Transcript Chapter 3_1 - UniMAP Portal

Fourier Representations of Signals &
Linear Time-Invariant Systems
Chapter 3
1
Introduction
• In the previous chapter, linearity property was
exploited to develop the convolution sum and
convolution integral.
• There, the basic idea of convolution is to break up
or decompose a signal into sum of elementary
function.
• Then, we find the response of the system to each
of those elementary function individually and add
the responses to get the overall response.
• In this chapter, we will express a signal as a sum
of real or complex sinusoids instead of sum of
impulses.
2
• The response of LTI system to sinusoids are also
sinusoids of the same frequency but with in
general, different amplitude and phase.
3
Complex Sinusoids & Frequency
Response of LTI System
• The response of an LTI system to a sinusoidal
input leads to a characterization of system
behaviour that is termed the ‘frequency response’
of the system.
4
Fourier Representation for Four
Signal Classes
• There are 4 distinct Fourier representation, each
applicable to a different class of signals.
• These 4 classes are defined by the periodicity
properties of a signal and whether it is continuous
or discrete.
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Relationship Between Time Properties of a Signal and the
Appropriate Fourier Representations
Time property
Periodic
Continuous-time Fourier Series
(CTFS)
Discrete-time
Fourier Series
(DTFS)
Nonperiodic
Fourier
Transform
(CTFT)
Fourier
Transform
(DTFT)
6
The Continuous-Time Fourier
Series
(CTFS)
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Objectives
• To develop methods of expressing periodic signals
as linear combination of sinusoids, real or
complex.
• To explore the general properties of these ways of
expressing signals.
• To apply these methods to find the responses of
systems to arbitrary periodic signals.
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Representing a Signal
• The Fourier series represents a signal as a linear
combination of complex sinusoids
• The responses of LTI system to sinusoids are also
sinusoids of the same frequency but with, in
general, different amplitude and phase.
• Expressing signals in this way leads to frequency
domain concept, thinking of signals as function of
frequency instead of time.
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Periodic Excitation and Response
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Aperiodic Excitation and Response
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Basic Concept & Development of
the Fourier Series
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Linearity and Superposition
If an excitation can be expressed as a sum of complex sinusoids
the response can be expressed as the sum of responses to
complex sinusoids (same frequency but different multiplying
constant).
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ContinuousTime
Fourier
Series
Concept
14
Conceptual Overview
The Fourier series represents a signal as a sum of sinusoids.
Consider original signal x(t), which we would like to present as a
linear combination of sinusoids as illustrated by the dash line.
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Conceptual Overview (cont…)
The best approximation to the dashed-line signal using a constant +
one sinusoid of the same fundamental frequency as the
dashed-line signal is the solid line.
+
=
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Conceptual Overview (cont…)
The best approximation to the dashed-line signal using a constant
+ one sinusoid of the same fundamental frequency as the
dashed-line signal + another sinusoid of twice the fundamental
frequency of the dashed-line signal is the solid line.
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Conceptual Overview (cont…)
The best approximation to the dashed-line signal using a constant
+ three sinusoids is the solid line. In this case, the third sinusoid has
zero amplitude, indicating that sinusoid at that frequency does not
help the approximation.
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Conceptual Overview (cont…)
The best approximation to the dashed-line signal using a constant
+ four sinusoids is the solid line (the forth fundamental frequency is
three times fundamental frequency of the dashed-line signal). This is
a good approximation which gets better with the addition of more
sinusoids at higher integer multiples of the fundamental frequency.
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Trigonometric Form of CTFS
• In the example above, each of the sinusoids used
in the approximation above is of the form
cos(2ПkfFt+θ) multiplied by a constant to set its
amplitude.
• So we can use trigonometry identity:
cos(a+b) = cos(a)cos(b) - sin(a)sin(b)
sin(a+b) = sin(a)cos(b) + cos(a)sin(b)
• Therefore, we can reformulate this functional form
into:
cos(2ПkfFt+θ)= cos(θ) cos(2ПkfFt) sin(θ)sin(2ПkfFt)
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Trigonometric Form of CTFS (cont…)
• The summation of all those sinusoids expressed as
cosines and sines are called the continuous-time
Fourier Series (CTFS).
• In the CTFS, the higher frequency sines and
cosines have frequencies that are integers
multiples of fundamental frequencies. The
multiple is called the harmonic number, k.
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Component of CTFS
• If we have function cos(2ПkfFt) or sin(2ПkfFt)
i) k is harmonic number
ii) kfF is highest frequency.
• If the signal to be represented is x(t), the amplitude
of the kth harmonic sine will be designed Xs[k] and
the amplitude of the kth harmonic cosine will be
designed Xc[k].
• Xs[k] and Xc[k] are called sine and cosine harmonic
function respectively.
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Complex Sinusoids form of CTFS
• Every sine and cosine can be replaced by a linear
combination of complex sinusoids
cos(2ПkfFt) = (ej2ПkfFt+ e-j2ПkfFt)/2
sin(2ПkfFt) = (ej2ПkfFt - e-j2ПkfFt)/j2
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Component of CTFS (cont…)
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CT Fourier Series Definition
The Fourier series representation x F  t  of a signal x(t )
over a time t0  t  t0  TF is
x F t  

j 2 kf F t
X
k
e



k 
where X[k] is the harmonic function, k is the harmonic
number and fF  1 / TF (pp. 240-242). The harmonic function
can be found from the signal as
1
X k  
TF
t0 TF

x  t  e  j 2 kf F t dt
t0
The signal and its harmonic function form a Fourier series
FS
pair indicated by the notation x  t  
 X  k .
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The Trigonometric CTFS
The fact that, for a real-valued function x(t)
X  k   X*  k 
also leads to the definition of an alternate form of the CTFS,
the so-called trigonometric form.

x F  t   Xc 0  X c  k  cos  2 kf F t   X s  k  sin  2 kf F t 
k 1
where
2
Xc  k  
TF
2
Xs k  
TF
t0 TF
 x  t  cos  2 kf t  dt
F
t0
t0 TF
 x  t  sin  2 kf t  dt
F
t0
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The Trigonometric CTFS
Since both the complex and trigonometric forms of the
CTFS represent a signal, there must be relationships
between the harmonic functions. Those relationships are


X c  0  X  0


X
0

0




s

 , k  1, 2,3,
*
X
k

X
k

X
k
    
 c 
 X k  j X k  X* k 
     
 s 


X  0  X c  0




X
k

j
X
k






s
X k   c

 , k  1, 2,3,
2



Xc k   j X s k  
*
X

k

X
k

 
  

2


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Periodicity of the CTFS
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The illustrations show how various kinds of signals are
represented by CTFS over a finite time.
The dash line are periodic continuations of the CTFS
representation
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The dash line are periodic continuations of the CTFS
representation
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Linearity of the CTFS
These relations hold only if the harmonic functions X of all
the component functions x are based on the same
representation time.
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Magnitude and Phase of X[k]
A graph of the magnitude and phase of the harmonic function
as a function of harmonic number is a good way of illustrating it.
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CTFS of Even and Odd Functions
For an even function, the complex CTFS harmonic function
X  k  is purely real and the sine harmonic function X s  k  is
zero.
For an odd function, the complex CTFS harmonic function
X  k  is purely imaginary and the cosine harmonic function
X c  k  is zero.
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Numerical Computation of the CTFS
How could we find the CTFS of this signal which has no
known functional description?
Numerically.
1
X k  
TF

TF
x  t  e  j 2 kf F t dt
Unknown
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Numerical Computation of the CTFS
We don’t know the function x(t), but if we set of NF samples
over one period starting at t=0, the time between the
samples is Ts TF/NF, and we can approximate the integral
by the sum of several integrals, each covering a time of
lenght Ts.
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Numerical Computation of the CTFS
(cont…)
 n1Ts


1
 j 2 kf F nTs
X k  
dt 
  x  nTs  e

TF n0  nTs

Samples from x(t)
N F 1
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Numerical Computation of the CTFS
(cont…)
X  k   1/ N F DFT x  nTs   , k  N F
where
D FT
 x  nTs   
N F 1
 j 2 nk / N F
x
nT
e
  s
n 0
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Convergence of the CTFS
• To examine how the CTFS summation approaches
the signal it represents as the number of terms
used in the sum approaches infinity.
• We do this by examining the partial sum.
x N t  
N

k  N
X k e
j 2  kf 0 t
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Convergence of the CTFS (cont…)
Partial CTFS Sums
x N t  
For continuous signals,
convergence is exact at
every point.
N

k  N
X k e
j 2  kf 0 t
A Continuous Signal
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Convergence of the CTFS (cont…)
Partial CTFS Sums
For discontinuous signals,
convergence is exact at
every point of continuity.
Discontinuous Signal
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Convergence of the CTFS
(cont…)
At points of discontinuity
the Fourier series
representation converges
to the mid-point of the
discontinuity.
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CTFS Properties
Let a signal x(t ) have a fundamental period T0 x and let a
signal y(t ) have a fundamental period T0 y . Let the CTFS
harmonic functions, each using a common period TF as the
representation time, be X[k ] and Y[k ]. Then the following
properties apply.
Linearity
FS
 x  t    y  t  
 X  k    Y  k 
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CTFS Properties
Time Shifting
FS
x  t  t0  
 e j 2 kf0t0 X  k 
FS
x  t  t0  
 e jk0t0 X  k 
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CTFS Properties (cont…)
Frequency Shifting
(Harmonic Number
Shifting)
FS
e j 2 k0 f0t x  t  
 X  k  k0 
FS
e jk00t x  t  
 X  k  k0 
A shift in frequency (harmonic number) corresponds to
multiplication of the time function by a complex exponential.
Time Reversal
FS
x  t  
 X  k 
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CTFS Properties (cont…)
Time Scaling
Let z  t   x  at  , a  0
Case 1. TF  T0 x / a  T0 z for z  t 
Zk   X k 
Case 2. TF  T0 x for z  t 
If a is an integer,
X  k / a  , k / a an integer
Zk   
, otherwise
0
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CTFS Properties (cont…)
Time Scaling (continued)
X  k / a  , k / a an integer
Zk   
, otherwise
0
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CTFS Properties (cont…)
Change of Representation Time
FS
With TF  T0 x , x  t  
 X k 
FS
With TF  mT0 x , x  t  
 Xm k 
X  k / m , k / m an integer
Xm k   
, otherwise
0
(m is any positive integer)
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CTFS Properties (cont…)
Change of Representation Time (cont..)
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CTFS Properties (cont…)
Time Differentiation
d
FS
x
t

 j 2 kf 0 X  k 




dt
d
FS
x
t

 jk0 X  k 
 

dt
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CTFS Properties (cont…)
Time Integration
Case 1. X  0  0
t
X k 

 x    d  
j 2  kf 
FS

0
t
X k 

 x    d  
j  k 
FS

0
Case 2. X  0  0
t
 x    d  is not periodic

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CTFS Properties (cont…)
Multiplication-Convolution Duality
FS
x  t  y  t  
 X k   Y k 
(The harmonic functions, X[ k ] and Y[ k ], must be based
on the same representation period TF .)
FS
x  t  # y  t  
T0 X  k  Y  k 
The symbol # indicates periodic convolution.
Periodic convolution is defined mathematically by
x  t  # y  t    x   y  t    d
T0
x t  y t   x ap t  y t  where x ap t  is any single period of x t 

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CTFS Properties (cont…)
Conjugation
FS
x*  t  
 X*  k 
Parseval’s Theorem
1
T0
 x t 
T0
2
dt 

 X k 
2
k 
The average power of a periodic signal is the sum of the
average powers in its harmonic components.
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