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1
ing
(fax)
DR. ROBERT A. SCHOWENGERDT [email protected]
ECE 425
Image Science and Engineer
Spring Semester 2000
Course Notes
Robert A. Schowengerdt
(520) 621-2706 (voice), (520) 621-8076
ECE402
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DR. ROBERT A. SCHOWENGERDT [email protected]
DEFINITIONS
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tection
DR. ROBERT A. SCHOWENGERDT [email protected]
Image science
• The theory of optical image formation and de
• Includes elements of:
• optics
• radiometry
• linear systems
• statistics
• vision
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ission, storage
DR. ROBERT A. SCHOWENGERDT [email protected]
Image engineering
• The technologies of image acquisition, transmand display
• Includes elements of:
• detectors
• signal processing
• data compression
• image processing
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rn life
ing, digital
oto scanners and
otics)
n monitoring)
l models,
DR. ROBERT A. SCHOWENGERDT [email protected]
OVERVIEW
Electronic imaging systems pervade mode
Examples
• Document processing (scanning, storage, printlibraries, WWW)
• Consumer products (HDTV, digital cameras, phprinters)
• Machine vision (quality control inspection, rob
• Medical imaging (disease diagnosis, medicatio
• Scientific visualization (complex mathematicainteractive graphics)
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monitoring,
g)
DR. ROBERT A. SCHOWENGERDT [email protected]
• Remote sensing (earth science, environmentalweather)
• Military (reconnaisance, surveillance, targetin
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7
of systems:
e total
e the
DR. ROBERT A. SCHOWENGERDT [email protected]
THE SYSTEMS APPROACH
A perceived image is the result of a chain
• optics
• detector
• coding/decoding
• display
• human vision
Each can be considered a subsystem of thelectronic imaging system
The engineering design goal is to optimiz
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ion to that of
ing system
DR. ROBERT A. SCHOWENGERDT [email protected]
performance of each subsystem in relatthe others and the total system
This course covers the tools used for imaganalysis, design and evaluation
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systems
ctronic
humanvision
subsystem
tina*
neuralnetwork
brain
DR. ROBERT A. SCHOWENGERDT [email protected]
An imaging system consists of several sub
• * points of signal transduction, optical <—> ele
lightsource scene
imageacquisitionsubsystem
transmissionsubsystem
displaysubsystem*
optics detector* electronics
coder decoder
optics
re
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10
with:
tics)
ystems)
DR. ROBERT A. SCHOWENGERDT [email protected]
For optical components, we’re concerned
• size and location of the image (geometrical op
• intensity of the image (radiometry)
• contrast and sharpness of the image (linear s
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11
ed with:
ers, A/D
DR. ROBERT A. SCHOWENGERDT [email protected]
For electronic components, we’re concern
• image sampling and quantization (analog filtconverters, coding)
• image processing (digital signal processing)
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12
L TOOLS
s
ier Transform
DR. ROBERT A. SCHOWENGERDT [email protected]
SECTION I – MATHEMATICAMathematics Background
Convolution and Fourier Transforms
Linear Filtering and Sampling
Two-dimensional Functions and Operation
Discrete Fourier Transform and Fast Four
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13
D
DR. ROBERT A. SCHOWENGERDT [email protected]
MATHEMATICS BACKGROUN
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ier analysis and
, joined by a
DR. ROBERT A. SCHOWENGERDT [email protected]
Complex Notation
• Complex arithmetic will be necessary for Fouroptics
• Complex numbers consist of two real numbersphasor relationship
• where
• c is a complex number,
• a is the real part of c
• b is the imaginary part of c
• j is
• Phasor relationship
c a jb+=
1–
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b
arealpart
imaginarypart
c
A
θ
DR. ROBERT A. SCHOWENGERDT [email protected]
• The amplitude A of c
• The phase θ of c
• Can write c as which, by Euler’s Theorem,
A a2
b2
+=
θ b a⁄( )atan=
c Aejθ
=
c A θcos j θsin+( )=
AaA--- j
bA---+
=
a jb+=
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ing relations:
DR. ROBERT A. SCHOWENGERDT [email protected]
• Using Euler’s Theorem, we can derive the follow
? Using Euler’s Theorem, show that
θ( )sin12 j----- e
jθe
jθ––( )=
θ( )cos12--- e
jθe
jθ–+( )=
θsin( )2 θcos( )2+ 1=
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ite and its
area
is ce in plots.
ea
n
n
x
x x0– b–( )]
DR. ROBERT A. SCHOWENGERDT [email protected]
Simple Functions
• Delta function and its relatives
• delta
• NOTE: The delta function’s amplitude is infin1. The amplitude is shown as 1 for convenien
? Write the equation that defines the arof a delta function as 1.
? Review the definition of delta functioin terms of the limit of conventional functions, such as the rectangle functio
• even delta pair
δ x x0–( )
1
x0x
1x0 = 0
δδx x0–
b--------------
b δ x x0– b+( ) δ+[=
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x+ b
δ x x0– b–( )– ]
x+ b
DR. ROBERT A. SCHOWENGERDT [email protected]
• odd delta pair
x
|b|
b- b
|b|
x0 x0 - b
x0 = 0 x0 ≠ 0
δδx x0–
b--------------
b δ x x0– b+( )[=
x
|b|
b- b
x0
x0 - b
x0 = 0
-|b|
|b|
-|b|
x0 ≠ 0
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xx0+2b
. . .
DR. ROBERT A. SCHOWENGERDT [email protected]
• comb (shah) combx x0–
b--------------
b δ x x0– nb–( )n ∞–=
∞
∑=
x
|b|
b- b
x0 = 0
0 2b- 2b
|b|
x0+bx0-b 0x0-2b x0
. . .. . . . . .x0 ≠ 0
Even delta pair, odd
delta pair and comb
functions are all scaled by b
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a particular shift
alue of the ble)
DR. ROBERT A. SCHOWENGERDT [email protected]
• Use of the δ function
• sifting
• NOTE: Sifting is a convolution, evaluated for
• Finds the value of a function at a specific vindependent variable (similar to a look-up ta
• sampling
f α( )δ α x0–( ) αd
∞–
∞
∫ f x0( ) constant= =
f x( )δ x x0–( ) f x0( )δ x x0–( )=
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21
mined by the value independent
x
)
DR. ROBERT A. SCHOWENGERDT [email protected]
• NOTE: Sampling is a mulitplication
• Output is a delta function, with area deterof the function at the specified value of the variable.
• uniform sampling
xx0 x0
x
x =f(x)
f(x0)
x0
1b----- f x( )comb
x x0–
b--------------
f x0 nb+( )δ x x0– nb–(n ∞–=
∞
∑=
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22
tain amplitude of
xb x0+2b
. . .
α f x x0–( )=
x
DR. ROBERT A. SCHOWENGERDT [email protected]
• NOTE: Must divide comb function by |b| to ref(x).
• NOTE: f(x) modulates the comb function.
• shifting
• replicating
xx0+bx0-b 0 x0+2bx0-2b x0
. . .. . .x0+x0-b 0x0-2b x0
. . .
1/|b|
g x( ) f x( ) ❉ δ x x0–( ) f α( )δ x x0– α–( )d
∞–
∞
∫= =
xx0
x
❉ =f(x)
x0
g(x)1
g x( ) 1b----- f x( ) ❉ comb
x x0–
b--------------
=
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23
of f(x)
xx0+bx0
. . .
x b⁄ 1 2⁄>x b⁄ 1 2⁄=
x b⁄ 1 2⁄<
x b⁄ 1≥b x b⁄ 1<
DR. ROBERT A. SCHOWENGERDT [email protected]
• NOTE: Must divide by |b| to retain amplitude
• Other 1-D functions
• rectangle (square pulse)
• triangle
xx0+bx0-b 0x0-2b x0
. . .. . .1/|b|
x
❉
f(x)
x0-b 0x0-2b
. . .
g(x)
=
rectxb---
0
1 2 ⁄1
=
trixb---
0
1 x ⁄–=
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n b, ction wide ect n
DR. ROBERT A. SCHOWENGERDT [email protected]
? What is the value of b in the above graph?
• sinc
• sinc-squared
-0.2
0
0.2
0.4
0.6
0.8
1
1.2
-60 -40 -20 0 20 40 60
recttri
f(x)
x
For a givethe tri funis twice as
as the rfunctio
sinc(x b )⁄ πx b⁄[ ]sinπx b⁄
--------------------------=
sinc2
x b⁄( )
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DR. ROBERT A. SCHOWENGERDT [email protected]
? What is the value of b in the above graph?
• gaus(sian)
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
1.2
-60 -40 -20 0 20 40 60
sincsinc squared
f(x)
x
gaus x b⁄( ) eπ x b⁄( )2–
=
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DR. ROBERT A. SCHOWENGERDT [email protected]
? What is the value of b in the above graph?
• cosine
• sine
-0.2
0
0.2
0.4
0.6
0.8
1
1.2
-60 -40 -20 0 20 40 60
gaus
f(x)
x
2πx b⁄( )cos ej2π x b⁄( )
ej– 2π x b⁄( )
+2
--------------------------------------------------------=
2πx b⁄( )sin ej2π x b⁄( )
ej– 2π x b⁄( )
–2 j
--------------------------------------------------------=
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DR. ROBERT A. SCHOWENGERDT [email protected]
? What is the value of b in the above graph?
-1.2
-0.8
-0.4
0
0.4
0.8
1.2
-60 -40 -20 0 20 40 60
cossin
f(x)
x
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RMS (1-D)
nt (LSI) system
m as,
, of the e, h(x)
f x( ) ❉ h x( )
DR. ROBERT A. SCHOWENGERDT [email protected]
CONVOLUTION AND FOURIER TRANSFO
Convolution (1-D)
• Why is it important?
• Describes the effect of a Linear Shift Invariaon input signals
• L is the system operator
• Can write a general description of any syste
• For an LSI system, L is a convolution, input signal and the system impulse respons
system
(operator L)
inputsignal f(x)
outputsignal g(x)
g x( ) L f x( )[ ]=
g x( ) =
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x or α
DR. ROBERT A. SCHOWENGERDT [email protected]
• Mathematical and graphical description
• Example
g x( ) f x( ) ❉ h x( ) f α( )h x α–( ) αd∞–
∞∫= =
2
0 3x or α
f(x) or f(α)
0 3
h(x) or h(α)
-1
1
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h(α)
)
DR. ROBERT A. SCHOWENGERDT [email protected]
Convolution
• 1. write both as a function of α f(α) and
• 2. flip h (or f) about α = 0 h(-α)
• 3. shift h (or f) by an amount x h(x - α)
• 4. multiply the two functions f(α)h(x -α
• 5. integrate the product function over all α g(x)
• 6. repeat steps 3 through 5 until done
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αarea = g(0)
α
area = g(3)
α
area = g(2)
αarea = g(1)
αarea = g(4)
integrate
DR. ROBERT A. SCHOWENGERDT [email protected]
α
h(-α)
1
x = 0
f(α)h(0 - α)
f(α)h(3 - α)
f(α)h(2 - α)
f(α)h(1 - α)
f(α)h(4 - α)
α
h(1 -α)
1
x = 1
α
h(2 -α)
1
x = 2
α
h(3 -α)
1
x = 3
α
h(4 -α)
1
x = 4
shift multiply
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ifts in xample integer s, for ration nience
DR. ROBERT A. SCHOWENGERDT [email protected]
• Plot g(x)
2
0
3
1
g(x)
x
The shthis e
are bystep
illustconve
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x)
DR. ROBERT A. SCHOWENGERDT [email protected]
1-D CONVOLUTION PROPERTIES
property
commutative
distributive
associative
f x( ) ❉ h x( ) h x( ) ❉ f x( )=
f x( ) ❉ h1 x( ) h2 x( )+[ ]
f x( ) ❉ h1 x( ) f x( ) ❉ h2(+=
f x( ) ❉ h1 x( ) ❉ h2 x( )[ ]
f x( ) ❉ h1 x( )[ ] ❉ h2 x( )=
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0)
0)
)
)
x
2-------
DR. ROBERT A. SCHOWENGERDT [email protected]
1-D CONVOLUTION EXAMPLES
f(x) h(x) g(x)
f(x) δ(x) f(x)
f(x-x0) h(x) g(x-x
f(x) h(x-x0) g(x-x
rect(x) rect(x) tri(x
sinc(x) sinc(x) sinc(x
gaus(x) gaus(x)1
2-------gaus
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ecomes a
quation,
DR. ROBERT A. SCHOWENGERDT [email protected]
Fourier Transforms (1-D)
• Why is it important?
• For an LSI system, the convolution operator bmultiplicative operator in the Fourier domain
• Taking the Fourier transfom of the system e
E C E 4 2 5 C L A S S N O T E S – 2 0 0 0
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nal, F(u) is the system transfer
stem in the
DR. ROBERT A. SCHOWENGERDT [email protected]
• where G(u) is the spectrum of the output sigspectrum of the input signal, and H(u) is thefunction
• In many cases, it is easier to analyze an LSI syFourier domain
• Forward transform
• Inverse transform
G u( ) F u( )H u( )=
F u( ) f x( )ej2πxu–
xd
∞–
∞
∫=
f x( ) F u( )ej2πxu
ud
∞–
∞
∫=
E C E 4 2 5 C L A S S N O T E S – 2 0 0 0
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37
l)
into its fferent erent
from F(u).
DR. ROBERT A. SCHOWENGERDT [email protected]
Properties for special functions
• f(x) and F(u) are in general, complex functions
• If f(x) real � F(u) = F*(-u)
• Hermitian: Re[F(u)] even, Im[F(u)] odd
• If f(x) real and even � Im[F(u)] = 0 (F(u) is rea
Forward transform is the analysis of f(x)spectrum F(u) of sines and cosines at difrequencies, in general, each with a diffamplitude and phase.
Inverse transform is the synthesis of f(x)
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DR. ROBERT A. SCHOWENGERDT [email protected]
1-D FOURIER TRANSFORM PAIRS
f(x) F(u)
1 δ(u)
δ(x) 1
rect(x) sinc(u)
sinc(x) rect(u)
comb(x) comb(u)
gaus(x) gaus(u)
tri(x) sinc2(u)
2πu0x( )cos1
2 u0------------δδ u
u0-----
12 x0-----------δδ x
x0-----
2πux0( )cos
2πu0x( )sinj
2 u0------------δδ u
u0-----
E C E 4 2 5 C L A S S N O T E S – 2 0 0 0
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39
)
u)
)
F u( )
0)
F u( )
)
a2F2 u( )
2 u( )
F2 u( )
DR. ROBERT A. SCHOWENGERDT [email protected]
1-D FOURIER TRANSFORM PROPERTIES
name f(x) F(u)
f(±x) F(±u
F(±x)
scaling f(x/b) |b|F(b
shifting
f(x ± x0)
derivative
linearity
convolution
f u+−(
ej2πx0u±
ej2πxu0±
f x( ) F u u+−(
fk( )
x( ) j2πu( )k
j2πx–( )kf x( ) F
k( )u(
a1 f 1 x( ) a2 f 2 x( )+ a1F1 u( ) +
f 1 x( ) ❉ f 2 x( ) F1 u( )F
f 1 x( ) f 2 x( ) F1 u( ) ❉
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40
u–( )
F2 u( )
DR. ROBERT A. SCHOWENGERDT [email protected]
correlation
1-D FOURIER TRANSFORM PROPERTIES
name f(x) F(u)
f 1 x( ) � f 2 x( ) F1 u( )F2
f 1 x( ) f 2 x–( ) F1 u( ) �
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41
EMS (1-D)
to the sum of
change over
(x) of an LSI
e
DR. ROBERT A. SCHOWENGERDT [email protected]
LINEAR, SHIFT-INVARIANT (LSI) SYST
Linear: output of a sum of inputs is equalthe individual outputs
shift-invariant: system response does notspace
Relation between input f(x) and output gsystem
• where h(x) is the system impulse respons
g1 x( ) af 1 x( ) ❉ h x( )=
g2 x( ) bf 2 x( ) ❉ h x( )=
g x( ) af 1 x( ) bf 2 x( )+[ ] ❉ h x( )=
af 1 x( ) ❉ h x( ) bf 2 x( ) ❉ h x( )+( )=
g1 x( ) g2 x( )+=
g x( ) f x( ) ❉ h x( )=
E C E 4 2 5 C L A S S N O T E S – 2 0 0 0
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42
trum F(u) of f(x)
odulation
for an LSI
DR. ROBERT A. SCHOWENGERDT [email protected]
Fourier transform of LSI system equation
• where H(u) is the system transfer function
• H(u) is a complex filter that modifies the spec
• In optics, the amplitude of H(u) is called the MTransfer Function (MTF)
The properties of complex functions give,system,
G u( ) F u( )H u( )=
ampl G u( )[ ] ampl F u( )[ ]ampl H u( )[ ]=
phase G u( )[ ] phase F u( )[ ] phase H u( )[ ]+=
E C E 4 2 5 C L A S S N O T E S – 2 0 0 0
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43
)
DR. ROBERT A. SCHOWENGERDT [email protected]
1-D CASCADED SYSTEMS
N cascaded LSI systems
. . .
f(x) g(x
h1(x) hN(x)
g x( ) f x( ) ❉ h1 x( )[ ] ❉ h2 x( ){ }… ❉ …hN x( )=
E C E 4 2 5 C L A S S N O T E S – 2 0 0 0
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44
e
DR. ROBERT A. SCHOWENGERDT [email protected]
Single system equivalent
• where hnet is the net system impulse respons
f(x)
hnet(x)
g(x)
hnet x( ) h1 x( ) ❉ h2 x( )… ❉ …hN x( )=
E C E 4 2 5 C L A S S N O T E S – 2 0 0 0
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45
LES
6rect 3u( )
u
2sinc(x/3)
DR. ROBERT A. SCHOWENGERDT [email protected]
FOURIER TRANSFORM EXAMP
Ex 1. Find sinc(x/2) ❉ sinc(x/3)
• Convolution in this case is very difficult!
• Take the Fourier transform
• Take the inverse Fourier transform
2rect 2u( ) 3rect 3u( )⋅ =
2
1/4-1/4
3
1/6-1/6
x
u u
6
1/6-1/6
=
63---sinc(x/3) =
E C E 4 2 5 C L A S S N O T E S – 2 0 0 0
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46
a DC bias
nc function
/period
x
. . .
x2---
❉ combx5---
DR. ROBERT A. SCHOWENGERDT [email protected]
Ex 2. Find spectrum of square wave with
• Write square wave as convolution
• Take the Fourier transform
• spectrum is comb function, modulated by si
• sampled at frequency interval ∆u = 1/5, i.e 1
• zeros at u = n/2, n = ± 1, ±2, . . .
+1-1 5 10-5
1
f(x)
. . .
f x( ) 15---rect=
F u( ) 15--- 2 5 sinc 2u( ) comb 5u( )⋅⋅ ⋅ ⋅=
2sinc 2u( )comb 5u( )=
E C E 4 2 5 C L A S S N O T E S – 2 0 0 0
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47
ave the classic
u
DR. ROBERT A. SCHOWENGERDT [email protected]
• if square wave period P = 2 x pulse width, we hsquare wave spectrum at
• u = 0, ±1/P, ±3/P, ±5/P, . . . ? Verify the above statement for an arbitrary period P
• sinc(2u) and comb(5u)
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
1.2
0 1/2 1 3/2-1/2-1-3/2
1/5 2/5 3/5 4/5 5/5 6/5 7/5 8/5-8/5-7/5-6/5-5/5-4/5-3/5-2/5-1/5
2
1/5
E C E 4 2 5 C L A S S N O T E S – 2 0 0 0
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48
comb, modulated by sinc function
DR. ROBERT A. SCHOWENGERDT [email protected]
• sinc(2u) times comb(5u)
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
1.2
0 1/2 1 3/2-1/2-1-3/2
1/5 2/53/5 4/5
5/5 6/5 7/58/5-8/5
-7/5-6/5-5/5-4/5-3/5
-2/5-1/5
2/5
E C E 4 2 5 C L A S S N O T E S – 2 0 0 0
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49
NSFORMS
DR. ROBERT A. SCHOWENGERDT [email protected]
SCALING PROPERTY OF FOURIER TRA
E C E 4 2 5 C L A S S N O T E S – 2 0 0 0
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50
width of f(x)
u
u
u
1
2
3
DR. ROBERT A. SCHOWENGERDT [email protected]
Width of F(u) is inversely proportional to
rect(x)
x
rect(x/2)
x
sinc(u)
2sinc(2u)
rect(x/3)
x
3sinc(3u)
F
1/2-1/2
1-1
3/2-3/2
E C E 4 2 5 C L A S S N O T E S – 2 0 0 0
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51
u
u
u
DR. ROBERT A. SCHOWENGERDT [email protected]
δδ(2x)
x
δδ(x)
x
δδ(2x/3)
x
cos(πu)
2cos(2πu)
3cos(3πu)
F
1/2-1/2
1-1
3/2-3/2
1/2
1
3/2
E C E 4 2 5 C L A S S N O T E S – 2 0 0 0
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52
u
u
u
1/2
1/2
1/2
1
2
3
DR. ROBERT A. SCHOWENGERDT [email protected]
δδ(u)/2
δδ(u/2)/4
δδ(u/3)/6
Fcos(2πx)
x
x
x
cos(4πx)
cos(6πx)
-1
-2
-3
E C E 4 2 5 C L A S S N O T E S – 2 0 0 0
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53
NSFORMS
sum of their
u
1/2
1
1
u
1/2
1 2
DR. ROBERT A. SCHOWENGERDT [email protected]
SUPERPOSITION PROPERTY OF FOURIER TRA
• Fourier transform of sum of functions equals individual Fourier transforms
1 + cos(2πx)
x
δ(u)+δδ(u)/2
F
-1
cos(2πx)+cos(4πx) δδ(u)/2+δδ(u/2)/4
-1-2x
E C E 4 2 5 C L A S S N O T E S – 2 0 0 0
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54
SFORM
ill-conditioned
ill-conditioned
x( )
u)H u( )
G u( )[ ]
) F u( )⁄
H u( )[ ]
) H u( )⁄
F u( )[ ]
DR. ROBERT A. SCHOWENGERDT [email protected]
SYSTEM ANALYSIS WITH THE FOURIER TRAN
• Variety of applications
• LSI system equation:
Application Given Find Spatial Domain Fourier Domain
system output f(x), h(x) g(x)
system
identificationf(x), g(x) h(x) NA
inversion h(x), g(x) f(x) NA
g x( ) f x( ) ❉ h=
g x( ) f x( ) ❉ h x( )=G u( ) F(=
g x( ) F 1–=
H u( ) G u(=
h x( ) F 1–=
F u( ) G u(=
f x( ) F 1–=
E C E 4 2 5 C L A S S N O T E S – 2 0 0 0
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55
ignal
tal values
amples) or
DR. ROBERT A. SCHOWENGERDT [email protected]
SAMPLING (1-D)
Two components to digitizing an analog s
• Sample analog signal at discrete values of x
• Quantize the sampled signal amplitude to digi
Sampling is either ideal (delta function snon-ideal (time-integrated samples)
E C E 4 2 5 C L A S S N O T E S – 2 0 0 0
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56
xb0 2b
. . .
fs(x)
DR. ROBERT A. SCHOWENGERDT [email protected]
IDEAL SAMPLING
Sampled function fs(x)
• Mathematically
• Sample interval = b, sample rate = 1/b
- b- 2b
. . .x
b- b 0 2b- 2b
. . .. . .=
f(x)
f s x( ) 1b----- f x( )comb
xb---
=
E C E 4 2 5 C L A S S N O T E S – 2 0 0 0
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57
u( )
u)
) ❉ comb bu( )
DR. ROBERT A. SCHOWENGERDT [email protected]
Fourier domain description
• Analog signal
• band limit of analog signal = ± uB
• (highest frequency component in signal)
• bandwidth of analog signal = 2uB
• Sampled signal
f x( ) F↔
F(u)
0
uB-uB
u
f s x( ) Fs(↔
1b----- f x( )comb
xb---
F u(↔
E C E 4 2 5 C L A S S N O T E S – 2 0 0 0
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58
. . .
u
(u-2u0)
u
DR. ROBERT A. SCHOWENGERDT [email protected]
• sampling frequency (rate) = us = 1/b
• folding frequency = uf = 1/2b
F(u)F(u+u0) F(u-u0)
0 1/b 2/b-1/b-2/b
uB-uB
. . .F(u-2u0) F
uf
-1/b+uB 1/b-uB
-uf
Fs(u)
1/b-1/b 2/b-2/b
E C E 4 2 5 C L A S S N O T E S – 2 0 0 0
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59
tra overlap
lower frequency
ore sampling) by s a lower
. . .
u
2u0)
DR. ROBERT A. SCHOWENGERDT [email protected]
Aliasing occurs where the individual spec
• Frequency components above uf appear to be components below uf in the sampled signal
• Once aliasing occurs, it cannot be removed
• Sometimes, analog signal is pre-filtered (befa low-pass filter so that the analog signal habandwidth and
F(u)F(u+u0) F(u-u0)
0 1/b 2/b-1/b-2/b
uB-uB
. . .F(u-2u0) F(u-
uf
-1/b+uB 1/b-uB
-uf
uB′ u f≤
E C E 4 2 5 C L A S S N O T E S – 2 0 0 0
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60
ite) and sample
n,
DR. ROBERT A. SCHOWENGERDT [email protected]
Can avoid aliasing if:
• original analog signal is band-limited (uB is finrate satisfies the following condition
• or, the sample interval satisfies the conditio
us 2uB≥
1b--- 2uB≥
b1
2uB---------≤