DEPARTMENT OF ELECTRICAL &ELECTRONICS
ENGINEERING
SIGNALS AND SYSTEMS
Assoc. Prof. Dr. Burak Kelleci
Spring 2018
OUTLINE
○ The Response of LTI Systems to Complex Exponentials
○ Fourier Series representation of continuous-time periodic
signals
○ Convergence of the Fourier series
○ Properties of continuous-time Fourier series
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Linearity
Time Shifting
Time Reversal
Time Scaling
Multiplication
Conjugation and Conjugate symmetry
Parseval Theorem
Signals and Systems - Week 4
2
FOURIER SERIES
○ Convolution is based on representing signals as linear combination of
shifted impulses.
○ Fourier series is based on representing signals as linear combination of a
set of basic signals, which are complex exponentials.
○ Because of the superposition property, the response of an LTI system to
any input consisting of a linear combination of basic signals is the same
linear combination of the individual responses to each of the basic
signals.
SYSTEM
Signals and Systems - Week 4
Read the history of
Fourier from the book
(Pages 178-182)
3
THE RESPONSE OF LTI SYSTEMS TO COMPLEX
EXPONENTIALS
○ Input Signal: Complex Exponential Signal
Continuous-Time: est (s: Complex Number)
● Discrete-Time: zn (z: Complex Number)
●
○ The response of LTI system
Continuous - Time : e st  H ( s)e st
Discrete - Time : z n  H ( z ) z n
○ H(s) and H(z) are functions of s and z, respectively.
Signals and Systems - Week 4
4
THE RESPONSE OF LTI SYSTEMS TO COMPLEX
EXPONENTIALS (CONT.)
○ Consider a continuous-time system with impulse response
h(t).
○ For an input x(t)=est, we can determine the output using
convolution integral.

y (t )   h( ) x(t   )d


  h( )e s ( t  ) d



 H ( s )   h( )e  s d

 e st  h( )e  s d

 H ( s )e st
Signals and Systems - Week 4
5
THE RESPONSE OF LTI SYSTEMS TO COMPLEX
EXPONENTIALS (CONT.)
○ Consider a discrete-time system with impulse response h[n].
○ For an input x[n]=zn, we can determine the output using
convolution sum.
y[n] 

 h[k ]x[n  k ]
k  


nk
h
[
k
]
z

k  
 zn
 H ( z) 

k
h
[
k
]
z


k
h
[
k
]
z

k  
k  
 H ( z) z n
Signals and Systems - Week 4
6
THE RESPONSE OF LTI SYSTEMS TO COMPLEX
EXPONENTIALS (CONT.)
○ Let x(t) is a combination of three complex exponentials
x(t )  a1e s1t  a2 e s2t  a3e s3t
The response to each separately is
a1e s1t  a1 H ( s1 )e s1t
a2 e s 2t  a2 H ( s2 )e s 2t
a3e s3t  a3 H ( s3 )e s3t
Because of superposit ion property the response is
y (t )  a1 H ( s1 )e s1t  a2 H ( s2 )e s2t  a3 H ( s3 )e s3t
Signals and Systems - Week 4
7
THE RESPONSE OF LTI SYSTEMS TO COMPLEX
EXPONENTIALS (CONT.)
○ As a general case, the input to a continuous-time LTI system
is represented as a linear combination of complex
exponentials
x(t )   ak e sk t  y (t )   ak H ( sk )e sk t
k
k
○ For Fourier Analysis, we will restrict the signal purely
imaginary, in other words s=jw
○ The same methodology can be used for discrete-time
signals
x[n]   ak zkn  y[n]   ak H ( zk ) zkn
k
k
○ For Fourier Analysis, we will restrict the signal purely
imaginary, in other words z=ejw
Signals and Systems - Week 4
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EXAMPLE
○ Consider an LTI system for which input and output are
related by a time shift of 3.
y (t )  x(t  3)
Apply the complex exponentia l signal x(t )  e j 2t
y (t )  e j 2 (t 3)  e  j 6 e j 2t
 H ( j 2)  e  j 6
The impulse response of the system is h(t )   (t  3)

H ( s )    (  3)e  s d  e 3 s

 H ( j 2)  e  j 6
Signals and Systems - Week 4
9
LINEAR COMBINATION OF HARMONICALLY RELATED
COMPLEX EXPONENTIALS
○ Consider the set of harmonically related complex exponentials with
fundamental frequency w0 and fundamental period T=2p/w0
k (t )  e jkw t  e jk 2p / T t , k  0,1,2, 
0


The fundamental frequency of these signals is integer multiple of w0.
Linear combination of harmonically related complex exponentials have
the following form.
x(t ) 

 ck e
k  




jkw0t


jk  2p / T t
c
e
 k
k  
The term k=0 is a constant
The term k=1 and k=-1 are fundamental components
The term k=2 and k=-2 are second harmonic components
The term k=N and k=-N are Nth harmonic components
Signals and Systems - Week 4
10
EXAMPLE
○ Consider a periodic signal with fundamental frequency 2p,
that is expressed as
x(t ) 
3
jk 2pt
c
e
 k
k  3
1
c1  c 1 
4
1
c 3  c 3 
3
c0  1
1
c2  c2 
2
1 j 2pt
1 j 4pt
1 j 6pt
 j 2pt
 j 4pt
x(t )  1  e
e
 e
e
 e
 e  j 6pt
4
2
3
1
2
x(t )  1  cos(2pt )  cos(4pt )  cos(6pt )
2
3



Signals and Systems - Week 4
 

11
EXAMPLE (CONT.)
Signals and Systems - Week 4
12
EXAMPLE (CONT.)
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13
LINEAR COMBINATION OF HARMONICALLY RELATED
COMPLEX EXPONENTIALS
○ If x(t) is real, x*(t)=x(t) then we obtain
x(t ) 

c e
k  
k
jkw0t


c e
k  
*
k
2p
w0 
T0
 jkw0t
Replace k by - k
x(t ) 

jkw0t
*
*
c
e

c

c
 k
k
k
k  
Let' s Rearrange x(t) and substitute c k* for c  k


x(t )  c0   c k e jkw0t  c  k e  jkw0t

The notation that is used
in these slides are
slightly different than the
notation in the
Oppenheim’s Book. In
the book, instead of ck ak
are used.
In the slides ak is the
coefficient of the cosines
in the Fourier Series.
k 1


x(t )  c0   c k e jkw0t  c k* e  jkw0t

k 1
Signals and Systems - Week 4
14
LINEAR COMBINATION OF HARMONICALLY RELATED
COMPLEX EXPONENTIALS
Since c k and c k* are complex conjugate


x(t )  c0   e c k e jkw0t

k 1
Let' s express c k in polar form
c k  C k e j k


x(t )  c0   e C k e j kw0t  k 

k 1
This is
commonly used
form for Fourier
Series of real
periodic signals

x(t )  c0   C k coskw 0 t   k 
k 1
Signals and Systems - Week 4
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LINEAR COMBINATION OF HARMONICALLY RELATED
COMPLEX EXPONENTIALS

x(t ) 
c e
k  
k
jkw0t


 c0   ck e jkw0t  c k e  jkw0t

k 1
using Euler' s Formula
e  jkw0t  coskw0t   j sin kw0t 

x(t )  c0   ck  c k  coskw0t   j ck  c k sin kw0t 
k 1
setting
a0
 c0 ak  ck  c k bk  j ck  c k 
2
a0 
x(t )    ak coskw0t   bk sin kw0t 
2 k 1
Signals and Systems - Week 4
If ck’s are real, so
that ck=Ck=ak and
both
representations
(polar and
rectangular)
reduce the same
form
16
DETERMINATION OF THE FOURIER SERIES REPRESENTATION
OF A CONTINUOUS-TIME PERIODIC SIGNAL
○ We need a procedure to determine the coefficients ck
x(t ) 

jkw0t
c
e
k
k  
Multiply both sides by e  jnw0t
x(t )e
 jnw0t


jkw0t  jnw0t
c
e
k e
k  
Integrate both sides from 0 to T  2π/ω0
T
 x(t )e
 jnw0t
T
dt  

jkw0t  jnw0t
c
e
 k e dt
0 k  
0
T is the fundamental period of x(t) and we are integratin g over one period.
Interchange the order of integratio n and summation
T
 x(t )e
0
 jnw0t
 T j k  n w0t 
dt   c k   e
dt 
k  
0


Signals and Systems - Week 4
17
DETERMINATION OF THE FOURIER SERIES REPRESENTATION
OF A CONTINUOUS-TIME PERIODIC SIGNAL
T
The evaluation of  e j k  n w0t dt is straightfo rward.
0
Rewrite it using Euler' s formula
T
e
j  k  n w0t
0
T
T
0
0
dt   cosk  n w 0 t dt  j  sin k  n w 0 t dt
For k  n, sin and cos terms are periodic with T/(k - n)
and integratin g them from 0 to T results in zero.
For k  n the integral equals to T
T
T
j  k  n w0t
dt  
0 e
0
kn
T
1
 c n   x(t )e  jnw0t dt
kn
T 0
Changing index n to k
T
1
c k   x(t )e  jkw0t dt
T 0
Signals and Systems - Week 4
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LINEAR COMBINATION OF HARMONICALLY RELATED
COMPLEX EXPONENTIALS
a0 
2p
x(t ) 
  a k coskw 0 t   bk sin kw 0 t  w 0 
2 k 1
T0
T
2
a k   x(t ) coskw 0 t dt
T 0
T
2
bk   x(t ) sin kw 0 t dt
T 0
Complex Fourier Coefficien ts c 0
a0
 c0 a k  c k  c  k bk  j c k  c  k 
2
1
1
c k  a k  jbk  c  k  a k  jbk 
2
2
when x(t) is real
a k  2 Rec k  bk  2 Imc k 
Signals and Systems - Week 4
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SUMMARY OF FOURIER SERIES
Synthesis Equation
x(t ) 

 ck e
k  
jkw0t


jk  2p / T t
c
e
 k
k  
Analysis Equation
T
T
1
1
 jkw0t
c k   x(t )e
dt   x(t )e  jk 2p / T t dt
T 0
T 0
These complex coefficients measure the portion of the
signal x(t) at each harmonic of the fundamental component.
 The coefficient c0 is the dc or constant component of x(t)

Signals and Systems - Week 4
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EXAMPLE
○ Calculate the Fourier Coefficients for the signal x(t)=sin(w0t)
x(t )  sin( w 0 t )
1 jw 0 t 1  jw 0 t
sin( w 0 t ) 
e

e
2j
2j
Compare this to the right side of synthesis equation
1
c1 
2j
ck  0
1
c -1  
2j
k  1 or  1
Signals and Systems - Week 4
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FOURIER SERIES OF EVEN AND ODD SIGNALS
○ If the periodic signal x(t) is even, then bk=0 and its Fouier
Series contains only cosine terms
a0 
x(t ) 
  a k coskw 0 t 
2 k 1
○ If x(t) is odd, then ak=0 and its Fourier series contains only
sine terms

x(t )   bk sin kw 0 t 
k 1
Signals and Systems - Week 4
22
CONVERGENCE OF THE FOURIER SERIES
○ In some cases, the integral in analysis equation may diverge
for some ck values. In other words, the calculated ck value is
infinite.
○ Although all the coefficients are finite, the synthesis equation
may not converge to the original signal x(t).
○ Therefore, we need a criteria to check the convergence of
Fourier Series.
○ A set of conditions are developed by P. L. Dirichlet to
guarantee that x(t) equals its Fourier series Representation.
○ Essentially all the physical signals that we have concerned
meets Dirichlet conditions.
Signals and Systems - Week 4
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DIRICHLET CONDITIONS
○ Condition 1. Over any period, x(t) must be absolutely
integrable
T
 x(t ) dt  
0
This guarantees that each coefficien t c k will be finite
1
ck 
T
T

x(t )e  jkw0t
0
1
dt 
T
T
 x(t ) dt
0
if
T
 x(t ) dt    c
k

0
Signals and Systems - Week 4
24
DIRICHLET CONDITIONS
○ Condition 2. In any interval of time, there are no more than a
finite number of maxima and minima during any single
period of the signal.
An example function t hat meets Condition 1 but not Condition 2
 2p 
x(t )  sin 
, 0  t  1
 t 
This function is periodic with T  1
1
 x(t)  1
0
The function has infinite number of maxima and minima.
○ Condition 3. In any finite interval of time, there are only a
finite number of discontinuities. Furtermore, each of these
discontinuities is finite.
Signals and Systems - Week 4
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DIRICHLET CONDITIONS

Signals that violate Dirichlet
Conditions
a. x(t)=1/t for 0<t≤1, a periodic
signal with period 1 violates
Condition 1
b. This periodic signal
(sin(2p/t)) violates
Condition 2
c. A signal periodic with
period 8 that violates
Condition 3. For 0≤t<8, the
value of x(t) decreases by a
factor of 2 whenever the
distance from t to 8
decreases by a factor of 2.
Signals and Systems - Week 4
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GIBBS PHENOMENON
○ When the Fourier series of a discontinuous signal is
truncated, the synthesized signal exhibit high-frequency
ripples and overshoot near the discontinuities.
○ The famous example is the Fourier Series of Square wave.
Since the Fourier series of square wave has infinite
components, truncating the series creates Gibbs
phenomenon.
Signals and Systems - Week 4
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GIBBS PHENOMENON
N=1
N=3
N=7
N=19
N=79
Signals and Systems - Week 4
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PROPERTIES OF CONTINUOUS-TIME FOURIER SERIES
○ Linearity
FS
x(t )  c x ,k
FS
y (t )  c y ,k
FS
z (t )  Ax(t )  By (t )  c z ,k  Ac x ,k  Bc y ,k
○ Time Shifting
FS
x(t )  c k
FS
x(t  t 0 )  e  jkw0t0 c k  e  jk 2p / T t0 c k
When a periodic signal is shifted in time, the magnitudes of
its Fourier Series Coefficients are unaltered
Signals and Systems - Week 4
29
PROPERTIES OF CONTINUOUS-TIME FOURIER SERIES
○ Time Reversal
FS
x(t )  c k
If x(t) is even x(-t)=x(t)  c-k=ck
If x(t) is odd x(-t)=-x(t)  c-k=-ck
FS
x(t )  c  k
○ Time Scaling
x(t ) 

jk w0 t
c
e
k
k  
While the Fourier Coefficients have not changed, the Fourier
series representation has changed because of the change in
the fundamental frequency
Signals and Systems - Week 4
30
PROPERTIES OF CONTINUOUS-TIME FOURIER SERIES
○ Multiplication
FS
x(t )  c x ,k
FS
y (t )  c y ,k
FS
x(t ) y (t )  c k 

c
l  
x ,l
c y , k l
○ Conjugation and Conjugate symmetry
FS
x(t )  c k
FS
x * (t )  c * k
If x(t) is real [x(t)=x*(t)], its Fourier Series
coefficients will be conjugate symmetric [c-k=c*k]
If x(t) is real and even, its Fourier coefficients are
also real and symmetric
If x(t) is real and odd, its Fourier coefficients are
purely imaginary and odd.
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31
PROPERTIES OF CONTINUOUS-TIME FOURIER SERIES
○ Differentiation in the Time Domain
FS
x(t )  c k
dx (t ) FS
 jkw 0 c k
dt
○ Integration in the Time Domain
FS
x(t )  c k
t
1
x(t )dt  jkw 0 ck
FS
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32
PROPERTIES OF CONTINUOUS-TIME FOURIER SERIES
○ Parseval Theorem
1
T

T
 x(t )
0
2
dt 

c
k  
2
k
The Parseval Theorem states that the total average power in
a periodic signal equals the sum of the average powers in all
of its harmonic components
Signals and Systems - Week 4
33