### Fourier Transforms

```Fourier Transform
Insights and Techniques
Synthesis Imaging School -- Narrabri, Sept. 2014
John Dickey
University of Tasmania
Including slides from
Bob Watson
Outline
1.
•
•
•
One dimensional functions
Fourier Series equations and examples
Fourier Transform examples and principles
Deconvolution and aliasing
2. Two dimensional functions
• deconvolution
• Combining single-dish and interferometer data
Reference: The Fourier Transform and its Applications, R.N. Bracewell (John Wiley)
Statement of Fourier’s Theorem
Suitable periodic functions, with a period
of 2p, may be represented as
f(q ) =
¥
1
2
a 0 + å (an cos nq + bn sin nq )
n=1
(called a Fourier series)
Here
a0 =
an =
bn =
1
p
1
p
1
p
q 0 +2 p
òq
0
q 0 +2 p
òq
f (q ) cos(nq ) dq
0
q 0 +2 p
òq
f (q ) dq
f (q ) sin(nq ) dq
0
The word “suitable” covers most
functions you meet in physics
Complex form of Fourier’s Theorem
Suitable periodic functions, with a
period of 2p, may be represented as
where j = √-1 and
Cn =
1
2p
ò
q0 +2 p
q0
f (q ) e
- jnq
dq
Triangular wave
h
-p
(
p
)
q
sin
q
sin
3
q
f (q ) = 8h 2 (
2
2
p
1
3
sin 5q
sin 7q
... +
...)
2
2
5
7
Triangular wave
First 3 series terms:
Waves: 5
Triangular wave
Sum of first 2 terms:
Waves: 5
Triangular wave
Sum of first 3 terms:
Waves: 5
Triangular wave
Sum of first 5 terms:
Waves: 5
Triangular wave
Sum of first 10 terms:
Waves: 5
Triangular wave
All partial series sums up to 20 terms:
20
15
10
5
Half-wave rectifier
h
-p
p
q
p
2
f (q ) = h ( 1 + sinq cos2q
p
2
1.3
2
2
... cos4q cos6q ...)
3.5
5.7
( )
Half-wave rectifier
First 3 series terms:
Waves: 5
Half-wave rectifier
Sum of first 2 terms:
Waves: 5
Half-wave rectifier
Sum of first 3 terms:
Waves: 5
Half-wave rectifier
Sum of first 5 terms:
Waves: 5
Half-wave rectifier
Sum of first 10 terms:
Waves: 5
Even square wave
h
-p
( )
p
q
1
4h
f (q ) =
(cosq - 3 cos3q
p
1
1
... + 5 cos5q - 7 cos7q ...)
Even square wave
First 3 series terms:
Waves: 5
Even square wave
Sum of first 2 terms:
Waves: 5
Even square wave
Sum of first 3 terms:
Waves: 5
Even square wave
Sum of first 5 terms:
Waves: 5
Even square wave
Sum of first 10 terms:
Gibbs’ phenomenon
Waves: 5
The Fourier Series Spectrum:
Even square wave
h
-p
q
p
a1 = 4h/p
an
a1/5
-a1/3
1
2
3
4
n
5
Example: Odd square wave
h
q
-p
p
b1 = 4h/p
b1/3
bn
b1/5
n
1
2
3
4
5
Spectra for complex transform coefficients
f (q ) =
+¥
åC e
jnq
n
n=-¥
Cn =
1
2p
ò
q0 +2 p
q0
f (q ) e
- jnq
dq
-5w1
-3w1
-w1
w1
3w1
5w1
w
Waves: 5
Temporally periodic case
If we have a function f(t) , with temporal
period T , which we wish to express as
a Fourier series, then change our f(q)
series using
So the Fourier series for f(t) =
where
Waves: 5
T is the fundamental period of f(t)
and w1 = 2p/T is the fundamental
angular frequency
Using w1:
f(t) =
where
Waves: 5
The corresponding complex form of
f(t) in terms of w1 is
where
Taking T
in the complex form
f (t ) =
+¥
åC e
n
jnw1t
n=-¥
=
å( ò
+¥
1
T
n=-¥
and substitute
+T /2
-T /2
f (t ) e
- jnw1t
ì w = nw1
ï
í dw
2
=
ïî
p
T
)
dt e
jnw1t
Hence
f (t ) =
As T 
å( ò
+¥
dw
2p
w =-¥
f (t ) =
1
2p
ò
+T /2
-T /2
+¥
-¥
f (t ) e
- jwt
F (w ) e
jw t
where
F (w ) =
ò
+¥
-¥
f (t ) e
- jw t
)
dt e
dt
dw
jwt
F(w) gives the continuous
spectrum of real and imaginary
parts (or amplitude and phase)
that must be present to
faithfully represent f(t) over an
infinite interval.
-5w1
-3w1
-w1
w1
3w1
5w1
w
Waves: 5
Fourier transform visualization
w
Waves: 5
FOURIER TRANSFORM DERIVATIONS
Example 1:
The rectangular or “slit” function
f(t)
h
-(1/2)
t
t
+(1/2)
t
Derivation:
Thus F(w) has the form
Thus F(w) has the form
F(w)
w
-2p
+2p
-p +p
-4p/t
+4p/t
-2p/t +2p/t
Waves: 2
Example 2:
Dirac delta-function
f(t)
h
t
( By definition
h-1
)
Derivation:
Take the rectangular pulse of Example 1 and
let t= h-1
Since first zero is at
Then F(w) = 1 (constant for all w).
Note:
1. F(w) is again real and even, as
expected for an even f(t)
F(w)
1
w
Example 3:
The Gauss function
f(t)
h
s
t
h e-1/2
Derivation:
F (w ) =
ò
+¥
-¥
(
he
- 12
(t s )
2
)
e
- jwt
dt
Convert this to the form e- s2 by the
change of variables
ì s = ( t s + jws ) 2
ï
ï 2
2
2ù
é
í s = ë( t s ) + 2 jw t - (ws ) û 2
ï
ï
î d s = dt 2s
Hence
and since
we have
i.e. If f(t) is Gaussian with [h , s]
then F(w) is Gaussian with
[h(2p)1/2s, s -1]
PROPERTIES OF
FOURIER TRANSFORMS
Fourier transforms are sometimes of
use in physics due to their direct
physical interpretation (see later), but
often their usefulness is more indirect
In particular, a great usefulness comes
from their help in solving awkward
differential equations
Here we list several general transform
properties of benefit in this and other
contexts
(Note that these properties are listed here
without proofs. Most proofs follow from
the defining equations)
1.
Linearity
If
then
2. Scale change
If
then
Thus compressing a time function expands its
spectrum in frequency and vice-versa
This expresses the idea of
“Time  Bandwidth” invariance
and is demonstrating in a general
way the Bandwidth Theorem
(i.e DwDt ~ 2p )
3. Delay and modulation
If
then
Thus :
... delaying a time function f(t)
simply adds a linear phase (-wt0)
to the original F(w) phase
... modulating f(t) by ejw0t shifts
the spectrum by w0
Similarly, if one modulates by the real
function cos(wot) we have
Thus this real modulation gives a
spectrum which is the average of two
copies of the original form, one shifted
by +w0 and the other by -w0
i.e. if f(t)
transforms to
w
then
[ cos(wot) f(t) ]
transforms to
-w0
+w0
w
4. Differentiation and integration
If
then
Thus taking the transform reduces
differentiation and integration to
the simpler operations of
multiplication or division by jw in
the frequency domain. Then
finding the inverse transform gives
the final result in the time domain
This is analogous to the taking of
logarithms to reduce “  and ” to
the simpler operations of
“ + and - ”. In that case finding the
anti-log gave the final result
5. Multiplication and convolution
If
then
Thus :
... a product of functions has a
transform that is the convolution of
the individual transforms
... conversely if you convolve
functions you multiply their
individual transforms
The delta function is the indentity
operation for convolution:
Next: A nicely illustrated example from
an old paper by Brault and White
(1971 Astron. & Astrophys. 13, 169).
•
•
•
•
sampling theorem
aliasing
apodizing
deconvolution
A continuous data function and its Fourier Transform.
Sampling (digitizing) the continuous
function at different intervals.
After removing the mean, multiply by an apodizing or masking
function to eliminate the discontinuity between the two ends.
Two Dimensional Fourier Transforms
For a function of two dimensions, the Fourier
Transform is done by transforming in each
dimension separately.
Example : Ponte Vecchio in Florence…
real
imag
Example : Ponte Vecchio in Florence… data compressed by 4 :
real
imag
Example : Ponte Vecchio in Florence… (raw image 512x512)
saving 16 x16 pixels in the centre of transform
saving 32 x 32
saving 64 x 64
saving 128 x128
Convolving the image with a Gaussian:
Convolving the image with a Gaussian:
Deconvolution with noise floors:
How can we
get from here
back to here?
Most of the rest of the workshop will describe
how to optimize the image making process.
Parkes
Compact Array
Combined
data
see McClure-Griffiths et al. 2000 and Stanimirovic 1999
Aperture or uv plane
Image or lm plane
Aperture
synthesis
mapping
Single
dish
mapping
Conclusions:
•
•
•
•
To understand aperture synthesis and
interferometry in general review your
Fourier transforms.
Understanding the telescope and how it
samples the sky requires visualization of
the uv plane sampling function.
Improving images by “deconvolution” of
unsampled parts of the uv plane requires
interpolation using some assumptions.
Writing new algorithms often involves
clever ways of using Fourier transforms.
The Fourier Transform is the
Radio Astronomer’s best friend.
```