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©2000, D. L. Jaggard 1EE 511EE 511
EE 511: Introduction to EE 511: Introduction to Fourier Optics and Fourier Optics and
Image UnderstandingImage UnderstandingVolume 1
I. History and BackgroundII. Fourier Transforms and Linear Systems
Dwight L. JaggardUniversity of Pennsylvania
308 Moore<jaggard@seas.upenn.edu>
215.898.4411
©2000, D. L. Jaggard 2EE 511EE 511
Course Goal
Understand the fundamentals of physical and ray optics and their application to
current science and technology
©2000, D. L. Jaggard 3EE 511EE 511
What good is optics?
©2000, D. L. Jaggard 4EE 511EE 511
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 5EE 511EE 511
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 6EE 511EE 511
Administrative Information
l Contactsl Dwight Jaggard, instructor
<jaggard@seas. upenn.edu> or 215.898.4411
l Lois Clearfield, secretary (appointments and handouts)
<lois@ee.upenn.edu>or 215.898.8241
l Office Hours: 4:30 - 5:30 Wl Review Sessions: as needed
©2000, D. L. Jaggard 7EE 511EE 511
Course Information
l Gradesl Homework ~15 - 20%l Midterm ~35 - 40%l Final ~35 - 40%l Mini-Project ~10%
l Late homework not acceptedl Midterm Exam: Wednesday, March 7l Guidelines on collaboration
©2000, D. L. Jaggard 8EE 511EE 511
Mini-Project
l Idea:Take topic related to course and
present topic to class Work in small groupsTurn in presentation (Power Point)
plus paper on topicl Talks given last two weeks of
classl Paper due April 30
©2000, D. L. Jaggard 9EE 511EE 511
Potential Mini -Project Topics - Il Image and pattern classificationl Signal reconstruction (from limited data)l Non-uniform or 2-D samplingl Rough surface scatteringl Electromagnetic scattering (physical optics
with polarization or exact methods)l Knife edge diffraction for both polarizationsl Low frequency diffraction by apertures
l Optical computing/neural nets
©2000, D. L. Jaggard 10EE 511EE 511
Potential Mini -Project Topics - IIl Diffraction by fractalsl Zernike polynomials and opticsl Diffuse beam propagation/imagingl Higher-order Gaussian beam propagationl Focused beams/“diffractionless” propagationl Applications of partial coherence theoryl Ray optics and lens designl Multilayers: application & designl Spectroscopyl Antenna radiation
©2000, D. L. Jaggard 11EE 511EE 511
Potential Mini -Project Topics - IIIl Phase retrieval probleml Holographyl Inverse problemsl Tomography & radon transforml Ultrasound imagingl X-ray diffraction/crystallographyl MRIl Adaptive opticsl Wavelets
©2000, D. L. Jaggard 12EE 511EE 511
Linksl Course website:
http://www.seas. upenn.edu/~ee511(homework posted here)
l Photonics information & newswww.optics.org
l Some interesting sites for physical opticswww.opticalimaging.org/fourieroptics.htmlhttp://dukemil.egr.duke.edu/Ultrasound/k -
space/bme265.htmhttp://wyant.optics.arizona.edu/fresnelZones/fresnelZone
s.htmhttp://hyperphysics.phy -
astr.gsu.edu/hbase/phyopt/diffracon.html#c1
©2000, D. L. Jaggard 13EE 511EE 511
Related Journalsl Journal of the Optical Society of America – Al Optics Lettersl Optics Communicationsl Applied Opticsl Optical Engineeringl Journal of Lightwave Technologyl Journal of Optics A: Pure and Applied Opticsl Optikl Journal of Modern Opticsl Optica Actal Applied Physics Lettersl Applied Physics B: Lasers and Opticsl Optics Express (online journal)
©2000, D. L. Jaggard 14EE 511EE 511
I. History and Background
A. History
B. Types of Optics
C. Applications of Physical Optics
©2000, D. L. Jaggard 15EE 511EE 511
History (I)l Classical Times & Greek Philosophersl Empedocles (circa 490-430 B.C.)l Euclid (circa 300 B.C.)
l The Golden 1600’sl Descartes (1596-1650)
l Considered the nature of lightl Light was pressure transmitted through the aether
l Galileo (1564-1642)l Experimental methods
l Snell (1621)l Refraction of light at interface
©2000, D. L. Jaggard 16EE 511EE 511
History (II)l More 1600’sl Fermat (1601-1665)
l “Principle of Least Time”l Refraction laws verified
l Father Grimaldi (1618-1663)l First noticed “diffraction”l Note: diffraction is the bending of light not caused
by refractionl Newton (1642-1727)
l Discovered basic qualities of colorl White light could be split up into colorsl Experiments with prisms and light and
“refrangibility ” or bending of light at an interface
©2000, D. L. Jaggard 17EE 511EE 511
History (III)l Still More 1600’sl Huygens (1629-1695)
l Wave propagation of lightl Polarization of lightl Laws of reflection and refraction
l Progress in the 1700’sl Young (1773-1829)
l Wave theoryl Interference (colors of thin films)
l Fresnel (1788-1827)l Confirmed wave theory of propagation and diffractionl Influence of earth’s motion of light propagationl Interference of polarized rays of light (light no
longitudinal)l Reflection and polarizationl Cause of dispersion
©2000, D. L. Jaggard 18EE 511EE 511
History (IV)
l The Maxwell Eral Faraday (1791-1867)
l Experiments in electricity and magnetisml Work independent of optics experiments
l Maxwell (1831-1879)l Theoretically unified electricity and magnetisml Showed possibility of electromagnetic waves propagating
with velocity that could be calculatedl Electrostatics, magnetostatics , induction, EM waves and
optics unified under single theory
l Lord Rayleigh (scientific work 1899-1920)l Investigated waves propagation and scatteringl Examined scattering from small particlesl Studied wave interactions with periodic structures
©2000, D. L. Jaggard 19EE 511EE 511
History (V)l Atomic Nature of Light - The Beginningl Fraunhofer (1787-1826)
l Discovered absorption lines in the solar spectruml Kirchhoff (1824-1887)
l Experimentally measured absorption lines of solar spectrum
l Plank, Bohr and Einstein (early 1900’s)l Quantum theory makes inroadsl Applications of quantum mechanics to atomic structure and line
spectra (materials have quantized atomic systems)l Photons postulatedl Certain effects (e.g., photo -electric effect) explained only by
photons
l Dirac (1927)l Field quantization (electromagnetic fields are quantized)l Quantum optics
©2000, D. L. Jaggard 20EE 511EE 511
What are the “brands” of
optics?
©2000, D. L. Jaggard 21EE 511EE 511
Types of Opticsl Ray or geometrical opticsl Wave/physical/Fourier opticsl Scalar theory (no polarization)l Vector or EM theory (polarization)l Beam optics
l Statistical opticsl Optics of atomic systems/materialsl Quantum optics
©2000, D. L. Jaggard 22EE 511EE 511
Hierarchy of Optics
Quantum Optics
Electromagnetic (Vector) Optics
Scalar Wave Optics
Geometrical (Ray) Optics
©2000, D. L. Jaggard 23EE 511EE 511
Applications of Physical Opticsl Remote sensing & inverse scatteringl Imaging & image systemsl Image processing, pattern discrimination
and classificationl Holography and non-destructive evaluation
and testing (NDE & NDT)l Rough surface scatteringl Antenna and array designl Spectroscopyl Inteferometryl Optical computing
©2000, D. L. Jaggard 24EE 511EE 511
Course OutlineI. History and BackgroundII. Fourier Transforms and Linear Systems
III. Scalar Diffraction Theory (Physical Optics)
IV. Fresnel and Fraunhofer Approximations
V. Vector Diffraction TheoryVI. Geometrical and Ray Optics
VII. Properties of Lenses
VIII. Coherent and Incoherent Imaging
IX. Partial Coherence TheoryX. Special Topics
©2000, D. L. Jaggard 25EE 511EE 511
II. Fourier Transforms and Linear Systems
A. RequirementsB. “Inventing” the Fourier TransformC. Dirac Delta FunctionD. Use of the F.T. in OpticsE. F.T. PropertiesF. Some Useful TransformsG. Two-Dimensional F.T.H. More Useful TransformsI. Sampling
©2000, D. L. Jaggard 26EE 511EE 511
A. Requirements
l To use Fourier Transforms (F.T.)
there are requirements on:
l System
l Signal
©2000, D. L. Jaggard 27EE 511EE 511
System Requirementsl To use F.T. the system must be:l Linear
l Nonlinear systems often use specialized
methods unique to each system
l No general theory exists
l Time invariant
l Memoryless
©2000, D. L. Jaggard 28EE 511EE 511
Signal Requirementsl To use F.T. the signal g(t) must:l Satisfy
l Must have finite number of discontinuities
e.g., cannot be the function
l Have a finite number of max and mine.g., cannot be the function
g( t) =+1
−1
for
for
t
t
rational
irrational
g(t) dt−∞
∞
∫ exists
g( t) = sin(t−1)
©2000, D. L. Jaggard 29EE 511EE 511
More on Signal Requirements
l Fourier transforms for signals notsatisfying three conditions can often be found:l 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
©2000, D. L. Jaggard 30EE 511EE 511
How can we discover the
Fourier Transform?
©2000, D. L. Jaggard 31EE 511EE 511
One Method
Joseph Fourier (1768-1839)
©2000, D. L. Jaggard 32EE 511EE 511
B. “Inventing” the F.T.
l The spectrum of g(t) is the amount of each frequency f contained in g(t)
l Mathematically the F.T. is the graph of the spectrum of g(t)
l Need a way to find out how much of each frequency f is in g(t)
©2000, D. L. Jaggard 33EE 511EE 511
Inner Product and F.T.l 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
l 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)
< g(t)h(t) > =∆
g(t )h* (t)dt−∞
∞
∫
©2000, D. L. Jaggard 34EE 511EE 511
Phasors and Signal of “Pure Frequency”
l An example of a signal of “pure frequency” is exp(j2πft)
l This is a rotating “phasor” of unit amplitude and angle 2πft and so rotates counterclockwise for f > 0
l Real and imaginary parts of this phasor are cos(2πft) and sin(2πft)
l A phasor exp(-j2πft) simply rotates clockwise for f > 0 [gives meaning to “negative frequency’]
©2000, D. L. Jaggard 35EE 511EE 511
Signal of “Pure Frequency” and g(t)
l 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)ej 2π ft >= g(t)e− j 2π ftdt
−∞
∞
∫
©2000, D. L. Jaggard 36EE 511EE 511
More on F.T. of g(t)l Fourier transform of g(t) gives
frequency-domain function G(f) which provides the graph of the spectrum of g(t)
l Note: the “spectrum” may also refer to |G(f)| 2 - so beware!
l Sometimes all we have is |G(f)|2 which does not give unique g(t) [“phase retrievelproblem”]
F g(t ){ }= G( f ) =< g(t)ej 2πft >= g(t)e− j2πft dt
−∞
∞
∫
©2000, D. L. Jaggard 37EE 511EE 511
F.T. and Its Inversel Fourier transform is not uniquel 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
l Many minor variations in notation (e.g., factors of 2 π, changes in sign of exponent)
F−1F g(t){ } = g(t)
©2000, D. L. Jaggard 38EE 511EE 511
Celebrated F.T. Pairl 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
−∞
∞
∫
l Note: we have only defined the first equation and postulated the second
l Here f and t are conjugate variables
©2000, D. L. Jaggard 39EE 511EE 511
Notes on F.T. Pairl We use the shorthand
to indicate a function g(t) and its transform G(f)
l There are many other F.T. conventions that change signs in the exponent and/or change the position of the factor (2π)
l If g(t) is discontinuous, F.T. construction will not give discontinuity but will converge to average value at discontinuity
g(t ) ⇔ G( f )
©2000, D. L. Jaggard 40EE 511EE 511
F.T. Inversion Formulag(t ) = G( f )e+ j2 πft df
−∞
∞
∫
= [g(t' )e− j 2 πft' dt'−∞
∞
∫ ]e+ j 2πftdf−∞
∞
∫
= [g(t' )−∞
∞
∫ e+ j2 πf (t − t ' )dt' df−∞
∞
∫
=f →∞lim g(t' )
e+ j2 πf (t −t ' ) − e− j2π f (t − t ' )
j2π (t − t' )
dt'−∞
∞
∫
= g(t' )f →∞lim
sin[2πf (t − t' )]π (t − t' )
dt'
−∞
∞
∫
= g(t' ) t ' = tf → ∞lim
sin[2πf (t − t' )]π (t − t' )
dt'
−∞
∞
∫= g(t )
[sin(t)/t] function issharply peaked and
“pulls out” or “sifts” g(t’) at t=t’
Integral has a value of unity
Assume inversionformula is correct,
now check
©2000, D. L. Jaggard 41EE 511EE 511
Sine and Cosine Integrals
Si(t) = sin(t' )t '0
t
∫ dt'
lim
t → ∞Si(t) = π
2
sin(2πft' )t '−∞
∞
∫ dt' = sin(t' )t '−∞
∞
∫ dt' = π
Ci(t) = γ + ln(t) +cos(t ' ) − 1
t '0
t
∫ dt'
where γ ≈ 0.5772156649...is Euler's constant
lim
t → ∞Ci(t) = 0
©2000, D. L. Jaggard 42EE 511EE 511
C. Dirac Delta Function
l Definition
l Candidates
l Some useful relations
©2000, D. L. Jaggard 43EE 511EE 511
Delta Function Definition
l Note:l Dirac delta function δ(t) is highly peaked where
its argument is zerol Its “weight” or “strength” is defined by its
integral (unity)l This is a “generalized function” or “distribution”
δ (t)dt = 1−∞
∞
∫and
δ (t) = 0 for t ≠ 0
This is not your usual function
©2000, D. L. Jaggard 44EE 511EE 511
Candidates for δ(t)l Functions that are sharply peaked
and have an area of unityl Example: define the rectangle
function or “rect function” of width T as rect(t/T):
rect(t / T ) =1 for t / T < 1
20 for t / T > 1
2
©2000, D. L. Jaggard 45EE 511EE 511
Delta Function andT–1 rect(t/T)
l Normalize rect function to give unit area and take limit:
δ(t) =lim
T → 0
1T
rect(t / T ) =1
T for t / T < 12
0 for t / T > 12
1/T
T
t
T–1rect(t/T)
Take limit T–> 0
©2000, D. L. Jaggard 46EE 511EE 511
Candidates for δ(t)l Candidates can
be continuous or discontinuous; monotonic or oscillatory; have finite support or infinite support
l All are peaked at the origin and are normalized to have unity area
δ (t) =lim
T → 0
1T
rect(t / T )
δ (t) =lim
α → ∞
sin(αt)πt
δ (t) =lim
α → ∞
απ
exp(−αt 2 )
δ (t) =lim
α → ∞
α2
exp(−α t )
δ (t) =lim
α → 0
απ(α 2 + t 2 )
©2000, D. L. Jaggard 47EE 511EE 511
Properties of Dirac Deltal Sifting Property
l Scaling Property
l Derivative
l Parity
f(t)δ(t − a)dt−∞
∞
∫ = f (a)
δ (at) = 1a
δ (t)
f ( t)δ ' (t − a)dt−∞
∞
∫ = − f ' (a )
δ(t) = δ(−t)
δ ' (t) = −δ ' (− t)
©2000, D. L. Jaggard 48EE 511EE 511
Additional Propertiesl Shift &
Scale
l Powers
l Roots
δ (at + b) =1a
δ (t +ba
) (a ≠ 0)
t nδ (n)(t) = (−1)n n!δ(t)
δ [ f ( t)] =δ ( t − t0 )
f ' (tn)n∑
where f (t) = 0 has roots tn
and f' (tn ) is non -zero and
and exists
©2000, D. L. Jaggard 49EE 511EE 511
Some Useful Relations
l Prove by using Euler identities and property of the Sine integral
l This implies F.T. and F.T. –1 of unity yields a delta function
exp(+ j2πft)−∞
∞
∫ df = δ (t)
exp(− j2πft)−∞
∞
∫ dt = δ ( f )
F δ(t){ }= 1
F 1{ } = δ( f )
©2000, D. L. Jaggard 50EE 511EE 511
Example: Show
exp(+ j2πft)−∞
∞
∫ df = [cos(2πft ) + j sin(2πft )]−∞
∞
∫ df
= 2 [cos( 2πft)]0
∞
∫ df
=lim
f → ∞2
sin(2πft )2πt
= δ(t)
exp(+ j2πft)−∞
∞
∫ df = δ(t)
sine function is odd and its
integral vanishes
©2000, D. L. Jaggard 51EE 511EE 511
II. Fourier Transforms and Linear Systems
A. RequirementsB. “Inventing” the Fourier TransformC. Dirac Delta FunctionD. Use of F.T. in OpticsE. F.T. PropertiesF. Some Useful TransformsG. Two-Dimensional F.T.H. More Useful TransformsI. Sampling
©2000, D. L. Jaggard 52EE 511EE 511
D. Use of the Fourier Transforms in Optics
l Spatial frequency
l Analogy between electrical and
optical quantities
l Essence of Fourier Optics
©2000, D. L. Jaggard 53EE 511EE 511
Spatial Frequency
l Conjugate variables for g(t)t (seconds) f (seconds–1)
l In space we have conjugate variables for g(x)
x(meters) fx (meters–1)
l fx is denoted spatial frequencyalong coordinate x
©2000, D. L. Jaggard 54EE 511EE 511
More Spatial Frequencyl For two-dimensional surfaces fx is the
spatial frequency along the x-axis while fyis the spatial frequency along the y-axis
l Note: l fx is the number of cycles per meter of
variation of a spatial signall fx = k x/2π (normalized wavenumber)l fy = k y/2π (normalized wavenumber)
©2000, D. L. Jaggard 55EE 511EE 511
Examples:Surface Height h(x,y)
h(x,y) = surface height
x
y
h(x,y) = surface height
x
y
Which fractal surface has more higher spatial frequencies?
©2000, D. L. Jaggard 56EE 511EE 511
Summary
l Signals in Timel g(t) signal
l t variablel f cyclic
frequencyl ω=2π f radian
frequency
l Signals in Spacel g(x) signal
l x variablel fx spatial
frequency
l kx=2π fx wavenumber
Note: Spatial frequency (=1/λ) is the normalized wavenumber k (=2π/λ)
©2000, D. L. Jaggard 57EE 511EE 511
Electrical/Optical Analog
l Electricall Fourier transforml Quadratic phase filterl Linear FM generatorl Filteringl Pulse shapingl Autocorrelationl Narrow band filter
l Opticall Fraunhofer diffractionl Fresnel diffractionl Lensl Contrast improvementl Apodizationl Coherencel Fabry-Perot cavity
Interferometer
©2000, D. L. Jaggard 58EE 511EE 511
Essence ofFourier Optics
l Far-zone field U(fx,fy) is the two-dimensional Fourier transform of the aperture field U0(x,y)
x’ fx~x
y’
z
fy~y
U(fx,fy)U0(x’,y’)
Diffracted field is the F.T. of the aperture field
©2000, D. L. Jaggard 59EE 511EE 511
E. Fourier Transform Properties
1. Linearity2. Oddness & Evenness
3. Scaling4. Shifting
5. Modulation6. Convolution7. Correlation
8. Rayleigh , Parseval and Power Theorems9. Differentiation
10. Moments11. Periodic Functions
©2000, D. L. Jaggard 60EE 511EE 511
Fourier Transform Properties - Linearity
Fourier transforms add and scale in amplitude as do the original functions
Ag(t ) + Bh(t) ⇔ AG( f ) + BH ( f )
Proof: Follows from definition
©2000, D. L. Jaggard 61EE 511EE 511
Example: rect function
F rect(t){ } = e− j2 πftdt−1 /2
1/ 2
∫
=e − jπ f − e+ jπf
− j2πf
=sin(πf )
πf
≡ sinc( f )or
rect( t) ⇔ sinc( f )
©2000, D. L. Jaggard 62EE 511EE 511
Fourier Transform Properties - Oddness
and EvennessSymmetry of function gives rise to symmetry of its transform
Note: g(t) = g*(–t)indicates Hermitianfunction
g(t) G(f)Real & even Real & even
Real & odd Imaginary & oddImaginary & even Imaginary & evenComp lex & even Complex & even
Comp lex & odd Complex & oddReal & asymmetrical Complex & hermitianImaginary and asymmet. Complex and antiherm.Real even plusimaginary odd
Real
Real odd plus imaginaryeven
Imaginary
Even EvenOdd Odd
©2000, D. L. Jaggard 63EE 511EE 511
©2000, D. L. Jaggard 64EE 511EE 511
Fourier Transform Properties - Scaling
There is an inverse scaling relation between functions and their
transforms
g(at)⇔1a
G( f / a)
Proof:
F g( at){ }= g( at)e −j 2πft dt let y = at−∞
∞
∫
= 1a
g( y)e− j2πfy / ady−∞
∞
∫ for a > 0
=1a
G( f / a)
similar proof for a < 0
©2000, D. L. Jaggard 65EE 511EE 511
Example: rect function
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)
F rect(t / T){ } = [1]e− j2 π ftdt−T/ 2
T/ 2
∫
=e− j2 π ft
− j2πf −T /2
T/ 2
= Tsin(πTf )
(πTf )
= T sinc(Tf )
©2000, D. L. Jaggard 66EE 511EE 511
Fourier Transform Properties - Shifting
Shifting in the time -domain leads to phase delay in the frequency -domain (no shift in frequency -domain) so F.T.
amplitude is unaltered
g( t − a) ⇔ G( f )exp(− j2πfa)
{ } 2
2 ( )
2 2
2
Proof:
( ) ( )
( )
( )
( )
j ft
j y a f
j fa j f y
j fa
g t a g t a e dt let y t a
g y e dy
e g y e dy
e G f
π
π
π π
π
∞−
−∞
∞− +
−∞
∞− −
−∞
−
− = − = −
=
=
=
∫
∫
∫
F
©2000, D. L. Jaggard 67EE 511EE 511
Two Time Shifts -Makes Difference
In Fourier optics this represents the interference for two-slit diffraction
{ } 2
2 2
Example:
( ) ( ) [ ( ) ( )]
( )[ ]
2 ( )cos(2 )
j ft
j fa j fa
g t a g t a g t a g t a e dt
G f e e
G f fa
π
π π
π
∞−
−∞
− +
− + + = − + +
=
= +
=
∫F
L
What does this mean for two -slit
diffraction?
©2000, D. L. Jaggard 68EE 511EE 511
Fourier Transform Properties - Modulation
Modulation in the time -domain leads to frequency shifting in the
frequency-domainUseful for modulated pulses
g(t)exp(+ j2πf0t)⇔ G( f − f0)
Proof :
F g(t )e+ j2πf0 t{ }= g(t )e+ j 2πf 0te− j2πft dt−∞
∞
∫
= g(t )e− j 2π ( f − f0 )t dt−∞
∞
∫= G( f − f0 )
©2000, D. L. Jaggard 69EE 511EE 511
Example: Modulated Pulsel Consider F. T. of rect(t/T) cos(2πf0t)
F rect(t / T)cos(2πf0t{ }=e+ j 2πf0t + e− j 2πf0t
2
e− j 2πftdt
−T / 2
T /2
∫
= e+ j 2π ( f0 − f ) t + e− j2π ( f0 + f )t
2
dt−T / 2
T /2
∫
=e+ j 2π ( f0 − f )t
j4π( f0 − f )
− T /2
T / 2
+e− j2π ( f0 + f )t
− j4π( f0 + f )
− T /2
T /2
=L
=T2
sinc[T( f − f0 )] +T2
sinc[T( f + f 0)]
How many cycles need to be in the pulse so that the relative bandwidth
of the transform is ~10%?
Half the transform is shifted to
positive frequency and half to
negative frequency
©2000, D. L. Jaggard 70EE 511EE 511
Fourier Transform Properties - Convolution
Convolution in the time -domain leads to multiplication in the frequency -domain
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 πf tdτdt−∞
∞
∫−∞
∞
∫integrate over t
= g(τ )H( f )e− j2 πfτdτ−∞
∞
∫= G( f )H( f )
This is an inner
product of a function and a shifted &
reversed function
©2000, D. L. Jaggard 71EE 511EE 511
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 Both f(t) and h(t) are absolutely integrable
on (–8,0); orl Both f(t) and h(t) are absolutely integrable
on (0, 8 ); orl Either f(t) or h(t) are absolutely integrable
on (–8, 8 )
©2000, D. L. Jaggard 72EE 511EE 511
Properties of Convolutionl Delta Function
l g(t) V d(n)(t) = g(n)(t) = nth derivative of g(t)
l Commutative Propertyl f(t) V h(t) = h(t) V f(t)
l Distributive Propertyl [av(t) + bw(t)] Vh(t) = a[v(t) V h(t)]
+ b[w(t) V h(t)]
l Shift Invariancel If f(t) V h(t) = g(t)
then f(t–t0) V h(t) = g(t–t0)
l Associative Propertyl [v(t) Vw(t)] Vh(t) = v(t) V [w(t) Vh(t)]
©2000, D. L. Jaggard 73EE 511EE 511
Physical Interpretation of Convolution
h(t)
H(f)
i(t)
I(f)
o(t)
O(f)
Input OutputLinear System
lO(f) = H(f) I(f) [system transfer function]l If we want O(f) = I(f) –> H(f) = 1lOr 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 74EE 511EE 511
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 75EE 511EE 511
Convolution ExampleLet
E(t )=e− t t > 00 t < 0
aE(αt )⊗ bE(βt) = ab E(ατ)−∞
∞
∫ E(βt − βτ)dτ
= abE(βt ) E (ατ − βτ)0
t
∫ dτ
= abE(βt )E (αt − βt)− 1
β −α
= abE(αt) − E(βt)
β −αConvolution calculation
t
τ
bE(βt − βτ)
aE(ατ )
aE(αt)⊗ bE(βt)
“Flip and Shift”
©2000, D. L. Jaggard 76EE 511EE 511
Result of Multiple Convolutions
l Convolution usually smoothes function (running average)
l Width of convolution is sum of widths of individual functions
l Central Limit Theorem yields Gaussian for manyconvolutions
Multiple convolutions of rect(t) and its F.T.
f
f
f
ft
t
t
t
rect(t) sinc(f)
©2000, D. L. Jaggard 77EE 511EE 511
Fourier Transform Properties - Correlation
Correlation in the time -domain leads to multiplication in the frequency -domain
Rgh(t) ≡ g( t)Ηh( t)≡ g(τ )h∗(−∞
∞
∫ τ − t)dτ
g(t)Η h(t) ⇔ G( f )H∗( f)This is an
inner product of a function and a shifted & conjugate function
• Proof is similar to convolutionrelation proof
• Definitions vary with authors• Autocorrelation is Rgg = g(t) é g(t)
é
é
©2000, D. L. Jaggard 78EE 511EE 511
Properties of Correlationl Cross-correlation does not commute
l g(t) é h(t) ? h(t) é g(t)
l Autocorrelation is Hermitianl Rgg (t) = Rgg*(–t)
l Maximum modulus of autocorrelation occurs at the origin
l |Rgg(t)| < Rgg (0)
l Autocorrelation decay provides “correlation time” or “correlation length” or characteristic scale of signals and surfaces
©2000, D. L. Jaggard 79EE 511EE 511
Fourier Transform Properties - Rayleigh/Parseval Relation
Area under the absolute value squared of a function is equal to area under the absolute
value squared of its transform
P roof:
g(t) g∗(t)dt−∞
∞
∫ = g(t)g∗(t)e −j2πf ' tdt−∞
∞
∫ for f ' = 0
= G( f ' ) ⊗ G∗(− f ' ) for f ' = 0
= G( f )G∗( f − f ' )df−∞
∞
∫ for f ' = 0
= G( f )G∗( f )df−∞
∞
∫
g(t) 2 dt−∞
∞
∫ = G( f ) 2 df−∞
∞
∫
©2000, D. L. Jaggard 80EE 511EE 511
Rayleigh Theorem Example
l Equal areas under |g(t)|2 and |G(f)|2 imply conservation of power for optical (and other wave) signals
g(t) G(f)
|g(t) |2 |G(f)|2
t
t f
f
©2000, D. L. Jaggard 81EE 511EE 511
Fourier Transform Properties - Differentiation
l Prove by usual meansl Differentiation in the time -domain
leads to multiplication by frequency in the frequency -domain
l Useful for solving D.E.sl Useful for finding F.T. of piecewise
continuous functions:l Take second derivativel Find inverse transform as sum of delta
functions
d n g(t)dtn = g ( n )(t) ⇔ ( j2πf )n G( f )
©2000, D. L. Jaggard 82EE 511EE 511
Fourier Transform Properties - Moments
l The moment of a function is related to the behavior of its transform near
the originl Therefore low frequency regime of
transform may be valuable in classification & identification
mn = tn
−∞
∞
∫ g( t)dt = m th moment of g(t)
mn =G(n) (0)
(− j2π )n
©2000, D. L. Jaggard 83EE 511EE 511
Properties of Moments
Moments can be used for signal or image classification
m0 = G(0 ) zeroth moment or area
m1 = G(1) (0)− j2π
first moment or centroid
m2 =G(2 ) (0)−4π 2 second moment or moment of inertia
m2
m0
=G( 2)(0)
−4π 2G(0)radius of gyration
©2000, D. L. Jaggard 84EE 511EE 511
Second Moment Examples
l Moments of g(t) affect G(f) at the origin
l Infinite moment of g(t) gives cusp at origin for G(f)
g(t) G(f)
t f
t
t
f
f
g(t) G(f)
©2000, D. L. Jaggard 85EE 511EE 511
Fourier Transform Properties -Periodic Functions
To find the F.T. of a periodic function, find the F.T. of its Fourier
series
If g(t) is periodic with period T = 1/ f0
g(t ) = Gn−∞
∞
∑ exp( j2nπf0t ) where Gn = g(t)− T /2
T / 2
∫ exp(− j2nπf0t )dt
F g(t ){ }= F Gn−∞
∞
∑ exp( j 2nπf0t )
= Gn−∞
∞
∑ F exp( j2nπf0 t){ }
= Gn−∞
∞
∑ δ ( f − nf0)
The spectrum of a periodic
function is a “line
spectrum”
This is the complex Fourier series
©2000, D. L. Jaggard 86EE 511EE 511
Some Notes
l F{F{g(t)}} = g(–t)l If we know a F.T. pair, we can invert this
pair by inverting the coordinatel If one lens in optics gives a F.T., then a
second lens can be used to provide an inverted image
l 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 87EE 511EE 511
II. Fourier Transforms and Linear Systems
A. RequirementsB. “Inventing” the Fourier TransformC. Dirac Delta FunctionD. Use of the F.T. in OpticsE. F.T. PropertiesF. Some Useful TransformsG. Two-Dimensional F.T.H. More Useful TransformsI. Sampling
©2000, D. L. Jaggard 88EE 511EE 511
F. Some Useful Transforms
l Unity and Delta Functionl Rectl Trianglel Gaussianl Signum and Stepl Combl Sine and Cosine
©2000, D. L. Jaggard 89EE 511EE 511
Some Useful Functionsrect(t / T ) =
1 t / T < 1 / 20 t / T > 1 / 2
Λ(t / T ) =1 − t / T t / T < 1
0 t / T > 1
Gaus(t / T ) = exp[−π (t / T )2 ]
comb(t / T ) = δ (t / T − n )−∞
∞
∑ = T δ (t − n / T )−∞
∞
∑
sgn( t) =1 t > 1−1 t < 1
u(t) =1 t > 10 t < 1
= 12
+ 12
sgn(t)
©2000, D. L. Jaggard 90EE 511EE 511
Some Useful F.T. Pairs1 ⇔ 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 )
exp(−t ) ⇔2
4π 2 f 2 +1
t−1 ⇔ − jπ sgn(−πf 2 )
sgn( t) ⇔ ( jπf )−1
u(t) ⇔ 2−1δ (t) + ( j2πf )−1
cos(2πf0t) ⇔ 2−1[δ( f − f0) + δ( f + f0)]
sin(2πf0t) ⇔ (2 j)−1[δ ( f − f0 )− δ( f + f0)]
©2000, D. L. Jaggard 91EE 511EE 511
G. Two-Dimensional F.T.
l Separable Function
l Circular Symmetry
l Hankel transforms
l Fourier-Bessel transforms
l Some Two -Dimensional F.T. Pairs
©2000, D. L. Jaggard 92EE 511EE 511
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:
lLet g(x,y) = j(x) h(y)
lThenF{g(x,y) = F{j(x)} F{ h(y)} = J(f) H(f)
©2000, D. L. Jaggard 93EE 511EE 511
Polar Separable Casel Let g(r,θ) be separable in polar
coordinates:l Let
g(r,θ) = gR(r)ejmθ
l Then I.C.B.S.T.F {g(r,θ)} = (–j)m ejmθ Hm{gR(r)}
whereHm{gR (r)} = 2π r gR(r) Jm(2πρr)dr
is the Hankel transform of order m and Jm is the Bessel function of order m
0
∞
∫
©2000, D. L. Jaggard 94EE 511EE 511
Case of Circular Symmetryl Let g(r,θ) = gR(r) = g(r) [circular
symmetry]:l Then I.C.B.S.T.
F {g(r)} = 2π r gR(r) J0(2πρr)dris the Fourier-Bessel transform of g (r) where J0 is the Bessel function of order 0
l This gives the Fourier-Bessel pair
0
∞
∫
G( ρ) = 2π rg(r)J00
∞
∫ (2πrρ)dr
g(r) = 2π ρG( ρ)J 00
∞
∫ (2πrρ)dρ
g( r) ⇔ G (ρ)
Also known as Hankel transform of order
zero
©2000, D. L. Jaggard 95EE 511EE 511
Fourier-Bessel Proof
G( fx , fy ) =−∞
∞
∫ g(x,y)e− j2π ( fx x+ fy y)
dxdy−∞
∞
∫
r = x 2 + y 2 ρ = fx2 + f y
2
θ = tan−1(y / x) φ = tan− 1( fy / fx)
yy
= rcosθsinθ
fx
f y
= ρcosφsinφ
F g(r){ } = G(ρ)
= g(r)0
∞
∫0
2π
∫ e− j2π rρ (cosθ cos φ +sin θ sin φ )rdrdθ
= g(r)0
∞
∫0
2π
∫ e− j2π rρ cos(θ −φ ) rdrdθ
Note: e− ja cos(θ −φ )dθ = 2πJ0 (a)0
2π
∫
F g(r){ } = G(ρ) = 2π rg(r )J 0(2πrρ)dr0
∞
∫
©2000, D. L. Jaggard 96EE 511EE 511
Some Useful RelationsyJ0 (y)dy = x
0
x
∫ J1 (x)
xν +1 exp(−αx2 )Jν (βx)dx =βν
2α ( ν +1)0
∞
∫ exp(−β 2 / 4α)
Jν (x) x→0 → (1 / 2)xν
Γ(ν +1)(v ≠ −1,−2, −3,...)
Jν (x) x→∞ → 2πx
cos[ x − (1 / 2)νπ − (1 / 4)π
where Γ (ν + 1) = ν! for ν = integer
Γ(ν + 1) =ν Γ (ν )
©2000, D. L. Jaggard 97EE 511EE 511
Bessel Functions J 0 & J1
l All Bessel functions except J0 are zero at the originl J0 is unity at the originl Bessel functions have significant value in the regime
where their order equals their argument
©2000, D. L. Jaggard 98EE 511EE 511
Two-Dimensional Functions
circ(r / a) ≡1 r / a <10 r / a >1
jinc(ρ) ≡ 2J1(2πρ)
2πρ
Gaus(ax,by) ≡ exp[−π (a2 x2 + b2y2 )]
Gaus(ar) ≡ exp[−πa2r2] = exp[−πa2 (x2 + y2)]
δ ( x, y) ≡ δ (x)δ( y) =δ (r)πr
comb( ax)comb(by) ≡ δ (ax − n,by − m)m= −∞
∞
∑n= −∞
∞
∑
©2000, D. L. Jaggard 99EE 511EE 511
“sinc” and “ jinc” Functionsjinc(r/2)
jinc2(r/2)sinc2(x)
sinc(x)
What are the differences between sinc and jinc functions?
©2000, D. L. Jaggard 100EE 511EE 511
Two-Dimensional F.T. Pairs
rect (x / a)rect (y / b) ⇔ absinc(afx )sinc(bfy )
circ(r / a) ⇔ π a 2 jinc(aρ)
δ (r)πr
⇔1
δ (ax,by ) ⇔1ab
1r
⇔1ρ
1 ⇔δ ( fx, fy )
cos(πr2 ) ⇔ sin(πρ2 )
exp(± jπr2 ) ⇔ ± j exp(mjπρ 2 )
Gaus(ax,by ) ⇔1ab
Gaus( fx / a, f y / b )
Gaus(ar) ⇔ 1a 2 Gaus( fρ / a)
comb(x / a)comb(y / b) ⇔ abcomb( afx )comb(bfy)
©2000, D. L. Jaggard 101EE 511EE 511
How Often Does a Signal Need to be
Sampled in Order to be Exactly Replicated?
©2000, D. L. Jaggard 102EE 511EE 511
Sampling of Signals
l Whittaker-Shannon Sampling TheoremA signal which is bandlimited (i.e., all
frequencies less than fmax) can be exactly reconstructed by accurate samples at times
l The frequency 2 fmax (number of samples per second) is known as the Nyquist frequency
l Requirement on sampling frequency is fs > 2 fmax
1max )2( −< fT
©2000, D. L. Jaggard 103EE 511EE 511
Miracle of Sampling
l Exact reconstruction if overlap avoided by T <1
2 fm
sinc functionis “interpolating
function” for exactreconstruction
©2000, D. L. Jaggard 104EE 511EE 511
Imperfect Samplingl What if samples of finite width are used
so that each sample is an average of a part of the signal?
l What if the signal is only sampled for a finite length of time?
l What if unequally spaced samples are used?
l What if the samples are descretized?
©2000, D. L. Jaggard 105EE 511EE 511
Two-Dimensional Sampling
l Various geometries can be usedl 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
lM = 16 LX LY BXBY
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