class_11_ft_convolution.pdf

Chengbin Ma UM-SJTU Joint Institute

Class#11

- Linearity and symmetry properties (3.9)

- Time- and frequency-shift properties (3.12)

- Scaling properties (3.15)

- Convolution property (3.10)

- Differentiation and integration properties (3.11)

Slide 1


Review of Previous Lecture

Slide 2

)()(

)()(

)(2

1)(

jXtx

dtetxjX

dejXtx

FT

tj

tj

A non-periodic signal can be

considered as a periodic

signal whose period is

infinite.

Extend FS to solve FT

Three Dirichlet Conditions


Typical FTs

Slide 3

2222

10),(

a

j

a

a

jaatue

FTat

1)(FT

t

0)( 0tj

FT

ett

)(2 00

FTtj

e

0

00

00

0

)sin()sinc(

sinc2otherwise ,0

|| ,1

T

TT

TTTt FT

Does not converge when a=0!


How about u(t) ?!

Slide 4

jaatue

FTat

10),(

)(1

)(

jtu

FT

Why not let a=0, then

jtu

FT 1)(

WRONG!

CORRECT

Because the real part of 1/(a+j)

has a spike at origin when a approaches 0,

whose integral is independent of a:

0

0 2222

22

arctan2

2

1Re

a

da

ad

a

a

a

a

ja

Strictly speaking u(t) has no FT

representation because u(t) is not absolutely

integrable. The FT here is actually an

approximation.

and

0 ,

0 ,0lim

220

a

a

a


Class#11






Slide 5


Summary of Properties (1)

Slide 6

)}(Im{)()},(Re{)( decomp.) odd-(Even

)()()( signals) odd and (Real

)()()( signals)even and (Real

)()( signals) (Real

:symmeteryn Conjugatio

)()( :nConjugatio

)()( :Reversal Time

)()()()()()( :Linearity

*

*

*

**

jXjtxjXtx

jXjXjX

jXjXjX

jXjX

jXtx

jXtx

jbYjaXjZtbytaxtz

FT

o

FT

e

FT

FT

FT

Purposes:

1) understand better the

nature of FT and FS

2) Simplify the calculation

of FT and FS


Summary of Properties (2)

Slide 7

)(||

1)( :Scaling

))(()( :Shift-Frequency

)()( :Shift-Time 00

ajX

aatx

jXtxe

jXettx

FT

FTtj

tjFT

Note: only FTs are shown as examples because they are more

general. For FSs, these properties obviously exist too.


Examples (1)

Problem 3.16 (a)

2 3

2 3

2

2

( ) 2 ( ) 3 ( ) ( ) ?

Solution:

1 1( ) , ( )

2 3

1 12 ( ) 3 ( ) 2 3

2 3 5 6

t t

t t

t t

x t e u t e u t X j

e u t e u tj j

je u t e u t

j j j j

jatue

FTat

1)(


Examples (2)

2 2

2 2

( ) ( ) ?

Solution:

( ) ( )

( ) ( ) ( )

( ) ( ) ( ) ( )2 2 2 ( )

2 2

1( ) Re ( ) Re

2( ) 2 ( )

a t

at

at at

at at

e

e

e

x t e X j

y t e u t

x t e u t e u t

e u t e u t y t y ty t

ay t Y j

a j a

ax t y t

a

)}(Re{)( jXtxe

jajX

atuetx at

1)(

0),()(

Note: u(t) is undefined when

t=0. This representation may

not be accurate.


Example (3)

Slide 10

2)cos( ,3

2 :sinusoidscomplex theof FT 2,

propertyLiearity 1,

:Hints

)3sin(3)cos(2)(

oftion representa FT theDerive

00

jtjt

FTtj

eet

)-(e

tttx


Linearity Property

FT ( ) ( ) ( ) ( ) ( ) ( )

FS ( ) ( ) ( ) [ ] [ ] [ ]

z t ax t by t Z j aX j bY j

z t ax t by t Z k aX k bY k

)()(

)()(

)(2

1)(

jXtx

dtetxjX

dejXtx

FT

tj

tj

T

tjk

k

tjk

dtetxT

kX

ekXtx

0

0

)(1

][

][)(


Time Reversal Property

FT ( ) ( )

FS ( ) [ ]

x t X j

x t X k

T

tjk

k

tjk

dtetxT

kX

ekXtx

0

0

)(1

][

][)(

)()(

)()(

)(2

1)(

jXtx

dtetxjX

dejXtx

FT

tj

tj

( )

( ) ( )

( ) ( )

tj t j

j

x t e dt x e d

x e d X j


Conjugation Property

* *

* *

FT ( ) ( )

FS ( ) [ ]

x t X j

x t X k

*

**

** *

* * *

( ) ( )

( ) ( ) ( )

( ) ( )

( ) ( ) FT{ ( )}

j t

j t j t

j t j t

j t

X j x t e dt

X j x t e dt x t e dt

x t e dt x t e dt

X j x t e dt x t

)()(

)()(

)(2

1)(

jXtx

dtetxjX

dejXtx

FT

tj

tj


Conjugate Symmetry

For real signals

*

*

If the signal is real, then the Fourier representation

is complex-conjugate symmetric.

FT ( ) ( )

FS [ ] [ ]

X j X j

X k X k

*

* *

*

*

( ) ( )

( ) ( ), ( ) ( )

( ) ( )

( ) ( )

x t x t

x t X j x t X j

X j X j

X j X j

* *

* *

FT ( ) ( )

FS ( ) [ ]

x t X j

x t X k


Symmetry Properties

For even signals

*

*

If the signal is real and even, then the Fourier

representation is real and even.

FT ( ) ( ) ( )

FS [ ] [ ] [ ]

X j X j X j

X k X k X k

* *

*

*

( ) ( ) ( ) ( )

( ) ( ) ( ) ( )

( ) ( ) ( )

( ) ( )

( ) ( )

x t x t X j X j

x t x t X j X j

X j X j X j

X j X j

X j X j


Symmetry Properties

For odd signals

*

*

If the signal is real and odd, then the Fourier

representation is purely imaginary and odd.

FT ( ) ( ) ( )

FS [ ] [ ] [ ]

X j X j X j

X k X k X k

Fig. 3.51, P259


Symmetry Properties (Proof)

For real and odd signals

* *

*

* *

( ) ( ) ( ) ( )

( ) ( ) ( ) ( )

( ) ( ) ( )

( ) ( ) ( ) ( )

( ) ( )

x t x t X j X j

x t x t X j X j

X j X j X j

X j X j X j X j

X j X j


Symmetry Properties

Even-odd decomposition of real signals

If the signal is real, then

FT ( ) Re{ ( )}, ( ) Im{ ( )}

FS ( ) Re{ [ ]}, ( ) Im{ [ ]}

e o

e o

x t X j x t j X j

x t X k x t j X k


Symmetry Properties (Proof)

Even-odd decomposition for real signals

* * *

*

*

( ) ( ) ( ) ( )( )

2 2

( ) ( ) ( ) ( )( )

2 2

( ) ( ) ( ) ( ) ( ) ( )

( ) ( )( ) Re{ ( )}

2

( ) ( )( ) Im{ ( )}

2

e

o

e

o

x t x t X j X jx t

x t x t X j X jx t

x t x t X j X j X j X j

X j X jx t X j

X j X jx t j X j

)()(* jXjx


Time-Shift property

0

0 0

0

0

FT ( ) ( )

FS ( ) [ ]

j t

jk t

x t t e X j

x t t e X k

0

0

0 0

( )

0( ) ( )

( ) ( )

t tj tj t

j t j tj

x t t e dt x e d

e x e d e X j

)()(

)()(

)(2

1)(

jXtx

dtetxjX

dejXtx

FT

tj

tj


Frequency-Shift Property

0 0

0

FT ( ) ( ( ))

FS ( ) [ ]

j t

jk t

e x t X j

e x t X k k

0 0

0

FT ( ) ( ( ))

FS ( ) [ ]

j t

jk t

e x t X j

e x t X k k

( )( ) ( ) ( ( ))j t j t j te x t e dt x t e d X j


Scaling Property

Time and frequency scaling property

1FT ( ) ( )

FS ( ), 0 [ ]

x at X ja a

x at a X k


Scaling Property (Proof)

Proof of time and frequency scaling property: FT

1 1( ) , 0 ( ) , 0

( )1 1

( ) , 0 ( ) , 0

1 1( ) ( )

j ja a

atj t

j ja a

ja

x e d a x e d aa a

x at e dt

x e d a x e d aa a

x e d X ja a a

)()(

)()(

)(2

1)(

jXtx

dtetxjX

dejXtx

FT

tj

tj


Scaling Property (Proof)

Proof of time and frequency scaling property: FS

0

0 0( ) ( )

0

( ) [ ]

( ) [ ] [ ]

2( 0, ( ) is periodic with period )

jk t

k

jk at jk a t

k k

x t X k e

x at X k e X k e

a x ata

The FS coefficients of x(t) and x(at) are identical;

the scaling operation simply changes the harmonic spacing from 0 to a0.


Class#11






Slide 25


Convolution Property

Perhaps the most important property of Fourier representations is the convolution property.

FT ( ) ( )* ( ) ( ) ( ) ( )

FS ( ) ( ) ( ) [ ] [ ] [ ]

where ( ) ( ) ( ) ( )

(periodic convolution)

T

y t h t x t Y j H j X j

y t h t x t Y k TH k X k

h t x t h x t d

= Very important!


Ratio of FT

The convolution property implies that the

frequency response of a system may be expressed

as the ratio of the Fourier transform of the output

to that of the input (does not need to depend on

any specific signal any more).

( ) ( )* ( ) ( ) ( ) ( )

( )( )

( )

y t x t h t Y j X j H j

Y jH j

X j


Proof (FT)

Proof of convolution property: FT

)()(

)()()()(

)()(

)()()()(

jHjX

dehjXdjXeh

ddtetxh

dtedtxhdtetyjY

jj

tj

tjtj

)()(

)()(

0

0

jXettx

dtetxjX

tj

tj


Proof (FS)

Proof of convolution property: FS

0 0

0 0

0

0 0

( )

( )

( ) ( ) ( ) ( ) ( )

[ ] [ ]

[ ] [ ]

[ ] [ ] [ ]

[ ] [ ] [ ]

[ ] [ ] [ ]

T

jk jr t

Tk r

jr t j k r

Tk r

jr t

k r

jk t jk t

k k

y t h t x t h x t d

H k e X r e d

H k X r e e d

H k X r e T k r

TH k X k e Y k e

Y k TH k X k

][

][)(

0

)( 0

0

rkTdte

ekXtx

Ttrkj

k

tjk


Example

)2()(2)2()(*)()(

)1()1()(

)()()sin(2

)sin(2

)(

:Solution

?)(

)(sin4

)( 22

trtrtrtztztx

tututz

jZjZjX

tx

jX

)sin(2

)()(

)()()(

)()()(

000 Tttuttu

jXjHjY

txthty


Filtering

The multiplication that occurs in the frequency-domain representation gives rise to the notion of filtering.

A system performs filtering on the input signal by presenting a different response to components of the input that are at different frequencies.

Typically, the term filtering implies that some frequency components of the input are eliminated while others are passed by the system unchanged.

-40

-30

-20

-10

0

Magnitu

de (

dB

)

101

102

103

104

105

-90

-45

0

Phase (

deg)

Bode Diagram

Frequency (rad/sec)

)(

)()()(

jXA

jXjHjY


Bode Plots

Slide 32

-40

-30

-20

-10

0M

agnitu

de (

dB

)

101

102

103

104

105

-90

-45

0

Phase (

deg)

Bode Diagram

Frequency (rad/sec)


Ideal Filters

Continuous-time Discrete-time

Ideal low-pass/high-pass/band-pass filters


Bands of Filters

The passband of a filter is the band of frequencies that are

passed by the system, while the stopband refers to the

range of frequencies that are attenuated by the system.

Realistic filters always have a gradual transition from the

passband to the stopband. The range of frequencies over

which this occurs is known as the transition band.

-40

-30

-20

-10

0

Magnitu

de (

dB

)

100

101

102

103

104

-90

-45

0

Phase (

deg)

Bode Diagram

Frequency (rad/sec)

1100

1

j

passband stopband

transition band


Class#11






Slide 35


Differentiation in time

0

FT ( ) ( )

FS ( ) [ ]

dx t j X j

dt

dx t jk X k

dt

Very important!


Proof (FT)

Proof of differentiation-in-time property: FT

1( ) ( )

2

1 1( ) ( ) ( )

2 2

1 1( ) ( )

2 2

( ) ( )

j t

j t j t

j t j t

x t X j e d

d d dx t X j e d X j e d

dt dt dt

X j j e d j X j e d

dx t j X j

dt


Proof (FS)

Proof of differentiation-in-time property: FS

0

0 0

0 0

0

0

( ) [ ]

( ) [ ] [ ]

[ ] [ ]

( ) [ ]

jk t

k

jk t jk t

k k

jk t jk t

k k

x t X k e

d d dx t X k e X k e

dt dt dt

dX k e jk X k e

dt

dx t jk X k

dt

0

FT ( ) ( )

FS ( ) [ ]

dx t j X j

dt

dx t jk X k

dt


Differential Equations

The differentiation property may be used to find

the frequency response of a continuous-time

system described by the differential equation

M

kk

k

k

N

kk

k

k txdt

dbty

dt

da

00

)()(


System Frequency Response

Find the frequency response of a continuous-time system

described by differential equation.

0 0

0 0

0

0

FT ( ) FT ( )

( ) ( ) ( ) ( )

( )( )

( )( )

( )

k kN M

k kk kk k

N Mk k

k k

k k

Mk

k

k

Nk

k

k

d da y t b x t

dt dt

a j Y j b j X j

b jY j

H jX j

a j


For a differential equation (n=10,000rad/s)

Slide 41

Example

)()()()(2

2

2

txtytydt

d

Qty

dt

dn

n

Its frequency response

22 )()(

1)(

nj

Qj

jHn

102

103

104

105

106

-250

-200

-150

-100

Magnitu

de (

dB

)

Bode Diagram

Frequency (rad/sec)

Q=200, 1, 2/5

In-class problem:

Passband?

0

FT ( ) ( )

FS ( ) [ ]

dx t j X j

dt

dx t jk X k

dt


Differentiation in Frequency

FT ( ) ( )d

jtx t X jd

Proof of differentiation-in-frequency: FT

( ) ( )

( ) ( ) ( )

( ) ( )

j t

j t j t

X j x t e dt

d dX j x t e dt jtx t e dt

d d

djtx t X j

d


Example

2

2

1)(

1)(

1)(

:Solution

?)()()(

jatute

ja

j

jad

dtujte

jatue

jXtutetx

at

at

at

at

)()(

)()(

jXd

dtjtx

dtetxjX tj


Integration Property

0

1FT ( ) ( ) ( ) ( ) ( 0) ( )

1FS ( ) ( ) [ ] [ ]

( ( ) is finite valued and periodic only if [0] 0)

t

t

y t x d Y j X j X jj

y t x d Y k X kjk

y t X

0

FT ( ) ( )

FS ( ) [ ]

dx t j X j

dt

dx t jk X k

dt

If not, the integration of x(t), i.e., y(t), will be infinite!


Proof (FT)

Proof of integration property: FT

( )* ( ) ( ) ( ) ( )

1( ) ( ) ( ) ( ) ( )

1( ) ( ) ( )

1( ) ( 0) ( )

t

t

x t u t x u t d x d

x d X j U j X jj

X j X jj

X j X jj

)(1

)(

)()()(

)(*)()(

jtu

jXjHjY

txthty


Proof (FS)

Proof of integration property: FS

0

0

( ) ( ) ( ) ( )

1[ ] [ ] [ ] [ ]

td

y t x d x t y tdt

X k jk Y k Y k X kjk

][)( 0 kXjktxdt

d


Homework

3.48(d), 3.49(d), 3.50(d) (f)

Prove Fourier Transform using IFT:

3.54(a)(c)(e)

3.58(a)(c)(e)

3.59(a)(c)(e)

3.67(c)

3.68(a)

3.70(b)

Due 2:00PM, Thursday of next week

Slide 47

dtetxjX tj )()(

class_11_ft_convolution.pdf

Documents

jz t ax t

tay t y

t z j ax j

j aax t y

scaling properties

symmetry properties

ft representation

integration properties