigenfunction
Eigenfunction in → same function out with a,gain”
From the superposition property of LTI systems:
Now the task of finding response of LTI systems is to determine λ
k
,
Complex Exponentials as the Eigenfunctions of any LTI Systems
eigenvalue eigenfunction
eigenvalue eigenfunction
DT:
What kinds of signals can we represent as
“sums” of complex exponentials?
For Now,Focus on restricted sets of complex exponentials
CT & DT Fourier Series and Transforms
CT:
DT:
Magnitude 1
Periodic Signals
Fourier Series Representation of CT Periodic Signals
ω
o
=
2π
T
-smallest such T is the fundamental period
-i te fundamental frequency
- periodic with period T
-{a
k
} are the Fourier (series) coefficients
- k = 0 DC
- k = ±1 first harmonic
- k = ±2 second harmonic
Question #1,How do we find the Fourier coefficients?
First,for simple periodic signals consisting of a few sinusoidal terms
0 – no dc component
0
0
Euler's relation
(memorize!)
For real periodic signals,there are two other commonly used
forms for CT Fourier series,
Because of the eigenfunction property of e
jωt
,we will usually
use the complex exponential form in 6.003,
- A consequence of this is that we need to include terms for
both positive and negative frequencies:
Now,the complete answer to Question #1
Ex,Periodic Square Wave
DC component
is just the
average
Convergence of CT Fourier Series
How can the Fourier series for the square wave possibly make
sense?
The key is,What do we mean by
One useful notion for engineers,there is no energy in the
difference
(just need x(t) to have finite energy per period)
Under a different,but reasonable set of conditions
(the Dirichlet conditions)
Condition 1,x(t) is absolutely integrable over one period,i,e.
Condition 3,In a finite time interval,x(t) has only a finite
number of discontinuities.
Ex,An example that violates
Condition 3.
And
Condition 2,In a finite time interval,
x(t) has a finite number
of maxima and minima.
Ex,An example that violates
Condition 2.
And
Dirichlet conditions are met for the signals we will
encounter in the real world,Then
- The Fourier series = x(t) at points where x(t) is continuous
- The Fourier series =,midpoint” at points of discontinuity
-As N → ∞,x
N
(t) exhibits Gibbs’ phenomenon at
points of discontinuity
Demo,Fourier Series for CT square wave (Gibbs phenomenon).
Still,convergence has some interesting characteristics:
