Fourier Series Fourier Transform The Basic Theorems and ... · The Fourier Transform 1 • Fourier Series • Fourier Transform • The Basic Theorems and Applications • Sampling

1 The Fourier Transform

• 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

2 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

3 Fourier Series

• Sine and cosine functions form a complete

orthogonal system over [-,]

• Basic relations (m,n 0):

• mn: Kronecker delta

• mn=0 for m n, mn=1 for m = n

nxnx cos,sin

mndxnxmx

sinsin

mndxnxmx

coscos

0cossin

dxnxmx

0sin

dxmx

0cos

dxmx

4 Fourier Series: Sawtooth

• Example: Sawtooth wave

x / (2L)

f(x)

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8

0

0.2

0.4

0.6

0.8

1

5 Fourier Series

• Fourier series is given by

• where

• 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

nxbnxaaxfn

n

n

n sincos2

1

11

0

dxxfa

10

dxnxxfan cos1

dxnxxfbn sin1

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

• Then

• and

L

xx

'

L

dxdx

'

L

xnb

L

xnaaxf

n

n

n

n

'sin

'cos

2

1

11

0

''

cos'1

dxL

xnxf

La

L

L

n

'

'sin'

1dx

L

xnxf

Lb

L

L

n

''

10 dxxf

La

L

L

7 Fourier Series

• For a function f(x) periodic on an interval [0,2L] instead of

[-,]:

''

cos'1

2

0

dxL

xnxf

La

L

n

''

sin'1

2

0

dxL

xnxf

Lb

L

n

''1

2

0

dxxfL

a

L

n



L

xxf

2

0 0.2 0.4 0.6 0.8 1 1.2

0

0.2

0.4

0.6

0.8

1

x / (2L)

f(x)

Finite Fourier series

12

12

0

0 dxL

x

La

L

dxL

xn

L

x

La

L

n

cos

2

12

0

0

sinsincos222

n

nnnn

dxL

xn

L

x

Lb

L

n

sin

2

12

0

n

n

nnn

1

2

2sin2cos222

L

xn

nxf

n

sin

11

2

1

1



• Fourier series:

L

xxf

2

x / (2L)

f(x)

m

n L

xn

nxmg

1

sin11

2

1,

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8

0

0.2

0.4

0.6

0.8

1


0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8

0

0.5

1

0 5 10 15 200

0.2

0.4

n : Frequency w / (/L)

Am

plit

ud

e a

0/2

, b

n

x / (2L)

x / (2L)

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.85

4

3

2

1

0

1

11 Example: Approximation by Fourier-Series

m=1

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 1.85 10

4

0

5 104

0.001

0.002

0.002

nxbnxaaxfm

n

n

m

n

n sincos2

1

11

0

12 Approximation by Fourier-Series

m=7

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 1.82 10

4

0

2 104

4 104

6 104

8 104

0.001

0.0012

0.0014

0.0016

0.0018

0.002

nxbnxaaxfm

n

n

m

n

n sincos2

1

11

0

13 Approximation by Fourier-Series

m=10

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 1.82 10

4

0

2 104

4 104

6 104

8 104

0.001

0.0012

0.0014

0.0016

0.0018

0.002

nxbnxaaxfm

n

n

m

n

n sincos2

1

11

0

14 Fourier Series: Properties

• Around points of discontinuity, a "ringing“, called Gibbs

phenomenon occurs

• If a function f(x) is even, is odd and thus

Even function: bn = 0 for all n

• If a function f(x) is odd, is odd and thus

Odd function: an = 0 for all n

nxxf sin 0sin

dxnxxf

nxxf cos 0cos

dxnxxf

15

inx

n

neAxf

Complex Fourier Series

• Fourier series with complex coefficients

• Calculation of coefficients:

• Coefficients may be expressed in terms of those in the real

Fourier series

dxexfA inx

n2

1

dxnxinxxfAn sincos2

1

nn

nn

iba

a

iba

2

1

2

1

2

1

0

for n < 0

for n = 0

for n > 0

16 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 or

• Inverse transform

• Symbolic notation

dxexfkF ikx2 kxfkF xF

dkekFxf ikx2 xkFxf k

1 F

xf kF

17 Forward and Inverse Transform

• If

1. 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

• For f(x) discontinuous at x

dxxf

dkdxexfexxfxf ikxikx

xk

221FF

xfxfxxfxk

2

11FF

18 Fourier Cosine Transform and Fourier Sine

Transform

• Any function may be split into an even and an odd function

• Fourier transform may be expressed in terms of the Fourier

cosine transform and Fourier sine transform

xOxExfxfxfxfxf 2

1

2

1

dxkxxOidxkxxEkF 2sin2cos

19 Real Even Function

• If a real function f(x) is even, O(x) = 0

• If a function is real and even, the Fourier transform is also

real and even

0

2cos22cos2cos dxkxxfdxkxxfdxkxxEkF

F(k) f(x)

Bracewell, R. The Fourier Transform and Its Applications,

3rd ed. New York: McGraw-Hill, 2000.

x k

20 Real Odd Function

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

• If a function is real and odd, the Fourier transform is

imaginary and odd

0

2sin22sin2sin dxkxxfidxkxxfidxkxxOikF

F(k) f(x)



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

21 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)

22 Symmetry Properties

F(k) f(x)

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

23 Fourier Transform: Properties and Basic Theorems

• Linearity

• Shift theorem

• Modulation theorem

• Derivative

0xxf kFekix02

xkxf 02cos 002

1kkFkkF

kbGkaF xbgxaf

xf kkFi 2

24 Fourier Transform: Properties and Basic Theorems

• Parseval's / Rayleigh’s theorem

• Fourier Transform of the complex conjugated

• Similarity (a 0 : real constant)

• Special case (a = -1):

xf * kF *

xf kF

axf

a

kFa

1

dkkFdxxf22

25

xf kF

Similarity Theorem

1

1

1

1

1

1

1

xf 2

22

1 kF

xf 5

55

1 kF

x k

1 x k

x k 1

26 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

27 Convolution

• Definition of convolution

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

dxgfgfxy

Eric W. Weisstein.

“Convolution. http://mathworld.

wolfram.com/Convolution.html

28 Properties of the Convolution

• Properties of the convolution

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

factors

dxxgdxxfdxgf

fggf

hgfhgf )(

hfgfhgf )(

agfgafgfa

dx

dgfg

dx

dfgf

dx

d

commutativity

associativity

distributivity

29 The Basic Theorems: Convolution

• 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)

• Convolution in the spectral domain

xgxf kGkF

dxdxxxgxfe ikx '''2

dxxxgedxxfe xxikikx ''' '2'2

'''''' ''2'2 dxxgedxxfe ikxikx

kGkF

xgxf

xgxf kGkF

30 Cross-Correlation

• Definition of cross-correlation

• !

• Fourier transform:

, (see FT rules)

dtgftgtftrfg

*)(

tgtftgtf *)(

fggf

tgtftgtf *)(

kGkF* tgtf )(

tf * kF *

31 Example: Cross-Correlation Radar (Low Noise)

Signal sent by radar antenna Signal received by radar antenna

Time 0 Time 0

Cross correlation of the signals

Time 0

32 Example: Cross-Correlation Radar

Signal sent by radar antenna Signal received by radar antenna

Time 0

Cross correlation of the signals

Time 0

Time 0

33 Example: Cross-Correlation Ultrasonic Distance Measurement

Signal received by ultrasonic transducer

Time

Signal sent by ultrasonic transducer

Time

Transmitter

Receiver

TimerObject

d

Pulse

Start

Stop

Transmitter

Receiver

TimerObject

d

Pulse

Transmitter

Receiver

TimerObject

d

Pulse

Start

Stop

2

flightvtd (0)

Cross correlation of the two signals

Time 0 tflight

34 Autocorrelation Theorem

• Autocorrelation

• Autocorrelation theorem:

• Proof

dtfftftf

*)(

2* kFkFkF tftf )(

tf * kF *

2

kF tftf )(

tftftftf *)( ,

35 Example: Autocorrelation Detection of a Signal in the Presence of Noise

Time

Sig

na

l

Signal: Noise

Autocorrelation function:

Peak at t = 0 : Noise

Time

r ff

0

36 Example: Autocorrelation Detection of a Signal in the Presence of Noise

Sig

na

l r f

f

Signal: Sinusoid plus noise

Autocorrelation function:

Peak at t = 0 : Noise

Period of rff reveals that signal is

periodic

Time

Time 0

37

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

38 The Delta Function

• 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) 0x 0x

1

dxx

for 1.

2.

39 The Delta Function

• The sifting (sampling) property of the delta function

• Sampling of f(x) at x = a

• Fourier transform:

0fdxxfx

afdxxfax

0xx 022

0

ikxikx edxexx

40

x

k

Fourier Transform Pairs

• Cosine function

• Sine function

1

1

1

1

1

1 x k

xcos xII

2

1

2

1

2

1II xxx

2

1

2

1

2

1II xxx

xi II xsin

41

x k

2xe 2ke

Fourier Transform Pairs

• Gaussian function

• Rectangle function of unit height and base P(x)

P

2

10

2

1

2

12

11

x

x

x

x

1

1

1

1

1

1

1

1

x k

x

xx

sinsinc

ksinc

42 Fourier Transform Pairs

• Triangle function of unit height and area

10

11

x

xxx

x k 1

1

1

1

k2sinc

• Constant function (unit height)

x 1

1

k 1

(k)

43 Sampling

• Sampling: “Noting” a function at finite intervals



44 Sampling of a Signal

Signal 1

Samples taken at equal

intervals of time

Sampled signal

45 Sampling

Sampled signal does

not contain the additional

Signal

Samples taken at equal

intervals of time

Signal 1 + Signal 2

Signal 2

(very short)

46 Aliasing

Sampling with a low

sampling frequency

Sampled signal

Apparent signal

47 Sampling

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

n

nxx III

n a

nx

aax

1III

• Multiplication of f(x) with III(x)

describes sampling at unit intervals

n

nxnfxfx III

• Sampling at intervals :

n

nxnfxfx

IIIBracewell, R. The Fourier Transform and Its Applications,


48 Sampling

xIII kIII

• Fourier transform of the sampling function

xIII

1 kIII xIII

2III

x k2III2

kIII

1

here: = 2



49 Sampling:

Band limited signal

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

x k kc

f(x) F(k)



50 Sampling in the Two Domains

kc

xIII k III

-1

kc

xfx

III kFk III

f(x) F(k)

|k| kc x

= =


51 Reconstruction of the Signal

kc

xfx

III kFk III

|k|

|k|

f(x) F(k)

|k| kc x

= =

Multiplication with rectangular

function in order to remove

additional signal components

Original signal in

frequency domain

1 Convolution

Fourier

Transform

52 Critical Sampling and Undersampling

kc

f(x) F(k)

|k| kc x

kFk

kk

c

c

2III2

1

ck2

1

xfxkc2III

kc

Undersampling

ck2

1


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 with and inverse

transformation

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

kFk III

P

ck

k

2

Documents

Fourier Series Fourier Transform The Basic Theorems and ... · The Fourier Transform 1 • Fourier Series • Fourier Transform • The Basic Theorems and Applications • Sampling