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

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Lecture 9 (Chapter 4)

Example (continued)

Amplitude modulation

(AM)

Drawn assuming:

17 Sharif University of Technology, Department of Computer Engineering, Signals & Systems