Fourier Series

Lecture 14: 27-Aug-12

Dr. P P Das

SourceThis presentation is lifted from:

http://www.ele.uri.edu/courses/ele436/FourierSeries.ppt

ContentPeriodic FunctionsFourier SeriesComplex Form of the Fourier Series Impulse TrainAnalysis of Periodic WaveformsHalf-Range ExpansionLeast Mean-Square Error Approximation

Fourier Series

Periodic Functions

The Mathematic Formulation

Any function that satisfies

( ) ( )f t f t T

where T is a constant and is called the period of the function.

Example:

4cos

3cos)(

tttf Find its period.

)()( Ttftf )(4

1cos)(

3

1cos

4cos

3cos TtTt

tt

Fact: )2cos(cos m

mT

23

nT

24

mT 6

nT 8

24T smallest T

Example:

tttf 21 coscos)( Find its period.

)()( Ttftf )(cos)(coscoscos 2121 TtTttt

mT 21

nT 22

n

m

2

1

2

1

must be a rational number

Example:

tttf )10cos(10cos)(

Is this function a periodic one?

10

10

2

1 not a rational number

Fourier Series

Fourier Series

Introduction

Decompose a periodic input signal into primitive periodic components.

A periodic sequenceA periodic sequence

T 2T 3T

t

f(t)

Synthesis

T

ntb

T

nta

atf

nn

nn

2sin

2cos

2)(

11

0

DC Part Even Part Odd Part

T is a period of all the above signals

)sin()cos(2

)( 01

01

0 tnbtnaa

tfn

nn

n

Let 0=2/T.

Orthogonal Functions

Call a set of functions {k} orthogonal on an interval a < t < b if it satisfies

nmr

nmdttt

n

b

a nm

0)()(

Orthogonal set of Sinusoidal Functions

Define 0=2/T.

0 ,0)cos(2/

2/ 0 mdttmT

T0 ,0)sin(

2/

2/ 0 mdttmT

T

nmT

nmdttntm

T

T 2/

0)cos()cos(

2/

2/ 00

nmT

nmdttntm

T

T 2/

0)sin()sin(

2/

2/ 00

nmdttntmT

T and allfor ,0)cos()sin(

2/

2/ 00

We now prove this one

Proof

dttntmT

T 2/

2/ 00 )cos()cos(

0

)]cos()[cos(2

1coscos

dttnmdttnmT

T

T

T

2/

2/ 0

2/

2/ 0 ])cos[(2

1])cos[(

2

1

2/

2/00

2/

2/00

])sin[()(

1

2

1])sin[(

)(

1

2

1 T

T

T

Ttnm

nmtnm

nm

m n

])sin[(2)(

1

2

1])sin[(2

)(

1

2

1

00

nmnm

nmnm

00

Proof

dttntmT

T 2/

2/ 00 )cos()cos(

0

)]cos()[cos(2

1coscos

dttmT

T 2/

2/ 02 )(cos

2/

2/

00

2/

2/

]2sin4

1

2

1T

T

T

T

tmm

t

m = n

2

T

]2cos1[2

1cos2

dttmT

T 2/

2/ 0 ]2cos1[2

1

nmT

nmdttntm

T

T 2/

0)cos()cos(

2/

2/ 00

Orthogonal set of Sinusoidal Functions

Define 0=2/T.

0 ,0)cos(2/

2/ 0 mdttmT

T0 ,0)sin(

2/

2/ 0 mdttmT

T

nmT

nmdttntm

T

T 2/

0)cos()cos(

2/

2/ 00

nmT

nmdttntm

T

T 2/

0)sin()sin(

2/

2/ 00

nmdttntmT

T and allfor ,0)cos()sin(

2/

2/ 00

,3sin,2sin,sin

,3cos,2cos,cos

,1

000

000

ttt

ttt

,3sin,2sin,sin

,3cos,2cos,cos

,1

000

000

ttt

ttt

an orthogonal set.an orthogonal set.

Decomposition

dttfT

aTt

t

0

0

)(2

0

,2,1 cos)(2

0

0

0

ntdtntfT

aTt

tn

,2,1 sin)(2

0

0

0

ntdtntfT

bTt

tn

)sin()cos(2

)( 01

01

0 tnbtnaa

tfn

nn

n

ProofUse the following facts:

0 ,0)cos(2/

2/ 0 mdttmT

T0 ,0)sin(

2/

2/ 0 mdttmT

T

nmT

nmdttntm

T

T 2/

0)cos()cos(

2/

2/ 00

nmT

nmdttntm

T

T 2/

0)sin()sin(

2/

2/ 00

nmdttntmT

T and allfor ,0)cos()sin(

2/

2/ 00

Example (Square Wave)

112

200

dta

,2,1 0sin1

cos2

200

nntn

ntdtan

,6,4,20

,5,3,1/2)1cos(

1 cos

1sin

2

200

n

nnn

nnt

nntdtbn

2 3 4 5--2-3-4-5-6

f(t)1

112

200

dta

,2,1 0sin1

cos2

200

nntn

ntdtan

,6,4,20

,5,3,1/2)1cos(

1 cos

1sin

2

100

n

nnn

nnt

nntdtbn

2 3 4 5--2-3-4-5-6

f(t)1

Example (Square Wave)

ttttf 5sin

5

13sin

3

1sin

2

2

1)(

ttttf 5sin

5

13sin

3

1sin

2

2

1)(

112

200

dta

,2,1 0sin1

cos2

200

nntn

ntdtan

,6,4,20

,5,3,1/2)1cos(

1 cos

1sin

2

100

n

nnn

nnt

nntdtbn

2 3 4 5--2-3-4-5-6

f(t)1

Example (Square Wave)

-0.5

0

0.5

1

1.5

ttttf 5sin

5

13sin

3

1sin

2

2

1)(

ttttf 5sin

5

13sin

3

1sin

2

2

1)(

Harmonics

T

ntb

T

nta

atf

nn

nn

2sin

2cos

2)(

11

0

DC Part Even Part Odd Part

T is a period of all the above signals

)sin()cos(2

)( 01

01

0 tnbtnaa

tfn

nn

n

Harmonics

tnbtnaa

tfn

nn

n 01

01

0 sincos2

)(

Tf

22 00Define , called the fundamental angular frequency.

0 nnDefine , called the n-th harmonic of the periodic function.

tbtaa

tf nn

nnn

n

sincos2

)(11

0

Harmonics

tbtaa

tf nn

nnn

n

sincos2

)(11

0

)sincos(2 1

0 tbtaa

nnnn

n

12222

220 sincos2 n

n

nn

nn

nn

nnn t

ba

bt

ba

aba

a

1

220 sinsincoscos2 n

nnnnnn ttbaa

)cos(1

0 nn

nn tCC

Amplitudes and Phase Angles

)cos()(1

0 nn

nn tCCtf

20

0

aC

22nnn baC

n

nn a

b1tan

harmonic amplitude phase angle

Fourier Series

Complex Form of the Fourier Series

Complex Exponentials

tnjtne tjn00 sincos0

tjntjn eetn 00

2

1cos 0

tnjtne tjn00 sincos0

tjntjntjntjn eej

eej

tn 0000

22

1sin 0

Complex Form of the Fourier Series

tnbtnaa

tfn

nn

n 01

01

0 sincos2

)(

tjntjn

nn

tjntjn

nn eeb

jeea

a0000

11

0

22

1

2

1

0 00 )(2

1)(

2

1

2 n

tjnnn

tjnnn ejbaejba

a

1

000

n

tjnn

tjnn ececc

)(2

1

)(2

12

00

nnn

nnn

jbac

jbac

ac

)(

2

1

)(2

12

00

nnn

nnn

jbac

jbac

ac

Complex Form of the Fourier Series

1

000)(

n

tjnn

tjnn ececctf

1

10

00

n

tjnn

n

tjnn ececc

n

tjnnec 0

)(2

1

)(2

12

00

nnn

nnn

jbac

jbac

ac

)(

2

1

)(2

12

00

nnn

nnn

jbac

jbac

ac

Complex Form of the Fourier Series

2/

2/

00 )(

1

2

T

Tdttf

T

ac

)(2

1nnn jbac

2/

2/ 0

2/

2/ 0 sin)(cos)(1 T

T

T

Ttdtntfjtdtntf

T

2/

2/ 00 )sin)(cos(1 T

Tdttnjtntf

T

2/

2/

0)(1 T

T

tjn dtetfT

2/

2/

0)(1

)(2

1 T

T

tjnnnn dtetf

Tjbac )(

2

1

)(2

12

00

nnn

nnn

jbac

jbac

ac

)(

2

1

)(2

12

00

nnn

nnn

jbac

jbac

ac

Complex Form of the Fourier Series

n

tjnnectf 0)(

n

tjnnectf 0)(

dtetfT

cT

T

tjnn

2/

2/

0)(1 dtetf

Tc

T

T

tjnn

2/

2/

0)(1

)(2

1

)(2

12

00

nnn

nnn

jbac

jbac

ac

)(

2

1

)(2

12

00

nnn

nnn

jbac

jbac

ac

If f(t) is real,*nn cc

nn jnnn

jnn ecccecc

|| ,|| *

22

2

1|||| nnnn bacc

n

nn a

b1tan

,3,2,1 n

00 2

1ac

Complex Frequency Spectra

nn jnnn

jnn ecccecc

|| ,|| *

22

2

1|||| nnnn bacc

n

nn a

b1tan ,3,2,1 n

00 2

1ac

|cn|

amplitudespectrum

n

phasespectrum

Example

2

T

2

T TT

2

d

t

f(t)

A

2

d

dteT

Ac

d

d

tjnn

2/

2/

0

2/

2/0

01

d

d

tjnejnT

A

2/

0

2/

0

0011 djndjn ejn

ejnT

A

)2/sin2(1

00

dnjjnT

A

2/sin1

002

1dn

nT

A

TdnT

dn

T

Adsin

TdnT

dn

T

Adcn

sin

82

5

1

T ,

4

1 ,

20

1

0

T

dTd

Example

40 80 120-40 0-120 -80

A/5

50 100 150-50-100-150

TdnT

dn

T

Adcn

sin

42

10

1

T ,

2

1 ,

20

1

0

T

dTd

Example

40 80 120-40 0-120 -80

A/10

100 200 300-100-200-300

Fourier Series & Fourier Transform

Lecture 15-16: 28-Aug-12

Dr. P P Das

Example

dteT

Ac

d tjnn

0

0

d

tjnejnT

A

00

01

00

110

jne

jnT

A djn

)1(1

0

0

djnejnT

A

2/0

sindjne

TdnT

dn

T

Ad

TT d

t

f(t)

A

0

)(1 2/2/2/

0

000 djndjndjn eeejnT

A

Fourier Series

Analysis ofPeriodic Waveforms

Waveform Symmetry

Even Functions

Odd Functions

)()( tftf

)()( tftf

Decomposition

Any function f(t) can be expressed as the sum of an even function fe(t) and an odd function fo(t).

)()()( tftftf oe

)]()([)( 21 tftftfe

)]()([)( 21 tftftfo

Even Part

Odd Part

Example

00

0)(

t

tetf

t

Even Part

Odd Part

0

0)(

21

21

te

tetf

t

t

e

0

0)(

21

21

te

tetf

t

t

o

Half-Wave Symmetry

)()( Ttftf and 2/)( Ttftf

TT/2T/2

Quarter-Wave Symmetry

Even Quarter-Wave Symmetry

TT/2T/2

Odd Quarter-Wave Symmetry

T

T/2T/2

Hidden Symmetry

The following is a asymmetry periodic function:

Adding a constant to get symmetry property.

A

TT

A/2

A/2

TT

Fourier Coefficients of Symmetrical Waveforms

The use of symmetry properties simplifies the calculation of Fourier coefficients.– Even Functions– Odd Functions– Half-Wave– Even Quarter-Wave– Odd Quarter-Wave– Hidden

Fourier Coefficients of Even Functions

)()( tftf

tnaa

tfn

n 01

0 cos2

)(

2/

0 0 )cos()(4 T

n dttntfT

a

Fourier Coefficients of Odd Functions

)()( tftf

tnbtfn

n 01

sin)(

2/

0 0 )sin()(4 T

n dttntfT

b

Fourier Coefficients for Half-Wave Symmetry

)()( Ttftf and 2/)( Ttftf

TT/2T/2

The Fourier series contains only odd harmonics.The Fourier series contains only odd harmonics.

Fourier Coefficients for Half-Wave Symmetry

)()( Ttftf and 2/)( Ttftf

)sincos()(1

00

n

nn tnbtnatf

odd for )cos()(4

even for 02/

0 0 ndttntfT

na T

n

odd for )sin()(4

even for 02/

0 0 ndttntfT

nb T

n

Fourier Coefficients forEven Quarter-Wave Symmetry

TT/2T/2

])12cos[()( 01

12 tnatfn

n

4/

0 012 ])12cos[()(8 T

n dttntfT

a

Fourier Coefficients forOdd Quarter-Wave Symmetry

])12sin[()( 01

12 tnbtfn

n

4/

0 012 ])12sin[()(8 T

n dttntfT

b

T

T/2T/2

ExampleEven Quarter-Wave Symmetry

4/

0 012 ])12cos[()(8 T

n dttntfT

a 4/

0 0 ])12cos[(8 T

dttnT

4/

0

00

])12sin[()12(

8T

tnTn

)12(

4)1( 1

nn

TT/2T/2

1

1T T/4T/4

ExampleEven Quarter-Wave Symmetry

4/

0 012 ])12cos[()(8 T

n dttntfT

a 4/

0 0 ])12cos[(8 T

dttnT

4/

0

00

])12sin[()12(

8T

tnTn

)12(

4)1( 1

nn

TT/2T/2

1

1T T/4T/4

ttttf 000 5cos

5

13cos

3

1cos

4)(

ttttf 000 5cos

5

13cos

3

1cos

4)(

Example

TT/2T/2

1

1

T T/4T/4

Odd Quarter-Wave Symmetry

4/

0 012 ])12sin[()(8 T

n dttntfT

b 4/

0 0 ])12sin[(8 T

dttnT

4/

0

00

])12cos[()12(

8T

tnTn

)12(

4

n

Example

TT/2T/2

1

1

T T/4T/4

Odd Quarter-Wave Symmetry

4/

0 012 ])12sin[()(8 T

n dttntfT

b 4/

0 0 ])12sin[(8 T

dttnT

4/

0

00

])12cos[()12(

8T

tnTn

)12(

4

n

ttttf 000 5sin

5

13sin

3

1sin

4)(

ttttf 000 5sin

5

13sin

3

1sin

4)(

Fourier Series

Half-Range Expansions

Non-Periodic Function Representation

A non-periodic function f(t) defined over (0, ) can be expanded into a Fourier series which is defined only in the interval (0, ).

Without Considering Symmetry

A non-periodic function f(t) defined over (0, ) can be expanded into a Fourier series which is defined only in the interval (0, ).

T

Expansion Into Even Symmetry

A non-periodic function f(t) defined over (0, ) can be expanded into a Fourier series which is defined only in the interval (0, ).

T=2

Expansion Into Odd Symmetry

A non-periodic function f(t) defined over (0, ) can be expanded into a Fourier series which is defined only in the interval (0, ).

T=2

Expansion Into Half-Wave Symmetry

A non-periodic function f(t) defined over (0, ) can be expanded into a Fourier series which is defined only in the interval (0, ).

T=2

Expansion Into Even Quarter-Wave Symmetry

A non-periodic function f(t) defined over (0, ) can be expanded into a Fourier series which is defined only in the interval (0, ).

T/2=2T=4

Expansion Into Odd Quarter-Wave Symmetry

A non-periodic function f(t) defined over (0, ) can be expanded into a Fourier series which is defined only in the interval (0, ).

T/2=2 T=4

Fourier Series

Least Mean-Square Error Approximation

Approximation a function

Use

k

nnnk tnbtna

atS

100

0 sincos2

)(

to represent f(t) on interval T/2 < t < T/2.

Define )()()( tStft kk

2/

2/

2)]([1 T

T kk dttT

E Mean-Square Error

Approximation a function

Show that using Sk(t) to represent f(t) has least mean-square property.

2/

2/

2)]([1 T

T kk dttT

E

2/

2/

2

100

0 sincos2

)(1 T

T

k

nnn dttnbtna

atf

T

Proven by setting Ek/ai = 0 and Ek/bi = 0.

Approximation a function

2/

2/

2)]([1 T

T kk dttT

E

2/

2/

2

100

0 sincos2

)(1 T

T

k

nnn dttnbtna

atf

T

0)(1

2

2/

2/

0

0

T

T

k dttfT

a

a

E 0)(1

2

2/

2/

0

0

T

T

k dttfT

a

a

E0cos)(

2 2/

2/ 0

T

Tnn

k tdtntfT

aa

E 0cos)(2 2/

2/ 0

T

Tnn

k tdtntfT

aa

E

0sin)(2 2/

2/ 0

T

Tnn

k tdtntfT

bb

E 0sin)(2 2/

2/ 0

T

Tnn

k tdtntfT

bb

E

Mean-Square Error

2/

2/

2)]([1 T

T kk dttT

E

2/

2/

2

100

0 sincos2

)(1 T

T

k

nnn dttnbtna

atf

T

2/

2/1

22202 )(

2

1

4)]([

1 T

T

k

nnnk ba

adttf

TE

2/

2/1

22202 )(

2

1

4)]([

1 T

T

k

nnnk ba

adttf

TE

Mean-Square Error

2/

2/

2)]([1 T

T kk dttT

E

2/

2/

2

100

0 sincos2

)(1 T

T

k

nnn dttnbtna

atf

T

2/

2/1

22202 )(

2

1

4)]([

1 T

T

k

nnn ba

adttf

T

2/

2/1

22202 )(

2

1

4)]([

1 T

T

k

nnn ba

adttf

T

Mean-Square Error

2/

2/

2)]([1 T

T kk dttT

E

2/

2/

2

100

0 sincos2

)(1 T

T

k

nnn dttnbtna

atf

T

2/

2/1

22202 )(

2

1

4)]([

1 T

Tn

nn baa

dttfT

2/

2/1

22202 )(

2

1

4)]([

1 T

Tn

nn baa

dttfT

Fourier Series

Impulse Train

Dirac Delta Function

0

00)(

t

tt and 1)(

dtt

0 t

Also called unit impulse function, but not actually a function in truesense.

Sifting Property

)0()()(

dttt )0()()(

dttt

)0()()0()0()()()(

dttdttdttt

(t): Test Function

Impulse Train

0 tT 2T 3TT2T3T

n

T nTtt )()(

n

T nTtt )()(

Fourier Series of the Impulse Train

n

T nTtt )()(

n

T nTtt )()(T

dttT

aT

T T

2)(

2 2/

2/0

Tdttnt

Ta

T

T Tn

2)cos()(

2 2/

2/ 0

0)sin()(2 2/

2/ 0 dttntT

bT

T Tn

n

T tnTT

t 0cos21

)(

n

T tnTT

t 0cos21

)(

)0()()(

dttt )0()()(

dttt

Complex FormFourier Series of the Impulse Train

Tdtt

T

ac

T

T T

1)(

1

2

2/

2/

00

Tdtet

Tc

T

T

tjnTn

1)(

1 2/

2/

0

n

tjnT e

Tt 0

1)(

n

tjnT e

Tt 0

1)(

n

T nTtt )()(

n

T nTtt )()(

)0()()(

dttt )0()()(

dttt

Fourier Transform

Impulse Train

Fourier Series / Transform

Fourier Series deals with discrete variable (harmonics of )

Fourier Transform deals with continuous frequency

00 2 f

Fourier Series / Transform

dtetfFtf tj 2)()()}({

deFFtf tj21 )()}({)(

n

tfjnnectf 02)(

2/

2/

2 0)(1 T

T

tfjnn dtetf

Tc

00 2 f

Example

ExampledtetfF tj

2)()(

2/

2/

2

2

1W

W

tjej

A

WjWj eej

A

2

W

WAW

sin

Example

Example: Unit Impulse

dtetF tj

2)()(

1002 ee j

dtettF tj

20 )()(

02 tje

Fourier Series of the Impulse Train

Tdtets

Tc

T

T

tjTn

Tn

1)(

1 2/

2/

2

n

tT

nj

T eT

ts21

)(

n

tT

nj

T eT

ts21

)(

nT Tntts )()(

nT Tntts )()(

Fourier Transform of the Impulse Train

n

n

tT

nj

T

T

n

T

eT

tsS

)(1

}1

{)}({)(2

n

n

tT

nj

T

T

n

T

eT

tsS

)(1

}1

{)}({)(2

)(}{2

T

ne

tT

nj

)(}{2

T

ne

tT

nj

Fourier Transform

Convolution

Convolution

A mathematical operator which computes the “amount of overlap” between two functions. Can be thought of as a general moving average

Discrete domain:

Continuous domain:

Discrete domain

Basic steps1. Flip (reverse) one of the digital functions.

2. Shift it along the time axis by one sample.

3. Multiply the corresponding values of the two digital functions.

4. Summate the products from step 3 to get one point of the digital convolution.

5. Repeat steps 1-4 to obtain the digital convolution at all times that the functions overlap.

Example

Continuous domain example

Continuous domain example

Convolution Theorem

)()()}()({ GFtgtf

)()()}()({ tgtfGF

Fourier Transform

Sampling

Convolution Theorem

Sampling

n

T

Tnttf

tstf

tf

)()(

)()(

)(~

nT Tntts )()(

nT Tntts )()(

Sampling & FT

n

T

T

nF

T

SF

tstftf

F

)(1

)()(

)}()({)}(~

{

)(~

n T

n

TS )(

1)(

n T

n

TS )(

1)(

Sampling & FT

Nyquist Rate

Reconstruction Filter

Aliasing

Discrete & Fast Fourier Transform

Lecture 17: 03-Sep-12

Dr. P P Das

Sampling & FT

n

T

T

nF

T

SF

tstftf

F

)(1

)()(

)}()({)}(~

{

)(~

n T

n

TS )(

1)(

n T

n

TS )(

1)(

Sampling & FT

DFT: How to compute FT?

Tnj

nn

n

tj

tj

n

tj

ef

dteTnttf

dteTnttf

dtetfF

2

2

2

2

)()(

)()(

)(~

)(~

)(

)()(

Tnf

dtTnttffn

Discrete Fourier Transform

Tnj

nnefF

2)(~ )( Tnffn

1,2,1,0,TM

m

T1/ to0 period

over the )(~

of samples spacedequally Obtain

DFTfor the basis theis period one Sampling

period oneover zedcharacteri be toneed )(~

T

1 period withperiodic infinitely and continuous is )(

~

function discretea is

Mm

FM

F

F

fn

Discrete Fourier Transform

Tnj

nnefF

2)(~

)( Tnffn

1,2,1,0,TM

m

Mm

1,2,1,0,

DFT1

0

/2

MmefFM

n

Mmnjnm

1,2,1,0,1

IDFT1

0

/2

MneFM

fM

m

Mmnjmn

DFT: Notations

D-2 in variables transform Discrete:,

D-2 in variablesfunction Discrete:,

D-2 in variables transformContinuous :,

D-2 in variablesfunction Continuous :,

D-1 in variable)(frequency Continuous :

D-1 in variable(time) Continuous :

vu

yx

zt

t

DFT Pair

1,2,1,0,)()(

DFT1

0

/2

MuexfuFM

x

Muxj

1,2,1,0,)(1

)(

IDFT1

0

/2

MxeuFM

xfM

u

Muxj

Discrete Fourier Transform Pair

1

0

/2)()(M

x

MuxjexfuF

1

0

/2)(1

)(M

u

MuxjeuFM

xf

uniformly takensamples discrete

ofset finiteany toapplicable Is

/1 Interval Frequencyoft Independen

Interval Sampling oft Independen

:pair DFT

T

T

DFT: Periodicity

,2,1,0),()(

DFT

kkMuFuF

,2,1,0),()(

IDFT

kkMxfxf

Sampling & Frequency Intervals

T interval sampling theon depends DFTthe

by spanned sfrequencie of Range

sampled. is )( over which duration theon

depends frequency in Resolution

1;

11;

tfT

uT

uMTTM

uTMT

DFT: Example

4)3(;4)2(;2)1(;1)0( ffff

DFT: Example

)23()3();01()2(

23)40()04()20(1

4421

)()1(

114421

)3()2()1()0()()0(

2/32/0

3

0

4/)1(2

3

0

jFjF

jjjj

eeee

exfF

ffffxfF

jjj

x

xj

x

DFT: Example

144

12312311

4

1

)(4

1

)(4

1)0(

3

0

3

0

)0(2

jj

uF

euFf

u

u

uj

M. Wu: ENEE631 Digital Image Processing (Spring'09)

1-D FFT in Matrix/Vector { z(n) } { Z(k) }

n, k = 0, 1, …, N-1, WN = exp{ - j2 / N } ~ complex conjugate of primitive Nth root of unity

)1(

)1(

)0(

)1(

)2(

)1(

)0(

1

1

1

1 1 11

)1(

)2(

)1(

)0(

110

/)1(2/)1(22/)1(2

/)1(22/42/22

/)1(2/22/2

2 NZ

Z

Z

aaa

NZ

Z

Z

Z

eee

eee

eee

Nz

z

z

z

N

NNjNNjNNj

NNjNjNj

NNjNjNj

1

0

1

0

)(1

)(

)(1

)(

N

k

nkN

N

n

nkN

WkZN

nz

WnzN

kZ

UM

CP

EN

EE

631

Sli

des

(cre

ated

by

M.W

u ©

200

1/20

04)

z

a

a

a

Nz

z

z

z

eee

eee

eee

NZ

Z

Z

Z

TN

T

T

NNjNNjNNj

NNjNjNj

NNjNjNj

*1

*1

*0

/)1(2/)1(22/)1(2

/)1(22/42/22

/)1(2/22/2

)1(

)2(

)1(

)0(

1

1

1

1 1 11

)1(

)2(

)1(

)0(

2

DFT: Example

4)3(;4)2(;2)1(;1)0( ffff

4

4

2

1

)3(

)2(

)1(

)0(

f

f

f

f

DFT: By Matrix

)3(

)2(

)1(

)0(

1

1

1

11 11

)3(

)2(

)1(

)0(

2/932/3

32

2/32/

f

f

f

f

eee

eee

eee

F

F

F

F

jjj

jjj

jjj

)3(

)2(

)1(

)0(

00101

010j1 011

00j1 01

11 11

)3(

)2(

)1(

)0(

f

f

f

f

jjj

jj

jj

F

F

F

F

DFT: By Matrix

j

j

jj

jj

jj

jj

f

f

f

f

jjj

jj

jj

F

F

F

F

23

1

23

11

4421

4421

4421

4421

4

4

2

1

11

11 11

1 1

11 11

)3(

)2(

)1(

0(

00101

010j1 011

00j1 01

11 11

)3(

)2(

)1(

)0(

4

4

2

1

)3(

)2(

)1(

)0(

f

f

f

f

IDFT: By Matrix

)3(

)2(

)1(

)0(

1

1

1

11 11

)3(

)2(

)1(

)0(

4

2/932/3

32

2/32/

F

F

F

F

eee

eee

eee

f

f

f

f

jjj

jjj

jjj

)3(

)2(

)1(

)0(

00101

010j1 011

00j1 01

11 11

)3(

)2(

)1(

)0(

4

F

F

F

F

jjj

jj

jj

f

f

f

f

IDFT: By Matrix

16

16

8

4

2312311

2312311

2312311

2312311

23

1

23

11

11

11 11

1 1

11 11

)3(

)2(

)1(

0(

00101

010j1 011

00j1 01

11 11

)3(

)2(

)1(

)0(

4

jj

jj

jj

jj

j

j

jj

jj

F

F

F

F

jjj

jj

jj

f

f

f

f

j

j

F

F

F

F

23

1

23

11

)3(

)2(

)1(

)0(

Inverse Discrete Fourier Transform

1,2,1,0,)()( :DFT1

0

/2

MuexfuFM

x

Muxj

))((1

)(

))((1

)(1

)(

:conjugatecomplex Taking

1,2,1,0,)(1

)( :IDFT

**

*1

0

/2**

1

0

/2

uFDFTM

xf

uFDFTM

euFM

xf

MxeuFM

xf

M

u

Muxj

M

u

Muxj

DFT Complexity

point DFT an in additions

complex ofNumber :)(

point DFT an in tionsmultiplica

complex ofNumber :)(

M

MAdd

M

MMult

DFT / IDFT Complexity

)()1()(

)()(

computed.-pre are that Assume

1,2,1,0,)()(

2

22

/2

1

0

/2

MOMMMAdd

MOMMMult

e

MuexfuF

Muxj

M

x

Muxj

Fast Fourier Transform

KMeW

MuWxfuF

nMjM

M

x

uxM

22

1,2,1,0,)()(

/2

1

0

uM

MuM

uM

MuM

uxK

uxK WWWWWW 222

2 ;;

:Properties

1,2,1,0,)()(1

0

/2

MuexfuFM

x

Muxj

Fast Fourier Transform

1

02

1

0

1

0

)12(2

1

0

)2(2

12

0

)12()2(

)12()2(

)()(

K

x

uK

uxK

K

x

uxK

K

x

xuK

K

x

xuK

K

x

uxM

WWxfWxf

WxfWxf

WxfuF

KMeW nMjM 22/2 ux

Kux

K WW 22

Fast Fourier Transform

uKoddeven

uKoddeven

K

x

uxKodd

K

x

uxKeven

WuFuFKuF

WuFuFuF

WxfuF

WxfuF

2

2

1

0

1

0

)()()(

)()()(

)12()(

)2()(

KM

eWn

MjM

22

/2

uM

MuM

uM

MuM WWWW 22;

FFT Complexity

point FFT 2 an in additions

complex ofNumber :)()(

point FFT 2 an in tionsmultiplica

complex ofNumber :)()(

n

n

M

nAddMAdd

M

nMultMMult

FFT

1

0

1

0

)12()(

)2()(

K

x

uxKodd

K

x

uxKeven

WxfuF

WxfuF

02

02

1

)0()0()1(

)0()0()0(

)1()0(

)0()0(

1;22;1

WFFF

WFFF

fF

fF

KMn

oddeven

oddeven

odd

even

uKoddeven

uKoddeven

WuFuFKuF

WuFuFuF

2

2

)()()(

)()()(

2)1(

1)1(

Add

Mult

FFT

1

0

1

0

)12()(

)2()(

K

x

uxKodd

K

x

uxKeven

WxfuF

WxfuF

14

04

14

04

2

)1()1()21()3(

)0()0()20()2(

)1()1()1(

)0()0()0(

)3(&)1( from point FFT-2)(

)2(&)0( from point FFT-2)(

2;42;2

WFFFF

WFFFF

WFFF

WFFF

ffuF

ffuF

KMn

oddeven

oddeven

oddeven

oddeven

odd

even

uKoddeven

uKoddeven

WuFuFKuF

WuFuFuF

2

2

)()()(

)()()(

4)1(2)2(

2)1(2)2(

AddAdd

MultMult

FFT Complexity

)log(log)()(

)log(log2

1)()(

0;;0

1;2)1(2)(

0;;0

1;2)1(2)(

22

22

1

MMOMMnAddMAdd

MMOMMnMultMMult

n

nnAddnAdd

n

nnMultnMult

n

n

2-D DFT & Images

Lecture 18-19: 04-Sep-12

Dr. P P Das

Kinect Demo in Lecture 19

2-D Impulse

0

0,00),(

zt

ztzt

and

1),(

dzdtzt

2-D Sifting Property

)0,0(),(),( fdzdtztztf

),(),(),( 0000 ztfdzdtzzttztf

2-D Discrete Impulse

01

0,00),(

yx

yxyx

2-D Discrete Sifting Property

)0,0(),(),( fyxyxfx y

),(),(),( 0000 yxfyyxxyxfx y

2-D Impulse

2-D Continuous Fourier Transform Pair

dzdteztfF ztj )(2),(),(

ddeFztf ztj )(2),(),(

2-D FT: Example

2-D FT: Example

)(

)sin(

)(

)sin(

),(),(

2/

2/

2/

2/

)(2

)(2

Z

Z

T

TATZ

dzdtAe

dzdteztfF

T

T

Z

Z

ztj

ztj

2-D Sampling

maxmax

maxmax

2

1

2

1

and for 0),(

limited-band is ),(

ZT

F

ztf

m nZT ZnzTmtzts ),(),(

2-D Sampling

Frequency Aliasing

Signals sampled below Nyquist rate (under-sampled) have overlapped periods.

In aliasing, high frequency components masquerade as low frequency components in sampled function. Hence, ‘aliasing’ or ‘false identity’.

Aliasing

Aliasing

Frequency Aliasing

Most band-limited signals have infinite frequency components in sampled signal due to finite duration of sampling.

n

TnTntcTnf

tfth

HFFtf

/)(sin)(

)(~

)(

)}()(~

{)}({)( 11

otherwise

Ttth

0

01)(

Frequency Aliasing

Aliasing is inevitable while working with sampled records of finite length

Aliasing can be reduced by smoothing the input function to attenuate higher frequencies (defocusing images)

This is known as ‘anti-aliasing’ and has to be done before the function is sampled

Aliasing & Interpolation

n

TnTntcTnftf /)(sin)()(

,function sinc of sum of ioninterpolat

0)(sin and 1)0(sin as ),()(

Tkt

mccTktTkftf

Aliasing in Images

Spatial Aliasing– Due to under-sampling in space

Temporal Aliasing– Due to slow time interval between frames in video– Wagon-Wheel Effect

Aliasing in Spectrum

Sampling: 96X96 Pixels

Interpolation Techniques

Nearest Neighbor Interpolation Bi-Linear Interpolation Bi-Cubic Interpolation

Interpolation & Re-sampling

Zooming – Over-sampling– Pixel Replication / Row-Column Duplication

Shrinking – Under-sampling – Row-Column Deletion– Blur slightly before shrinking

Aliasing Artifact: Jaggies

Block-like image component Common in images with string edge content

Aliasing Artifact: Moiré Patterns

Beat patterns produced between two gratings of approximately equal spacing

Common in images with periodic or nearly periodic content

2-D DFT Properties

Lecture 20: 10-Sep-12

Dr. P P Das

2-D DFT Pair

1

0

1

0

)//(2),(),(

DFTM

x

N

y

NvyMuxjeyxfvuF

1

0

1

0

)//(2),(1

)(

IDFTM

u

N

v

NvyMuxjevuFMN

xf

2-D: Separability

1

0

/2

1

0

/2

1

0

1

0

/2/2

1

0

1

0

)//(2

),(),( where

),(

),(

),(),(

N

y

Nvyj

M

x

Muxj

M

x

N

y

NvyjMuxj

M

x

N

y

NvyMuxj

eyxfvxF

evxF

eyxfe

eyxfvuF

Properties of 2-D DFT:Spatial & Frequency Intervals

ZNv

TMu

1

1

Properties of 2-D DFT:Translation & Rotation

),(),(

sincossincos

:scoordinatepolar In

),(),(

),(),(

00

)//(200

)//(200

00

00

Frf

vuryrx

evuFyyxxf

eyxfvuuFNvyMuxj

NyvMxuj

Properties of 2-D DFT:Periodicity

),(

),(

),(

),(

21

2

1

NkvMkuF

NkvuF

vMkuF

vuF

),(

),(

),(

),(

21

2

1

NkyMkxf

Nkyxf

yMkxf

yxf

Periodicity: Centering the Transform

)2/,2/()1(,

)2/()1(

2/For

)(

D-1 In

0

0/)(2 0

NvMuFy)f(x

MuFf(x)

Mu

uuFf(x)e

yx

x

Mxujπ

Properties of 2-D DFT:Symmetry

),(2

),(),(),(

),(2

),(),(),(

),(),(),(

yxwyxwyxw

yxw

yxwyxwyxw

yxw

yxwyxwyxw

oo

ee

oe

Properties of 2-D DFT:Symmetry

),(),(

),(),(

0),(),(1

0

1

0

yNxMwyxw

yNxMwyxw

yxwyxw

oo

ee

o

M

x

N

ye

Symmetry

),(),(

symmetric-anti conjugate is FT,),(imaginary For

),(),(

symmetric conjugate is FT,),( realFor

*

*

vuFvuF

yxf

vuFvuF

yxf

Fourier Spectrum & Phase Angle

),(),(

),(),( :SpectrumPower

),(

),(arctan :Angle Phase

),(),(),( :SpectrumFourier

),(),(),(),(

22

2

),(

2/122

),(

vuIvuR

vuFvuP

vuR

vuIe

vuIvuRvuF

evuFvujIvuRvuF

vuj

vuj

Fourier Spectrum & Phase Angle

),(),()0,0(

),(),(

symmetry odd has Angle Phase

),(),(

symmetry even has Spectrum

symmetric conjugate is FT,),( realFor

1

0

1

0

yxfMNyxfF

vuvu

vuFvuF

yxf

M

x

N

y

yxyxf )1)(,( ),(log1 vuF

2-D Convolution

)()(),(),(

)()(),(),(

),(),(

),(),(1

0

1

0

HFyxhyxf

HFyxhyxf

nymxhnmf

yxhyxfM

m

N

n

Effect of DFT on Convolution

399

0

)()()()(m

mxhmfxhxf

Wraparound Error needs Zero Padding

2-D: Separability

1

0

/2

1

0

1

0

/2/2

1

0

1

0

)//(2

),(

),(

),(),(

DFT

M

x

Muxj

M

x

N

y

NvyjMuxj

M

x

N

y

NvyMuxj

evxF

eyxfe

eyxfvuF

Thank you