Astronom

ische Waarneem

technieken(Astronom

ical Observing T

echniques)

5thLecture

: 12 Octob

er 2

011

1.

Fourie

r Serie

s

2.

Fourie

r Tra

nsfo

rm

Source

s: Lena b

ook, Brace

wellbook, W

ikipedia

2.

Fourie

r Tra

nsfo

rm

3.

FT

Exam

ple

s in

•1

D•

2D

4.

Importa

nt T

heore

ms

Jean B

aptiste

Jose

ph Fourie

r

From

Wikiped

ia:Jean B

aptisteJoseph

Fourier (2

1 March

1768 –

16 May 18

30) was F

rench math

ematician and

physicist b

est known for initiating th

e investigation of F

ourier series and their

applications to problem

s of heat transfer and

vibrations.

A Fourie

r series decom

poses any pe

riodic function or

period

ic signal into the sum

of a (possibly infinite

) set of

simple oscillating functions, nam

ely sine

sand

cosines (or

complex expone

ntials).

Application: h

armonic analysis of a function f(x

,t)to

study spatial or

temporal fre

quencie

s.

Fourie

r Serie

s

Fourier analysis = d

ecomposition using sin() and

cos() as basis set.

Consid

er a periodic function:

()

()

[]

∑∞

++

0sin

cos

nx

bn

xa

a

()

()

π2+

=x

fx

f

The F

ourier series for f(x)is given b

y:()

()

()

()d

xn

xx

fb

dx

nx

xf

a

n n

sin1

cos

1

∫ ∫− −

= =

ππ ππ

π π ()

()

[]

∑=

++

1

0sin

cos

2n

nn

nx

bn

xa

a

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:

()

!1

==

∫ π

and hence:

()

()

()n

dx

nx

xb

dx

nx

xa

n

n n

1

!

12

sin1

0)

aro

un

d

sym

metric

is

(cos()

0

cos

1

+

− −

−=

=

==

∫ ∫ππ π

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

Wikipe

dia: L

eonh

ard Euler (17

07 –1783) was a pione

ering

Swiss m

athematician and

physicist. H

e made important

discove

ries in fie

lds as d

iverse

as infinitesim

al calculus and

graph theory. H

e also introd

uced much

of the modern

math

ematical te

rminology and

notation.

Euler’s form

ula describ

es the relationsh

ip betw

een the trigonom

etric functions and the com

plex

()

()

πθ

πθ

πθ

2sin

2co

s2

ie

i+

=

the trigonom

etric functions and the com

plex

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−

+∞∞

−

⋅=∫

+∞

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:

()

()

ds

es

Fx

fxs

iπ

2⋅

=∫ +∞∞

−

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

tremasand

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

xP

sF

xQ

xP

xf

od

deven

∫ ∞+

=⇒

+=

2co

s2

If

π(

)(

)(

)

()

()d

xxs

xQ

i

dx

xsx

Ps

F

∫ ∫∞

+

− =⇒

0

0

2sin

2

2co

s2

π π

Propertie

s of the Fourie

r Transform

(2)

()

()

⇔→

a sF

aa

xf

xf

1

SIM

ILARITY:

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:

LINEARITY:

TRANSLATION:

DERIVATIVE:

()

()

()s

Fs

ix

fn

n

π2⇔

∂

()

()s

Fa

as

F⋅

=

DERIVATIVE:

ADDITION:

()

()

()s

Fs

ix

xf

n

nπ2

⇔∂

∂

Importa

nt 1-D Fourie

r Pairs

Importa

nt 1-D Fourie

r Pairs

Spe

cial 1

-D Pa

irs (1): th

e Box

Function

Consid

er the b

ox function:

<

<=

Π

elsew

here

0

22

for

1

ax

a-

a x

()

()

()s

sx

sin

c

sin≡

⇔Π

πWith the F

ourier pairs

a-

a

()

()

()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

(as)

a-

a

Spe

cial 1

-D Pa

irs (2): th

e D

irac C

omb

Consid

er Dirac’s d

elta “function”:

∞∞

Fo

urier

1

()

()

()

{}

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

:

()

()

∑∑

∞−∞

=

∞−∞

=

∆∆

=∆

−=

Ξn

Tn

xi

Fo

urier

seriesk

xe

xx

kx

x/

21

πδ

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 impulse train or sam

pling function.

Side note

: Sampling (1

)Sampling m

eans reading off th

e value of the signal at d

iscrete 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 .

()

()s

Fx

f⇔

()

()

∆Ξ⋅

→x x

xf

xf

[]

ss

+−

,bound

ed support .

Then, a sam

pled distrib

ution of the form

with a sam

pling rate of:

is enough to reconstruct f(x

)for all x

.

()

()

∆Ξ⋅

=x x

xf

xg

ms

x2

1=

∆

[]m

ms

s+

−,

Side note

: Sampling (2

)

Sampling at any rate ab

ove or below

the critical sam

pling is called

oversampling

or undersam

pling, respectively.

Oversam

pling: red

undant m

easurements, often low

ering the S

/N

Undersam

pling: measurem

ent depend

ent on “single pixel” or aliasing

A fam

ily of sinusoids at th

e critical fre

quency, all h

aving the sam

e sam

ple seque

nces

of alternating +1 and

–1. That is, th

ey all are

aliases of e

ach oth

er, e

ven th

ough their

freque

ncy is not above

half th

e sam

ple rate

.

Side note

: Besse

l Functions (1

)

Friedrich

Wilhelm Besse

l (1784 –1846) was a G

erman

math

ematician, astronom

er, and

systematize

rof th

e

Besse

l functions. “His” functions w

ere first d

efine

d by

the math

ematician D

aniel Bernoulli and

then ge

neralize

d

by F

riedrich

Besse

l.

The Besse

l functionsare

canonical solutions y(x)of

Besse

l's diffe

rential e

quation:

for an arbitrary re

al or complex num

ber n, th

e so-calle

d

order of th

e Besse

l function. ()

02

2

2

22

=−

+∂ ∂

+∂ ∂

yn

xx y

xx

yx

Side note

: Besse

l Functions (2

)

The solutions

to Besse

l's diffe

rential e

quation are calle

d

Besse

l functions:

()

()(

)∑

∞=

+

+

−

=0

2! !

21

k

nk

k

nn

kk

x

xJ

Bessel functions are also know

n as cylinder

functions or cylindrical

harmonics

because they

are found in the solution

to Laplace's equation in cylind

rical coordinates.

Spe

cial 2

-D Pa

irs (1): th

e Box

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 : ω

2⇔

ΠUsing th

e Bessel function J

1 :

and using th

e similarity relation :

()

ω

ωπ

aJ

aa r

2

2

1⋅

⇔

Π

Example: 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 inte

gral of the

product of th

e tw

o functions after one

is reverse

d and

shifte

d:

()

()

()

()

()d

uu

xg

uf

xg

xf

xh

∫ +∞∞

−

−⋅

=∗

=Convolution (2

)

()

()

()

()

()

()

()

()

()

()s

Hs

Gs

Fx

gx

fx

hs

Gx

g

sF

xf

=⋅

⇔∗

=→

⇔ ⇔

Note

: The convolution of tw

o functions (distrib

utions) is equivale

nt to the prod

uct of their F

ourier transform

s:

Convolution (3

)

Example:

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=

∗

Example:

Convolution of f(x

)with a sm

ooth kernel g(x

) can be used

to smooth

en Convolution of f(x

)with a sm

ooth kernel g(x

) can be used

to smooth

en f(x

)

Example:

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-corre

lation (or covariance) is a m

easure

of sim

ilarity of two w

aveform

s as a function of a time-lag

applied to one

of them.

()

()

()

()

()d

uu

xg

uf

xg

xf

xk

∫ +∞∞

−

+⋅

=⊗

=

The d

ifferencebetw

een cross-correlation and convolution is:

•Convolution reverses th

e signal (‘-’ sign)•

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

)

Convolution a

nd Cross-

Corre

lation

The cross-corre

lationis a

measure

of similarity of tw

o wave

forms as a function of

an offset (e

.g., a time-lag)

between th

em.

The convolution

is similar in

nature to th

e cross-corre

lation but th

e convolution first

reverse

s the signal (“m

irrors the function”) prior to

calculating the ove

rlap.(

)(

)(

)(

)(

)d

uu

xg

uf

xg

xf

xk

∫ +∞

+⋅

=⊗

=

()

()

()

()

()

∫ +∞

−⋅

=∗

=

Example: se

arch a long

duration signal for a sh

orter,

known fe

ature.

Example: th

e measure

d signal

is the intrinsic signal convolve

d

with the response

function

()

()

()

()

()

∫∞−

()

()

()

()

()d

uu

xg

uf

xg

xf

xh

∫∞

−

−⋅

=∗

=

Whereas convolution involves reversing a signal, th

en shifting

it and multiplying b

y another signal, correlation only involves

shifting it and

multiplying (no reversing).

Auto-

Corre

lation

The auto-corre

lation is a cross-correlation of a

function with itse

lf:(

)(

)(

)(

)(

)d

uu

xf

uf

xf

xf

xk

∫ +∞∞

−

+⋅

=⊗

=

+

+

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 S

pectrum

Sfof f(x

)(or th

e Pow

er S

pectral

Density, PS

D) describ

es how the pow

er of a signal is

distrib

uted with fre

quency.

The pow

er is often defined

as the squared

value of the signal:

()

()

2s

Fs

Sf

=

The pow

er spe

ctrum ind

icates w

hat fre

quencie

s 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’s

theore

m (or R

ayleigh

’s Energy T

heore

m) state

s that th

e sum

of the square

of a function is the sam

e as

the sum

of the square

of transform:

()

()

ds

sF

dx

xf

∫∫

+∞∞

−

+∞∞

−

=2

2

Interpre

tation:The total e

nergy containe

d in a signal

f(t), summed ove

r all times t

is equal to th

e total e

nergy

of the signal’s F

ourier transform

F(v)

summed ove

r all fre

quencie

s v.

Wiene

r-Khinch

inTheore

m

The Wiener–K

hinch

in(also W

iener–K

hintch

ine) th

eore

m

states th

at the pow

er spe

ctral density S

fof a function

f(x)is th

e Fourie

r transform of its auto-corre

lation function:

()

()

()

{}

xf

xf

FT

sF

2

⊗=

b()

()s

Fs

F*

⋅

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.

Fourie

r Filte

ring –an E

xample

Example take

n from http://te

rpconnect.um

d.edu/~

toh/spe

ctrum/Fourie

rFilte

r.html

To

p le

ft: sig

na

l –is

I just ra

nd

om

no

ise

?

To

p rig

ht:

po

we

r sp

ectru

m: h

igh

-freq

ue

ncy c

om

po

ne

nts

do

min

ate

the

sig

na

l

Bo

ttom

left:

po

we

r sp

ectru

m e

xp

an

de

d in

X a

nd

Y to

em

ph

asiz

e th

e lo

w-fre

qu

en

cy re

gio

n.

Th

en

: use

Fo

urie

r filter fu

nctio

n to

de

lete

all h

arm

on

ics h

igh

er th

an

20

Bo

ttom

righ

t: reco

nstru

cte

d s

ign

al �

sig

na

l co

nta

ins tw

o b

an

ds a

t x=

20

0 a

nd

x=

30

0.

Overvie

w

Convolution

Cross-

correlation

Auto-

correlation

()

()

()

()

()d

uu

xg

uf

xg

xf

xh

∫ +∞∞

−

−⋅

=∗

=

()

()

()

()

()d

uu

xg

uf

xg

xf

xk

∫ +∞∞

−

+⋅

=⊗

=

()

()

()

()

()d

uu

xf

uf

xf

xf

xk

∫ +∞∞

−

+⋅

=⊗

=

Power spe

ctrum

Parse

val’s

theore

m

Wiene

r-Khinch

intheore

m

()

()

2s

Fs

Sf

=

()

()

ds

sF

dx

xf

∫∫

+∞∞

−

+∞∞

−

=2

2

()

()

()

{}

()

()s

Fs

F

xf

xf

FT

sF

*

2

⋅

⊗=

b