Electronically Scanned Arrays (ESAs) and K-Space Gain Formulation

114 Pages·2019·7.07 MB·English

by  Roger A. Dana

Roger A. Dana
Electronically
Scanned Arrays
(ESAs) and K-Space
Gain Formulation
Electronically Scanned Arrays (ESAs) and K-Space
Gain Formulation
Roger A. Dana
Electronically Scanned
Arrays (ESAs) and K-Space
Gain Formulation
RogerA.Dana
AdvancedTechnologyCenterofRockwellCollins
CedarRapids,IA,USA
ISBN978-3-030-04677-4 ISBN978-3-030-04678-1 (eBook)
https://doi.org/10.1007/978-3-030-04678-1
LibraryofCongressControlNumber:2018964714
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Acknowledgments
The electronically scanned array (ESA) simulation based on the theory described
here has been developed over a number of years with the financial support and
guidanceofJamesWestofRockwellCollinsAdvancedTechnologyCenter(ATC)
AdvancedRadioSystems(ARS)Department.Muchofthematerialinthisbookhas
beenwrittenwiththefinancialsupportofAndersWalkeroftheATCCommunica-
tions&ElectronicWarfareGroup.Withouttheirsupport,thisbookwouldnothave
been possible. Special thanks go to the ATC/ARS antenna group, and to Jeremiah
WolfandDr.MatildaLivadaruinparticular,foraskinglotsofinterestingquestions,
manyofwhichmotivatedthismaterial.
v
Contents
1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1 KeyAssumptions,AnalysisLimitations,andAntenna
PerformanceMetrics. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.2 PlaneWavesasSolutionstoMaxwell’sEquations. . . . . . . . . . . 3
1.3 FourierTransformsandPlaneWaves. . . . . . . . . . . . . . . . . . . . . 5
1.4 Organization. . . . . . .. . . . . . . . . .. . . . . . . . . .. . . . . . . . . .. . 7
2 SomeBasicPrinciplesofRFElectronicSystemsandAntennas. . . . . 9
2.1 SomeBasicPrinciplesthatGovernElectronicSystems
andApertureAntennas. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.2 Friis’LinkMarginEquation. . . . . . . . . . . . . . . . . . . . . . . . . . . 14
2.3 Friis’FormulaforNoiseFactor. . . . . . . . . . . . . . . . . . . . . . . . . 15
3 K-SpaceGainandAntennaMetrics. . . . . . . . . . . . . . . . . . . . . . . . . 17
3.1 RelationshipBetweenK-andAngular-SpaceGains. . . . . . . . . . . 17
3.2 DFTImplementationofK-SpaceGain. . . . . . . . . . . . . . . . . . . . 21
3.3 AffineTransformationandForeshorteningEffects. . . . . . . . . . . . 24
3.4 CosineTaperofElementGain. . . . . . . . . . . . . . . . . . . . . . . . . . 26
3.5 FrequencyandForeshorteningEffectsonPerformance
Metrics. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
3.6 PeakGainandEIRPforanESA. . . . . . . . . . . . . . . . . . . . . . . . 27
3.7 Phase-ComparisonMonopulse. . . . . . . . . . . . . . . . . . . . . . . . . . 28
3.8 ComputingDirectivityDirectlyinK-Space. . . . . . . . . . . . . . . . . 30
3.9 IntegratedSidelobeLevel(ISL). . . . . . . . . . . . . . . . . . . . . . . . . 32
3.9.1 SomePropertiesofISL. . . . . . . . . . . . . . . . . . . . . . . . . 32
3.9.2 ISLof1-Dand2-DArrays. . . . . . . . . . . . . . . . . . . . . . . 35
4 EffectofSkyNoiseonAntennaTemperature. . . . . . . . . . . . . . . . . . 37
4.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
4.2 TotalAntennaTemperature. . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
4.3 ApplicationtoModernESAs. . . . . . . . . . . . . . . . . . . . . . . . . . . 42
vii
viii Contents
5 SidelobeControlandMonopulseWeighting. . . . . . . . . . . . . . . . . . . 45
5.1 TaylorWeights. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
5.2 OctagonalShapedESAs. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
5.3 SidelobeRotation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
5.3.1 MathematicalGainofUniformlyWeighted
Parallelograms. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
5.3.2 SidelobeRotationwithParallelograms. . . . . . . . . . . . . . 55
5.4 BaylissandOtherWeightingSchemesforPhase-Comparison
Monopulse. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
6 DigitalBeamformingandAdaptiveProcessing. . . . . . . . . . . . . . . . . 61
6.1 One-DimensionalGainofUniformlyWeightedArray
ofArrays. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
6.2 DigitalBeamformingatSubarrayLevel. . . . . . . . . . . . . . . . . . . 62
6.3 MinimumVarianceDistortionlessResponse(MVDR). . . . . . . . . 65
6.4 Space-TimeAdaptiveProcessing(STAP). . . . . . . . . . . . . . . . . . 67
6.5 EquivalenceofIdealizedSTAPandMVDR. . . . . . . . . . . . . . . . 67
6.6 ExamplesofDigitalBeamformingandAdaptiveProcessing
on32(cid:1)32ESAs. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
6.6.1 VariationwithNumberofJammersandDistance
ofJammersfromSignal. . . . . . . . . . . . . . . . . . . . . . . . . 70
6.6.2 AngleofArrivalofSignal. . . . . . . . . . . . . . . . . . . . . . . 73
6.6.3 NumberofAdaptiveSubarrays. . . . . . . . . . . . . . . . . . . . 75
6.6.4 Summary. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
Appendix1:Far-FieldDemarcation. . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
Appendix2:DiscreteFourierTransforms. . . . . . . . . . . . . . . . . . . . . . . 79
Appendix3:AntennaPointingwithDirectionCosineMatrices. . . . . . . 89
Appendix4:TranslationofPositionandAttitudeErrors
intoPointingDirectionErrors. . . . . . . . . . . . . . . . . . . . . . 99
Appendix5:LossofESAGainbyNoiseCorrelationorSignal
Decorrelation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
Index. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111
Chapter 1
Introduction
Modern engineering literature often starts where others have left off, stating well-
knownresultswithlittleornoattributionordiscussionastotheirrelationshipwith
first principles. This seems particularly true of literature on antenna theory. Alter-
natively,someantennabooksbeginwithlongdiscussionsofMaxwell’sequations,
always a good starting point, that tend to obscure the simple set of principles on
which antenna gain calculations depend. So when this author was asked by the
design team engineers to make predictions on the effect of imperfections on the
performance of an electronically scanned array (ESA) under development, his
approachasaphysiciststartedwithbasicconcepts,havingfailedtofindanantenna
referenceto his liking that did thesame. Thiswork isa compilationof discoveries
made,mostofwhichhavebeenlostinthepast,onthetheoryofESAsstartingfrom
firstprinciplesandafewbasicconceptsusedtocomputeormeasureantennagains.
Akeyconceptisthatonceatransmittedsignal,forexample,propagatesbeyond
the far-field boundary, one can consider the received field at a point to be a plane
wave. So why not eliminate the middle man, the near field, and consider the
transmittedorreceivedsignalatthefaceoftheaperturetobeaplanewave?
Just as time domain functions can be expanded in terms of sine waves using
Fourier transforms, spatial domain functions can be expanded in terms of plane
waves also using a Fourier transform. The Fourier domain of physical space is k-
space,andantennagainink-spaceisexpressedasaFouriertransformoftheaperture
distribution. But k-space gain is not the same as the more familiar angular-space
gain, and relating the two provides in a straightforward manner the relationship
betweenantennagainandeffectiveaperturearea:G¼4πA/λ2.
So the goal for the book is to allow readers to understand the firm theoretical
foundationofantennagaincalculationsevenwhentheymuststartfromwell-known
formulations rather than first principles. Applications such as sidelobe control and
adaptiveprocessingareanattempttoshowtheunityofideasinthesetopicsandthat
adaptive processing, which is often described in gruesome detail in the literature,
also has unifying concepts that connect techniques such as minimum variance
distortionlessresponse(MVDR)andspace-timeadaptiveprocessing(STAP).
©SpringerNatureSwitzerlandAG2019 1
R.A.Dana,ElectronicallyScannedArrays(ESAs)andK-SpaceGain
Formulation,https://doi.org/10.1007/978-3-030-04678-1_1
2 1 Introduction
Thisisabookaboutmathematical modelingESAs topredict howthey perform
ideally;itisnotaboutactuallyconstructingthemorabouthowimperfections(e.g.,
mutual coupling, process tolerances, failed elements, dynamic range, and/or quan-
tization)degradeperformancefromtheideal.
OthertopicsdiscussedinthisbookincludethediscreteFouriertransform(DFT)
formulationofantennagainandwhatismissinginthisformulation,theeffectofsky
temperature on the often specified G/T ratio of antennas, sidelobe control using
conventional and novel techniques, and ESA digital beamforming versus adaptive
processingtolimitinterference.Theappendicesdescribe(1)theDFTinmoredetail
andshowhowtocomputethepowerspectraldensityofarandomprocess,wherethe
1/N term goes most naturally and why, and why zero padding is used to compute
antenna gain; (2) how to point an ESA at a known target using direction cosine
matrices(DCMs)andwhyDCMshavetheparticularformtheydo;and(3)theeffect
ofnoisecorrelationandsignaldecorrelationonthemaximumgainofanESA.
1.1 Key Assumptions, Analysis Limitations, and Antenna
Performance Metrics
Thereareanumberofassumptionsthatwemakeinthemathematicaldevelopments
reportedhere, somemade explicitandsome notsomuch. Unless stated otherwise,
thekeyassumptionsincludedinthemathematicaldevelopmentsare:
(cid:129) Planararrayofomnidirectionalelements,uniformlysampled
(cid:129) PassiveESAinreceivingmode
(cid:129) Narrowband,singlebeamwithlinearpolarization
(cid:129) Arraypatternobservedinthefarfield
Asthisworkisnotabouthardwareimplementationsorlimitations,keyeffectsnot
includedare,forexample:
(cid:129) Elementmutualcoupling
(cid:129) Impedancemodulationasafunctionofscanangle
(cid:129) Surfacewaveinitiation
(cid:129) Quantizationandsummationnetworkimperfections
The far-field demarcation is often mentioned in the literature but rarely is
described as arbitrary. So consider a linear array of length D, and an observation
pointatadistanceRfromthecenterofthearrayandalongalinenormaltothearray.
ThenthepropagationdistancedifferenceΔRfromawavelaunchedfromthecenter
elementandthatofawavefromanedgeelementis
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
(cid:3) (cid:4)
ΔR¼ R2þðD=2Þ2(cid:2)R(cid:3)R 1þD2=8R2 (cid:2)R¼D2=8R:
1.2 PlaneWavesasSolutionstoMaxwell’sEquations 3
(cid:4)
Ifwearbitrarilyallowthetwopathstodifferinphasebynomorethan22.5 ,thenit
mustbethecasethatΔR<λ/16,and
R>2D2=λ ðFarFieldRangeLimitÞ:
Equivalently,ifwetransmitasphericalwavefromthepointR,thephasevariationof
thesphericalelectricfieldincidentontheapertureoflengthDisnomorethan22.5(cid:4).
Ofcourse,wecouldchooseaphasedifferenceofλ/8orλ/32orsomeotherarbitrary
fraction ofawavelengthandgetadifferentexpressionforthefar-fieldrange limit.
ThisissueisdiscussedfurtherinAppendix1.
When discussing examples, we are compelled to compare performance metrics
for various configurations. We focus on four out of a large number of possible
metrics:
(cid:129) Peakgainandpeaksidelobelevel
(cid:129) Beamwidth
(cid:129) Integratedsidelobelevel(ISL)
TheISLasaperformancemetricwarrantsfurtherdiscussion,asmoreofteninthe
antennaliteraturemetricssuchasaverageorRMSsidelobelevelsarequoted.Weuse
ISL in this book as this metric compares integrated main beam gain to integrated
gain in the sidelobes, something that is not apparent from sidelobe-only based
metrics. Formally, ISL is the ratio of the integrated gain outside of the main beam
extenttothatinsidethemainbeamandassuchrepresentstheratiooftheintegrated
gaininthesideloberegiontothatinthemainlobe.Itisreportedindecibelssolarger
negativenumbersarebetter–moresignalpowerinthemainlobecomparedtothatin
the sidelobes. We show that an ideal, uniformly weighted rectangular array has an
ISL of (cid:2)6.44 dB, the gold standard for schemes that seek to improve the sidelobe
structureofanarray.
1.2 Plane Waves as Solutions to Maxwell’s Equations
A simplified approach to the demarcation between the near and far fields of an
antenna, as described above, is 2D2/λ, where D is a representative aperture size.
Beyondthisdistance,thesphericalwavesemanatingfromtheelementssumtogether
inamannerthatshowslittlecurvatureacrossadistanceD,essentiallylookinglikea
planewave.Soweeliminatethemiddleman,thenearfield,andassumethatplane
wavesaretransmittedbyeachelementgivingusthefar-fieldgainfromthesumma-
tionofallcontributions.
Maxwell’s equations for propagation in free space, written in physics units, are
(e.g.,Jackson1962;orStratton1941)