Fourier Transform Signal Theory Zdzisław Papir Periodicity of Fourier series Limiting behaviour of...

Fourier Transform

„Signal Theory” Zdzisław Papir

•Periodicity of Fourier series

•Limiting behaviour of Fourier series

•Limiting form of Fourier series

•Fourier transform pairs

•Existence of Fourier transform

Periodicity of Fourier series

„Signal Theory” Zdzisław Papir

TeXTtxeXtxn

Ttjnn

n

tjnn 2, o

oo

sawtooth signal

time t

period T = 1

Limiting behaviourof Fourier series

„Signal Theory” Zdzisław Papir

x(t)

time t-T/2

xT(t)

Periodic extension of a signal window xT(t)through Fourier series

+T/2

txtx TT

Limiting behaviour of Fourier series

„Signal Theory” Zdzisław Papir

„Signal Theory” Zdzisław Papir

TejnT

TX Tn

2,1

1

11o

o

tettx 1

Limiting behaviour of Fourier series

„Signal Theory” Zdzisław Papir

TejnT

TX Tn

2,1

1

11o

o

0

02o

Tn

T

TX

T

Limiting behaviour of Fourier series

„Signal Theory” Zdzisław Papir

Limiting behaviour of Fourier series

04

1221

T

T

T

eTX

amplitude of the 1st spectrum line of an exponential puls

Fourier series window

04

1221

T

T

T

eTX

„Signal Theory” Zdzisław Papir

Limiting behaviour of Fourier series

222 4

1

nT

eTX

T

n

Fourier series window T

amplitude spectrum – exponential pulse

„Signal Theory” Zdzisław Papir

Limiting behaviour of Fourier series

222 4

1

nT

eTX

T

n

Fourier series window 3T

amplitude spectrum – exponential pulse

„Signal Theory” Zdzisław Papir

Limiting behaviour of Fourier series

222 4

1

nT

eTX

T

n

amplitude spectrum – exponential pulse

Fourier series window 10T

„Signal Theory” Zdzisław Papir

Limiting behaviour of Fourier series

222 4

1

nT

eTX

T

n

amplitude spectrum – exponential pulse

Fourier series window 100T

„Signal Theory” Zdzisław Papir

j

Xj

eTTX T

n

T

n

1

1

1

1

Squeezing Fourier series coefficients in FREQUENCY:

n

Tn

T

n j

eTTX

jn

eTTX

n

n

1

1

1

1o

o

Limiting behaviour of Fourier series

Squeezing Fourier series coefficients in AMPLITUDE:

„Signal Theory” Zdzisław Papir

Riemann integral

a b

x

f(x)

nx

nxf

b

an

x

n

nn dxxfxxfS n 0max

Limiting form of Fourier series

„Signal Theory” Zdzisław Papir

Fourier series coefficients:

2

2

2

2

o1

T

T

tj

T

T

tjnn

dtetxTTX

dtetxT

TX

n

n

dtetxXTTX tj

T n

lim

FORWARD FOURIER TRANSFORM:

Limiting form of Fourier series

„Signal Theory” Zdzisław Papir

Fourier series:

deXtxtx tjT

T

1lim

INVERSE FOURIER TRANSFORM:

n

n

n

tjT

n

tjnn

n

tjnnT

eTTXtx

eTTXeTXtx

2

1

2

1o

oo

„Signal Theory” Zdzisław Papir

n

Tt

t

tj

n

tjnTt

t

jn

n

tjnTt

t

jn

dextx

edextx

edexTtx

n0

0

00

0

0

00

0

0

2

1

2

1

1

0

n

tjnneXtx 0)(

Tt

t

tjnn dtetXTX

0

0

0)(1

Fourier Integral Theorem

„Signal Theory” Zdzisław Papir

Fourier transform

n

Tt

t

tj dextx n0

02

1

ddextx tj

2

1

dtetxX tj

dedextx tjj

2

1

Fourier integral theorem

ForwardFouriertransform

„Signal Theory” Zdzisław Papir

Inverse Fourier transform

ddexetx jtj

2

1

dtetxX tj

InverseFouriertransform

deXtx tj

2

1

Fourier transform pairs

„Signal Theory” Zdzisław Papir

TRANSFORM

dtetxX tj

deXtx tj1

INVERSEFORWARD

TRANSFORMPAIRS

Xtx

Xtx

txX1

FF

Fourier transform pairs

„Signal Theory” Zdzisław Papir

jdtedteeX

ette

ttx

tjtjt

tt

1

1

0,

0,0

o

1

o

1

FORWARD FOURIER TRANSFORM:

j

et t

1

11

Fourier transform pairs

„Signal Theory” Zdzisław Papir

j

et t

1

11

FOURIER TRANSFORM:

tettx 1

time t

jdtedteeX

ette

ttx

tjtjt

tt

1

1

0,

0,0

o

1

o

1

Fourier transform pairs

„Signal Theory” Zdzisław Papir

xxxT

TdteX

tTt

Tttx

T

T

tj

T

sinSa,2

Sa

2,1

2,0

2

2-

2

SaT

TtT

FOURIER TRANSFORM:

T/2-T/2

1 tT

fTT

tx Sa

2Sa

frequency f

„Signal Theory” Zdzisław Papir

xxxT

TdtetX

tTtTt

Tttx

tjT

T

sinSa,4

Sa2

1

2,21

2,0

2

-

4

Sa2

1 2 TTtT

FOURIER TRANSFORM:

T/2-T/2

tT

Fourier transform pairs

2

Sa4

Sa 22 fTTtx

frequency f

„Signal Theory” Zdzisław Papir

Existence of Fourier transformDirichlet conditions are necessaryfor Fourier transform existence.

• Signal x(t) must have only a finite number of maxima and minima, as well as a finite number of discontinuities over the entire range [–, + ].• Signal x(t) is also allowed to be unbounded provided that it is absolutely integrable:

dttx

„Signal Theory” Zdzisław Papir

Summary• Fourier series is a spectral decomposition of periodic signal or produces a periodic extension of signal window.• Fourier transform is a tool for spectral decomposition of nonperiodic signals.• Fourier transform is a limiting case of Fourier series with signal window being extended up to infinity.• Dirichlet conditions are necessary for Fourier transform existence.• In engineering applications it is commonly assumed that signals of limited energy are Fourier transformable.