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Fourier TransformationFourier TransformationFourier TransformationFourier Transformation
FourierTransformasjon
FourierTransformasjonf(x)f(x) F(u)F(u)
22
Continuous Fourier TransformContinuous Fourier TransformDefDefContinuous Fourier TransformContinuous Fourier TransformDefDef
The Fourier transform of a one-dimentional function f(x)
dxexfuf uxj 2)()(ˆ
The Inverse Fourier Transform
dueufxf uxj 2)(ˆ)(
33
Continuous Fourier TransformContinuous Fourier TransformDef - NotationDef - NotationContinuous Fourier TransformContinuous Fourier TransformDef - NotationDef - Notation
The Fourier transform of a one-dimentional function f(x)
dxxfexfFxfeuFuf uxjuxj )()()()()(ˆ 22
The inverse Fourier Transform of F(u)
duufeufFufexf uxjuxj )(ˆ)(ˆ)(ˆ)( 212
44
Continuous Fourier TransformContinuous Fourier TransformAlternative DefAlternative DefContinuous Fourier TransformContinuous Fourier TransformAlternative DefAlternative Def
dxexfxfFF xj )()()(
deufFFxf xj)(ˆ2
1)()( 1
dxexfxfFuF uxj 2)()()(
dueufuFFxf uxj 21 )(ˆ)()(
dxexfxfFF xj
)(
2
1)()(
deufFFxf xj)(ˆ2
1)()( 1
55
Continuous Fourier TransformContinuous Fourier TransformExample - cos(2Example - cos(2ft)ft)Continuous Fourier TransformContinuous Fourier TransformExample - cos(2Example - cos(2ft)ft)
66
Continuous Fourier TransformContinuous Fourier TransformExample - cos(Example - cos(t)t)Continuous Fourier TransformContinuous Fourier TransformExample - cos(Example - cos(t)t)
77
Continuous Fourier TransformContinuous Fourier TransformExample - sin(Example - sin(t)t)Continuous Fourier TransformContinuous Fourier TransformExample - sin(Example - sin(t)t)
88
Continuous Fourier TransformContinuous Fourier TransformExample - Delta-functionExample - Delta-functionContinuous Fourier TransformContinuous Fourier TransformExample - Delta-functionExample - Delta-function
99
Continuous Fourier TransformContinuous Fourier TransformExample - Gauss functionExample - Gauss functionContinuous Fourier TransformContinuous Fourier TransformExample - Gauss functionExample - Gauss function
1010
Signals and Fourier TransformSignals and Fourier TransformFrequency InformationFrequency InformationSignals and Fourier TransformSignals and Fourier TransformFrequency InformationFrequency Information
)sin( 11 ty
)sin( 22 ty
)sin()sin( 213 tty
FT
FT
FT
1111
Stationary / Non-stationary signalsStationary / Non-stationary signalsStationary / Non-stationary signalsStationary / Non-stationary signals
60 hvis )sin(
60 hvis )sin(
2
14 tt
tty
)sin()sin( 213 tty
FT
FT
Stationary
Non stationary
The stationary and the non-stationary signal both have the same FT.FT is not suitable to take care of non-stationary signals to give information about time.
1212
Constant function in [-3,3].Dominating frequency = 0and some freequency because of edges.
Transient signalresulting in extra frequencies > 0.
Narrower transient signalresulting in extra higher frequenciespushed away from origin.
Transient SignalTransient SignalFrequency InformationFrequency InformationTransient SignalTransient SignalFrequency InformationFrequency Information
1313
Transient SignalTransient SignalNo Information about PositionNo Information about PositionTransient SignalTransient SignalNo Information about PositionNo Information about Position
Moving the transient part of the signal to a new position does not resultin any change in the transformed signal.
Conclusion: The Fourier transformationcontains information of a transient partof a signal, but only the frequencynot the position.
1414
Inverse Fourier Transform [1/3]Inverse Fourier Transform [1/3]Inverse Fourier Transform [1/3]Inverse Fourier Transform [1/3]
0 4
2
2
edtee tjt
4
44
4)
2(
2
2
2
2
2
22
22
1
)(
edtee
jy
edtee
dtedtedteeyf
tjt
yt
y
yyt
yttytt
Theorem:
Proof:
dxexfuf uxj 2)()(ˆ
dueufxf uxj 2)(ˆ)(
1515
Inverse Fourier Transform [2/3]Inverse Fourier Transform [2/3]Inverse Fourier Transform [2/3]Inverse Fourier Transform [2/3]
)(, ˆˆ 1 RLgf(ay)g(y)dyf(ay)dygf(y)--
Theorem:
Proof:
dxexfuf uxj 2)()(ˆ
dueufxf uxj 2)(ˆ)(
dyygayf
dxxgaxf
dxxgdyeyf
dxdyexgyf
dydxexgyf
dydxexgf(y)(ay)dygf(y)
ayxj
ayxj
ayxj
-
ayxj
-
)()(ˆ
)()(ˆ
)()(
)()(
)()(
)(ˆ
2
2
2
2
1616
Inverse Fourier Transform [3/3]Inverse Fourier Transform [3/3]Inverse Fourier Transform [3/3]Inverse Fourier Transform [3/3]
1)(
)(ˆ
2
1)(
2
2
)2(
4
dxxg
eug
exg
u
x
dxexfuf uxj 2)()(ˆ
dueufxf uxj 2)(ˆ)(
)2(2
1
2
1
2
1)(ˆ
2
1)(
4
)2(
)2(
2
2
2
2
2
yxge
dtee
dteeeyg
eetg
xy
txyjt
ytjtjtx
tjtx
)](ˆ[
)(ˆ22
1
2ˆ
2
1
2ˆ
2
1lim
)(2
ˆlim
2ˆ)(lim
)(*)(lim )()(lim
)0()(
1
2
0
0
0
00
2
xfF
dyeyf
dyey
f
dyeey
f
dyygy
f
dyy
gyf
xgxfdyyxgyf
xfxf
yxj
jyx
yjyx
1717
PropertiesPropertiesPropertiesProperties
0
1)(1
)(
11
11111
)(][ )]([ )]([
)]([1
))](([
)]([ )]([
][)( ][
][)( ][
][)( ][
][ ][
][][ ][][ ][
][][ ][][ ][
dtetffLjfLfF
atfF
aatfF
tfFeatfF
fFjtfF
fFjfF
fFdt
djfF
fFd
djftF
fcFcfFgFfFgfF
fcFcfFgFfFgfF
ts
aj
nn
nn
n
nnn
n
nnn
1818
Fourier Transforms of Fourier Transforms of Harmonic and Constant FunctionHarmonic and Constant FunctionFourier Transforms of Fourier Transforms of Harmonic and Constant FunctionHarmonic and Constant Function
)()(2
)2sin(
)(1 )()(2
1)2cos(
- )2sin(2
)2cos(2
)()(
)()()()()(
000
000
0
022
20
20
20000
1
uuuuj
xuF
uFuuuxuF
xuj
xuee
dueuudueuu
dueuuuuuuuuFxf
uxjuxj
uxjuxj
uxj
1919
Fourier Transforms of Fourier Transforms of Some Common FunctionsSome Common FunctionsFourier Transforms of Fourier Transforms of Some Common FunctionsSome Common Functions
22
0
)(2
1 )(
)(
)(sin )(
)(sin )(
1
)(
2 x)u(2sin
)()(2
1 x)u(2cos
)( )(
2
2
2
02
000
000
ux
xuj
ee
u
juxu
u
ux
u
ux
δ(x)
uue
)uδ(u)uδ(uj
uuuu
uFxf
2020
)( )(
)( )(
tftf
tftf
oddodd
eveneven
)()(2
1 )(
)()(2
1 )(
)()( )(
tftftf
tftftf
tftftf
odd
even
oddeven
)()(
)()(
)()( )(
,,
,,
tftf
tftf
tftftf
imagoddimageven
realoddrealeven
oddeven
Even and Odd Functions [1/Even and Odd Functions [1/3]3]Even and Odd Functions [1/Even and Odd Functions [1/3]3]
Def
Every function can be splitin an even and an odd part
Every function can be splitin an even and an odd partand each of this can in turn be split in a real and an imaginary part
2121
)()(
)2sin()()2cos()(
)2sin()()2sin()()2cos()()2cos()(
)2sin()()2cos()(
)()( 2
ujFuF
dxuxxfjdxuxxf
dxuxxfjdxuxxfjdxuxxfdxuxxf
dxuxtfjdxuxxf
dxexfuF
oddeven
oddeven
oddevenoddeven
uxj
Even and Odd Functions [2/Even and Odd Functions [2/33]]Even and Odd Functions [2/Even and Odd Functions [2/33]]
1. Even component in f produces an even component in F2. Odd component in f produces an odd component in F3. Odd component in f produces an coefficient -j
2222
Even and Odd Functions [Even and Odd Functions [33/3]/3]Even and Odd Functions [Even and Odd Functions [33/3]/3]
Imag Even Imag plus Odd Real
Real Odd Imag plusEven Real
Hermite Real
Odd Complex Odd Complex
Even Complex Even Complex
Even Imag Even Imag
Odd Imag Odd Real
Even Real Even Real
Odd Odd
Even Even
)F( f(t)
u
)()(
Hermite* uFuF
2323
The Shift TheoremThe Shift TheoremThe Shift TheoremThe Shift Theorem
)(
)(
)(
)(
)()(
2
2
22
)(2
2
uFe
xfFe
dxexfe
dxexf
dxeaxfaxfF
uaj
uaj
uxjuaj
axuj
uxj
)(
)()(2
2
uFe
xfFeaxfFuaj
uaj
2424
The Similarity TheoremThe Similarity TheoremThe Similarity TheoremThe Similarity Theorem
1
)(1
)()(
2
2
a
uF
a
dxexfa
dxeaxfaxfF
xj
uxj
au
a
uF
aaxfF
1)(
2525
The Convolution TheoremThe Convolution TheoremThe Convolution TheoremThe Convolution Theorem
gfuGuF
uGdyeyfdyuGeyf
dydteyxgyf
dxexgxfxgxfF
duutguftgtf
uyjuyj
uxj
uxj
ˆˆ)()(
)()()()(
)()(
)(*)()(*)(
)()()(*)(
22
2
2
gfgfF
gfgfF
*ˆˆ
ˆˆ*
1
2626
ConvolutionConvolutionEdge detectionEdge detectionConvolutionConvolutionEdge detectionEdge detection
2727
The Adjoint of the Fourier TransformThe Adjoint of the Fourier TransformThe Adjoint of the Fourier TransformThe Adjoint of the Fourier Transform
22
1
LLgFfgfF
2
2
1
1
2
2
)()(
)()(
)()(
)()(ˆ
L
uxj
uxj
L
gFf
dxxgFxf
dxdueugxf
duugdtexf
duugufgfF
Theorem: Suppose f and g er are square integrable. Then:
Proof:
2828
Plancherel Formel - The Parselval’s TheoremPlancherel Formel - The Parselval’s TheoremPlancherel Formel - The Parselval’s TheoremPlancherel Formel - The Parselval’s Theorem
22
2222
paricular In
11
LL
LLLL
gfgFfF
f F[f]gfgFfF
Theorem: Suppose f and g are square integrable. Then:
Proof: 222
222
111
1
LLL
LLL
gfgfFFgFfF
gfgFFfgFfF
2929
dxxf2
)(energy
duuFdxxf22
)()(
dxxfxfdxxf )()()( *2
22
22
LL
LL
f F[f]
gfgFfF
The Rayleigh’s TheoremThe Rayleigh’s TheoremConConsservation of Energyervation of EnergyThe Rayleigh’s TheoremThe Rayleigh’s TheoremConConsservation of Energyervation of Energy
The energy of a signal in the time domain
is the same as the energy in the frequency domain
2Lf
2
ˆL
f
3030
The Fourier Series ExpansionThe Fourier Series Expansionu a discrete variable - Forward transformu a discrete variable - Forward transformThe Fourier Series ExpansionThe Fourier Series Expansionu a discrete variable - Forward transformu a discrete variable - Forward transform
Tudxexfunff
dxexfdxexfuf
T
T
ujnnn
T
T
uxjuxj
1 )()(ˆˆ
)()()(ˆ
2/
2/
2
2/
2/
22
Suppose f(t) is a transient function that is zero outside the interval [-T/2,T/2] or is considered to be one cycle of a periodic function.We can obtain a sequence of coefficients by making a discrete variableand integrating only over the interval.
3131
The Fourier Series ExpansionThe Fourier Series Expansionu a discrete variable - Inverse transformu a discrete variable - Inverse transformThe Fourier Series ExpansionThe Fourier Series Expansionu a discrete variable - Inverse transformu a discrete variable - Inverse transform
The inverse transform becomes:
n
xT
jn
nn
xT
jn
nn
uxjnn
uxj
efTT
efueunf
duexfxf
222
2
ˆ11ˆ)(ˆ
)(ˆ)(
Tudxexfunff
T
T
ujnnn
1 )()(ˆˆ
2/
2/
2
3232
The Fourier Series ExpansionThe Fourier Series Expansionccnn coefficients coefficientsThe Fourier Series ExpansionThe Fourier Series Expansionccnn coefficients coefficients
n
xT
jn
nn
xT
jn
nuxj ecef
Tduexfxf
222 ˆ1
)(ˆ)(
2/
2/
2
2
)(1
)(
T
T
xT
nj
n
n
xT
jn
n
dxexfT
c
ecxf
Tudxexfunff
T
T
ujnnn
1 )()(ˆˆ
2/
2/
2
3333
The Fourier Series ExpansionThe Fourier Series Expansionzznn, a, ann, b, bnn coefficients coefficientsThe Fourier Series ExpansionThe Fourier Series Expansionzznn, a, ann, b, bnn coefficients coefficients
2/
2/
2
2
)(1
)(
T
T
xT
nj
n
n
xT
jn
n
exfT
c
ecxf
2/
2/
2
2222/
2/
222/
2/
2
1
0
1
22/
2/
222/
2/
20
0
22/
2/
20
22/
2/
22
)(2
)()(2
1)()(
1
2)()(
1
2
)(1
2
)(1
)(
T
T
T
xnj
nn
T
xnj
nnT
xnj
nn
xT
nj
T
T
tT
njx
T
nj
T
T
xT
nj
n
nn
n
xT
nj
T
T
xT
njx
T
nj
T
T
xT
nj
nn
xT
nj
T
T
xT
nj
n
xT
nj
T
T
xT
nj
n
xT
jn
n
etfT
iba
eibaeibaedxexfedxexfT
z
za
edxexfedxexfT
a
edxexfT
a
edxexfT
ecxf
3434
The Fourier Series ExpansionThe Fourier Series Expansionaann,b,bnn coefficients coefficientsThe Fourier Series ExpansionThe Fourier Series Expansionaann,b,bnn coefficients coefficients
2/
2/
2
22
1
0
)(2
)()(2
1
2)(
T
T
T
xnj
nn
T
xnj
nnT
xnj
nnn
nn
etfT
iba
eibaeibaz
za
tf
2/
2/
2/
2/
1
0
2sin)(
2
2cos)(
2
2sin
2cos
2)(
T
T
n
T
T
n
nnn
dxT
xnxf
Tb
dxT
xnxf
Ta
T
xnb
T
xna
axf
3535
Fourier SeriesFourier SeriesPulse trainPulse trainFourier SeriesFourier SeriesPulse trainPulse train
N = 1
N
i
xii
xf1
2)12(sin
12
14)(
N = 2
N = 5
N = 10
Pulse train approximated by Fourier Serie
3636
Fourier SeriesFourier SeriesPulse trainPulse train – Java program – Java programFourier SeriesFourier SeriesPulse trainPulse train – Java program – Java program
3737
Pulse Train approximated by Fourier SeriePulse Train approximated by Fourier SeriePulse Train approximated by Fourier SeriePulse Train approximated by Fourier Serie
f(x) square wave (T=2)
N=2
N=10
1
1
0
])12sin[(12
14
2sin
2cos
2)(
n
nnn
xnn
T
xnb
T
xna
axf
N
n
xnn
xf1
])12sin[(12
14)(
N=1
3838
2
)sin(1
)1(2
)(1
1
kikxik
xfN
i
i
N = 1
N = 2
N = 5
N = 10
Zig tag approximated by Fourier Serie
Fourier SeriesFourier SeriesZig tagZig tagFourier SeriesFourier SeriesZig tagZig tag
3939
2
cos(ikx))(
(-1)4
23
1)(
N
1i2
i2
kik
xf
N = 1
N = 2
N = 5
N = 10
Negative sinus function approximated by Fourier Serie
Fourier SeriesFourier SeriesNegative sinus functionNegative sinus functionFourier SeriesFourier SeriesNegative sinus functionNegative sinus function
4040
2
cos(2ikx)1)2(
12)sin(
2
11)(
N
1i2
ki
kxxf
N = 1
N = 2
N = 5
N = 10
Truncated sinus function approximated by Fourier Serie
Fourier SeriesFourier SeriesTruncated sinus functionTruncated sinus functionFourier SeriesFourier SeriesTruncated sinus functionTruncated sinus function
4141
L
L
j
L
L
j
N
j
N
j
dxjkxxfL
b
LdxjkxxfL
a
Lkjkxjkx
axf
)sin()(1
)cos()(1
)sin(b)cos(a2
)(0
j0
j0
N = 1
N = 2
N = 5
N = 10 N = 50
Lineapproximated by Fourier Serie
Fourier SeriesFourier SeriesLineLineFourier SeriesFourier SeriesLineLine
4242
Approximate functions by adjusting Fourier coefficients (Java program)
Fourier SeriesFourier SeriesJava program for approximating Fourier coefficientsJava program for approximating Fourier coefficientsFourier SeriesFourier SeriesJava program for approximating Fourier coefficientsJava program for approximating Fourier coefficients
4343
The Discrete Fourier Transform - DFTThe Discrete Fourier Transform - DFTDiscrete Fourier Transform - Discretize both time and frequencyDiscrete Fourier Transform - Discretize both time and frequencyThe Discrete Fourier Transform - DFTThe Discrete Fourier Transform - DFTDiscrete Fourier Transform - Discretize both time and frequencyDiscrete Fourier Transform - Discretize both time and frequency
dueufxf uxj 2)(ˆ)(
ContinuousFourier transform
n
xT
jn
nefT
xf2
ˆ1)(
2/
2/
2)()(ˆT
T
uxj dxexfuf
Tunu
1u
N
Tttit
n
nN
ij
ni efT
xiff2ˆ1
)(
2/
2/
2)()(ˆˆT
T
ujnn dxexfunff
iN
njN
Niin ef
N
Tunff
22/
2/
)(ˆˆ
Discrete frequencyFourier Serie
Discrete frequency and timeDiscrete Fourier Transform
4444
The Discrete Fourier Transform - DFTThe Discrete Fourier Transform - DFTDiscrete Fourier Transform - Discretize both time and frequencyDiscrete Fourier Transform - Discretize both time and frequencyThe Discrete Fourier Transform - DFTThe Discrete Fourier Transform - DFTDiscrete Fourier Transform - Discretize both time and frequencyDiscrete Fourier Transform - Discretize both time and frequency
n
nN
ij
ni efT
xiff2ˆ1
)(
iN
njN
Niin ef
N
Tunff
22/
2/
)(ˆˆ
{ fi } sequence of length N, taking samples of a continuous function at equal intervals
iN
njN
iin ef
Nf
21
0
1ˆ
nN
ijN
nni ef
Nf
21
0
ˆ1
4545
Continuous Fourier Transform in two DimensionsContinuous Fourier Transform in two DimensionsDefDefContinuous Fourier Transform in two DimensionsContinuous Fourier Transform in two DimensionsDefDef
The Fourier transform of a two-dimentional function f(x,y)
dydxeyxfvuf vyuxj )(2),(),(ˆ
The Inverse Fourier Transform
dvduevufyxf vyuxj )(2),(ˆ),(
4646
The Two-Dimensional DFT and Its InverseThe Two-Dimensional DFT and Its InverseThe Two-Dimensional DFT and Its InverseThe Two-Dimensional DFT and Its Inverse
1
0
1
0
)(2),(
1),(ˆ
M
x
N
y
yN
vx
M
uj
eyxfMN
vuf
1
0
1
0
)(2),(ˆ1
),(M
x
N
y
yN
vx
M
uj
evufMN
yxf
4747
Fourier Transform in Fourier Transform in TTwo Dimensionswo DimensionsExample 1Example 1Fourier Transform in Fourier Transform in TTwo Dimensionswo DimensionsExample 1Example 1
4848
Fourier Transform in Two DimensionsFourier Transform in Two DimensionsExample 2Example 2Fourier Transform in Two DimensionsFourier Transform in Two DimensionsExample 2Example 2
4949
EndEnd
