Fourier
Transforms
John Reynolds
Joseph Fourier
1768-1830
Outline
• Basic properties of the Fourier transform
• Discrete form and the FFT
• Simple example applications
• Applications in radio astronomy;
–
–
–
–
–
Synthesis imaging and the u-v plane
Frequency conversion
The Sampling Theorem
Advanced signal processing (DSP)
filter-banks, spectroscopy
Fourier Integral Transform

Fourier integral transform
h(t )   H ( f )e2iftdf


Inverse transform
H ( f )   h(t )e

Mutant forms;
ij
(Engineering)
2πf  ω
(Pure maths or theoretical physics)
f,t  x,y
Individual cosine, sine transforms
2ift
dt
Basic Properties I
a . h(t)
h(t) + g(t)


a . H(f)
H(f) + G(f)
h(t) is real
h(t) is imag’ry
h(-t) = h(t)
h(t) real,even




H(-f) = H(f)* symmetry
H(-f) = -H(f)*
H(-f) = H(f)
H(f) real,even
linearity
linearity
Basic Properties II
Scaling;
“broad   narrow”
h(at)   H(f/a) / |a|
Shifting; “shift   phase roll/gradient”
h(t-t0)   H(f) * exp(2πi f t0)
Convolution;
“convolution   multiplication”
h(t) * g(t)   H(f) G(f)
Some well-known examples
The Dirac delta
The humble sinusoid
it
cos(t )  (e
1
2
it
e
)
Dirac comb or “shah” Ш
dt
df=1/dt
Basic Properties II
Scaling;
“broad   narrow”
h(at)   H(f/a) / |a|
Shifting; “shift   phase roll/gradient”
h(t-t0)   H(f) * exp(2πi f t0)
Convolution;
“convolution   multiplication”

h(t ) * g (t )   h(t  t ' ) g (t ' )dt'

  H(f) G(f)
Convolution – a simple example
*
=
More convoluted example
After J. J. Condon and S. M. Ransom
ESSENTIAL RADIO ASTRONOMY
http://www.cv.nrao.edu/course/astr534
Parseval and correlation theorems
Correlation function:

Corr( g , h)   g (  t )h( )d

Corr (g,h)  
G(f)H(-f)
= G(f)H(f)*
Corr (g,g)   |G(f) |2



2

(Wiener-Khinchin)
2
h(t ) dt   H ( f ) df

for real g(t),h(t)
(Parseval)
Parseval



2

2
h(t ) dt   H ( f ) df

Energy is conserved!
|H(f)|2 := power spectral density
PSD
Consumer applications
Fourier Transform Processing With ImageMagick
Introduction
One of the hardest concepts to comprehend in
image processing is Fourier Transforms.
There are two reasons for this. First, it is
mathematically advanced and second, resulting
images, which do not resemble the original image,
are hard to interpret.
2-D and beyond
H (u , v)   e 2iul dl  h(l , m)e 2imv dm
   h(l , m)e
2i ( ul  vm)
“Top Hat”   Airy disk
dl .dm
Practical realisation
Periodic
  Discrete (“Fourier series”)

df = 1/period
period
Periodic & Discrete   Periodic & Discrete
period = N.dt
N.dt.df = 1
period = N.df
DFT: Discrete Fourier Transform
Periodic, discretely sampled functions with;
t = k.dt, f = n.df, (where N.dt.df = 1)
Replace indefinite integral with summation over N values;
H n  k 0 hk e
N 1
2ikn / N
hk 
1
N

N 1
n 0
H ne
2ikn/ N
All aforementioned properties of Fourier integrals carry over, e.g.;
k 0 hk 
N 1
2
1
N

N 1
n0
2
Hn
Discrete form of Parseval
* One or other of h(t), H(f) function is generally “band-limited”
FFT – the Fast Fourier Transform
Simple DFT requires ~N2 multiplications
Gets very slow with large N
Decompose the NxN matrix into
a product of N sparse matrices
Have reduced to 2 DFTs of order N/2
Keep going until you get to order 1.
Number of mults now ~N.logN
Why phase is important
Original
image
2D (3D) Transform 
Spatial
Frequency
domain
Filter:
Filter:
F (u , v )
F (u, v)
F (u, v)
Amplitude only
Phase only
error
correction
by spatial
masking
Applications in radio astronomy
• Aperture synthesis imaging
• Frequency conversion
• Sampling theorem
• Signal processing (spectrometers, PFBs)
u-v plane
Synthesis interferometer: we cross-correlate each pair of antennas
spatial
auto-correlation 
1
2
3
East 
aperture plane
1-1, 2-2 etc excluded!
3-1 3-2
2-1 1-2
2-3 1-3
u
u-v plane
Distribution function A(x,y) in antennas  Transfer function W(u,v)
For n antennas  n(n-1)/2 baselines (points) in u-v plane
ASKAP – Australian SKA Pathfinder
ASKAP u-v coverage
Fourier transform of sky brightness
is a function in the u-v plane

λ/a
Complex visibility
V (u , v)   A(l , m) B(l , m)e
2i ( ul  mv )
B(l,m) := sky brightness in direction l,m
A(l,m) := antenna reception pattern
dl .dm
Mixing it down – Frequency
Conversion
Mixer (Multiplier)
Signal 1
Signal 1 × Signal 2
Signal 2
cos(ω1t)cos(ω2t)=½[cos((ω1+ω2)t)+ cos((ω1-ω2)t)]
cos(ωt) = ½[exp(iwt)/2 + exp(-iwt)]
Power
1*2
Difference
Frequency 
Δf
Δf
Sum
Frequency 
Mixing it down II– Frequency Conversion
(aka superheterodyne principle)
Mixer (Multiplier)
Signal 1
Local
Oscillator
Band pass
filter
flo
Frequency
Frequency
Δf
Δf
Image rejection
Unwanted image response
Frequency
*
-flo
-2flo
Δf
flo
2flo
Frequency
CSIRO. Receiver Systems for Radio Astronomy
Negative frequencies: learn to love them!
cos(t )  12 eit  eit 
-ω
Analytic signal of real f(t);
h(t)  h(t) + i.H(f)(t)
H(f) := Hilbert transform
cos(ωt)  cos(ωt) + i.sin(ωt)
ω
Single Sideband Mixers
√2cos[(ωLO- ω1)t] (USB)
0
(LSB)
2√2cos(ω1t)
Signal
Upper
sideband
Local
Oscillator
Signal
CSIRO. Receiver Systems for Radio Astronomy
Lower
sideband
Sampling Theorem – History
The theorem is commonly called the Nyquist sampling theorem;
since it was also discovered independently by E. T. Whittaker, by
Vladimir Kotelnikov, and by others, it is also known as Nyquist
Shannon–Kotelnikov, Whittaker–Shannon–Kotelnikov,
Whittaker–Nyquist–Kotelnikov–Shannon, WKS, etc., sampling
theorem, as well as the Cardinal Theorem of Interpolation
Theory.
It is often referred to simply as the sampling theorem.
(From Wikipedia)
Sampling Theorem (Shannon)
If a function x(t) contains no frequencies
higher than B hertz, it is completely
determined by giving its ordinates at a
series of points spaced 1/(2B) seconds
apart.
tsamp < 1 / 2B
Sampling Theorem
“band-limited”
ts=1/2B
=1/B (x2
(Nyquist)
undersampled)
 “aliased response”, or “aliasing”
Sampling Theorem continued
Also;
Radiotelescopes
– Christiansen and Högbom
Radio Astronomy
– J.D. Kraus
Principles of Interferometry and Synthesis
in Radio Astronomy
- Thompson, Moran, Swenson
Aliased sampling
3rd Nyquist
zone
Baseband
Frequency
B
*
-2fs
-fs
Sampling theorem: fs = 1/tsamp > 2B
fs = 1/tsamp
2fs
3fs
Recent Trends
• Faster, cheaper, samplers
• Faster, cheaper processing, data storage
Wider sampled bandwidths
Fewer downconversion stages
“direct conversion” (no downconversion)
e.g. DRAO receiver at Parkes)
CABB signal path
This talk ends here!