Fourier Transform

Fourier Series and Transform

Overview

Why Fourier transform

Trigonometric functions

Who is Fourier

Fourier series

Fourier transform

Discrete Fourier transform

Fast Fourier transform

2D Fourier transform

Tips


Fourier not being noble could not enter the artillery although he was a second Newton

⎯ Francois Jean Dominique Arago

For signal processing Fourier transform is the tool to connect the time domain and frequency domain

Why would we do the exchange between time domain and frequency domain Because we can do all kinds of useful analytical tricks in the frequency domain that are just too hard to do computationally with the original time series in the time domain Simplify the calculation


)502sin()( ttf π=


This example is a sound record analysis The left picture is thesound signal changing with time However we have no any idea about this sound by the time record By the Fourier transform we know that this sound is generated at 50Hz and 120Hz mixed with other noises

Trigonometric functions (ex1)

Trigonometric system is the periodic functions as

1 sinx cosx sin2x cos2x hellip sinnx cosnx hellip

The properties of trigonometric system

Trigonometric system is the orthogonal system

2)cos(2)cos(coscos φθφθφθ ++minus=2)cos(2)cos(sinsin φθφθφθ +minusminus=

(sin ) cos (cos ) sinθ θ θ θ= = minus

int =int =minusminus

π

π

π

π0sin0cos nxdxnxdx int =

minus

π

π0cossin nxdxmx

int⎩⎨⎧

=ne

=minus

π

π π nmnm

nxdxmx0

coscos int⎩⎨⎧

=ne

=minus

π

π π nmnm

nxdxmx0

sinsin

In Matlab

Int(f)Int(fab)

if f is a symbolic expression

q=quad(funab)q=quad(funabtol)[qfcnt]=quad(funabhellip)

Quadrature is a numerical method used to find the area under the graph of a function that is to compute a definite integral

q=quad(funab) tries to approximate the integral of function fun from a to b to within an error of 1e-6 using recursive adaptive Simpson quadrature fun is a function handle for either an M-file function or an anonymous function The function y=fun(x) should accept a vector argument x and return a vector result y the integrand evaluated at each element of x

intb

adxxf )(

int=b

adxxfq )(

Who is Fourier

Fourier is one of the Francersquos greatest administrators historians and mathematicians

He graduated with honors from the military school in Auxerre and became a teach of math when he was 16 years old

Later he joined the faculty at Ecole Normale and then the Polytechnique in Paris when he is 27

He went to Egypt with Napoleon as the Governor of Lower Egypt after the 1798 Expedition

He was secretary of the Academy of Sciences in 1816 and Fellow in 1817

Donrsquot believe it

Neither did Lagrange Laplace Poisson and other big wigs

Not translated into English until 1878

But itrsquos true

Jean Baptiste Joseph Fourier(France 1768~1830)

Fourierrsquos basic idea

Trigonometric functions sin(x) and cos(x) has the period 2π

sin(nx) and cos(nx) have period 2πn

The linear combination of these functions or multiply each by a constant the adding result still has a period 2π

Fourier series

For any function f(x) with period 2π (f(x) = f(2π +x)) we can describe the f(x) in terms of an infinite sum of sines and cosines

To find the coefficients a b and a we multiply above equation by cosmx or sinmx and integrate it over interval -πltxltπ By the orthogonality relations of sin and cos functions we can get

mxdxxfam cos)(1int=minus

π

ππ

)sincos(2

)(1

0 sum ++=infin

=mmm mxbmxaaxf

mxdxxfbm sin)(1int=minus

π

ππ

dxxfa int=minus

π

ππ)(1

0

Fourier series example (ex2)

⎩⎨⎧

ltltminusminusltlt

=01

01)(

xx

xfπ

π

0

sin1sin1

cos)1(1cos1

0

0

0

01

=

minusminus=

int minus+int=

minus

minus

π

π

π

π

ππ

ππ

xx

dxxdxxa

0

sin1sin1

cos)1(1cos1

0

0

0

0

=

+minus=

int minus+int=

minus

minus

π

π

π

π

ππ

ππ

m

m

m

mxm

mxm

dxmxdxmxa

Period function

The parameters are 0)1(111 0

00 =int minus+int=

minusdxdxa

π

π

ππ

5314== n

nbn π

6420 == kbk


π

ππ

)55sin

33sin

1sin(4)( +++=

xxxxfπ

Fourier series example (conthellip)

⎩⎨⎧

ltltminusminusltlt

=01

01)(

xx

xfπ

πPeriod function

The Fourier series is

)55sin

33sin

1sin(4)( +++=

xxxxfπ


⎩⎨⎧

ltltminusminusltlt

=01

01)(

xx

xfπ

πPeriod function


)55sin

33sin

1sin(4)( +++=

xxxxfπ

=+


⎩⎨⎧

ltltminusminusltlt

=01

01)(

xx

xfπ

πPeriod function


)55sin

33sin

1sin(4)( +++=

xxxxfπ

=+


⎩⎨⎧

ltltminusminusltlt

=01

01)(

xx

xfπ

πPeriod function


)55sin

33sin

1sin(4)( +++=

xxxxfπ

+ =


⎩⎨⎧

ltltminusminusltlt

=01

01)(

xx

xfπ

πPeriod function


)55sin

33sin

1sin(4)( +++=

xxxxfπ

Fourier series example

π20)( ltlt= xxxfPeriod function

The parameters are

)33sin

22sin

1sin(2)( +++minus=

xxxxf π

Fourier series general

For any function f(xrsquo) with arbitrary period T a simple change of variables can be used to transform the interval of integration from [-π π] to [-T2T2] as

The f(xrsquo) can be described by the Fourier series as

where

Replace ω 2πT and xrsquo t

22 dxT

dxxT

x ππ==

21))2sin()2cos((2

)(1

0 =sum ++=infin

=mx

Pmbx

Tmaaxf

mmm

ππ

)2cos()(2 2

2dxx

Tmxf

Ta

T

Tm

πint=

minus

)(2 2

20 dxxf

Ta

T

Tint=

minus

)2sin()(2 2

2dxx

Tmxf

Tb

T

Tm

πint=

minus

21)sincos(2

)(1

0 =sum ++=infin

=mtmbtmaatf

mmm ωω

The complex form of Fourier series

))(2

)(2

(2

)(1

0 sum minus+++=infin

=

minusminus

m

timtimmtimtimm eei

beeaatf ωωωω

Euler formulae

Fourier seriesieex

eexxixexixe

ixix

ixix

ix

ix

2)(sin2)(cos

sincossincos

minus

minus

minus minus=+=

rArr⎭⎬⎫

minus=+=

For certain m=k

Denoting that as


tikkktikkk

tiktik

k

tiktik

kkk

eibaeibaieebeeatkbtka

ωω

ωωωω

ωω

minus

minusminus

++

minus=

minus+

+=+

22

22sincos

2

2

20

0kk

kkk

k

ibacibacac +=

minus== minus

int=

sumsum =++=

minus

minus

infin

minusinfin=

infin

=

minusminus

2

2

10

)(2

)()(

T

T

tikk

k

tikk

k

tikk

tikk

dtetfT

c

ecececctf

ω

ωωω

Fourier transform

For any non-periodic function and assume T infin rewrite previous general Fourier series equation and get

Define

Here F(ω) is called as the Fourier Transform of f(t) Equation of f(t) is called the inverse Fourier Transform

int=

int=

infin

infinminus

infin

infinminus

minus

ωωπ

ω

ω

ω

deFtf

dtetfF

ti

ti

)(21)(

)()(

intintrarr

sum intrarr

sum int=

infin

infinminus

minusinfin

infinminus

infin

minusinfin= minus

minus=

infin

minusinfin= minus

minus

ξξωπ

ξξωπ

ωξω

ξωωπ

ωω

defde

def

edtetfT

tf

iti

k

T

T

tikT

k

tikT

T

tik

)(21

)(1

))(2()(

2

2

)(2

2

2

Fourier transform Parsevalrsquos law

The time signal squared f2(t) represents how the energy contained in the signal distributes over time t while its spectrum squared F2(ω) represents how the energy distributes over frequency (therefore the term power density spectrum) Obviously the sameamount of energy is contained in either time or frequency domain as indicated by Parsevalrsquos formula

int=intinfin

infinminus

infin

infinminusωω dFdttf 22 )()(

Fourier transform properties

The properties of Fourier transfrom

- Linearity property given f(x) g(x)

- Similarity property g(x)=f(ax)

-Shift formula given g(x) = f(x+b)

- Derivative formula

)()())()(( ωω bGaFxbgxafFT +=+

)(1)(a

Fa

G ωω =

)())(( ωωFixfFT =

)()( ωω ω FeG bi=

In Matlab

F = fourier(f)

This is the Fourier transform of the symbolic scalar f with default independent variable x The default return is a function of ω This represents

int=infin

infinminus

minus dxexfF xiωω )()(

f = ifourier(F)

This is the inverse Fourier transform of the symbolic scalar F with default independent variable ω The default return is a function of x This represents

int=infin

infinminusωω

πω deFxf xi)(

21)(

Discrete Fourier Transform (DFT)

Discrete Fourier Transform can be understood as a numerical approximation to the Fourier transform

This is used in the case where both the time and the frequency variables are discrete (which they are if digital computers are being used to perform the analysis)

To convert the integral Fourier Transform (FT) into the DiscreteFourier Transform (DFT) we can do following steps

1) Assume the sampling window is T The number of sampling points is N Define the sample interval ∆T=Ts=TN

2) Define the sample points tk = k(∆T) for k = 0 hellip (N-1)

3) Define the signal values at each sampling points as fk=f(tk)

4) Define the frequency sampling points ωn=2πnT where 2πnT is termed as the fundamental frequency


5) Consider the problem of approximating the FT of f at the points ωn=2πnT The answer is

6) Approximate this integral by Riemann sum approximation using the points tk since f~0 for tgtT

int minus==infin

infinminus

minus )1(0)()( NndttfeF tnin

ωω

1210)()(1

0minus=sum=

minus

=

minus NnetfFN

k

ktnikn

ωω

This is the Discrete Fourier Transform

The inverse Discrete Fourier Transform is defined as

1

0

1( ) ( ) 012 1n k

Ni t

k nn

f t F e k NN

ωωminus

=

= = minussum

Fast Fourier Transform (FFT)

Fast Fourier Transform (FFT) is a effective algorithm of Discrete Fourier Transform (DFT) and developed by Cooley and Tukey at 1965

This algorithm reduces the computation time of DFT for N points from N2 to Nlog2(N) (This algorithm is called Butterfly algorithm)

The only requirement of this algorithm is that number of point in the series have to be a power of 2 (2n points) such as 32 1024 4096

Zero padding at the end of the data set if the sampling number is not equal to the exact the power of 2

In Matlab

Y = fft(X)

This command returns the discrete Fourier transform (DFT) of X computed with a fast Fourier transform (FFT) algorithm

Y = fft(Xn)

This command returns the n-point DFT of X If the length of X is less than n X is padded with trailing zeros to length n If the length of X is greater than n the sequence X is truncated

Y = ifft(X)

This command returns the inverse discrete Fourier transform (DFT) of X computed with a fast Fourier transform (FFT) algorithm

Y = ifft(Xn)

This command returns the n-point inverse DFT of X

Fourier transform Delta function

Delta function

⎩⎨⎧ =

==otherst

ttf0

01)()( δ

Fourier transform Uniform function

Unit function1)( =tf

Fourier transform Sin function

example g(t) = sin(2πf t) + (13)sin(2π3f t)

= +


example g(t) = sin(2πf t) + (13)sin(2π 3f t)

= +

Fourier transform Cos function

cos functions)502cos()( ttf π=

Applications convolution

The time domain recorded waveform is a convolution product

Simplify the complex convolution product into the direct multiply in the frequency domain by Fourier transform

int minus=infin

infinminusτττ dstvtr )()()( 0

)()()( 0 ωωω SVRFFT

=rArr


The simulated DI water waveform and ethanol waveform at room temperature by FFT

2D Integral Fourier transform

2D integral Fourier transform is

Inverse 2D Fourier transform is

)()( )(int int=infin

infinminus

infin

infinminus

+minus dxdyeyxfvuF vyuxi

)(21)( )(int int=

infin

infinminus

infin

infinminus

+ dudvevuFyxf vyuxi

π

2D Discrete Fourier transformFor 2D function f(xy) DFT is

Rewrite above equation

We can implement 2D Fourier transform as a sequence of 1-D Fourier transform operations

12101210

)()(1

0

1

0

)(

minus=minus=

sum sum=minus

=

minus

=

+minus

NnMm

eyxfvuFM

xk

N

yk

yvxuiykxknm

yknxkm

sum=

sum ⎥⎦⎤

⎢⎣⎡ sum=

minus

=

minus

minus

=

minusminus

=

minus

1

0

1

0

1

0

)()(

)()(

M

m

xiunm

M

m

xiuN

n

yivykxknm

xkm

xkmykn

evxFvuF

eeyxfvuF

2D Inverse Discrete Fourier transformFor 2D function f(xy) inverse DFT is


We can implement 2D inverse Fourier transform as a sequence of 1-D inverse Fourier transform operations

12101210

)()(1

0

1

0

)(

minus=minus=

sum sum=minus

=

minus

=

+minus

NykMxk

evuFyxfM

m

N

n

ykynvxkxmuinmykxk

[ ]sum=

sum sum=

minus

=

minus

minus

=

minus

=

1

0

1

0

1

0

)(1)(

)(1)(

M

m

xkxmiunxkykxk

M

m

xkxmiuN

n

ykynivnmykxk

evxFMN

yxf

eevuFMN

yxf

2D Fourier transform properties


- Linearity property given f(xy) g(xy)

- Shift formula given g(xy) = f(x+ay+b)

- Similarity property g(xy)=f(axby)

- Convolution

)()())()(( vubGvuaFyxbgyxafFT +=+

)()( )( vuFevuG vbuai +=

)()()()( vuHvuGyxhyxg sdot=lowast

)(1)(bv

auF

abvuG =

In Matlab

Y = fft2(X)

This command returns the discrete Fourier transform (DFT) of X(xy) computed with a fast Fourier transform (FFT) algorithm

Y = fft2(Xmn)

This command returns the n-point DFT of X(xy) with the certain length of x at m and y at n

Y = ifft2(X)


Y = ifft2(Xmn)

This command returns the m-point of x and n-point of y inverse DFT of X(xy)

Image Compression (JPEG)

JPEG compression comparison

89k 12k

Tips Nyquist frequency

Nyquist frequency is called the highest frequency that can be coded at a given sampling rate in order to be able to fully reconstruct the signal

The total sampling period is T Then the base frequency is 2T This represent the lowest frequency of the signal we can see in the frequency domain

On the other hand Nyquist frequency represents the highest frequency of the signal we can see in the frequency domain

tfNF ∆

=21

Tips Sampling ratesampling total time

Sampling rate is the sampling interval This will control the highest frequency band

Sampling total time is the total time period we looked to sampling the data This will control the lowest frequency band

Please notice that we can extend the sampling total time by padding zero for FFT This will change the lower frequency resolution However it can not change the highest frequency

Tips windows

Any sampling data range is limitedfinite

( ) sin(2 5 )f t tπ=

+=

Tips windows

( ) sin(2 5 )f t tπ=

The true sampling signal is frsquo(t) = f(t)win(t)

After the Fourier transform the transformed the signal is the convolution products of F(ω) and WIN(ω)

For the true transformed the signal we have to de-convolution of the transformed the results

)()()( ωωω WINFF lowast=

The Homework of Fourier transform is using Matlab

Do the Fourier transform of one simple harmonic function

f1(x)=sin(2pi500t)

and

f2(x) = cos(2pi500t)+2sin(2pi1000t)+05cos(2pi200t)

Please practice with

1) choosing two different sampling rate (how many points you sampled in total in time domain N)

2) choosing two different sampling interval (delta_t)

3) choosing two different window length to get the sense of how these different sampling will influence on your Fourier transform results in frequency domain


Overview




In Matlab

Who is Fourier


Fourier series










Fourier transform



In Matlab




In Matlab









2D Discrete Fourier transform

2D Inverse Discrete Fourier transform


In Matlab





Tips windows

Tips windows

Overview


Trigonometric functions

Who is Fourier

Fourier series

Fourier transform

Discrete Fourier transform

Fast Fourier transform

2D Fourier transform

Tips







)502sin()( ttf π=











π

π

π


minus

π

π0cossin nxdxmx

int⎩⎨⎧

=ne

=minus

π

π π nmnm

nxdxmx0

coscos int⎩⎨⎧

=ne

=minus

π

π π nmnm

nxdxmx0

sinsin

In Matlab

Int(f)Int(fab)





intb

adxxf )(

int=b

adxxfq )(

Who is Fourier









But itrsquos true






Fourier series




π

ππ

)sincos(2

)(1

0 sum ++=infin

=mmm mxbmxaaxf


π

ππ

dxxfa int=minus

π

ππ)(1

0


⎩⎨⎧

ltltminusminusltlt

=01

01)(

xx

xfπ

π

0

sin1sin1

cos)1(1cos1

0

0

0

01

=

minusminus=

int minus+int=

minus

minus

π

π

π

π

ππ

ππ

xx

dxxdxxa

0

sin1sin1

cos)1(1cos1

0

0

0

0

=

+minus=

int minus+int=

minus

minus

π

π

π

π

ππ

ππ

m

m

m

mxm

mxm

dxmxdxmxa

Period function


00 =int minus+int=

minusdxdxa

π

π

ππ

5314== n

nbn π

6420 == kbk


π

ππ

)55sin

33sin

1sin(4)( +++=

xxxxfπ


⎩⎨⎧

ltltminusminusltlt

=01

01)(

xx

xfπ

πPeriod function


)55sin

33sin

1sin(4)( +++=

xxxxfπ


⎩⎨⎧

ltltminusminusltlt

=01

01)(

xx

xfπ

πPeriod function


)55sin

33sin

1sin(4)( +++=

xxxxfπ

=+


⎩⎨⎧

ltltminusminusltlt

=01

01)(

xx

xfπ

πPeriod function


)55sin

33sin

1sin(4)( +++=

xxxxfπ

=+


⎩⎨⎧

ltltminusminusltlt

=01

01)(

xx

xfπ

πPeriod function


)55sin

33sin

1sin(4)( +++=

xxxxfπ

+ =


⎩⎨⎧

ltltminusminusltlt

=01

01)(

xx

xfπ

πPeriod function


)55sin

33sin

1sin(4)( +++=

xxxxfπ



The parameters are

)33sin

22sin

1sin(2)( +++minus=

xxxxf π




where


22 dxT

dxxT

x ππ==

21))2sin()2cos((2

)(1

0 =sum ++=infin

=mx

Pmbx

Tmaaxf

mmm

ππ

)2cos()(2 2

2dxx

Tmxf

Ta

T

Tm

πint=

minus

)(2 2

20 dxxf

Ta

T

Tint=

minus

)2sin()(2 2

2dxx

Tmxf

Tb

T

Tm

πint=

minus

21)sincos(2

)(1

0 =sum ++=infin

=mtmbtmaatf

mmm ωω


))(2

)(2

(2

)(1


=

minusminus

m

timtimmtimtimm eei

beeaatf ωωωω

Euler formulae

Fourier seriesieex

eexxixexixe

ixix

ixix

ix

ix

2)(sin2)(cos

sincossincos

minus

minus

minus minus=+=

rArr⎭⎬⎫

minus=+=

For certain m=k

Denoting that as


tikkktikkk

tiktik

k

tiktik

kkk


ωω

ωωωω

ωω

minus

minusminus

++

minus=

minus+

+=+

22

22sincos

2

2

20

0kk

kkk

k

ibacibacac +=

minus== minus

int=

sumsum =++=

minus

minus

infin

minusinfin=

infin

=

minusminus

2

2

10

)(2

)()(

T

T

tikk

k

tikk

k

tikk

tikk

dtetfT

c

ecececctf

ω

ωωω

Fourier transform


Define


int=

int=

infin

infinminus

infin

infinminus

minus

ωωπ

ω

ω

ω

deFtf

dtetfF

ti

ti

)(21)(

)()(

intintrarr

sum intrarr

sum int=

infin

infinminus

minusinfin

infinminus

infin

minusinfin= minus

minus=

infin

minusinfin= minus

minus

ξξωπ

ξξωπ

ωξω

ξωωπ

ωω

defde

def

edtetfT

tf

iti

k

T

T

tikT

k

tikT

T

tik

)(21

)(1

))(2()(

2

2

)(2

2

2



int=intinfin

infinminus

infin









)(1)(a

Fa

G ωω =

)())(( ωωFixfFT =


In Matlab

F = fourier(f)


int=infin

infinminus


f = ifourier(F)


int=infin

infinminusωω

πω deFxf xi)(

21)(












int minus==infin

infinminus


ωω

1210)()(1

0minus=sum=

minus

=

minus NnetfFN

k

ktnikn

ωω



1

0

1( ) ( ) 012 1n k

Ni t

k nn

f t F e k NN

ωωminus

=

= = minussum






In Matlab

Y = fft(X)


Y = fft(Xn)


Y = ifft(X)


Y = ifft(Xn)



Delta function

⎩⎨⎧ =

==otherst

ttf0

01)()( δ





= +



= +






int minus=infin



=rArr







infinminus

infin

infinminus


)(21)( )(int int=

infin

infinminus

infin

infinminus

+ dudvevuFyxf vyuxi

π




12101210

)()(1

0

1

0

)(

minus=minus=

sum sum=minus

=

minus

=

+minus

NnMm

eyxfvuFM

xk

N

yk

yvxuiykxknm

yknxkm

sum=

sum ⎥⎦⎤

⎢⎣⎡ sum=

minus

=

minus

minus

=

minusminus

=

minus

1

0

1

0

1

0

)()(

)()(

M

m

xiunm

M

m

xiuN

n

yivykxknm

xkm

xkmykn

evxFvuF

eeyxfvuF




12101210

)()(1

0

1

0

)(

minus=minus=

sum sum=minus

=

minus

=

+minus

NykMxk

evuFyxfM

m

N

n

ykynvxkxmuinmykxk

[ ]sum=

sum sum=

minus

=

minus

minus

=

minus

=

1

0

1

0

1

0

)(1)(

)(1)(

M

m

xkxmiunxkykxk

M

m

xkxmiuN

n

ykynivnmykxk

evxFMN

yxf

eevuFMN

yxf






- Convolution




)(1)(bv

auF

abvuG =

In Matlab

Y = fft2(X)


Y = fft2(Xmn)


Y = ifft2(X)


Y = ifft2(Xmn)




89k 12k





tfNF ∆

=21





Tips windows


( ) sin(2 5 )f t tπ=

+=

Tips windows

( ) sin(2 5 )f t tπ=







f1(x)=sin(2pi500t)

and







Overview




In Matlab

Who is Fourier


Fourier series










Fourier transform



In Matlab




In Matlab












In Matlab





Tips windows

Tips windows







)502sin()( ttf π=











π

π

π


minus

π

π0cossin nxdxmx

int⎩⎨⎧

=ne

=minus

π

π π nmnm

nxdxmx0

coscos int⎩⎨⎧

=ne

=minus

π

π π nmnm

nxdxmx0

sinsin

In Matlab

Int(f)Int(fab)





intb

adxxf )(

int=b

adxxfq )(

Who is Fourier









But itrsquos true






Fourier series




π

ππ

)sincos(2

)(1

0 sum ++=infin

=mmm mxbmxaaxf


π

ππ

dxxfa int=minus

π

ππ)(1

0


⎩⎨⎧

ltltminusminusltlt

=01

01)(

xx

xfπ

π

0

sin1sin1

cos)1(1cos1

0

0

0

01

=

minusminus=

int minus+int=

minus

minus

π

π

π

π

ππ

ππ

xx

dxxdxxa

0

sin1sin1

cos)1(1cos1

0

0

0

0

=

+minus=

int minus+int=

minus

minus

π

π

π

π

ππ

ππ

m

m

m

mxm

mxm

dxmxdxmxa

Period function


00 =int minus+int=

minusdxdxa

π

π

ππ

5314== n

nbn π

6420 == kbk


π

ππ

)55sin

33sin

1sin(4)( +++=

xxxxfπ


⎩⎨⎧

ltltminusminusltlt

=01

01)(

xx

xfπ

πPeriod function


)55sin

33sin

1sin(4)( +++=

xxxxfπ


⎩⎨⎧

ltltminusminusltlt

=01

01)(

xx

xfπ

πPeriod function


)55sin

33sin

1sin(4)( +++=

xxxxfπ

=+


⎩⎨⎧

ltltminusminusltlt

=01

01)(

xx

xfπ

πPeriod function


)55sin

33sin

1sin(4)( +++=

xxxxfπ

=+


⎩⎨⎧

ltltminusminusltlt

=01

01)(

xx

xfπ

πPeriod function


)55sin

33sin

1sin(4)( +++=

xxxxfπ

+ =


⎩⎨⎧

ltltminusminusltlt

=01

01)(

xx

xfπ

πPeriod function


)55sin

33sin

1sin(4)( +++=

xxxxfπ



The parameters are

)33sin

22sin

1sin(2)( +++minus=

xxxxf π




where


22 dxT

dxxT

x ππ==

21))2sin()2cos((2

)(1

0 =sum ++=infin

=mx

Pmbx

Tmaaxf

mmm

ππ

)2cos()(2 2

2dxx

Tmxf

Ta

T

Tm

πint=

minus

)(2 2

20 dxxf

Ta

T

Tint=

minus

)2sin()(2 2

2dxx

Tmxf

Tb

T

Tm

πint=

minus

21)sincos(2

)(1

0 =sum ++=infin

=mtmbtmaatf

mmm ωω


))(2

)(2

(2

)(1


=

minusminus

m

timtimmtimtimm eei

beeaatf ωωωω

Euler formulae

Fourier seriesieex

eexxixexixe

ixix

ixix

ix

ix

2)(sin2)(cos

sincossincos

minus

minus

minus minus=+=

rArr⎭⎬⎫

minus=+=

For certain m=k

Denoting that as


tikkktikkk

tiktik

k

tiktik

kkk


ωω

ωωωω

ωω

minus

minusminus

++

minus=

minus+

+=+

22

22sincos

2

2

20

0kk

kkk

k

ibacibacac +=

minus== minus

int=

sumsum =++=

minus

minus

infin

minusinfin=

infin

=

minusminus

2

2

10

)(2

)()(

T

T

tikk

k

tikk

k

tikk

tikk

dtetfT

c

ecececctf

ω

ωωω

Fourier transform


Define


int=

int=

infin

infinminus

infin

infinminus

minus

ωωπ

ω

ω

ω

deFtf

dtetfF

ti

ti

)(21)(

)()(

intintrarr

sum intrarr

sum int=

infin

infinminus

minusinfin

infinminus

infin

minusinfin= minus

minus=

infin

minusinfin= minus

minus

ξξωπ

ξξωπ

ωξω

ξωωπ

ωω

defde

def

edtetfT

tf

iti

k

T

T

tikT

k

tikT

T

tik

)(21

)(1

))(2()(

2

2

)(2

2

2



int=intinfin

infinminus

infin









)(1)(a

Fa

G ωω =

)())(( ωωFixfFT =


In Matlab

F = fourier(f)


int=infin

infinminus


f = ifourier(F)


int=infin

infinminusωω

πω deFxf xi)(

21)(












int minus==infin

infinminus


ωω

1210)()(1

0minus=sum=

minus

=

minus NnetfFN

k

ktnikn

ωω



1

0

1( ) ( ) 012 1n k

Ni t

k nn

f t F e k NN

ωωminus

=

= = minussum






In Matlab

Y = fft(X)


Y = fft(Xn)


Y = ifft(X)


Y = ifft(Xn)



Delta function

⎩⎨⎧ =

==otherst

ttf0

01)()( δ





= +



= +






int minus=infin



=rArr







infinminus

infin

infinminus


)(21)( )(int int=

infin

infinminus

infin

infinminus

+ dudvevuFyxf vyuxi

π




12101210

)()(1

0

1

0

)(

minus=minus=

sum sum=minus

=

minus

=

+minus

NnMm

eyxfvuFM

xk

N

yk

yvxuiykxknm

yknxkm

sum=

sum ⎥⎦⎤

⎢⎣⎡ sum=

minus

=

minus

minus

=

minusminus

=

minus

1

0

1

0

1

0

)()(

)()(

M

m

xiunm

M

m

xiuN

n

yivykxknm

xkm

xkmykn

evxFvuF

eeyxfvuF




12101210

)()(1

0

1

0

)(

minus=minus=

sum sum=minus

=

minus

=

+minus

NykMxk

evuFyxfM

m

N

n

ykynvxkxmuinmykxk

[ ]sum=

sum sum=

minus

=

minus

minus

=

minus

=

1

0

1

0

1

0

)(1)(

)(1)(

M

m

xkxmiunxkykxk

M

m

xkxmiuN

n

ykynivnmykxk

evxFMN

yxf

eevuFMN

yxf






- Convolution




)(1)(bv

auF

abvuG =

In Matlab

Y = fft2(X)


Y = fft2(Xmn)


Y = ifft2(X)


Y = ifft2(Xmn)




89k 12k





tfNF ∆

=21





Tips windows


( ) sin(2 5 )f t tπ=

+=

Tips windows

( ) sin(2 5 )f t tπ=







f1(x)=sin(2pi500t)

and







Overview




In Matlab

Who is Fourier


Fourier series










Fourier transform



In Matlab




In Matlab












In Matlab





Tips windows

Tips windows











π

π

π


minus

π

π0cossin nxdxmx

int⎩⎨⎧

=ne

=minus

π

π π nmnm

nxdxmx0

coscos int⎩⎨⎧

=ne

=minus

π

π π nmnm

nxdxmx0

sinsin

In Matlab

Int(f)Int(fab)





intb

adxxf )(

int=b

adxxfq )(

Who is Fourier









But itrsquos true






Fourier series




π

ππ

)sincos(2

)(1

0 sum ++=infin

=mmm mxbmxaaxf


π

ππ

dxxfa int=minus

π

ππ)(1

0


⎩⎨⎧

ltltminusminusltlt

=01

01)(

xx

xfπ

π

0

sin1sin1

cos)1(1cos1

0

0

0

01

=

minusminus=

int minus+int=

minus

minus

π

π

π

π

ππ

ππ

xx

dxxdxxa

0

sin1sin1

cos)1(1cos1

0

0

0

0

=

+minus=

int minus+int=

minus

minus

π

π

π

π

ππ

ππ

m

m

m

mxm

mxm

dxmxdxmxa

Period function


00 =int minus+int=

minusdxdxa

π

π

ππ

5314== n

nbn π

6420 == kbk


π

ππ

)55sin

33sin

1sin(4)( +++=

xxxxfπ


⎩⎨⎧

ltltminusminusltlt

=01

01)(

xx

xfπ

πPeriod function


)55sin

33sin

1sin(4)( +++=

xxxxfπ


⎩⎨⎧

ltltminusminusltlt

=01

01)(

xx

xfπ

πPeriod function


)55sin

33sin

1sin(4)( +++=

xxxxfπ

=+


⎩⎨⎧

ltltminusminusltlt

=01

01)(

xx

xfπ

πPeriod function


)55sin

33sin

1sin(4)( +++=

xxxxfπ

=+


⎩⎨⎧

ltltminusminusltlt

=01

01)(

xx

xfπ

πPeriod function


)55sin

33sin

1sin(4)( +++=

xxxxfπ

+ =


⎩⎨⎧

ltltminusminusltlt

=01

01)(

xx

xfπ

πPeriod function


)55sin

33sin

1sin(4)( +++=

xxxxfπ



The parameters are

)33sin

22sin

1sin(2)( +++minus=

xxxxf π




where


22 dxT

dxxT

x ππ==

21))2sin()2cos((2

)(1

0 =sum ++=infin

=mx

Pmbx

Tmaaxf

mmm

ππ

)2cos()(2 2

2dxx

Tmxf

Ta

T

Tm

πint=

minus

)(2 2

20 dxxf

Ta

T

Tint=

minus

)2sin()(2 2

2dxx

Tmxf

Tb

T

Tm

πint=

minus

21)sincos(2

)(1

0 =sum ++=infin

=mtmbtmaatf

mmm ωω


))(2

)(2

(2

)(1


=

minusminus

m

timtimmtimtimm eei

beeaatf ωωωω

Euler formulae

Fourier seriesieex

eexxixexixe

ixix

ixix

ix

ix

2)(sin2)(cos

sincossincos

minus

minus

minus minus=+=

rArr⎭⎬⎫

minus=+=

For certain m=k

Denoting that as


tikkktikkk

tiktik

k

tiktik

kkk


ωω

ωωωω

ωω

minus

minusminus

++

minus=

minus+

+=+

22

22sincos

2

2

20

0kk

kkk

k

ibacibacac +=

minus== minus

int=

sumsum =++=

minus

minus

infin

minusinfin=

infin

=

minusminus

2

2

10

)(2

)()(

T

T

tikk

k

tikk

k

tikk

tikk

dtetfT

c

ecececctf

ω

ωωω

Fourier transform


Define


int=

int=

infin

infinminus

infin

infinminus

minus

ωωπ

ω

ω

ω

deFtf

dtetfF

ti

ti

)(21)(

)()(

intintrarr

sum intrarr

sum int=

infin

infinminus

minusinfin

infinminus

infin

minusinfin= minus

minus=

infin

minusinfin= minus

minus

ξξωπ

ξξωπ

ωξω

ξωωπ

ωω

defde

def

edtetfT

tf

iti

k

T

T

tikT

k

tikT

T

tik

)(21

)(1

))(2()(

2

2

)(2

2

2



int=intinfin

infinminus

infin









)(1)(a

Fa

G ωω =

)())(( ωωFixfFT =


In Matlab

F = fourier(f)


int=infin

infinminus


f = ifourier(F)


int=infin

infinminusωω

πω deFxf xi)(

21)(












int minus==infin

infinminus


ωω

1210)()(1

0minus=sum=

minus

=

minus NnetfFN

k

ktnikn

ωω



1

0

1( ) ( ) 012 1n k

Ni t

k nn

f t F e k NN

ωωminus

=

= = minussum






In Matlab

Y = fft(X)


Y = fft(Xn)


Y = ifft(X)


Y = ifft(Xn)



Delta function

⎩⎨⎧ =

==otherst

ttf0

01)()( δ





= +



= +






int minus=infin



=rArr







infinminus

infin

infinminus


)(21)( )(int int=

infin

infinminus

infin

infinminus

+ dudvevuFyxf vyuxi

π




12101210

)()(1

0

1

0

)(

minus=minus=

sum sum=minus

=

minus

=

+minus

NnMm

eyxfvuFM

xk

N

yk

yvxuiykxknm

yknxkm

sum=

sum ⎥⎦⎤

⎢⎣⎡ sum=

minus

=

minus

minus

=

minusminus

=

minus

1

0

1

0

1

0

)()(

)()(

M

m

xiunm

M

m

xiuN

n

yivykxknm

xkm

xkmykn

evxFvuF

eeyxfvuF




12101210

)()(1

0

1

0

)(

minus=minus=

sum sum=minus

=

minus

=

+minus

NykMxk

evuFyxfM

m

N

n

ykynvxkxmuinmykxk

[ ]sum=

sum sum=

minus

=

minus

minus

=

minus

=

1

0

1

0

1

0

)(1)(

)(1)(

M

m

xkxmiunxkykxk

M

m

xkxmiuN

n

ykynivnmykxk

evxFMN

yxf

eevuFMN

yxf






- Convolution




)(1)(bv

auF

abvuG =

In Matlab

Y = fft2(X)


Y = fft2(Xmn)


Y = ifft2(X)


Y = ifft2(Xmn)




89k 12k





tfNF ∆

=21





Tips windows


( ) sin(2 5 )f t tπ=

+=

Tips windows

( ) sin(2 5 )f t tπ=







f1(x)=sin(2pi500t)

and







Overview




In Matlab

Who is Fourier


Fourier series










Fourier transform



In Matlab




In Matlab












In Matlab





Tips windows

Tips windows

In Matlab

Int(f)Int(fab)





intb

adxxf )(

int=b

adxxfq )(

Who is Fourier









But itrsquos true






Fourier series




π

ππ

)sincos(2

)(1

0 sum ++=infin

=mmm mxbmxaaxf


π

ππ

dxxfa int=minus

π

ππ)(1

0


⎩⎨⎧

ltltminusminusltlt

=01

01)(

xx

xfπ

π

0

sin1sin1

cos)1(1cos1

0

0

0

01

=

minusminus=

int minus+int=

minus

minus

π

π

π

π

ππ

ππ

xx

dxxdxxa

0

sin1sin1

cos)1(1cos1

0

0

0

0

=

+minus=

int minus+int=

minus

minus

π

π

π

π

ππ

ππ

m

m

m

mxm

mxm

dxmxdxmxa

Period function


00 =int minus+int=

minusdxdxa

π

π

ππ

5314== n

nbn π

6420 == kbk


π

ππ

)55sin

33sin

1sin(4)( +++=

xxxxfπ


⎩⎨⎧

ltltminusminusltlt

=01

01)(

xx

xfπ

πPeriod function


)55sin

33sin

1sin(4)( +++=

xxxxfπ


⎩⎨⎧

ltltminusminusltlt

=01

01)(

xx

xfπ

πPeriod function


)55sin

33sin

1sin(4)( +++=

xxxxfπ

=+


⎩⎨⎧

ltltminusminusltlt

=01

01)(

xx

xfπ

πPeriod function


)55sin

33sin

1sin(4)( +++=

xxxxfπ

=+


⎩⎨⎧

ltltminusminusltlt

=01

01)(

xx

xfπ

πPeriod function


)55sin

33sin

1sin(4)( +++=

xxxxfπ

+ =


⎩⎨⎧

ltltminusminusltlt

=01

01)(

xx

xfπ

πPeriod function


)55sin

33sin

1sin(4)( +++=

xxxxfπ



The parameters are

)33sin

22sin

1sin(2)( +++minus=

xxxxf π




where


22 dxT

dxxT

x ππ==

21))2sin()2cos((2

)(1

0 =sum ++=infin

=mx

Pmbx

Tmaaxf

mmm

ππ

)2cos()(2 2

2dxx

Tmxf

Ta

T

Tm

πint=

minus

)(2 2

20 dxxf

Ta

T

Tint=

minus

)2sin()(2 2

2dxx

Tmxf

Tb

T

Tm

πint=

minus

21)sincos(2

)(1

0 =sum ++=infin

=mtmbtmaatf

mmm ωω


))(2

)(2

(2

)(1


=

minusminus

m

timtimmtimtimm eei

beeaatf ωωωω

Euler formulae

Fourier seriesieex

eexxixexixe

ixix

ixix

ix

ix

2)(sin2)(cos

sincossincos

minus

minus

minus minus=+=

rArr⎭⎬⎫

minus=+=

For certain m=k

Denoting that as


tikkktikkk

tiktik

k

tiktik

kkk


ωω

ωωωω

ωω

minus

minusminus

++

minus=

minus+

+=+

22

22sincos

2

2

20

0kk

kkk

k

ibacibacac +=

minus== minus

int=

sumsum =++=

minus

minus

infin

minusinfin=

infin

=

minusminus

2

2

10

)(2

)()(

T

T

tikk

k

tikk

k

tikk

tikk

dtetfT

c

ecececctf

ω

ωωω

Fourier transform


Define


int=

int=

infin

infinminus

infin

infinminus

minus

ωωπ

ω

ω

ω

deFtf

dtetfF

ti

ti

)(21)(

)()(

intintrarr

sum intrarr

sum int=

infin

infinminus

minusinfin

infinminus

infin

minusinfin= minus

minus=

infin

minusinfin= minus

minus

ξξωπ

ξξωπ

ωξω

ξωωπ

ωω

defde

def

edtetfT

tf

iti

k

T

T

tikT

k

tikT

T

tik

)(21

)(1

))(2()(

2

2

)(2

2

2



int=intinfin

infinminus

infin









)(1)(a

Fa

G ωω =

)())(( ωωFixfFT =


In Matlab

F = fourier(f)


int=infin

infinminus


f = ifourier(F)


int=infin

infinminusωω

πω deFxf xi)(

21)(












int minus==infin

infinminus


ωω

1210)()(1

0minus=sum=

minus

=

minus NnetfFN

k

ktnikn

ωω



1

0

1( ) ( ) 012 1n k

Ni t

k nn

f t F e k NN

ωωminus

=

= = minussum






In Matlab

Y = fft(X)


Y = fft(Xn)


Y = ifft(X)


Y = ifft(Xn)



Delta function

⎩⎨⎧ =

==otherst

ttf0

01)()( δ





= +



= +






int minus=infin



=rArr







infinminus

infin

infinminus


)(21)( )(int int=

infin

infinminus

infin

infinminus

+ dudvevuFyxf vyuxi

π




12101210

)()(1

0

1

0

)(

minus=minus=

sum sum=minus

=

minus

=

+minus

NnMm

eyxfvuFM

xk

N

yk

yvxuiykxknm

yknxkm

sum=

sum ⎥⎦⎤

⎢⎣⎡ sum=

minus

=

minus

minus

=

minusminus

=

minus

1

0

1

0

1

0

)()(

)()(

M

m

xiunm

M

m

xiuN

n

yivykxknm

xkm

xkmykn

evxFvuF

eeyxfvuF




12101210

)()(1

0

1

0

)(

minus=minus=

sum sum=minus

=

minus

=

+minus

NykMxk

evuFyxfM

m

N

n

ykynvxkxmuinmykxk

[ ]sum=

sum sum=

minus

=

minus

minus

=

minus

=

1

0

1

0

1

0

)(1)(

)(1)(

M

m

xkxmiunxkykxk

M

m

xkxmiuN

n

ykynivnmykxk

evxFMN

yxf

eevuFMN

yxf






- Convolution




)(1)(bv

auF

abvuG =

In Matlab

Y = fft2(X)


Y = fft2(Xmn)


Y = ifft2(X)


Y = ifft2(Xmn)




89k 12k





tfNF ∆

=21





Tips windows


( ) sin(2 5 )f t tπ=

+=

Tips windows

( ) sin(2 5 )f t tπ=







f1(x)=sin(2pi500t)

and







Overview




In Matlab

Who is Fourier


Fourier series










Fourier transform



In Matlab




In Matlab












In Matlab





Tips windows

Tips windows

Who is Fourier









But itrsquos true






Fourier series




π

ππ

)sincos(2

)(1

0 sum ++=infin

=mmm mxbmxaaxf


π

ππ

dxxfa int=minus

π

ππ)(1

0


⎩⎨⎧

ltltminusminusltlt

=01

01)(

xx

xfπ

π

0

sin1sin1

cos)1(1cos1

0

0

0

01

=

minusminus=

int minus+int=

minus

minus

π

π

π

π

ππ

ππ

xx

dxxdxxa

0

sin1sin1

cos)1(1cos1

0

0

0

0

=

+minus=

int minus+int=

minus

minus

π

π

π

π

ππ

ππ

m

m

m

mxm

mxm

dxmxdxmxa

Period function


00 =int minus+int=

minusdxdxa

π

π

ππ

5314== n

nbn π

6420 == kbk


π

ππ

)55sin

33sin

1sin(4)( +++=

xxxxfπ


⎩⎨⎧

ltltminusminusltlt

=01

01)(

xx

xfπ

πPeriod function


)55sin

33sin

1sin(4)( +++=

xxxxfπ


⎩⎨⎧

ltltminusminusltlt

=01

01)(

xx

xfπ

πPeriod function


)55sin

33sin

1sin(4)( +++=

xxxxfπ

=+


⎩⎨⎧

ltltminusminusltlt

=01

01)(

xx

xfπ

πPeriod function


)55sin

33sin

1sin(4)( +++=

xxxxfπ

=+


⎩⎨⎧

ltltminusminusltlt

=01

01)(

xx

xfπ

πPeriod function


)55sin

33sin

1sin(4)( +++=

xxxxfπ

+ =


⎩⎨⎧

ltltminusminusltlt

=01

01)(

xx

xfπ

πPeriod function


)55sin

33sin

1sin(4)( +++=

xxxxfπ



The parameters are

)33sin

22sin

1sin(2)( +++minus=

xxxxf π




where


22 dxT

dxxT

x ππ==

21))2sin()2cos((2

)(1

0 =sum ++=infin

=mx

Pmbx

Tmaaxf

mmm

ππ

)2cos()(2 2

2dxx

Tmxf

Ta

T

Tm

πint=

minus

)(2 2

20 dxxf

Ta

T

Tint=

minus

)2sin()(2 2

2dxx

Tmxf

Tb

T

Tm

πint=

minus

21)sincos(2

)(1

0 =sum ++=infin

=mtmbtmaatf

mmm ωω


))(2

)(2

(2

)(1


=

minusminus

m

timtimmtimtimm eei

beeaatf ωωωω

Euler formulae

Fourier seriesieex

eexxixexixe

ixix

ixix

ix

ix

2)(sin2)(cos

sincossincos

minus

minus

minus minus=+=

rArr⎭⎬⎫

minus=+=

For certain m=k

Denoting that as


tikkktikkk

tiktik

k

tiktik

kkk


ωω

ωωωω

ωω

minus

minusminus

++

minus=

minus+

+=+

22

22sincos

2

2

20

0kk

kkk

k

ibacibacac +=

minus== minus

int=

sumsum =++=

minus

minus

infin

minusinfin=

infin

=

minusminus

2

2

10

)(2

)()(

T

T

tikk

k

tikk

k

tikk

tikk

dtetfT

c

ecececctf

ω

ωωω

Fourier transform


Define


int=

int=

infin

infinminus

infin

infinminus

minus

ωωπ

ω

ω

ω

deFtf

dtetfF

ti

ti

)(21)(

)()(

intintrarr

sum intrarr

sum int=

infin

infinminus

minusinfin

infinminus

infin

minusinfin= minus

minus=

infin

minusinfin= minus

minus

ξξωπ

ξξωπ

ωξω

ξωωπ

ωω

defde

def

edtetfT

tf

iti

k

T

T

tikT

k

tikT

T

tik

)(21

)(1

))(2()(

2

2

)(2

2

2



int=intinfin

infinminus

infin









)(1)(a

Fa

G ωω =

)())(( ωωFixfFT =


In Matlab

F = fourier(f)


int=infin

infinminus


f = ifourier(F)


int=infin

infinminusωω

πω deFxf xi)(

21)(












int minus==infin

infinminus


ωω

1210)()(1

0minus=sum=

minus

=

minus NnetfFN

k

ktnikn

ωω



1

0

1( ) ( ) 012 1n k

Ni t

k nn

f t F e k NN

ωωminus

=

= = minussum






In Matlab

Y = fft(X)


Y = fft(Xn)


Y = ifft(X)


Y = ifft(Xn)



Delta function

⎩⎨⎧ =

==otherst

ttf0

01)()( δ





= +



= +






int minus=infin



=rArr







infinminus

infin

infinminus


)(21)( )(int int=

infin

infinminus

infin

infinminus

+ dudvevuFyxf vyuxi

π




12101210

)()(1

0

1

0

)(

minus=minus=

sum sum=minus

=

minus

=

+minus

NnMm

eyxfvuFM

xk

N

yk

yvxuiykxknm

yknxkm

sum=

sum ⎥⎦⎤

⎢⎣⎡ sum=

minus

=

minus

minus

=

minusminus

=

minus

1

0

1

0

1

0

)()(

)()(

M

m

xiunm

M

m

xiuN

n

yivykxknm

xkm

xkmykn

evxFvuF

eeyxfvuF




12101210

)()(1

0

1

0

)(

minus=minus=

sum sum=minus

=

minus

=

+minus

NykMxk

evuFyxfM

m

N

n

ykynvxkxmuinmykxk

[ ]sum=

sum sum=

minus

=

minus

minus

=

minus

=

1

0

1

0

1

0

)(1)(

)(1)(

M

m

xkxmiunxkykxk

M

m

xkxmiuN

n

ykynivnmykxk

evxFMN

yxf

eevuFMN

yxf






- Convolution




)(1)(bv

auF

abvuG =

In Matlab

Y = fft2(X)


Y = fft2(Xmn)


Y = ifft2(X)


Y = ifft2(Xmn)




89k 12k





tfNF ∆

=21





Tips windows


( ) sin(2 5 )f t tπ=

+=

Tips windows

( ) sin(2 5 )f t tπ=







f1(x)=sin(2pi500t)

and







Overview




In Matlab

Who is Fourier


Fourier series










Fourier transform



In Matlab




In Matlab












In Matlab





Tips windows

Tips windows





Fourier series




π

ππ

)sincos(2

)(1

0 sum ++=infin

=mmm mxbmxaaxf


π

ππ

dxxfa int=minus

π

ππ)(1

0


⎩⎨⎧

ltltminusminusltlt

=01

01)(

xx

xfπ

π

0

sin1sin1

cos)1(1cos1

0

0

0

01

=

minusminus=

int minus+int=

minus

minus

π

π

π

π

ππ

ππ

xx

dxxdxxa

0

sin1sin1

cos)1(1cos1

0

0

0

0

=

+minus=

int minus+int=

minus

minus

π

π

π

π

ππ

ππ

m

m

m

mxm

mxm

dxmxdxmxa

Period function


00 =int minus+int=

minusdxdxa

π

π

ππ

5314== n

nbn π

6420 == kbk


π

ππ

)55sin

33sin

1sin(4)( +++=

xxxxfπ


⎩⎨⎧

ltltminusminusltlt

=01

01)(

xx

xfπ

πPeriod function


)55sin

33sin

1sin(4)( +++=

xxxxfπ


⎩⎨⎧

ltltminusminusltlt

=01

01)(

xx

xfπ

πPeriod function


)55sin

33sin

1sin(4)( +++=

xxxxfπ

=+


⎩⎨⎧

ltltminusminusltlt

=01

01)(

xx

xfπ

πPeriod function


)55sin

33sin

1sin(4)( +++=

xxxxfπ

=+


⎩⎨⎧

ltltminusminusltlt

=01

01)(

xx

xfπ

πPeriod function


)55sin

33sin

1sin(4)( +++=

xxxxfπ

+ =


⎩⎨⎧

ltltminusminusltlt

=01

01)(

xx

xfπ

πPeriod function


)55sin

33sin

1sin(4)( +++=

xxxxfπ



The parameters are

)33sin

22sin

1sin(2)( +++minus=

xxxxf π




where


22 dxT

dxxT

x ππ==

21))2sin()2cos((2

)(1

0 =sum ++=infin

=mx

Pmbx

Tmaaxf

mmm

ππ

)2cos()(2 2

2dxx

Tmxf

Ta

T

Tm

πint=

minus

)(2 2

20 dxxf

Ta

T

Tint=

minus

)2sin()(2 2

2dxx

Tmxf

Tb

T

Tm

πint=

minus

21)sincos(2

)(1

0 =sum ++=infin

=mtmbtmaatf

mmm ωω


))(2

)(2

(2

)(1


=

minusminus

m

timtimmtimtimm eei

beeaatf ωωωω

Euler formulae

Fourier seriesieex

eexxixexixe

ixix

ixix

ix

ix

2)(sin2)(cos

sincossincos

minus

minus

minus minus=+=

rArr⎭⎬⎫

minus=+=

For certain m=k

Denoting that as


tikkktikkk

tiktik

k

tiktik

kkk


ωω

ωωωω

ωω

minus

minusminus

++

minus=

minus+

+=+

22

22sincos

2

2

20

0kk

kkk

k

ibacibacac +=

minus== minus

int=

sumsum =++=

minus

minus

infin

minusinfin=

infin

=

minusminus

2

2

10

)(2

)()(

T

T

tikk

k

tikk

k

tikk

tikk

dtetfT

c

ecececctf

ω

ωωω

Fourier transform


Define


int=

int=

infin

infinminus

infin

infinminus

minus

ωωπ

ω

ω

ω

deFtf

dtetfF

ti

ti

)(21)(

)()(

intintrarr

sum intrarr

sum int=

infin

infinminus

minusinfin

infinminus

infin

minusinfin= minus

minus=

infin

minusinfin= minus

minus

ξξωπ

ξξωπ

ωξω

ξωωπ

ωω

defde

def

edtetfT

tf

iti

k

T

T

tikT

k

tikT

T

tik

)(21

)(1

))(2()(

2

2

)(2

2

2



int=intinfin

infinminus

infin









)(1)(a

Fa

G ωω =

)())(( ωωFixfFT =


In Matlab

F = fourier(f)


int=infin

infinminus


f = ifourier(F)


int=infin

infinminusωω

πω deFxf xi)(

21)(












int minus==infin

infinminus


ωω

1210)()(1

0minus=sum=

minus

=

minus NnetfFN

k

ktnikn

ωω



1

0

1( ) ( ) 012 1n k

Ni t

k nn

f t F e k NN

ωωminus

=

= = minussum






In Matlab

Y = fft(X)


Y = fft(Xn)


Y = ifft(X)


Y = ifft(Xn)



Delta function

⎩⎨⎧ =

==otherst

ttf0

01)()( δ





= +



= +






int minus=infin



=rArr







infinminus

infin

infinminus


)(21)( )(int int=

infin

infinminus

infin

infinminus

+ dudvevuFyxf vyuxi

π




12101210

)()(1

0

1

0

)(

minus=minus=

sum sum=minus

=

minus

=

+minus

NnMm

eyxfvuFM

xk

N

yk

yvxuiykxknm

yknxkm

sum=

sum ⎥⎦⎤

⎢⎣⎡ sum=

minus

=

minus

minus

=

minusminus

=

minus

1

0

1

0

1

0

)()(

)()(

M

m

xiunm

M

m

xiuN

n

yivykxknm

xkm

xkmykn

evxFvuF

eeyxfvuF




12101210

)()(1

0

1

0

)(

minus=minus=

sum sum=minus

=

minus

=

+minus

NykMxk

evuFyxfM

m

N

n

ykynvxkxmuinmykxk

[ ]sum=

sum sum=

minus

=

minus

minus

=

minus

=

1

0

1

0

1

0

)(1)(

)(1)(

M

m

xkxmiunxkykxk

M

m

xkxmiuN

n

ykynivnmykxk

evxFMN

yxf

eevuFMN

yxf






- Convolution




)(1)(bv

auF

abvuG =

In Matlab

Y = fft2(X)


Y = fft2(Xmn)


Y = ifft2(X)


Y = ifft2(Xmn)




89k 12k





tfNF ∆

=21





Tips windows


( ) sin(2 5 )f t tπ=

+=

Tips windows

( ) sin(2 5 )f t tπ=







f1(x)=sin(2pi500t)

and







Overview




In Matlab

Who is Fourier


Fourier series










Fourier transform



In Matlab




In Matlab












In Matlab





Tips windows

Tips windows


⎩⎨⎧

ltltminusminusltlt

=01

01)(

xx

xfπ

π

0

sin1sin1

cos)1(1cos1

0

0

0

01

=

minusminus=

int minus+int=

minus

minus

π

π

π

π

ππ

ππ

xx

dxxdxxa

0

sin1sin1

cos)1(1cos1

0

0

0

0

=

+minus=

int minus+int=

minus

minus

π

π

π

π

ππ

ππ

m

m

m

mxm

mxm

dxmxdxmxa

Period function


00 =int minus+int=

minusdxdxa

π

π

ππ

5314== n

nbn π

6420 == kbk


π

ππ

)55sin

33sin

1sin(4)( +++=

xxxxfπ


⎩⎨⎧

ltltminusminusltlt

=01

01)(

xx

xfπ

πPeriod function


)55sin

33sin

1sin(4)( +++=

xxxxfπ


⎩⎨⎧

ltltminusminusltlt

=01

01)(

xx

xfπ

πPeriod function


)55sin

33sin

1sin(4)( +++=

xxxxfπ

=+


⎩⎨⎧

ltltminusminusltlt

=01

01)(

xx

xfπ

πPeriod function


)55sin

33sin

1sin(4)( +++=

xxxxfπ

=+


⎩⎨⎧

ltltminusminusltlt

=01

01)(

xx

xfπ

πPeriod function


)55sin

33sin

1sin(4)( +++=

xxxxfπ

+ =


⎩⎨⎧

ltltminusminusltlt

=01

01)(

xx

xfπ

πPeriod function


)55sin

33sin

1sin(4)( +++=

xxxxfπ



The parameters are

)33sin

22sin

1sin(2)( +++minus=

xxxxf π




where


22 dxT

dxxT

x ππ==

21))2sin()2cos((2

)(1

0 =sum ++=infin

=mx

Pmbx

Tmaaxf

mmm

ππ

)2cos()(2 2

2dxx

Tmxf

Ta

T

Tm

πint=

minus

)(2 2

20 dxxf

Ta

T

Tint=

minus

)2sin()(2 2

2dxx

Tmxf

Tb

T

Tm

πint=

minus

21)sincos(2

)(1

0 =sum ++=infin

=mtmbtmaatf

mmm ωω


))(2

)(2

(2

)(1


=

minusminus

m

timtimmtimtimm eei

beeaatf ωωωω

Euler formulae

Fourier seriesieex

eexxixexixe

ixix

ixix

ix

ix

2)(sin2)(cos

sincossincos

minus

minus

minus minus=+=

rArr⎭⎬⎫

minus=+=

For certain m=k

Denoting that as


tikkktikkk

tiktik

k

tiktik

kkk


ωω

ωωωω

ωω

minus

minusminus

++

minus=

minus+

+=+

22

22sincos

2

2

20

0kk

kkk

k

ibacibacac +=

minus== minus

int=

sumsum =++=

minus

minus

infin

minusinfin=

infin

=

minusminus

2

2

10

)(2

)()(

T

T

tikk

k

tikk

k

tikk

tikk

dtetfT

c

ecececctf

ω

ωωω

Fourier transform


Define


int=

int=

infin

infinminus

infin

infinminus

minus

ωωπ

ω

ω

ω

deFtf

dtetfF

ti

ti

)(21)(

)()(

intintrarr

sum intrarr

sum int=

infin

infinminus

minusinfin

infinminus

infin

minusinfin= minus

minus=

infin

minusinfin= minus

minus

ξξωπ

ξξωπ

ωξω

ξωωπ

ωω

defde

def

edtetfT

tf

iti

k

T

T

tikT

k

tikT

T

tik

)(21

)(1

))(2()(

2

2

)(2

2

2



int=intinfin

infinminus

infin









)(1)(a

Fa

G ωω =

)())(( ωωFixfFT =


In Matlab

F = fourier(f)


int=infin

infinminus


f = ifourier(F)


int=infin

infinminusωω

πω deFxf xi)(

21)(












int minus==infin

infinminus


ωω

1210)()(1

0minus=sum=

minus

=

minus NnetfFN

k

ktnikn

ωω



1

0

1( ) ( ) 012 1n k

Ni t

k nn

f t F e k NN

ωωminus

=

= = minussum






In Matlab

Y = fft(X)


Y = fft(Xn)


Y = ifft(X)


Y = ifft(Xn)



Delta function

⎩⎨⎧ =

==otherst

ttf0

01)()( δ





= +



= +






int minus=infin



=rArr







infinminus

infin

infinminus


)(21)( )(int int=

infin

infinminus

infin

infinminus

+ dudvevuFyxf vyuxi

π




12101210

)()(1

0

1

0

)(

minus=minus=

sum sum=minus

=

minus

=

+minus

NnMm

eyxfvuFM

xk

N

yk

yvxuiykxknm

yknxkm

sum=

sum ⎥⎦⎤

⎢⎣⎡ sum=

minus

=

minus

minus

=

minusminus

=

minus

1

0

1

0

1

0

)()(

)()(

M

m

xiunm

M

m

xiuN

n

yivykxknm

xkm

xkmykn

evxFvuF

eeyxfvuF




12101210

)()(1

0

1

0

)(

minus=minus=

sum sum=minus

=

minus

=

+minus

NykMxk

evuFyxfM

m

N

n

ykynvxkxmuinmykxk

[ ]sum=

sum sum=

minus

=

minus

minus

=

minus

=

1

0

1

0

1

0

)(1)(

)(1)(

M

m

xkxmiunxkykxk

M

m

xkxmiuN

n

ykynivnmykxk

evxFMN

yxf

eevuFMN

yxf






- Convolution




)(1)(bv

auF

abvuG =

In Matlab

Y = fft2(X)


Y = fft2(Xmn)


Y = ifft2(X)


Y = ifft2(Xmn)




89k 12k





tfNF ∆

=21





Tips windows


( ) sin(2 5 )f t tπ=

+=

Tips windows

( ) sin(2 5 )f t tπ=







f1(x)=sin(2pi500t)

and







Overview




In Matlab

Who is Fourier


Fourier series










Fourier transform



In Matlab




In Matlab












In Matlab





Tips windows

Tips windows


⎩⎨⎧

ltltminusminusltlt

=01

01)(

xx

xfπ

πPeriod function


)55sin

33sin

1sin(4)( +++=

xxxxfπ


⎩⎨⎧

ltltminusminusltlt

=01

01)(

xx

xfπ

πPeriod function


)55sin

33sin

1sin(4)( +++=

xxxxfπ

=+


⎩⎨⎧

ltltminusminusltlt

=01

01)(

xx

xfπ

πPeriod function


)55sin

33sin

1sin(4)( +++=

xxxxfπ

=+


⎩⎨⎧

ltltminusminusltlt

=01

01)(

xx

xfπ

πPeriod function


)55sin

33sin

1sin(4)( +++=

xxxxfπ

+ =


⎩⎨⎧

ltltminusminusltlt

=01

01)(

xx

xfπ

πPeriod function


)55sin

33sin

1sin(4)( +++=

xxxxfπ



The parameters are

)33sin

22sin

1sin(2)( +++minus=

xxxxf π




where


22 dxT

dxxT

x ππ==

21))2sin()2cos((2

)(1

0 =sum ++=infin

=mx

Pmbx

Tmaaxf

mmm

ππ

)2cos()(2 2

2dxx

Tmxf

Ta

T

Tm

πint=

minus

)(2 2

20 dxxf

Ta

T

Tint=

minus

)2sin()(2 2

2dxx

Tmxf

Tb

T

Tm

πint=

minus

21)sincos(2

)(1

0 =sum ++=infin

=mtmbtmaatf

mmm ωω


))(2

)(2

(2

)(1


=

minusminus

m

timtimmtimtimm eei

beeaatf ωωωω

Euler formulae

Fourier seriesieex

eexxixexixe

ixix

ixix

ix

ix

2)(sin2)(cos

sincossincos

minus

minus

minus minus=+=

rArr⎭⎬⎫

minus=+=

For certain m=k

Denoting that as


tikkktikkk

tiktik

k

tiktik

kkk


ωω

ωωωω

ωω

minus

minusminus

++

minus=

minus+

+=+

22

22sincos

2

2

20

0kk

kkk

k

ibacibacac +=

minus== minus

int=

sumsum =++=

minus

minus

infin

minusinfin=

infin

=

minusminus

2

2

10

)(2

)()(

T

T

tikk

k

tikk

k

tikk

tikk

dtetfT

c

ecececctf

ω

ωωω

Fourier transform


Define


int=

int=

infin

infinminus

infin

infinminus

minus

ωωπ

ω

ω

ω

deFtf

dtetfF

ti

ti

)(21)(

)()(

intintrarr

sum intrarr

sum int=

infin

infinminus

minusinfin

infinminus

infin

minusinfin= minus

minus=

infin

minusinfin= minus

minus

ξξωπ

ξξωπ

ωξω

ξωωπ

ωω

defde

def

edtetfT

tf

iti

k

T

T

tikT

k

tikT

T

tik

)(21

)(1

))(2()(

2

2

)(2

2

2



int=intinfin

infinminus

infin









)(1)(a

Fa

G ωω =

)())(( ωωFixfFT =


In Matlab

F = fourier(f)


int=infin

infinminus


f = ifourier(F)


int=infin

infinminusωω

πω deFxf xi)(

21)(












int minus==infin

infinminus


ωω

1210)()(1

0minus=sum=

minus

=

minus NnetfFN

k

ktnikn

ωω



1

0

1( ) ( ) 012 1n k

Ni t

k nn

f t F e k NN

ωωminus

=

= = minussum






In Matlab

Y = fft(X)


Y = fft(Xn)


Y = ifft(X)


Y = ifft(Xn)



Delta function

⎩⎨⎧ =

==otherst

ttf0

01)()( δ





= +



= +






int minus=infin



=rArr







infinminus

infin

infinminus


)(21)( )(int int=

infin

infinminus

infin

infinminus

+ dudvevuFyxf vyuxi

π




12101210

)()(1

0

1

0

)(

minus=minus=

sum sum=minus

=

minus

=

+minus

NnMm

eyxfvuFM

xk

N

yk

yvxuiykxknm

yknxkm

sum=

sum ⎥⎦⎤

⎢⎣⎡ sum=

minus

=

minus

minus

=

minusminus

=

minus

1

0

1

0

1

0

)()(

)()(

M

m

xiunm

M

m

xiuN

n

yivykxknm

xkm

xkmykn

evxFvuF

eeyxfvuF




12101210

)()(1

0

1

0

)(

minus=minus=

sum sum=minus

=

minus

=

+minus

NykMxk

evuFyxfM

m

N

n

ykynvxkxmuinmykxk

[ ]sum=

sum sum=

minus

=

minus

minus

=

minus

=

1

0

1

0

1

0

)(1)(

)(1)(

M

m

xkxmiunxkykxk

M

m

xkxmiuN

n

ykynivnmykxk

evxFMN

yxf

eevuFMN

yxf






- Convolution




)(1)(bv

auF

abvuG =

In Matlab

Y = fft2(X)


Y = fft2(Xmn)


Y = ifft2(X)


Y = ifft2(Xmn)




89k 12k





tfNF ∆

=21





Tips windows


( ) sin(2 5 )f t tπ=

+=

Tips windows

( ) sin(2 5 )f t tπ=







f1(x)=sin(2pi500t)

and







Overview




In Matlab

Who is Fourier


Fourier series










Fourier transform



In Matlab




In Matlab












In Matlab





Tips windows

Tips windows


⎩⎨⎧

ltltminusminusltlt

=01

01)(

xx

xfπ

πPeriod function


)55sin

33sin

1sin(4)( +++=

xxxxfπ

=+


⎩⎨⎧

ltltminusminusltlt

=01

01)(

xx

xfπ

πPeriod function


)55sin

33sin

1sin(4)( +++=

xxxxfπ

=+


⎩⎨⎧

ltltminusminusltlt

=01

01)(

xx

xfπ

πPeriod function


)55sin

33sin

1sin(4)( +++=

xxxxfπ

+ =


⎩⎨⎧

ltltminusminusltlt

=01

01)(

xx

xfπ

πPeriod function


)55sin

33sin

1sin(4)( +++=

xxxxfπ



The parameters are

)33sin

22sin

1sin(2)( +++minus=

xxxxf π




where


22 dxT

dxxT

x ππ==

21))2sin()2cos((2

)(1

0 =sum ++=infin

=mx

Pmbx

Tmaaxf

mmm

ππ

)2cos()(2 2

2dxx

Tmxf

Ta

T

Tm

πint=

minus

)(2 2

20 dxxf

Ta

T

Tint=

minus

)2sin()(2 2

2dxx

Tmxf

Tb

T

Tm

πint=

minus

21)sincos(2

)(1

0 =sum ++=infin

=mtmbtmaatf

mmm ωω


))(2

)(2

(2

)(1


=

minusminus

m

timtimmtimtimm eei

beeaatf ωωωω

Euler formulae

Fourier seriesieex

eexxixexixe

ixix

ixix

ix

ix

2)(sin2)(cos

sincossincos

minus

minus

minus minus=+=

rArr⎭⎬⎫

minus=+=

For certain m=k

Denoting that as


tikkktikkk

tiktik

k

tiktik

kkk


ωω

ωωωω

ωω

minus

minusminus

++

minus=

minus+

+=+

22

22sincos

2

2

20

0kk

kkk

k

ibacibacac +=

minus== minus

int=

sumsum =++=

minus

minus

infin

minusinfin=

infin

=

minusminus

2

2

10

)(2

)()(

T

T

tikk

k

tikk

k

tikk

tikk

dtetfT

c

ecececctf

ω

ωωω

Fourier transform


Define


int=

int=

infin

infinminus

infin

infinminus

minus

ωωπ

ω

ω

ω

deFtf

dtetfF

ti

ti

)(21)(

)()(

intintrarr

sum intrarr

sum int=

infin

infinminus

minusinfin

infinminus

infin

minusinfin= minus

minus=

infin

minusinfin= minus

minus

ξξωπ

ξξωπ

ωξω

ξωωπ

ωω

defde

def

edtetfT

tf

iti

k

T

T

tikT

k

tikT

T

tik

)(21

)(1

))(2()(

2

2

)(2

2

2



int=intinfin

infinminus

infin









)(1)(a

Fa

G ωω =

)())(( ωωFixfFT =


In Matlab

F = fourier(f)


int=infin

infinminus


f = ifourier(F)


int=infin

infinminusωω

πω deFxf xi)(

21)(












int minus==infin

infinminus


ωω

1210)()(1

0minus=sum=

minus

=

minus NnetfFN

k

ktnikn

ωω



1

0

1( ) ( ) 012 1n k

Ni t

k nn

f t F e k NN

ωωminus

=

= = minussum






In Matlab

Y = fft(X)


Y = fft(Xn)


Y = ifft(X)


Y = ifft(Xn)



Delta function

⎩⎨⎧ =

==otherst

ttf0

01)()( δ





= +



= +






int minus=infin



=rArr







infinminus

infin

infinminus


)(21)( )(int int=

infin

infinminus

infin

infinminus

+ dudvevuFyxf vyuxi

π




12101210

)()(1

0

1

0

)(

minus=minus=

sum sum=minus

=

minus

=

+minus

NnMm

eyxfvuFM

xk

N

yk

yvxuiykxknm

yknxkm

sum=

sum ⎥⎦⎤

⎢⎣⎡ sum=

minus

=

minus

minus

=

minusminus

=

minus

1

0

1

0

1

0

)()(

)()(

M

m

xiunm

M

m

xiuN

n

yivykxknm

xkm

xkmykn

evxFvuF

eeyxfvuF




12101210

)()(1

0

1

0

)(

minus=minus=

sum sum=minus

=

minus

=

+minus

NykMxk

evuFyxfM

m

N

n

ykynvxkxmuinmykxk

[ ]sum=

sum sum=

minus

=

minus

minus

=

minus

=

1

0

1

0

1

0

)(1)(

)(1)(

M

m

xkxmiunxkykxk

M

m

xkxmiuN

n

ykynivnmykxk

evxFMN

yxf

eevuFMN

yxf






- Convolution




)(1)(bv

auF

abvuG =

In Matlab

Y = fft2(X)


Y = fft2(Xmn)


Y = ifft2(X)


Y = ifft2(Xmn)




89k 12k





tfNF ∆

=21





Tips windows


( ) sin(2 5 )f t tπ=

+=

Tips windows

( ) sin(2 5 )f t tπ=







f1(x)=sin(2pi500t)

and







Overview




In Matlab

Who is Fourier


Fourier series










Fourier transform



In Matlab




In Matlab












In Matlab





Tips windows

Tips windows


⎩⎨⎧

ltltminusminusltlt

=01

01)(

xx

xfπ

πPeriod function


)55sin

33sin

1sin(4)( +++=

xxxxfπ

=+


⎩⎨⎧

ltltminusminusltlt

=01

01)(

xx

xfπ

πPeriod function


)55sin

33sin

1sin(4)( +++=

xxxxfπ

+ =


⎩⎨⎧

ltltminusminusltlt

=01

01)(

xx

xfπ

πPeriod function


)55sin

33sin

1sin(4)( +++=

xxxxfπ



The parameters are

)33sin

22sin

1sin(2)( +++minus=

xxxxf π




where


22 dxT

dxxT

x ππ==

21))2sin()2cos((2

)(1

0 =sum ++=infin

=mx

Pmbx

Tmaaxf

mmm

ππ

)2cos()(2 2

2dxx

Tmxf

Ta

T

Tm

πint=

minus

)(2 2

20 dxxf

Ta

T

Tint=

minus

)2sin()(2 2

2dxx

Tmxf

Tb

T

Tm

πint=

minus

21)sincos(2

)(1

0 =sum ++=infin

=mtmbtmaatf

mmm ωω


))(2

)(2

(2

)(1


=

minusminus

m

timtimmtimtimm eei

beeaatf ωωωω

Euler formulae

Fourier seriesieex

eexxixexixe

ixix

ixix

ix

ix

2)(sin2)(cos

sincossincos

minus

minus

minus minus=+=

rArr⎭⎬⎫

minus=+=

For certain m=k

Denoting that as


tikkktikkk

tiktik

k

tiktik

kkk


ωω

ωωωω

ωω

minus

minusminus

++

minus=

minus+

+=+

22

22sincos

2

2

20

0kk

kkk

k

ibacibacac +=

minus== minus

int=

sumsum =++=

minus

minus

infin

minusinfin=

infin

=

minusminus

2

2

10

)(2

)()(

T

T

tikk

k

tikk

k

tikk

tikk

dtetfT

c

ecececctf

ω

ωωω

Fourier transform


Define


int=

int=

infin

infinminus

infin

infinminus

minus

ωωπ

ω

ω

ω

deFtf

dtetfF

ti

ti

)(21)(

)()(

intintrarr

sum intrarr

sum int=

infin

infinminus

minusinfin

infinminus

infin

minusinfin= minus

minus=

infin

minusinfin= minus

minus

ξξωπ

ξξωπ

ωξω

ξωωπ

ωω

defde

def

edtetfT

tf

iti

k

T

T

tikT

k

tikT

T

tik

)(21

)(1

))(2()(

2

2

)(2

2

2



int=intinfin

infinminus

infin









)(1)(a

Fa

G ωω =

)())(( ωωFixfFT =


In Matlab

F = fourier(f)


int=infin

infinminus


f = ifourier(F)


int=infin

infinminusωω

πω deFxf xi)(

21)(












int minus==infin

infinminus


ωω

1210)()(1

0minus=sum=

minus

=

minus NnetfFN

k

ktnikn

ωω



1

0

1( ) ( ) 012 1n k

Ni t

k nn

f t F e k NN

ωωminus

=

= = minussum






In Matlab

Y = fft(X)


Y = fft(Xn)


Y = ifft(X)


Y = ifft(Xn)



Delta function

⎩⎨⎧ =

==otherst

ttf0

01)()( δ





= +



= +






int minus=infin



=rArr







infinminus

infin

infinminus


)(21)( )(int int=

infin

infinminus

infin

infinminus

+ dudvevuFyxf vyuxi

π




12101210

)()(1

0

1

0

)(

minus=minus=

sum sum=minus

=

minus

=

+minus

NnMm

eyxfvuFM

xk

N

yk

yvxuiykxknm

yknxkm

sum=

sum ⎥⎦⎤

⎢⎣⎡ sum=

minus

=

minus

minus

=

minusminus

=

minus

1

0

1

0

1

0

)()(

)()(

M

m

xiunm

M

m

xiuN

n

yivykxknm

xkm

xkmykn

evxFvuF

eeyxfvuF




12101210

)()(1

0

1

0

)(

minus=minus=

sum sum=minus

=

minus

=

+minus

NykMxk

evuFyxfM

m

N

n

ykynvxkxmuinmykxk

[ ]sum=

sum sum=

minus

=

minus

minus

=

minus

=

1

0

1

0

1

0

)(1)(

)(1)(

M

m

xkxmiunxkykxk

M

m

xkxmiuN

n

ykynivnmykxk

evxFMN

yxf

eevuFMN

yxf






- Convolution




)(1)(bv

auF

abvuG =

In Matlab

Y = fft2(X)


Y = fft2(Xmn)


Y = ifft2(X)


Y = ifft2(Xmn)




89k 12k





tfNF ∆

=21





Tips windows


( ) sin(2 5 )f t tπ=

+=

Tips windows

( ) sin(2 5 )f t tπ=







f1(x)=sin(2pi500t)

and







Overview




In Matlab

Who is Fourier


Fourier series










Fourier transform



In Matlab




In Matlab












In Matlab





Tips windows

Tips windows


⎩⎨⎧

ltltminusminusltlt

=01

01)(

xx

xfπ

πPeriod function


)55sin

33sin

1sin(4)( +++=

xxxxfπ

+ =


⎩⎨⎧

ltltminusminusltlt

=01

01)(

xx

xfπ

πPeriod function


)55sin

33sin

1sin(4)( +++=

xxxxfπ



The parameters are

)33sin

22sin

1sin(2)( +++minus=

xxxxf π




where


22 dxT

dxxT

x ππ==

21))2sin()2cos((2

)(1

0 =sum ++=infin

=mx

Pmbx

Tmaaxf

mmm

ππ

)2cos()(2 2

2dxx

Tmxf

Ta

T

Tm

πint=

minus

)(2 2

20 dxxf

Ta

T

Tint=

minus

)2sin()(2 2

2dxx

Tmxf

Tb

T

Tm

πint=

minus

21)sincos(2

)(1

0 =sum ++=infin

=mtmbtmaatf

mmm ωω


))(2

)(2

(2

)(1


=

minusminus

m

timtimmtimtimm eei

beeaatf ωωωω

Euler formulae

Fourier seriesieex

eexxixexixe

ixix

ixix

ix

ix

2)(sin2)(cos

sincossincos

minus

minus

minus minus=+=

rArr⎭⎬⎫

minus=+=

For certain m=k

Denoting that as


tikkktikkk

tiktik

k

tiktik

kkk


ωω

ωωωω

ωω

minus

minusminus

++

minus=

minus+

+=+

22

22sincos

2

2

20

0kk

kkk

k

ibacibacac +=

minus== minus

int=

sumsum =++=

minus

minus

infin

minusinfin=

infin

=

minusminus

2

2

10

)(2

)()(

T

T

tikk

k

tikk

k

tikk

tikk

dtetfT

c

ecececctf

ω

ωωω

Fourier transform


Define


int=

int=

infin

infinminus

infin

infinminus

minus

ωωπ

ω

ω

ω

deFtf

dtetfF

ti

ti

)(21)(

)()(

intintrarr

sum intrarr

sum int=

infin

infinminus

minusinfin

infinminus

infin

minusinfin= minus

minus=

infin

minusinfin= minus

minus

ξξωπ

ξξωπ

ωξω

ξωωπ

ωω

defde

def

edtetfT

tf

iti

k

T

T

tikT

k

tikT

T

tik

)(21

)(1

))(2()(

2

2

)(2

2

2



int=intinfin

infinminus

infin









)(1)(a

Fa

G ωω =

)())(( ωωFixfFT =


In Matlab

F = fourier(f)


int=infin

infinminus


f = ifourier(F)


int=infin

infinminusωω

πω deFxf xi)(

21)(












int minus==infin

infinminus


ωω

1210)()(1

0minus=sum=

minus

=

minus NnetfFN

k

ktnikn

ωω



1

0

1( ) ( ) 012 1n k

Ni t

k nn

f t F e k NN

ωωminus

=

= = minussum






In Matlab

Y = fft(X)


Y = fft(Xn)


Y = ifft(X)


Y = ifft(Xn)



Delta function

⎩⎨⎧ =

==otherst

ttf0

01)()( δ





= +



= +






int minus=infin



=rArr







infinminus

infin

infinminus


)(21)( )(int int=

infin

infinminus

infin

infinminus

+ dudvevuFyxf vyuxi

π




12101210

)()(1

0

1

0

)(

minus=minus=

sum sum=minus

=

minus

=

+minus

NnMm

eyxfvuFM

xk

N

yk

yvxuiykxknm

yknxkm

sum=

sum ⎥⎦⎤

⎢⎣⎡ sum=

minus

=

minus

minus

=

minusminus

=

minus

1

0

1

0

1

0

)()(

)()(

M

m

xiunm

M

m

xiuN

n

yivykxknm

xkm

xkmykn

evxFvuF

eeyxfvuF




12101210

)()(1

0

1

0

)(

minus=minus=

sum sum=minus

=

minus

=

+minus

NykMxk

evuFyxfM

m

N

n

ykynvxkxmuinmykxk

[ ]sum=

sum sum=

minus

=

minus

minus

=

minus

=

1

0

1

0

1

0

)(1)(

)(1)(

M

m

xkxmiunxkykxk

M

m

xkxmiuN

n

ykynivnmykxk

evxFMN

yxf

eevuFMN

yxf






- Convolution




)(1)(bv

auF

abvuG =

In Matlab

Y = fft2(X)


Y = fft2(Xmn)


Y = ifft2(X)


Y = ifft2(Xmn)




89k 12k





tfNF ∆

=21





Tips windows


( ) sin(2 5 )f t tπ=

+=

Tips windows

( ) sin(2 5 )f t tπ=







f1(x)=sin(2pi500t)

and







Overview




In Matlab

Who is Fourier


Fourier series










Fourier transform



In Matlab




In Matlab












In Matlab





Tips windows

Tips windows


⎩⎨⎧

ltltminusminusltlt

=01

01)(

xx

xfπ

πPeriod function


)55sin

33sin

1sin(4)( +++=

xxxxfπ



The parameters are

)33sin

22sin

1sin(2)( +++minus=

xxxxf π




where


22 dxT

dxxT

x ππ==

21))2sin()2cos((2

)(1

0 =sum ++=infin

=mx

Pmbx

Tmaaxf

mmm

ππ

)2cos()(2 2

2dxx

Tmxf

Ta

T

Tm

πint=

minus

)(2 2

20 dxxf

Ta

T

Tint=

minus

)2sin()(2 2

2dxx

Tmxf

Tb

T

Tm

πint=

minus

21)sincos(2

)(1

0 =sum ++=infin

=mtmbtmaatf

mmm ωω


))(2

)(2

(2

)(1


=

minusminus

m

timtimmtimtimm eei

beeaatf ωωωω

Euler formulae

Fourier seriesieex

eexxixexixe

ixix

ixix

ix

ix

2)(sin2)(cos

sincossincos

minus

minus

minus minus=+=

rArr⎭⎬⎫

minus=+=

For certain m=k

Denoting that as


tikkktikkk

tiktik

k

tiktik

kkk


ωω

ωωωω

ωω

minus

minusminus

++

minus=

minus+

+=+

22

22sincos

2

2

20

0kk

kkk

k

ibacibacac +=

minus== minus

int=

sumsum =++=

minus

minus

infin

minusinfin=

infin

=

minusminus

2

2

10

)(2

)()(

T

T

tikk

k

tikk

k

tikk

tikk

dtetfT

c

ecececctf

ω

ωωω

Fourier transform


Define


int=

int=

infin

infinminus

infin

infinminus

minus

ωωπ

ω

ω

ω

deFtf

dtetfF

ti

ti

)(21)(

)()(

intintrarr

sum intrarr

sum int=

infin

infinminus

minusinfin

infinminus

infin

minusinfin= minus

minus=

infin

minusinfin= minus

minus

ξξωπ

ξξωπ

ωξω

ξωωπ

ωω

defde

def

edtetfT

tf

iti

k

T

T

tikT

k

tikT

T

tik

)(21

)(1

))(2()(

2

2

)(2

2

2



int=intinfin

infinminus

infin









)(1)(a

Fa

G ωω =

)())(( ωωFixfFT =


In Matlab

F = fourier(f)


int=infin

infinminus


f = ifourier(F)


int=infin

infinminusωω

πω deFxf xi)(

21)(












int minus==infin

infinminus


ωω

1210)()(1

0minus=sum=

minus

=

minus NnetfFN

k

ktnikn

ωω



1

0

1( ) ( ) 012 1n k

Ni t

k nn

f t F e k NN

ωωminus

=

= = minussum






In Matlab

Y = fft(X)


Y = fft(Xn)


Y = ifft(X)


Y = ifft(Xn)



Delta function

⎩⎨⎧ =

==otherst

ttf0

01)()( δ





= +



= +






int minus=infin



=rArr







infinminus

infin

infinminus


)(21)( )(int int=

infin

infinminus

infin

infinminus

+ dudvevuFyxf vyuxi

π




12101210

)()(1

0

1

0

)(

minus=minus=

sum sum=minus

=

minus

=

+minus

NnMm

eyxfvuFM

xk

N

yk

yvxuiykxknm

yknxkm

sum=

sum ⎥⎦⎤

⎢⎣⎡ sum=

minus

=

minus

minus

=

minusminus

=

minus

1

0

1

0

1

0

)()(

)()(

M

m

xiunm

M

m

xiuN

n

yivykxknm

xkm

xkmykn

evxFvuF

eeyxfvuF




12101210

)()(1

0

1

0

)(

minus=minus=

sum sum=minus

=

minus

=

+minus

NykMxk

evuFyxfM

m

N

n

ykynvxkxmuinmykxk

[ ]sum=

sum sum=

minus

=

minus

minus

=

minus

=

1

0

1

0

1

0

)(1)(

)(1)(

M

m

xkxmiunxkykxk

M

m

xkxmiuN

n

ykynivnmykxk

evxFMN

yxf

eevuFMN

yxf






- Convolution




)(1)(bv

auF

abvuG =

In Matlab

Y = fft2(X)


Y = fft2(Xmn)


Y = ifft2(X)


Y = ifft2(Xmn)




89k 12k





tfNF ∆

=21





Tips windows


( ) sin(2 5 )f t tπ=

+=

Tips windows

( ) sin(2 5 )f t tπ=







f1(x)=sin(2pi500t)

and







Overview




In Matlab

Who is Fourier


Fourier series










Fourier transform



In Matlab




In Matlab












In Matlab





Tips windows

Tips windows



The parameters are

)33sin

22sin

1sin(2)( +++minus=

xxxxf π




where


22 dxT

dxxT

x ππ==

21))2sin()2cos((2

)(1

0 =sum ++=infin

=mx

Pmbx

Tmaaxf

mmm

ππ

)2cos()(2 2

2dxx

Tmxf

Ta

T

Tm

πint=

minus

)(2 2

20 dxxf

Ta

T

Tint=

minus

)2sin()(2 2

2dxx

Tmxf

Tb

T

Tm

πint=

minus

21)sincos(2

)(1

0 =sum ++=infin

=mtmbtmaatf

mmm ωω


))(2

)(2

(2

)(1


=

minusminus

m

timtimmtimtimm eei

beeaatf ωωωω

Euler formulae

Fourier seriesieex

eexxixexixe

ixix

ixix

ix

ix

2)(sin2)(cos

sincossincos

minus

minus

minus minus=+=

rArr⎭⎬⎫

minus=+=

For certain m=k

Denoting that as


tikkktikkk

tiktik

k

tiktik

kkk


ωω

ωωωω

ωω

minus

minusminus

++

minus=

minus+

+=+

22

22sincos

2

2

20

0kk

kkk

k

ibacibacac +=

minus== minus

int=

sumsum =++=

minus

minus

infin

minusinfin=

infin

=

minusminus

2

2

10

)(2

)()(

T

T

tikk

k

tikk

k

tikk

tikk

dtetfT

c

ecececctf

ω

ωωω

Fourier transform


Define


int=

int=

infin

infinminus

infin

infinminus

minus

ωωπ

ω

ω

ω

deFtf

dtetfF

ti

ti

)(21)(

)()(

intintrarr

sum intrarr

sum int=

infin

infinminus

minusinfin

infinminus

infin

minusinfin= minus

minus=

infin

minusinfin= minus

minus

ξξωπ

ξξωπ

ωξω

ξωωπ

ωω

defde

def

edtetfT

tf

iti

k

T

T

tikT

k

tikT

T

tik

)(21

)(1

))(2()(

2

2

)(2

2

2



int=intinfin

infinminus

infin









)(1)(a

Fa

G ωω =

)())(( ωωFixfFT =


In Matlab

F = fourier(f)


int=infin

infinminus


f = ifourier(F)


int=infin

infinminusωω

πω deFxf xi)(

21)(












int minus==infin

infinminus


ωω

1210)()(1

0minus=sum=

minus

=

minus NnetfFN

k

ktnikn

ωω



1

0

1( ) ( ) 012 1n k

Ni t

k nn

f t F e k NN

ωωminus

=

= = minussum






In Matlab

Y = fft(X)


Y = fft(Xn)


Y = ifft(X)


Y = ifft(Xn)



Delta function

⎩⎨⎧ =

==otherst

ttf0

01)()( δ





= +



= +






int minus=infin



=rArr







infinminus

infin

infinminus


)(21)( )(int int=

infin

infinminus

infin

infinminus

+ dudvevuFyxf vyuxi

π




12101210

)()(1

0

1

0

)(

minus=minus=

sum sum=minus

=

minus

=

+minus

NnMm

eyxfvuFM

xk

N

yk

yvxuiykxknm

yknxkm

sum=

sum ⎥⎦⎤

⎢⎣⎡ sum=

minus

=

minus

minus

=

minusminus

=

minus

1

0

1

0

1

0

)()(

)()(

M

m

xiunm

M

m

xiuN

n

yivykxknm

xkm

xkmykn

evxFvuF

eeyxfvuF




12101210

)()(1

0

1

0

)(

minus=minus=

sum sum=minus

=

minus

=

+minus

NykMxk

evuFyxfM

m

N

n

ykynvxkxmuinmykxk

[ ]sum=

sum sum=

minus

=

minus

minus

=

minus

=

1

0

1

0

1

0

)(1)(

)(1)(

M

m

xkxmiunxkykxk

M

m

xkxmiuN

n

ykynivnmykxk

evxFMN

yxf

eevuFMN

yxf






- Convolution




)(1)(bv

auF

abvuG =

In Matlab

Y = fft2(X)


Y = fft2(Xmn)


Y = ifft2(X)


Y = ifft2(Xmn)




89k 12k





tfNF ∆

=21





Tips windows


( ) sin(2 5 )f t tπ=

+=

Tips windows

( ) sin(2 5 )f t tπ=







f1(x)=sin(2pi500t)

and







Overview




In Matlab

Who is Fourier


Fourier series










Fourier transform



In Matlab




In Matlab












In Matlab





Tips windows

Tips windows




where


22 dxT

dxxT

x ππ==

21))2sin()2cos((2

)(1

0 =sum ++=infin

=mx

Pmbx

Tmaaxf

mmm

ππ

)2cos()(2 2

2dxx

Tmxf

Ta

T

Tm

πint=

minus

)(2 2

20 dxxf

Ta

T

Tint=

minus

)2sin()(2 2

2dxx

Tmxf

Tb

T

Tm

πint=

minus

21)sincos(2

)(1

0 =sum ++=infin

=mtmbtmaatf

mmm ωω


))(2

)(2

(2

)(1


=

minusminus

m

timtimmtimtimm eei

beeaatf ωωωω

Euler formulae

Fourier seriesieex

eexxixexixe

ixix

ixix

ix

ix

2)(sin2)(cos

sincossincos

minus

minus

minus minus=+=

rArr⎭⎬⎫

minus=+=

For certain m=k

Denoting that as


tikkktikkk

tiktik

k

tiktik

kkk


ωω

ωωωω

ωω

minus

minusminus

++

minus=

minus+

+=+

22

22sincos

2

2

20

0kk

kkk

k

ibacibacac +=

minus== minus

int=

sumsum =++=

minus

minus

infin

minusinfin=

infin

=

minusminus

2

2

10

)(2

)()(

T

T

tikk

k

tikk

k

tikk

tikk

dtetfT

c

ecececctf

ω

ωωω

Fourier transform


Define


int=

int=

infin

infinminus

infin

infinminus

minus

ωωπ

ω

ω

ω

deFtf

dtetfF

ti

ti

)(21)(

)()(

intintrarr

sum intrarr

sum int=

infin

infinminus

minusinfin

infinminus

infin

minusinfin= minus

minus=

infin

minusinfin= minus

minus

ξξωπ

ξξωπ

ωξω

ξωωπ

ωω

defde

def

edtetfT

tf

iti

k

T

T

tikT

k

tikT

T

tik

)(21

)(1

))(2()(

2

2

)(2

2

2



int=intinfin

infinminus

infin









)(1)(a

Fa

G ωω =

)())(( ωωFixfFT =


In Matlab

F = fourier(f)


int=infin

infinminus


f = ifourier(F)


int=infin

infinminusωω

πω deFxf xi)(

21)(












int minus==infin

infinminus


ωω

1210)()(1

0minus=sum=

minus

=

minus NnetfFN

k

ktnikn

ωω



1

0

1( ) ( ) 012 1n k

Ni t

k nn

f t F e k NN

ωωminus

=

= = minussum






In Matlab

Y = fft(X)


Y = fft(Xn)


Y = ifft(X)


Y = ifft(Xn)



Delta function

⎩⎨⎧ =

==otherst

ttf0

01)()( δ





= +



= +






int minus=infin



=rArr







infinminus

infin

infinminus


)(21)( )(int int=

infin

infinminus

infin

infinminus

+ dudvevuFyxf vyuxi

π




12101210

)()(1

0

1

0

)(

minus=minus=

sum sum=minus

=

minus

=

+minus

NnMm

eyxfvuFM

xk

N

yk

yvxuiykxknm

yknxkm

sum=

sum ⎥⎦⎤

⎢⎣⎡ sum=

minus

=

minus

minus

=

minusminus

=

minus

1

0

1

0

1

0

)()(

)()(

M

m

xiunm

M

m

xiuN

n

yivykxknm

xkm

xkmykn

evxFvuF

eeyxfvuF




12101210

)()(1

0

1

0

)(

minus=minus=

sum sum=minus

=

minus

=

+minus

NykMxk

evuFyxfM

m

N

n

ykynvxkxmuinmykxk

[ ]sum=

sum sum=

minus

=

minus

minus

=

minus

=

1

0

1

0

1

0

)(1)(

)(1)(

M

m

xkxmiunxkykxk

M

m

xkxmiuN

n

ykynivnmykxk

evxFMN

yxf

eevuFMN

yxf






- Convolution




)(1)(bv

auF

abvuG =

In Matlab

Y = fft2(X)


Y = fft2(Xmn)


Y = ifft2(X)


Y = ifft2(Xmn)




89k 12k





tfNF ∆

=21





Tips windows


( ) sin(2 5 )f t tπ=

+=

Tips windows

( ) sin(2 5 )f t tπ=







f1(x)=sin(2pi500t)

and







Overview




In Matlab

Who is Fourier


Fourier series










Fourier transform



In Matlab




In Matlab












In Matlab





Tips windows

Tips windows


))(2

)(2

(2

)(1


=

minusminus

m

timtimmtimtimm eei

beeaatf ωωωω

Euler formulae

Fourier seriesieex

eexxixexixe

ixix

ixix

ix

ix

2)(sin2)(cos

sincossincos

minus

minus

minus minus=+=

rArr⎭⎬⎫

minus=+=

For certain m=k

Denoting that as


tikkktikkk

tiktik

k

tiktik

kkk


ωω

ωωωω

ωω

minus

minusminus

++

minus=

minus+

+=+

22

22sincos

2

2

20

0kk

kkk

k

ibacibacac +=

minus== minus

int=

sumsum =++=

minus

minus

infin

minusinfin=

infin

=

minusminus

2

2

10

)(2

)()(

T

T

tikk

k

tikk

k

tikk

tikk

dtetfT

c

ecececctf

ω

ωωω

Fourier transform


Define


int=

int=

infin

infinminus

infin

infinminus

minus

ωωπ

ω

ω

ω

deFtf

dtetfF

ti

ti

)(21)(

)()(

intintrarr

sum intrarr

sum int=

infin

infinminus

minusinfin

infinminus

infin

minusinfin= minus

minus=

infin

minusinfin= minus

minus

ξξωπ

ξξωπ

ωξω

ξωωπ

ωω

defde

def

edtetfT

tf

iti

k

T

T

tikT

k

tikT

T

tik

)(21

)(1

))(2()(

2

2

)(2

2

2



int=intinfin

infinminus

infin









)(1)(a

Fa

G ωω =

)())(( ωωFixfFT =


In Matlab

F = fourier(f)


int=infin

infinminus


f = ifourier(F)


int=infin

infinminusωω

πω deFxf xi)(

21)(












int minus==infin

infinminus


ωω

1210)()(1

0minus=sum=

minus

=

minus NnetfFN

k

ktnikn

ωω



1

0

1( ) ( ) 012 1n k

Ni t

k nn

f t F e k NN

ωωminus

=

= = minussum






In Matlab

Y = fft(X)


Y = fft(Xn)


Y = ifft(X)


Y = ifft(Xn)



Delta function

⎩⎨⎧ =

==otherst

ttf0

01)()( δ





= +



= +






int minus=infin



=rArr







infinminus

infin

infinminus


)(21)( )(int int=

infin

infinminus

infin

infinminus

+ dudvevuFyxf vyuxi

π




12101210

)()(1

0

1

0

)(

minus=minus=

sum sum=minus

=

minus

=

+minus

NnMm

eyxfvuFM

xk

N

yk

yvxuiykxknm

yknxkm

sum=

sum ⎥⎦⎤

⎢⎣⎡ sum=

minus

=

minus

minus

=

minusminus

=

minus

1

0

1

0

1

0

)()(

)()(

M

m

xiunm

M

m

xiuN

n

yivykxknm

xkm

xkmykn

evxFvuF

eeyxfvuF




12101210

)()(1

0

1

0

)(

minus=minus=

sum sum=minus

=

minus

=

+minus

NykMxk

evuFyxfM

m

N

n

ykynvxkxmuinmykxk

[ ]sum=

sum sum=

minus

=

minus

minus

=

minus

=

1

0

1

0

1

0

)(1)(

)(1)(

M

m

xkxmiunxkykxk

M

m

xkxmiuN

n

ykynivnmykxk

evxFMN

yxf

eevuFMN

yxf






- Convolution




)(1)(bv

auF

abvuG =

In Matlab

Y = fft2(X)


Y = fft2(Xmn)


Y = ifft2(X)


Y = ifft2(Xmn)




89k 12k





tfNF ∆

=21





Tips windows


( ) sin(2 5 )f t tπ=

+=

Tips windows

( ) sin(2 5 )f t tπ=







f1(x)=sin(2pi500t)

and







Overview




In Matlab

Who is Fourier


Fourier series










Fourier transform



In Matlab




In Matlab












In Matlab





Tips windows

Tips windows

Fourier transform


Define


int=

int=

infin

infinminus

infin

infinminus

minus

ωωπ

ω

ω

ω

deFtf

dtetfF

ti

ti

)(21)(

)()(

intintrarr

sum intrarr

sum int=

infin

infinminus

minusinfin

infinminus

infin

minusinfin= minus

minus=

infin

minusinfin= minus

minus

ξξωπ

ξξωπ

ωξω

ξωωπ

ωω

defde

def

edtetfT

tf

iti

k

T

T

tikT

k

tikT

T

tik

)(21

)(1

))(2()(

2

2

)(2

2

2



int=intinfin

infinminus

infin









)(1)(a

Fa

G ωω =

)())(( ωωFixfFT =


In Matlab

F = fourier(f)


int=infin

infinminus


f = ifourier(F)


int=infin

infinminusωω

πω deFxf xi)(

21)(












int minus==infin

infinminus


ωω

1210)()(1

0minus=sum=

minus

=

minus NnetfFN

k

ktnikn

ωω



1

0

1( ) ( ) 012 1n k

Ni t

k nn

f t F e k NN

ωωminus

=

= = minussum






In Matlab

Y = fft(X)


Y = fft(Xn)


Y = ifft(X)


Y = ifft(Xn)



Delta function

⎩⎨⎧ =

==otherst

ttf0

01)()( δ





= +



= +






int minus=infin



=rArr







infinminus

infin

infinminus


)(21)( )(int int=

infin

infinminus

infin

infinminus

+ dudvevuFyxf vyuxi

π




12101210

)()(1

0

1

0

)(

minus=minus=

sum sum=minus

=

minus

=

+minus

NnMm

eyxfvuFM

xk

N

yk

yvxuiykxknm

yknxkm

sum=

sum ⎥⎦⎤

⎢⎣⎡ sum=

minus

=

minus

minus

=

minusminus

=

minus

1

0

1

0

1

0

)()(

)()(

M

m

xiunm

M

m

xiuN

n

yivykxknm

xkm

xkmykn

evxFvuF

eeyxfvuF




12101210

)()(1

0

1

0

)(

minus=minus=

sum sum=minus

=

minus

=

+minus

NykMxk

evuFyxfM

m

N

n

ykynvxkxmuinmykxk

[ ]sum=

sum sum=

minus

=

minus

minus

=

minus

=

1

0

1

0

1

0

)(1)(

)(1)(

M

m

xkxmiunxkykxk

M

m

xkxmiuN

n

ykynivnmykxk

evxFMN

yxf

eevuFMN

yxf






- Convolution




)(1)(bv

auF

abvuG =

In Matlab

Y = fft2(X)


Y = fft2(Xmn)


Y = ifft2(X)


Y = ifft2(Xmn)




89k 12k





tfNF ∆

=21





Tips windows


( ) sin(2 5 )f t tπ=

+=

Tips windows

( ) sin(2 5 )f t tπ=







f1(x)=sin(2pi500t)

and







Overview




In Matlab

Who is Fourier


Fourier series










Fourier transform



In Matlab




In Matlab












In Matlab





Tips windows

Tips windows



int=intinfin

infinminus

infin









)(1)(a

Fa

G ωω =

)())(( ωωFixfFT =


In Matlab

F = fourier(f)


int=infin

infinminus


f = ifourier(F)


int=infin

infinminusωω

πω deFxf xi)(

21)(












int minus==infin

infinminus


ωω

1210)()(1

0minus=sum=

minus

=

minus NnetfFN

k

ktnikn

ωω



1

0

1( ) ( ) 012 1n k

Ni t

k nn

f t F e k NN

ωωminus

=

= = minussum






In Matlab

Y = fft(X)


Y = fft(Xn)


Y = ifft(X)


Y = ifft(Xn)



Delta function

⎩⎨⎧ =

==otherst

ttf0

01)()( δ





= +



= +






int minus=infin



=rArr







infinminus

infin

infinminus


)(21)( )(int int=

infin

infinminus

infin

infinminus

+ dudvevuFyxf vyuxi

π




12101210

)()(1

0

1

0

)(

minus=minus=

sum sum=minus

=

minus

=

+minus

NnMm

eyxfvuFM

xk

N

yk

yvxuiykxknm

yknxkm

sum=

sum ⎥⎦⎤

⎢⎣⎡ sum=

minus

=

minus

minus

=

minusminus

=

minus

1

0

1

0

1

0

)()(

)()(

M

m

xiunm

M

m

xiuN

n

yivykxknm

xkm

xkmykn

evxFvuF

eeyxfvuF




12101210

)()(1

0

1

0

)(

minus=minus=

sum sum=minus

=

minus

=

+minus

NykMxk

evuFyxfM

m

N

n

ykynvxkxmuinmykxk

[ ]sum=

sum sum=

minus

=

minus

minus

=

minus

=

1

0

1

0

1

0

)(1)(

)(1)(

M

m

xkxmiunxkykxk

M

m

xkxmiuN

n

ykynivnmykxk

evxFMN

yxf

eevuFMN

yxf






- Convolution




)(1)(bv

auF

abvuG =

In Matlab

Y = fft2(X)


Y = fft2(Xmn)


Y = ifft2(X)


Y = ifft2(Xmn)




89k 12k





tfNF ∆

=21





Tips windows


( ) sin(2 5 )f t tπ=

+=

Tips windows

( ) sin(2 5 )f t tπ=







f1(x)=sin(2pi500t)

and







Overview




In Matlab

Who is Fourier


Fourier series










Fourier transform



In Matlab




In Matlab












In Matlab





Tips windows

Tips windows








)(1)(a

Fa

G ωω =

)())(( ωωFixfFT =


In Matlab

F = fourier(f)


int=infin

infinminus


f = ifourier(F)


int=infin

infinminusωω

πω deFxf xi)(

21)(












int minus==infin

infinminus


ωω

1210)()(1

0minus=sum=

minus

=

minus NnetfFN

k

ktnikn

ωω



1

0

1( ) ( ) 012 1n k

Ni t

k nn

f t F e k NN

ωωminus

=

= = minussum






In Matlab

Y = fft(X)


Y = fft(Xn)


Y = ifft(X)


Y = ifft(Xn)



Delta function

⎩⎨⎧ =

==otherst

ttf0

01)()( δ





= +



= +






int minus=infin



=rArr







infinminus

infin

infinminus


)(21)( )(int int=

infin

infinminus

infin

infinminus

+ dudvevuFyxf vyuxi

π




12101210

)()(1

0

1

0

)(

minus=minus=

sum sum=minus

=

minus

=

+minus

NnMm

eyxfvuFM

xk

N

yk

yvxuiykxknm

yknxkm

sum=

sum ⎥⎦⎤

⎢⎣⎡ sum=

minus

=

minus

minus

=

minusminus

=

minus

1

0

1

0

1

0

)()(

)()(

M

m

xiunm

M

m

xiuN

n

yivykxknm

xkm

xkmykn

evxFvuF

eeyxfvuF




12101210

)()(1

0

1

0

)(

minus=minus=

sum sum=minus

=

minus

=

+minus

NykMxk

evuFyxfM

m

N

n

ykynvxkxmuinmykxk

[ ]sum=

sum sum=

minus

=

minus

minus

=

minus

=

1

0

1

0

1

0

)(1)(

)(1)(

M

m

xkxmiunxkykxk

M

m

xkxmiuN

n

ykynivnmykxk

evxFMN

yxf

eevuFMN

yxf






- Convolution




)(1)(bv

auF

abvuG =

In Matlab

Y = fft2(X)


Y = fft2(Xmn)


Y = ifft2(X)


Y = ifft2(Xmn)




89k 12k





tfNF ∆

=21





Tips windows


( ) sin(2 5 )f t tπ=

+=

Tips windows

( ) sin(2 5 )f t tπ=







f1(x)=sin(2pi500t)

and







Overview




In Matlab

Who is Fourier


Fourier series










Fourier transform



In Matlab




In Matlab












In Matlab





Tips windows

Tips windows

In Matlab

F = fourier(f)


int=infin

infinminus


f = ifourier(F)


int=infin

infinminusωω

πω deFxf xi)(

21)(












int minus==infin

infinminus


ωω

1210)()(1

0minus=sum=

minus

=

minus NnetfFN

k

ktnikn

ωω



1

0

1( ) ( ) 012 1n k

Ni t

k nn

f t F e k NN

ωωminus

=

= = minussum






In Matlab

Y = fft(X)


Y = fft(Xn)


Y = ifft(X)


Y = ifft(Xn)



Delta function

⎩⎨⎧ =

==otherst

ttf0

01)()( δ





= +



= +






int minus=infin



=rArr







infinminus

infin

infinminus


)(21)( )(int int=

infin

infinminus

infin

infinminus

+ dudvevuFyxf vyuxi

π




12101210

)()(1

0

1

0

)(

minus=minus=

sum sum=minus

=

minus

=

+minus

NnMm

eyxfvuFM

xk

N

yk

yvxuiykxknm

yknxkm

sum=

sum ⎥⎦⎤

⎢⎣⎡ sum=

minus

=

minus

minus

=

minusminus

=

minus

1

0

1

0

1

0

)()(

)()(

M

m

xiunm

M

m

xiuN

n

yivykxknm

xkm

xkmykn

evxFvuF

eeyxfvuF




12101210

)()(1

0

1

0

)(

minus=minus=

sum sum=minus

=

minus

=

+minus

NykMxk

evuFyxfM

m

N

n

ykynvxkxmuinmykxk

[ ]sum=

sum sum=

minus

=

minus

minus

=

minus

=

1

0

1

0

1

0

)(1)(

)(1)(

M

m

xkxmiunxkykxk

M

m

xkxmiuN

n

ykynivnmykxk

evxFMN

yxf

eevuFMN

yxf






- Convolution




)(1)(bv

auF

abvuG =

In Matlab

Y = fft2(X)


Y = fft2(Xmn)


Y = ifft2(X)


Y = ifft2(Xmn)




89k 12k





tfNF ∆

=21





Tips windows


( ) sin(2 5 )f t tπ=

+=

Tips windows

( ) sin(2 5 )f t tπ=







f1(x)=sin(2pi500t)

and







Overview




In Matlab

Who is Fourier


Fourier series










Fourier transform



In Matlab




In Matlab












In Matlab





Tips windows

Tips windows












int minus==infin

infinminus


ωω

1210)()(1

0minus=sum=

minus

=

minus NnetfFN

k

ktnikn

ωω



1

0

1( ) ( ) 012 1n k

Ni t

k nn

f t F e k NN

ωωminus

=

= = minussum






In Matlab

Y = fft(X)


Y = fft(Xn)


Y = ifft(X)


Y = ifft(Xn)



Delta function

⎩⎨⎧ =

==otherst

ttf0

01)()( δ





= +



= +






int minus=infin



=rArr







infinminus

infin

infinminus


)(21)( )(int int=

infin

infinminus

infin

infinminus

+ dudvevuFyxf vyuxi

π




12101210

)()(1

0

1

0

)(

minus=minus=

sum sum=minus

=

minus

=

+minus

NnMm

eyxfvuFM

xk

N

yk

yvxuiykxknm

yknxkm

sum=

sum ⎥⎦⎤

⎢⎣⎡ sum=

minus

=

minus

minus

=

minusminus

=

minus

1

0

1

0

1

0

)()(

)()(

M

m

xiunm

M

m

xiuN

n

yivykxknm

xkm

xkmykn

evxFvuF

eeyxfvuF




12101210

)()(1

0

1

0

)(

minus=minus=

sum sum=minus

=

minus

=

+minus

NykMxk

evuFyxfM

m

N

n

ykynvxkxmuinmykxk

[ ]sum=

sum sum=

minus

=

minus

minus

=

minus

=

1

0

1

0

1

0

)(1)(

)(1)(

M

m

xkxmiunxkykxk

M

m

xkxmiuN

n

ykynivnmykxk

evxFMN

yxf

eevuFMN

yxf






- Convolution




)(1)(bv

auF

abvuG =

In Matlab

Y = fft2(X)


Y = fft2(Xmn)


Y = ifft2(X)


Y = ifft2(Xmn)




89k 12k





tfNF ∆

=21





Tips windows


( ) sin(2 5 )f t tπ=

+=

Tips windows

( ) sin(2 5 )f t tπ=







f1(x)=sin(2pi500t)

and







Overview




In Matlab

Who is Fourier


Fourier series










Fourier transform



In Matlab




In Matlab












In Matlab





Tips windows

Tips windows






In Matlab

Y = fft(X)


Y = fft(Xn)


Y = ifft(X)


Y = ifft(Xn)



Delta function

⎩⎨⎧ =

==otherst

ttf0

01)()( δ





= +



= +






int minus=infin



=rArr







infinminus

infin

infinminus


)(21)( )(int int=

infin

infinminus

infin

infinminus

+ dudvevuFyxf vyuxi

π




12101210

)()(1

0

1

0

)(

minus=minus=

sum sum=minus

=

minus

=

+minus

NnMm

eyxfvuFM

xk

N

yk

yvxuiykxknm

yknxkm

sum=

sum ⎥⎦⎤

⎢⎣⎡ sum=

minus

=

minus

minus

=

minusminus

=

minus

1

0

1

0

1

0

)()(

)()(

M

m

xiunm

M

m

xiuN

n

yivykxknm

xkm

xkmykn

evxFvuF

eeyxfvuF




12101210

)()(1

0

1

0

)(

minus=minus=

sum sum=minus

=

minus

=

+minus

NykMxk

evuFyxfM

m

N

n

ykynvxkxmuinmykxk

[ ]sum=

sum sum=

minus

=

minus

minus

=

minus

=

1

0

1

0

1

0

)(1)(

)(1)(

M

m

xkxmiunxkykxk

M

m

xkxmiuN

n

ykynivnmykxk

evxFMN

yxf

eevuFMN

yxf






- Convolution




)(1)(bv

auF

abvuG =

In Matlab

Y = fft2(X)


Y = fft2(Xmn)


Y = ifft2(X)


Y = ifft2(Xmn)




89k 12k





tfNF ∆

=21





Tips windows


( ) sin(2 5 )f t tπ=

+=

Tips windows

( ) sin(2 5 )f t tπ=







f1(x)=sin(2pi500t)

and







Overview




In Matlab

Who is Fourier


Fourier series










Fourier transform



In Matlab




In Matlab












In Matlab





Tips windows

Tips windows


Delta function

⎩⎨⎧ =

==otherst

ttf0

01)()( δ





= +



= +






int minus=infin



=rArr







infinminus

infin

infinminus


)(21)( )(int int=

infin

infinminus

infin

infinminus

+ dudvevuFyxf vyuxi

π




12101210

)()(1

0

1

0

)(

minus=minus=

sum sum=minus

=

minus

=

+minus

NnMm

eyxfvuFM

xk

N

yk

yvxuiykxknm

yknxkm

sum=

sum ⎥⎦⎤

⎢⎣⎡ sum=

minus

=

minus

minus

=

minusminus

=

minus

1

0

1

0

1

0

)()(

)()(

M

m

xiunm

M

m

xiuN

n

yivykxknm

xkm

xkmykn

evxFvuF

eeyxfvuF




12101210

)()(1

0

1

0

)(

minus=minus=

sum sum=minus

=

minus

=

+minus

NykMxk

evuFyxfM

m

N

n

ykynvxkxmuinmykxk

[ ]sum=

sum sum=

minus

=

minus

minus

=

minus

=

1

0

1

0

1

0

)(1)(

)(1)(

M

m

xkxmiunxkykxk

M

m

xkxmiuN

n

ykynivnmykxk

evxFMN

yxf

eevuFMN

yxf






- Convolution




)(1)(bv

auF

abvuG =

In Matlab

Y = fft2(X)


Y = fft2(Xmn)


Y = ifft2(X)


Y = ifft2(Xmn)




89k 12k





tfNF ∆

=21





Tips windows


( ) sin(2 5 )f t tπ=

+=

Tips windows

( ) sin(2 5 )f t tπ=







f1(x)=sin(2pi500t)

and







Overview




In Matlab

Who is Fourier


Fourier series










Fourier transform



In Matlab




In Matlab












In Matlab





Tips windows

Tips windows





= +



= +






int minus=infin



=rArr







infinminus

infin

infinminus


)(21)( )(int int=

infin

infinminus

infin

infinminus

+ dudvevuFyxf vyuxi

π




12101210

)()(1

0

1

0

)(

minus=minus=

sum sum=minus

=

minus

=

+minus

NnMm

eyxfvuFM

xk

N

yk

yvxuiykxknm

yknxkm

sum=

sum ⎥⎦⎤

⎢⎣⎡ sum=

minus

=

minus

minus

=

minusminus

=

minus

1

0

1

0

1

0

)()(

)()(

M

m

xiunm

M

m

xiuN

n

yivykxknm

xkm

xkmykn

evxFvuF

eeyxfvuF




12101210

)()(1

0

1

0

)(

minus=minus=

sum sum=minus

=

minus

=

+minus

NykMxk

evuFyxfM

m

N

n

ykynvxkxmuinmykxk

[ ]sum=

sum sum=

minus

=

minus

minus

=

minus

=

1

0

1

0

1

0

)(1)(

)(1)(

M

m

xkxmiunxkykxk

M

m

xkxmiuN

n

ykynivnmykxk

evxFMN

yxf

eevuFMN

yxf






- Convolution




)(1)(bv

auF

abvuG =

In Matlab

Y = fft2(X)


Y = fft2(Xmn)


Y = ifft2(X)


Y = ifft2(Xmn)




89k 12k





tfNF ∆

=21





Tips windows


( ) sin(2 5 )f t tπ=

+=

Tips windows

( ) sin(2 5 )f t tπ=







f1(x)=sin(2pi500t)

and







Overview




In Matlab

Who is Fourier


Fourier series










Fourier transform



In Matlab




In Matlab












In Matlab





Tips windows

Tips windows



= +






int minus=infin



=rArr







infinminus

infin

infinminus


)(21)( )(int int=

infin

infinminus

infin

infinminus

+ dudvevuFyxf vyuxi

π




12101210

)()(1

0

1

0

)(

minus=minus=

sum sum=minus

=

minus

=

+minus

NnMm

eyxfvuFM

xk

N

yk

yvxuiykxknm

yknxkm

sum=

sum ⎥⎦⎤

⎢⎣⎡ sum=

minus

=

minus

minus

=

minusminus

=

minus

1

0

1

0

1

0

)()(

)()(

M

m

xiunm

M

m

xiuN

n

yivykxknm

xkm

xkmykn

evxFvuF

eeyxfvuF




12101210

)()(1

0

1

0

)(

minus=minus=

sum sum=minus

=

minus

=

+minus

NykMxk

evuFyxfM

m

N

n

ykynvxkxmuinmykxk

[ ]sum=

sum sum=

minus

=

minus

minus

=

minus

=

1

0

1

0

1

0

)(1)(

)(1)(

M

m

xkxmiunxkykxk

M

m

xkxmiuN

n

ykynivnmykxk

evxFMN

yxf

eevuFMN

yxf






- Convolution




)(1)(bv

auF

abvuG =

In Matlab

Y = fft2(X)


Y = fft2(Xmn)


Y = ifft2(X)


Y = ifft2(Xmn)




89k 12k





tfNF ∆

=21





Tips windows


( ) sin(2 5 )f t tπ=

+=

Tips windows

( ) sin(2 5 )f t tπ=







f1(x)=sin(2pi500t)

and







Overview




In Matlab

Who is Fourier


Fourier series










Fourier transform



In Matlab




In Matlab












In Matlab





Tips windows

Tips windows






int minus=infin



=rArr







infinminus

infin

infinminus


)(21)( )(int int=

infin

infinminus

infin

infinminus

+ dudvevuFyxf vyuxi

π




12101210

)()(1

0

1

0

)(

minus=minus=

sum sum=minus

=

minus

=

+minus

NnMm

eyxfvuFM

xk

N

yk

yvxuiykxknm

yknxkm

sum=

sum ⎥⎦⎤

⎢⎣⎡ sum=

minus

=

minus

minus

=

minusminus

=

minus

1

0

1

0

1

0

)()(

)()(

M

m

xiunm

M

m

xiuN

n

yivykxknm

xkm

xkmykn

evxFvuF

eeyxfvuF




12101210

)()(1

0

1

0

)(

minus=minus=

sum sum=minus

=

minus

=

+minus

NykMxk

evuFyxfM

m

N

n

ykynvxkxmuinmykxk

[ ]sum=

sum sum=

minus

=

minus

minus

=

minus

=

1

0

1

0

1

0

)(1)(

)(1)(

M

m

xkxmiunxkykxk

M

m

xkxmiuN

n

ykynivnmykxk

evxFMN

yxf

eevuFMN

yxf






- Convolution




)(1)(bv

auF

abvuG =

In Matlab

Y = fft2(X)


Y = fft2(Xmn)


Y = ifft2(X)


Y = ifft2(Xmn)




89k 12k





tfNF ∆

=21





Tips windows


( ) sin(2 5 )f t tπ=

+=

Tips windows

( ) sin(2 5 )f t tπ=







f1(x)=sin(2pi500t)

and







Overview




In Matlab

Who is Fourier


Fourier series










Fourier transform



In Matlab




In Matlab












In Matlab





Tips windows

Tips windows







infinminus

infin

infinminus


)(21)( )(int int=

infin

infinminus

infin

infinminus

+ dudvevuFyxf vyuxi

π




12101210

)()(1

0

1

0

)(

minus=minus=

sum sum=minus

=

minus

=

+minus

NnMm

eyxfvuFM

xk

N

yk

yvxuiykxknm

yknxkm

sum=

sum ⎥⎦⎤

⎢⎣⎡ sum=

minus

=

minus

minus

=

minusminus

=

minus

1

0

1

0

1

0

)()(

)()(

M

m

xiunm

M

m

xiuN

n

yivykxknm

xkm

xkmykn

evxFvuF

eeyxfvuF




12101210

)()(1

0

1

0

)(

minus=minus=

sum sum=minus

=

minus

=

+minus

NykMxk

evuFyxfM

m

N

n

ykynvxkxmuinmykxk

[ ]sum=

sum sum=

minus

=

minus

minus

=

minus

=

1

0

1

0

1

0

)(1)(

)(1)(

M

m

xkxmiunxkykxk

M

m

xkxmiuN

n

ykynivnmykxk

evxFMN

yxf

eevuFMN

yxf






- Convolution




)(1)(bv

auF

abvuG =

In Matlab

Y = fft2(X)


Y = fft2(Xmn)


Y = ifft2(X)


Y = ifft2(Xmn)




89k 12k





tfNF ∆

=21





Tips windows


( ) sin(2 5 )f t tπ=

+=

Tips windows

( ) sin(2 5 )f t tπ=







f1(x)=sin(2pi500t)

and







Overview




In Matlab

Who is Fourier


Fourier series










Fourier transform



In Matlab




In Matlab












In Matlab





Tips windows

Tips windows




12101210

)()(1

0

1

0

)(

minus=minus=

sum sum=minus

=

minus

=

+minus

NnMm

eyxfvuFM

xk

N

yk

yvxuiykxknm

yknxkm

sum=

sum ⎥⎦⎤

⎢⎣⎡ sum=

minus

=

minus

minus

=

minusminus

=

minus

1

0

1

0

1

0

)()(

)()(

M

m

xiunm

M

m

xiuN

n

yivykxknm

xkm

xkmykn

evxFvuF

eeyxfvuF




12101210

)()(1

0

1

0

)(

minus=minus=

sum sum=minus

=

minus

=

+minus

NykMxk

evuFyxfM

m

N

n

ykynvxkxmuinmykxk

[ ]sum=

sum sum=

minus

=

minus

minus

=

minus

=

1

0

1

0

1

0

)(1)(

)(1)(

M

m

xkxmiunxkykxk

M

m

xkxmiuN

n

ykynivnmykxk

evxFMN

yxf

eevuFMN

yxf






- Convolution




)(1)(bv

auF

abvuG =

In Matlab

Y = fft2(X)


Y = fft2(Xmn)


Y = ifft2(X)


Y = ifft2(Xmn)




89k 12k





tfNF ∆

=21





Tips windows


( ) sin(2 5 )f t tπ=

+=

Tips windows

( ) sin(2 5 )f t tπ=







f1(x)=sin(2pi500t)

and







Overview




In Matlab

Who is Fourier


Fourier series










Fourier transform



In Matlab




In Matlab












In Matlab





Tips windows

Tips windows




12101210

)()(1

0

1

0

)(

minus=minus=

sum sum=minus

=

minus

=

+minus

NykMxk

evuFyxfM

m

N

n

ykynvxkxmuinmykxk

[ ]sum=

sum sum=

minus

=

minus

minus

=

minus

=

1

0

1

0

1

0

)(1)(

)(1)(

M

m

xkxmiunxkykxk

M

m

xkxmiuN

n

ykynivnmykxk

evxFMN

yxf

eevuFMN

yxf






- Convolution




)(1)(bv

auF

abvuG =

In Matlab

Y = fft2(X)


Y = fft2(Xmn)


Y = ifft2(X)


Y = ifft2(Xmn)




89k 12k





tfNF ∆

=21





Tips windows


( ) sin(2 5 )f t tπ=

+=

Tips windows

( ) sin(2 5 )f t tπ=







f1(x)=sin(2pi500t)

and







Overview




In Matlab

Who is Fourier


Fourier series










Fourier transform



In Matlab




In Matlab












In Matlab





Tips windows

Tips windows






- Convolution




)(1)(bv

auF

abvuG =

In Matlab

Y = fft2(X)


Y = fft2(Xmn)


Y = ifft2(X)


Y = ifft2(Xmn)




89k 12k





tfNF ∆

=21





Tips windows


( ) sin(2 5 )f t tπ=

+=

Tips windows

( ) sin(2 5 )f t tπ=







f1(x)=sin(2pi500t)

and







Overview




In Matlab

Who is Fourier


Fourier series










Fourier transform



In Matlab




In Matlab












In Matlab





Tips windows

Tips windows

In Matlab

Y = fft2(X)


Y = fft2(Xmn)


Y = ifft2(X)


Y = ifft2(Xmn)




89k 12k





tfNF ∆

=21





Tips windows


( ) sin(2 5 )f t tπ=

+=

Tips windows

( ) sin(2 5 )f t tπ=







f1(x)=sin(2pi500t)

and







Overview




In Matlab

Who is Fourier


Fourier series










Fourier transform



In Matlab




In Matlab












In Matlab





Tips windows

Tips windows



89k 12k





tfNF ∆

=21





Tips windows


( ) sin(2 5 )f t tπ=

+=

Tips windows

( ) sin(2 5 )f t tπ=







f1(x)=sin(2pi500t)

and







Overview




In Matlab

Who is Fourier


Fourier series










Fourier transform



In Matlab




In Matlab












In Matlab





Tips windows

Tips windows





tfNF ∆

=21





Tips windows


( ) sin(2 5 )f t tπ=

+=

Tips windows

( ) sin(2 5 )f t tπ=







f1(x)=sin(2pi500t)

and







Overview




In Matlab

Who is Fourier


Fourier series










Fourier transform



In Matlab




In Matlab












In Matlab





Tips windows

Tips windows





Tips windows


( ) sin(2 5 )f t tπ=

+=

Tips windows

( ) sin(2 5 )f t tπ=







f1(x)=sin(2pi500t)

and







Overview




In Matlab

Who is Fourier


Fourier series










Fourier transform



In Matlab




In Matlab












In Matlab





Tips windows

Tips windows

Tips windows

( ) sin(2 5 )f t tπ=







f1(x)=sin(2pi500t)

and







Overview




In Matlab

Who is Fourier


Fourier series










Fourier transform



In Matlab




In Matlab












In Matlab





Tips windows

Tips windows



f1(x)=sin(2pi500t)

and







Overview




In Matlab

Who is Fourier


Fourier series










Fourier transform



In Matlab




In Matlab












In Matlab





Tips windows

Tips windows

Fourier Transform

Documents

Transcript of Fourier Transform