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