International Dotoral Shool in Information
and Communiation Tehnology
DISI - University of Trento
Prediting the Tolerane Effets on the
Radiation Pattern
of Refletarray Antennas through Interval
Analysis
Nasim Ebrahimi
Advisor:
Andrea Massa, Professor
Università degli Studi di Trento
I would like to express my thanks to Prof. Massa and the whole ELEDIA
group forhelping and supporting me duringthese three years.
I would like to express my sinere gratitude to Prof. Paolo Roa, for his
suggestions and indiations for ompleting this thesis. I am truly thankful for
hiskindnessand support. Hispatiene andenouragementwere keymotivations
throughoutmy Ph.D.
I want to thank my friends for listening, disussing, giving me advie and
supporting meduringtheseyears. Youwere always besideme duringhappy and
hard moments.
The greatest thankyouisformybeloved family,my biggestsupporter inmy
life. Thanks for their love, enouragement and valuable prayers. And Finally
I wish to express my heartfelt love to my nephews Radmehr and Kohyar and
my niee Kianaz. For their ute and lovely alling and supporting their Auntie
The thesis fouses on prediting tolerane eets on the radiation pattern of
re-etarray antennas through Interval Analysis. In fat, the unertainty on the
atualsizeof allparametersunder fabriation toleranessuhas element
dimen-sionsand dieletri properties, are modeled with intervalvalues. Afterwards,the
rules of Interval Arithmeti are exploited to ompute the bounds of deviation in
the resonane frequeny of eah element, the phase response of the element and
the radiated power pattern. Due to the redundany problems of using Interval
Cartesian(
IA
−
CS
) foromplexstruture, theintervalboundsareoverestimated and thereasons are the Dependenyand Wrapping eets of using intervalanal-ysisforomplexstrutures. Dierenttehniquesare proposedandassessedin
or-derto eliminatethedependenyeetsuhasreformulatingtheintervalfuntion
and the Enumerative interval analysis. Moreover, the Minkowski sum approah
is used to eliminate the wrapping eet. In numerial validation, a set of
rep-resentative results, show the power bounds omputations withInterval Cartesian
method (
IA
−
CS
), a modied Interval Cartesian method (IA
−
CS
∗
),
Inter-val Enumerative method (
IA
−
ENUM
) and Interval Enumerative Minkowski method (IA
−
ENUM
−
MS
) and a omparative study is reported in order to assess the eetiveness of the proposed approah (IA
−
ENUM
−
MS
) with respet to the other methods. Furthermore, dierent toleranes in path width,length,substrate thikness anddieletri permittivity are onsideredwhih shows
that the higher unertainty produes the larger deviation of the pattern bounds
and the larger deviation inlude the smaller deviation and the nominal one. To
validate the inlusion properties of the interval bounds, the results are ompared
with Monte Carlo simulation results. Then, a numerial study is devoted to
analyze the dependeny of the degradation of the pattern features to steering
an-gle and the bandwidth. Finally, the eet of feed displaement errors on the
power pattern of reetarray antennas is onsidered with Interval Enumerative
Minkowski method. The maximal deviations from the nominal power pattern
(error free) and its features are analyzed for several reetarray strutures with
dierentfoal-length-to-diameterratios to provethe eetivenessof theproposed
method.
Keywords
ReetarrayAntennas,SensitivityAnalysis,AntennaUnertainties,Interval
1 Introdution and State-of-the-Art 1
2 Radiation Analysis for Reetarray Antenna 7
2.1 Introdution . . . 8
2.2 Overview of AnalysisTehniques . . . 8
2.2.1 Array-Theory Method . . . 8
2.2.2 Aperture-FieldMethod . . . 9
2.2.3 RadiationPatterns . . . 9
2.2.4 UnitCell Design . . . 11
2.2.4.1 Phase-ShiftDistributionTehnique . . . 11
2.2.4.2 Tehniques for Analysisof the Unit Cell . . . 12
2.2.4.3 Analytial ApproahBased onQ Fator Analysis 13 2.2.4.4 Floquet Harmonis . . . 13
2.2.4.5 Theoretial Expression for The Reetion Coef-ient . . . 14
3 Fundamentals of Interval Analysis 17 3.1 Introdution . . . 18
3.2 IntervalAnalysis . . . 19
3.2.1 Interval ElementaryOperations . . . 20
3.2.2 Interval Arithmeti . . . 21
3.2.3 Properties of the IntervalArithmeti . . . 22
3.2.4 AlgebraiProperties . . . 22
3.2.5 InlusionProperty of Interval Arithmeti . . . 23
3.2.6 Interval Funtion . . . 23
3.2.7 Interval FuntionProperty . . . 24
3.2.8 InlusionMonotoniity . . . 24
3.2.9 Dependeny Problem . . . 25
3.2.9.1 Dependeny Problemin Interval Funtion . . . . 26
3.3 Complex Intervals . . . 27
3.3.1 ComplexIntervalFuntion . . . 29
4.2 MathematialFormulation . . . 35
4.2.1 Cartesian
(
IA
−
CS
)
. . . 374.2.2 Cartesian
(
IA
−
CS
∗
)
. . . 404.2.3 Enumerative Strategy
(
IA
−
ENUM
)
. . . 414.2.4 Minkowski
(
IA
−
MS
)
. . . 445 Interval Method Validation with Numerial Results 47 5.1 Introdution . . . 48
5.1.1 NominalPattern Computation. . . 48
5.2 IntervalComputation . . . 52
5.2.1 Reetor Error . . . 52
5.2.1.1 Tolerane Analysis AgainstPathError . . . 52
5.2.1.2 Method Validation . . . 55
5.2.1.3 Tolerane Analysis AgainstSubstrate Error . . . 63
5.3 Feed Error . . . 63
5.3.1 MathematialRepresentation of Feed LoationDistortion. 67 5.3.2 IA-based Approah . . . 71
5.3.3 NumerialResults . . . 72
5.3.3.1 ComparativeAssessment . . . 72
5.3.3.2 Tolerane Analysis Feed Error . . . 72
5.3.3.3 PerformaneAnalysisVersusDierentF oal-length-to-diameterRatio
(
F/D
)
. . . 765.1 AnalysisoftheIA-based patternpreditionvs. pathwidtherrors
in
u
= 0
andv
= 0
planes,△
w
=
{
5
,
10
,
20
,
50
}{
µm
}
- Interval pattern features[
p
(
u, v
)
, SLL, BW
]
and patterntolerane index∆
575.2 AnalysisoftheIA-basedpatternpreditionvs. pathlengtherrors
in
u
= 0
andv
= 0
planes,△
l
=
{
5
,
10
,
20
,
50
}{
µm
}
- Interval pattern features[
p
(
u, v
)
, SLL, BW
]
and patterntolerane index∆
605.3 Analysis vs. bandwidth
f
=
{
28
.
5
,
29
.
25
,
30
.
75
,
31
.
5
}
for path length errors inv
= 0
plane,△
l
=
{
10
}{
µm
}
- Interval pattern features[
p
(
u, v
)
, SLL, BW
]
and pattern tolerane index∆
. . . . 605.4 Analysis vs. bandwidth
f
=
{
28
,
5
,
29
.
25
,
30
.
75
,
31
.
5
}
for path length errors inv
= 0
plane,△
l
=
{
10
}{
µm
}
- Interval pattern features[
p
(
u, v
)
, SLL, BW
]
and pattern tolerane index∆
. . . . 635.5 AnalysisoftheIA-basedpatternpreditionvs. substratethikness
errors in
u
= 0
andv
= 0
planes,△
d
=
{
5
,
10
,
20
,
50
}{
µm
}
-Intervalpatternfeatures[
p
(
u, v
)
, SLL, BW
]
andpatterntolerane index∆
. . . 665.6 Analysis of the IA-based patternpredition vs. dieletri
permit-tivityerrorsin
u
= 0
andv
= 0
planes,△
ε
=
{
0
.
003
,
0
.
005
,
0
.
007
}{
µm
}
-Intervalpatternfeatures[
p
(
u, v
)
, SLL, BW
]
andpatterntolerane index∆
. . . 665.7 Analysis of the IA-based
co
-polar pattern predition vs. feed displaement errors errors inu
= 0
andv
= 0
planes,△
z
f
=
{
λ/
200
,
100
,
50
,
20
,
10
}
-Intervalpatternfeatures[
p
(
u, v
)
, SLL, BW
]
and patterntolerane index
∆
. . . 745.8 Analysisofthe IA-based
co
−
polarpatternpreditionvs. feeddis-plaementerrorsin
u
= 0
andv
= 0
planes,△
x
f
=
{
λ/
200
,
100
,
50
,
20
,
10
}
-Intervalpattern features[
p
(
u, v
)]
. . . 765.9 Analysis of the IA-based
co
−
polar pattern predition vs. feed displaement errors errors inu
= 0
andv
= 0
planes,△
y
f
=
{
λ/
200
,
100
,
50
,
20
,
10
}
.Intervalpatternfeatures[
p
(
u, v
)
, SLL, BW
]
3.1 Sketh of the table and itsdimension . . . 18
3.2 Intervalend points . . . 20
3.3 Intervalmidpoint and width . . . 20
3.4 Intervalfuntion . . . 24
3.5 Inlusion Monotoniity . . . 25
3.6 Complex Interval Value. . . 28
3.7 WrappingEet of The Complex Interval. . . 30
3.8 Cartesian IntervalRepresentation . . . 31
3.9 PolarInterval Representation . . . 31
4.1 Sketh of the reetarray antennawith its parameters. . . 36
4.2 Dependeny assessment - inf and sup of the eetive eletrial length with dependeny and dependenyfree intervalfuntion. . . 41
4.3 Enumerativestrategy
(
a
)
Samplingofthe amplitudeof the ree-tion oeient(
b
)
Samplingof the reetion phase. . . 434.4
IA
-Minkowski approah -Minkowski Sum of two intervalphasors. 45 5.1 Phase distribution on the aperture surfae for reetarray with 749 elements . . . 495.2 H-Plane radiationpatternwith omparison with Karnati2014 . . 50
5.3 E-Plane radiationpattern with omparison with Karnati2014 . . 51
5.4 Phase distributionon the aperture surfae . . . 53
5.5 Phase behaviourversuspathlengthinretangularmirostrippath 54 5.6 Comparative Assessment
△
w
= 50
µm
; Plot of the interval power patternpredited withtheIA
−
CS
, theIA
−
CS
∗
,IA
−
ENUM
,IA
−
ENUM
−
MS
together with the nominal power patter in H-Plane(
φ
= 0
◦
)
. . . 555.7 Method Validation - Comparison of
IA
−
CS
,IA
−
ENUM
,IA
−
ENUM
−
MS IA
−
CS
∗
, together with the Monte Carlo patterns△
w
= 50
µm
. . . 565.9 Inlusionpropertyvalidation againstpathlengtherror-Nominal
powerpatternand
IA
−
ENUM
−
MS
intervalpowerpatternfor△
l
=
{
5
,
10
,
20
,
50
}
(
a
)
inv
= 0
plane(
b
)
inu
= 0
plane. . . 59 5.10 Analysis versus steering angle, IA-pattern features(
a
)
P
(
u
0
) (
b
)
SLL
(
c
)
BW against mainlobe steering angle. . . 61 5.11 Analysisversusfrequeny, IA-patternfeatures(
a
)
P
(
u
0
)(
b
)
SLL(
c
)
BWagainstfrequeny
(
a
)
f
= 28
.
5 (
b
)
f
= 29
.
25 (
c
)
f
= 30
.
75(
d
)
f
=
31
.
5
. . . 625.12 Inlusionproperty validation againstpathdieletri permittivity
error - Nominal power pattern and
IA
−
ENUM
−
MS
interval powerpatternfor△
ε
=
{
0
.
003
,
0
.
005
,
0
.
007
}
(
a
)
inv
= 0
plane(
b
)
inu
= 0
plane. . . 64 5.13 Inlusionpropertyvalidation againstpathsubstrate thiknesser-ror - Nominal power pattern and
IA
−
ENUM
−
MS
interval power pattern for△
d
=
{
5
,
10
,
20
,
50
}
(
a
)
inv
= 0
plane(
b
)
inu
= 0
plane. . . 655.14 Antenna struture with the feed displaement . . . 68
5.15 Inlusionpropertyvalidation againstfeedloationerror-Nominal
power pattern and
IA
−
MS co
-polar interval power pattern for△
z
f
=
{
λ/
200
,
100
,
50
,
20
,
10
}
(
a
)
inv
= 0
plane(
b
)
inu
= 0
plane. 735.16 MonteCarlopowerpatternover
IA
−
MS
bounds with△
z
f
=
λ/
20
. 74 5.17 Inlusionpropertyvalidation againstfeedloationerror-Nominalpower pattern and
IA
−
MS co
-polar interval power pattern for△
x
f
=
{
λ/
200
,
100
,
50
,
20
,
10
}
(
a
)
inv
= 0
plane(
b
)
inu
= 0
plane. 75 5.18 Inlusionpropertyvalidation againstfeedloationerror-Nominalpower pattern and
IA
−
MS co
-polar interval power pattern for△
y
f
=
{
λ/
200
,
100
,
50
,
20
,
10
}
(
a
)
inv
= 0
plane(
b
)
inu
= 0
plane. 77 5.19 Analysis of the IA-basedco
−
polar pattern predition vs. feedThe thesisis struturedinhaptersaording tothe organizationdetailedin
the following.
The rsthapter(hapter1)deals withanintrodutiontothe thesisand the
state-of-the-artinvestigation, fousingonthe introdutory remarks on
reetar-rays andthe mainmotivationsof usingintervalanalysis fortoleraneanalysis of
the reetarray antenna.
Chapter 2 provides the dierent approahes used for the analysis of the
ra-diation pattern of reetarrays , fousing on the Aperture Field Method and
Floquet model expansions. Tehniques for omputing the phase distribution on
the reetarray aperture surfae are provided in this hapter. Approahes for
designing the unit ellis alsoovered inthis hapter. The analytialexpression
for the reetion oeient is explained. The radiation patternand its relation
tothe physial parameters of the unit ellis expressed.
Chapter 3isdevoted tothe fundamentalofthe IntervalAnalysismethod,
fo-usingonthedenition,propertiesandthekeyfeaturesoftheintervalarithmeti
rules. Interval funtions with inlusion theory are dened. Complex interval is
explained. The twomain problems of Dependeny and Wrapping relatedto the
useofIntervalAnalysisandArithmetiofIntervalsinomplexstruturearefully
explained. These problems produe the redundany in the interval bounds.
Re-formulatingoftheintervalexpressionforsolvingthedependenyeetisproperly
determined.
Mathematial formulation of the reetarray analysis and itsinterval
exten-sion are desribed in hapter 4. A mirostrip reetarray antenna is onsidered
as an illustration. Then Aperture Field method together with Floquet model
onsider the eet of the geometrial parameters error on the radiation
perfor-mane of the antenna, interval analysis is applied. This hapter deals with the
interval extension of the reetarray radiation pattern expression. Geometrial
parameters suh path length, width, substrate thikness and dieletri
permit-tivity aremodeled withintervalvalues, thenthe intervalfuntionof the fareld
isomputed. Dierent tehniques foreliminatingdependenyeet in
reetar-rayantennaformulationsareexpressed. MinkowskiSumapproahtoremovethe
wrapping eet is explained.
Several representativeresultsare presented inhapter5. The inlusion
prop-erties are heked by omparing the resulting bounds with the Monte Carlo
simulation results. A omparative study heks the improvement of Interval
EnumerativeMinkowski(
IA
−
ENUM
−
MS
)toIntervalCartesian(IA
−
CS
). Pattern feature analysis versus steering angle, bandwidth are assessed andde-sribed. The eet of feed displaement errors on the radiation performane of
the reetarrays is alsoonsidered inthis hapter. The bounds of the deviation
asaresultof theaxialerrorsareomputedby IntervalMinkowskimethods. The
inlusionpropertiesare heked by omparingthe resultswith the MonteCarlo
simulation results. Analysisversus dierent foal-length-to-diameter
(
F/D
)
ra-tios in dierent tolerane errors in the feed positions is evaluated. The resultsare explained and ompared together.
Introdution and State-of-the-Art
In the introdution, the motivation of the thesis is pointed out starting from a
brief overview on the reetarrays and tehniques presented in the
appliations. Reetor and array antennas are traditionallytwo main antennas
for highgain appliations [1℄. Curve surfae of the reetor antennas makes the
manufaturing proess more diult. Furthermore, high mass and volume of
reetor antenna inrease the launhost speially inspae ommuniation. In
reentyears, phasedarrayantennashavebeenused asanappropriateoptionfor
satelliteommuniationdue to the advantages of low prole,low ost,lowmass
and high gain radiation patterns [2℄. Despite the previously mentioned
advan-tages,thefeedingsystemofthe phasedarrayantennasisquiteompliated. The
most ideal antennas for spae ommuniation are the ones whih an ombine
the best features of the reetor and array antennas. Over the past few years,
reetarray antennas have proved tobean exellent alternativetoreetor and
arrayantennas. Reetarrayantennasrstintroduedin1960sbyBerry,Maleh
and Kennedy [3℄. They were short-endedwaveguide withvariable-length
waveg-uide. Then in mid-1970s spiraphase reetarray was presented by Phelan [4℄.
In the 1980s, mirostrip reetarray antennas were developed [5℄.
Favorablefeaturesoflowprole,lowmass,lowostandhigheienyaswell
astheabilityofbeingfoldedinspaehavemadethe reetarraysthemost
appli-ableantennasforspae ommuniations [6℄,[7℄. Reetarray antennastruture
inludeseveralradiatingelementsloatedinareetivesurfaewhihare
illumi-natedby a feedantenna. Mirostrippathes, dipoles andrings are the radiating
elements in the reetarray antenna [8℄, [9℄ . These radiating elements produe
the required phases to form a planar phase front in the far-eld [10℄. Dierent
approahes an be used to produe the required phases[13 ℄. These approahes
arevariablephase delay linesattahedtoelement[11℄,variable-sizepathes [12℄,
dipoles orrings and the elementrotations. Among themvariable-sizeapproah
has the disadvantages of the the limited realizablephase range. The ahievable
phase range by this approah is less that
360
[deg℄. This unattainable phaserange leads to phase error. To take a phase variation near to
360
[deg℄, pathsize should hange signiantly about 40 perent whih leads to an ineieny
[10℄. By using element rotation better eieny ould be obtained due to the
lak of speular reetion of the o-broadside inident rays [10℄. Despite of all
advantagesofthe reetarray antenna,ithasone maindisadvantagewhihisits
narrowbandwidth. Itisusuallybeyondtenperentdependingonitselement
de-sign,aperturesize andthefoallength[14℄. Thisbandwidthismainlylimitedby
the element geometry and dierential spatial phase delay. For ahieving wider
bandwidth thik substrate, staking multiple pathes and sequentially rotated
subarrays areproposed. More than15perentbandwidth isgainedby these
ap-proahes [15℄, [16℄. Reetarray with larger foal-length-to-diameter (f/D) ratio
havewiderbandwidth. Curvedreetarraywithpieewiseatsurfaeshaslarger
bandwidththan aatreetarrayantenna. Despite ofthe bandwidthlimitation
for reetarray antenna are still an ongoing proess. Several development and
innovation tehniques are used in reetarray antenna design whih is worth to
mention. Using multi-layer staked path inrease the bandwidth from a few
perent toten perent [15℄, [16℄. Thisstruture an improvethe phase rangefar
in exess of
360
[deg℄. By varying the dimensions of three staked pathes, over600 [deg℄ phase ranges an be ahieved. In [17℄, the eletrially largest
ree-tarray in the mirowave and millimeter-wavespetra is introdued. It is a 3-m
Ka-bandirularlypolarizedinatablereetarrayonsistingof200000elements.
In [3℄, an amplifying reetarray antenna was developed in whih eah element
reeives the signal from feed, then goes to amplier and retransmit the signal.
It angiveveryhigh equivalentisotropiradiatedpower. Another improvement
in reetarray antenna design is applying optimization algorithm to synthesis
theantennapattern. Thereare dierentparametersinreetarrayantennasuh
as substrate thikness, path size, inident angle, main beam and bandwidth.
These parameters an be optimized inorder toahieve the high gain, eieny
and the diretivity. Geneti Algorithm (GA)and Partile Swarm Optimization
(PSO) are properly used in[20℄ and [19℄ tosynthesize the reetarray antenna.
A novelreetarray antenna integrated with solar ells forsatellite
ommunia-tion is developed in [21℄. Over all of these innovations and developments, there
is one main hallenge whih is not onsidered eiently in reetarray antenna
design. Reetarrayantenna an beaetedby surfae deviationsandthe
man-ufaturingtoleranes due toits reetion mehanism, path dimensions and the
eletrial phases [10℄. The phase response of the path element in mirostrip
reetarray depends on itsphysial parameters [22℄. Due to the inauraiesin
manufaturingproess, the dimension of the single element and the position of
the feed deviate from their atual values. This deviation auses a onsiderable
hange in phase response of the single element and eventually the degradation
in the radiation pattern. To derease the sensitivity toward manufaturing
er-rors, two layer struture is suggested. As an example, a tolerane error of 0.1
mminpath dimension willprodue only6.5 error inphase, whihindiate the
low sensitivity to manufaturing toleranes rather than a single layer
reetar-ray antenna [10℄. The eet of manufaturing errors is more sensible when the
frequeny is inreasing.
To improve the robustness of the system, any hange in the phase response
shouldbeavoided. Dierentmehanismshavebeenappliedtoestimatethephase
errors and the pattern degradation. Tolerane analysis has been applied to the
reetarray antennainthe workof Pozar etal. [24℄ wherestatistialapproahes
areimplemented toestimatethedeviationinphaseresponse ofmirostrip path
elementwhiletheroot-mean-squareerrorofpathdimensionsareknown. In[23℄,
numerialanalysis is presented to ompute radiationdisrepany of metalli
re-etarray antenna experiening manufaturing distortion at millimeter waves.
Errors are modeled with normal distribution. Sine reetarray antennas are
unavoid-position ofthe elementdue to mehanial errors[25℄. A probabilistianalysis is
exploited in[26℄ and [27℄ to alulatethe maximum tolerane inarray elements
tosatisfythespeionstrains. In[47℄,toleraneanalysisbasedonMonteCarlo
method isexploited topredit the eets of errors inthe exitation and the
po-sition of eah element. The above-mentioned methods are based on statistial
approahesinwhihthea-prioriknowledgeofthe errordistributionisneessary.
The main problem assoiated with Monte Carlo method is the lengthy
ompu-tations of the innite number of error ombinations. Sine it is not plausible
to realize the innite number of errors, the Monte Carlo results are not totally
reliable[47℄.
To overome the urrent limitations of statistial approahes in tolerane
analysis, Arithmeti Interval is applied to perform operations between interval
values [29℄. Interval Analysis was rst used to solve the linear and nonlinear
funtions[29℄ and optimizationproblems [30℄. Its usage ineletromagneti eld
was initiated with the robust design of the magneti devies [31℄ and reliable
systems for target traking radar [32℄. Reently Interval Analysis was used to
model the manufaturing toleranes in exitation and position of linear array
antenna. Interval arithmeti was then exploited to ompute the bounds of the
radiationpatterndegradationovertheintervalerrors[33℄. Alosedform
expres-sion has been presented forthe upper and lowerboundsof the powerpattern in
the reetor antennawith bump-likesurfae by the featuresof IntervalAnalysis
and the rules of arithmeti for intervals [49℄. Aording to the state of the art,
thesignianeofapplyingmanufaturingtoleraneanalysisinantennadesignis
quitewell-known. In orderto apply intervalanalysis inreetarrayantenna, we
need toapply proper analysis method. In the next hapter, dierent tehniques
Radiation Analysis for Reetarray
Antenna
In this hapter dierent approahes for analysis of the radiation pattern of the
reetarrayantennaarepresented. Comparativestudiesamongtheseapproahes
are provided. There are several approximations for feed antenna pattern and
the element reetion oeient. These approximations are explained in this
setion. The aurate method for analysis of the o- and ross-omponents of
2.1 Introdution
To ompute the radiation harateristi of the reetarray antenna, dierent
approahes suh as Array-Theory method and Aperture-Field method an be
applied. Advantages and disadvantages of using these methodsare desribed in
this setion. One of the mostruial partof the reetarray analysisand design
istheaurateevaluationoftheunitellelementwhihprovidearequired
phase-shift. Thephase shiftdistributionofthe reetarray surfae andunit elldesign
are lariedin this setion.
2.2 Overview of Analysis Tehniques
2.2.1 Array-Theory Method
Conventional array theory is applied to ompute the far eld radiation pattern
ofthereetarrayantenna. Consideringthearrayantennawith
M
∗
N
elements. The total eletri eld of the array antenna is the multipliationof the elementpatternand the elementexitation as[35℄:
E
(ˆ
v
) =
M
X
m
=1
N
X
n
=1
−
→
b
mn
(ˆ
v
)
• −
→
a
mn
(
−
→
r
mn
)
,
(2.1)ˆ
v
= ˆ
xsinθcosφ
+ ˆ
ysinθsinφ
+ ˆ
zcosθ
(2.2)Where
−
→
r
mn
is the position vetor andbmn
,amn
are the element fator and the exitation vetor funtion, respetively. For the sake of simpliity elementfatorsand the exitationvetor areapproximatedby salarfuntions. Aosine
q
modelis onsidered for the element patternas:bmn
(ˆ
v
)
≈
cos
qe
(
θ
)
e
jk
(
−
→
r
mn
•
ˆ
v
)
(2.3)
The exitationvetor
−
→
a
mn
is approximated as:amn
≈
cos
qf
θf
(
m, n
)
|−
→
r
mn
− −
→
r
f
|
e
−
jk
(
|−
→
r
mn
−−
→
r
f
|
)
|
Γ
mn
|
e
jφmn
(2.4)The element exitation is the multipliation of the feed-horn pattern funtion
and the reeiving mode pattern of the element
(Γ
mn
)
. Feed horn pattern is approximated by osineq
model and taking into aount the distane between the feed horn and the element.θf
is the spherialangel and−
→
r
f
is the position vetor offeed. The reeiving mode patternof the elementis as follows:E
(
θ, φ
) =
M
X
m
=1
N
X
n
=1
cos
qe
θ
cos
qf
θf
(
m, n
)
|−
→
r
mn
− −
→
r
f
|
e
−
jk
(
|−
→
r
mn
−−
→
r
f
|−−
→
r
mn
•
v
ˆ
)
cos
qe
θe
(
m, n
)
e
jφmn
(2.6)
where
φmn
is the required phase delay of mn-th
element.Advantages and disadvantages of this methodare asfollows [35℄:
•
Advantage: simpliityof the formulationand the programdevelopment•
Disadvantage: the ross-polarizationis not modeled.2.2.2 Aperture-Field Method
In this method,rst the tangentialeletrield onthe aperture surfae is
om-putedbyonsideringthepolarizationoftheeldhorn. Ahornantennaisusually
usedas afeedinthereetarray antenna. Theradiationpatternof thehorn
an-tenna isgiven [10℄:
For a
x
-polarized feedE
F x
(
θ, φ
) =
jke
−
jkr
2
πr
(ˆ
θCE
(
θ
)
cosφ
−
φCH
ˆ
(
θ
)
sinφ
)
(2.7) For ay
-polarizedfeedE
F y
(
θ, φ
) =
jke
−
jkr
2
πr
(ˆ
θCE
(
θ
)
sinφ
+ ˆ
φCH
(
θ
)
cosφ
)
(2.8)CH
andCE
are theH-plane andE-planeradiationpatterns ofthe hornantenna.Theyaremodeledas
cos
q
(
θ
)
funtions.
q
isthevaluewhihisomputedfromthe aperture eieny and the feedhorn data. In (2.7) and (2.8), the radiatedeldof the feed in the spherial oordinate is omputed. The spherial omponents
of the eletri eld is transformed to Cartesian omponents from the following
matrix transformation.
E
F
x
E
F
y
E
F
z
=
sinθsinφ cosθsinφ
sinθcosφ cosθcosφ
−
cosφ
sinφ
cosθ
−
sinθ
E
θ
F
E
F
φ
Thenthis omponents should onvert fromfeed oordinatesystem tothe
ree-tarray oordinatesystem by a propertransformation matrix.
2.2.3 Radiation Patterns
After omputing the tangential eletri eld, the radiated far eld is obtained
by anasymptotievaluationofthe integrals. Theradiatedfar eldare asfollow
E
(
θ, φ
) =
jk
[(ˆ
θcosφ
−
φsinφcosθ
ˆ
) ˜
ERx
(
u, v
) + (ˆ
θsinφ
+ ˆ
φcosφcosθ
) ˜
ERy
(
u, v
)]
e
−
jkr
2
πr
(2.9)
Where
E
˜
Rx
(
u, v
)
andE
˜
Ry
(
u, v
)
are the Fourier transformof the Cartesian om-ponents of the tangential eletri eldERx
(
u, v
)
andERy
(
u, v
)
, expressed as follows:˜
ERx/y
(
u, v
) =
Z Z
RA
ERx/y
(
x, y
)
e
jk
0
(
ux
+
vy
)
dxdy
(2.10)
Where
u
andv
are the angular oordinates as:u
=
sinθcosφ
(2.11)v
=
sinθsinφ
(2.12)To ompute the (2.10) element by element,variablehange inthe oordinateis
used for the oordinate
(
x, y
)
:x
=
x
′
+
mp
x
−
(
N
x
−
1)
p
x
2
;
m
= 0
,
1
,
2
, ..., N
x
−
1
(2.13)y
=
y
′
+
np
y
−
(
Ny
−
1)
py
2
;
n
= 0
,
1
,
2
, ..., N
y
−
1
(2.14)Central point of the element
(
m, n
)
are(
mpx
−
(
Nx
−
1)
px
2
, npy
−
(
Ny
−
1)
py
2
)
.x
′
and
y
′
are withingthe following bounds [10℄:
−
px
2
≤
x
′
≤
px
2
(2.15)−
py
2
≤
y
′
≤
py
2
(2.16)where
px
andpy
are theperiodiityalongx
andy
,diretions. Maximumnumber of element inx
andy
diretion areNx
andNy
, respetively. By substituting (2.13),(2.14) in (2.10),the spetral funtion is as:˜
ERx/y
(
u, v
) =
K
1
M
X
−
1
m
=0
N
X
−
1
n
=0
[
e
jk
0
(
umpx
+
vnpy
)
Z
−
px
2
−
px
2
Z
py
2
−
py
2
E
Rx/y
m,n
(
x
′
, y
′
)
e
jk
0
(
ux
′
+
vy
′
)
dx
′
dy
′
]
(2.17)
K
1
=
e
−
j
k
0
2
[
u
(
M
−
1)
dx
+
v
(
N
−
1)
dy
]
(2.18)Tangential eld omponents in eah ell of the reetarray is shown with the
omplex oeient ofthe reeted eld.
E
Rx/y
m,n
(
x
′
, y
′
) =
ax/Y
(
m, n
) =
Ax/y
(
m, n
)
e
jφ
x/y
(
m,n
)
(2.19)
Bysubstituting(2.19)in(2.17),theintegrationanbewrittenbythesummation
as:
˜
ERx/y
(
u, v
) =
K
1
pxpy
sinc
(
k
0
upx
2
)
sinc
(
k
0
vpy
2
)
M
X
m
=0
N
X
n
=0
Ax/y
(
m, n
)
e
jφ
x/y
(
m,n
)
e
jk
0
(
umdx
+
vndy
)
(2.20)
In every reetarray element, Cartesian omponents of the reeted and the
inident eld are relatedto eahother by sattering matrix as:
a
x
(
m, n
)
ay
(
m, n
)
=
S
11
S
12
S
21
S
22
E
F
x
E
F
y
(2.21)This sattering matrix an beomputed by the Method of Moment inthe
spe-tral domain. The element of the sattering matrix an also be replaed by the
reetion oeient.
ax
(
m, n
)
ay
(
m, n
)
=
Γ
xx
Γ
xy
Γ
yx
Γ
yy
E
F
x
E
F
y
(2.22)As it is obviousin (2.21) and (2.22), sattering matrix and the reetion
oe-ientare the mainparts for omputing the radiationpattern. These oeients
relate to element performane of the unit ell. In the following part, dierent
tehniques for analysis of the unit ellelement are proposed.
2.2.4 Unit Cell Design
Eahelementinreetarrayantennashouldproduetherequiredphasesinorder
toompensatedierent spatialdistanes fromfeed toelement. This phase shift
distributionis omputed fromthe following expressions.
2.2.4.1 Phase-Shift Distribution Tehnique
Eahelementmustprodueaphase-shifttoprovideaollimatedbeaminagiven
diretion. From the array theory, the phase distribution to produe a beam in
the main beam diretion
(
θb
, φb
)
isas [10℄:(
xmn, ymn
)
is the loation of the mn-th
element in the reetarray surfae.k
0
is the propagation onstant. Phase of the reeted eletri eld an also be
omputed from the following way. Phase of the reeted eld is equal to the
phase of the inident eld plus phase-shiftprodued by eah element as:
φ
(
xmn, ymn
) =
−
k
0
Rmn
+
φR
(
xmn, ymn
)
(2.24)R
mn
is the distane from phase enter of the feed toelement.φ
R
(
x
mn
, y
mn
)
is aphase-shift for mn-
th
element. From (2.23) and (2.24), the required phase shift for eah element is:φR
(
xmn, ymn
) =
k
0(
Rmn
−
(
xmncosφb
+
ymnsinφb
)
sinθb
)
(2.25) In reetarray antenna phase of the reetion oeient should hange inordertomaththesephases. Dierenttehniques anbeused toprovidethesephases.
These tehniques are asfollows:
•
Connetingvariable-lengthstubesto element•
Path with variable sizes•
Element loadedwith MEMS, varators and liquidristal polymersPerformaneoftheantennaelementinreetarrayantennarelatedtoitsphysial
parameters suh as
•
path dimensions length (l
) and width(w
)•
substrate thikness (d
)•
dieletri onstant (εr
)•
dieletri and ondutor losses (tanδ
,σ
)•
spaingbetween elements (px, py
)In the following setion, reent tehniques for analysis and design of the single
element are provided.
2.2.4.2 Tehniques for Analysis of the Unit Cell
In this setion, dierentmethodsfor analysis of the unit ell elements are
men-tioned. Andthemostreenttheoretialmethodforanalysisofthesingleelement
is explained with more details. Tehniques of the analysis of the single element
are as follows:
•
Commerialpakages suh as HFSS,CST, FEKO•
Theoretialmodel basedon full-wavesimulations•
Analytial approahbased onQ fator analysisSine in this thesis we want to ompute the radiation pattern analytialy,
ana-lytialapproahbasedonQfatorisexplainedwithmoredetailsinthefollowing
part.
2.2.4.3 Analytial Approah Based on Q Fator Analysis
A omplete analytial approah based on Q fator is introdued to extrat the
reetionpropertiesoftheunitell. Inordertoextratthelosed-formformulas,
we need to express the inident and the reeted eld in terms of orthogonal
Floquet modes. Let us provide a short explanation about Floquet harmonis
and itsexpression.
2.2.4.4 Floquet Harmonis
An arbitrary inident eld an be dened as a summation of the TE and TM
Floquet spae harmonis with omplex amplitude. The inident eletri eld
propagatingtoward -
z
with Floquet harmoni expansionis writtenas [10℄:E
1
i
=
2
L
X
l
=1
d
1
e
1
exp
(
j
(
kxmx
+
kyny
+
kzlz
))
(2.26)The transverse reeted eletri eld propagate toward
z
with the Floquet har-moni expansionis:E
1
r
=
2
L
X
l
=1
a
1
e
1
exp
(
j
(
kxmx
+
kyny
−
kzlz
))
(2.27)The normalized modalelds for TE and TM Floquet harmonis
e
l
are as: for (T E
)1
≤
l
≤
L
el
=
1
kcl
(
−
kyn
x
ˆ
+
kxm
y
ˆ
)
(2.28) for (T M
)L
+ 1
≤
l
≤
2
L
e
l
=
1
kcl
(
k
xm
x
ˆ
+
k
yn
y
ˆ
)
(2.29)with
kcl
=
q
k
2
xm
+
k
yn
2
(2.30)k
xm
=
k
0
sinθcosφ
+
2
mπ
px
=
k
x
0
+
2
mπ
k
yn
=
k
0
sinθcosφ
+
2
nπ
py
=
k
y
0
+
2
nπ
py
(2.32)Complex amplitude of the inident eld are
(
d
x
)
and(
d
y
)
. They an be on-verted todl
(
T E
)
anddL
+1(
T M
)
omponentsby a matrix transformation. This transformationis as follows:dl
dL
+1
=
1
k
cl
−
ky
0
kx
0
kx
0
ky
0
dx
dy
(2.33)
Similarly,the
T E
/T M
omponentsof the reeted eldal
(
T E
)
andaL
+1(
T M
)
an onvert tox/y
omponents as:ax
ay
=
1
kcl
−
ky
0
kx
0
kx
0
ky
0
al
aL
+1
(2.34)
TheinidentandreetedeldsofthereetarrayarerepresentedbyFloquet
modes. They are onneted to eah other by the reetion oeient. The
expression for the reetion oeient is explained using oupled-mode theory.
Theoretialexpressionforthereetionoeientaredesribed inthenextpart.
2.2.4.5 Theoretial Expression for The Reetion Coeient
In order toextrat the reetion oeient expression for a single element, the
antennaelementisonsideredinsideawaveguidesupportingFloquet modes. By
onsidering the waveguide with two orthogonal polarized fundamental Floquet
modes(
T E
00
,T M
00
),theo-ouplingandross-ouplingreetionoeientare asfollows [22℄:When the inident eld is TE , the fration of the reeted power into TE
mode is alledTE o-oupledreetion oeient with the followingexpression
[22℄:
Γ
T Eco
(
f
) =
1
QradT E
−
(
1
QradT M
+
1
Q
0
)
−
2
j
(
f
−
f
0
)
f
0
1
QradT E
+
1
Q
radT M
+
1
Q
0
+
2
j
(
f
−
f
0
)
f
0
(2.35)
whentheinidenteldisTM,the frationofthe reetedpowerintoTMmode
isalled TM o-oupled reetion oeient with the following expression [22℄:
Γ
T M co
(
f
) =
1
QradT M
−
(
1
QradT E
+
1
Q
0
)
−
2
j
(
f
−
f
0
)
f
0
1
QradT E
+
1
QradT M
+
1
Q
0
+
2
j
(
f
−
f
0
)
f
0
(2.36)
Cross-oupling is a fration of the reeted power into
(
T E/T M
)
when the in-ident eld is(
T M/T E
)
. The ross-oupling expression is as[22℄:Γ
cross
(
f
) =
2
√
QradT EQradT M
1
QradT E
+
1
Q
radT M
+
1
Q
0
+
2
j
(
f
−
f
0
)
f
0
where,
QradT E
andQradT M
are the radiationQ
fators for theT E
andT M
modes, respetively.f
is the working frequeny.f
0
is the resonant frequeny.Q
0
isa ombinedQ
fators of ondutor and dieletri losses as:Qc
=
d
p
πf µσ
(2.38)Qd
=
1
tanδ
(2.39)Q
0
=
QcQd
Qc
+
Qd
(2.40)where substrate thikness is
(
d
)
and the loss tangent istanδ
. The eet of the inidentangleandthephysialparametersareobviousintheQ
fatorexpression. As anexample, the losed formexpression for a retangularmirostrip pathinterm of unit ell's physialparameters and the inidentangle as [22℄:
QradT E
=
f
0
πε
4
d
l
w
pxpy
η
0
cosθcos
2
φ
(2.41)QradT M
=
f
0
πε
4
d
l
w
pxpy
η
0
cosθ
sin
2
φ
(2.42)These expressions help us to properly investigate the eet of physial
pa-rameters errors on the reetion oeient and the radiation pattern. By
sub-stituting (5.24),(2.42), (2.38), (2.39) and (2.40) in (5.22),(2.36)and (2.37), the
reetion oeient of the unit ellare obtained.
If we onsider inident eld of the reetarray in terms of the
T E
andT M
Floquet harmonis (2.22), the reetedT E
andT M
omponentsrelated to the inident Floquet harmonis by the reetion oeients as:E
T E
ref
E
T M
ref
=
Γ
T Eco
Γ
cross
Γ
cross
Γ
T M co
E
T E
inc
E
T M
inc
(2.43)
Bysubstituting(2.43) in(2.22)andonsequently inthe(2.17) and(2.9), the
total power pattern expression an be ahieved. By using analytial expression
for the reetion oeient,the analytiallyexpression for the total power
pat-tern is obtained. In this total power pattern, the diret relation between the
radiation pattern and the physial parameters are desribed. The physial
pa-rameters deviate from their nominal values due to manufaturingunertainties.
This random manufaturing unertainties produe dierent radiation patterns.
Ourgoalistodeneaneientstrategytoomputeinlusivepatternboundsfor
a given maximum toleranes on the reetarray geometrial parameters. After
providingthe analytialexpression for the radiationpattern,the tolerane
anal-ysis shouldbe appliedinorder toompute the powerpattern deviationbounds.
tools for the tolerane analysis of the antenna. In the next hapter, Interval
Fundamentals of Interval Analysis
In this hapter, an overview for learning Interval Analysis is introdued. First
the interval values and their need in our real life are desribed by examples.
Then Interval Analysis approahes are dened. The rules for the arithmeti
Intervaloperationssuh asaddition,subtration,multipliationanddivision are
provided. Then the properties of Interval Arithmeti are presented. Interval
funtion and their features are fully desribed in this hapter. Dependeny and
3.1 Introdution
Unertainty is part of our every-day life. The need to enlose the number is
obviousinmanydierentappliations.Inthefollowingexample,wewanttoshow
the appearane of the unertainty inour daily life.
Suppose we want tomeasure the dimension of the table. The table with the
dimensions isshown inthe Fig (3.1).
l
w
Figure3.1: Sketh of the tableand itsdimension
Dierent measurement instruments suh as tailor or aliper an be used to
measure the dimensions. Firstweuse aatailor tomeasure thedimensions. The
dimension are shown as follows:
l
= 1
.
2 +
/
−
0
.
1
m
(3.1)w
= 0
.
8 +
/
−
0
.
1
m
(3.2) Thenwith aaliperormore preisedevies, thedimension values areasfollows:l
= 1
.
20
m
+
/
−
0
.
01
m
(3.3)w
= 0
.
80
m
+
/
−
0
.
01
m
(3.4) Asitislearfromthemeasurement,unertainty isalwaysavailableregardlessofthe auray of the instrument. By using aliper, the orret length/width lies
withinertain ranges.
0
.
19
< l <
1
.
21
→
l
∈
[1
.
19
,
1
.
21]
(3.5)Clearly, the length and width belongs to the intervals. As it is mentioned
before, unertainty isunavoidablepart ofour life. We need toapply the proper
mathematialtoolstodealwithunertainvalues. Iwouldliketoexplaintheneed
forIntervalAnalysisby theotherexample fromphysialsiene. Byonsidering
Newtons lawas [36℄:
F
=
ma
(3.7)Ifthe quantity of the fore
F
and the massm
liein the ertainranges as:F
0
−
∆
F
≤
F
≤
F
0
+ ∆
F
(3.8)Then the aeleration
a
isdened with the bounds as follows:al
≤
a
≤
au
(3.9)lower and upper range of the the aeleration (
a
l
,a
u
) depends onF
0
,m
0
,∆
F
,∆
m
. Sine the quantity of fore is not an exat value and an be dened in a ertain range then the aelerations is also desribe within the range whoseupper and lower bounds depends on the upper and lower bounds of the fore.
Oneofthe strongmathematialtooltoopewithunertainworld istheInterval
Analysismethod. Inthefollowingsetion,thedenitionofintervalnumbers and
the intervalarithmeti rules willbedesribed.
3.2 Interval Analysis
A real interval
[
X
]
is a non-empty ompat set of real numbers between and inludingthe endpointsxinf
andxsup
[36℄.[
X
] =
{
x
∈
R
:
xinf
≤
x
≤
xsup
}
(3.10)As it is shown in Fig (3.2), left end point inmum of
[
X
]
and the right end point supremum of[
X
]
are the maximum and minimum of all points in the interval:xinf
=
min
{
x
∈
[
X
]
}
(3.11)inf
x
R
0
[
]
sup
x
Figure 3.2: Interval end points
Twointervals
[
X
]
and[
Y
]
are equaliftheirendpointsareequal. Theabsolute valueoftheinterval|
[
X
]
|
isthe maximumofthe absolutevaluesof itsendpoints.|
[
X
]
|
=
max
{|
xinf
|
,
|
xsup
|}
(3.13)Width and the midpoint of the interval
[
X
]
are shown in gure (3.3). The denition of the width of the interval[
X
]
based onthe endpoints isas:w
([
X
]) =
xsup
−
xinf
(3.14)Midpointof
[
x
]
is relatedto the endpoints given as:m
([
X
]) =
x
inf
+
x
sup
2
Width of the interval with the intervalmid-pointare shown inFig. 3.3
inf
x
R
0
[
]
sup
x
m ([X])
w ([X])
m (
Figure3.3: Interval midpointand width
3.2.1 Interval Elementary Operations
Elementary operations an also be applied for the interval numbers. This
ele-mentary operations inlude sum, dierene, produt , inverse and the division.
Inintervaldomain,operationsaredealingwithsets thanavalue. Byperforming
an operation between two intervals, the resulting is a set ontaining all pairs
[
X
] + [
Y
] =
{
x
+
y
:
x
∈
[
X
]
, y
∈
[
Y
]
}
[
X
]
−
[
Y
] =
{
x
−
y
:
x
∈
[
X
]
, y
∈
[
Y
]
}
[
X
]
.
[
Y
] =
{
xy
:
x
∈
[
X
]
, y
∈
[
Y
]
}
[
X
]
[
Y
]
=
{
x
y
;
x
∈
[
X
]
, y
∈
[
Y
]
}
(3.15)
Whateverthe operation is,the resultingintervalenlose allthe possibleresults.
3.2.2 Interval Arithmeti
Interval arithmeti is aset of rules for performing elementary arithmeti
opera-tions onintervals.
Endpoint Formulas for Arithmeti of Intervals
Letusshowthe operationalformulas forthe elementaryoperationrelated to
boundary of intervals. For the sum of two intervals
[
X
]
and[
Y
]
, the operation is as[36℄:[
X
] + [
Y
] = [
xinf
+
yinf
, xsup
+
ysup
]
(3.16)The operationalformulafor interval subtrationin term of endpoints is as:
[
X
]
−
[
Y
] = [
xinf
−
ysup, xsup
−
yinf
]
(3.17)The relationof the produt of twointervals
[
X
]
and[
Y
]
to their endpoints is:[
X
][
Y
] =
[
min
(
xinf
yinf
, xinf
ysup, xsupyinf
, xsupysup
)
,
max
(
xinf
yinf
, xinf
ysup, xsupyinf
, xsupysup
)]
(3.18)The inverse of the interval is:
1
[
X
]
= [
1
xsup
,
1
xinf
]; 0
∈
/
[
X
]
(3.19)Divisionof twointervals an be aomplishedby using the multipliation of the
intervaland the inverse of the intervalas:
[
X
][
Y
] =
[
min
(
x
inf
/y
inf
, x
inf
/y
sup
, x
sup
/y
inf
, x
sup
/y
sup
)
,
max
(
xinf
/yinf
, xinf
/ysup, xsup/yinf
, xsup/ysup
)]
(3.20)0
∈
/
[
Y
]
Thekeyfeature ofthe operationisthat theoperationsinvolve theboundariesof
3.2.3 Properties of the Interval Arithmeti
3.2.4 Algebrai Properties
Interval addition and multipliation are ommutative and assoiative. Let us
onsider three intervals
[
X
]
and[
Y
]
and[
Z
]
, the ommutative and assoiative features are shown as [36℄:[
X
] + [
Y
] = [
Y
] + [
X
]
[
X
] + ([
Y
] + [
Z
]) = ([
X
] + [
Y
]) + [
Z
]
[
X
]
.
[
Y
] = [
Y
]
.
[
X
]
[
X
]
.
([
Y
]
.
[
Z
]) = ([
X
]
.
[
Y
])
.
[
Z
]
(3.21)
0
and1
are additive and multipliativeidentity elementin the intervaldomain.0 + [
X
] = [
X
] + 0
(3.22)1
.
[
X
] = [
X
]
.
1 = [
X
]
(3.23)0
.
[
X
] = [
X
]
.
0 = 0
(3.24)Inarealnumbers,
−
x
isanadditiveinverse forx
. Butthis isnot trueininterval domain. In intervalsystems forany interval[
X
]
,we have:[
X
] + (
−
[
X
]) = [
xinf
, xsup
] + [
−
xsup,
−
xinf
] = [
xinf
−
xsup, xsup
−
xinf
]
(3.25)If
x
xup
=
x
inf
then this equals[0
,
0]
. Otherwise[
X
]
−
[
X
] =
w
[
X
][
−
1
,
1]
(3.26) There isno multipliativeinverses exeptw
[
X
] = 0
,in generalwe have[
X
]
[
X
]
=
(
[
x
inf
xsup
,
xsup
xinf
]
if
0
< xinf
[
xsup
xinf
,
x
inf
xsup
]
if xsup
<
0
(3.27)
Forthe interval systems we havethe following inequality:
[
X
]([
Y
] + [
Z
])
6
= [
X
][
Y
] + [
X
][
Z
]
(3.28) This rule an be shown by onsidering threefollowing intervals as[
X
] = [1
,
2]
,
[
Y
] = [1
,
1]
,
[
Z
] =
−
[1
,
1]
(3.29) Left side of the (3.28) by onsidering the values of 3.29 isas:Whereas, the rightside by using intervalarithmeti rules is as:
[
X
][
Y
] + [
X
][
Z
] = [1
,
2]
.
[1
,
1]
−
[1
,
2]
.
[1
,
1] =
[
min
(1
,
2)
, max
(1
,
2)]
−
[
min
(1
,
2)
, max
(1
,
2)] = [1
,
2]
−
[1
,
1] = [
−
1
,
1]
(3.31)
As it is shown in (3.30) and (3.31), the right and left side are not equal. If
[
Y
][
Z
]
>
0
then[
X
]([
Y
] + [
Z
]) = [
X
][
Y
] + [
X
][
Z
]
holdtrue. Ingeneral,followingrule hold true for the interval.
[
X
]([
Y
] + [
Z
])
⊆
[
X
][
Y
] + [
X
][
Z
]
(3.32)A real number an multiplyto the summation of two intervalsas:
x
([
Y
] + [
Z
]) =
x
[
Y
] +
x
[
Z
]
(3.33)Canellationlawis alsovalidin intervalsystems:
[
X
] + [
Z
] = [
Y
] + [
Z
]
⇒
[
X
] = [
Y
]
(3.34)3.2.5 Inlusion Property of Interval Arithmeti
Ifweonsider twointervals
[
x
] = [
xinf
, xsup
]
,[
y
] = [
yinf
, ysup
]
andperformthe op-eration between two intervals,the resulting interval[
Z
] = [
X
]
op
[
Y
]
from what-ever operation between two intervals inludes all valuesz
=
{
xopy
}
(belongs to[
Z
]
(i.e.,z
∈
[
Z
]
))whih is the resultingz
=
{
xopy
}
value fromthe same opera-tion onthe real numbers of thex
∈
[
X
]
andy
∈
[
Y
]
.3.2.6 Interval Funtion
Intervalfuntionisanintervalvaluedfuntionofoneormoreintervalsarguments.
Letusonsider
f
asareal-valuedfuntion of avariablex
. The range off
(
x
)
asx
represent by interval[
X
]
is the interval funtion. In more general ase for agiven funtion
f
=
f
(
x
1
, ..., xn
)
of several variables,the intervaloff
is as:f
([
X
1]
, ...,
[
Xn
]) =
{
f
(
x
1
, ..., xn
) :
x
1
∈
[
X
1]
, ..., xn
∈
[
Xn
]
}
(3.35)x
f
(
x
)
f
([
X
]) = [
F
]
0
f
(
x
0
)
x
0
[
X
]
Figure3.4: Interval funtion
Let usonsider elementaryfuntion of intervalsby providingfollowing
fun-tion
f
(
x
) =
x
2
, if
[
X
] = [
x
inf
, x
sup
]
then the interval off
[
X
]
an be expressed as follows:f
([
X
]) =
[
x
2
inf
, s
2
sup
]
,
0
≤
x
inf
≤
x
sup
[
x
2
inf
, s
2
sup
]
, xinf
≤
xsup
≤
0
[0
, max
{
x
2
inf
, x
2
sup}
]
, xinf
≤
0
≤
xsup
(3.36)
Monotoni Funtions
If
f
(
x
)
is a monotoni, it maps the interval[
X
] = [
xinf
, xsup
]
into intervalf
([
X
]) = [
f
(
xinf
)
, f
(
xsup
)]
. As an example,f
(
x
) =
exp
(
x
) =
e
x
(
x
∈
R
)
then
exp
[
X
] = [
exp
(
xinf
)
, exp
(
xsup
)]
. Similarlyforlogarithmiisamonotonifuntionand itsinterval is,
f
(
x
) =
logx
(
x >
0)
log
[
X
] = [
logxinf
, logxsup
]
(3.37)The expression forthe square rootof interval isas:
p
[
X
] = [
√
xinf
,
√
xsup
]
(3.38)3.2.7 Interval Funtion Property
If
[
F
] =
f
([
X
])
isanintervalextensionofthefuntionf
thentheintervalfuntionf
([
X
]) = [
F
]
must inlude allthe valuesf
(
X
)
forx
∈
[
X
]
.3.2.8 Inlusion Monotoniity
of values of
f
(
x
1
, x
2
, ..., xN
)
for allxn
∈
[
xn
]
n
= 1
, ..., N
. If we onsiderf
([
X
]) = [
F
]
and[
X
′
]
⊆
[
X
]
then[
F
′
] =
f
([
X
′
])
⊆
[
F
] =
f
([
x
])
[37℄. Inlu-sion monotoniityproperty is shown inFig (3.5) .x
f
(
x
)
f
([
X
]) = [
F
]
0
]
'
[
X
]
[
])
([
X'
F
'
f
]
[
X
Figure 3.5: Inlusion Monotoniity
As it isobvious, if
[
X
′
]
⊆
[
X
]
then[
F
′
] =
f
([
X
′
])
⊆
[
F
] =
f
([
x
])
.3.2.9 Dependeny Problem
In general, eah ourrene of a given variable in an interval omputation is
treatedasadierentvariable. Thisausewideningofomputedsharpnumerial
bounds. This unwanted extra intervalwidth isalled the dependeny problem.
Letussupposewewanttoevaluatetheintervalperimeter
[
P
]
ofaretangular table whihhas a intervallength[
L
] = [1
.
19
,
1
.
21]
and width[
W
] = [0
.
79
,
0
.
81]
. We apply twodierent ways to ompute:•
Firstadding intervalsof the[
L
]
and[
W
]
:[
P
] = [
L
] + [
W
] = [1
.
98
,
2
.
02]
m
•
Subtrating[
W
]
and[
L
]
from[2
P
]
:[
P
] = [2
P
]
−
[
W
]
−
[
L
] = [1
.
94
,
2
.
06]
m
As it is shown, two dierent results are ahieved from the same quantity. The
[
P
] = [2
P
]
−
[
W
]
−
[
L
] = [
W
] + [
W
] + [
L
] + [
L
]
−
[
W
]
−
[
L
] =
+[
L
] + ([
W
]
−
[
W
]) + ([
L
]
−
[
L
]) =
+[1
.
19
,
1
.
21] + ([0
.
79
,
0
.
81]
−
[0
.
79
,
0
.
81]) + ([1
.
19
,
1
.
21]
−
[1
.
19
,
1
.
21]) =
[1
.
98
,
2
.
02] + ([
−
0
.
02
,
0
.
02] + [
−
0
.
02
,
0
.
02]) =
[1
.
98
,
2
.
02] + [
−
0
.
04
,
0
.
04] = [1
.
94
,
2
.
06]
m
(3.39)
As we an see in (3.39) ,
[
W
]
−
[
W
]
= 0
6
and[
L
]
−
[
L
]
6
= 0
, this is beause of the dependeny eet in the interval analysis. Every ourrene of an intervalvariable isonsidered as anindependent variable
[
W
]
−
[
W
] = [
W
1]
−
[
W
2]
even if[
W
] = [
W
1
] = [
W
2
]
are the same interval. The dependeny probleminrease the width of the resulting interval. Following rules an be applied to avoiddependeny problem:
•
Reduethenumberofourreneofeahvariable: asanexampleofinterval perimeterof the table[
P
] = [
W
] + [
W
] + [
L
] + [
L
]
−
[
W
]
−
[
L
]
→
[
P
] = [
W
] + [
L
]
•
Redene interval operations/funtions[
W
]
−
[
W
] = 0
instead of[
winf
−
w
sup
, w
sup
−
w
inf
]
6
= 0
Byusingproperintervalfuntiondenition,the optimalintervalsolutionwillbe
obtained.
3.2.9.1 Dependeny Problem in Interval Funtion
Assume that we have the funtions
f
(
x
) =
f
1(
x
) =
f
2(
x
)
then we want to evaluate interval extension off
. Interval extension off
1([
X
])
andf
2([
X
])
has the bounds of[
F
1
]
and[
F
2
]
. These two bounds are not the same.[
F
1
]
and[
F
2
]
inlude all values off
(
x
)
forx
∈
[
X
]
butw
([
F
1])
is larger thanw
([
F
2])
. As it is lear,f
1(
x
) =
f
2(
x
)
for same[
X
]
, butf
1(
