Fourier Seriesksrao/pdf/ip-19/rcg-ch4a.pdf · The Mathematic Formulation Any function that...

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:

3cos)(

tttf Find its period.

)()( Ttftf )(4

1cos)(

3cos TtTt

Fact: )2cos(cos m

24T smallest T

Example:

tttf 21 coscos)( Find its period.

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

must be a rational number

Example:

tttf )10cos(10cos)(

Is this function a periodic one?

1 not a rational number

Fourier Series

Introduction

Decompose a periodic input signal into primitive periodic components.

A periodic sequenceA periodic sequence

T 2T 3T

Synthesis

DC Part Even Part Odd Part

T is a period of all the above signals

)sin()cos(2

0 tnbtnaa

Let 0=2/T.

Orthogonal Functions

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

nmdttt

Orthogonal set of Sinusoidal Functions

Define 0=2/T.

0 ,0)cos(2/

2/ 0 mdttmT

T0 ,0)sin(

2/ 0 mdttmT

nmdttntm

0)cos()cos(

nmdttntm

0)sin()sin(

nmdttntmT

T and allfor ,0)cos()sin(

We now prove this one

dttntmT

2/ 00 )cos()cos(

)]cos()[cos(2

1coscos

dttnmdttnmT

2/ 0 ])cos[(2

1])cos[(

])sin[()(

1])sin[(

])sin[(2)(

1])sin[(2

dttntmT

2/ 00 )cos()cos(

)]cos()[cos(2

1coscos

2/ 02 )(cos

]2sin4

]2cos1[2

2/ 0 ]2cos1[2

nmdttntm

0)cos()cos(

Orthogonal set of Sinusoidal Functions

Define 0=2/T.

0 ,0)cos(2/

2/ 0 mdttmT

T0 ,0)sin(

2/ 0 mdttmT

nmdttntm

0)cos()cos(

nmdttntm

0)sin()sin(

nmdttntmT

,3sin,2sin,sin

,3cos,2cos,cos

,3sin,2sin,sin

,3cos,2cos,cos

an orthogonal set.an orthogonal set.

Decomposition

,2,1 cos)(2

ntdtntfT

,2,1 sin)(2

ntdtntfT

)sin()cos(2

0 tnbtnaa

ProofUse the following facts:

0 ,0)cos(2/

2/ 0 mdttmT

T0 ,0)sin(

2/ 0 mdttmT

nmdttntm

0)cos()cos(

nmdttntm

0)sin()sin(

nmdttntmT

Example (Square Wave)

,2,1 0sin1

ntdtan

,6,4,20

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

nntdtbn

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

,2,1 0sin1

ntdtan

,6,4,20

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

nntdtbn

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

ttttf 5sin

,2,1 0sin1

ntdtan

,6,4,20

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

nntdtbn

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

ttttf 5sin

Harmonics

DC Part Even Part Odd Part

T is a period of all the above signals

)sin()cos(2

0 tnbtnaa

Harmonics

tnbtnaa

0 sincos2

22 00Define , called the fundamental angular frequency.

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

sincos2

Harmonics

sincos2

)sincos(2 1

0 tbtaa

220 sincos2 n

220 sinsincoscos2 n

nnnnnn ttbaa

)cos(1

nn tCC

Amplitudes and Phase Angles

)cos()(1

nn tCCtf

22nnn baC

harmonic amplitude phase angle

Fourier Series

Complex Form of the Fourier Series

Complex Exponentials

tnjtne tjn00 sincos0

tjntjn eetn 00

1cos 0

tnjtne tjn00 sincos0

tjntjntjntjn eej

tn 0000

1sin 0

tnbtnaa

0 sincos2

tjntjn

nn eeb

0 00 )(2

tjnnn ejbaejba

tjnn ececc

tjnn ececctf

tjnn ececc

tjnnec 0

1nnn jbac

2/ 0 sin)(cos)(1 T

Ttdtntfjtdtntf

2/ 00 )sin)(cos(1 T

Tdttnjtntf

0)(1 T

tjn dtetfT

tjnnnn dtetf

Tjbac )(

tjnnectf 0)(

dtetfT

0)(1 dtetf

If f(t) is real,*nn cc

nn jnnn

jnn ecccecc

|| ,|| *

1|||| nnnn bacc

,3,2,1 n

Complex Frequency Spectra

nn jnnn

jnn ecccecc

|| ,|| *

1|||| nnnn bacc

b1tan ,3,2,1 n

amplitudespectrum

phasespectrum

Example

tjnejnT

0011 djndjn ejn

)2/sin2(1

dnjjnT

2/sin1

Example

40 80 120-40 0-120 -80

50 100 150-50-100-150

Example

40 80 120-40 0-120 -80

100 200 300-100-200-300

Fourier Series & Fourier Transform

Lecture 15-16: 28-Aug-12

Dr. P P Das

Example

d tjnn

tjnejnT

djnejnT

sindjne

)(1 2/2/2/

000 djndjndjn eeejnT

Fourier Series

Analysis ofPeriodic Waveforms

Waveform Symmetry

Even Functions

Odd Functions

)()( 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

Even Part

Odd Part

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/2T/2

Hidden Symmetry

The following is a asymmetry periodic function:

Adding a constant to get symmetry property.

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

0 cos2

0 0 )cos()(4 T

n dttntfT

Fourier Coefficients of Odd Functions

)()( tftf

tnbtfn

0 0 )sin()(4 T

n dttntfT

Fourier Coefficients for Half-Wave Symmetry

TT/2T/2

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

Fourier Coefficients for Half-Wave Symmetry

)sincos()(1

nn tnbtnatf

odd for )cos()(4

even for 02/

0 0 ndttntfT

odd for )sin()(4

even for 02/

0 0 ndttntfT

Fourier Coefficients forEven Quarter-Wave Symmetry

TT/2T/2

])12cos[()( 01

12 tnatfn

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

n dttntfT

Fourier Coefficients forOdd Quarter-Wave Symmetry

])12sin[()( 01

12 tnbtfn

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

n dttntfT

T/2T/2

ExampleEven Quarter-Wave Symmetry

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

n dttntfT

0 0 ])12cos[(8 T

])12sin[()12(

4)1( 1

TT/2T/2

1T T/4T/4

ExampleEven Quarter-Wave Symmetry

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

n dttntfT

0 0 ])12cos[(8 T

])12sin[()12(

4)1( 1

TT/2T/2

1T T/4T/4

ttttf 000 5cos

Example

TT/2T/2

T T/4T/4

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

n dttntfT

0 0 ])12sin[(8 T

])12cos[()12(

Example

TT/2T/2

T T/4T/4

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

n dttntfT

0 0 ])12sin[(8 T

])12cos[()12(

ttttf 000 5sin

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

Expansion Into Even Symmetry

Expansion Into Odd Symmetry

Expansion Into Half-Wave Symmetry

Expansion Into Even Quarter-Wave Symmetry

T/2=2T=4

Expansion Into Odd Quarter-Wave Symmetry

T/2=2 T=4

Fourier Series

Least Mean-Square Error Approximation

Approximation a function

nnnk tnbtna

0 sincos2

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

Define )()()( tStft kk

2)]([1 T

T kk dttT

E Mean-Square Error

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

2)]([1 T

T kk dttT

0 sincos2

nnn dttnbtna

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

2)]([1 T

T kk dttT

0 sincos2

nnn dttnbtna

k dttfT

E 0)(1

k dttfT

E0cos)(

k tdtntfT

E 0cos)(2 2/

k tdtntfT

0sin)(2 2/

k tdtntfT

E 0sin)(2 2/

k tdtntfT

Mean-Square Error

2)]([1 T

T kk dttT

0 sincos2

nnn dttnbtna

22202 )(

nnnk ba

22202 )(

nnnk ba

Mean-Square Error

2)]([1 T

T kk dttT

0 sincos2

nnn dttnbtna

22202 )(

nnn ba

22202 )(

nnn ba

Mean-Square Error

2)]([1 T

T kk dttT

0 sincos2

nnn dttnbtna

22202 )(

nn baa

22202 )(

nn baa

Fourier Series

Impulse Train

Dirac Delta Function

tt and 1)(

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

Sifting Property

)0()()(

dttt )0()()(

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

dttdttdttt

(t): Test Function

Impulse Train

0 tT 2T 3TT2T3T

T nTtt )()(

Fourier Series of the Impulse Train

T nTtt )()(

T nTtt )()(T

Tdttnt

2)cos()(

0)sin()(2 2/

2/ 0 dttntT

T tnTT

t 0cos21

T tnTT

t 0cos21

)0()()(

dttt )0()()(

Complex FormFourier Series of the Impulse Train

tjnT e

T nTtt )()(

)0()()(

dttt )0()()(

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 )()}({)(

tfjnnectf 02)(

2 0)(1 T

tfjnn dtetf

00 2 f

Example

ExampledtetfF tj

WjWj eej

Example

Example: Unit Impulse

dtetF tj

1002 ee j

dtettF tj

20 )()(

02 tje

Fourier Series of the Impulse Train

Tdtets

nT Tntts )()(

Fourier Transform of the Impulse Train

{)}({)(2

Fourier Transform

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

Convolution Theorem

)()()}()({ GFtgtf

)()()}()({ tgtfGF

Fourier Transform

Sampling

Convolution Theorem

Sampling

nT Tntts )()(

Sampling & FT

tstftf

)}()({)}(~

Sampling & FT

Nyquist Rate

Reconstruction Filter

Aliasing

Discrete & Fast Fourier Transform

Lecture 17: 03-Sep-12

Dr. P P Das

Sampling & FT

tstftf

)}()({)}(~

Sampling & FT

DFT: How to compute FT?

dteTnttf

dtetfF

dtTnttffn

Discrete Fourier Transform

2)(~ )( Tnffn

1,2,1,0,TM

T1/ to0 period

over the )(~

of samples spacedequally Obtain

DFTfor the basis theis period one Sampling

period oneover zedcharacteri be toneed )(~

1 period withperiodic infinitely and continuous is )(

function discretea is

Discrete Fourier Transform

)( Tnffn

1,2,1,0,TM

1,2,1,0,

MmefFM

Mmnjnm

1,2,1,0,1

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 :

DFT Pair

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

MuexfuFM

1,2,1,0,)(1

MxeuFM

Discrete Fourier Transform Pair

/2)()(M

MuxjexfuF

MuxjeuFM

uniformly takensamples discrete

ofset finiteany toapplicable Is

/1 Interval Frequencyoft Independen

Interval Sampling oft Independen

:pair DFT

DFT: Periodicity

,2,1,0),()(

kkMuFuF

,2,1,0),()(

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

DFT: Example

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

DFT: Example

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

23)40()04()20(1

114421

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

2/32/0

4/)1(2

ffffxfF

DFT: Example

12312311

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 11

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

/)1(22/42/22

/)1(2/22/2

NNjNNjNNj

NNjNjNj

NNjNNjNNj

NNjNjNj

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

/)1(22/42/22

/)1(2/22/2

1 1 11

DFT: Example

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

DFT: By Matrix

2/932/3

010j1 011

00j1 01

DFT: By Matrix

010j1 011

00j1 01

IDFT: By Matrix

2/932/3

010j1 011

00j1 01

IDFT: By Matrix

2312311

010j1 011

00j1 01

Inverse Discrete Fourier Transform

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

MuexfuFM

:conjugatecomplex Taking

1,2,1,0,)(1

)( :IDFT

uFDFTM

MxeuFM

DFT Complexity

point DFT an in additions

complex ofNumber :)(

point DFT an in tionsmultiplica

complex ofNumber :)(

DFT / IDFT Complexity

)()1()(

computed.-pre are that Assume

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

MOMMMAdd

MOMMMult

MuexfuF

Fast Fourier Transform

MuWxfuF

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

uxK WWWWWW 222

:Properties

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

MuexfuFM

)12()2(

WWxfWxf

WxfWxf

KMeW nMjM 22/2 ux

K WW 22

uKoddeven

uxKodd

uxKeven

WuFuFKuF

WuFuFuF

)()()(

)12()(

MuM WWWW 22;

FFT Complexity

point FFT 2 an in additions

complex ofNumber :)()(

point FFT 2 an in tionsmultiplica

complex ofNumber :)()(

nAddMAdd

nMultMMult

)12()(

uxKodd

uxKeven

)0()0()1(

)0()0()0(

)1()0(

)0()0(

1;22;1

oddeven

uKoddeven

WuFuFKuF

WuFuFuF

)()()(

)12()(

uxKodd

uxKeven

)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

oddeven

uKoddeven

WuFuFKuF

WuFuFuF

)()()(

4)1(2)2(

2)1(2)2(

AddAdd

MultMult

FFT Complexity

)log(log)()(

)log(log2

1;2)1(2)(

MMOMMnAddMAdd

MMOMMnMultMMult

nnAddnAdd

nnMultnMult

2-D DFT & Images

Lecture 18-19: 04-Sep-12

Dr. P P Das

Kinect Demo in Lecture 19

2-D Impulse

0,00),(

dzdtzt

2-D Sifting Property

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

),(),(),( 0000 ztfdzdtzzttztf

2-D Discrete Impulse

0,00),(

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

),(),(

dzdtAe

dzdteztfF

2-D Sampling

maxmax

and for 0),(

limited-band is ),(

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

Frequency Aliasing

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

TnTntcTnf

/)(sin)(

)}()(~

{)}({)( 11

otherwise

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

TnTntcTnftf /)(sin)()(

,function sinc of sum of ioninterpolat

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

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

)//(2),(),(

NvyMuxjeyxfvuF

)//(2),(1

NvyMuxjevuFMN

2-D: Separability

),(),( where

),(),(

NvyjMuxj

NvyMuxj

eyxfvxF

eyxfvuF

Properties of 2-D DFT:Spatial & Frequency Intervals

Properties of 2-D DFT:Translation & Rotation

),(),(

sincossincos

:scoordinatepolar In

),(),(

)//(200

vuryrx

evuFyyxxf

eyxfvuuFNvyMuxj

NyvMxuj

Properties of 2-D DFT:Periodicity

NkvMkuF

NkyMkxf

Periodicity: Centering the Transform

)2/,2/()1(,

)2/()1(

D-1 In

0/)(2 0

NvMuFy)f(x

MuFf(x)

uuFf(x)e

Mxujπ

Properties of 2-D DFT:Symmetry

),(),(),(

yxwyxwyxw

Properties of 2-D DFT:Symmetry

),(),(

0),(),(1

yNxMwyxw

yxwyxw

Symmetry

),(),(

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

),(),(

symmetric conjugate is FT,),( realFor

vuFvuF

Fourier Spectrum & Phase Angle

),(),(

),(),( :SpectrumPower

),(arctan :Angle Phase

),(),(),( :SpectrumFourier

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

vuIvuR

vuFvuP

vuIvuRvuF

evuFvujIvuRvuF

Fourier Spectrum & Phase Angle

),(),()0,0(

),(),(

symmetry odd has Angle Phase

),(),(

symmetry even has Spectrum

symmetric conjugate is FT,),( realFor

yxfMNyxfF

vuFvuF

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

2-D Convolution

)()(),(),(

),(),(

),(),(1

HFyxhyxf

nymxhnmf

yxhyxfM

Effect of DFT on Convolution

)()()()(m

mxhmfxhxf

Wraparound Error needs Zero Padding

2-D: Separability

),(),(

NvyjMuxj

NvyMuxj

eyxfvuF

Thank you

Fourier Seriesksrao/pdf/ip-19/rcg-ch4a.pdf · The Mathematic Formulation Any function that...

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