IEEE Transactions on Signal Processing (May 2005)

351 Pages·2005·16.897 MB·English

MAY 2005 VOLUME53 NUMBER 5 ITPRED (ISSN1053-587X)
PAPERS
MultichannelSignalProcessingApplications
StatisticalResolutionLimitsandtheComplexifiedCramér–RaoBound.. ... ... ... .... ..... ... ... ... .. S.T.Smith 1597
Covariance,Subspace,andIntrinsicCramér–RaoBounds.. ... .... ... ... ... ...... ... ... ... ... ... .. S.T.Smith 1610
OptimalDimensionalityReductionofSensorDatainMultisensorEstimationFusion. ... ... ... ... ... ... ... ....
.. ... ... .... ... ... ... ... ... ... ... ... ... ... ... .... ... ... ... ..Y.Zhu,E.Song,J.Zhou,andZ.You 1631
Fourth-OrderBlindIdentificationofUnderdeterminedMixturesofSources(FOBIUM) .. ... ... ... ... ... ... ....
.. ... ... .... ... ... ... ... ... ... ... ... ... ... ... .... ... ... ...A.Ferréol,L.Albera,andP.Chevalier 1640
PenaltyFunction-BasedJointDiagonalizationApproachforConvolutiveBlindSeparationofNonstationarySources....
.. ... ... .... ... ... ... ... ... ... ... ... ... ... ... .... ... ... ...W.Wang,S.Sanei,andJ.A.Chambers 1654
AnEmpiricalBayesEstimatorforIn-ScaleAdaptiveFiltering . .... ... ... ... ...... ... ... ... ... ... P.J.Gendron 1670
MethodsofSensorArrayandMultichannelProcessing
RobustMinimumVarianceBeamforming . ... ... ... ... ... ....... ... ... ... ... ... ..R.G.LorenzandS.P.Boyd 1684
BlindSpatialSignatureEstimationviaTime-VaryingUserPowerLoadingandParallelFactorAnalysis... ... ... ....
.. ... ... .... ... ... ... ... ... ... ... ... .. Y.Rong,S.A.Vorobyov,A.B.Gershman,andN.D.Sidiropoulos 1697
SourceLocalizationbySpatiallyDistributedElectronicNosesforAdvectionandDiffusion .. ... ... ... ... ... ....
.. ... ... .... ... ... ... ... ... ... ... ... ... ... ... .... ... ... ... .J.Matthes,L.Gröll,andH.B.Keller 1711
SignalandSystemModeling
DampedandDelayedSinusoidalModelforTransientSignals.. .... ... ...... ... ... ...R.BoyerandK.Abed-Meraim 1720
FilterDesignandTheory
Robust FilteringforUncertain2-DContinuousSystems .. ...... .... .. S.Xu,J.Lam,Y.Zou,Z.Lin,andW.Paszke 1731
TheoryandDesignofMultirateSensorArrays. ... ... ... ... ..... ..... ... ... ... ... ..O.S.JahromiandP.Aarabi 1739
ArmletsandBalancedMultiwavelets:FlippingFilterConstruction .. ... ... ... ... ..... .... ... ... ... ... ..J.Lian 1754
OptimizationofTwo-DimensionalIIRFiltersWithNonseparableandSeparableDenominator ... ...... ...B.Dumitrescu 1768
ANewApproachforEstimationofStatisticallyMatchedWavelet... .... ..... ... ..A.Gupta,S.D.Joshi,andS.Prasad 1778
PhaseletsofFramelets ... ... ... ... ... ... ... ... ... ..... ..... ... ... ... ... ... ... ... ... .. R.A.Gopinath 1794
(ContentsContinuedonBackCover)
TLFeBOOK
(ContentsContinuedfromFrontCover)
NonlinearSignalProcessing
DirectProjectionDecodingAlgorithmforSigma-DeltaModulatedSignals .. ... ...... ... ...I.WiemerandW.Schwarz 1807
AlgorithmsandApplications
TestingforStochasticIndependence:ApplicationtoBlindSourceSeparation ... .... ..... ... .. C.-J.KuandT.L.Fine 1815
NeuralNetworksMethodsandApplications
NonlinearAdaptivePredictionofComplex-ValuedSignalsbyComplex-ValuedPRNN .... ...S.L.GohandD.P.Mandic 1827
SignalProcessingforCommunications
EqualizationWithOversamplinginMultiuserCDMASystems. .... ... ...... ... ... . B.VrceljandP.P.Vaidyanathan 1837
Noise-PredictiveDecision-FeedbackDetectionforMultiple-InputMultiple-OutputChannels. ... ... ... ... ... ....
.. ... ... .... ... ... ... ... ... ... ... ... ... ... ... .... ... ... ... ... ... .D.W.WatersandJ.R.Barry 1852
BlindEqualizationforCorrelatedInputSymbols:ABussgangApproach.... ... ... ... ... ... ... ... ... ... ....
.. ... ... .... ... ... ... ... ... ... ... ... ... ... ... ... G.Panci,S.Colonnese,P.Campisi,andG.Scarano 1860
LowNoiseReversibleMDCT(RMDCT)andItsApplicationinProgressive-to-LosslessEmbeddedAudioCoding. ....
.. ... ... .... ... ... ... ... ... ... ... ... ... ... ... .... ... ... ... ... ... ... ... ... ... ... ... .J.Li 1870
SignalTransmission,Propagation,andRecovery
ConvolutionalCodesUsingFinite-FieldWavelets:Time-VaryingCodesandMore... ... ... ... ... ... ... ... ....
.. ... ... .... ... ... ... ... ... ... ... ... ... ... ...F.Fekri,M.Sartipi,R.M.Mersereau,andR.W.Schafer 1881
BERSensitivitytoMistiminginUltra-WidebandImpulseRadios—PartII:FadingChannels . ... ... ... ... ... ....
.. ... ... .... ... ... ... ... ... ... ... ... ... ... ... .... ... ... ... ... ... ..Z.TianandG.B.Giannakis 1897
DesignandAnalysisofFeedforwardSymbolTimingEstimatorsBasedontheConditionalMaximumLikelihoodPrinciple
.. ... ... .... ... ... ... ... ... ... .... ..... ... ... ... .... ... ... ... ... ... ... .Y.-C.WuandE.Serpedin 1908
CORRESPONDENCE
MethodsofSensorArrayandMultichannelProcessing
ABayesianApproachtoArrayGeometryDesign. . ... ... ... .... ..... .... ... ... ... ... .Ü.OktelandR.L.Moses 1919
ComputationofSpectralandRootMUSICThroughRealPolynomialRooting. .. ... ... .... ..... ... ... ... . J.Selva 1923
SignalDetectionandEstimation
EstimateofAliasingErrorforNon-SmoothSignalsPrefilteredbyQuasi-ProjectionsIntoShift-InvariantSpaces .. ....
.. ... ... .... ... ... ... ... ... ... ... ... ... ... ... .... ... ... ... ... .. W.Chen,B.Han,andR.-Q.Jia 1927
RobustSuper-ExponentialMethodsforDeflationaryBlindSourceSeparationofInstantaneousMixtures.. ... ... ....
.. ... ... .... ... ... ... ... ... ... ... ... ... ... ... .... ... ... .. M.Kawamoto,K.Kohno,andY.Inouye 1933
SignalProcessingforCommunications
IterativeDecodingofWrappedSpace-TimeCodes.... ... ... .... .... ..... ... ... ... ... ..A.SezginandH.Boche 1937
ANNOUNCEMENTS
CallforPapers—2006IEEEInternationalSymposiumonBiomedicalEngineering:FromNanotoMacro . ... ...... .... 1942
CallforPapers—IEEETRANSACTIONSONSPEECHANDAUDIOPROCESSINGSpecialIssueonProgressinRichTranscription
.. ... ... .... ... ... ... ... ... ... ... ... ... .... ..... .... ... ... ... ... ... ... ... ... ... ... ... .... 1943
CallforPapers—IEEETRANSACTIONSONINFORMATIONFORENSICSANDSECURITY . ... ... ... ..... .... ... ... .... 1944
CallforPapers—IEEETRANSACTIONSONSIGNALPROCESSINGSpecialIssueonGenomicSignalProcessing . ..... ..... 1945
EDICS—Editor’sInformationClassificationScheme.. ... ... .... ... ... ... .... ..... ... ... ... ... ... ... .... 1946
InformationforAuthors.. ... ... ... ... ... ... ... ... ... .... ...... ... ... ... ... ... ... ... ... ... ... .... 1947
TLFeBOOK
IEEETRANSACTIONSONSIGNALPROCESSING,VOL.53,NO.5,MAY2005 1597
Statistical Resolution Limits and the Complexified
Cramér–Rao Bound
Steven Thomas Smith, Senior Member, IEEE
Abstract—Arrayresolution limitsand accuracy boundson the I. INTRODUCTION
multitudeofsignalparameters(e.g.,azimuth,elevation,Doppler,
RESOLUTIONandaccurateestimationofsignalsandtheir
range, cross-range, depth, frequency, chirp, polarization, ampli-
tude,phase,etc.)estimatedbyarrayprocessingalgorithmsarees- parametersarecentralcapabilitiesofadaptivesensorarray
sentialtoolsintheevaluationofsystemperformance.Thecasein processing.TheCramér–Raobound(CRB)[29],[34],[44],[54]
which the complex amplitudes of the signals are unknown is of providesthebestaccuracyachievablebyanyunbiasedestimator
particularpracticalinterest.Acomputationallyefficientformula- of the signal parameters and therefore provides a fundamental
tionofthesebounds(fromtheperspectiveofderivationsandanal-
physical limit on system accuracy. In this paper, deterministic
ysis)ispresentedforthecaseofdeterministicandunknownsignal
CRBs are derived for a general signal model with unknown
amplitudes. A new derivation is given using the unknown com-
plexsignalparametersandtheircomplexconjugates.Thenewfor- signalamplitudes,andnewclosed-formexpressionsforstatis-
mulaisreadilyapplicabletoobtainingeithersymbolicornumer- ticalresolutionlimitsofarraysarederivedfromthesebounds.
icalsolutionstoestimationboundsforaverywideclassofprob- The statistical resolution limit is defined as the source sepa-
lems encountered in adaptive sensor array processing. This for- ration that equals its own CRB, providing an algorithm-inde-
mulaisshowntoyieldseveralofthestandardCramér–Raoresults
pendent resolution bound. Multiparameter CRBs for adaptive
for array processing, along with new results of fundamental in-
sensorarrayprocessingproblemsarewellknown[8],[10],[27],
terest.Specifically,anewclosed-formexpressionforthestatistical
[40], [43], [45], [55]. In this paper, a new formula and new
resolutionlimitofanapertureforanyasymptoticallyunbiasedsu-
perresolutionalgorithm(e.g.,MUSIC,ESPRIT)isprovided.The derivationforCRBsinvolvingacomplexparameterspace[56]
statisticalresolutionlimitisdefinedasthesourceseparationthat areshowntobebothverygeneralandveryconvenient.
equalsitsownCramér–Raobound,providinganalgorithm-inde- Thenewformula[see(29)]fordeterministicCRBshasbeen
pendent bound on the resolution of any high-resolution method.
usedinawidevarietyofapplicationstoobtainbothnumerical
It is shown thatthe statistical resolution limit ofan array orco-
herentintegrationwindowisabout12 SNR 1 4 relativetothe andanalyticalCRBresultsquicklyandtransparently.Whatdis-
tinguishestheCRBresultsinthispaperfromthegreatmajority
Fourier resolution limit of 2 radians (large number of
arrayelements).Thatis,thehighestachievableresolutionispro- of existing derivations is a “complexified” approach to the
portionaltothereciprocalofthefourthrootofthesignal-to-noise generalproblemofestimatingmultiplesignalparametersfrom
ratio(SNR),incontrasttothesquare-root(SNR 1 2)dependence multiple signals in (proper [26]) complex noise, each signal
of standard accuracy bounds. These theoretical results are con- withanunknownamplitudeandphase.Whereasthe“realified”
sistentwithpreviouslypublishedboundsforspecificsuperresolu-
approach to estimating complex parameters involves their
tion algorithms derived by other methods. It is also shown that
real and imaginary components, the complexified approach
thepotentialresolutionimprovementobtainedbyseparatingtwo
uses the complex parameters and their complex conjugates
collineararrays(syntheticultra-wideband),eachwithafixedaper-
ture wavelengthsby wavelengths(assumedlarge),isapprox- directly, oftentimes resulting in a greatly simplified analysis.
imately( )1 2,incontrasttotheresolutionimprovementof (See Appendix A for background and algebraic definitions of
forafullaperture.Exactclosed-formresultsfortheseprob- realified and complexified vector spaces.) The complexified
lemswiththeirasymptoticapproximationsarepresented. approach and the efficient expressions for CRBs it yields are
Index Terms—Adaptive arrays, adaptive estimation, adaptive describedbyYauandBresler[56]andvandenBos[53].New
signal processing, amplitude estimation, direction-of-arrival esti- results employing the complexified approach have begun to
mation, error analysis, ESPRIT, estimation, Fisher information, appear [20], [40]. The derivation of the complexified CRB
image resolution, maximum likelihood estimation, MUSIC, pa-
presented here is greatly simplified from previous work, and
rameter estimation, parameter space methods, phase estimation,
the resulting formula is more general than Yau and Bresler’s
radarresolution,signalresolution,spectralanalysis,superresolu-
[56, Th. 2] because a more general steering vector model
tion,ultra-widebandprocessing.
and parameter space are considered, as well as an arbitrary
interference-plus-noisecovariance.
TheCRBformulaeareappliedtocomputingnewclosed-form
expressions for the statistical resolution limit of both a single
Manuscript received January 5, 2004; revised May 25, 2004. This work uniformlineararray(ULA)[see(61)]andasplit-apertureused
was supported by the United States Air Force under Air Force Contract
in synthetic ultra-wideband processing [see (73)]. The case of
F19628-00-C-0002.Opinions,interpretations,conclusions,andrecommenda-
a single data snapshot or nonfluctuating (deterministic) target
tionsarethoseoftheauthorandarenotnecessarilyendorsedbytheUnited
States Government. The associate editor coordinating the review of this model is considered. Synthetic ultra-wideband processing
manuscriptandapprovingitforpublicationwasDr.FulvioGini. [1], [5], [6], [22], [40], [58] refers to a family of methods
TheauthoriswiththeLincolnLaboratory,MassacusettsInstituteofTech-
using model-based interpolation/extrapolation across widely
nology,Lexington,MA02420USA(e-mail:[email protected]).
DigitalObjectIdentifier10.1109/TSP.2005.845426 separated subbands/apertures, usually over large fractional
1053-587X/$20.00©2005IEEE
TLFeBOOK
1598 IEEETRANSACTIONSONSIGNALPROCESSING,VOL.53,NO.5,MAY2005
bandwidths. Because the statistical resolution limit is based where is a proper complex [26] Gaussian interference-plus-
ontheCRB,itrepresentsahighSNRboundontheresolution noisevectorwith -by- covariancematrix
achievable by any asymptotically unbiased superresolution
method,suchasMUSIC[24],[30],[35],[46],[48]orESPRIT (2)
[32], [48]. CRB analysis of the source separation to bound
isacomplex -by- “steering”matrixwhosecolumnsdefine
the resolution has been used by Clark [4], Smith [40], and
the responses of signals impinging upon the sensor,
Ying et al. [57] in a mathematically equivalent treatment. It
isthe -vectorofrealsignalparameters,and
is shown that the statistical resolution limit of an aperture is
isthe(unknown,deterministic)complex -vectorofampli-
about SNR rad relative to the Fourier resolution limit
tudescorrespondingtothe signals.Throughoutthepaper,it
rad (large number of array elements). That is, the
statistical resolution limit is proportional to the reciprocal of is assumed that the unknown amplitudes are fixed (determin-
thefourthrootofthesignal-to-noiseratio(SNR),incontrastto istic), although the extension to the stochastic case is straight-
the square-root SNR dependence of standard accuracy forward.AsexplainedinAppendixB,deterministicCRBresults
bounds. For the application of array processing, this provides donotdependonthelackofknowledgeofthecovariancema-
the best possible fraction of the Rayleigh resolution limit trix . Note that this signal model is identical to that of Yau
achievablebyanysuperresolutionmethod. and Bresler’s [56] in the frequently encountered case, where
The single-aperture results are comparable in many ways ( ,i.e., parametersper
with the deterministic resolution limit found by Lee [23], the eachofthe signals).Theprobabilitydistributionof is
stochastic CRB case explored by Swingler [49]–[52], and
the resolvability detectors considered by Shahram and Mi-
lanfar [37]–[39]. The analysis of ULA resolution is applied (3)
to the problem of computing resolution limits for synthetic andthelog-likelihoodfunctionis
ultra-wideband processing, in which widely separated aper-
tures(eitherinthefrequencyorspatialdomains)arecoherently (4)
combinedusingamodel-basedproceduretoprovideenhanced
resolution.Itisshownthatsyntheticultra-widebandprocessing Bounds on the estimation error of the real parameters ,
does indeed improve the resolution above that obtained by which are typically composed of a collection of angles of
using each aperture individually. It is also shown that the arrival, Doppler frequencies, ranges, frequencies, damping
potential resolution improvement obtained by separating two factors, etc., are to be computed. If the unknown complex
collinear arrays, each with a fixed aperture wavelengths by amplitudes are in fact known, then the Cramér–Rao theorem
wavelengths(assumedlarge),isapproximately , [29], [34], [54] asserts that the covariance of any unbiased
in contrast to the resolution improvement of for a full estimatorofthe (real-valued)unknownparametersisbounded
aperture. belowbytheinverseoftheFIM:
Itshouldbekeptinmindthattheaccuracyandresolutionre-
sultsdescribedinthispaperareallderivedfromCRBs.There- (5)
fore,theyinheritthepropertiesofsuchbounds:Theyarelocal
(inthesensethatforsymmetricmatrices and , if
(notglobal)boundsanddonotreflectthelargeerrorsthatoccur
andonlyif ispositivesemidefinite),where
atlowerSNR’s,wheretheambiguityfunctionhashighsidelobe
levels.
(6)
Thepaper’sbodyisorganizedintothreesections.SectionII
containsthedescriptionofthecomplexifiedFisherinformation istheFIM,and
matrix(FIM)andthecorrespondingcomplexifiedCRBs.These
bounds are applied to the key example of pole model estima-
(7)
tioninSectionIII.Thepolemodelaccuracyboundsobtainedin
SectionIIIarethenappliedtotheproblemofcomputingreso-
(8)
lution limits inSectionIV, which includesa treatment ofboth
thesingle-apertureandsplit-aperturecases,aswellasacompar- are the Jacobian and Hessian matrices (1-by- and -by-
isonoftheresolutionimprovementbetweenthetwo.Thereare matrices, respectively) of derivatives of with respect to the
alsoappendicesdefiningthealgebraicconceptofcomplexified elementsoftheparametervector .Recallthatif isan
vectorspaces,derivingthecomplexifiedFIM,andapplyingthe
arbitrarymappingfrom to ,thentheJacobianmatrixand,
complexifiedCRBtoasimpleangle-angleestimationproblem.
consequently,theFIMtransformcontravariantly,i.e.,
(thechainrule).Therefore
II. COMPLEXIFIEDCRB
(9)
A. Preliminaries
Themeasurementvector fromasensorarraycontaining
where is the Jacobian matrix of the
elementsisgivenbytheequation
transformation. Furthermore, if there are real nuisance pa-
rameters thatmustbeestimated(such
(1) asthesignalamplitudes)alongwiththedesiredparameters ,
TLFeBOOK
SMITH:STATISTICALRESOLUTIONLIMITSANDTHECOMPLEXIFIEDCRAMÉR-RAOBOUND 1599
thentheCramér–Raolowerboundontheestimationerrorof brieflydiscussed.Thereal-valuedFIM ispositivedefi-
isgivenbytheinverseoftheSchurcomplementof inthe
nitewithrespecttothemetric ,i.e.,alleigenvaluesof
full FIM
the matrix pencil are positive. By (12),
(10)
the metrics
ThisCRBontheerrorcovariance is
(11)
(14)
which increases the CRB due to the lack of knowledge of .
(RecallthattheSchurcomplementofthesquarematrix inthe
are equivalent. Therefore, the complexified FIM
blockmatrix is .Theinverse
of (13) is positive definite with respect to the metric
oftheSchurcomplementisdenotedhereas .)Note
, i.e., the
that the CRB depends on the unknown parameter vector
generalized eigenvalues of the matrix pencil
.Equation(11)isinvarianttoarbitrarylineartransformations
ofthenuisanceparameters ,where isaninvertible (whichequaltheeigenvaluesof )
matrixthatmaypossiblyhavecomplex-valuedelements.By(9),
areallpositive.Comparethesefactspertainingtotheeffectof
thechangeofvariables inducesthefollowingtrans-
lineartransformationsonestimationaccuracywiththeproper-
formationsin : and ;
tiesofgeneralcomplexrandomvariables[36];seealsoKrantz
therefore, for an arbitrary
[19]forageneraltreatmentoffunctionsofcomplexvariables.
invertiblematrix .
Note that the change of variables in (12) and the chain rule
Consider the unknown complex amplitude vector
definesderivativeswithrespectto and [19]:
Re Im appearingin(1).Traditionally,CRB
results are derived in terms of the real and imaginary parts
and .However,duetotheinvarianceof(11),theCRBremains (15)
unchangedifthecomplexifiedvariables
(16)
(12)
Thesederivativesareconsistentwiththedesiredformulae
areused.Becausethemostcomputationallyconvenientformof
(17)
thelog-likelihoodfunctiondependsonthecomplexifiedparam-
eters and andnottherealparameters and ,acomplexi-
Asanaside,if isafunctionofseveralcomplexvariables,
fiedapproachisgenerallypreferred[53],[56].By(9),theFIM
isanalyticifandonlyif ,i.e.,theCauchy-Riemann
oftherealandimaginarycomponents and isrelatedtothe
equations.Becausethelog-likelihoodfunctionisreal-valuedev-
complexifiedcoordinates and bytheequation
erywhereandthereforedependson , isnotanalytic,andin-
deed,thetheoryofanalyticfunctionsisirrelevantfortheresults
ofthispaper.
B. ComplexifiedFIMandCRB
(13)
TheFIM
(Equation (13) is equivalent to Theorem 1 of Yau and Bresler
(18)
[56]). The log-likelihood function of many array processing
problems[e.g.,(4)]dependslinearlyon and (i.e.,sesquilin-
willbeusedwith(11)todeterminetheCRBonthecovarianceof
earlyon ),oftentimesgreatlysimplifyingthederivationofthe
foranyunbiasedestimator .Thedetailsarestraightforward
complexifiedFIM of(13).Notethatin
andprovidedinAppendixB,asisthereasonthedeterministic
contrast to the complex Hessian matrix encountered in
CRBisnotaffectedbytheinclusionoftheknownorunknown
thestudyofplurisubharmonicfunctions[19],thecomplexified
covariancematrix intheparametervector.ThefullFIMis
FIMrepresentstheexpectationofthefullHessianmatrixwith
respect to all the elements of the complexified vector .
However, the diagonal blocks and usually vanish. (19)
Therefore, only the cross terms typically appear in the
resultingCRBs.
Some additional nonstandard properties of complexified where is an -by- complex matrix (the
FIM’s that do not appear elsewhere in the literature will be negative expectation of the complex Hessian matrix
TLFeBOOK
1600 IEEETRANSACTIONSONSIGNALPROCESSING,VOL.53,NO.5,MAY2005
[19]), Re isa -by- realmatrix, In the simplest case of a single source and a uniform
linear array, , and letting
is a -by- complex matrix, and the
be the array center’s phase, (23)–(28) reduce
subscripted notation “ ” denotes the substitution of the first
to SNR , SNR , yielding
derivative with respect to wherever the th index of is
thewell-knownresult
required(seeAppendixBforspecificexamples).Theelements
ofthematrices and aregivenby
rad (31)
SNR
Re thelement (20)
throw (21) which is independent of the phase center . It also is in-
structive to compare the deterministic CRB of (29) to the
where stochastic CRB [2], [14], [49], [50], where the signal am-
plitudes are random with unknown covariance matrix
(22) , and the output probability distribution of
(1) is with co-
isthe -by- complexmatrixofthederivativeofthesteering variance matrix , and data matrix
matrix withrespectto . . In this case, the full FIM is given by
Applying(11)to(19),thecomplexifiedCRBisobtained: . This is an appropriate
data model for multiple data snapshots with fluctuating target
-by- (23) amplitudes.Matveyevetal.[25]provideadetailedcomparison
ofthedeterministicandstochasticCRBs.
where -by- (24)
The bounds shown in (29) must be evaluated for a partic-
Re -by- ularchoiceofamplitudevector andparametervector .These
vectorsaresetatsomenominalvaluesofinterest.Themagni-
(25)
tude of the vector may be set to achieve a given array-level
Re -by- (26) signal-to-noiseratio(SNR)viatheequation
-by- (27)
SNR (32)
-by- (28)
where is the covariance matrix assumed for the noise
The somewhat obvious notation for a subscripted “ ” is ex-
model. Note that . In the specific
plainedinthepreviousparagraph,andtheformulaefor , ,
caseconsideredinSectionIVoftwosourceswithidentical(but
and are repeatedfor convenience. Equations(24)–(28) may
unknown) amplitudes, the array SNR when the sources coa-
becombinedtoobtaintheexpression
lesceisgivenbySNR SNR ,whichisobtainedfrom
Re , SNR , and , and SNR
denotestheelement-levelSNR.
(29) The standard application of computing bounds on azimuth
andelevationmeasurementsisprovidedinAppendixC.
In the frequently encountered case where the signal model is
,(29)isreadilyseen
III. EXAMPLE:POLEMODELESTIMATION
tocorrespondto
Considertheproblemofmodelingfrequencydatausingaset
ofdampedexponentials:
Re
(30)
(33)
where“ ”and“ ”denotetheSchur–Hadamard(component-
wise)andKronecker(tensor)products,respectively, denotes
where the are unknown complex numbers, and the are
the -by- matrixhavingunityelements,and,byabuseofno-
samples in frequency. This may also be viewed as a time-se-
tation, [in riesmodelingproblemofsinusoidsincomplexnoise[43],[45],
(30)only].ThisissimplyYauandBresler’s[56]resultforthe [48], [49], [52]; however, the mathematically equivalent fre-
case . The matrix may be interpreted as an array of quency-domainversion used by Cuomo et al. [5], [6] for syn-
“crossdifferenceSNR”terms;thematrix maybeinterpreted theticultra-widebandprocessingisconsideredhere.Steedlyand
astheextralossofaccuracyduetotheunknownamplitudes.Ap- Moses[43]computedeterministicCRBsforthecomplexpoles
plicationofthiscomplexifiedapproachisstraightforwardfora andtheunknownamplitudesusingthestandardreal/imaginary
largeclassofapplications;compare(29)and(30)tootherdeter- approach. In this section, the complexified approach will be
ministicCRBformulaereportedintheliterature[8],[25],[27], usedtoobtainformulaefortheidentical CRboundsofsignif-
[43], [45]–[47], [55]. If is an unbiased estimator of , the icantlygreatersimplicity.Theseformulaewillthenbeapplied
CRB holds. tocomputingclosed-formresolutionlimitsinSectionIV.
TLFeBOOK
SMITH:STATISTICALRESOLUTIONLIMITSANDTHECOMPLEXIFIEDCRAMÉR-RAOBOUND 1601
Equation(33)canbeexpressedcompactlyusing(1)andthe itshouldbeverifiedthat(39)–(46)areconsistentwiththeinter-
steeringmatrix pretationof givenin(21)and(22).Toseetherelativesim-
plicityachievedbythecomplexifiedCRBs,compare(38)–(46)
totheequivalent,real/imaginarybounds[43,pp.1306–1309].
InSectionIV,theseCRBsontheestimatesofthepoles will
. . . (34) beusedtoderiveboundsonthefundamentalresolutionachiev-
. . .
. . .
ablebyanaperture,i.e.,thesmallestpoleseparationmeasurable
byanyunbiasedestimatorofthepoles.
Because the unknowns are complex numbers ,
thetreatmentofSectionIImustbeslightlymodifiedtohandle IV. STATISTICALRESOLUTIONLIMITS
thiscase.Thereisachoice:Usethecomplexifiedapproachfor
The CRB quantifies a lower bound on estimation error, but
and directly,orusethemorecommonparameterizations
itdoesnotdirectlyindicatethebestresolutionachievablebyan
oftherealandimaginarypartsorthemoduliandphasesofthe
unbiasedestimator.Nevertheless,theCRBcanbeusedtodefine
poles .Inthisexample,the steering
an absolute limit on resolution. The minimum requirement to
matrix dependsonlyonthecomplexpolevalues (andnot
resolvetwosignalsis
theircomplexconjugates ),andderivativeswithrespectto
and ,whichareeasilyobtainedfrom(15)and(16),are
standarddeviation
sourceseparation (47)
(35) ofsourceseparation
(36)
Thestatisticalresolutionlimitisdefinedasthesourceseparation
atwhichequalityin(47)isachieved,i.e.,thesourceseparation
whichmaybeextractedfromtherealandimaginarypartsofthe
thatequalsitsownCRB.Becausethisconceptofresolutionis
derivativesof withrespectto .Therefore,itwillbesimplest
based on the CRB, it has the advantage of bounding the reso-
tochoosetheparameterization
lutionofallsuper-resolutionmethodsintheregimeswherethe
CRBholds:highSNRs,unbiasedestimators,andnomodeling
(37)
orsignalmismatch.Inthissense,thestatisticalresolutionlimit
providesthebest-caseresolutionboundforanyalgorithm.Due
i.e., the standard real/imaginary approach, and proceed as de-
totheso-calledSNRthresholdeffect[17],[31]causedbyesti-
scribedinSectionIIusingahybridoftherealandcomplexified
mationoutliersinthethresholdregion,theresolutionofspecific
approachestocomputeboundsontheunknowncomplexpoles
algorithmstypicallyexceedsthestatisticalresolutionlimit,al-
withtheunknowncomplexamplitudes .
thoughtheirqualitativebehaviorsmaybesimilar.Bystatistical
Given(37),theCRBforanyunbiasedestimator of from
considerations alone, Swingler [51] concludes that the practi-
(23)–(28)isgivenby
callyachievableresolutionlimitisaboutonetenthtoonefourth
of the Rayleigh resolution limit. Furthermore, mismatch and
modelingerrorscangreatlydegraderesolution[11],[17],[31],
butnoneoftheseeffectsaretakenintoaccountbythestatistical
(38)
resolutionlimit.
where Re (39) The CRB of the source separation has been used to bound
resolutioninpreviousworkbyClark[4],Smith[40],andYing
(40) etal.[57],whoprovideamathematicallyequivalenttreatment
of(47).TheCRBofthesourceaccuraciesthemselves(notthe
Re (41) CRB of the difference) has been used by Lee [23], Swingler
(42) [49]–[52],Steedlyetal.[42],andseveralresearchersintheop-
tics community (see the references cited in the survey by den
(43)
DekkerandvandenBos[7],especiallythepaperbyHelstrom
[16],aswellasLucy[21]andBettensetal.[3]).Aswillbeseen,
(44)
theresultsderivedinthispaperarequalitativelyconsistentwith
manyotherpublishedresults.
(45)
Equation (47) is an appealing definition of resolution in a
practical sense. In the regime where the sources are resolved,
thcolumnof (46) the estimation accuracy is bounded by the CRB. If the poles
are too close, they cannot be resolved because the two spec-
Notethatthetensorproductsin(40)and(44)appearintheoppo- tral peaks of each signal coalesce into one; this fact is behind
siteorderthanthatin(30)becauseofthedifferentlexicographic much of the analysis of the resolution limits for several algo-
orderofthe parametervectorin(37).Becausethe thcolumn rithms. In this case, the two poles are estimated to lie at the
of dependsonlyon and ,thetensor same location somewhere between the two true locations, the
maybeefficientlyrepresentedusingan -by- matrix; difference between the poles is estimated to be zero, and the
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1602 IEEETRANSACTIONSONSIGNALPROCESSING,VOL.53,NO.5,MAY2005
root-mean-squareerror(RMSE)oftheestimateddifferenceap- Equation(57)isVanTrees’sproperty5ofCRBsofnonrandom
proximatelyequalsthepoleseparation. variables[54,sec.2.4].
Symbolicalgebrapackages,suchasMapleorMathematica,
A. Single-ApertureResolutionLimit areusefulforcomputingtheLaurentseries
Theresolutionlimitfortwosignalssampledoveranequally
Var (58)
spacedapertureof pointswillbecomputed.Assumethatsig-
nals are modeled by the poles and (
small),thateachsignalhasidenticalbutunknownamplitudes, SNR
and that the noise background is white . For this ex- (59)
ampleproblem,(38)–(46)canbeexpressedcompactlyusingthe
Dirichletkernel for small, where SNR SNR is the array SNR
whenthepolescoalesce,andSNR istheelement-levelSNR.
As discussed above, it is desired to compute the separation
(48)
thatyieldsatleastastandarddeviationof ,i.e.,
Because of the special form of (33), it will be convenient to
takederivativeswithrespecttothepoles andthen SNR
(60)
transformthemtoderivativeswithrespecttotherealandimag- Solving(60)forthis (assumedsmall)determinesthestatis-
inarypartsofthepolesvia(35)and(36).Expressing(38)–(46) ticalresolutionlimit
intermsoftheexample,itisfoundthat
(61)
Re (49) SNR
for large (62)
SNR
(50)
Dividing(62)bythestandardFourierresolutionlimit rad
yieldsausefulrule-of-thumbfortheabsoluteresolutionlimit:
(51)
(52) resolution re.Fourierres. (63)
SNR
(53) Inaddition,notethat(61)impliesthatmorethanthreesensors
arerequiredtosuperresolvethetwocomplexpoles.
Re (54) Equation(61)indicatesthattodoubletheresolutionusinga
fixedaperture,theSNRmustincreaseby12dB,ortoincrease
where , and for some unknown the resolution tenfold, the SNR must increase by 40 dB. This
propertyisshowninFig.1,whichfortheexampleof
complex parameter . In the assumed case of unit-variance
shows the RMSEs and the CRBs of the modulus of the dif-
whitenoise, SNR ,whereSNR istheelement-level
ference of the estimated poles versus the pole separation. The
SNR, and is an unknown phase. Applying (32) when the
“ringing”patternobservedinFig.1whenthepolesareresolved
polescoalesce ,SNR SNR .Tosimplify
occurs as the sources move in and out of the nulls and peaks
thecomputations,aphase-centeredapertureisused.
of their respective array responses. The error is lowest when
Asdiscussedabove,aboundisdesiredfortheRMSEofthe
onepoleisnearanulloftheotherarray’sresponseandhighest
difference
whenitisnearapeak.Equation(61)isverifiedbyMonteCarlo
complexified (55) simulations (Figs. 2 and 4) of the maximum-likelihood (ML)
methodforpoleestimationwithunknownamplitudes.Fig.2il-
or real (56) lustrates an example with 50 samplesand SNRof 48 dB. The
maximum-likelihoodmethodachievestheCRBwhenthepoles
are resolved but breaks away from the CRB curve at the pre-
where , and ,
dicted statistical resolution limit of about 0.08 of the Fourier
as above. Therefore, the FIM must be computed in terms of resolution rad. The resolution performance of both the
the variable or .Becausereal/imaginarycoordinatesare MUSICandESPRITalgorithmsisalsoshown,usingadatama-
used in (38)–(46), the parameter vector from (56) is used, trixconstructedfromaslidingwindowof samples.
resultingin Both of these superresolution methods achieve resolutions of
about0.3–0.4oftheFourierresolution rad,withESPRIT
achieving a slightly better resolution than MUSIC due to the
(57) invariance properties of pole estimation that ESPRIT exploits.
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SMITH:STATISTICALRESOLUTIONLIMITSANDTHECOMPLEXIFIEDCRAMÉR-RAOBOUND 1603
Fig.1. Cramér-RaoboundofthepoledifferenceforN = 50andseveral
SNRs,itsasymptotecomputedusing(60),andtheresolutionlimitscomputed
using(61)expressedusingastandardFourierresolutioncell(FRC)of2(cid:25)=N (cid:25)
1=8rad.NotethathalvingtheresolutionrequiresincreasingtheSNRby12dB.
Inthisexample,thestandarddeviationofthedifferencebetweenpolelocations Fig. 2. Full aperture predicted versus measured resolution with two poles.
iscomputedassumingtwosignalswithpolesat1ande ((cid:1)(cid:30)(cid:17)separation) TheRMSEofthedifferenceofthepoleestimatesversuspoleseparationfor
andwithequalbutunknownamplitudes.The“ringing”patternobservedresults themaximum-likelihood(ML)method(reddashedcurvewith(cid:14)s),theMUSIC
fromthesourcesmovinginandoutofthenullsandpeaksoftheirrespective algorithm(purpledashedcurvewith+s),theESPRITalgorithm(bluedashed
arrayresponses. curvewithxs),andtheCRB(darkgreensolidcurve)areallshownusingthe
unitofaFourierresolutioncell(FRC)2(cid:25)=N (cid:25) 1=8rad,whereN = 50,
and SNR = 48dB in this example. For both the MUSIC and ESPRIT
Beneath their resolution limits, MUSIC and ESPRIT provide algorithms,aslidingwindowofN=2 = 25samplesisusedtogeneratethe
the identical estimate for the two different poles. Hence, the datamatrices.Themaximum-likelihoodmethodfailstoresolvethetwopoles
at the resolution predicted by (63) and breaks away from the CRB at this
RMSEofthedifferenceequalsthepoleseparationoftheunre- point;thestatisticalresolutionlimit(separation=RMSE)isthedashedblack
solvedpoles.Incontrast,beneaththeresolvabilityoftheMLal- linetotheupperright.TheSNRthresholdbehaviorofboththeMUSICand
gorithm,therandombehaviorofpeaksintheeight-dimensional ESPRIT algorithms is observed, resulting in an achieved resolution greater
thanthestatisticalresolutionlimitforbothofthesemethods.Asexpected,the
log-likelihoodsurface(magnitudeandphasefortwopolesand
resolutionoftheESPRITmethodisslightlygreaterthanthatoftheMUSIC
twocomplexamplitudes)yieldsanearlyflatRMSEcurvewhen methodbecauseESPRITexploitsinvarianceinformationnotusedbyMUSIC.
thepolesareunresolved. AMonteCarlosimulationwith1000trialsisusedtocomputetheRMSEs.The
“ringing”patternobservedisthesourcesmovinginandoutofthenullsand
These resolution results are qualitatively consistent with peaksoftheirrespectivearrayresponses.
manyresolutionboundsreportedintheliterature[40].Lee[23,
Eqs.(62)and (63)]hypothesizes anecessaryconditionforthe SNR , which is a good match to the theoretical result
resolvability of signals (deterministicCRB assumption) and SNR . Steedly and Moses [43] (deterministic assumption)
deduces a dependence of SNR or SNR for . numerically compute the CRB for the angle separation of two
Swingler [49]–[52] (stochastic assumption) uses an approxi- polesontheunitcircleandillustrateaplot(Fig.5onp.1312)
mationofthestochasticCRBtoobtainresolutionbounds.For thatisqualitativelysimilartoFig.1.Thisresultisalsoderived
large , high SNR, and a single data snapshot, Swingler’s by Kaveh and Lee [17, Eq. (5.118)] (stochastic case), using
approximation [51, Eq. (3)] implies that the reso- asymptotic properties of the Wishart distribution. Stoica and
lution (normalized by the Fourier resolution of ) is at Nehorai [46], [47] (deterministic and stochastic assumptions)
least SNR SNR .Uptotheconstant illustrate numerical evidence that the standard deviation is
multiple of 1.7, this is comparable to the statistical resolution proportional to the inverse of the separation as in (60). Stoica
limit of SNR given in (63). Ying et al. [57] define a and Söderström (stochastic case) observe in [48, p. 1842] that
resolutionlimitintheir(5)thatismathematicallyequivalentto the covariance elements become large when two frequencies
thestatisticalresolutionlimitdefinedin(47),andtheyillustrate are close but do not quantify this increase. Zhang [59, Fig. 1]
resolution proportional to SNR in their Fig. 1, as well illustrates numerical results for the probability of resolution
provide as an alternate geometric interpretation of this limit. thatshowaclosematchtothepredictedSNR dependence.
ShahramandMilanfar[37]–[39]useadetectiontheoreticcon- Incontrasttothisbodyofwork,Kosarev[18]usesaninfor-
ceptofresolvabilitytoarriveataresolutionlimitthatdepends mation-theoreticapproachtoarriveatasuperresolutionbound
onSNR .LeeandWengrovitz[24](stochasticassumption) thatdependslogarithmicallyontheSNR.
showintheirEq.(91)thattheresolutionthresholdforMUSIC Ifthemagnitudesofthetwopolesarefixedontheunitcircle,
isproportionaltoSNR .Farina[9,Sec.5.7]confirmsthere- a similar analysis shows that the resolution is also determined
sultofGabriel[12]thattheachievablesuperresolutionisabout by(61)iftheresolutionlimitismultipliedby .
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1604 IEEETRANSACTIONSONSIGNALPROCESSING,VOL.53,NO.5,MAY2005
B. Split-ApertureResolutionLimit
Split-aperturearraysrefertothecasewhereinsteadofhaving
a full aperture, there are two apertures separated by a large
gap. The analysis of the preceding section can be modified
to compute the achievable resolution of split-aperture arrays
with so-called synthetic ultra-wideband coherent processing
[5], [6], [40], [58]. CRBs on source direction of arrivals are
well known for arbitrary array geometries [13], [28]. The
CRB of the source separation for a split-aperture is consid-
ered here. Let the synthetic ultra-wideband coherent aperture
consist of samples (as above), but with the middle
samples removed(for convenience, assume thatthe difference
isevensothatthetwowidelyseparatedapertureshave
equal length). The frequency samples occur at the samples
, deletedband , ,
. The results of the pre-
vioussectioncarryoveriftheDirichletkernel of(48)is
replacedwiththemodifiedDirichletkernel
Fig.3. BandgapCRBforB = 25,M = 500(yieldingN = 550),and
several SNRs expressed using a standard Fourier resolution cell (FRC) of
2(cid:25)=N (cid:25)0:011rad.Theasymptotesfrom(71)andtheresolutionlimitsfrom
(73)arealsoshown.The“ringing”patternobservedisthesourcesmovingin
andoutofthenullsandpeaksoftheirrespectivearrayresponses.
(64)
Intermsofthesubaperturebandwidth andthebandsepara-
tion ,(65)becomes
ComputingtheLaurentseriesofthevarianceforsmall
Var Var
SNR SNR
(65) (71)
(72)
where SNR
(66)
(for large, fixed), where
.
Applying(47)to(65)[or(71)]yieldsthestatisticalresolution
limitforasplit-bandaperture
(67)
(68) SNR
(73)
(74)
SNR
(radians, large, fixed).
(69) Fig.3illustratestheCRBandresolutionachievableforesti-
matingtwopolepositionswhosesignalamplitudesareassumed
SNR SNR isthearraySNRwhenthepoles equal(with36dBarraySNR)butunknownfortwofrequency
coalesce, and SNR is the element-level SNR. Note that this bandsof25sampleseachseparatedby100samples.Fig.4com-
equationissymmetricin and andthatitreducesto(59)if parestheperformanceofthemaximum-likelihoodestimatorto
(asdesired).Theultrawidebandwidth ismoreconve- thecomputeCRBs.Notethatattheresolutionlimit,theRMSE
nientlyparameterizedintermsofthetwoseparatedbandwidths ofthemaximum-likelihoodestimatebreaksawayfromthereso-
andthebandgap as lutionboundbecausethepolesarenolongerresolvedandthere-
fore cannot be estimated individually as the CRB analysis as-
(70) sumes. Comparing Figs. 1 and 3, it is seen that splitting and
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