2.2 Fourier Transform, Convolution, Cross Correlation ... - TUHH

The Fourier Transform

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• Fourier Series • Fourier Transform • The Basic Theorems and Applications • Sampling Bracewell, R. The Fourier Transform and Its Applications, 3rd ed. New York: McGraw-Hill, 2000. Eric W. Weisstein. "Fourier Transform.“ http://mathworld.wolfram.com/FourierTransform.html

Fourier Series • Fourier series is an expansion (German: Entwicklung) of a periodic function f(x) (period 2L, angular frequency /L) in terms of a sum of sine and cosine functions with angular frequencies that are integer multiples of /L

• Any piecewise continuous function f(x) in the interval [-L,L] may be approximated with the mean square error converging to zero

2

3

Fourier Series • Sine and cosine functions sin nx , cosnx  form a complete orthogonal system over [-,]  • Basic relations (m,n  0): sinmx sinnx dx   mn







 cosmxcosnx dx  

mn





 sinmxcosnx dx  0



• mn: Kronecker delta • mn=0 for m  n, mn=1 for m = n



 sinmxdx  0









cosmx dx  0

4

Fourier Series: Sawtooth • Example: Sawtooth wave 1

0.8

f(x)

0.6

0.4

0.2

0 0

0.2

0.4

0.6

0.8

1

x / (2L)

1.2

1.4

1.6

1.8

Fourier Series • Fourier series is given by

  1 f x   a0  an cosnx  bn sin nx 2 n 1 n 1



• where



a0  an  bn 

1

 1





1





 f x dx



 f x cosnx dx





 f x sinnx dx



• If the function f(x) has a finite number of discontinuities and a finite number of extrema (Dirichlet conditions): The Fourier series converges to the original function at points of continuity or to the average of the two limits at points of discontinuity

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6

Fourier Series • For a function f(x) periodic on an interval [-L,L] instead of [-,] change of variables can be used to transform the interval of integration

x

x' L

dx 

dx' L

• Then  1  nx'    nx'  f x   a0   an cos  b sin   n   2 L L   n1   n 1

• and L

1 a0   f x' dx' L L

1  nx'  an   f x' cos dx' L L  L  L

1  nx'  bn   f x' sin dx' L L  L  L

Fourier Series • For a function f(x) periodic on an interval [0,2L] instead of [-,]: 1 an  L 1 an  L

2L

1 bn  L

2L

 0

 0

2L

 f x'dx' 0

 nx'  f x' cos dx'  L   nx'  f x' sin dx'  L 

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8

Fourier Series: Sawtooth • Example: Sawtooth wave 1 a0  L

f x  

2L

x 0 2Ldx  1

x 2L

1 1  1  nx  f x     sin  2  n1 n  L 

1 x  nx  an   cos dx L 0 2L L   2L

2n cosn   sin n sin n  

Finite Fourier series

n 2 2

1

0

0.8

x  nx  sin 0 2L  L dx

2L

 2n cos 2n   sin 2n  2n 2 2 1  n 

f(x)

1 bn  L

0.6 0.4 0.2 0 0

0.2

0.4

0.6

x / (2L)

0.8

1

1.2

Fourier Series: Sawtooth f x  

x 2L 1 1 m 1  nx  g m, x     sin  2  n 1 n  L 

• Example: Sawtooth wave • Fourier series: 1

f(x)

0.8

0.6

0.4

0.2

0 0

0.2

0.4

0.6

0.8

1

x / (2L)

1.2

1.4

1.6

1.8

9

10

Fourier Series: Sawtooth 1

0 0

0.2

0.4

0.6

0.8

1 x / (2L)

1.2

1.4

1.6

1.8

1 0

Amplitude a0/2, bn

0.5

0.4

0.2

1

0

2 3 4 5

0

0.2

0.4

0.6

0.8 1 x / (2L)

1.2

1.4

1.6

1.8

0

5

10

15

n : Frequency w / (/L)

20

Example: Approximation by Fourier-Series m=1 0.002

0.002

0.001

5 10

4

0

5 10

4 0

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

1

1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8

m m 1 f  x   a0   an cosnx    bn sin nx  2 n 1 n 1

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Approximation by Fourier-Series m=7 0.002 0.0018 0.0016 0.0014 0.0012

8 6 4 2

0.001 4 10 4 10 4 10 4 10 0

2 10

4 0

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

1

1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8

m m 1 f x   a0   an cosnx    bn sin nx  2 n 1 n 1

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Approximation by Fourier-Series m=10 0.002 0.0018 0.0016 0.0014 0.0012

8 6 4 2

0.001 4 10 4 10 4 10 4 10 0

2 10

4 0

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

1

1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8

m m 1 f x   a0   an cosnx    bn sin nx  2 n 1 n 1

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Fourier Series: Properties • Around points of discontinuity, a "ringing“, called Gibbs phenomenon occurs • If a function f(x) is even, f x sinnx is odd and thus



 f x sinnx dx  0



 Even function: bn = 0 for all n • If a function f(x) is odd, f x cosnx  is odd and thus  Odd function: an = 0 for all n



 f x cosnx dx  0



Complex Fourier Series • Fourier series with complex coefficients f x  



inx A e  n

n  

• Calculation of coefficients:

1 An  2





f x e inxdx



• Coefficients may be expressed in terms of those in the real Fourier series 1



1 An  2









 2 a n  ib n for n < 0  1 for n = 0 f x cosnx   i sinnx dx   a0 2 1  2 an  ibn  for n > 0 

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Fourier Transform • Generalization of the complex Fourier series for infinite L and continuous variable k • L  , n/L  k, An  F(k)dk

• Forward transform F k  





f x e 2ikx dx

or F k   Fx  f x k 



• Inverse transform f x  



2ikx   F k e dk 



• Symbolic notation

f x 

F k 

f x   Fk

1

F k x

Forward and Inverse Transform •

If 1.

17



 f x  dx

exists



2. f(x) has a finite number of discontinuities 3. The variation of the function is bounded

•

Forward and inverse transform:

•

For f(x) continuous at x f x   Fk Fx f x x    e



1



•

2ikx

    f x e 2ikx dx dk     

For f(x) discontinuous at x Fk 1 Fx f x x    f x   f x  1 2

Fourier Cosine Transform and Fourier Sine Transform • Any function may be split into an even and an odd function f x  

1  f x   f  x   1  f x   f  x   E x   Ox  2 2

• Fourier transform may be expressed in terms of the Fourier cosine transform and Fourier sine transform F k  









 Ex cos2kxdx  i  Ox sin2kxdx

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Real Even Function • If a real function f(x) is even, O(x) = 0 F k  











0

 Excos2kxdx   f xcos2kxdx 2 f x cos2kxdx

• If a function is real and even, the Fourier transform is also real and even F(k)

f(x)

x

k Bracewell, R. The Fourier Transform and Its Applications, 3rd ed. New York: McGraw-Hill, 2000.

20

Real Odd Function • If a real function f(x) is odd, E(x) = 0

sin(..-k..) = -sin(..k..)











0

F k   i  Ox sin 2kxdx   i  f x sin 2kxdx   2i  f x sin 2kxdx

• If a function is real and odd, the Fourier transform is imaginary and odd f(x)

F(k)

Bracewell, R. The Fourier Transform and Its Applications, 3rd ed. New York: McGraw-Hill, 2000.

Symmetry Properties f(x)

F(k)

Real and even

Real and even

Real and odd

Imaginary and odd

Imaginary and even

Imaginary and even

Imaginary and odd

Real and odd

Real asymmetrical

Complex hermitian

Imaginary asymmetrical

Complex antihermitian

Even

Even

Odd

Odd Hermitian: f(x) = f*(-x) Antihermitian: f(x) = -f*(-x)

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22

Symmetry Properties f(x)

F(k)

Bracewell, R. The Fourier Transform and Its Applications, 3rd ed. New York: McGraw-Hill, 2000.

23

Fourier Transform: Properties and Basic Theorems • Linearity af x   bg x 

aF k   bGk 

• Shift theorem f x  x0 

e 2ix0k F k 

• Modulation theorem f x cos2k0 x 

1 F k  k0   F k  k0  2

• Derivative f x 

i 2kFk 

24

Fourier Transform: Properties and Basic Theorems • Parseval's / Rayleigh’s theorem 





f x  dx   F k  dk 2



2



• Fourier Transform of the complex conjugated f * x 

F *  k 

• Similarity (a  0 : real constant)

f ax 

1  k  a F  a

• Special case (a = -1): f  x 

F  k 

25

Similarity Theorem 1

f x 

1

1

1

x

1

f 2 x 

1

1

x

1

x

f 5 x 

1

F k 

k

1 k F  2 2

k

1 k F  5 5

k

Convolution • Convolution describes an action of an observing instrument (linear instrument) when it takes a weighted mean of a physical quantity over a narrow range of a variable • Examples: • Photo camera: “Measured” photo is described by real image convolved with a function describing the apparatus • Spectrometer: Measured spectrum is given by real spectrum convolved with a function describing limited resolution of the spectrometer

• Other terms for convolution: Folding (German: Faltung), composition product • System theory: Linear time (or space) invariant systems are described by convolution

26

Convolution • Definition of convolution

yx   f  g 

27



 f   g x   d



Eric W. Weisstein. “Convolution. http://mathworld. wolfram.com/Convolution.html

• JAVA Applets: http://www.jhu.edu/~signals/

Properties of the Convolution • Properties of the convolution

f g  g f

commutativity

f  ( g  h)   f  g  h

associativity

f  ( g  h)   f  g    f  h

distributivity

a f  g   af  g  f  ag  d  f  g   df  g  f  dg dx dx dx

• The integral of a convolution is the product of integrals of the    factors

  f  g dx   f xdx  g x dx







28

The Basic Theorems: Convolution

29

• The Fourier transform of the convolution of f(x) and g(x) is given by the product of the individual transforms F(k) and G(k) F k Gk 

f x   g x  f x   g x 

 

 e

 2ikx

  



  e

f x'g x  x'dx' dx

2ikx '



f x' dx' e 2ik  x  x '  g x  x' dx



  2ikx '    2ikx ''    e f x' dx'   e g x' ' dx' '      

 F k Gk 

• Convolution in the spectral domain f x g x 

F k  Gk 



Cross-Correlation • Definition of cross-correlation rfg t   f (t )g t  





f *  g t   d



f (t )g t   f *  t  g t 

•

f g  g f

!

• Fourier transform: f (t )g t   f *  t  g t  

f (t )g t 

, f *  t  F * k Gk 

F * k  (see FT rules)

30

Example: Cross-Correlation Radar (Low Noise) Signal sent by radar antenna

0

Time

Signal received by radar antenna

0

Time

Cross correlation of the signals

0

Time

31

Example: Cross-Correlation Radar Signal sent by radar antenna

0

Time

Signal received by radar antenna

0

Time

Cross correlation of the signals

0

Time

32

Example: Cross-Correlation

33

Ultrasonic Distance Measurement Signal sent by ultrasonic transducer

Signal received by ultrasonic transducer

Time

Time

Cross correlation of the two signals Pulse Start 

Transmitter

Timer

Object

Stop Receiver

d

d

vt flight 2

(0) 0

tflight

Time

Autocorrelation Theorem • Autocorrelation f (t ) f t  





f *   f t   d



• Autocorrelation theorem: F k 

f (t ) f t 

2

• Proof f (t ) f t   f *  t  f t 

f (t ) f t 

,

f *  t 

F * k F k   F k 

2

F * k 

34

Example: Autocorrelation

Signal: Noise

Signal

Detection of a Signal in the Presence of Noise

Autocorrelation function: Peak at t = 0 : Noise

rff

Time

0 Time

35

Example: Autocorrelation

Signal: Sinusoid plus noise

Signal

Detection of a Signal in the Presence of Noise

Autocorrelation function: Peak at t = 0 : Noise Period of rff reveals that signal is periodic

rff

Time

0 Time

36

37

http://en.wikipedia.org/wiki/Cross-correlation

The Delta Function

38

• Function for the description of point sources, point masses, point charges, surface charges etc. • Also called impulse function: Brief (unit area) impulse so that all measuring equipment is unable to resolve pulse • Important attribute is not value at a given time (or position) but the integral • Definition of the delta function (distribution, also called Dirac function)  x   0 for x  0 1. 

2.

  x dx  1



39

The Delta Function • The sifting (sampling) property of the delta function 

  x  f x dx  f 0



• Sampling of f(x) at x = a 

  x  a  f x dx  f a 



• Fourier transform:  x  x0 



 2ikx  2ikx    x  x e dx  e 0 



0

40

Fourier Transform Pairs • Cosine function 1

1  1  1  IIx      x      x    2  2  2 

cosx 

1

IIx  x 1

• Sine function 1

k

1  1  1  I I x      x      x    2  2  2 

sin x  1

x

i I I x  1

k

41

Fourier Transform Pairs • Gaussian function 1

e

x 2

1

1

e

k 2

x

1

k

• Rectangle function of unit height and base P(x)  1  1 P x    2 0 

1 2 1 x 2 1 x 2 x

sincx   1

1

1

x

sinck 

1

k

sin x x

42

Fourier Transform Pairs • Triangle function of unit height and area 1  x  x     0

x 1

1

1

x 1

1

x

sinc 2 k 

k

1

• Constant function (unit height)

(k)

1

1

x

1

k

Sampling

43

• Sampling: “Noting” a function at finite intervals

Bracewell, R. The Fourier Transform and Its Applications, 3rd ed. New York: McGraw-Hill, 2000.

Sampling of a Signal

Signal 1

Samples taken at equal intervals of time

Sampled signal

44

45

Sampling

Signal 1 + Signal 2 Signal 2 (very short)

Samples taken at equal intervals of time

Sampled signal does not contain the additional Signal

Aliasing Sampling with a low sampling frequency

Sampled signal

Apparent signal

46

47

Sampling • Sampling function (sampling comb) III(x) “Shah” IIIx  



  x  n 

n  

1 IIIax   a



n  x   a n   

• Multiplication of f(x) with III(x) describes sampling at unit intervals IIIx  f x  



 f n x  n

n  

• Sampling at intervals :   x III  f x     f n  x  n    n  

Bracewell, R. The Fourier Transform and Its Applications, 3rd ed. New York: McGraw-Hill, 2000.

48

Sampling • Fourier transform of the sampling function IIIx 

IIIk 

 x III    

IIIk 

1

IIIx 

IIIk 

 x III  2

2 III2k 

 here:  = 2 Bracewell, R. The Fourier Transform and Its Applications, 3rd ed. New York: McGraw-Hill, 2000.

1

Sampling: Band limited signal

49

• Band limited signal: F(k) = 0 for |k|>kc

f(x)

F(k)

x

kc

k

Bracewell, R. The Fourier Transform and Its Applications, 3rd ed. New York: McGraw-Hill, 2000.

50

Sampling in the Two Domains f(x)

 

 x III   

F(k)

x

kc  IIIk 



|k|

-1

=

 x III  f x   

kc  IIIk  F k 

=

kc Bracewell, R. The Fourier Transform and Its Applications, 3rd ed. New York: McGraw-Hill, 2000.

51

Reconstruction of the Signal  IIIk  F k 

 x III  f x   



kc

|k|

Multiplication with rectangular function in order to remove additional signal components

Convolution

1

=f(x)



=

F(k)

|k|

Original signal in frequency domain x

Fourier Transform

kc

|k|

Critical Sampling and Undersampling f(x)

F(k)

x III2kc x  f x 

1 2k c

52

kc

|k|  k    F k  2 k  c

2kc 1 III

kc Undersampling 

1 2k c

kc Bracewell, R. The Fourier Transform and Its Applications, 3rd ed. New York: McGraw-Hill, 2000.

53

Sampling • Sampling theorem • A function whose Fourier transform is zero (F(k) = 0) for |k|>kc is fully specified by values spaced at equal intervals  < 1/(2kc).

• Alternative (simplified) statement: The sampling frequency must be higher than twice the highest frequency in the signal • Remark: This does not imply that a reconstruction is impossible for all cases if the sampling frequency is lower. If additional knowledge about the signal is available (e.g. lower limit of Fourier transform, single frequency signal), reconstruction may be possible.

• Description of sampling at intervals : Multiplication of f(x) with III(x/ ) • Reconstruction of the signal: Multiplication of the Fourier transform  IIIk  F k  with

 k   P  2k c 

and inverse

transformation Or: Convolution of the sampled function with a sinc-function