Astronom

ische Waarneem

technieken

(Astronom

ical Observing T

echniques)

5thLecture

: 13 Octob

er 2

010

Source

s: Lena b

ook, Brace

wellbook, W

ikipedia

Jean B

aptiste

Jose

ph Fourie

r

From

Wikiped

ia:Jean B

aptisteJoseph

Fourier (2

1 March

1768 –

16 May 18

30) w

as French

math

ematician and

physicist b

est known for initiating th

e investigation of F

ourier series and th

eir applications to prob

lems of h

eat transfer and

vibrations.

A Fourier series d

ecomposes any period

ic function or period

ic signal into the sum

of a (possibly infinite) set of

simple oscillating functions, nam

ely sinesand

cosines (or com

plex ex

ponentials).

Application: h

armonic analysis of a function f(x

,t)to

study spatial or

temporal frequencies.

Fourie

r Serie

s

Fourier analysis = d

ecomposition using sin() and

cos() as basis set.

Consid

er a periodic function:

()

()

()

()d

xn

xx

fb

dx

nx

xf

a

n n

sin1

cos

1

∫ ∫− −

= =

ππ ππ

π π ()

()

[]

∑∞=

++

1

0sin

cos

2n

nn

nx

bnx

aa

()

()

π2+

=x

fx

f

The F

ourier series for f(x)is given b

y:

with

the tw

o Fourier coefficients:

Example

: Sawtooth

Function

Consid

er the saw

toothfunction:

()

()

()x

fx

f

xx

xf

=+

<<

−=

π

ππ

2

for

Then th

e Fourier coefficients are:

and hence:

()

()

()n

dx

nx

xb

dx

nx

xa

n

n n

1

!

12

sin1

0)

aro

und

sy

mm

etric

is

(cos()

0

cos

1

+

− −

−=

=

==

∫ ∫ππ ππ

π π

()

()

()

[]

()

()

nx

nn

xb

nx

aa

xf

n

n

n

nn

sin1

2sin

cos

21

1

1

0∑

∑∞=

+∞=

−=

++

=

()

()

()

nx

nx

fn

n

sin1

21

1

∑∞=

+−

=

Example

: Sawtooth

Function (2

)

Side note

: Eule

r’s Form

ula

Wikiped

ia: Leonh

ard Euler (17

07 –1783) w

as a pioneering Swiss m

athem

atician and ph

ysicist. He m

ade im

portant discoveries in field

s as diverse as infinitesim

al calculus and

graph th

eory. He also introd

uced much

of the m

odern

math

ematical term

inology and notation.

()

()

πθ

πθ

πθ

2sin

2co

s2

ie

i+

=

Euler’s form

ula describ

es the relationsh

ip betw

een the trigonom

etric functions and th

e complex

exponential function:

With

that w

e can rewrite th

e Fourier series

in terms of th

e basic w

avesπ

θ2i

e

Definition of th

e Fourie

r Transform

The functions f(x

)and

F(s)

are called Fourier pairs if:

()

()

dx

ex

fs

Fxs

iπ

2−

+∞∞

−

⋅=∫

()

()

ds

es

Fx

fxs

iπ

2⋅

=∫ +∞∞

−

For sim

plicity we use x

but it can b

e generalized to m

ore dimensions.

The F

ourier transform is reciprocal, i.e., th

e back-transform

ation is:

Requirem

ents:•

f(x) is b

ounded

•f(x

) is square-integrable

•f(x

) has a finite num

ber of ex

tremas

and discontinuities

()

∫ +∞∞

−

dx

xf

2

Note that m

any mathem

atical functions (incl. trigonometric functions)

are not square integrable, b

ut essentially all physical quantities are.

Propertie

s of the Fourie

r Transform

(1)

SYMMETRY:

The F

ourier transform is sym

metric:

()

()

()

()

()

()

()

()d

xxs

xQ

i

dx

xsx

Ps

F

xQ

xP

xf

od

deven

∫ ∫∞

+ ∞+

− =⇒

+=

0

0

2sin

2

2co

s2

If

π π

Propertie

s of the Fourie

r Transform

(2)

()

()

⇔→

a sF

aa

xf

xf

1

SIM

ILARIT

Y:

The d

ilatation (or expansion) of a function f(x

)causes a contraction

of its transform F(s):

Propertie

s of the Fourie

r Transform

(3)

()

()s

Fe

ax

fa

si

π2

−⇔

−

More properties:

LIN

EARIT

Y:

TRANSLATIO

N:

DERIVATIVE:

ADDIT

ION:

()

()

()s

Fs

ix

xf

n

n

n

π2⇔

∂

∂

()

()s

Fa

as

F⋅

=

Importa

nt 1-D Fourie

r Pairs

Spe

cial 1

-D Pa

irs (1): th

e B

ox Function

Consid

er the b

ox function:

<

<=

Π

elsew

here

0

22

for

1

ax

a-

a x

()

()

()s

s

sx

sin

c

sin≡

⇔Π

π

πWith

the F

ourier pairs

and using th

e similarity relation w

e get:

()

as

aa x

sin

c⋅

⇔

Π

a-

a

(as)

a-

a

Spe

cial 1

-D Pa

irs (2): th

e D

irac C

omb

Consid

er Dirac’s d

elta “function”:

()

()

∑∑

∞−∞

=

∞−∞

=

∆∆

=∆

−=

Ξn

Tn

xi

Fo

urier

seriesk

xe

xx

kx

x/

21

πδ

()

()

()

{}

1

2

=→

==

∫ +∞∞

−

xF

Td

xe

xx

fsx

iδ

δπ

Now construct th

e “Dirac com

b” from

an infinite series of d

elta-functions, spaced at intervals of T

:

Ξ(x

)

Ξ(x

)⋅f(x)

Note:

•the F

ourier transform of a D

irac comb is also

a Dirac com

b•

Because of its sh

ape, the D

irac comb is also

called im

pulse train or sampling function.

Side note

: Sampling (1

)Sam

pling means read

ing off the value of th

e signal at discrete values

of the variab

le on the x

-axis.

The interval b

etween tw

o successive readings is th

e sampling rate.

The critical sam

pling is given by th

e Nyquist-S

hannon th

eorem:

Consid

er a function , where F

(s) has

bound

ed support .

Then, a sam

pled distrib

ution of the form

with

a sampling rate of:

is enough to reconstruct f(x

)for all x

.

()

()

∆Ξ⋅

=x x

xf

xg

ms

x2

1=

∆

()

()s

Fx

f⇔

()

()

∆Ξ⋅

→x x

xf

xf

[]m

ms

s+

−,

Side note

: Sampling (2

)

A fam

ily of sinusoids at th

e critical fre

quency, all h

aving the sam

e sam

ple se

quence

s of alte

rnating +1 and –1. T

hat is, th

ey all are

aliases of e

ach oth

er, e

ven th

ough th

eir

freque

ncy is not above

half th

e sam

ple rate

.

Sam

pling at any rate above or b

elow th

e critical sampling is called

oversam

plingor und

ersampling, respectively.

Oversam

pling: red

undant m

easurements, often low

ering the S

/N

Undersam

pling: measurem

ent depend

ent on “single pixel” or aliasing

Side note

: Besse

l Functions (1

)

Fried

rich W

ilhelm

Bessel (17

84 –1846) w

as a Germ

an math

ematician, astronom

er, and system

atizerof th

e Bessel functions. “H

is” functions were first d

efined by

the m

athem

atician Daniel B

ernoulli and th

en generalized

by F

riedrich

Bessel.

The B

essel functionsare canonical solutions y(x

)of

Bessel's d

ifferential equation:

for an arbitrary real or com

plex num

ber n, th

e so-called

order of th

e Bessel function. (

)0

22

2

22

=−

+∂ ∂

+∂ ∂

yn

xx y

xx

yx

Side note

: Besse

l Functions (2

)

The solutions

to Bessel's d

ifferential equation are called

Bessel functions:

Bessel functions are also

known as cylind

er functions or cylind

rical harm

onicsbecause they

are found in the solution

to Laplace's equation in cylind

rical coordinates. (

)(

)()

∑∞=

+

+

−

=0

2! !

21

k

nk

k

nn

kk

x

xJ

Spe

cial 2

-D Pa

irs (1): th

e B

ox Function

Consid

er the 2

-D box function

with

r2= x

2+ y

2:

≥ <=

Π1

for

0

1fo

r

1

2r r

r

()

ω πω

2

2

1J

r⇔

ΠUsing th

e Bessel function J

1 :

and using th

e similarity relation :

()

ω

ωπ

aJ

aa r

2

2

1⋅

⇔

Π

Exam

ple: optical telescope

Aperture (pupil):

Focal plane:

()

()

ω πω

21

J

rΠ

Spe

cial 2

-D Pa

irs (2): th

e G

auss F

unction

Consid

er a 2-D Gauss

functionwith

r2= x

2+ y

2:

() 2

2

22

sim

ilarityω

ππ

πω

πa

a r

re

ae

ee

−

−

−−

⋅⇔

→⇔

Note: T

he G

auss function is preserved und

er Fourier transform

!

Importa

nt 2-D

Fourie

r Pairs

Convolution (1

)The convolution

of two functions, ƒ

∗g, is the integral of th

e prod

uct of the tw

o functions after one is reversed and

shifted

: (

)(

)(

)(

)(

)du

ux

gu

fx

gx

fx

h

∫ +∞∞

−

−⋅

=∗

=Convolution (2

)

()

()

()

()

()

()

()

()

()

() s

Hs

Gs

Fx

gx

fx

hs

Gx

g

sF

xf

=⋅

⇔∗

=→

⇔ ⇔

Note: T

he convolution of tw

o functions (distrib

utions) is equivalent to th

e product of th

eir Fourier transform

s:

Convolution (3

)

Exam

ple:f(x

): star

g(x): telescope transfer function

Then h(x

)is th

e point spread function (PS

F)of th

e system

()

()

()x

hx

gx

f=

∗

Exam

ple:Convolution of f(x

)with

a smooth

kernel g(x) can b

e used to sm

oothen

f(x)

Exam

ple:The inverse step (d

econvolution) can be used

to “disentangle” tw

o com

ponents, e.g., removing th

e spherical ab

erration of a telescope.

Cross-

Corre

lation

The cross-correlation (or covariance) is a m

easure of sim

ilarity of two w

aveforms as a function of a tim

e-lag applied

to one of them

.

()

()

()

()

()du

ux

gu

fx

gx

fx

k

∫ +∞∞

−

+⋅

=⊗

=

The d

ifferencebetw

een cross-correlation and convolution is:

•Convolution reverses th

e signal (‘-’ sign)•

Cross-correlation sh

ifts the signal and

multiplies it w

ith anoth

er

Interpretation: By h

ow much

(x) m

ust g(u)be sh

ifted to m

atch f(u)?

The answ

er is given by th

e maximum

of k(x)

Auto-

Corre

lation

The auto-correlation is a cross-correlation of a

function with

itself:(

)(

)(

)(

)(

)du

ux

fu

fx

fx

fx

k

∫ +∞∞

−

+⋅

=⊗

=

Wikiped

ia: The auto-correlation yield

s the similarity

betw

een observations

as a function of the time

separation betw

een them.

It is a mathem

atical tool for finding repeating patterns,

such as the presence of a periodic signal w

hich has been

buried

under noise.

+

+

Power S

pectrum

The Pow

er Spectrum

Sfof f(x

)(or th

e Power S

pectral Density, PS

D) d

escribes h

ow th

e power of a signal is

distrib

uted with

frequency.

The pow

er is often defined

as the squared

value of the signal:

()

()

2s

Fs

Sf

=

The pow

er spectrum ind

icates what frequencies carry

most of th

e energy .

The total energy of a signal is:

Applications:

spectrum analyzers, calorim

eters of light sources, …

()

∫ +∞∞

−

ds

sS

f

Parse

val’s T

heore

m

Parseval’stheorem

(or Rayleigh

’s Energy T

heorem

) states that th

e sum of th

e square of a function is the sam

e as the sum

of the square of transform

:

()

()

ds

sF

dx

xf

∫∫

+∞∞

−

+∞∞

−

=2

2

Interpretation:The total energy contained

in a signal f(t), sum

med over all tim

es tis equal to th

e total energy of th

e signal’s Fourier transform

F(v)

summed over all

frequencies v.

Wiene

r-Khinch

in Theore

m

The W

iener–Khinch

in(also W

iener–Khintch

ine) theorem

states th

at the pow

er spectral density S

fof a function

f(x)is th

e Fourier transform

of its auto-correlation function:

()

()

()

{}

()

()s

Fs

F

xf

xf

FT

sF

*

2

⋅

⊗=

b

Applications:

E.g. in th

e analysis of linear time-invariant system

s, when th

e inputs and outputs are not square integrab

le, i.e. their

Fourier transform

s do not ex

ist.