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ing

(fax)

DR. ROBERT A. SCHOWENGERDT [email protected]

ECE 425

Image Science and Engineer

Spring Semester 2000

Course Notes

Robert A. Schowengerdt

[email protected]

(520) 621-2706 (voice), (520) 621-8076

ECE402

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DEFINITIONS

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tection


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


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,


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)


• Remote sensing (earth science, environmentalweather)

• Military (reconnaisance, surveillance, targetin

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of systems:

e total

e the


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


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


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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with:

tics)

ystems)


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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ed with:

ers, A/D


For electronic components, we’re concern

• image sampling and quantization (analog filtconverters, coding)

• image processing (digital signal processing)

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L TOOLS

s

ier Transform


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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D


MATHEMATICS BACKGROUN

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ier analysis and

, joined by a


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


• 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:


• 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–( )]


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


• 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

. . .


• 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)


• 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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mined by the value independent

x

)


• 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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tain amplitude of

xb x0+2b

. . .

α f x x0–( )=

x


• 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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of f(x)

xx0+bx0

. . .

x b⁄ 1 2⁄>x b⁄ 1 2⁄=

x b⁄ 1 2⁄<

x b⁄ 1≥b x b⁄ 1<


• 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


? 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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• 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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• 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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-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( )


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


• More on this later

• 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)

)


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


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


• Plot g(x)

2

0

3

1

g(x)

x

The shthis e

are bystep

illustconve

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x)


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-------


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,


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

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nal, F(u) is the system transfer

stem in the


• 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

∞–

∞

∫=

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l)

into its fferent erent

from F(u).


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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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-----

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)

u)

)

F u( )

0)

F u( )

)

a2F2 u( )

2 u( )

F2 u( )


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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u–( )

F2 u( )


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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EMS (1-D)

to the sum of

change over

(x) of an LSI

e


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( )=

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trum F(u) of f(x)

odulation

for an LSI


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( )[ ]+=

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)


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( )=

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e


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( )=

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LES

6rect 3u( )

u

2sinc(x/3)


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) =

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a DC bias

nc function

/period

x

. . .

x2---

❉ combx5---


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( )=

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ave the classic

u


• 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

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comb, modulated by sinc function


• 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

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NSFORMS


SCALING PROPERTY OF FOURIER TRA

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width of f(x)

u

u

u

1

2

3


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

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u

u

u


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

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u

u

u

1/2

1/2

1/2

1

2

3


δδ(u)/2

δδ(u/2)/4

δδ(u/3)/6

Fcos(2πx)

x

x

x

cos(4πx)

cos(6πx)

-1

-2

-3

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53

NSFORMS

sum of their

u

1/2

1

1

u

1/2

1 2


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

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SFORM

ill-conditioned

ill-conditioned

x( )

u)H u( )

G u( )[ ]

) F u( )⁄

H u( )[ ]

) H u( )⁄

F u( )[ ]


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–=

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ignal

tal values

amples) or


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)

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xb0 2b

. . .

fs(x)


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---

=

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u( )

u)

) ❉ comb bu( )


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(↔

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. . .

u

(u-2u0)

u


• 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

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tra overlap

lower frequency

ore sampling) by s a lower

. . .

u

2u0)


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≤

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ite) and sample

n,


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---------≤

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