EE 511: Introduction to
Fourier Optics and
Image Understanding
Volume 1
I. History and Background
II. Fourier Transforms and Linear Systems
Dwight L. Jaggard
University of Pennsylvania
308 Moore
<jaggard@seas.upenn.edu>
215.898.4411
©2000, D. L. Jaggard
EE 511
1
Course Goal
Understand the fundamentals
of physical and ray optics
and their application to
current science and
technology
©2000, D. L. Jaggard
EE 511
2
What good
is optics?
©2000, D. L. Jaggard
EE 511
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Course Outline (I)
I.
History and Background
A. History
B. Types of Optics
C. Applications of Physical Optics
II.
Fourier Transforms and Linear Systems
III.
Scalar Diffraction Theory (Physical Optics)
IV.
Fresnel and Fraunhofer Approximations
V.
Vector Diffraction Theory
VI.
Geometrical and Ray Optics
©2000, D. L. Jaggard
EE 511
4
Course Outline (II)
VII.
Properties of Lenses
VIII. Coherent and Incoherent Imaging
IX.
Partial Coherence Theory
X.
Special Topics (topics selected according to
time and interest of class)
A. Image classification and understanding
B. Non-destructive evaluation and testing
C. Fractal antennas and arrays
D.
“Electromagnetic bullets” or focused beams
E. Inverse problems
©2000, D. L. Jaggard
EE 511
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Administrative Information
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Contacts
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Dwight Jaggard , instructor
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Lois Clearfield, secretary (appointments
and handouts)
<jaggard@seas. upenn.edu>
or 215.898.4411
<lois@ee.upenn.edu>
or 215.898.8241
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Office Hours: 4:30 - 5:30 W
Review Sessions: as needed
©2000, D. L. Jaggard
EE 511
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Course Information
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Grades
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Homework
Midterm
Final
Mini-Project
~15 - 20%
~35 - 40%
~35 - 40%
~10%
Late homework not accepted
Midterm Exam: Wednesday, March 7
Guidelines on collaboration
©2000, D. L. Jaggard
EE 511
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Mini-Project
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Idea:
Take topic related to course and
present topic to class
Work in small groups
Turn in presentation (Power Point)
plus paper on topic
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Talks given last two weeks of
class
l Paper due April 30
©2000, D. L. Jaggard
EE 511
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Potential Mini -Project Topics - I
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Image and pattern classification
Signal reconstruction (from limited data)
Non-uniform or 2-D sampling
Rough surface scattering
Electromagnetic scattering (physical optics
with polarization or exact methods)
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Knife edge diffraction for both polarizations
Low frequency diffraction by apertures
Optical computing/neural nets
©2000, D. L. Jaggard
EE 511
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Potential Mini -Project Topics - II
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Diffraction by fractals
Zernike polynomials and optics
Diffuse beam propagation/imaging
Higher-order Gaussian beam propagation
Focused beams/“diffractionless” propagation
Applications of partial coherence theory
Ray optics and lens design
Multilayers: application & design
Spectroscopy
Antenna radiation
©2000, D. L. Jaggard
EE 511
10
Potential Mini -Project Topics - III
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Phase retrieval problem
Holography
Inverse problems
Tomography & radon transform
Ultrasound imaging
X-ray diffraction/crystallography
MRI
Adaptive optics
Wavelets
©2000, D. L. Jaggard
EE 511
11
Links
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Course website:
http://www.seas. upenn.edu/~ee511
(homework posted here)
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Photonics information & news
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Some interesting sites for physical optics
www.optics.org
www.opticalimaging.org/fourieroptics.html
http://dukemil.egr.duke.edu/Ultrasound/k space/bme265.htm
http://wyant.optics.arizona.edu/fresnelZones/fresnelZone
s.htm
http://hyperphysics.phy astr.gsu.edu/hbase/phyopt/diffracon.html#c1
©2000, D. L. Jaggard
EE 511
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Related Journals
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Journal of the Optical Society of America – A
Optics Letters
Optics Communications
Applied Optics
Optical Engineering
Journal of Lightwave Technology
Journal of Optics A: Pure and Applied Optics
Optik
Journal of Modern Optics
Optica Acta
Applied Physics Letters
Applied Physics B: Lasers and Optics
Optics Express (online journal)
©2000, D. L. Jaggard
EE 511
13
I.
History and Background
A. History
B.
Types of Optics
C. Applications of Physical Optics
©2000, D. L. Jaggard
EE 511
14
History (I)
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Classical Times & Greek Philosophers
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Empedocles (circa 490-430 B.C.)
Euclid (circa 300 B.C.)
The Golden 1600’s
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Descartes (1596-1650)
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Considered the nature of light
Light was pressure transmitted through the aether
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Galileo (1564-1642)
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Snell (1621)
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Experimental methods
Refraction of light at interface
©2000, D. L. Jaggard
EE 511
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History (II)
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More 1600’s
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Fermat (1601-1665)
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Father Grimaldi (1618-1663)
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“Principle of Least Time”
Refraction laws verified
First noticed “diffraction”
Note: diffraction is the bending of light not caused
by refraction
Newton (1642-1727)
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Discovered basic qualities of color
White light could be split up into colors
Experiments with prisms and light and
“refrangibility ” or bending of light at an interface
©2000, D. L. Jaggard
EE 511
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History (III)
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Still More 1600’s
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Huygens (1629-1695)
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Wave propagation of light
Polarization of light
Laws of reflection and refraction
Progress in the 1700’s
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Young (1773-1829)
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Fresnel (1788-1827)
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Wave theory
Interference (colors of thin films)
Confirmed wave theory of propagation and diffraction
Influence of earth’s motion of light propagation
Interference of polarized rays of light (light no
longitudinal)
Reflection and polarization
Cause of dispersion
©2000, D. L. Jaggard
EE 511
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History (IV)
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The Maxwell Era
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Faraday (1791-1867)
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Maxwell (1831-1879)
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Experiments in electricity and magnetism
Work independent of optics experiments
Theoretically unified electricity and magnetism
Showed possibility of electromagnetic waves propagating
with velocity that could be calculated
Electrostatics, magnetostatics , induction, EM waves and
optics unified under single theory
Lord Rayleigh (scientific work 1899-1920)
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Investigated waves propagation and scattering
Examined scattering from small particles
Studied wave interactions with periodic structures
©2000, D. L. Jaggard
EE 511
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History (V)
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Atomic Nature of Light - The Beginning
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Fraunhofer (1787-1826)
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Kirchhoff (1824-1887)
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Plank, Bohr and Einstein (early 1900’s)
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Discovered absorption lines in the solar spectrum
Experimentally measured absorption lines of solar spectrum
Quantum theory makes inroads
Applications of quantum mechanics to atomic structure and line
spectra (materials have quantized atomic systems)
Photons postulated
Certain effects (e.g., photo -electric effect) explained only by
photons
Dirac (1927)
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Field quantization (electromagnetic fields are quantized)
Quantum optics
©2000, D. L. Jaggard
EE 511
19
What are the
“brands” of
optics?
©2000, D. L. Jaggard
EE 511
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Types of Optics
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Ray or geometrical optics
l Wave/physical/Fourier optics
Scalar theory (no polarization)
Vector or EM theory (polarization)
l Beam optics
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Statistical optics
Optics of atomic systems/materials
l Quantum optics
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©2000, D. L. Jaggard
EE 511
21
Hierarchy of Optics
Quantum Optics
Electromagnetic
(Vector) Optics
Scalar Wave
Optics
Geometrical (Ray)
Optics
©2000, D. L. Jaggard
EE 511
22
Applications of Physical Optics
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Remote sensing & inverse scattering
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Imaging & image systems
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Image processing, pattern discrimination
and classification
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Holography and non -destructive evaluation
and testing (NDE & NDT)
Rough surface scattering
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Antenna and array design
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Spectroscopy
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Inteferometry
Optical computing
©2000, D. L. Jaggard
EE 511
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Course Outline
I.
II.
History and Background
Fourier Transforms and Linear Systems
III.
Scalar Diffraction Theory (Physical Optics)
IV.
Fresnel and Fraunhofer Approximations
V.
VI.
Vector Diffraction Theory
Geometrical and Ray Optics
VII.
Properties of Lenses
VIII. Coherent and Incoherent Imaging
IX.
X.
Partial Coherence Theory
Special Topics
©2000, D. L. Jaggard
EE 511
24
II. Fourier Transforms
and Linear Systems
A.
B.
C.
D.
E.
F.
G.
H.
I.
Requirements
“Inventing” the Fourier Transform
Dirac Delta Function
Use of the F.T. in Optics
F.T. Properties
Some Useful Transforms
Two-Dimensional F.T.
More Useful Transforms
Sampling
©2000, D. L. Jaggard
EE 511
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A. Requirements
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To use Fourier Transforms (F.T.)
there are requirements on:
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System
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Signal
©2000, D. L. Jaggard
EE 511
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System Requirements
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To use F.T. the system must be:
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Linear
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Nonlinear systems often use specialized
methods unique to each system
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No general theory exists
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Time invariant
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Memoryless
©2000, D. L. Jaggard
EE 511
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Signal Requirements
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To use F.T. the signal g(t) must:
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Satisfy
∞
∫ g(t) dt
exists
−∞
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Must have finite number of
discontinuities
e.g., cannot be the function
+1 for t rational
g( t) = 
−1 for t irrational
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Have a finite number of max and min
e.g., cannot be the function
g(t) = sin(t−1)
©2000, D. L. Jaggard
EE 511
28
More on Signal
Requirements
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Fourier transforms for signals not
satisfying three conditions can
often be found:
For signals whose absolute value has
infinite area, one can use a damping
function and take the limit
l For signals with discontinuities
impose a Lipschitz condition
l Signals from real systems are most
often well-behaved
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©2000, D. L. Jaggard
EE 511
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How can we
discover the
Fourier
Transform?
©2000, D. L. Jaggard
EE 511
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One Method
Joseph
©2000, D. L. Jaggard
Fourier
(1768-1839)
EE 511
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B. “Inventing” the F.T.
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The spectrum of g(t) is the amount
of each frequency f contained in g(t)
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Mathematically the F.T. is the graph
of the spectrum of g(t)
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Need a way to find out how much of
each frequency f is in g(t)
©2000, D. L. Jaggard
EE 511
32
Inner Product and F.T.
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The inner product is a measure of
how much a signal is like another
signal
l Define the inner product of g(t) and
∞
h(t) as
∆
*
< g(t)h(t) > = ∫ g(t )h (t)dt
−∞
Clearly this is max when g(t) = h(t)
l If inner product is zero, g(t) and h(t)
are orthogonal (just like vector dot
product)
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©2000, D. L. Jaggard
EE 511
33
Phasors and Signal of
“Pure Frequency”
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An example of a signal of “pure
frequency” is exp(j2πft)
This is a rotating “phasor” of unit
amplitude and angle 2πft and so rotates
counterclockwise for f > 0
Real and imaginary parts of this phasor
are cos(2πft) and sin(2πft)
A phasor exp(-j2πft) simply rotates
clockwise for f > 0 [gives meaning to
“negative frequency’]
©2000, D. L. Jaggard
EE 511
34
Signal of “Pure
Frequency” and g(t)
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Let h(t) = exp(j2πft) to find how
much of frequency “f” is in signal
“g(t)” through the inner product
l Define the F.T. as:
F {g(t )} =< g(t)e
j 2πft
∞
>=
∫ g(t)e
− j 2πft
dt
−∞
©2000, D. L. Jaggard
EE 511
35
More on F.T. of g(t)
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Fourier transform of g(t) gives
frequency-domain function G(f) which
provides the graph of the spectrum of
g(t)
∞
F {g(t )} = G( f ) =< g(t)e j 2πft >= ∫ g(t)e − j2 πft dt
−∞
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Note: the “spectrum” may also refer to
|G(f)| 2 - so beware!
Sometimes all we have is |G(f)| 2 which
does not give unique g(t) [“phase retrievel
problem”]
©2000, D. L. Jaggard
EE 511
36
F.T. and Its Inverse
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Fourier transform is not unique
l Only combined pair of the Fourier
transform F{g(t)} = G(f) and its
associated inverse transform F–1 {G(f)}
= g(t) are unique where
F
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−1
F {g(t)} = g(t)
Many minor variations in notation
(e.g., factors of 2 π, changes in sign of
exponent)
©2000, D. L. Jaggard
EE 511
37
Celebrated F.T. Pair
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The Fourier transform pair used
here is:
F {g(t )} = G( f ) =
∞
∫ g(t)e
− j 2πft
dt
−∞
∞
F
−1
{G( f ) } = g(t) = ∫ G( f )e + j 2 πftdf
−∞
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Note: we have only defined the first
equation and postulated the second
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Here f and t are conjugate variables
©2000, D. L. Jaggard
EE 511
38
Notes on F.T. Pair
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We use the shorthand
g(t ) ⇔ G( f )
to indicate a function g(t) and its transform G(f)
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There are many other F.T. conventions that
change signs in the exponent and/or change the
position of the factor (2 π)
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If g(t) is discontinuous, F.T. construction will not
give discontinuity but will converge to average
value at discontinuity
©2000, D. L. Jaggard
EE 511
39
F.T. Inversion Formula
∞
g(t ) =
∫ G( f )e
+ j2 πft
df
Assume inversion
formula is correct,
now check
−∞
=
∞ ∞
∫ ∫ [g(t' )e
− j 2 πft'
+ j 2πft
dt']e
df
−∞ −∞
=
∞ ∞
∫ ∫ [g(t' )e
+ j2 πf (t − t )
'
dt' df
−∞ −∞
∞
 e + j2 πf (t −t ) − e − j2π f (t − t ) 
= lim ∫ g(t' )
dt'
j2π (t − t' )
f →∞ −∞


=
'
∞
'
 sin[ 2πf (t − t' )]
dt'
π (t − t' ) 
∫ g(t' ) lim 
−∞
f →∞
∞
= g(t' ) t' = t
sin[ 2πf (t − t' )] 
dt'
π (t − t' )

∫ lim 
−∞
f→∞
[sin(t)/t] function is
sharply peaked and
“pulls out” or “sifts”
g(t’) at t=t’
Integral has a
value of unity
= g(t )
©2000, D. L. Jaggard
EE 511
40
Sine and Cosine Integrals
t
Si(t) = ∫
sin(t' )
dt'
t'
Si(t) = π
2
0
lim
t→∞
∞
∞
sin(t' )
sin(2 πft' )
dt' = ∫
dt' = π
t'
t'
−∞
−∞
∫
t
Ci(t) = γ + ln(t ) + ∫
0
cos(t ' ) − 1
dt'
t'
where γ ≈ 0.5772156649
...
is Euler's constant
lim
t →∞
Ci(t) = 0
©2000, D. L. Jaggard
EE 511
41
C. Dirac Delta Function
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Definition
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Candidates
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Some useful relations
©2000, D. L. Jaggard
EE 511
42
Delta Function Definition
∞
This is not
your usual
function
∫ δ (t)dt = 1
−∞
and
δ (t) = 0
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for
t≠0
Note:
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Dirac delta function δ(t) is highly peaked where
its argument is zero
Its “weight” or “strength” is defined by its
integral (unity)
This is a “generalized function” or “distribution”
©2000, D. L. Jaggard
EE 511
43
Candidates for δ(t)
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Functions that are sharply peaked
and have an area of unity
l Example: define the rectangle
function or “rect function” of
width T as rect(t/T):
1 for

rect(t / T ) = 
0 for

©2000, D. L. Jaggard
1
2
1
t /T >
2
t /T <
EE 511
44
Delta Function and
T–1 rect(t/T)
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Normalize rect function to give
unit area and take limit:
δ(t) =
lim
T → 0
1
1
 T for t / T <
1
2
rect(t / T ) = 
1
T
 0 for t / T >
2
T–1rect(t/T)
T
1/T
Take limit T–> 0
t
©2000, D. L. Jaggard
EE 511
45
Candidates for δ(t)
δ (t) =
δ (t) =
δ (t) =
δ (t) =
δ (t) =
lim
T→ 0
1
rect(t / T )
T
lim
α →∞
lim
α →∞
lim
α →∞
lim
α →0
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sin(αt)
πt
α
2
exp(−αt )
π
α
exp( −α t )
2
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α
2
2
π(α + t )
©2000, D. L. Jaggard
Candidates can
be continuous or
discontinuous;
monotonic or
oscillatory; have
finite support or
infinite support
All are peaked at
the origin and
are normalized
to have unity
area
EE 511
46
Properties of Dirac Delta
∞
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Sifting Property
∫ f(t)δ(t − a)dt =
f (a)
−∞
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Scaling Property
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Derivative
δ (at) =
1
δ (t)
a
∞
∫ f ( t)δ' (t − a)dt = − f ' (a )
−∞
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δ(t) = δ( −t)
Parity
δ' (t) = −δ' (− t)
©2000, D. L. Jaggard
EE 511
47
Additional Properties
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Shift & δ (at + b) = 1a δ (t + ba )
Scale
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Powers
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Roots
(a ≠ 0)
t δ (t) = (−1) n!δ(t)
n ( n)
n
δ [ f ( t)] =
∑
n
δ ( t − t0 )
f ' (t n )
where f (t) = 0 has
and
f' (tn ) is
and
exists
©2000, D. L. Jaggard
roots tn
non - zero and
EE 511
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Some Useful Relations
∞
∫ exp(+ j2πft)df = δ (t)
−∞
∞
∫ exp(− j2πft)dt = δ ( f )
−∞
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Prove by using Euler identities and
property of the Sine integral
This implies F.T. and F.T. –1 of unity yields
a delta function
F {δ(t)} = 1
F {1} = δ( f )
©2000, D. L. Jaggard
EE 511
49
∞
Example: Show ∫ exp( + j2πft)df = δ(t)
−∞
∞
∞
∫ exp( + j2πft) df = ∫ [cos( 2πft ) + j sin(2πft )]df
−∞
−∞
∞
= 2∫ [cos( 2πft)]df
sine function is
odd and its
integral vanishes
0
=
lim
f → ∞
2
sin(2πft )
2πt
= δ(t)
©2000, D. L. Jaggard
EE 511
50
II. Fourier Transforms
and Linear Systems
A.
B.
C.
D.
E.
F.
G.
H.
I.
Requirements
“Inventing” the Fourier Transform
Dirac Delta Function
Use of F.T. in Optics
F.T. Properties
Some Useful Transforms
Two-Dimensional F.T.
More Useful Transforms
Sampling
©2000, D. L. Jaggard
EE 511
51
D. Use of the Fourier
Transforms in Optics
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Spatial frequency
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Analogy between electrical and
optical quantities
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Essence of Fourier Optics
©2000, D. L. Jaggard
EE 511
52
Spatial Frequency
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Conjugate variables for g(t)
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In space we have conjugate
variables for g(x)
t (seconds)
x(meters)
f (seconds –1)
f x (meters –1)
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is denoted spatial frequency
along coordinate x
©2000, D. L. Jaggard
EE 511
53
More Spatial Frequency
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For two-dimensional surfaces f x is the
spatial frequency along the x-axis while f y
is the spatial frequency along the y-axis
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Note:
l fx
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is the number of cycles per meter of
variation of a spatial signal
fx = k x /2π
(normalized wavenumber)
fy = k y /2π
(normalized wavenumber)
©2000, D. L. Jaggard
EE 511
54
Examples:
Surface Height h(x,y)
h(x,y) = surface height
h(x,y) = surface height
y
x
y
x
Which fractal surface has more higher
spatial frequencies?
©2000, D. L. Jaggard
EE 511
55
Summary
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Signals in Time
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Signals in Space
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g(t)
signal
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g(x)
signal
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x
fx
variable
spatial
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ω=2π f
variable
cyclic
frequency
radian
frequency
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t
f
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kx=2π fx
frequency
wavenumber
Note: Spatial frequency (=1/λ) is the
normalized wavenumber k (=2π/λ)
©2000, D. L. Jaggard
EE 511
56
Electrical/Optical Analog
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Electrical
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Fourier transform
Quadratic phase filter
Linear FM generator
Filtering
Pulse shaping
Autocorrelation
Narrow band filter
Optical
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©2000, D. L. Jaggard
Fraunhofer diffraction
Fresnel diffraction
Lens
Contrast improvement
Apodization
Coherence
Fabry-Perot cavity
Interferometer
EE 511
57
Essence of
Fourier Optics
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Far-zone field U(f x,f y) is the twodimensional Fourier transform of the
aperture field U 0(x,y)
x’
fx ~x
U0 (x’,y’)
U(fx,fy)
z
y’
fy ~y
Diffracted field is the F.T. of the aperture field
©2000, D. L. Jaggard
EE 511
58
E. Fourier Transform
Properties
1.
2.
Linearity
Oddness & Evenness
3.
4.
Scaling
Shifting
5.
6.
7.
Modulation
Convolution
Correlation
8.
9.
Rayleigh , Parseval and Power Theorems
Differentiation
10.
11.
Moments
Periodic Functions
©2000, D. L. Jaggard
EE 511
59
Fourier Transform
Properties - Linearity
Ag(t ) + Bh(t) ⇔ AG( f ) + BH ( f )
Proof:
Follows from definition
Fourier transforms add and scale in
amplitude as do the original functions
©2000, D. L. Jaggard
EE 511
60
Example: rect function
1/ 2
F {rect(t)} =
∫e
− j 2 πft
dt
−1 / 2
e − jπ f − e+ jπf
− j2πf
sin(πf )
=
πf
=
≡ sinc( f )
or
rect( t) ⇔ sinc( f )
©2000, D. L. Jaggard
EE 511
61
Fourier Transform
Properties - Oddness
and Evenness
g(t)
G(f)
Real & ev en
Real & ev en
Real & odd
Imaginary & odd
Imaginary & even
Imaginary & even
Comp lex & even
Complex & even
Comp lex & odd
Complex & odd
Real & asymmetrical
Complex & hermitian
Symmetry of
function gives
rise to
symmetry of
its transform
Imaginary and asymmet. Complex and antiherm.
Real even plus
imaginary odd
Real odd plus imaginary
even
Even
Real
Odd
Odd
Imaginary
Even
©2000, D. L. Jaggard
Note: g(t) = g*(–t)
indicates Hermitian
function
EE 511
62
©2000, D. L. Jaggard
EE 511
63
Fourier Transform
Properties - Scaling
g( at) ⇔
1
G( f / a)
a
Proof:
F {g( at)} =
∞
∫ g( at)e
−j 2 πft
dt
let
y = at
−∞
1
=
a
∞
∫ g( y)e
− j2 πfy / a
dy
for
a>0
−∞
1
G( f / a)
a
similar proof
=
for a < 0
There is an inverse scaling relation
between functions and their
transforms
©2000, D. L. Jaggard
EE 511
64
Example: rect function
F {rect(t / T)} =
T/ 2
∫ [1]e
− j2 πft
dt
−T/ 2
=
− j2 πft T/ 2
e
− j2πf
=T
−T /2
sin(πTf )
(πTf )
= T sinc(Tf )
If rect function is stretched by
factor “T” then its transform is
compressed by same factor “T” and
its amplitude is also scaled (to
conserve power)
©2000, D. L. Jaggard
EE 511
65
Fourier Transform
Properties - Shifting
g( t − a) ⇔ G( f )exp(− j2πfa)
Proof:
F {g (t − a )} =
∞
∫ g (t − a )e
− j 2 π ft
dt
let
y =t − a
−∞
=
∞
− j 2 π ( y +a ) f
dy
∫ g (y )e
−∞
∞
= e− j 2 π fa ∫ g ( y )e − j 2 π f y dy
−∞
= e− j 2 π fa G ( f )
Shifting in the time -domain leads to
phase delay in the frequency -domain
(no shift in frequency -domain) so F.T.
amplitude is unaltered
©2000, D. L. Jaggard
EE 511
66
Two Time Shifts Makes Difference
Example:
F { g (t − a) +g (t + a )} =
∞
∫ [g (t − a) + g (t + a )]e
− j 2 π ft
dt
−∞
=L
= G ( f )[e− j 2π fa + e+ j 2π fa ]
= 2G( f )cos(2π fa)
What does this
mean for two -slit
diffraction?
In Fourier optics this represents the
interference for two-slit diffraction
©2000, D. L. Jaggard
EE 511
67
Fourier Transform
Properties - Modulation
g(t) exp(+ j2πf0t) ⇔ G( f − f0 )
Proof :
F {g(t )e + j2πf 0 t }=
∞
∫ g(t )e
∞
=
+ j 2πf 0t − j2πft
e
dt
−∞
∫ g(t )e
− j 2π ( f − f 0 )t
dt
−∞
= G( f − f0 )
Modulation in the time -domain leads
to frequency shifting in the
frequency -domain
Useful for modulated pulses
©2000, D. L. Jaggard
EE 511
68
Example: Modulated Pulse
l
Consider F. T. of rect(t/T) cos(2πf0t)
F {rect(t / T)cos(2πf0t } =
T /2
∫
−T / 2
T /2
∫
=
−T / 2
Half the
transform is
shifted to
positive frequency
and half to
negative
frequency
=
 e+ j 2πf 0t + e − j 2πf 0t  − j 2πft
e
dt


2
 e+ j 2π ( f 0 − f ) t + e− j2π (
2

T/ 2
f0 + f )t

dt

T /2
e

 e

+
 j4π( f0 − f ) 
 − j4π( f 0 + f ) 
− T /2
− T /2
+ j 2π ( f 0 − f )t
− j2π ( f0 + f )t
=L
=
T
T
sinc[T( f − f0 )] + sinc[T( f + f 0)]
2
2
How many cycles need to be in the
pulse so that the relative bandwidth
of the transform is ~10%?
©2000, D. L. Jaggard
EE 511
69
Fourier Transform
Properties - Convolution
g(t) ⊗ h(t) ≡
∞
∫ g(τ )h(t −τ )dτ
−∞
g( t) ⊗ h(t ) ⇔ G( f )H ( f )
Proof:
F {g(t )⊗ h(t )} =
∞ ∞
∫ ∫ g(τ )h(t − τ)e
− j2 πft
dτdt
−∞−∞
This is an
inner
product of a
function and
a shifted &
reversed
function
integrate over t
=
∞
∫ g(τ )H(f )e
− j2 πfτ
dτ
−∞
= G( f )H( f )
Convolution in the time -domain leads to
multiplication in the frequency -domain
©2000, D. L. Jaggard
EE 511
70
Sufficient Conditions for
Convolution
l
For g(t) = f(t) V h(t) to exist (assuming
f and h are reasonably well-behaved and
single valued):
l
l
l
Both f(t) and h(t) are absolutely integrable
on (–8 ,0); or
Both f(t) and h(t) are absolutely integrable
on (0, 8 ); or
Either f(t) or h(t) are absolutely integrable
on (–8 , 8 )
©2000, D. L. Jaggard
EE 511
71
Properties of Convolution
l
Delta Function
l
Commutative Property
l
Distributive Property
l
Shift Invariance
l
l
l
l
l
g(t) V
d(n)(t) = g(n)(t) = nth derivative of g(t)
f(t) V h(t) = h(t) V f(t)
[av(t) + bw(t)] V h(t) = a[v(t) V h(t)]
+ b[w(t) V h(t)]
If f(t) V h(t) = g(t)
then f(t –t 0) V h(t) = g(t –t0 )
Associative Property
l
[v(t) V w(t)] V h(t) = v(t) V [w(t) V h(t)]
©2000, D. L. Jaggard
EE 511
72
Physical Interpretation
of Convolution
i(t)
h(t)
o(t)
I(f)
H(f)
O(f)
Input
Linear System
Output
l O(f)
= H(f) I(f) [system transfer function]
l If we want O(f) = I(f) –> H(f) = 1
l Or h(t) = δ(t) since F–1{1} = δ(t)
l o(t) = i(t) V h(t) so h(t) = δ(t) = F–1{H(f)}
l Therefore, h(t) is called the “impulse
response”
©2000, D. L. Jaggard
EE 511
73
More on Convolution
l
What happens if the linear system
distorts input?
l Convolution is the natural
operation to find the time -domain
response o(t) to the system for
arbitrary input i(t)
©2000, D. L. Jaggard
EE 511
74
Convolution Example
“Flip and
Shift”
Let
e − t
E(t )= 
0
aE(ατ )
bE(βt − βτ)
τ
t> 0
t <0
∞
aE(αt )⊗ bE(βt) = ab ∫ E(ατ)E(βt − βτ)dτ
−∞
t
= abE(βt )∫ E (ατ − βτ)dτ
aE(αt) ⊗ bE( βt)
0
t
Convolution
calculation
©2000, D. L. Jaggard
= abE(βt )
= ab
E (αt − βt)− 1
β −α
E(αt) − E(βt)
β −α
EE 511
75
Result of Multiple
Convolutions
l
sinc(f)
rect(t)
t
f
t
f
t
f
t
f
l
l
Convolution usually
smoothes function
(running average)
Width of
convolution is sum
of widths of
individual functions
Central Limit
Theorem yields
Gaussian for many
convolutions
Multiple convolutions of
rect(t) and its F.T.
©2000, D. L. Jaggard
EE 511
76
Fourier Transform
Properties - Correlation
Rgh (t) ≡ g( t)Ηh(
é t) ≡
∞
∫ g(τ )h (τ − t)dτ
∗
−∞
∗
éh(t) ⇔ G( f )H ( f)
g(t)Η
This is an
inner
product of a
function and
a shifted &
conjugate
function
• Proof is similar to convolution
relation proof
• Definitions vary with authors
• Autocorrelation is Rgg = g(t) é g(t)
Correlation in the time -domain leads to
multiplication in the frequency -domain
©2000, D. L. Jaggard
EE 511
77
Properties of Correlation
l
Cross-correlation does not commute
l
Autocorrelation is Hermitian
l
Maximum modulus of autocorrelation
occurs at the origin
l
Autocorrelation decay provides
“correlation time” or “correlation length”
or characteristic scale of signals and
surfaces
l g(t)
é h(t) ? h(t) é g(t)
l Rgg (t) =
l
Rgg*(–t)
|Rgg(t)| < R gg (0)
©2000, D. L. Jaggard
EE 511
78
Fourier Transform Properties
- Rayleigh/Parseval Relation
∞
∫ g(t)
∞
2
dt =
−∞
∫ G( f )
2
df
−∞
P roof:
∞
∞
∫ g(t) g (t)dt = ∫ g(t)g (t)e
∗
∗
−∞
−j 2 πf ' t
dt
f' = 0
for
−∞
= G( f ' ) ⊗ G∗ (− f ' )
=
∞
∫ G( f )G ( f − f ' )df
∗
for
f' = 0
for
f' = 0
−∞
=
∞
∫ G( f )G∗ ( f )df
−∞
Area under the absolute value squared of a
function is equal to area under the absolute
value squared of its transform
©2000, D. L. Jaggard
EE 511
79
Rayleigh Theorem
Example
g(t)
G(f)
t
|g(t)|2
f
|G(f)|2
t
l
f
Equal areas under |g(t)| 2 and
|G(f)|2 imply conservation of
power for optical (and other
wave) signals
©2000, D. L. Jaggard
EE 511
80
Fourier Transform
Properties - Differentiation
d n g(t)
= g ( n )(t) ⇔ ( j2πf )n G( f )
n
dt
l
l
l
l
Prove by usual means
Differentiation in the time -domain
leads to multiplication by frequency in
the frequency -domain
Useful for solving D.E.s
Useful for finding F.T. of piecewise
continuous functions:
l
l
Take second derivative
Find inverse transform as sum of delta
functions
©2000, D. L. Jaggard
EE 511
81
Fourier Transform
Properties - Moments
∞
m n = ∫ t n g(t)dt = m th moment
of
g(t)
−∞
mn =
( n)
G (0)
(− j2π )n
The moment of a function is related
to the behavior of its transform near
the origin
l Therefore low frequency regime of
transform may be valuable in
classification & identification
l
©2000, D. L. Jaggard
EE 511
82
Properties of Moments
m0 = G(0 )
zeroth moment
or area
(1)
m1 =
G (0)
− j2π
G (2 ) (0)
m2 =
−4π 2
m2
=
m0
first
moment
or centroid
second moment
G ( 2) (0)
2
−4π G(0)
radius
or
moment of
inertia
of gyration
Moments can be used for signal or
image classification
©2000, D. L. Jaggard
EE 511
83
Second Moment Examples
G(f)
g(t)
l
t
f
t
f
l
g(t)
Moments of
g(t) affect
G(f) at the
origin
G(f)
t
f
©2000, D. L. Jaggard
Infinite
moment of g(t)
gives cusp at
origin for G(f)
EE 511
84
Fourier Transform Properties Periodic Functions
If g(t) is periodic
g(t ) =
with period
∞
∑G
−∞
exp( j2nπf0t )
n
where
Gn =
∞
∑ G F {exp( j2 nπf t)}
n
0
−∞
=
∞
∑ G δ ( f − nf )
n
T/2
∫ g(t) exp( − j2 nπf t )dt
0
− T /2
∞

F {g(t )}= F ∑ Gn exp( j 2nπf0t )
 −∞

=
T = 1 / f0
0
The
spectrum of
a periodic
function is a
“line
spectrum”
This is the
complex
Fourier
series
−∞
To find the F.T. of a periodic
function, find the F.T. of its Fourier
series
©2000, D. L. Jaggard
EE 511
85
Some Notes
l
F{F{g(t)}} = g(–t)
l
l
l
If we know a F.T. pair, we can invert this
pair by inverting the coordinate
If one lens in optics gives a F.T., then a
second lens can be used to provide an
inverted image
If g(t) and its first (n–1) derivatives
are continuous, its F.T. decays as
least as rapidly as |f| –(n+1)
©2000, D. L. Jaggard
EE 511
86
II. Fourier Transforms
and Linear Systems
A.
B.
C.
D.
E.
F.
G.
H.
I.
Requirements
“Inventing” the Fourier Transform
Dirac Delta Function
Use of the F.T. in Optics
F.T. Properties
Some Useful Transforms
Two-Dimensional F.T.
More Useful Transforms
Sampling
©2000, D. L. Jaggard
EE 511
87
F. Some Useful
Transforms
l
Unity and Delta Function
Rect
l Triangle
l Gaussian
l Signum and Step
l Comb
l Sine and Cosine
l
©2000, D. L. Jaggard
EE 511
88
Some Useful Functions
t / T <1 / 2
t / T >1 / 2
1
rect(t / T ) = 
0
1 − t / T t / T < 1
Λ(t / T ) = 
t /T >1
 0
Gaus(t / T ) = exp[−π (t / T )2 ]
∞
comb(t / T ) = ∑ δ (t / T − n ) = T
−∞
1
sgn( t) = 
−1
1
u(t) = 
0
∞
∑δ (t − n / T )
−∞
t >1
t <1
t >1
t <1
©2000, D. L. Jaggard
=
1 1
+ sgn(t)
2 2
EE 511
89
Some Useful F.T. Pairs
1⇔ δ( f )
δ (t) ⇔ 1
rect(t / T) ⇔ T sinc (Tf )
Λ(t / T) ⇔ T sinc 2 (Tf )
comb(t / T) ⇔ T comb(Tf )
Gaus(t / T) ⇔ T Gaus(Tf )
2
exp(−t ) ⇔
4π 2 f 2 +1
t −1 ⇔ − jπ sgn(−πf 2 )
sgn( t) ⇔ ( jπf )−1
u(t) ⇔ 2 −1 δ (t) + ( j2πf )−1
cos(2πf0t) ⇔ 2 −1 [δ( f − f0 ) + δ( f + f0 )]
sin(2πf0t) ⇔ (2 j)−1 [δ ( f − f0 ) − δ( f + f 0 )]
©2000, D. L. Jaggard
EE 511
90
G. Two-Dimensional F.T.
l
Separable Function
l
Circular Symmetry
l
l
Hankel transforms
l
Fourier-Bessel transforms
Some Two -Dimensional F.T. Pairs
©2000, D. L. Jaggard
EE 511
91
Separable Functions -- Easy
Fourier Transforms
l
If the two -dimensional function is
separable, its F.T. is the product of
two one-dimensional F.T.s:
l Let
g(x,y) = j(x) h(y)
l Then
F{g(x,y) = F{j(x)} F{ h(y)} = J(f) H(f)
©2000, D. L. Jaggard
EE 511
92
Polar Separable Case
l
Let g(r,θ) be separable in polar
coordinates:
l Let
g(r,θ) = g R(r)ejm θ
l Then I.C.B.S.T.
F {g(r,θ)} = (–j)m ejm θ Hm {g R(r)}
where
∞
Hm {g R (r)} = 2π ∫ r g R(r) Jm (2πρr)dr
0
is the Hankel transform of order m and Jm is
the Bessel function of order m
©2000, D. L. Jaggard
EE 511
93
Case of Circular Symmetry
l
Let g(r,θ) = gR(r) = g(r) [circular
symmetry]:
l Then I.C.B.S.T.
∞
F {g(r)} = 2 π ∫ r g R(r) J0(2πρr)dr
0
is the Fourier-Bessel transform of g (r)
where J 0 is the Bessel function of order 0
l
This gives the Fourier-Bessel pair
∞
G( ρ) = 2π ∫ rg(r)J0 (2πrρ)dr
0
∞
g(r) = 2π ∫ ρG( ρ)J 0 (2πrρ)dρ
0
g( r) ⇔ G (ρ)
©2000, D. L. Jaggard
Also known
as Hankel
transform
of order
zero
EE 511
94
Fourier-Bessel Proof
∞
G( f x , f y ) =
∞
∫ ∫ g(x,y)e
− j2π ( f x x + fy y)
dxdy
−∞ −∞
r= x +y
2
ρ=
2
θ = tan −1 (y / x)
fx 2 + f y 2
φ = tan − 1 (f y / f x )
 fx 
 cosφ
 f  = ρ

 y
 sinφ 
y 
cosθ 
  = r

y 
sinθ 
F {g(r)} = G(ρ)
2π ∞
=
∫ ∫ g(r)e
− j2π rρ (cosθ cos φ +sin θ sin φ )
rdrdθ
0 0
=
2π ∞
∫ ∫ g(r)e
− j2π rρ cos(θ −φ )
rdrdθ
0 0
2π
∫e
Note:
− ja cos(θ −φ )
dθ = 2πJ0 (a)
0
∞
©2000,
L. Jaggard
F {g(r)} = G(ρ)
= 2πD.∫rg(r
)J 0 (2πrρ)dr
0
EE 511
95
Some Useful Relations
x
∫ yJ
0
(y)dy = x J1 (x)
0
∞
∫x
ν +1
exp( −αx )J ν (βx)dx =
2
0
Jν (x) 
→
x→0
(1 / 2 )x ν
Γ(ν +1)
βν
2
exp( −β / 4α)
2α ( ν +1)
(v ≠ −1,−2, −3,...)
2
cos[ x − (1 / 2 )νπ − (1 / 4)π
Jν (x) 
→
x→∞
πx
where Γ (ν + 1) = ν!
for ν = integer
Γ(ν + 1) =ν Γ (ν )
©2000, D. L. Jaggard
EE 511
96
Bessel Functions J 0 & J 1
l
l
l
All Bessel functions except J 0 are zero at the origin
J0 is unity at the origin
Bessel functions have significant value in the regime
where their order equals their argument
©2000, D. L. Jaggard
EE 511
97
Two-Dimensional Functions
r / a <1
1
circ(r / a) ≡ 
r / a >1
0
J (2π ρ)
jinc(ρ) ≡ 2 1
2πρ
Gaus(ax, by) ≡ exp[−π (a 2 x2 + b 2 y2 )]
Gaus(ar) ≡ exp[−π a2 r 2 ] = exp[−πa 2 (x2 + y2 )]
δ ( x, y) ≡ δ (x)δ( y) =
comb( ax)comb(by) ≡
∞
δ (r)
πr
∞
∑ ∑δ (ax − n, by − m)
n= −∞ m= −∞
©2000, D. L. Jaggard
EE 511
98
“sinc” and “ jinc” Functions
sinc(x)
jinc(r/2)
sinc2(x)
jinc2(r/2)
What are the differences between
sinc and jinc functions?
©2000, D. L. Jaggard
EE 511
99
Two-Dimensional F.T. Pairs
rect (x / a)rect (y / b) ⇔ absinc(afx )sinc(bfy )
circ(r / a) ⇔ π a jinc(aρ)
2
δ (r)
⇔1
πr
1
ab
1
1
⇔
r
ρ
δ (ax,by ) ⇔
1 ⇔δ ( f x , f y )
cos(πr2 ) ⇔ sin(πρ2 )
exp( ± jπr2 ) ⇔ ± j exp( mjπρ 2 )
Gaus(ax,by ) ⇔
Gaus(ar) ⇔
1
Gaus( f x / a, f y / b )
ab
1
Gaus( f ρ / a)
2
a
©2000, D. L. Jaggard
comb(x / a)comb(y / b) ⇔ abcomb( afx )comb(bfy )
EE 511
100
How Often Does a
Signal Need to be
Sampled in Order to
be Exactly Replicated?
©2000, D. L. Jaggard
EE 511
101
Sampling of Signals
l
Whittaker-Shannon Sampling Theorem
A signal which is bandlimited (i.e., all
frequencies less than f max) can be exactly
reconstructed by accurate samples at times
T < ( 2 f max ) −1
l
l
The frequency 2 fmax (number of samples per
second) is known as the Nyquist frequency
Requirement on sampling frequency is fs > 2
fmax
©2000, D. L. Jaggard
EE 511
102
Miracle of Sampling
sinc function
is “interpolating
function” for exact
reconstruction
l
Exact reconstruction if overlap avoided by T < 1
2 fm
©2000, D. L. Jaggard
EE 511
103
Imperfect Sampling
l
l
l
l
What if samples of finite width are used
so that each sample is an average of a
part of the signal?
What if the signal is only sampled for a
finite length of time?
What if unequally spaced samples are
used?
What if the samples are descretized?
©2000, D. L. Jaggard
EE 511
104
Two-Dimensional Sampling
l
l
Various geometries can be used
Space-bandwidth product:
l If g(x,y) has significant value in the
region |x|<LX and |y|<LY, and if g(x,y) is
sampled on a rectangular lattice
(Nyquist rate) with spacing (2B X)–1 and
(2BY) –1 in the x and y directions, then
the total number of (possibly complex)
samples needed to represent g(x,y) is
l M = 16 LX LY BX BY
©2000, D. L. Jaggard
EE 511
105