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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.
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