Signals and Systems Prof. H. Sameti Chapter 4: The Continuous Time Fourier Transform • Derivation of the CT Fourier Transform pair • Examples of Fourier Transforms Topic three • Fourier Transforms of Periodic Signals • Properties of the CT Fourier Transform • The Convolution Property of the CTFT • Frequency Response and LTI Systems Revisited • Multiplication Property and Parseval’s Relation • The DT Fourier Transform
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Signals and Systems Prof. H. Sameti Chapter 4: The Continuous Time Fourier Transform Derivation of the CT Fourier Transform pair Examples of Fourier Transforms.
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Signals and SystemsProf. H. Sameti
Chapter 4: The Continuous Time Fourier Transform• Derivation of the CT Fourier Transform pair• Examples of Fourier Transforms Topic three• Fourier Transforms of Periodic Signals• Properties of the CT Fourier Transform• The Convolution Property of the CTFT • Frequency Response and LTI Systems Revisited • Multiplication Property and Parseval’s Relation• The DT Fourier Transform
Book Chapter4: Section1
2
Fourier’s Derivation of the CT Fourier Transform
x(t) - an aperiodic signalview it as the limit of a periodic signal as T → ∞
For a periodic signal, the harmonic components are spaced ω0 = 2π/T apart ...
As T → ∞, ω0 → 0, and harmonic components are spaced closer and closer in frequency
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Fourier series Fourier integral
Book Chapter4: Section1
3
Motivating Example: Square wave
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increaseskept fixed
Discrete frequency points become denser in ω as T increases
0
0 1
0
1
2sin( )
2sin( )
k
k
k
k Ta
k T
TTa
Book Chapter4: Section1
4
So, on with the derivation ...
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For simplicity, assumex(t) has a finite duration.
( ),2 2( )
,2
T Tx t t
x tT
periodic t
As , ( ) ( ) for all T x t x t t
Book Chapter4: Section1
5
Derivation (continued)
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0
0 0
0
0
2 2
2 2
0
2( ) ( )
1 1( ) ( )
( ) ( ) in this interval
1( ) (1)
If we define
( ) ( )
then Eq.(1)
( )
jk tk
k
T T
jk t jk tk
T T
jk t
j t
k
x t a eT
a x t e dt x t e dtT T
x t x t
x t e dtT
X j x t e dt
X jka
T
Book Chapter4: Section1
6
Derivation (continued)
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0
0
0
0 0
0
Thus, for 2 2
1( ) ( ) ( )
1( )
2
As , , we get the CT Fourier Transform pair
1( ) ( ) Synthesis equation
2
( ) ( ) A
k
jk t
k
a
jk t
k
j t
j t
T Tt
x t x t X jk eT
X jk e
T d
x t X j e d
X j x t e dt
nalysis equation
Book Chapter4: Section1
7
For what kinds of signals can we do this?
(1) It works also even if x(t) is infinite duration, but satisfies:a) Finite energy
In this case, there is zero energy in the error
b) Dirichlet conditions
c) By allowing impulses in x(t)or in X(jω), we can represent even more Signals
E.g. It allows us to consider FT for periodic signals
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2( )x t dt
21( ) ( ) ( ) Then ( ) 0
2j te t x t X j e d e t dt
1 (i) ( ) ( ) at points of continuity
21
(ii) ( ) midpoint at discontinuity2
(iii) Gibb's phenomenon
j t
j t
X j e d x t
X j e d
Book Chapter4: Section1
8
Example #1
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0
0
0
( ) ( ) ( )
( ) ( ) 1
1( ) Synthesis equation for ( )
2( ) ( ) ( )
( ) ( )
j t
j t
j t
j t
a x t t
X j t e dt
t e d t
b x t t t
X j t t e dt
e
Book Chapter4: Section1
9
Example #2: Exponential function
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( )0
( )
( ) ( ), 0
( ) ( )
1 1( )
0
a j t
at
j t at j t
e
a j t
x t e u t a
X j x t e dt e e dt
ea j a j
Even symmetry Odd symmetry
Book Chapter4: Section1
10
Example #3: A square pulse in the time-domain
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1
1
12sin( )
T j t
T
TX j e dt
Note the inverse relation between the two widths Uncertainty principle⇒
Book Chapter4: Section1
11
Useful facts about CTFT’s
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1
(0) ( )
1(0) ( )
2
Example above: ( ) 2 (0)
1Ex. above: (0) ( )
21
(Area of the triangle)2
X x t dt
x X j d
x t dt T X
x X j d
Book Chapter4: Section1
12
Example #4:
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2
A Gaussian, important in probability, optics, etc.( ) atx t e2
2 2 2
22
2
[ ( ) ] ( )2 2
( )2 4
4
( )
[ ].
at j t
j ja t j t a
a a a
ja t
a a
a
a
X j e e dt
e dt
e dt e
ea
Also a Gaussian!
(Pulse width in t)•(Pulse width in ω) ∆⇒ t•∆ω ~ (1/a1/2)•(a1/2) = 1
Uncertainty Principle! Cannot make both ∆t and ∆ω arbitrarily small.
Book Chapter4: Section1
13
CT Fourier Transforms of Periodic Signals
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0
0
0
0
0
0 0
periodic in with freq
Suppose
( ) ( )
1 1( ) ( )
2 2That i
All the energy is concentrated in one fr
ue
s
2 ( )
More generall
equency
ncy
y
j tj t
j t
X j
x t t e d e
e
x
00( ) ( ) 2 ( )jk t
k kk k
t a e X j a k
Book Chapter4: Section1
14
Example #5:
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0 00
0 0
1 1( ) cos
2 2
( ) ( ) ( )
j t j tx t t e e
X j
“Line Spectrum”
Book Chapter4: Section1
15
Example #6:
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Sampling function( ) ( )n
x t t nT
0
0
2
2
2
1 1( ) ( )
2 2( ) ( )
k
T jk tk T
na k
x t a x t e dtT T
kX j
T T
x(t)
Same function in the frequency-domain!
Note: (period in t) T ⇔ (period in ω) 2π/T Inverse relationship again!
Book Chapter4: Section1
16
Properties of the CT Fourier Transform
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0
0
0
0
( )
1) Linearity ( ) ( ) ( ) ( )
2) Time Shifting ( ) ( )
Proof: ( ) ( )
magnitude unchanged
j t
j tj t j t
tX j
ax t by t aX j bY j
x t t e X j
x t t e dt e x t e dt
FT
0
00
( ) ( )
Linear change in phase
( ( )) ( )
j t
j t
e X j X j
FT
e X j X j t
Book Chapter4: Section1
17
Properties (continued)
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*
Conjugate Symmetry
( ) real ( ) ( )
( ) ( )
( ) ( )
{ ( )} { ( )}
{ ( )} { ( )}
Even
x t X j X j
X j X j
X j X j
Re X j Re X j
Im X j Im X j
Odd
Od
n
d
Eve
Book Chapter4: Section1
18
The Properties Keep on Coming ...
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1Time-Scaling ( ) ( )
1 E.g. 1
( ) ( ) compressed in time stretched in frequency
a) ( ) real and even
x at X ja a
a a at t
x t X j
x t
*
*
( ) ( )
( ) ( ) ( ) Real & even
b) ( ) real and odd ( ) ( )
( ) ( ) ( ) Purely imaginary &:
c) ( ) { ( )}+ { ( )}
x t x t
X j X j X j
x t x t x t
X j X j X j
X j Re X j jIm X j
( ) { ( )} { ( )}x t Ev x t Od x t
For real
Book Chapter4: Section2
19
The CT Fourier Transform Pair
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─ FT(Analysis Equation)
─ Inverse FT(Synthesis Equation)
Last lecture: some propertiesToday: further exploration
Book Chapter4: Section2
20
Coefficient a
Y(jω)
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Convolution Property
A consequence of the eigenfunction property :
h(t)
H(jω).a
Synthesis equation for y(t)
Book Chapter4: Section2
21
The Frequency Response Revisited
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h(t)
¿impulse response
¿frequency response
The frequency response of a CT LTI system is simply the Fourier transform of its impulse response
Example #1: ¿H(jω)
¿Recall
¿⇓𝑦 (𝑡)=𝐻 ( 𝑗 𝜔0)𝑒
𝑗 𝜔0𝑡inverse FT
Book Chapter4: Section2
22
Example #2 A differentiator
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