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Dr Martin Hendry,School of Physics and AstronomyUniversity of Glasgow, UK
Astronomical Data
Analysis I
11 lectures, beginning autumn 2010
5. Fourier Methods
As we remarked in Section 1, a lot of astronomical data is
collected or processed as Fourier components.
In this section we briefly discuss the mathematics of Fourier
series and Fourier transforms. Some of these methods will be
applied to astronomical problems in ADA II.
5.1 Fourier Series
Any ‘well-behaved’ function can be
expanded in terms of an infinite sum of sines
and cosines. The expansion takes the form:
Joseph Fourier(1768-1830)
)(xf
1 1
021 sincos)(
n nnn nxbnxaaxf (5.1)
The Fourier coefficients are given by the formulae:
These formulae follow from the orthogonality properties of sin and cos:
dxxfa )(
10 (5.2)
nxdxxfan cos)(
1
nxdxxfbn sin)(
1
(5.3)
(5.4)
mnnxdxmx
sinsin mnnxdxmx
coscos
0cossin nxdxmx
“Fourier's Theorem is not only one of the most beautiful results of modern analysis, but it is said to furnish an indispensable instrument in the treatment of nearly every recondite question in modern physics”
The Fourier series can also be written in complex form:
where
and recall that
n
inxneAxf )( (5.5)
dxexfA inx
n )(2
1(5.6)
nxinxeinx sincos
nxinxe inx sincos (5.7)
The Fourier transform can be thought of simply as extending the
idea of a Fourier series from an infinite sum over discrete, integer
Fourier modes to an infinite integral over continuous Fourier modes.
Consider, for example, a physical process that is varying in the time
domain,
i.e. it is described by some function of time .
Alternatively we can describe the physical process in the frequency
domain by defining the Fourier Transform function .
It is useful to think of and as two different
representations of the same function; the information they convey
about the underlying physical process should be equivalent.
)(th
5.2 Fourier Transform: Basic Definition
)( fH
)(th )( fH
We define the Fourier transform as
and the corresponding inverse Fourier transform as
If time is measured in seconds then frequency is measured in cycles
per second, or Hertz.
dtethfH tif2)()( (5.8)
dfefHth tif2)()( (5.9)
In many astrophysical applications it is common to define the
frequency domain behaviour of the function in terms of angular
frequency
This changes eqs. (5.8) and (5.9) accordingly:
Thus the symmetry of eqs. (5.8) and (5.9) is broken.
In this course we will adopt the definitions given in (5.8) and (5.9)
f 2
dtethH ti 2)()( (5.10)
deHth ti2)(2
1)( (5.11)
5.3 Fourier Transform: Further properties
The FT is a linear operation:
(1) the FT of the sum of two functions is equal to the sum of their FTs
(2) the FT of a constant times a function is equal to the constant times
the FT of the function.
If the time domain function is a real function, then its FT is
complex.
However, more generally we can consider the case where is also
a
complex function – i.e. we can write
may also possess certain symmetries: even function
odd function
)(th
)(th
)()()( tihthth IR (5.12)
Real partImaginary part
)(th )()( thth
)()( thth
The following properties then hold:
Note that in the above table a star (*) denotes the complex
conjugate, i.e. if z = x + i y then z* = x − i y
See Numerical Recipes, Section 12.0
For convenience we will denote the FT pair by
It is then straightforward to show that
)()( fHth
)(1
)( afHa
ath (5.13) “time scaling”
)()(1
fbHbthb
(5.14) “frequency scaling”
020 )()( tfiefHtth (5.15)
“frequency scaling”
“time shifting”
(5.16))()( 02 0 ffHeth tfi
Suppose we have two functions and
Their convolution is defined as
We can prove the Convolution Theorem
i.e. the FT of the convolution of the two functions is equal to the product of their individual FTs.
Also their correlation, which is also a function of t , is defined as
)(tg )(th
dssthsgthg )()()( (5.17)
)()()( fHfGthg (5.18)
dsshtsghg )()(),(Corr (5.19)
Known as the lag
We can prove the Correlation Theorem
i.e. the FT of the first time domain function, multiplied by the complex conjugate of the FT of the second time domain function, is equal to the FT of their correlation.
The correlation of a function with itself is called the auto-correlation
In this case
The function is known as the power spectral density,
or (more loosely) as the power spectrum.
Hence, the power spectrum is equal to the Fourier Transform of the
auto-correlation function for the time domain function
)()(),(Corr fHfGhg
(5.21)
(5.20)
2)(),(Corr fGgg
2)( fG
)(tg
5.4 The power spectral density
The power spectral density is analogous to the pdf we defined in
previous sections.
In order to know how much power is contained in a given interval of
frequency, we need to integrate the power spectral density over
that interval.
The total power in a signal is the same, regardless of whether we
measure it in the time domain or the frequency domain:
- -
22PowerTotal dfH(f)dth(t) (5.22)
Parseval’s Theorem
We can, therefore, think of moving between the time and frequency domain as analogous to the change of variables we employed for pdfs in Section 4
Often we will want to know how much power is contained in a
frequency interval without distinguishing between positive and
negative values.
In this case we define the one-sided power spectral density:
And
When is a real function
With the proper normalisation, the total power (i.e. the integrated
area under the relevant curve) is the same regardless of whether
we are working with the time domain signal, the power spectral
density or the one-sided power spectral density.
ffHfHfPh 0)()()(22
(5.23)
0
)(PowerTotal dffPh (5.24)
)(th 2)(2)( fHfPh (5.25)
const.)( th
t
)(th
)0()( DfH
f
)( fH
0 0
Dirac Delta function
tfieth 02)(
t
)(th
0
Imaginary partReal part
)()( 0D fffH
f
)( fH
0
(1)
(2)
5.5 Examples
xy sinc
x
1y
xatzero1stxatzero1st
otherwise0
for1)( 2
121 t
th
t
)(th
0
(3)
)(sinc)( ffH
f
)( fH
0
Imaginary part = 0
The sinc function occurs frequently in many areas of astrophysics
The function has a maximum at and the zeros occur at
for positive integer m
x
xx
sinsinc
0xmx
1/2 t0
1
-1/2
(4) )()( tth
f0
1
)(sinc)( 2 ffH
(5)
Real part
Imaginary part
ateth )(
2222
2
222
2
4
2)(Im
4)( Re
fa
fafH
fa
afH
)( fH
)( fH
t
222
2
4)( Re
fa
afH
This function is a Lorentzian
and is commonly modelled as the
shape of spectral line profiles in
astronomy.
One can also show that the
Power Spectrum corresponding
to this FT is also a Lorentzian.
(See examples).
)( fH
f
i.e. the FT of a Gaussian function in the time domain is also a
Gaussian in the frequency domain.
The broader the Gaussian is in the time domain, then the narrower
the Gaussian FT in the frequency domain, and vice versa.
(6) 2exp)( atth
t0
f0
222 /exp)( affH
Although we have discussed FTs so far in the context of a
continuous, analytic function, , in many practical situations we
must work instead with observational data which are sampled at a
discrete set of times.
Suppose that we sample in total times at evenly
spaced time intervals , i.e. (for even)
[ If is non-zero over only a finite interval of time, then we
suppose that the sampled points contain this interval. Or if
has an infinite range, then we at least suppose that the
sampled points cover a sufficient range to be representative of the
behaviour of ].
5.6 Discrete Fourier Transforms
)(th
)(th
(5.26)2,...,0,...,2, NNkktk )( kk thh where
1N
)(th
1N )(th
)(th
N
We therefore approximate eq. (5.8) as
Since we are sampling at discrete timesteps, in view of
the symmetry of the FT and inverse FT it makes sense also to
compute
only at a set of discrete frequencies:
(The frequency is known as the Nyquist (critical)
frequency and it is a very important value. We discuss its
significance in Section 6).
2
2
22)()(Nk
Nk
tifk
tif kehdtethfH (5.27)
)(th 1N)( fH
1N
2,...,0,...,2, NNnN
nfn
(5.28)
21cf
Then
Note that
Hence, in eq. (5.29) there are only independent values.
Also, note that
So we can re-define eq. (5.29) as:
n
N
k
Nnkikn HehfH
1
0
/2)(
nini ee (5.30)
N
NkNniNnkiniNnki eee /)(2/22/2
(5.31)
2
2
/22
2
2)(Nk
Nk
Nnkik
Nk
Nk
tifkn ehehfH kn
(5.29)
Discrete Fourier Transform of the kh
The discrete inverse FT, which recovers the set of from the
set of
is
Parseval’s theorem for discrete FTs takes the form
There are also discrete analogues to the convolution and correlation
theorems.
s'khs'nH
1
0
/21 N
k
Nnkink eH
Nh
(5.32)
1
0
21
0
2 1 N
nn
N
kk H
Nh (5.33)
Consider again the formula for the discrete FT. We can write it as
This is a matrix equation: we compute the vector of
by multiplying the matrix by the vector of
.
In general, this requires of order multiplications (and the
may be complex numbers).
e.g. suppose . Even if a computer can
perform (say) 1 billion multiplications per second, it would still
require more than 115 days to calculate the FT.
5.7 Fast Fourier Transforms
(5.34)
1
0
1
0
/2N
kk
nkN
kk
Nnkin hWheH
s'kh)1( N s'nH
)( NN nkW )1( N
2N s'kh
1628 1010 NN
Fortunately, there is a way around this problem.
Suppose (as we assumed before) is an even number. Then we can
write
where
So we have turned an FT with points into the weighted sum of two
FTs with points. This would reduce our computing time by a
factor of two.
N
12/
012
/)12(212/
02
/)2(21
0
/2N
jj
NnjiN
jj
NnjiN
kk
Nnkin heheheH
(5.35)
Even values of k Odd values of k
1
012
/21
02
/2M
jj
MnjinM
jj
Mnji heWhe
2/NM
N2/N
(5.36)
Why stop there, however?...
If is also even, we can repeat the process and re-write the FTs of
length as the weighted sum of two FTs of length .
If is a power of two (e.g. 1024, 2048, 1048576 etc) then we
can repeat iteratively the process of splitting each longer FT into two
FTs half as long.
The final step in this iteration consists of computing FTs of length unity:
i.e. the FT of each discretely sampled data value is just the data value
itself.
MM 2/M
……
N
00
020 hheH
kk
ki
(5.37)
This iterative process converts multiplications into
operations.
So our operations are reduced to about
operations.
Instead of 100 days of CPU time, we can perform the
FT in less than 3 seconds.
The Fast Fourier Transform (FFT) has revolutionised our ability to tackle
problems in Fourier analysis on a desktop PC which would otherwise be
impractical, even on the largest supercomputers.
)( 2NO)log( 2 NNO
This notation means ‘of the order of’
16109107.2
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