Fourier Transform: Important Propertieseeweb.poly.edu/~yao/EE3054/Chap11.5_11.10_FT.pdf · LECTURE...

EE3054

Signals and Systems

Fourier Transform: Important Properties

Yao Wang

Polytechnic University

Some slides included are extracted from lecture presentations prepared by McClellan and Schafer

4/4/2008 © 2003, JH McClellan & RW Schafer 2

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LECTURE OBJECTIVES

� Basic properties of Fourier transforms

� Duality, Delay, Freq. Shifting, Scaling

� Convolution property

� Multiplication property

� Differentiation property

� Freq. Response of Differential Equation System


Fourier Transform Defined

� For non-periodic signals

Fourier Synthesis

Fourier Analysis

∫∞

∞−

−= dtetxjXtjωω )()(

∫∞

∞−

= ωω ωπ

dejXtxtj)()(

21

Table of Fourier Transforms

)()()()cos()( ccc jXttx ωωπδωωπδωω ++−=⇔=

1)()()( =⇔= ωδ jXttx

>

<=⇔=

b

bb jX

t

ttx

ωω

ωωω

π

ω

0

1)(

)sin()(

2/

)2/sin()(

2/0

2/1)(

ω

ωω

TjX

Tt

Tttx =⇔

>

<=

ωω

jjXtuetx

t

+=⇔= −

1

1)()()(

)(2)()( ctj

jXetx c ωωπδωω−=⇔=

)()()()sin()( ccc jjjXttx ωωπδωωπδωω ++−−=⇔=

0)()()( 0tj

ejXtttxωωδ −

=⇔−=


Duality of FT Pairs

∫∞

∞−

−= dtetxjXtjωω )()(∫

∞

∞−

= ωω ωπ

dejXtxtj)()(

21

( ))(2)( Then

)( If

ωπ

ω

−⇔

⇔

xtg

gtx


Fourier Transform of a

General Periodic Signal

� If x(t) is periodic with period T0 ,

∫∑ −∞

−∞=

==0

00

00

)(1

)(

T

tjkk

k

tjkk dtetx

Taeatx

ωω

)(2 since Therefore, 00 ωωπδω

ketjk

−⇔

∑∞

−∞=

−=k

k kajX )(2)( 0ωωδπω


Square Wave Signal

x(t) = x(t + T0 )

T0−2T0 −T0 2T00 t

ak =e

− jω0kt

− jω0kT0 0

T0 / 2

−e

− jω 0kt

− jω0kT0 T0 /2

T0

=1− e− jπk

jπk

ak =1

T0

(1)e− jω0 kt

dt +1

T0

(−1)e− jω 0kt

dtT0 / 2

T0

∫0

T0 / 2

∫


Square Wave Fourier Transform

X( jω ) = 2π akδ(ω − kω0 )k =−∞

∞

∑

x(t) = x(t + T0 )

T0−2T0 −T0 2T00 t


FT of Impulse Train

� The periodic impulse train is

p(t) = δ (t − nT0 ) =n=−∞

∞

∑ akejkω0t

n=−∞

∞

∑

ak =1

T0δ (t)e

− jω0tdt =

−T0 /2

T0 /2

∫1

T0 for all k

∴ P( jω) =2π

T0

δ (

k = −∞

∞∑ ω − kω0 )

ω0 = 2π / T0

Plot of impulse train in time

and frequency


Table of Easy FT Properties

ax1(t) + bx2 (t) ⇔ aX1( jω) + bX2 ( jω )

x(t − td ) ⇔ e− jωtd X( jω )

x(t)ejω0t

⇔ X( j(ω − ω0 ))

Delay Property

Frequency Shifting

Linearity Property

x(at) ⇔ 1|a | X( j(ω

a ))Scaling

Duality


Delay Property


x(t − td )e− jωtdt−∞

∞

∫ = x(τ )e− jω(τ +td )

dτ−∞

∞

∫

= e− jω td X( jω )

For example, e−a(t−5)

u(t − 5) ⇔e− jω 5

a + jω


Multiply by e^jw0

x(t)ejω0t

⇔ X( j(ω − ω0 ))

))((

)()(

0

)( 00

ωω

ωωωω

−=

= ∫∫∞

∞−

−−∞

∞−

−

jX

dtetxdtetxetjtjtj


Multiply by cos(w0)?

( )

( )))(())((2

1

)()(2

1)cos()(

))(()(

))(()(

00

0

0

0

00

0

0

ωωωω

ω

ωω

ωω

ωω

ω

ω

++−

⇔+=

+⇔

−⇔

−

−

jXjX

etxetxttx

jXetx

jXetx

tjtj

tj

tj


Shifting in frequency by

multiply by cos()

= (Amplitude Modulation)

� Illustrate the spectrum in class


y(t) = x(t)cos(ω0t) ⇔

Y( jω ) =1

2X( j(ω − ω0 )) +

1

2X( j(ω + ω0 ))

x(t)

x(t) =1 t < T / 2

0 t > T / 2

⇔ X( jω ) =sin(ωT / 2)

ω / 2( )


Another example

� x(t)=cos (w0 t)

� What is y(t)=x(t) * cos (w1 t)

� Consider w1 >w0 and w1<w0

� Verify by trigonometric identities


What about multiply by sin( )?


Scaling Property

expands)(shrinks;)2(22

1 ωjXtx

)(

)()(

1

)/(

aa

adajtj

jX

exdteatx

ω

λλωω λ

=

= ∫∫∞

∞−

−∞

∞−

−

)()( 1aa

jXatx ω⇔


Scaling Property

)()( 1aa

jXatx ω⇔

)2()( 12 txtx =


Uncertainty Principle

� Try to make x(t) shorter

� Then X(jωωωω) will get wider

� Narrow pulses have wide bandwidth

� Try to make X(jωωωω) narrower

� Then x(t) will have longer duration

�� Cannot simultaneously reduce time Cannot simultaneously reduce time

duration and bandwidthduration and bandwidth



ax1(t) + bx2 (t) ⇔ aX1( jω) + bX2 ( jω )


x(t)ejω0t

⇔ X( j(ω − ω0 ))

Delay Property

Frequency Shifting

Linearity Property

x(at) ⇔ 1|a | X( j(ω

a ))Scaling


Significant FT Properties

x(t) ∗h(t) ⇔ H( jω )X( jω )

x(t)ejω0t

⇔ X( j(ω − ω0 ))

x(t)p(t) ⇔1

2πX( jω )∗ P( jω )

dx(t)

dt⇔ ( jω)X( jω)

Differentiation Property

Duality


Convolution Property

� Convolution in the time-domain

corresponds to MULTIPLICATIONMULTIPLICATION in the

frequency-domain

y(t) = h(t) ∗ x(t) = h(τ )−∞

∞

∫ x(t − τ )dτ

Y( jω ) = H( jω )X( jω )

y(t) = h(t) ∗ x(t)x(t)

Y( jω ) = H( jω )X( jω )X( jω)


Proof (in class)


Convolution Example

� Bandlimited Input Signal� “sinc” function

� Ideal LPF (Lowpass Filter)� h(t) is a “sinc”

� Output is Bandlimited� Convolve “sincs”


Ideally Bandlimited Signal

>

<=⇔=

πω

πωω

π

π

1000

1001)(

)100sin()( jX

t

ttx

πω 100=b


Ex: x(t) and y(t) are both sinc

sin(100π t)

πt∗

sin(200πt)

π t=

x(t) ∗h(t) ⇔ H( jω )X( jω )

sin(100π t)

πt


Ex. x(t) and y(t) are both rect.

pulse

Y( jω ) =sin(ω / 2)

ω / 2

2

y(t) = x(t) ∗ h(t)

Y( jω ) = H( jω )X( jω )


Cosine Input to LTI System

Y (jω) = H( jω )X( jω)

= H( jω )[πδ(ω − ω0 ) +πδ(ω +ω 0)]

= H( jω0 )πδ (ω −ω0 ) + H(− jω0 )πδ (ω +ω0 )

y(t) = H (jω0 ) 12 e

jω0t+ H(− jω0 ) 1

2 e− jω 0t

= H( jω0 ) 12 e

jω0t+ H

*( jω 0)

12 e

− jω0t

= H( jω0 ) cos(ω 0t +∠H( jω0 ))

)cos(*)()( 0tthty ω=


Ideal Lowpass Filter

Hlp( jω)

ωco−ωco

y(t) = x(t) if ω0 < ωco

y(t) = 0 if ω0 > ωco


Ideal Lowpass Filter

y(t) =4

πsin 50πt( ) +

4

3πsin 150πt( )

fco "cutoff freq."

H( jω ) =1 ω < ωco

0 ω > ωco


Multiplier

� Multiplication in the time-domain corresponds to convolution in the frequency-domain.

Y( jω ) =1

2πX( jω) ∗ P( jω )

y(t) = p(t)x(t)

X( jω )

x(t)

p(t)

Y( jω ) =1

2πX( jθ )

−∞

∞

∫ P( j(ω −θ ))dθ


p(t) = cos(ω0t) ⇔ P( jω) = πδ (ω − ω0 )

+ πδ (ω + ω0 )

y(t) = x(t)p(t) ⇔ Y( jω ) =1

2πX( jω )∗ P( jω)

y(t) = x(t)cos(ω0t) ⇔

Y( jω ) =1

2πX( jω ) ∗[πδ (ω − ω0 ) + πδ (ω + ω0 )]

Y( jω ) =1

2X( j(ω − ω0 )) +

1

2X( j(ω + ω0 ))

Multiply by cos(w0 t)


Differentiation Property

dx (t )

dt=

d

dt

1

2πX ( jω )e

jω tdω

−∞

∞

∫

=1

2π( jω ) X( jω )e

jω tdω

−∞

∞

∫

Multiply by jωωωωdx(t)

dt⇔ ( jω)X( jω)


Example

d

dte

−atu(t)( )= −ae

−atu(t) + e

−atδ (t)

= δ (t) − ae−atu(t)

ωjatue

at

+⇔− 1

)(

)(1)( ωωω

ω

ωω jXj

ja

j

ja

ajY =

+=

+−=


High order differentiation?

dx(t)

dt⇔ ( jω)X( jω ) ( ) ( )ωω jXj

dx

txd k

k

k

⇔)(

Proof in class

System of Differential

Equation

( ) ( )

( )

( )∑

∑

∑∑

∑∑

=

=

==

==

==

=

=

N

k

kk

M

k

kk

M

k

kk

N

k

kk

M

kk

k

k

N

kk

k

k

ja

jb

jX

jYjH

jXjbjYja

dt

txdb

dt

tyda

0

0

00

00

)(

)()(

)()(

)()(

ω

ω

ω

ωω

ωωωω

c


Recall Difference Equation?

( ) ( )

( )

( )∑

∑

∑∑

∑∑

=

=

==

==

==

=

=

N

k

kk

M

k

kk

M

k

kk

N

k

kk

M

kk

k

k

N

kk

k

k

ja

jb

jX

jYjH

jXjbjYja

dt

txdb

dt

tyda

0

0

00

00

)(

)()(

)()(

)()(

ω

ω

ω

ωω

ωωωω

c

∑

∑

∑∑

∑∑

=

−

=

−

=

−

=

−

==

==

=

−=−

N

k

kk

M

k

kk

M

k

kk

N

k

kk

M

k

k

N

k

k

za

zb

zX

zYzH

zXzbzYza

knxbknya

0

0

00

00

)(

)()(

)()(

][][

c

Discrete time system

(Difference equation)

Continuous time system

(Differentiation equation)

Example systems

� Example systems described by low order differential equations

� How to determine the frequency response

� How to determine the impulse response


Strategy for using the FT

� Develop a set of known Fourier transform pairs.

� Develop a set of “theorems” or properties of the Fourier transform.

� Develop skill in formulating the problem in either the time-domain or the frequency-domain, which ever leads to the simplest solution.

Table of Fourier Transforms

)()()()cos()( ccc jXttx ωωπδωωπδωω ++−=⇔=

1)()()( =⇔= ωδ jXttx

>

<=⇔=

b

bb jX

t

ttx

ωω

ωωω

π

ω

0

1)(

)sin()(

2/

)2/sin()(

2/0

2/1)(

ω

ωω

TjX

Tt

Tttx =⇔

>

<=

ωω

jjXtuetx

t

+=⇔= −

1

1)()()(

)(2)()( ctj

jXetx c ωωπδωω−=⇔=

)()()()sin()( ccc jjjXttx ωωπδωωπδωω ++−−=⇔=

0)()()( 0tj

ejXtttxωωδ −

=⇔−=



ax1(t) + bx2 (t) ⇔ aX1( jω) + bX2 ( jω )


x(t)ejω0t

⇔ X( j(ω − ω0 ))

Delay Property

Frequency Shifting

Linearity Property

x(at) ⇔ 1|a | X( j(ω

a ))Scaling

Significant FT Properties

x(t) ∗h(t) ⇔ H( jω )X( jω )

x(t)ejω0t

⇔ X( j(ω − ω0 ))

x(t)p(t) ⇔1

2πX( jω )∗ P( jω )

dx(t)

dt⇔ ( jω)X( jω)

Duality

( ) ( )ωω jXjdx

txd k

k

k

⇔)(

READING ASSIGNMENTS

� This Lecture:

� Chapter 11, Sects. 11-5 to 11-10

� Tables in Section 11-9

� Other Reading:

� Entire chap 11

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