Lecture 9 (Chapter 4) The Continuous-Time Fourier Transform Adapted from: Lecture notes from MIT Dr. Hamid R. Rabiee Fall 2012 Signals & Systems
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Lecture 9 (Chapter 4)
The Continuous-Time Fourier Transform
Adapted from: Lecture notes from MIT
Dr. Hamid R. Rabiee
Fall 2012
Signals & Systems
Lecture 9 (Chapter 4)
� The Convolution Property of the CTFT
� Frequency Response and LTI Systems Revisited
� Multiplication Property and Parseval’s Relation
� The DT Fourier Transform
Content
2
Last lecture: some properties of the CTFT
Today: further exploration
Sharif University of Technology, Department of Computer Engineering, Signals & Systems
Lecture 9 (Chapter 4)
The CT Fourier Transform Pair
(Analysis Equation)
(Synthesis Equation)
FT
Inverse FT
3
( ) ( )x t X jω→
( ) ( )
1( ) ( )
2
j t
j t
X j x t e dt
x t X j e d
ω
ω
ω
ω ωπ
∞
−
−∞
∞
−∞
=
=
∫
∫
Sharif University of Technology, Department of Computer Engineering, Signals & Systems
Lecture 9 (Chapter 4)
Convolution Property
4
impulse response
frequency response
Proof on next slide
( ) ( )* ( ) ( ) ( ) ( )
( ) ( )
y t h t x t Y j H j X j
where h t H j
ω= ←→ = ω ω
↔ ω
F
( ) ( ) ( )Y j H j X jω ω ω=
)(tx ( ) ( ) * ( )y t x t h t=)(th
Sharif University of Technology, Department of Computer Engineering, Signals & Systems
Lecture 9 (Chapter 4)
5
Synthesis equation for y(t)
A consequence of the eigen function property:
Convolution Property(continued)
1( ) ( ( ) )
2
j tx t X j d e
ωωπ
∞
−∞
= ω∫
1 1( ) ( ( ) ( ) ) ( ) ( )
2 2
j t j ty t H j X j d e H j X j e dω
π π
∞ ∞
ω ω
−∞ −∞
= ω ω = ω ω ω∫ ∫
j tae
ω( ) j t
H j aeωω( )H jω
Sharif University of Technology, Department of Computer Engineering, Signals & Systems
Lecture 9 (Chapter 4)
Example#1:
Recall
inverse FT
6
The frequency response of a CT LTI system is simply the
Fourier transform of its impulse response
02 ( )j t
e π0ω ←→ δ ω− ωF
( )
0
0
( ) ( ) ( ) ( ) 2 ) 2 ( ) )
( ) ( )j t
Y j H j X j H j H j
y t H j e
π π0 0 0
ω
ω = ω ω = ω δ(ω − ω = ω δ(ω − ω
⇓
= ω
0j tae
ω ( )y t( )H jω
Sharif University of Technology, Department of Computer Engineering, Signals & Systems
Lecture 9 (Chapter 4)
2) +π/2 phase shift ( j = ejπ/2)
Differentiation property:
1) Amplifies high frequencies (enhances sharp edges)
Example #2: A differentiator( )
7
( )( )
dx ty t
dt=
( ) ( )y j j X jω = ω ω
0
0
0 0 0 0
0 0 0 0
sin cos sin( )2
cos sin cos( )2
dt t t
dt
dt t t
dt
π
π
ω = ω ω = ω ω +
ω = −ω ω = ω ω +
( )H j jω = ω
Sharif University of Technology, Department of Computer Engineering, Signals & Systems
Lecture 9 (Chapter 4)
Example #3: Impulse Response of an Ideal Lowpass Filter
8
Questions:1) Is this a causal system? No.
2)What is h(0)?
3)What is the steady-state value of
the step response, i.e. s(∞)?
sin1( )
2
sin ( )
c
c
j t c
c c
th t e d
t
tc
π π
π π
ω
ω
−ω
ω= ω =
ω ω=
∫
s inD e f in e : s in c ( )=
π θθ
π θ
( )12
2( 0 )
2
c ch H j dπ
ω ωω ω
π π
∞
− ∞= = =∫
( ) ( )
( ) ( ) ( 0 ) 1
t
s t h t d t
s h t d t H j
− ∞
∞
− ∞
=
∞ = = =
∫
∫
Sharif University of Technology, Department of Computer Engineering, Signals & Systems
Lecture 9 (Chapter 4)
Example #4: Cascading filtering operations
H(jω)
9
1( )H jω 2 ( )H jω
1 2( ) ( ) ( )H j H j H jω ω ω=
Sharif University of Technology, Department of Computer Engineering, Signals & Systems
Lecture 9 (Chapter 4)
Example #5:
Example #6:
Gaussian × Gaussian = Gaussian ⇒ Gaussian ∗ Gaussian = Gaussian
10
sin 4 sin8* ?
t t
t t
π π
π π=
Sharif University of Technology, Department of Computer Engineering, Signals & Systems
Lecture 9 (Chapter 4)
Example #2 from last lecture
11 Sharif University of Technology, Department of Computer Engineering, Signals & Systems
Lecture 9 (Chapter 4)
Example #7:
- a rational function of jω, ratio of polynomials of jω
Partial fraction expansion
inverse FT
12 Sharif University of Technology, Department of Computer Engineering, Signals & Systems
Lecture 9 (Chapter 4)
Using the Differentiation Property
1) Rational, can use
PFE to get h(t)
2) If X(jω) is rational
e.g. x(t)=Σcie-at u(t)
then Y(jω) is also rational
Transform both sides of the equation
Example #8: LTI Systems Described by LCCDE’s
(Linear-constant-coefficient differential equations)
13 Sharif University of Technology, Department of Computer Engineering, Signals & Systems
Lecture 9 (Chapter 4)
Total energy
in the time-domain
Total energy
in the frequency-domain - Spectral density
Parseval’s Relation
14
2 21| ( ) | | ( ) |
2x t dt X j d
π
∞ ∞
−∞ −∞
= ω ω∫ ∫ ( )21
2X jω
π
Sharif University of Technology, Department of Computer Engineering, Signals & Systems
Lecture 9 (Chapter 4)
Multiplication Property
We already know that:
Then it isn’t a
surprise that:
Convolution in ω
A consequence of Duality
FT is highly symmetric,
Sharif University of Technology, Department of Computer Engineering, Signals & Systems
Lecture 9 (Chapter 4)
Examples of the Multiplication Property
For any s(t) ...
16
0 0
1 1( ) ( ( )) ( ( ))
2 2R j S j S jω = ω − ω + ω + ω
1( ) ( ). ( ) ( ) [ ( )* ( )]
2r t s t p t R j S j P j
π= ↔ ω = ω ω
Sharif University of Technology, Department of Computer Engineering, Signals & Systems
Lecture 9 (Chapter 4)
Example (continued)
Amplitude modulation
(AM)
Drawn assuming:
17 Sharif University of Technology, Department of Computer Engineering, Signals & Systems
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