research article statistical angular resolution limit for ultrawideband mimo...

Research ArticleStatistical Angular Resolution Limit forUltrawideband MIMO Noise Radar

Xiaoli Zhou Hongqiang Wang Yongqiang Cheng Yuliang Qin and Haowen Chen

School of Electronic Science and Engineering National University of Defense Technology Changsha 410073 China

Correspondence should be addressed to Hongqiang Wang oliverwhq1970gmailcom

Received 23 April 2014 Accepted 15 October 2014

Academic Editor Michelangelo Villano

Copyright copy 2015 Xiaoli Zhou et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

The two-dimensional angular resolution limit (ARL) of elevation and azimuth for MIMO radar with ultrawideband (UWB) noisewaveforms is investigated using statistical resolution theory First the signal model of monostatic UWB MIMO noise radar isestablished in a 3D reference frameThen the statistical angular resolution limits (SARLs) of two closely spaced targets are derivedusing the detection-theoretic and estimation-theoretic approaches respectively The detection-theoretic approach is based onthe generalized likelihood ratio test (GLRT) with given probabilities of false alarm and detection while the estimation-theoreticapproach is based on Smithrsquos criterionwhich involves the Cramer-Rao lower bound (CRLB) Furthermore the relationship betweenthe two approaches is presented and the factors affecting the SARL that is detection parameters transmit waveforms arraygeometry signal-to-noise ratio (SNR) and parameters of target (ie radar cross section (RCS) and direction) are analyzedCompared with the conventional radar resolution theory defined by the ambiguity function the SARL reflects the practicalresolution ability of radar and can provide an optimization criterion for radar system design

1 Introduction

As a novel class of radar system multiple-input multiple-output (MIMO) radar has gained popularity and attractedattention [1] MIMO radar can offer additional degreesof freedom by employing multiple transmit and receiveantennas and it possess significant potentials for fadingmitigation resolution enhancement and interference andjamming suppression Fully exploiting these potentials canimprove target detection parameter estimation and targettracking and recognition performance [2] MIMO radar usesmultiple antennas to simultaneously transmit noncoherentwaveforms while the ultrawideband (UWB) noise waveformis an ideal candidate for MIMO operation [3] Recently therehas been some active research on using UWB noise radar inMIMOconfiguration [3ndash5] AUWBnoise radar transmittinga noise or noise-like waveform with a fractional bandwidthof over 25 or an absolute bandwidth of over 500MHz [3]has high range resolution uncoupled ambiguity function(AF) low probabilities of interception and detection andimmunity from interference [5 6] Combing the benefitsof MIMO radar and UWB noise radar UWB MIMO noise

radar satisfies some important military requirements and isconsidered to be a promising technique for high resolutiontarget detection and estimation [4]

For UWB MIMO noise radar a typical function is targetlocalization [2 7] and there exists performance improvementin this application while it is necessary to know the perfor-mance level of target localization expected from a particularradar configuration and resolution limit is an importantperformance measure indicating the capability to resolveclosely spaced targets Among the tools characterizing theresolution limit the classic AF is commonly used to describethe inherent resolution properties of time delay and Dopplerfor a specific waveform [8] Early AF concentrated on themonostatic narrowband applications Owing to the potentialadvantages of the UWB waveform and MIMO radar systemwidebandUWBAF [6 9 10] andMIMOAF [10ndash12] are theninvestigated

Although AF describes the resolving ability of radarwaveform and MIMO AF can simultaneously characterizethe effects of array geometry transmitted waveform andtarget scattering on resolution performance [2] the effect ofenvironment noise is neglected Moreover the conventional

Hindawi Publishing CorporationInternational Journal of Antennas and PropagationVolume 2015 Article ID 906313 12 pageshttpdxdoiorg1011552015906313

2 International Journal of Antennas and Propagation

resolution limit (ie AF) is limited by the Rayleigh limitwhereas the super-resolution theory indicates that resolutionbeyond theRayleigh limit is possible under certain conditionsrelated to signal-to-noise ratio (SNR)

The statistical resolution limit (SRL) [13] deduced fromthe statistical characteristic of radar measurement and wherethe uncertainty of measurement is taken into account hasattracted much attention recently The SRL can be usedto analyze the statistical resolution capability beyond theRayleighrsquos criterion and reflect the practical resolution abilityof radarThere aremainly three approaches to define the SRLThe earliest approach is based on themean null spectrum [14]which is quite simple whereas its application is limited to thespecific estimation algorithm In practical applications thereexist another two commonly used approaches independentof the algorithm to define the SRL (I) One is based ondetection theory the resolution problem is modeled as ahypothesis test to decide if the two sources are resolvable Andthe resolution limit is defined by the probability of error orright criterion [15ndash21] (II) The other is based on estimationaccuracy the SRL is given with the Cramer-Rao lower bound(CRLB) [13 15 16 22ndash24] There are two main criteria basedon the CRLB Leersquos criterion [25] and Smithrsquos criterion [13]

In recent years the SRLs of DOD (direction of departure)and DOA (direction of arrival) for MIMO radar have beendeeply researched [21 24] whereas the majority of thispublished work has assumed narrowband signal scenarioswhich cannot be applied to the widebandUWB situationdirectly The SRL of widebandUWB MIMO radar has notbeen investigated not to mention UWBMIMO noise radar

In this paper the 2D statistical angular resolution limits(SARLs) of elevation and azimuth for UWB MIMO noiseradar in a 3D reference frame are derived The prespecifieddetection-theoretic and estimation-theoretic approaches areused in the derivations which are based on GLRT with theconstraints on probabilities of false alarm and detection andCRLB with the constraint on Smithrsquos criterion respectivelyThe closed-form expressions of SARL are achieved whichare compact and can provide useful qualitative informationon the behavior of resolution limit Moreover the rela-tionships between SARL and different factors (ie SNRdetection parameters array geometry and the radar crosssection (RCS) and direction of target) are analyzed As forthe application the SARL presented in this paper can beused to evaluate the performance of sensor arrays in targetlocalization (such as in radar and sonar) and the approachesmay be applied to general detection andparameter estimationproblems

This paper is organized as follows In Section 2 weintroduce the signal model and the related assumptionswhich are used in the derivations In Section 3 detection-theoretic approach is presented to investigate the SARLThenthe SARL is further derived based on CRLB with the Smithrsquoscriterion and the relationship between the two approachesis discussed in Section 4 In Section 5 the results obtainedin Sections 3 and 4 are discussed to provide further insightinto the SARL and some insightful properties revealed by theresults of simulations are given Finally some comments andconclusions are presented in Section 6

A comment on notation is as follows we use boldfacelowercase letters for vectors and boldface uppercase lettersfor matrices (sdot)119867 (sdot)lowast (sdot)119879 and (sdot)minus1 denote the conjugatetranspose conjugate transpose and inverse of a matrixrespectively 119864sdot is used for expectation with respect to therandomvariableRsdot andIsdot denote the real and imaginaryparts respectively Finally sdot denotes the Frobenius normof a vector and I

119873is the119873th-order identity matrix

2 Problem Setup

21 Assumptions Throughout the paper it is assumed thatthe following assumptions hold

(1) The radar is operated inmonostatic configuration andthen the DOD is approximately equal to the DOA

(2) The targets are point targets located in the far-field ofthe radar array

(3) The background noise is an independent additivewide-sense stationary (WSS) complex circular whiteGaussian random process with zero-mean and vari-ance 1205902

119899

(4) All transmit waveforms are mutually orthogonalband-limited noise waveforms with equal power 1205902

119904

and bandwidth 119861119908 Meanwhile the signal spectral

components at different frequencies are also mutuallyorthogonal

(5) The channel attenuation is ignored(6) The coherent processing time for signal and noise is

sufficient

22 Signal Model Consider two closely spaced targets inthe far-field for MIMO radar Target T1 is assumed as thereference target with a known direction 120579

1= (1205931 1206011) and

RCS while target T2 whose direction 1205792

= (1205932 1206012) is

unknown is close to T1 120593119894isin [0 1205872] and 120601

119894isin [0 2120587] denote

the elevation and azimuth of the 119894th target (forall119894 isin 1 2)respectively

TheMIMO radar geometry is shown in Figure 1 Both the119873-element transmitters and119872-element receivers which arelocated in the same plane are circular arrays (but not limitedto circular array) with radii 119903

119905and 119903119903 respectively The origin

119874 of the Cartesian coordinate system is set up as the center ofthe arrays wherein the radar array and targets are consideredas point objects

In this paper a single snapshot is considered Then thedemodulated baseband signal received by the 119898th receivercan be expressed as [7]

119910119898 (119905) =

2

sum119894=1

120572119894

119873

sum119899=1

119904119899(119905 minus 120591

119894

119898119899) 119890minus1198952120587119891

119888120591119894

119898119899 + 119908119898 (119905) (1)

where 119904119899(119905) is the baseband stochastic signal transmitted

by the 119899th transmitter with the bandwidth 119861119908 119908119898

is thenoise received by the 119898th receiver and 119891

119888is the carrier

frequency of the transmit waveforms 120572119894denotes the com-

plex amplitude proportional to the RCS of the 119894th target

International Journal of Antennas and Propagation 3

z

x

yOrt

rr

ReceiverTransmitter

T1T2

12059321205931

12060121206011

Figure 1 Geometry ofmonostaticMIMOradarwith circular arrays

120591119894119898119899

= sin120593119894(119903119903cos(120601119894minus120599119903119898) + 119903119905cos(120601119894minus120599119905119899))119888 is the relative

propagation delay corresponding to the 119899th transmitter(located at 120579

119905119899= (1205872 120599

119905119899)) and the 119898th receiver (located at

120579119903119898

= (1205872 120599119903119898)) with respect to the reference point119874 where

119888 is the speed of wave propagationSince 119904

119899(119905) is an UWB stochastic signal which is not

convenient to analysis thus 119904119899(119905) is expressed as the spectral

representation 119904119899(119905) = int

1198911198901198952120587119891119905119889119878

119899(119891) in the frequency

domain 119889119878119899(119891) denotes a measure of the signal spectrum

at frequency 119891 which is independent of variable 119905 and willsimplify the derivations involving the stochastic signal

3 SARL Based on Detection-TheoreticApproach

In this section the 2D SARL based on the hypothesis test(more precisely GLRT) is derived using the signal model inSection 2

31 Statistic of the Observation From the statistical signalprocessing theory we know that the sufficient statistic is theeffective summarization and makes the probability densityfunction (PDF) be independent with the unknown param-eters For UWB MIMO noise radar one of the sufficientstatistics for targets resolving is

E ≜ y (119905) = x (119905) + w (119905) (2)

where y(119905) = [1199101(119905) 119910

119872(119905)]119879 x(119905) = [119909

1(119905) 119909

119872(119905)]119879

w(119905) = [1199081(119905) 119908

119872(119905)]119879 119909

119898(119905) = sum

2

119894=1120572119894sum119873

119899=1119904119899(119905 minus

120591119894119898119899)119890minus1198952120587119891119888120591

119894

119898119899

32 Linear Form of the Signal Model The 2D SARL can beexpressed as 120575 ≜ [120575120593 120575

120601]119879 where 120575

120593≜ 1205932minus 1205931and

120575120601≜ 1206012minus 1206011 The derivation of 120575

120593and 120575

120601is a nonlinear

optimization problem which is analytically intractable Wetherefore approximate the model to be linear wrt theunknown parameters

Substituting 119904119899(119905) = int

1198911198901198952120587119891119905119889119878

119899(119891) into 119909

119898(119905) the first

order Taylor expansion of 119909119898(119905) around 120575 = 0 is given as

119909119898 (119905) asymp (120572

1+ 1205722)

119873

sum119899=1

int119891

1198901198952120587(119891(119905minus120591

1

119898119899)minus1198911198881205911

119898119899)119889119878119899(119891)

minus 11989521205871205722120575120593cos1205931

119873

sum119899=1

int119891

119886119898119899(119891 + 119891

119888)

times 1198901198952120587(119891(119905minus120591

1

119898119899)minus1198911198881205911

119898119899)119889119878119899(119891)

+ 11989521205871205722120575120601sin1205931

119873

sum119899=1

int119891

119887119898119899(119891 + 119891

119888)

times 1198901198952120587(119891(119905minus120591

1

119898119899)minus1198911198881205911

119898119899)119889119878119899(119891)

(3)

Then the sufficient statistic can be expressed as

y = G120577 + w (4)

120577 = [1205721+ 12057221205722120575119879]119879

G = [B C D]

119861119898=

119873

sum119899=1

int119891

1198901198952120587(119891(119905minus120591

1

119898119899)minus1198911198881205911

119898119899)119889119878119899(119891)

119862119898= minus1198952120587 cos120593

1

119873

sum119899=1

int119891

119886119898119899(119891 + 119891

119888)

times 1198901198952120587(119891(119905minus120591

1

119898119899)minus1198911198881205911

119898119899)119889119878119899(119891)

119863119898= 1198952120587 cos120593

1

119873

sum119899=1

int119891

119887119898119899(119891 + 119891

119888)

times 1198901198952120587(119891(119905minus120591

1

119898119899)minus1198911198881205911

119898119899)119889119878119899(119891)

(5)

where119861119898119862119898 and119863

119898represent the119898th element ofBC and

D respectively 119886119898119899

and 119887119898119899

are denoted by

119886119898119899

=(119903119903cos (120601

1minus 120599119903119898) + 119903119905cos (120601

1minus 120599119905119899))

119888

119887119898119899

=(119903119903sin (120601

1minus 120599119903119898) + 119903119905sin (120601

1minus 120599119905119899))

119888

(6)

33 Derivation of the SARL It is a common sense thatthe targets can be resolved easily and the probability ofdetection is higher if the spacing between the two targets islarger In this sense the probability of detection can be usedto represent the resolvability of two closely spaced targetsConsequently the resolution problem can be modeled as ahypothesis test problem and the relationships between theminimal resolvable 120575 and the factors affecting the resolutionlimit can be obtained Denoting that P = [0 I2] as P120577 = 120572

2120575


and 1205722

= 0 the corresponding hypothesis for the resolutionproblem is given by

H0 P120577 = 0

H1 P120577 = 0

(7)

whereH1andH

0indicate that the two targets are resolvable

or not respectivelyA basic method for hypothesis test is the classical

approach based on Neyman-Pearson (NP) lemma [26]According to the NP lemma the optimum solution to theabove hypothesis test problem is the likelihood ratio test(LRT) However since 120575 is unknown it is impossible to givethe PDF exactly under H

1and design an optimal detector

in the NP sense A common approach is the GLRT in thiscase GLRTfirst computes themaximum likelihood estimates(MLEs) of the unknown parameters and then uses theestimated values to form the standard NP detector AlthoughGLRT is not optimal it is an asymptotically uniformly mostpowerful (UMP) test among all the invariant statistical tests[26] and its performance is very close to that of an idealdetector to which the values of all the parameters in themodel are known Hence the performance of the describeddetector can be reasonably considered as an approximate per-formance bound in practice According to (7) the expressionof GLRT is

119871 (y) =max120575119901 (y 120577H1)max120575119901 (yH0)

=119901 (y H

1)

119901 (yH0)

H1

≷H0

120578 (8)

where 119901(y 120577H1) and 119901(yH

0) are the PDFs under H

1and

H0 respectively 120578 is detection threshold = (GHG)minus1GHy

is the MLE of 120585 when 119901(y H1) is maximized Equivalently

the GLRT in (8) can be rewritten as

119879 (y) = 2 ln 119871 (y)H1

≷H0

1205781015840= 2 ln 120578 (9)

Then the GLRT statistic for the approximated modelyields [26]

119879 (y) = 2

1205902119899

(P)H(P (GHG)

minus1

PH)minus1

P (10)

To choose the value of 1205781015840 the asymptotic performance ofthe test [26] is formed as (11) as the expression of GLRT maybe generally complicated and its performancemay be difficultto analyze Consider

119879 (y) sim

12059422119872 H

0

120594101584022119872

(120582 (119901119891119886 119901119889)) H

1

(11)

where 1205942

2119872and 12059410158402

2119872(120582(119901119891119886 119901119889)) are the central and non-

central chi-square distribution with 2119872 degrees of freedomrespectivelyThe noncentrality parameter is 120582(119901

119891119886 119901119889)which

is critical in illuminating the relationship between the SNRand SARL Then 120578

1015840 is conditioned by the probabilities of

false alarm and detection which are defined as 119901119891119886

=

1198761205942

2119872

(1205781015840) and 119901119889= 11987612059410158402

2119872(120582(119901119891119886119901119889))(1205781015840) respectively 119876

1205942

2119872

(sdot)

and 11987612059410158402

2119872(120582(119901119891119886119901119889))(sdot) denote the right tail probabilities of 1205942

2119872

and 120594101584022119872(120582(119901119891119886 119901119889)) respectively Then 120582(119901

119891119886 119901119889) can be

computed as the solution of

119876minus1

1205942

2119872

(119901119891119886) = 119876

minus1

1205942

2119872(120582(119901119891119886119901119889))(119901119889) (12)

On the other hand 120582(119901119891119886 119901119889) can be given as

120582 (119901119891119886 119901119889) =

2

1205902119899

(P120577)H (P(GHG)minus1PH)minus1

(P120577) (13)

From the viewpoint of statistics it is impossible to resolvethe two closely spaced targets at 100 percent whereas theyare resolvable at a given probability Since the constraints on119901119891119886and 119901

119889for test (7) are assigned the SRL can be rigorously

defined by the constraints otherwise the resolution limit canbe arbitrarily low and the result may be meaningless [20]Based on the constraints we may claim that the two targetsare statistically resolved under the constraints on 119901

119891119886and

119901119889 In other words the probability to resolve both targets is

119901119889for a given 119901

119891119886 There definitely exists a minimum 120575 so

that both constraints can be satisfied If 120575 is lower than theminimumvalue the constraints on119901

119891119886and119901119889will be broken

The minimum 120575 therefore can be taken as the SRLWhen computing (P(GHG)minus1PH)minus1 it is necessary to

obtain BHB BHC BHD CHC CHD and DHD Since 119904119899(119905)

is a stochastic signal the expectation operator is appliedto calculate (P(GHG)minus1PH)minus1 As we have assumed that thespectral components at different frequencies are mutuallyorthogonal in Section 2 then [27]

119864 [119889119878119899(119891) 119889119878

lowast

119899(1198911015840)] =

120588119899(119891) 119889119891 119891 = 1198911015840

0 119891 = 1198911015840(14)

where 120588119899(119891) is the power spectrum density (PSD) of 119904

119899(119905) at

the frequency 119891 From (5) and (14) we can obtain

119864 BHB = 1198721198731205902

119904 (15)

where 1205902119904= int119891120588119899(119891)119889119891 is the power of each transmitter The

same way is used to obtain 119864BHC 119864BHD and so onThe total SNR is defined as SNR = (1198731205902

119904)(1198721205902

119899)

where 1205902119899can be obtained from (13) Finally the relationship

between the SARL 120575 and the minimum SNR which isrequired to resolve two closely spaced targets is given by

SNR =119873120582 (119901

119891119886 119901119889)

8120587211987210038171003817100381710038171205722

10038171003817100381710038172120575119879(1198912119888(1205761minus 1205762119872119873) + 1198612rms1205761) 120575

(16)

where 119861rms = radicint1198911198912120588(119891)119889119891 int

119891120588(119891)119889119891 is the root mean

square (RMS) bandwidth The RMS bandwidth which is notequivalent to bandwidth 119861

119908 is commonly used as a basic


quantitative characteristic to describe the feature of powerspectra Consider

1205761= [

[

(cos1205931)21205761

minus sin1205931cos12059311205763

minus sin1205931cos12059311205763

(sin1205931)21205762

]

]

1205762= [

[

(cos1205931)212057624

minus sin1205931cos120593112057641205765

minus sin1205931cos120593112057641205765

(sin1205931)212057625

]

]

1205761=

119872

sum119898=1

119873

sum119899=1

1198862

119898119899 120576

2=

119872

sum119898=1

119873

sum119899=1

1198872

119898119899 120576

3=

119872

sum119898=1

119873

sum119899=1

119886119898119899119887119898119899

1205764=

119872

sum119898=1

119873

sum119899=1

119886119898119899 120576

5=

119872

sum119898=1

119873

sum119899=1

119887119898119899

(17)

From (16) the SARL depends not only on SNR butalso on the detection parameters transmit waveforms arraygeometry RCS and direction of target In Particular 120576

1=

1205762= 119872119873(1199032

119905+ 1199032119903)(21198882) and 120576

3= 1205764= 1205765= 0 for uniform

circular array (UCA) when119872119873 = 1 Then the SARL can begiven by

cos212059311205752

120593+ sin2120593

11205752

120601

=1198882120582 (119901

119891119886 119901119889)

41205872119872210038171003817100381710038171205722

10038171003817100381710038172(1199032119905+ 1199032119903) (1198912119888+ 1198612rms) SNR

(18)

Then we can conclude that the SARL is independent ofthe azimuth of target for UCA

To complement the results in this section we carryout the Fisher information matrix (FIM) derivation for thegeneral signal model to achieve the SARL expression formthe parameter estimation perspective in the following

4 SARL Based on Estimation-TheoreticApproach

In this section we derive and analyze the SARL for UWBMIMO noise radar based on the Smithrsquos criterion whichinvolves the CRLB The SARL can be defined as the solutionof the Smith equation 120575 = radicCRLB(120575) To solve the equationwe need to have a closed-form expression of CRLB(120575) Thederivation process is as follows First the FIM derivation forthe general signal model is carried out Then the closed-form expression of the CRLB for the considered problem isachieved Finally we derive in closed form the analytic SARLunder the assumption of UCA

41 FIM Derivation Since T1 is considered as the referencetarget 120572

1 1205931 and 120601

1are assumed to be known Thus the

vector of unknown parameters in (2) is

120585 = [120572119877

2120572119868

2⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟120572

120575120593120575120601⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟120575

]

119879

(19)

where 1205721198772

= R1205722 and 120572119868

2= I120572

2 denote the real and

imaginary parts of 1205722 that is 120572

2= 1205721198772+ 1198951205721198682 In (19) we

identify two sets of parameters parameters of interest 120575 ≜[120575120593 120575

120601]119879 and nuisance parameters 120572 ≜ [120572

119877

2120572119868

2]119879

The FIM J(120585) with respect to the vector 120585 is given by [7]

J (120585) = 119864y|120585 [120597

120597120585log119901 (y | 120585)] [ 120597

120597120585log119901 (y | 120585)]

119867

(20)

where 119901(y | 120585) is the joint PDF of y conditioned on 120585 and119864y|120585sdot is the conditional expectation of y given 120585

The received signals y are parameterized by the unknownparameters 120585 The conditional joint PDF of the observationsat the receivers given by (1) is

119901 (y | 120585) prop expminus 1

1205902119899

119872

sum119898=1

int119879

1003817100381710038171003817119910119898 (119905) minus 119909119898 (119905)10038171003817100381710038172119889119905 (21)

Then we develop the FIM for 120585 based on the conditionalPDF in (21) The expression for (20) is obtained using

[J (120585)]119894119895 = [J (120585)]119895119894 = minus119864y|120585 1205972 log119901 (y | 120585)

120597120585119894120597120585119895

119894 119895 = 1 2 3 4

(22)

Before the derivation continues some conclusions shouldbemade Denote 119904

119899(119905minus120591119897119898119899) = 120597119904

119899(119905minus120591119897119898119899)120597120591119897119898119899 as described

in Section 2 119904119899(119905) = int

1198911198901198952120587119891119905

119889119878119899(119891) then 119904

119899(119905 minus 120591

119897

119898119899) can be

expressed as

119904119899(119905 minus 120591

119897

119898119899) = int119891

minus11989521205871198911198901198952120587119891(119905minus120591

119897

119898119899)119889119878119899(119891) (23)

According to the conclusion described in (14) the con-clusions can be drawn as follows

119864int119879

119904119899(119905 minus 120591

1

119898119899) 119904lowast

119899(119905 minus 120591

1

119898119899) 119889119905 = int

119891

minus1198952120587119891120588119899119889119891 = 0

119864 int119879

119904119899(119905 minus 120591

1

119898119899) 119904lowast

119899(119905 minus 120591

1

119898119899) 119889119905 = int

119891

(2120587119891)2120588119899119889119891

= 412058721198612

rms1205902

119904

(24)

Based on (14) and (24) the expression for the FIM J(120585) isderived in the Appendix

42 CRLB Derivation TheCRLB provides a lower bound forthe MSE of any unbiased estimator for unknown parametersGiven a vector parameter 120585 the CRLB is defined as

CRLB (120585) = [J (120585)]minus1 (25)

To give an insight into the terms in the expression of FIMJ(120585) defined by (20) can be partitioned as

[J (120585)] = [J120572120572 J119867120572120575

J120572120575 J120575120575] (26)


where the elements J120572120572 J120572120575 and J120575120575 are defined as

J120572120572 = 2119872119873snrI2

J120572120575 = J119867120572120575= 4120587119891

119888snr[

1205721198682cos12059311205764

minus1205721198772cos12059311205764

minus1205721198682sin12059311205765

1205721198772sin12059311205765

]

J120575120575 = 81205872 10038171003817100381710038171205722

10038171003817100381710038172(1198612

rms + 1198912

119888)

times snr[cos2120593

11205761

minus sin1205931cos12059311205763

minus sin1205931cos12059311205763

sin212059311205762

]

(27)

where snr = 12059021199041205902119899and I2is the identity matrix Then the

CRLB of the parameters of interest 120585 can be expressed as

CRLB (120575) = [J120575120575 minus J120572120575Jminus1

120572120572J119867120572120575]minus1

=119873

8120587210038171003817100381710038171205722

10038171003817100381710038172119872SNR

[[[[[

[

((1198612

rms + 1198912

119888) 1205761minus119891211988812057624

119872119873) cos2120593

1minus((1198612rms + 119891

2

119888) 1205763minus119891211988812057641205765

119872119873) sin120593

1cos1205931

minus((1198612rms + 1198912

119888) 1205763minus119891211988812057641205765

119872119873) sin120593

1cos1205931((1198612rms + 119891

2

119888) 1205762minus119891211988812057625

119872119873) sin2120593

1

]]]]]

]

minus1

(28)

According to the Smithrsquos criterion the 2D SARLs in thiscase are

120575120593= radicCRLB (120593) = radic[CRLB (120575)]11


(29)

As is shown in (28) the derivation of SARL in closed formis difficult in the context ofMIMO radar However in the caseof UCA 120576

1= 1205762= 119872119873(1199032

119905+ 1199032119903)(21198882) and 120576

3= 1205764= 1205765= 0

Then the CRLB can be simplified as

CRLB (120575) = 1198882

4120587210038171003817100381710038171205722

100381710038171003817100381721198722

1003817100381710038171003817120572210038171003817100381710038172(1199032119905+ 1199032119903) (1198612rms + 119891

2

119888) SNR

times[[[

[

1

cos21205931

0

01

sin21205931

]]]

]

(30)

Consequently the SARLs of elevation and azimuth are

120575120593=

119888

212058711987210038171003817100381710038171205722

1003817100381710038171003817 cos1205931radic

1

(1199032119905+ 1199032119903) (1198912119888+ 1198612rms) SNR

(31)

120575120601=

119888

212058711987210038171003817100381710038171205722

1003817100381710038171003817 sin1205931radic

1

(1199032119905+ 1199032119903) (1198912119888+ 1198612rms) SNR

(32)

43 Relationship between the SARL Based on Detection-Theoretic and Estimation-Theoretic Approaches The SARLshave been derived based on the detection-theoretic and

estimation-theoretic approaches respectively As describedin [26] the noncentrality parameter 120582(119901

119891119886 119901119889) is given as

120582 (119901119891119886 119901119889) = (120575 minus 0)

119879([Jminus1 (120585)]

120575120575)minus1

(120575 minus 0) (33)

where [Jminus1(120585)]120575120575 denotes the J120575120575 defined in (26) As theobservation time is sufficiently long (33) is equivalent to

120582 (119901119891119886 119901119889) cong (120575 minus 0)

119879([Jminus1 (120585)]

120575120575

10038161003816100381610038161003816120575=0)minus1

(120575 minus 0)

= 120575119879(CRLB (120575)|120575=0)

minus1120575

(34)

where ldquocongrdquo stands for ldquoasymptotically equalrdquoThe conclusion in (34) can be verified by (16) and (28)

Moreover radic120582(119901119891119886 119901119889) the proportionality constant corre-sponding to a given couple (119875

119891119886 119875119889) is called ldquotranslation

factorrdquo (from CRLB to the SRL) in [20] Thus the asymptoticSRL based on the hypothesis test approach is consistent withthe SRL based on the CRLB approach (ie using the Smithrsquoscriterion) Furthermore the SRL based on Smithrsquos criterion isequivalent to an asymptotically UMP test among all invariantstatistical tests when 120582(119901

119891119886 119901119889) = 1 Consequently (34)

shows that the detection-based approach is unified with theestimation-based approach [20]

5 Discussion and Numerical Analysis

The 2D SARLs for UWB MIMO noise radar have beenderived in Sections 3 and 4 In this section the resultsexpressed in (16) and (28) are discussed Then some numeri-cal simulations are made to analysis the factors impacting the2D SARLs


Table 1 RMS bandwidth for different signals

Types of spectra Expression RMS bandwidthRectangularlow-frequencyspectrum

120588 (119891) =

1205880

10038161003816100381610038161198911003816100381610038161003816 le 1198611199082

010038161003816100381610038161198911003816100381610038161003816 gt 1198611199082

1198611199082radic3

Gaussianspectrum 120588 (119891) = 120588

0exp (minus2119891212059021198612

119908) 120590119861

1199082

51 Discussion The following comments are intended toprovide further insight into the results obtained in Sections3 and 4

(A1) Since the SARL is inversely proportional to RMSbandwidth 119861rms and carrier frequency 119891

119888 increas-

ing 119861rms and 119891119888will improve the SARL while for

narrowband signals 119861rms119891119888 ≪ 1 the SARL isinversely proportional to 119891

119888and independent of 119861rms

As119861rms is the integration over the range of frequencieswith nonzero signal content 119861rms is determined byboth the bandwidth 119861

119908and the distribution of PSD

As is shown in Table 1 the RMS bandwidth ofnoise waveforms for different distributions of PSD isdifferent which leads to different SARL

(A2) As is shown in (28) the off-diagonal elements of the2 times 2-order CRLB(120575) corresponding to 120575

120593and 120575

120601

are not equivalent to zero which means the CRLBis coupled and the SARL of one direction parameter(120575120593or 120575120601) is degraded by the uncertainty of another

The uncoupling is important which means that theSARL of one parameter is independent of others [28]Since both diagonal and off-diagonal elements of J120575120575depend on the array geometry the optimum arraygeometry can be found to get the best resolution limitAs the off-diagonal elements of J120575120575 are zero the arraygeometry for monostatic MIMO radar with UCAconfiguration satisfies the uncoupling conditions in[1]

(A3) The SARL is strongly reliant on the relative geometryof antennas versus target locationThis dependency isincorporated in the terms of 120601

1and 120593

1 Moreover it

is apparent from (28) that there is a tradeoff betweenthe SARLs of the azimuth and elevation For examplea set of antennasrsquo locations that increases the SARLof azimuth may result in a low SARL of elevation aswe expect intuitivelyThis is caused by the fact that 120575

120601

is summation of sine functions and 120575120593is summation

of cosine functions In order to truly determine theminimum achievable SARLs in both azimuth andelevation we need to minimize the overall SARLdefined as the total SARL in the next part

(A4) As the SARLs in (16) and (28) are achieved for thecoherent processing approach the phase informationis used which leads to the resolution limit improve-ment comparing with the noncoherent processingapproach However both time synchronization andphase synchronization between the transmitters andreceivers are necessary in this case while only time

synchronization is necessary in the noncoherent case[7] which is a challenge for the radar system

(A5) The SARL as expressed by CRLB provides a tightbound at high SNR while the CRLB is not tight atlow-SNR case thus a more rigid bound is necessaryfor the estimation accuracy in this case which isbeyond this paper

52 Numerical Simulation The factors impacting the 2DSARLs of MIMO radar with UWB noise waveforms areanalyzed by numerical simulations as follows The transmitsignals are set up as ideal bandlimited Gaussian noise wave-forms with 119891

119888= 9GHz and 119861

119908= 02119891

119888 then the RMS

bandwidth is 119861rms = 1198611199082radic3 And we consider the case that

the number of antennas is119872 = 119873 = 4 and all antennas lie ona disk of radius 119903

119905= 119903119903= 119903 = 1m

First the SARLs for different target directions and arraygeometry configurations are simulated to verify the com-ments A2 and A3 Assume that the transmitters and receiversare configured in the same array geometry Consider fourrelative geometries of antennas versus target location inFigure 2 where ldquotimesrdquo represents the target location with theazimuth 120601

1from the vertical view

It can be shown from Figure 3 that the SARL is stronglyrelated to array geometry and target location As for UCA intype (d) 120576

3= 1205764= 1205765= 0 satisfies the uncoupling conditions

described in [1] while the array geometries described inFigures 2(a)sim2(c) are coupling configurations As discussedabove the coupling increases the parameter uncertaintywhich degrades the SARL and the conclusion is verifiedin Figure 3 Meanwhile the aperture size of the array fortype (d) is bigger than the other types of array geometryFurthermore comparing Figure 3(a) with Figure 3(b) theSARL of elevation degrades with the elevation of target whilethe SARL of azimuth is enhanced by the increasing elevationThe contradictionmakes it hard to design the array geometryversus the target direction

As described in [20] the SARL (in the two-target case)is asymptotically proportional to the square root of the lowerbound of the MSAE (in the single-target case) The MSAE isdefined as

MSAECRLB ≜ sin2120593CRLB (120601) + CRLB (120593) (35)

where CRLB(120601) and CRLB(120593) are the CRLBs on 120601 and 120593

(in the single-target case 120601 = 1206011= 1206012and 120593 = 120593

1=

1205932) respectively radicMSAECRLB contains the CRLBs of the

azimuth and elevation and provides an overall measure ofthe resolution limit thus radicMSAECRLB can be used as theldquototal SARLrdquoMoreover it is also a very useful qualitymeasurefor gaining physical insight into the SARL in 3D spaceFigure 4 shows thatradicMSAECRLB varieswith the elevation andazimuth of target for different types of array geometry Thereare some lowest points in each subfigure which can be chosenas the tradeoff between the SARLs of azimuth and elevation incertain applications And the tradeoff elevation and azimuthdiffer for different array geometry configurations Whendesigning the radar system the radar plane can be adjusted to


AntennaTarget

(a)

AntennaTarget

(b)

AntennaTarget

(c)

AntennaTarget

(d)

Figure 2 Different types of antenna geometry (a) All elements concentrated at locations 1199031198901198951206011 or 119903119890119895(1206011+120587) (b) All elements concentrated atlocations 119903119890119895(1206011+1205872) or 119903119890119895(1206011minus1205872) (c) The elements are randomly located at the disk (d) The elements are uniformly located on the disk

10 20 30 40 50 60 70 80Elevation of target (deg)

SARL

of e

leva

tion

(deg

)

Type (a)Type (b)

Type (c)Type (d)

10minus3

10minus4

10minus5

(a) SARL of elevation versus the elevation of target


SARL

of a

zim

uth

(deg

)

Type (a)Type (b)

Type (c)Type (d)

10minus3

10minus4

10minus5

(b) SARL of azimuth versus the elevation of target

Figure 3 SARLs of elevation and azimuth versus the elevation of target for different types of array geometry

make the target located at the tradeoff elevation and azimuthfrom the plane which will lead to the best SARL

Next we focus on the effects of other factors on the SARLIn the simulations both the transmitters and receivers areassumed to be arranged in UCA and the SARL based ondetection approach is considered

The effect of RCS on the resolution limit can be concludeddirectly from (16) and (28) The SARL is improved as theRCS increases Although RCS does not affect the transmitwaveform the received signal power and the probability ofdetection increase as the RCS increases which induces abetter resolution limit Thus the target characteristic indeedcould affect the resolution limit which is ignored by theconventional radar resolution theory In the frameworkof our derivations target T1 is assumed as the referencetarget with a known direction and RCS while target T2 is

unknown and is close to T1 thus the SARL is independentof the RCS of T2 However the RCS values of both targetscould affect the SARL as the received signal power andSNR will increase if the RCS values increase In our futurework we consider the resolution of two targets in casethat both the targets are unknown although the derivationsbecome complex Some related work has been published in[29]

Figure 5 shows the effect of detection parameters whereonly 120575

120593is considered (the same way is done for 120575

120601) From

Figure 5 the resolution limit decreases as 119901119889increases and

119901119891119886

decreases since increasing 119901119889and decreasing 119901

119891119886lead

to a more selective decision Then the resolution limit canbe improved by increasing 119901

119891119886and calling for a lower 119901

119889

However the performance of detection will be depressed inthis case


2

15

05

1

0

0

300200

100

Elevation (deg)

Azimuth (deg)

8060

4020

2

18

16

14

12

08

06

04

02

1

times10minus3

times10minus3radic

MSA

E CRL

B

(a)

radicM

SAE C

RLB

2

15

05

1

0

0

300200

100

Elevation (deg)

Azimuth (deg)

8060

4020

2

18

16

14

12

08

06

04

02

1

times10minus3

times10minus3

(b)

0

0

300200

100

Elevation (deg)

Azimuth (deg)

8060

4020

times10minus4

radicM

SAE C

RLB

5

4

3

2

1

times10minus4

5

4

3

2

1

(c)

0

0

300200

100Elevation (deg)

Azimuth (deg)

8060

4020

times10minus4

times10minus4

2

1

radicM

SAE C

RLB

2

15

1

05

(d)

Figure 4radicMSAECRLB versus the elevation and azimuth of target for different types of array geometry

Figure 6 shows the resolution cells for different targetdirections and SNR with (119901

119891119886 119901119889) = (001 099) For visual

convenience 120575120593and 120575120601are scaled up to 120 times Obviously

increasing SNR can improve the SARLMeanwhile the SARLvaries with the target direction because of the spatial diversityof radar which means that the SARL is ldquononuniformrdquo on thewhole space Moreover the resolution cell is ellipse whichmeans that the SARL varies at different directions for a giventarget location

Consider the SARL of elevation 120575120593as shown in (31)

the relationship between SARL and the radius of the arraysis simulated in Figure 7 The simulation parameters areconfigured as 119872 = 119873 = 4 119903

119905= 119903119903= 119903 isin 1m 2m 3m

and 1205931= 1∘ Figure 7 represents that increasing the radius of

the arrays will lead to the resolution limit improvement sincea lager virtual aperture is achieved in this case according tothe MIMO radar virtual aperture theory

6 Conclusion

Resolution limit can be interpreted as the minimal differenceto resolve two closely spaced targets and is a critical quantityto evaluate the performance of a radar system In this paperwe have derived the explicit expressions for 2D SARLsof elevation and azimuth for UWB MIMO noise radarwhich are defined using GLRT based on hypothesis test andestimation approach based on Smithrsquos criterion respectivelyThe relationship between the two approaches has been provedand verified Meanwhile we have analyzed the effects ofSNR detection parameters transmit waveforms and arraygeometry and the parameters of target on the SAR and somesignificant conclusions are achieved as follows (1)The SARLsbased on the detection and estimation approaches are unified(2) The array geometry is an important factor that impactsthe SARL (3) There is a tradeoff between the SARLs of


01 02 03 04 05 06 07 08 09

051

152

253

354

455

55

SARL

times10minus4

pfa = 001

pfa = 005

pfa = 01

pfa = 02

pd

Figure 5 SARL as a function of detection parameters (119901119891119886 119901119889) for

SNR = 20 dB and 1205931= 1∘

Elevation (rad)

Azi

mut

h (r

ad)

0 05 1 150

05

1

15

SNR = 15dBSNR = 20dBSNR = 25dB



Figure 6 Resolution cell versus target direction for elevation 1205931isin

10∘ 30∘ 60∘ and (119901119891119886 119901119889) = (001 099)

elevation and azimuth which can be determined by the totalSARL radicMSAECRLB (4) The SARLs of MIMO radar on thewhole space and at different directions are ldquononuniformrdquo(5) The target characteristic and detection parameters (ifthe detection approach is used) can impact the resolutionlimit (6) Increasing the bandwidth and SNR will result inthe SARL improvement Compared with the conventionalradar resolution theory the SRL reflects not only the effectof transmit waveforms but also SNR parameters of targetandmeasurement approachTherefore the SRL describes thepractical resolution ability of radar effectively and providesthe optimization criterion for radar system design

10 15 20 25 30 35 40 45 50SNR (dB)

SARL

(rad

)

10minus3

10minus4

10minus5

r = 1mr = 2mr = 3m

Figure 7 SARL as a function of 119903 and SNR (in dB)

Appendix

Derivation of the FIM in (20) is as followsThe first derivative of log119901(y | 120585) with respect to the

elements of 120585 is

120597 log119901 (y | 120585)1205971205721198772

=2

1205902119899

119872

sum119898=1

int119879

Re(119910119898 (119905) minus

2

sum119897=1

120572119897

119873

sum119899=1

119904119899(119905 minus 120591

119897

119898119899) 119890minus1198952120587119891

119888120591119897

119898119899)

times (

119873

sum119899=1

119904119899(119905 minus 120591

2

119898119899) 119890minus1198952120587119891

1198881205912

119898119899)

lowast

119889119905

(A1)120597 log119901 (y | 120585)

1205971205721198682

=2

1205902119899

119872

sum119898=1

int119879

Re(119910119898 (119905) minus

2

sum119897=1

120572119897

119873

sum119899=1

119904119899(119905 minus 120591

119897

119898119899) 119890minus1198952120587119891

119888120591119897

119898119899)

times (119895

119873

sum119899=1

119904119899(119905 minus 120591

2

119898119899) 119890minus1198952120587119891

1198881205912

119898119899)

lowast

119889119905

(A2)120597 log119901 (y | 120585)

120597120575120593

=minus2 cos120593

1

1205902119899

times

119872

sum119898=1

int119879

Re

times (119910119898 (119905) minus

2

sum119897=1

120572119897

119873

sum119899=1

119904119899(119905 minus 120591

119897

119898119899) 119890minus1198952120587119891

119888120591119897

119898119899)


times (1205722

119873

sum119899=1

119886119898119899( 119904119899(119905minus1205912

119898119899)+1198952120587119891

119888119904119899(119905minus1205912

119898119899))

times119890minus1198952120587119891

1198881205912

119898119899)

lowast

119889119905

(A3)

120597 log119901 (y | 120585)120597120575120601

=2 sin120593

1

1205902119899

times

119872

sum119898=1

int119879

Re(119910119898 (119905) minus

2

sum119897=1

120572119897

119873

sum119899=1

119904119899(119905 minus 120591

119897

119898119899) 119890minus1198952120587119891

119888120591119897

119898119899)

times (1205722

119873

sum119899=1

119887119898119899( 119904119899(119905 minus 120591

2

119898119899)

+ 1198952120587119891119888119904119899(119905 minus 120591

2

119898119899))

times 119890minus1198952120587119891

1198881205912

119898119899)

lowast

119889119905

(A4)

Denote snr = 12059021199041205902119899and use the conclusions in (24) and

then apply the second derivative to (A1)sim(A4) to define FIMJ(120585) with the following elements

[J (120585)]11 = minus 119864y|120585 1205972 log119901 (y | 120585)

(1205971205721198772)2

1003816100381610038161003816100381610038161003816100381610038161003816120575120593=0 120575120601=0

= 2119872119873snr

(A5)

[J (120585)]22 = minus 119864y|120585 1205972 log119901 (y | 120585)

(1205971205721198682)2

1003816100381610038161003816100381610038161003816100381610038161003816120575120593=0 120575120601=0

= 2119872119873snr

(A6)

[J (120585)]21 = [J (120585)]12

= minus 119864y|120585 1205972 log119901 (y | 120585)

12059712057211987721205971205721198682

100381610038161003816100381610038161003816100381610038161003816120575120593=0 120575120601=0

=2

1205902119899

Re minus1198951198721198731205902

119904

= 0

(A7)

[J (120585)]31 = [J (120585)]13

= minus 119864y|120585 1205972 log119901 (y | 120585)

1205971205721198772120597120575120593

1003816100381610038161003816100381610038161003816100381610038161003816120575120593=0 120575120601=0

= 41205871198911198881205722snr cos120593

11205764

(A8)

[J (120585)]41 = [J (120585)]14

= minus 119864y|120585 1205972 log119901 (y | 120585)

1205971205721198772120597120575120593

1003816100381610038161003816100381610038161003816100381610038161003816120575120593=0 120575120601=0

= minus4120587119891119888120572119868

2snr sin120593

11205765

(A9)

[J (120585)]32 = [J (120585)]23

= minus 119864y|120585 1205972 log119901 (y | 120585)

1205971205721198682120597120575120593

1003816100381610038161003816100381610038161003816100381610038161003816120575120593=0120575120601=0

= minus4120587119891119888120572119877

2snr cos120593

11205764

(A10)

[J (120585)]42 = [J (120585)]24

= minus 119864y|120585 1205972 log119901 (y | 120585)

1205971205721198682120597120575120601

1003816100381610038161003816100381610038161003816100381610038161003816120575120593=0 120575120601=0

= 4120587119891119888120572119877

2snr sin120593

11205765

(A11)

[J (120585)]33 = minus119864y|120585

1205972 log119901 (y | 120585)

(120597120575120593)2

100381610038161003816100381610038161003816100381610038161003816100381610038161003816120575120593=0 120575120601=0

= 81205872 10038171003817100381710038171205722

10038171003817100381710038172(1198612

rms + 1198912

119888) snr cos2120593

11205761

(A12)

[J (120585)]44 = minus119864y|120585

1205972 log119901 (y | 120585)

(120597120575120601)2

100381610038161003816100381610038161003816100381610038161003816100381610038161003816120575120593=0120575120601=0

= 81205872 10038171003817100381710038171205722

10038171003817100381710038172(1198612

rms + 1198912

119888) snr sin2120593

11205762

(A13)

[J (120585)]34 = [J (120585)]34

= minus119864y|120585 1205972 log119901 (y | 120585)

120597120575120593120597120575120601

1003816100381610038161003816100381610038161003816100381610038161003816120575120593=0120575120601=0

= minus81205872 10038171003817100381710038171205722

10038171003817100381710038172(1198612

rms + 1198912

119888) snr sin120593

1cos12059311205763

(A14)

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This work was supported by the National Natural ScienceFoundation of China (no 61302149 and no 61101182) andthe Research Fund for the Doctoral Program of HigherEducation of China (20124307110013) The authors wouldlike to thank the editors and reviewers for their insightfulcomments


References

[1] H Chen X Li HWang and Z Zhuang ldquoPerformance boundsof direction finding and its applications for multiple-inputmultiple-output radarrdquo IET Radar Sonar amp Navigation vol 8no 3 pp 251ndash263 2014

[2] J Li and P Stoica Eds MIMO Radar Signal Processing JohnWiley amp Sons New York NY USA 2008

[3] W J Chen and R M Narayanan ldquoComparison of the esti-mation performance of coherent and non-coherent ambiguityfunctions for an ultrawideband multi-input-multi-output noiseradarrdquo IET Radar Sonar and Navigation vol 6 no 1 pp 49ndash592012

[4] W-J Chen and R M Narayanan ldquoCGLRT plus TDL beam-forming for ultrawideband MIMO noise radarrdquo IEEE Trans-actions on Aerospace and Electronic Systems vol 48 no 3 pp1858ndash1869 2012

[5] W-J Chen and R M Narayanan ldquoAntenna placement forminimizing target localization error in UWB MIMO noiseradarrdquo IEEE Antennas and Wireless Propagation Letters vol 10pp 135ndash138 2011

[6] M Dawood and R M Narayanan ldquoGeneralised widebandambiguity function of a coherent ultrawideband random noiseradarrdquo IEE Proceedings Radar Sonar and Navigation vol 150no 5 pp 379ndash386 2003

[7] H Godrich A M Haimovich and R S Blum ldquoTarget local-ization accuracy gain in MIMO radar-based systemsrdquo IEEETransactions on Information Theory vol 56 no 6 pp 2783ndash2803 2010

[8] P Woodward Probability and Information Theory With Appli-cations to Radar Pergamon New York NY USA 1957

[9] J M Speiser ldquoWideband ambiguity functionsrdquo IEEE Transac-tions on Information Theory vol 13 no 1 pp 122ndash123 1967

[10] H Yan G Shen R Zetik O Hirsch and R S Thoma ldquoUltra-widebandMIMOambiguity function and its factorabilityrdquo IEEETransactions onGeoscience and Remote Sensing vol 51 no 1 pp504ndash519 2013

[11] G S Antonio D R Fuhrmann and F C Robey ldquoMIMO radarambiguity functionsrdquo IEEE Journal on Selected Topics in SignalProcessing vol 1 no 1 pp 167ndash177 2007

[12] Y I Abramovich and G J Frazer ldquoBounds on the volume andheight distributions for the MIMO radar ambiguity functionrdquoIEEE Signal Processing Letters vol 15 pp 505ndash508 2008

[13] S T Smith ldquoStatistical resolution limits and the complexifiedCramer-Rao boundrdquo IEEE Transactions on Signal Processingvol 53 no 5 pp 1597ndash1609 2005

[14] H Cox ldquoResolving power and sensitivity to mismatch ofoptimum array processorsrdquo Journal of the Acoustical Society ofAmerica vol 54 no 3 pp 771ndash785 1973

[15] M N El Korso R Boyer A Renaux and S Marcos ldquoStatisticalresolution limit for the multidimensional harmonic retrievalmodel- hypothesis test and Cramer-Rao bound approachesrdquoEURASIP Journal on Advances in Signal Processing vol 2011article 12 2011

[16] M Shahram and P Milanfar ldquoStatistical and information-theoretic analysis of resolution in imagingrdquo IEEE Transactionson Information Theory vol 52 no 8 pp 3411ndash3437 2006

[17] M N El Korso R Boyer A Renaux and S Marcos ldquoOn theasymptotic resolvability of twopoint sources in known subspaceinterference using a GLRT-based frameworkrdquo Signal Processingvol 92 no 10 pp 2471ndash2483 2012

[18] A Amar and A J Weiss ldquoFundamental limitations on theresolution of deterministic signalsrdquo IEEE Transactions on SignalProcessing vol 56 no 11 pp 5309ndash5318 2008

[19] M Shahram and P Milanfar ldquoOn the resolvability of sinusoidswith nearby frequencies in the presence of noiserdquo IEEE Trans-actions on Signal Processing vol 53 no 7 pp 2579ndash2588 2005

[20] Z Liu and A Nehorai ldquoStatistical angular resolution limit forpoint sourcesrdquo IEEE Transactions on Signal Processing vol 55no 11 pp 5521ndash5527 2007

[21] M N El Korso R Boyer A Renaux and S Marcos ldquoStatisticalresolution limit for source localization with clutter interferencein aMIMO radar contextrdquo IEEE Transactions on Signal Process-ing vol 60 no 2 pp 987ndash992 2012

[22] ND Tran R Boyer SMarcos and P Larzabal ldquoOn the angularresolution limit for array processing in the presence ofmodelingerrorsrdquo IEEETransactions on Signal Processing vol 61 no 19 pp4701ndash4706 2013

[23] M N El Korso R Boyer A Renaux and S Marcos ldquoStatisticalresolution limit of the uniform linear cocentered orthogonalloop and dipole arrayrdquo IEEE Transactions on Signal Processingvol 59 no 1 pp 425ndash431 2011

[24] R Boyer ldquoPerformance bounds and angular resolution limitfor the moving colocated MIMO radarrdquo IEEE Transactions onSignal Processing vol 59 no 4 pp 1539ndash1552 2011

[25] H B Lee ldquoThe Cramer-Rao bound on frequency estimates ofsignals closely spaced in frequencyrdquo IEEETransactions on SignalProcessing vol 40 no 6 pp 1508ndash1517 1992

[26] S M Kay Fundamentals of Statistical Signal Processing VolumeII DetectionTheory chapter 6 Prentice-Hall Englewood CliffsNJ USA 1993

[27] D C Bell and R M Narayanan ldquoTheoretical aspects of radarimaging using stochastic waveformsrdquo IEEE Transactions onSignal Processing vol 49 no 2 pp 394ndash400 2001

[28] M Hawkes ldquoEffects of sensor placement on acoustic vector-sensor array performancerdquo IEEE Journal of Oceanic Engineeringvol 24 no 1 pp 33ndash40 1999

[29] H Chen W Zhou J Yang Y Peng and X Li ldquoManifoldstudies on fundamental limits of direction-finding multiple-input multiple-output radar systemsrdquo IET Radar Sonar andNavigation vol 6 no 8 pp 708ndash718 2012

International Journal of

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Control Scienceand Engineering

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Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design



Shock and Vibration


Civil EngineeringAdvances in

Acoustics and VibrationAdvances in



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Journal of

Advances inOptoElectronics


Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of


Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Chemical EngineeringInternational Journal of Antennas and

Propagation




Navigation and Observation



DistributedSensor Networks



resolution limit (ie AF) is limited by the Rayleigh limitwhereas the super-resolution theory indicates that resolutionbeyond theRayleigh limit is possible under certain conditionsrelated to signal-to-noise ratio (SNR)

The statistical resolution limit (SRL) [13] deduced fromthe statistical characteristic of radar measurement and wherethe uncertainty of measurement is taken into account hasattracted much attention recently The SRL can be usedto analyze the statistical resolution capability beyond theRayleighrsquos criterion and reflect the practical resolution abilityof radarThere aremainly three approaches to define the SRLThe earliest approach is based on themean null spectrum [14]which is quite simple whereas its application is limited to thespecific estimation algorithm In practical applications thereexist another two commonly used approaches independentof the algorithm to define the SRL (I) One is based ondetection theory the resolution problem is modeled as ahypothesis test to decide if the two sources are resolvable Andthe resolution limit is defined by the probability of error orright criterion [15ndash21] (II) The other is based on estimationaccuracy the SRL is given with the Cramer-Rao lower bound(CRLB) [13 15 16 22ndash24] There are two main criteria basedon the CRLB Leersquos criterion [25] and Smithrsquos criterion [13]

In recent years the SRLs of DOD (direction of departure)and DOA (direction of arrival) for MIMO radar have beendeeply researched [21 24] whereas the majority of thispublished work has assumed narrowband signal scenarioswhich cannot be applied to the widebandUWB situationdirectly The SRL of widebandUWB MIMO radar has notbeen investigated not to mention UWBMIMO noise radar

In this paper the 2D statistical angular resolution limits(SARLs) of elevation and azimuth for UWB MIMO noiseradar in a 3D reference frame are derived The prespecifieddetection-theoretic and estimation-theoretic approaches areused in the derivations which are based on GLRT with theconstraints on probabilities of false alarm and detection andCRLB with the constraint on Smithrsquos criterion respectivelyThe closed-form expressions of SARL are achieved whichare compact and can provide useful qualitative informationon the behavior of resolution limit Moreover the rela-tionships between SARL and different factors (ie SNRdetection parameters array geometry and the radar crosssection (RCS) and direction of target) are analyzed As forthe application the SARL presented in this paper can beused to evaluate the performance of sensor arrays in targetlocalization (such as in radar and sonar) and the approachesmay be applied to general detection andparameter estimationproblems

This paper is organized as follows In Section 2 weintroduce the signal model and the related assumptionswhich are used in the derivations In Section 3 detection-theoretic approach is presented to investigate the SARLThenthe SARL is further derived based on CRLB with the Smithrsquoscriterion and the relationship between the two approachesis discussed in Section 4 In Section 5 the results obtainedin Sections 3 and 4 are discussed to provide further insightinto the SARL and some insightful properties revealed by theresults of simulations are given Finally some comments andconclusions are presented in Section 6

A comment on notation is as follows we use boldfacelowercase letters for vectors and boldface uppercase lettersfor matrices (sdot)119867 (sdot)lowast (sdot)119879 and (sdot)minus1 denote the conjugatetranspose conjugate transpose and inverse of a matrixrespectively 119864sdot is used for expectation with respect to therandomvariableRsdot andIsdot denote the real and imaginaryparts respectively Finally sdot denotes the Frobenius normof a vector and I

119873is the119873th-order identity matrix

2 Problem Setup

21 Assumptions Throughout the paper it is assumed thatthe following assumptions hold

(1) The radar is operated inmonostatic configuration andthen the DOD is approximately equal to the DOA

(2) The targets are point targets located in the far-field ofthe radar array

(3) The background noise is an independent additivewide-sense stationary (WSS) complex circular whiteGaussian random process with zero-mean and vari-ance 1205902

119899

(4) All transmit waveforms are mutually orthogonalband-limited noise waveforms with equal power 1205902

119904

and bandwidth 119861119908 Meanwhile the signal spectral

components at different frequencies are also mutuallyorthogonal

(5) The channel attenuation is ignored(6) The coherent processing time for signal and noise is

sufficient

22 Signal Model Consider two closely spaced targets inthe far-field for MIMO radar Target T1 is assumed as thereference target with a known direction 120579

1= (1205931 1206011) and

RCS while target T2 whose direction 1205792

= (1205932 1206012) is

unknown is close to T1 120593119894isin [0 1205872] and 120601

119894isin [0 2120587] denote

the elevation and azimuth of the 119894th target (forall119894 isin 1 2)respectively

TheMIMO radar geometry is shown in Figure 1 Both the119873-element transmitters and119872-element receivers which arelocated in the same plane are circular arrays (but not limitedto circular array) with radii 119903

119905and 119903119903 respectively The origin

119874 of the Cartesian coordinate system is set up as the center ofthe arrays wherein the radar array and targets are consideredas point objects

In this paper a single snapshot is considered Then thedemodulated baseband signal received by the 119898th receivercan be expressed as [7]

119910119898 (119905) =

2

sum119894=1

120572119894

119873

sum119899=1

119904119899(119905 minus 120591

119894

119898119899) 119890minus1198952120587119891

119888120591119894

119898119899 + 119908119898 (119905) (1)

where 119904119899(119905) is the baseband stochastic signal transmitted

by the 119899th transmitter with the bandwidth 119861119908 119908119898

is thenoise received by the 119898th receiver and 119891

119888is the carrier

frequency of the transmit waveforms 120572119894denotes the com-

plex amplitude proportional to the RCS of the 119894th target


z

x

yOrt

rr

ReceiverTransmitter

T1T2

12059321205931

12060121206011


120591119894119898119899



119905119899= (1205872 120599


120579119903119898






1198911198901198952120587119891119905119889119878







E ≜ y (119905) = x (119905) + w (119905) (2)

where y(119905) = [1199101(119905) 119910

119872(119905)]119879 x(119905) = [119909

1(119905) 119909

119872(119905)]119879

w(119905) = [1199081(119905) 119908

119872(119905)]119879 119909

119898(119905) = sum

2

119894=1120572119894sum119873

119899=1119904119899(119905 minus

120591119894119898119899)119890minus1198952120587119891119888120591

119894

119898119899


120601]119879 where 120575

120593≜ 1205932minus 1205931and


120593and 120575




1198911198901198952120587119891119905119889119878

119899(119891) into 119909

119898(119905) the first


119909119898 (119905) asymp (120572

1+ 1205722)

119873

sum119899=1

int119891

1198901198952120587(119891(119905minus120591

1

119898119899)minus1198911198881205911

119898119899)119889119878119899(119891)

minus 11989521205871205722120575120593cos1205931

119873

sum119899=1

int119891

119886119898119899(119891 + 119891

119888)

times 1198901198952120587(119891(119905minus120591

1

119898119899)minus1198911198881205911

119898119899)119889119878119899(119891)

+ 11989521205871205722120575120601sin1205931

119873

sum119899=1

int119891

119887119898119899(119891 + 119891

119888)

times 1198901198952120587(119891(119905minus120591

1

119898119899)minus1198911198881205911

119898119899)119889119878119899(119891)

(3)


y = G120577 + w (4)

120577 = [1205721+ 12057221205722120575119879]119879

G = [B C D]

119861119898=

119873

sum119899=1

int119891

1198901198952120587(119891(119905minus120591

1

119898119899)minus1198911198881205911

119898119899)119889119878119899(119891)

119862119898= minus1198952120587 cos120593

1

119873

sum119899=1

int119891

119886119898119899(119891 + 119891

119888)

times 1198901198952120587(119891(119905minus120591

1

119898119899)minus1198911198881205911

119898119899)119889119878119899(119891)

119863119898= 1198952120587 cos120593

1

119873

sum119899=1

int119891

119887119898119899(119891 + 119891

119888)

times 1198901198952120587(119891(119905minus120591

1

119898119899)minus1198911198881205911

119898119899)119889119878119899(119891)

(5)

where119861119898119862119898 and119863


D respectively 119886119898119899

and 119887119898119899

are denoted by

119886119898119899

=(119903119903cos (120601

1minus 120599119903119898) + 119903119905cos (120601

1minus 120599119905119899))

119888

119887119898119899

=(119903119903sin (120601

1minus 120599119903119898) + 119903119905sin (120601

1minus 120599119905119899))

119888

(6)


2120575


and 1205722


H0 P120577 = 0

H1 P120577 = 0

(7)

whereH1andH






119871 (y) =max120575119901 (y 120577H1)max120575119901 (yH0)

=119901 (y H

1)

119901 (yH0)

H1

≷H0

120578 (8)

where 119901(y 120577H1) and 119901(yH


1and




119879 (y) = 2 ln 119871 (y)H1

≷H0

1205781015840= 2 ln 120578 (9)


119879 (y) = 2

1205902119899

(P)H(P (GHG)

minus1

PH)minus1

P (10)


119879 (y) sim

12059422119872 H

0

120594101584022119872

(120582 (119901119891119886 119901119889)) H

1

(11)

where 1205942

2119872and 12059410158402



119891119886 119901119889)which




=

1198761205942

2119872

(1205781015840) and 119901119889= 11987612059410158402

2119872(120582(119901119891119886119901119889))(1205781015840) respectively 119876

1205942

2119872

(sdot)

and 11987612059410158402


2119872


119891119886 119901119889) can be


119876minus1

1205942

2119872

(119901119891119886) = 119876

minus1

1205942

2119872(120582(119901119891119886119901119889))(119901119889) (12)


120582 (119901119891119886 119901119889) =

2

1205902119899


(P120577) (13)




119891119886and


119901119889for a given 119901







119864 [119889119878119899(119891) 119889119878

lowast

119899(1198911015840)] =

120588119899(119891) 119889119891 119891 = 1198911015840

0 119891 = 1198911015840(14)


119899(119905) at


119864 BHB = 1198721198731205902

119904 (15)



119904)(1198721205902

119899)



SNR =119873120582 (119901

119891119886 119901119889)

8120587211987210038171003817100381710038171205722

10038171003817100381710038172120575119879(1198912119888(1205761minus 1205762119872119873) + 1198612rms1205761) 120575

(16)


119891120588(119891)119889119891 is the root mean





1205761= [

[

(cos1205931)21205761

minus sin1205931cos12059311205763

minus sin1205931cos12059311205763

(sin1205931)21205762

]

]

1205762= [

[

(cos1205931)212057624

minus sin1205931cos120593112057641205765

minus sin1205931cos120593112057641205765

(sin1205931)212057625

]

]

1205761=

119872

sum119898=1

119873

sum119899=1

1198862

119898119899 120576

2=

119872

sum119898=1

119873

sum119899=1

1198872

119898119899 120576

3=

119872

sum119898=1

119873

sum119899=1

119886119898119899119887119898119899

1205764=

119872

sum119898=1

119873

sum119899=1

119886119898119899 120576

5=

119872

sum119898=1

119873

sum119899=1

119887119898119899

(17)


1=

1205762= 119872119873(1199032

119905+ 1199032119903)(21198882) and 120576

3= 1205764= 1205765= 0 for uniform


cos212059311205752

120593+ sin2120593

11205752

120601

=1198882120582 (119901

119891119886 119901119889)

41205872119872210038171003817100381710038171205722

10038171003817100381710038172(1199032119905+ 1199032119903) (1198912119888+ 1198612rms) SNR

(18)






1 1205931 and 120601



120585 = [120572119877

2120572119868

2⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟120572

120575120593120575120601⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟120575

]

119879

(19)

where 1205721198772

= R1205722 and 120572119868

2= I120572



2= 1205721198772+ 1198951205721198682 In (19) we



119877

2120572119868

2]119879


J (120585) = 119864y|120585 [120597

120597120585log119901 (y | 120585)] [ 120597

120597120585log119901 (y | 120585)]

119867

(20)



119901 (y | 120585) prop expminus 1

1205902119899

119872

sum119898=1

int119879

1003817100381710038171003817119910119898 (119905) minus 119909119898 (119905)10038171003817100381710038172119889119905 (21)


[J (120585)]119894119895 = [J (120585)]119895119894 = minus119864y|120585 1205972 log119901 (y | 120585)

120597120585119894120597120585119895

119894 119895 = 1 2 3 4

(22)


119899(119905minus120591119897119898119899) = 120597119904


in Section 2 119904119899(119905) = int

1198911198901198952120587119891119905

119889119878119899(119891) then 119904

119899(119905 minus 120591

119897

119898119899) can be

expressed as

119904119899(119905 minus 120591

119897

119898119899) = int119891

minus11989521205871198911198901198952120587119891(119905minus120591

119897

119898119899)119889119878119899(119891) (23)


119864int119879

119904119899(119905 minus 120591

1

119898119899) 119904lowast

119899(119905 minus 120591

1

119898119899) 119889119905 = int

119891

minus1198952120587119891120588119899119889119891 = 0

119864 int119879

119904119899(119905 minus 120591

1

119898119899) 119904lowast

119899(119905 minus 120591

1

119898119899) 119889119905 = int

119891

(2120587119891)2120588119899119889119891

= 412058721198612

rms1205902

119904

(24)



CRLB (120585) = [J (120585)]minus1 (25)


[J (120585)] = [J120572120572 J119867120572120575

J120572120575 J120575120575] (26)



J120572120572 = 2119872119873snrI2

J120572120575 = J119867120572120575= 4120587119891

119888snr[

1205721198682cos12059311205764

minus1205721198772cos12059311205764

minus1205721198682sin12059311205765

1205721198772sin12059311205765

]

J120575120575 = 81205872 10038171003817100381710038171205722

10038171003817100381710038172(1198612

rms + 1198912

119888)


11205761

minus sin1205931cos12059311205763

minus sin1205931cos12059311205763

sin212059311205762

]

(27)




120572120572J119867120572120575]minus1

=119873

8120587210038171003817100381710038171205722

10038171003817100381710038172119872SNR

[[[[[

[

((1198612

rms + 1198912

119888) 1205761minus119891211988812057624

119872119873) cos2120593

1minus((1198612rms + 119891

2

119888) 1205763minus119891211988812057641205765

119872119873) sin120593

1cos1205931

minus((1198612rms + 1198912

119888) 1205763minus119891211988812057641205765

119872119873) sin120593

1cos1205931((1198612rms + 119891

2

119888) 1205762minus119891211988812057625

119872119873) sin2120593

1

]]]]]

]

minus1

(28)




(29)


1= 1205762= 119872119873(1199032

119905+ 1199032119903)(21198882) and 120576

3= 1205764= 1205765= 0


CRLB (120575) = 1198882

4120587210038171003817100381710038171205722

100381710038171003817100381721198722

1003817100381710038171003817120572210038171003817100381710038172(1199032119905+ 1199032119903) (1198612rms + 119891

2

119888) SNR

times[[[

[

1

cos21205931

0

01

sin21205931

]]]

]

(30)


120575120593=

119888

212058711987210038171003817100381710038171205722

1003817100381710038171003817 cos1205931radic

1

(1199032119905+ 1199032119903) (1198912119888+ 1198612rms) SNR

(31)

120575120601=

119888

212058711987210038171003817100381710038171205722

1003817100381710038171003817 sin1205931radic

1

(1199032119905+ 1199032119903) (1198912119888+ 1198612rms) SNR

(32)



119891119886 119901119889) is given as

120582 (119901119891119886 119901119889) = (120575 minus 0)

119879([Jminus1 (120585)]

120575120575)minus1

(120575 minus 0) (33)


120582 (119901119891119886 119901119889) cong (120575 minus 0)

119879([Jminus1 (120585)]

120575120575

10038161003816100381610038161003816120575=0)minus1

(120575 minus 0)

= 120575119879(CRLB (120575)|120575=0)

minus1120575

(34)





119891119886 119901119889) = 1 Consequently (34)







120588 (119891) =

1205880

10038161003816100381610038161198911003816100381610038161003816 le 1198611199082

010038161003816100381610038161198911003816100381610038161003816 gt 1198611199082

1198611199082radic3


0exp (minus2119891212059021198612

119908) 120590119861

1199082



119888 increas-








120593and 120575

120601




1and 120593

1 Moreover it


120601







119888= 9GHz and 119861

119908= 02119891

119888 then the RMS



119905= 119903119903= 119903 = 1m










1=




AntennaTarget

(a)

AntennaTarget

(b)

AntennaTarget

(c)

AntennaTarget

(d)



SARL

of e

leva

tion

(deg

)

Type (a)Type (b)

Type (c)Type (d)

10minus3

10minus4

10minus5



SARL

of a

zim

uth

(deg

)

Type (a)Type (b)

Type (c)Type (d)

10minus3

10minus4

10minus5









120601) From


119901119891119886


119891119886lead



119889



2

15

05

1

0

0

300200

100

Elevation (deg)

Azimuth (deg)

8060

4020

2

18

16

14

12

08

06

04

02

1

times10minus3

times10minus3radic

MSA

E CRL

B

(a)

radicM

SAE C

RLB

2

15

05

1

0

0

300200

100

Elevation (deg)

Azimuth (deg)

8060

4020

2

18

16

14

12

08

06

04

02

1

times10minus3

times10minus3

(b)

0

0

300200

100

Elevation (deg)

Azimuth (deg)

8060

4020

times10minus4

radicM

SAE C

RLB

5

4

3

2

1

times10minus4

5

4

3

2

1

(c)

0

0

300200

100Elevation (deg)

Azimuth (deg)

8060

4020

times10minus4

times10minus4

2

1

radicM

SAE C

RLB

2

15

1

05

(d)



119891119886 119901119889) = (001 099) For visual





119905= 119903119903= 119903 isin 1m 2m 3m



6 Conclusion



01 02 03 04 05 06 07 08 09

051

152

253

354

455

55

SARL

times10minus4

pfa = 001

pfa = 005

pfa = 01

pfa = 02

pd


SNR = 20 dB and 1205931= 1∘

Elevation (rad)

Azi

mut

h (r

ad)

0 05 1 150

05

1

15





10∘ 30∘ 60∘ and (119901119891119886 119901119889) = (001 099)


10 15 20 25 30 35 40 45 50SNR (dB)

SARL

(rad

)

10minus3

10minus4

10minus5

r = 1mr = 2mr = 3m


Appendix



120597 log119901 (y | 120585)1205971205721198772

=2

1205902119899

119872

sum119898=1

int119879

Re(119910119898 (119905) minus

2

sum119897=1

120572119897

119873

sum119899=1

119904119899(119905 minus 120591

119897

119898119899) 119890minus1198952120587119891

119888120591119897

119898119899)

times (

119873

sum119899=1

119904119899(119905 minus 120591

2

119898119899) 119890minus1198952120587119891

1198881205912

119898119899)

lowast

119889119905

(A1)120597 log119901 (y | 120585)

1205971205721198682

=2

1205902119899

119872

sum119898=1

int119879

Re(119910119898 (119905) minus

2

sum119897=1

120572119897

119873

sum119899=1

119904119899(119905 minus 120591

119897

119898119899) 119890minus1198952120587119891

119888120591119897

119898119899)

times (119895

119873

sum119899=1

119904119899(119905 minus 120591

2

119898119899) 119890minus1198952120587119891

1198881205912

119898119899)

lowast

119889119905

(A2)120597 log119901 (y | 120585)

120597120575120593

=minus2 cos120593

1

1205902119899

times

119872

sum119898=1

int119879

Re

times (119910119898 (119905) minus

2

sum119897=1

120572119897

119873

sum119899=1

119904119899(119905 minus 120591

119897

119898119899) 119890minus1198952120587119891

119888120591119897

119898119899)


times (1205722

119873

sum119899=1

119886119898119899( 119904119899(119905minus1205912

119898119899)+1198952120587119891

119888119904119899(119905minus1205912

119898119899))

times119890minus1198952120587119891

1198881205912

119898119899)

lowast

119889119905

(A3)

120597 log119901 (y | 120585)120597120575120601

=2 sin120593

1

1205902119899

times

119872

sum119898=1

int119879

Re(119910119898 (119905) minus

2

sum119897=1

120572119897

119873

sum119899=1

119904119899(119905 minus 120591

119897

119898119899) 119890minus1198952120587119891

119888120591119897

119898119899)

times (1205722

119873

sum119899=1

119887119898119899( 119904119899(119905 minus 120591

2

119898119899)

+ 1198952120587119891119888119904119899(119905 minus 120591

2

119898119899))

times 119890minus1198952120587119891

1198881205912

119898119899)

lowast

119889119905

(A4)



[J (120585)]11 = minus 119864y|120585 1205972 log119901 (y | 120585)

(1205971205721198772)2

1003816100381610038161003816100381610038161003816100381610038161003816120575120593=0 120575120601=0

= 2119872119873snr

(A5)

[J (120585)]22 = minus 119864y|120585 1205972 log119901 (y | 120585)

(1205971205721198682)2

1003816100381610038161003816100381610038161003816100381610038161003816120575120593=0 120575120601=0

= 2119872119873snr

(A6)

[J (120585)]21 = [J (120585)]12

= minus 119864y|120585 1205972 log119901 (y | 120585)

12059712057211987721205971205721198682

100381610038161003816100381610038161003816100381610038161003816120575120593=0 120575120601=0

=2

1205902119899

Re minus1198951198721198731205902

119904

= 0

(A7)

[J (120585)]31 = [J (120585)]13

= minus 119864y|120585 1205972 log119901 (y | 120585)

1205971205721198772120597120575120593

1003816100381610038161003816100381610038161003816100381610038161003816120575120593=0 120575120601=0

= 41205871198911198881205722snr cos120593

11205764

(A8)

[J (120585)]41 = [J (120585)]14

= minus 119864y|120585 1205972 log119901 (y | 120585)

1205971205721198772120597120575120593

1003816100381610038161003816100381610038161003816100381610038161003816120575120593=0 120575120601=0

= minus4120587119891119888120572119868

2snr sin120593

11205765

(A9)

[J (120585)]32 = [J (120585)]23

= minus 119864y|120585 1205972 log119901 (y | 120585)

1205971205721198682120597120575120593

1003816100381610038161003816100381610038161003816100381610038161003816120575120593=0120575120601=0

= minus4120587119891119888120572119877

2snr cos120593

11205764

(A10)

[J (120585)]42 = [J (120585)]24

= minus 119864y|120585 1205972 log119901 (y | 120585)

1205971205721198682120597120575120601

1003816100381610038161003816100381610038161003816100381610038161003816120575120593=0 120575120601=0

= 4120587119891119888120572119877

2snr sin120593

11205765

(A11)

[J (120585)]33 = minus119864y|120585

1205972 log119901 (y | 120585)

(120597120575120593)2

100381610038161003816100381610038161003816100381610038161003816100381610038161003816120575120593=0 120575120601=0

= 81205872 10038171003817100381710038171205722

10038171003817100381710038172(1198612

rms + 1198912

119888) snr cos2120593

11205761

(A12)

[J (120585)]44 = minus119864y|120585

1205972 log119901 (y | 120585)

(120597120575120601)2

100381610038161003816100381610038161003816100381610038161003816100381610038161003816120575120593=0120575120601=0

= 81205872 10038171003817100381710038171205722

10038171003817100381710038172(1198612

rms + 1198912

119888) snr sin2120593

11205762

(A13)

[J (120585)]34 = [J (120585)]34

= minus119864y|120585 1205972 log119901 (y | 120585)

120597120575120593120597120575120601

1003816100381610038161003816100381610038161003816100381610038161003816120575120593=0120575120601=0

= minus81205872 10038171003817100381710038171205722

10038171003817100381710038172(1198612

rms + 1198912

119888) snr sin120593

1cos12059311205763

(A14)



Acknowledgments



References
































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










z

x

yOrt

rr

ReceiverTransmitter

T1T2

12059321205931

12060121206011


120591119894119898119899



119905119899= (1205872 120599


120579119903119898






1198911198901198952120587119891119905119889119878







E ≜ y (119905) = x (119905) + w (119905) (2)

where y(119905) = [1199101(119905) 119910

119872(119905)]119879 x(119905) = [119909

1(119905) 119909

119872(119905)]119879

w(119905) = [1199081(119905) 119908

119872(119905)]119879 119909

119898(119905) = sum

2

119894=1120572119894sum119873

119899=1119904119899(119905 minus

120591119894119898119899)119890minus1198952120587119891119888120591

119894

119898119899


120601]119879 where 120575

120593≜ 1205932minus 1205931and


120593and 120575




1198911198901198952120587119891119905119889119878

119899(119891) into 119909

119898(119905) the first


119909119898 (119905) asymp (120572

1+ 1205722)

119873

sum119899=1

int119891

1198901198952120587(119891(119905minus120591

1

119898119899)minus1198911198881205911

119898119899)119889119878119899(119891)

minus 11989521205871205722120575120593cos1205931

119873

sum119899=1

int119891

119886119898119899(119891 + 119891

119888)

times 1198901198952120587(119891(119905minus120591

1

119898119899)minus1198911198881205911

119898119899)119889119878119899(119891)

+ 11989521205871205722120575120601sin1205931

119873

sum119899=1

int119891

119887119898119899(119891 + 119891

119888)

times 1198901198952120587(119891(119905minus120591

1

119898119899)minus1198911198881205911

119898119899)119889119878119899(119891)

(3)


y = G120577 + w (4)

120577 = [1205721+ 12057221205722120575119879]119879

G = [B C D]

119861119898=

119873

sum119899=1

int119891

1198901198952120587(119891(119905minus120591

1

119898119899)minus1198911198881205911

119898119899)119889119878119899(119891)

119862119898= minus1198952120587 cos120593

1

119873

sum119899=1

int119891

119886119898119899(119891 + 119891

119888)

times 1198901198952120587(119891(119905minus120591

1

119898119899)minus1198911198881205911

119898119899)119889119878119899(119891)

119863119898= 1198952120587 cos120593

1

119873

sum119899=1

int119891

119887119898119899(119891 + 119891

119888)

times 1198901198952120587(119891(119905minus120591

1

119898119899)minus1198911198881205911

119898119899)119889119878119899(119891)

(5)

where119861119898119862119898 and119863


D respectively 119886119898119899

and 119887119898119899

are denoted by

119886119898119899

=(119903119903cos (120601

1minus 120599119903119898) + 119903119905cos (120601

1minus 120599119905119899))

119888

119887119898119899

=(119903119903sin (120601

1minus 120599119903119898) + 119903119905sin (120601

1minus 120599119905119899))

119888

(6)


2120575


and 1205722


H0 P120577 = 0

H1 P120577 = 0

(7)

whereH1andH






119871 (y) =max120575119901 (y 120577H1)max120575119901 (yH0)

=119901 (y H

1)

119901 (yH0)

H1

≷H0

120578 (8)

where 119901(y 120577H1) and 119901(yH


1and




119879 (y) = 2 ln 119871 (y)H1

≷H0

1205781015840= 2 ln 120578 (9)


119879 (y) = 2

1205902119899

(P)H(P (GHG)

minus1

PH)minus1

P (10)


119879 (y) sim

12059422119872 H

0

120594101584022119872

(120582 (119901119891119886 119901119889)) H

1

(11)

where 1205942

2119872and 12059410158402



119891119886 119901119889)which




=

1198761205942

2119872

(1205781015840) and 119901119889= 11987612059410158402

2119872(120582(119901119891119886119901119889))(1205781015840) respectively 119876

1205942

2119872

(sdot)

and 11987612059410158402


2119872


119891119886 119901119889) can be


119876minus1

1205942

2119872

(119901119891119886) = 119876

minus1

1205942

2119872(120582(119901119891119886119901119889))(119901119889) (12)


120582 (119901119891119886 119901119889) =

2

1205902119899


(P120577) (13)




119891119886and


119901119889for a given 119901







119864 [119889119878119899(119891) 119889119878

lowast

119899(1198911015840)] =

120588119899(119891) 119889119891 119891 = 1198911015840

0 119891 = 1198911015840(14)


119899(119905) at


119864 BHB = 1198721198731205902

119904 (15)



119904)(1198721205902

119899)



SNR =119873120582 (119901

119891119886 119901119889)

8120587211987210038171003817100381710038171205722

10038171003817100381710038172120575119879(1198912119888(1205761minus 1205762119872119873) + 1198612rms1205761) 120575

(16)


119891120588(119891)119889119891 is the root mean





1205761= [

[

(cos1205931)21205761

minus sin1205931cos12059311205763

minus sin1205931cos12059311205763

(sin1205931)21205762

]

]

1205762= [

[

(cos1205931)212057624

minus sin1205931cos120593112057641205765

minus sin1205931cos120593112057641205765

(sin1205931)212057625

]

]

1205761=

119872

sum119898=1

119873

sum119899=1

1198862

119898119899 120576

2=

119872

sum119898=1

119873

sum119899=1

1198872

119898119899 120576

3=

119872

sum119898=1

119873

sum119899=1

119886119898119899119887119898119899

1205764=

119872

sum119898=1

119873

sum119899=1

119886119898119899 120576

5=

119872

sum119898=1

119873

sum119899=1

119887119898119899

(17)


1=

1205762= 119872119873(1199032

119905+ 1199032119903)(21198882) and 120576

3= 1205764= 1205765= 0 for uniform


cos212059311205752

120593+ sin2120593

11205752

120601

=1198882120582 (119901

119891119886 119901119889)

41205872119872210038171003817100381710038171205722

10038171003817100381710038172(1199032119905+ 1199032119903) (1198912119888+ 1198612rms) SNR

(18)






1 1205931 and 120601



120585 = [120572119877

2120572119868

2⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟120572

120575120593120575120601⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟120575

]

119879

(19)

where 1205721198772

= R1205722 and 120572119868

2= I120572



2= 1205721198772+ 1198951205721198682 In (19) we



119877

2120572119868

2]119879


J (120585) = 119864y|120585 [120597

120597120585log119901 (y | 120585)] [ 120597

120597120585log119901 (y | 120585)]

119867

(20)



119901 (y | 120585) prop expminus 1

1205902119899

119872

sum119898=1

int119879

1003817100381710038171003817119910119898 (119905) minus 119909119898 (119905)10038171003817100381710038172119889119905 (21)


[J (120585)]119894119895 = [J (120585)]119895119894 = minus119864y|120585 1205972 log119901 (y | 120585)

120597120585119894120597120585119895

119894 119895 = 1 2 3 4

(22)


119899(119905minus120591119897119898119899) = 120597119904


in Section 2 119904119899(119905) = int

1198911198901198952120587119891119905

119889119878119899(119891) then 119904

119899(119905 minus 120591

119897

119898119899) can be

expressed as

119904119899(119905 minus 120591

119897

119898119899) = int119891

minus11989521205871198911198901198952120587119891(119905minus120591

119897

119898119899)119889119878119899(119891) (23)


119864int119879

119904119899(119905 minus 120591

1

119898119899) 119904lowast

119899(119905 minus 120591

1

119898119899) 119889119905 = int

119891

minus1198952120587119891120588119899119889119891 = 0

119864 int119879

119904119899(119905 minus 120591

1

119898119899) 119904lowast

119899(119905 minus 120591

1

119898119899) 119889119905 = int

119891

(2120587119891)2120588119899119889119891

= 412058721198612

rms1205902

119904

(24)



CRLB (120585) = [J (120585)]minus1 (25)


[J (120585)] = [J120572120572 J119867120572120575

J120572120575 J120575120575] (26)



J120572120572 = 2119872119873snrI2

J120572120575 = J119867120572120575= 4120587119891

119888snr[

1205721198682cos12059311205764

minus1205721198772cos12059311205764

minus1205721198682sin12059311205765

1205721198772sin12059311205765

]

J120575120575 = 81205872 10038171003817100381710038171205722

10038171003817100381710038172(1198612

rms + 1198912

119888)


11205761

minus sin1205931cos12059311205763

minus sin1205931cos12059311205763

sin212059311205762

]

(27)




120572120572J119867120572120575]minus1

=119873

8120587210038171003817100381710038171205722

10038171003817100381710038172119872SNR

[[[[[

[

((1198612

rms + 1198912

119888) 1205761minus119891211988812057624

119872119873) cos2120593

1minus((1198612rms + 119891

2

119888) 1205763minus119891211988812057641205765

119872119873) sin120593

1cos1205931

minus((1198612rms + 1198912

119888) 1205763minus119891211988812057641205765

119872119873) sin120593

1cos1205931((1198612rms + 119891

2

119888) 1205762minus119891211988812057625

119872119873) sin2120593

1

]]]]]

]

minus1

(28)




(29)


1= 1205762= 119872119873(1199032

119905+ 1199032119903)(21198882) and 120576

3= 1205764= 1205765= 0


CRLB (120575) = 1198882

4120587210038171003817100381710038171205722

100381710038171003817100381721198722

1003817100381710038171003817120572210038171003817100381710038172(1199032119905+ 1199032119903) (1198612rms + 119891

2

119888) SNR

times[[[

[

1

cos21205931

0

01

sin21205931

]]]

]

(30)


120575120593=

119888

212058711987210038171003817100381710038171205722

1003817100381710038171003817 cos1205931radic

1

(1199032119905+ 1199032119903) (1198912119888+ 1198612rms) SNR

(31)

120575120601=

119888

212058711987210038171003817100381710038171205722

1003817100381710038171003817 sin1205931radic

1

(1199032119905+ 1199032119903) (1198912119888+ 1198612rms) SNR

(32)



119891119886 119901119889) is given as

120582 (119901119891119886 119901119889) = (120575 minus 0)

119879([Jminus1 (120585)]

120575120575)minus1

(120575 minus 0) (33)


120582 (119901119891119886 119901119889) cong (120575 minus 0)

119879([Jminus1 (120585)]

120575120575

10038161003816100381610038161003816120575=0)minus1

(120575 minus 0)

= 120575119879(CRLB (120575)|120575=0)

minus1120575

(34)





119891119886 119901119889) = 1 Consequently (34)







120588 (119891) =

1205880

10038161003816100381610038161198911003816100381610038161003816 le 1198611199082

010038161003816100381610038161198911003816100381610038161003816 gt 1198611199082

1198611199082radic3


0exp (minus2119891212059021198612

119908) 120590119861

1199082



119888 increas-








120593and 120575

120601




1and 120593

1 Moreover it


120601







119888= 9GHz and 119861

119908= 02119891

119888 then the RMS



119905= 119903119903= 119903 = 1m










1=




AntennaTarget

(a)

AntennaTarget

(b)

AntennaTarget

(c)

AntennaTarget

(d)



SARL

of e

leva

tion

(deg

)

Type (a)Type (b)

Type (c)Type (d)

10minus3

10minus4

10minus5



SARL

of a

zim

uth

(deg

)

Type (a)Type (b)

Type (c)Type (d)

10minus3

10minus4

10minus5









120601) From


119901119891119886


119891119886lead



119889



2

15

05

1

0

0

300200

100

Elevation (deg)

Azimuth (deg)

8060

4020

2

18

16

14

12

08

06

04

02

1

times10minus3

times10minus3radic

MSA

E CRL

B

(a)

radicM

SAE C

RLB

2

15

05

1

0

0

300200

100

Elevation (deg)

Azimuth (deg)

8060

4020

2

18

16

14

12

08

06

04

02

1

times10minus3

times10minus3

(b)

0

0

300200

100

Elevation (deg)

Azimuth (deg)

8060

4020

times10minus4

radicM

SAE C

RLB

5

4

3

2

1

times10minus4

5

4

3

2

1

(c)

0

0

300200

100Elevation (deg)

Azimuth (deg)

8060

4020

times10minus4

times10minus4

2

1

radicM

SAE C

RLB

2

15

1

05

(d)



119891119886 119901119889) = (001 099) For visual





119905= 119903119903= 119903 isin 1m 2m 3m



6 Conclusion



01 02 03 04 05 06 07 08 09

051

152

253

354

455

55

SARL

times10minus4

pfa = 001

pfa = 005

pfa = 01

pfa = 02

pd


SNR = 20 dB and 1205931= 1∘

Elevation (rad)

Azi

mut

h (r

ad)

0 05 1 150

05

1

15





10∘ 30∘ 60∘ and (119901119891119886 119901119889) = (001 099)


10 15 20 25 30 35 40 45 50SNR (dB)

SARL

(rad

)

10minus3

10minus4

10minus5

r = 1mr = 2mr = 3m


Appendix



120597 log119901 (y | 120585)1205971205721198772

=2

1205902119899

119872

sum119898=1

int119879

Re(119910119898 (119905) minus

2

sum119897=1

120572119897

119873

sum119899=1

119904119899(119905 minus 120591

119897

119898119899) 119890minus1198952120587119891

119888120591119897

119898119899)

times (

119873

sum119899=1

119904119899(119905 minus 120591

2

119898119899) 119890minus1198952120587119891

1198881205912

119898119899)

lowast

119889119905

(A1)120597 log119901 (y | 120585)

1205971205721198682

=2

1205902119899

119872

sum119898=1

int119879

Re(119910119898 (119905) minus

2

sum119897=1

120572119897

119873

sum119899=1

119904119899(119905 minus 120591

119897

119898119899) 119890minus1198952120587119891

119888120591119897

119898119899)

times (119895

119873

sum119899=1

119904119899(119905 minus 120591

2

119898119899) 119890minus1198952120587119891

1198881205912

119898119899)

lowast

119889119905

(A2)120597 log119901 (y | 120585)

120597120575120593

=minus2 cos120593

1

1205902119899

times

119872

sum119898=1

int119879

Re

times (119910119898 (119905) minus

2

sum119897=1

120572119897

119873

sum119899=1

119904119899(119905 minus 120591

119897

119898119899) 119890minus1198952120587119891

119888120591119897

119898119899)


times (1205722

119873

sum119899=1

119886119898119899( 119904119899(119905minus1205912

119898119899)+1198952120587119891

119888119904119899(119905minus1205912

119898119899))

times119890minus1198952120587119891

1198881205912

119898119899)

lowast

119889119905

(A3)

120597 log119901 (y | 120585)120597120575120601

=2 sin120593

1

1205902119899

times

119872

sum119898=1

int119879

Re(119910119898 (119905) minus

2

sum119897=1

120572119897

119873

sum119899=1

119904119899(119905 minus 120591

119897

119898119899) 119890minus1198952120587119891

119888120591119897

119898119899)

times (1205722

119873

sum119899=1

119887119898119899( 119904119899(119905 minus 120591

2

119898119899)

+ 1198952120587119891119888119904119899(119905 minus 120591

2

119898119899))

times 119890minus1198952120587119891

1198881205912

119898119899)

lowast

119889119905

(A4)



[J (120585)]11 = minus 119864y|120585 1205972 log119901 (y | 120585)

(1205971205721198772)2

1003816100381610038161003816100381610038161003816100381610038161003816120575120593=0 120575120601=0

= 2119872119873snr

(A5)

[J (120585)]22 = minus 119864y|120585 1205972 log119901 (y | 120585)

(1205971205721198682)2

1003816100381610038161003816100381610038161003816100381610038161003816120575120593=0 120575120601=0

= 2119872119873snr

(A6)

[J (120585)]21 = [J (120585)]12

= minus 119864y|120585 1205972 log119901 (y | 120585)

12059712057211987721205971205721198682

100381610038161003816100381610038161003816100381610038161003816120575120593=0 120575120601=0

=2

1205902119899

Re minus1198951198721198731205902

119904

= 0

(A7)

[J (120585)]31 = [J (120585)]13

= minus 119864y|120585 1205972 log119901 (y | 120585)

1205971205721198772120597120575120593

1003816100381610038161003816100381610038161003816100381610038161003816120575120593=0 120575120601=0

= 41205871198911198881205722snr cos120593

11205764

(A8)

[J (120585)]41 = [J (120585)]14

= minus 119864y|120585 1205972 log119901 (y | 120585)

1205971205721198772120597120575120593

1003816100381610038161003816100381610038161003816100381610038161003816120575120593=0 120575120601=0

= minus4120587119891119888120572119868

2snr sin120593

11205765

(A9)

[J (120585)]32 = [J (120585)]23

= minus 119864y|120585 1205972 log119901 (y | 120585)

1205971205721198682120597120575120593

1003816100381610038161003816100381610038161003816100381610038161003816120575120593=0120575120601=0

= minus4120587119891119888120572119877

2snr cos120593

11205764

(A10)

[J (120585)]42 = [J (120585)]24

= minus 119864y|120585 1205972 log119901 (y | 120585)

1205971205721198682120597120575120601

1003816100381610038161003816100381610038161003816100381610038161003816120575120593=0 120575120601=0

= 4120587119891119888120572119877

2snr sin120593

11205765

(A11)

[J (120585)]33 = minus119864y|120585

1205972 log119901 (y | 120585)

(120597120575120593)2

100381610038161003816100381610038161003816100381610038161003816100381610038161003816120575120593=0 120575120601=0

= 81205872 10038171003817100381710038171205722

10038171003817100381710038172(1198612

rms + 1198912

119888) snr cos2120593

11205761

(A12)

[J (120585)]44 = minus119864y|120585

1205972 log119901 (y | 120585)

(120597120575120601)2

100381610038161003816100381610038161003816100381610038161003816100381610038161003816120575120593=0120575120601=0

= 81205872 10038171003817100381710038171205722

10038171003817100381710038172(1198612

rms + 1198912

119888) snr sin2120593

11205762

(A13)

[J (120585)]34 = [J (120585)]34

= minus119864y|120585 1205972 log119901 (y | 120585)

120597120575120593120597120575120601

1003816100381610038161003816100381610038161003816100381610038161003816120575120593=0120575120601=0

= minus81205872 10038171003817100381710038171205722

10038171003817100381710038172(1198612

rms + 1198912

119888) snr sin120593

1cos12059311205763

(A14)



Acknowledgments



References
































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










and 1205722


H0 P120577 = 0

H1 P120577 = 0

(7)

whereH1andH






119871 (y) =max120575119901 (y 120577H1)max120575119901 (yH0)

=119901 (y H

1)

119901 (yH0)

H1

≷H0

120578 (8)

where 119901(y 120577H1) and 119901(yH


1and




119879 (y) = 2 ln 119871 (y)H1

≷H0

1205781015840= 2 ln 120578 (9)


119879 (y) = 2

1205902119899

(P)H(P (GHG)

minus1

PH)minus1

P (10)


119879 (y) sim

12059422119872 H

0

120594101584022119872

(120582 (119901119891119886 119901119889)) H

1

(11)

where 1205942

2119872and 12059410158402



119891119886 119901119889)which




=

1198761205942

2119872

(1205781015840) and 119901119889= 11987612059410158402

2119872(120582(119901119891119886119901119889))(1205781015840) respectively 119876

1205942

2119872

(sdot)

and 11987612059410158402


2119872


119891119886 119901119889) can be


119876minus1

1205942

2119872

(119901119891119886) = 119876

minus1

1205942

2119872(120582(119901119891119886119901119889))(119901119889) (12)


120582 (119901119891119886 119901119889) =

2

1205902119899


(P120577) (13)




119891119886and


119901119889for a given 119901







119864 [119889119878119899(119891) 119889119878

lowast

119899(1198911015840)] =

120588119899(119891) 119889119891 119891 = 1198911015840

0 119891 = 1198911015840(14)


119899(119905) at


119864 BHB = 1198721198731205902

119904 (15)



119904)(1198721205902

119899)



SNR =119873120582 (119901

119891119886 119901119889)

8120587211987210038171003817100381710038171205722

10038171003817100381710038172120575119879(1198912119888(1205761minus 1205762119872119873) + 1198612rms1205761) 120575

(16)


119891120588(119891)119889119891 is the root mean





1205761= [

[

(cos1205931)21205761

minus sin1205931cos12059311205763

minus sin1205931cos12059311205763

(sin1205931)21205762

]

]

1205762= [

[

(cos1205931)212057624

minus sin1205931cos120593112057641205765

minus sin1205931cos120593112057641205765

(sin1205931)212057625

]

]

1205761=

119872

sum119898=1

119873

sum119899=1

1198862

119898119899 120576

2=

119872

sum119898=1

119873

sum119899=1

1198872

119898119899 120576

3=

119872

sum119898=1

119873

sum119899=1

119886119898119899119887119898119899

1205764=

119872

sum119898=1

119873

sum119899=1

119886119898119899 120576

5=

119872

sum119898=1

119873

sum119899=1

119887119898119899

(17)


1=

1205762= 119872119873(1199032

119905+ 1199032119903)(21198882) and 120576

3= 1205764= 1205765= 0 for uniform


cos212059311205752

120593+ sin2120593

11205752

120601

=1198882120582 (119901

119891119886 119901119889)

41205872119872210038171003817100381710038171205722

10038171003817100381710038172(1199032119905+ 1199032119903) (1198912119888+ 1198612rms) SNR

(18)






1 1205931 and 120601



120585 = [120572119877

2120572119868

2⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟120572

120575120593120575120601⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟120575

]

119879

(19)

where 1205721198772

= R1205722 and 120572119868

2= I120572



2= 1205721198772+ 1198951205721198682 In (19) we



119877

2120572119868

2]119879


J (120585) = 119864y|120585 [120597

120597120585log119901 (y | 120585)] [ 120597

120597120585log119901 (y | 120585)]

119867

(20)



119901 (y | 120585) prop expminus 1

1205902119899

119872

sum119898=1

int119879

1003817100381710038171003817119910119898 (119905) minus 119909119898 (119905)10038171003817100381710038172119889119905 (21)


[J (120585)]119894119895 = [J (120585)]119895119894 = minus119864y|120585 1205972 log119901 (y | 120585)

120597120585119894120597120585119895

119894 119895 = 1 2 3 4

(22)


119899(119905minus120591119897119898119899) = 120597119904


in Section 2 119904119899(119905) = int

1198911198901198952120587119891119905

119889119878119899(119891) then 119904

119899(119905 minus 120591

119897

119898119899) can be

expressed as

119904119899(119905 minus 120591

119897

119898119899) = int119891

minus11989521205871198911198901198952120587119891(119905minus120591

119897

119898119899)119889119878119899(119891) (23)


119864int119879

119904119899(119905 minus 120591

1

119898119899) 119904lowast

119899(119905 minus 120591

1

119898119899) 119889119905 = int

119891

minus1198952120587119891120588119899119889119891 = 0

119864 int119879

119904119899(119905 minus 120591

1

119898119899) 119904lowast

119899(119905 minus 120591

1

119898119899) 119889119905 = int

119891

(2120587119891)2120588119899119889119891

= 412058721198612

rms1205902

119904

(24)



CRLB (120585) = [J (120585)]minus1 (25)


[J (120585)] = [J120572120572 J119867120572120575

J120572120575 J120575120575] (26)



J120572120572 = 2119872119873snrI2

J120572120575 = J119867120572120575= 4120587119891

119888snr[

1205721198682cos12059311205764

minus1205721198772cos12059311205764

minus1205721198682sin12059311205765

1205721198772sin12059311205765

]

J120575120575 = 81205872 10038171003817100381710038171205722

10038171003817100381710038172(1198612

rms + 1198912

119888)


11205761

minus sin1205931cos12059311205763

minus sin1205931cos12059311205763

sin212059311205762

]

(27)




120572120572J119867120572120575]minus1

=119873

8120587210038171003817100381710038171205722

10038171003817100381710038172119872SNR

[[[[[

[

((1198612

rms + 1198912

119888) 1205761minus119891211988812057624

119872119873) cos2120593

1minus((1198612rms + 119891

2

119888) 1205763minus119891211988812057641205765

119872119873) sin120593

1cos1205931

minus((1198612rms + 1198912

119888) 1205763minus119891211988812057641205765

119872119873) sin120593

1cos1205931((1198612rms + 119891

2

119888) 1205762minus119891211988812057625

119872119873) sin2120593

1

]]]]]

]

minus1

(28)




(29)


1= 1205762= 119872119873(1199032

119905+ 1199032119903)(21198882) and 120576

3= 1205764= 1205765= 0


CRLB (120575) = 1198882

4120587210038171003817100381710038171205722

100381710038171003817100381721198722

1003817100381710038171003817120572210038171003817100381710038172(1199032119905+ 1199032119903) (1198612rms + 119891

2

119888) SNR

times[[[

[

1

cos21205931

0

01

sin21205931

]]]

]

(30)


120575120593=

119888

212058711987210038171003817100381710038171205722

1003817100381710038171003817 cos1205931radic

1

(1199032119905+ 1199032119903) (1198912119888+ 1198612rms) SNR

(31)

120575120601=

119888

212058711987210038171003817100381710038171205722

1003817100381710038171003817 sin1205931radic

1

(1199032119905+ 1199032119903) (1198912119888+ 1198612rms) SNR

(32)



119891119886 119901119889) is given as

120582 (119901119891119886 119901119889) = (120575 minus 0)

119879([Jminus1 (120585)]

120575120575)minus1

(120575 minus 0) (33)


120582 (119901119891119886 119901119889) cong (120575 minus 0)

119879([Jminus1 (120585)]

120575120575

10038161003816100381610038161003816120575=0)minus1

(120575 minus 0)

= 120575119879(CRLB (120575)|120575=0)

minus1120575

(34)





119891119886 119901119889) = 1 Consequently (34)







120588 (119891) =

1205880

10038161003816100381610038161198911003816100381610038161003816 le 1198611199082

010038161003816100381610038161198911003816100381610038161003816 gt 1198611199082

1198611199082radic3


0exp (minus2119891212059021198612

119908) 120590119861

1199082



119888 increas-








120593and 120575

120601




1and 120593

1 Moreover it


120601







119888= 9GHz and 119861

119908= 02119891

119888 then the RMS



119905= 119903119903= 119903 = 1m










1=




AntennaTarget

(a)

AntennaTarget

(b)

AntennaTarget

(c)

AntennaTarget

(d)



SARL

of e

leva

tion

(deg

)

Type (a)Type (b)

Type (c)Type (d)

10minus3

10minus4

10minus5



SARL

of a

zim

uth

(deg

)

Type (a)Type (b)

Type (c)Type (d)

10minus3

10minus4

10minus5









120601) From


119901119891119886


119891119886lead



119889



2

15

05

1

0

0

300200

100

Elevation (deg)

Azimuth (deg)

8060

4020

2

18

16

14

12

08

06

04

02

1

times10minus3

times10minus3radic

MSA

E CRL

B

(a)

radicM

SAE C

RLB

2

15

05

1

0

0

300200

100

Elevation (deg)

Azimuth (deg)

8060

4020

2

18

16

14

12

08

06

04

02

1

times10minus3

times10minus3

(b)

0

0

300200

100

Elevation (deg)

Azimuth (deg)

8060

4020

times10minus4

radicM

SAE C

RLB

5

4

3

2

1

times10minus4

5

4

3

2

1

(c)

0

0

300200

100Elevation (deg)

Azimuth (deg)

8060

4020

times10minus4

times10minus4

2

1

radicM

SAE C

RLB

2

15

1

05

(d)



119891119886 119901119889) = (001 099) For visual





119905= 119903119903= 119903 isin 1m 2m 3m



6 Conclusion



01 02 03 04 05 06 07 08 09

051

152

253

354

455

55

SARL

times10minus4

pfa = 001

pfa = 005

pfa = 01

pfa = 02

pd


SNR = 20 dB and 1205931= 1∘

Elevation (rad)

Azi

mut

h (r

ad)

0 05 1 150

05

1

15





10∘ 30∘ 60∘ and (119901119891119886 119901119889) = (001 099)


10 15 20 25 30 35 40 45 50SNR (dB)

SARL

(rad

)

10minus3

10minus4

10minus5

r = 1mr = 2mr = 3m


Appendix



120597 log119901 (y | 120585)1205971205721198772

=2

1205902119899

119872

sum119898=1

int119879

Re(119910119898 (119905) minus

2

sum119897=1

120572119897

119873

sum119899=1

119904119899(119905 minus 120591

119897

119898119899) 119890minus1198952120587119891

119888120591119897

119898119899)

times (

119873

sum119899=1

119904119899(119905 minus 120591

2

119898119899) 119890minus1198952120587119891

1198881205912

119898119899)

lowast

119889119905

(A1)120597 log119901 (y | 120585)

1205971205721198682

=2

1205902119899

119872

sum119898=1

int119879

Re(119910119898 (119905) minus

2

sum119897=1

120572119897

119873

sum119899=1

119904119899(119905 minus 120591

119897

119898119899) 119890minus1198952120587119891

119888120591119897

119898119899)

times (119895

119873

sum119899=1

119904119899(119905 minus 120591

2

119898119899) 119890minus1198952120587119891

1198881205912

119898119899)

lowast

119889119905

(A2)120597 log119901 (y | 120585)

120597120575120593

=minus2 cos120593

1

1205902119899

times

119872

sum119898=1

int119879

Re

times (119910119898 (119905) minus

2

sum119897=1

120572119897

119873

sum119899=1

119904119899(119905 minus 120591

119897

119898119899) 119890minus1198952120587119891

119888120591119897

119898119899)


times (1205722

119873

sum119899=1

119886119898119899( 119904119899(119905minus1205912

119898119899)+1198952120587119891

119888119904119899(119905minus1205912

119898119899))

times119890minus1198952120587119891

1198881205912

119898119899)

lowast

119889119905

(A3)

120597 log119901 (y | 120585)120597120575120601

=2 sin120593

1

1205902119899

times

119872

sum119898=1

int119879

Re(119910119898 (119905) minus

2

sum119897=1

120572119897

119873

sum119899=1

119904119899(119905 minus 120591

119897

119898119899) 119890minus1198952120587119891

119888120591119897

119898119899)

times (1205722

119873

sum119899=1

119887119898119899( 119904119899(119905 minus 120591

2

119898119899)

+ 1198952120587119891119888119904119899(119905 minus 120591

2

119898119899))

times 119890minus1198952120587119891

1198881205912

119898119899)

lowast

119889119905

(A4)



[J (120585)]11 = minus 119864y|120585 1205972 log119901 (y | 120585)

(1205971205721198772)2

1003816100381610038161003816100381610038161003816100381610038161003816120575120593=0 120575120601=0

= 2119872119873snr

(A5)

[J (120585)]22 = minus 119864y|120585 1205972 log119901 (y | 120585)

(1205971205721198682)2

1003816100381610038161003816100381610038161003816100381610038161003816120575120593=0 120575120601=0

= 2119872119873snr

(A6)

[J (120585)]21 = [J (120585)]12

= minus 119864y|120585 1205972 log119901 (y | 120585)

12059712057211987721205971205721198682

100381610038161003816100381610038161003816100381610038161003816120575120593=0 120575120601=0

=2

1205902119899

Re minus1198951198721198731205902

119904

= 0

(A7)

[J (120585)]31 = [J (120585)]13

= minus 119864y|120585 1205972 log119901 (y | 120585)

1205971205721198772120597120575120593

1003816100381610038161003816100381610038161003816100381610038161003816120575120593=0 120575120601=0

= 41205871198911198881205722snr cos120593

11205764

(A8)

[J (120585)]41 = [J (120585)]14

= minus 119864y|120585 1205972 log119901 (y | 120585)

1205971205721198772120597120575120593

1003816100381610038161003816100381610038161003816100381610038161003816120575120593=0 120575120601=0

= minus4120587119891119888120572119868

2snr sin120593

11205765

(A9)

[J (120585)]32 = [J (120585)]23

= minus 119864y|120585 1205972 log119901 (y | 120585)

1205971205721198682120597120575120593

1003816100381610038161003816100381610038161003816100381610038161003816120575120593=0120575120601=0

= minus4120587119891119888120572119877

2snr cos120593

11205764

(A10)

[J (120585)]42 = [J (120585)]24

= minus 119864y|120585 1205972 log119901 (y | 120585)

1205971205721198682120597120575120601

1003816100381610038161003816100381610038161003816100381610038161003816120575120593=0 120575120601=0

= 4120587119891119888120572119877

2snr sin120593

11205765

(A11)

[J (120585)]33 = minus119864y|120585

1205972 log119901 (y | 120585)

(120597120575120593)2

100381610038161003816100381610038161003816100381610038161003816100381610038161003816120575120593=0 120575120601=0

= 81205872 10038171003817100381710038171205722

10038171003817100381710038172(1198612

rms + 1198912

119888) snr cos2120593

11205761

(A12)

[J (120585)]44 = minus119864y|120585

1205972 log119901 (y | 120585)

(120597120575120601)2

100381610038161003816100381610038161003816100381610038161003816100381610038161003816120575120593=0120575120601=0

= 81205872 10038171003817100381710038171205722

10038171003817100381710038172(1198612

rms + 1198912

119888) snr sin2120593

11205762

(A13)

[J (120585)]34 = [J (120585)]34

= minus119864y|120585 1205972 log119901 (y | 120585)

120597120575120593120597120575120601

1003816100381610038161003816100381610038161003816100381610038161003816120575120593=0120575120601=0

= minus81205872 10038171003817100381710038171205722

10038171003817100381710038172(1198612

rms + 1198912

119888) snr sin120593

1cos12059311205763

(A14)



Acknowledgments



References
































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











1205761= [

[

(cos1205931)21205761

minus sin1205931cos12059311205763

minus sin1205931cos12059311205763

(sin1205931)21205762

]

]

1205762= [

[

(cos1205931)212057624

minus sin1205931cos120593112057641205765

minus sin1205931cos120593112057641205765

(sin1205931)212057625

]

]

1205761=

119872

sum119898=1

119873

sum119899=1

1198862

119898119899 120576

2=

119872

sum119898=1

119873

sum119899=1

1198872

119898119899 120576

3=

119872

sum119898=1

119873

sum119899=1

119886119898119899119887119898119899

1205764=

119872

sum119898=1

119873

sum119899=1

119886119898119899 120576

5=

119872

sum119898=1

119873

sum119899=1

119887119898119899

(17)


1=

1205762= 119872119873(1199032

119905+ 1199032119903)(21198882) and 120576

3= 1205764= 1205765= 0 for uniform


cos212059311205752

120593+ sin2120593

11205752

120601

=1198882120582 (119901

119891119886 119901119889)

41205872119872210038171003817100381710038171205722

10038171003817100381710038172(1199032119905+ 1199032119903) (1198912119888+ 1198612rms) SNR

(18)






1 1205931 and 120601



120585 = [120572119877

2120572119868

2⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟120572

120575120593120575120601⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟120575

]

119879

(19)

where 1205721198772

= R1205722 and 120572119868

2= I120572



2= 1205721198772+ 1198951205721198682 In (19) we



119877

2120572119868

2]119879


J (120585) = 119864y|120585 [120597

120597120585log119901 (y | 120585)] [ 120597

120597120585log119901 (y | 120585)]

119867

(20)



119901 (y | 120585) prop expminus 1

1205902119899

119872

sum119898=1

int119879

1003817100381710038171003817119910119898 (119905) minus 119909119898 (119905)10038171003817100381710038172119889119905 (21)


[J (120585)]119894119895 = [J (120585)]119895119894 = minus119864y|120585 1205972 log119901 (y | 120585)

120597120585119894120597120585119895

119894 119895 = 1 2 3 4

(22)


119899(119905minus120591119897119898119899) = 120597119904


in Section 2 119904119899(119905) = int

1198911198901198952120587119891119905

119889119878119899(119891) then 119904

119899(119905 minus 120591

119897

119898119899) can be

expressed as

119904119899(119905 minus 120591

119897

119898119899) = int119891

minus11989521205871198911198901198952120587119891(119905minus120591

119897

119898119899)119889119878119899(119891) (23)


119864int119879

119904119899(119905 minus 120591

1

119898119899) 119904lowast

119899(119905 minus 120591

1

119898119899) 119889119905 = int

119891

minus1198952120587119891120588119899119889119891 = 0

119864 int119879

119904119899(119905 minus 120591

1

119898119899) 119904lowast

119899(119905 minus 120591

1

119898119899) 119889119905 = int

119891

(2120587119891)2120588119899119889119891

= 412058721198612

rms1205902

119904

(24)



CRLB (120585) = [J (120585)]minus1 (25)


[J (120585)] = [J120572120572 J119867120572120575

J120572120575 J120575120575] (26)



J120572120572 = 2119872119873snrI2

J120572120575 = J119867120572120575= 4120587119891

119888snr[

1205721198682cos12059311205764

minus1205721198772cos12059311205764

minus1205721198682sin12059311205765

1205721198772sin12059311205765

]

J120575120575 = 81205872 10038171003817100381710038171205722

10038171003817100381710038172(1198612

rms + 1198912

119888)


11205761

minus sin1205931cos12059311205763

minus sin1205931cos12059311205763

sin212059311205762

]

(27)




120572120572J119867120572120575]minus1

=119873

8120587210038171003817100381710038171205722

10038171003817100381710038172119872SNR

[[[[[

[

((1198612

rms + 1198912

119888) 1205761minus119891211988812057624

119872119873) cos2120593

1minus((1198612rms + 119891

2

119888) 1205763minus119891211988812057641205765

119872119873) sin120593

1cos1205931

minus((1198612rms + 1198912

119888) 1205763minus119891211988812057641205765

119872119873) sin120593

1cos1205931((1198612rms + 119891

2

119888) 1205762minus119891211988812057625

119872119873) sin2120593

1

]]]]]

]

minus1

(28)




(29)


1= 1205762= 119872119873(1199032

119905+ 1199032119903)(21198882) and 120576

3= 1205764= 1205765= 0


CRLB (120575) = 1198882

4120587210038171003817100381710038171205722

100381710038171003817100381721198722

1003817100381710038171003817120572210038171003817100381710038172(1199032119905+ 1199032119903) (1198612rms + 119891

2

119888) SNR

times[[[

[

1

cos21205931

0

01

sin21205931

]]]

]

(30)


120575120593=

119888

212058711987210038171003817100381710038171205722

1003817100381710038171003817 cos1205931radic

1

(1199032119905+ 1199032119903) (1198912119888+ 1198612rms) SNR

(31)

120575120601=

119888

212058711987210038171003817100381710038171205722

1003817100381710038171003817 sin1205931radic

1

(1199032119905+ 1199032119903) (1198912119888+ 1198612rms) SNR

(32)



119891119886 119901119889) is given as

120582 (119901119891119886 119901119889) = (120575 minus 0)

119879([Jminus1 (120585)]

120575120575)minus1

(120575 minus 0) (33)


120582 (119901119891119886 119901119889) cong (120575 minus 0)

119879([Jminus1 (120585)]

120575120575

10038161003816100381610038161003816120575=0)minus1

(120575 minus 0)

= 120575119879(CRLB (120575)|120575=0)

minus1120575

(34)





119891119886 119901119889) = 1 Consequently (34)







120588 (119891) =

1205880

10038161003816100381610038161198911003816100381610038161003816 le 1198611199082

010038161003816100381610038161198911003816100381610038161003816 gt 1198611199082

1198611199082radic3


0exp (minus2119891212059021198612

119908) 120590119861

1199082



119888 increas-








120593and 120575

120601




1and 120593

1 Moreover it


120601







119888= 9GHz and 119861

119908= 02119891

119888 then the RMS



119905= 119903119903= 119903 = 1m










1=




AntennaTarget

(a)

AntennaTarget

(b)

AntennaTarget

(c)

AntennaTarget

(d)



SARL

of e

leva

tion

(deg

)

Type (a)Type (b)

Type (c)Type (d)

10minus3

10minus4

10minus5



SARL

of a

zim

uth

(deg

)

Type (a)Type (b)

Type (c)Type (d)

10minus3

10minus4

10minus5









120601) From


119901119891119886


119891119886lead



119889



2

15

05

1

0

0

300200

100

Elevation (deg)

Azimuth (deg)

8060

4020

2

18

16

14

12

08

06

04

02

1

times10minus3

times10minus3radic

MSA

E CRL

B

(a)

radicM

SAE C

RLB

2

15

05

1

0

0

300200

100

Elevation (deg)

Azimuth (deg)

8060

4020

2

18

16

14

12

08

06

04

02

1

times10minus3

times10minus3

(b)

0

0

300200

100

Elevation (deg)

Azimuth (deg)

8060

4020

times10minus4

radicM

SAE C

RLB

5

4

3

2

1

times10minus4

5

4

3

2

1

(c)

0

0

300200

100Elevation (deg)

Azimuth (deg)

8060

4020

times10minus4

times10minus4

2

1

radicM

SAE C

RLB

2

15

1

05

(d)



119891119886 119901119889) = (001 099) For visual





119905= 119903119903= 119903 isin 1m 2m 3m



6 Conclusion



01 02 03 04 05 06 07 08 09

051

152

253

354

455

55

SARL

times10minus4

pfa = 001

pfa = 005

pfa = 01

pfa = 02

pd


SNR = 20 dB and 1205931= 1∘

Elevation (rad)

Azi

mut

h (r

ad)

0 05 1 150

05

1

15





10∘ 30∘ 60∘ and (119901119891119886 119901119889) = (001 099)


10 15 20 25 30 35 40 45 50SNR (dB)

SARL

(rad

)

10minus3

10minus4

10minus5

r = 1mr = 2mr = 3m


Appendix



120597 log119901 (y | 120585)1205971205721198772

=2

1205902119899

119872

sum119898=1

int119879

Re(119910119898 (119905) minus

2

sum119897=1

120572119897

119873

sum119899=1

119904119899(119905 minus 120591

119897

119898119899) 119890minus1198952120587119891

119888120591119897

119898119899)

times (

119873

sum119899=1

119904119899(119905 minus 120591

2

119898119899) 119890minus1198952120587119891

1198881205912

119898119899)

lowast

119889119905

(A1)120597 log119901 (y | 120585)

1205971205721198682

=2

1205902119899

119872

sum119898=1

int119879

Re(119910119898 (119905) minus

2

sum119897=1

120572119897

119873

sum119899=1

119904119899(119905 minus 120591

119897

119898119899) 119890minus1198952120587119891

119888120591119897

119898119899)

times (119895

119873

sum119899=1

119904119899(119905 minus 120591

2

119898119899) 119890minus1198952120587119891

1198881205912

119898119899)

lowast

119889119905

(A2)120597 log119901 (y | 120585)

120597120575120593

=minus2 cos120593

1

1205902119899

times

119872

sum119898=1

int119879

Re

times (119910119898 (119905) minus

2

sum119897=1

120572119897

119873

sum119899=1

119904119899(119905 minus 120591

119897

119898119899) 119890minus1198952120587119891

119888120591119897

119898119899)


times (1205722

119873

sum119899=1

119886119898119899( 119904119899(119905minus1205912

119898119899)+1198952120587119891

119888119904119899(119905minus1205912

119898119899))

times119890minus1198952120587119891

1198881205912

119898119899)

lowast

119889119905

(A3)

120597 log119901 (y | 120585)120597120575120601

=2 sin120593

1

1205902119899

times

119872

sum119898=1

int119879

Re(119910119898 (119905) minus

2

sum119897=1

120572119897

119873

sum119899=1

119904119899(119905 minus 120591

119897

119898119899) 119890minus1198952120587119891

119888120591119897

119898119899)

times (1205722

119873

sum119899=1

119887119898119899( 119904119899(119905 minus 120591

2

119898119899)

+ 1198952120587119891119888119904119899(119905 minus 120591

2

119898119899))

times 119890minus1198952120587119891

1198881205912

119898119899)

lowast

119889119905

(A4)



[J (120585)]11 = minus 119864y|120585 1205972 log119901 (y | 120585)

(1205971205721198772)2

1003816100381610038161003816100381610038161003816100381610038161003816120575120593=0 120575120601=0

= 2119872119873snr

(A5)

[J (120585)]22 = minus 119864y|120585 1205972 log119901 (y | 120585)

(1205971205721198682)2

1003816100381610038161003816100381610038161003816100381610038161003816120575120593=0 120575120601=0

= 2119872119873snr

(A6)

[J (120585)]21 = [J (120585)]12

= minus 119864y|120585 1205972 log119901 (y | 120585)

12059712057211987721205971205721198682

100381610038161003816100381610038161003816100381610038161003816120575120593=0 120575120601=0

=2

1205902119899

Re minus1198951198721198731205902

119904

= 0

(A7)

[J (120585)]31 = [J (120585)]13

= minus 119864y|120585 1205972 log119901 (y | 120585)

1205971205721198772120597120575120593

1003816100381610038161003816100381610038161003816100381610038161003816120575120593=0 120575120601=0

= 41205871198911198881205722snr cos120593

11205764

(A8)

[J (120585)]41 = [J (120585)]14

= minus 119864y|120585 1205972 log119901 (y | 120585)

1205971205721198772120597120575120593

1003816100381610038161003816100381610038161003816100381610038161003816120575120593=0 120575120601=0

= minus4120587119891119888120572119868

2snr sin120593

11205765

(A9)

[J (120585)]32 = [J (120585)]23

= minus 119864y|120585 1205972 log119901 (y | 120585)

1205971205721198682120597120575120593

1003816100381610038161003816100381610038161003816100381610038161003816120575120593=0120575120601=0

= minus4120587119891119888120572119877

2snr cos120593

11205764

(A10)

[J (120585)]42 = [J (120585)]24

= minus 119864y|120585 1205972 log119901 (y | 120585)

1205971205721198682120597120575120601

1003816100381610038161003816100381610038161003816100381610038161003816120575120593=0 120575120601=0

= 4120587119891119888120572119877

2snr sin120593

11205765

(A11)

[J (120585)]33 = minus119864y|120585

1205972 log119901 (y | 120585)

(120597120575120593)2

100381610038161003816100381610038161003816100381610038161003816100381610038161003816120575120593=0 120575120601=0

= 81205872 10038171003817100381710038171205722

10038171003817100381710038172(1198612

rms + 1198912

119888) snr cos2120593

11205761

(A12)

[J (120585)]44 = minus119864y|120585

1205972 log119901 (y | 120585)

(120597120575120601)2

100381610038161003816100381610038161003816100381610038161003816100381610038161003816120575120593=0120575120601=0

= 81205872 10038171003817100381710038171205722

10038171003817100381710038172(1198612

rms + 1198912

119888) snr sin2120593

11205762

(A13)

[J (120585)]34 = [J (120585)]34

= minus119864y|120585 1205972 log119901 (y | 120585)

120597120575120593120597120575120601

1003816100381610038161003816100381610038161003816100381610038161003816120575120593=0120575120601=0

= minus81205872 10038171003817100381710038171205722

10038171003817100381710038172(1198612

rms + 1198912

119888) snr sin120593

1cos12059311205763

(A14)



Acknowledgments



References
































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











J120572120572 = 2119872119873snrI2

J120572120575 = J119867120572120575= 4120587119891

119888snr[

1205721198682cos12059311205764

minus1205721198772cos12059311205764

minus1205721198682sin12059311205765

1205721198772sin12059311205765

]

J120575120575 = 81205872 10038171003817100381710038171205722

10038171003817100381710038172(1198612

rms + 1198912

119888)


11205761

minus sin1205931cos12059311205763

minus sin1205931cos12059311205763

sin212059311205762

]

(27)




120572120572J119867120572120575]minus1

=119873

8120587210038171003817100381710038171205722

10038171003817100381710038172119872SNR

[[[[[

[

((1198612

rms + 1198912

119888) 1205761minus119891211988812057624

119872119873) cos2120593

1minus((1198612rms + 119891

2

119888) 1205763minus119891211988812057641205765

119872119873) sin120593

1cos1205931

minus((1198612rms + 1198912

119888) 1205763minus119891211988812057641205765

119872119873) sin120593

1cos1205931((1198612rms + 119891

2

119888) 1205762minus119891211988812057625

119872119873) sin2120593

1

]]]]]

]

minus1

(28)




(29)


1= 1205762= 119872119873(1199032

119905+ 1199032119903)(21198882) and 120576

3= 1205764= 1205765= 0


CRLB (120575) = 1198882

4120587210038171003817100381710038171205722

100381710038171003817100381721198722

1003817100381710038171003817120572210038171003817100381710038172(1199032119905+ 1199032119903) (1198612rms + 119891

2

119888) SNR

times[[[

[

1

cos21205931

0

01

sin21205931

]]]

]

(30)


120575120593=

119888

212058711987210038171003817100381710038171205722

1003817100381710038171003817 cos1205931radic

1

(1199032119905+ 1199032119903) (1198912119888+ 1198612rms) SNR

(31)

120575120601=

119888

212058711987210038171003817100381710038171205722

1003817100381710038171003817 sin1205931radic

1

(1199032119905+ 1199032119903) (1198912119888+ 1198612rms) SNR

(32)



119891119886 119901119889) is given as

120582 (119901119891119886 119901119889) = (120575 minus 0)

119879([Jminus1 (120585)]

120575120575)minus1

(120575 minus 0) (33)


120582 (119901119891119886 119901119889) cong (120575 minus 0)

119879([Jminus1 (120585)]

120575120575

10038161003816100381610038161003816120575=0)minus1

(120575 minus 0)

= 120575119879(CRLB (120575)|120575=0)

minus1120575

(34)





119891119886 119901119889) = 1 Consequently (34)







120588 (119891) =

1205880

10038161003816100381610038161198911003816100381610038161003816 le 1198611199082

010038161003816100381610038161198911003816100381610038161003816 gt 1198611199082

1198611199082radic3


0exp (minus2119891212059021198612

119908) 120590119861

1199082



119888 increas-








120593and 120575

120601




1and 120593

1 Moreover it


120601







119888= 9GHz and 119861

119908= 02119891

119888 then the RMS



119905= 119903119903= 119903 = 1m










1=




AntennaTarget

(a)

AntennaTarget

(b)

AntennaTarget

(c)

AntennaTarget

(d)



SARL

of e

leva

tion

(deg

)

Type (a)Type (b)

Type (c)Type (d)

10minus3

10minus4

10minus5



SARL

of a

zim

uth

(deg

)

Type (a)Type (b)

Type (c)Type (d)

10minus3

10minus4

10minus5









120601) From


119901119891119886


119891119886lead



119889



2

15

05

1

0

0

300200

100

Elevation (deg)

Azimuth (deg)

8060

4020

2

18

16

14

12

08

06

04

02

1

times10minus3

times10minus3radic

MSA

E CRL

B

(a)

radicM

SAE C

RLB

2

15

05

1

0

0

300200

100

Elevation (deg)

Azimuth (deg)

8060

4020

2

18

16

14

12

08

06

04

02

1

times10minus3

times10minus3

(b)

0

0

300200

100

Elevation (deg)

Azimuth (deg)

8060

4020

times10minus4

radicM

SAE C

RLB

5

4

3

2

1

times10minus4

5

4

3

2

1

(c)

0

0

300200

100Elevation (deg)

Azimuth (deg)

8060

4020

times10minus4

times10minus4

2

1

radicM

SAE C

RLB

2

15

1

05

(d)



119891119886 119901119889) = (001 099) For visual





119905= 119903119903= 119903 isin 1m 2m 3m



6 Conclusion



01 02 03 04 05 06 07 08 09

051

152

253

354

455

55

SARL

times10minus4

pfa = 001

pfa = 005

pfa = 01

pfa = 02

pd


SNR = 20 dB and 1205931= 1∘

Elevation (rad)

Azi

mut

h (r

ad)

0 05 1 150

05

1

15





10∘ 30∘ 60∘ and (119901119891119886 119901119889) = (001 099)


10 15 20 25 30 35 40 45 50SNR (dB)

SARL

(rad

)

10minus3

10minus4

10minus5

r = 1mr = 2mr = 3m


Appendix



120597 log119901 (y | 120585)1205971205721198772

=2

1205902119899

119872

sum119898=1

int119879

Re(119910119898 (119905) minus

2

sum119897=1

120572119897

119873

sum119899=1

119904119899(119905 minus 120591

119897

119898119899) 119890minus1198952120587119891

119888120591119897

119898119899)

times (

119873

sum119899=1

119904119899(119905 minus 120591

2

119898119899) 119890minus1198952120587119891

1198881205912

119898119899)

lowast

119889119905

(A1)120597 log119901 (y | 120585)

1205971205721198682

=2

1205902119899

119872

sum119898=1

int119879

Re(119910119898 (119905) minus

2

sum119897=1

120572119897

119873

sum119899=1

119904119899(119905 minus 120591

119897

119898119899) 119890minus1198952120587119891

119888120591119897

119898119899)

times (119895

119873

sum119899=1

119904119899(119905 minus 120591

2

119898119899) 119890minus1198952120587119891

1198881205912

119898119899)

lowast

119889119905

(A2)120597 log119901 (y | 120585)

120597120575120593

=minus2 cos120593

1

1205902119899

times

119872

sum119898=1

int119879

Re

times (119910119898 (119905) minus

2

sum119897=1

120572119897

119873

sum119899=1

119904119899(119905 minus 120591

119897

119898119899) 119890minus1198952120587119891

119888120591119897

119898119899)


times (1205722

119873

sum119899=1

119886119898119899( 119904119899(119905minus1205912

119898119899)+1198952120587119891

119888119904119899(119905minus1205912

119898119899))

times119890minus1198952120587119891

1198881205912

119898119899)

lowast

119889119905

(A3)

120597 log119901 (y | 120585)120597120575120601

=2 sin120593

1

1205902119899

times

119872

sum119898=1

int119879

Re(119910119898 (119905) minus

2

sum119897=1

120572119897

119873

sum119899=1

119904119899(119905 minus 120591

119897

119898119899) 119890minus1198952120587119891

119888120591119897

119898119899)

times (1205722

119873

sum119899=1

119887119898119899( 119904119899(119905 minus 120591

2

119898119899)

+ 1198952120587119891119888119904119899(119905 minus 120591

2

119898119899))

times 119890minus1198952120587119891

1198881205912

119898119899)

lowast

119889119905

(A4)



[J (120585)]11 = minus 119864y|120585 1205972 log119901 (y | 120585)

(1205971205721198772)2

1003816100381610038161003816100381610038161003816100381610038161003816120575120593=0 120575120601=0

= 2119872119873snr

(A5)

[J (120585)]22 = minus 119864y|120585 1205972 log119901 (y | 120585)

(1205971205721198682)2

1003816100381610038161003816100381610038161003816100381610038161003816120575120593=0 120575120601=0

= 2119872119873snr

(A6)

[J (120585)]21 = [J (120585)]12

= minus 119864y|120585 1205972 log119901 (y | 120585)

12059712057211987721205971205721198682

100381610038161003816100381610038161003816100381610038161003816120575120593=0 120575120601=0

=2

1205902119899

Re minus1198951198721198731205902

119904

= 0

(A7)

[J (120585)]31 = [J (120585)]13

= minus 119864y|120585 1205972 log119901 (y | 120585)

1205971205721198772120597120575120593

1003816100381610038161003816100381610038161003816100381610038161003816120575120593=0 120575120601=0

= 41205871198911198881205722snr cos120593

11205764

(A8)

[J (120585)]41 = [J (120585)]14

= minus 119864y|120585 1205972 log119901 (y | 120585)

1205971205721198772120597120575120593

1003816100381610038161003816100381610038161003816100381610038161003816120575120593=0 120575120601=0

= minus4120587119891119888120572119868

2snr sin120593

11205765

(A9)

[J (120585)]32 = [J (120585)]23

= minus 119864y|120585 1205972 log119901 (y | 120585)

1205971205721198682120597120575120593

1003816100381610038161003816100381610038161003816100381610038161003816120575120593=0120575120601=0

= minus4120587119891119888120572119877

2snr cos120593

11205764

(A10)

[J (120585)]42 = [J (120585)]24

= minus 119864y|120585 1205972 log119901 (y | 120585)

1205971205721198682120597120575120601

1003816100381610038161003816100381610038161003816100381610038161003816120575120593=0 120575120601=0

= 4120587119891119888120572119877

2snr sin120593

11205765

(A11)

[J (120585)]33 = minus119864y|120585

1205972 log119901 (y | 120585)

(120597120575120593)2

100381610038161003816100381610038161003816100381610038161003816100381610038161003816120575120593=0 120575120601=0

= 81205872 10038171003817100381710038171205722

10038171003817100381710038172(1198612

rms + 1198912

119888) snr cos2120593

11205761

(A12)

[J (120585)]44 = minus119864y|120585

1205972 log119901 (y | 120585)

(120597120575120601)2

100381610038161003816100381610038161003816100381610038161003816100381610038161003816120575120593=0120575120601=0

= 81205872 10038171003817100381710038171205722

10038171003817100381710038172(1198612

rms + 1198912

119888) snr sin2120593

11205762

(A13)

[J (120585)]34 = [J (120585)]34

= minus119864y|120585 1205972 log119901 (y | 120585)

120597120575120593120597120575120601

1003816100381610038161003816100381610038161003816100381610038161003816120575120593=0120575120601=0

= minus81205872 10038171003817100381710038171205722

10038171003817100381710038172(1198612

rms + 1198912

119888) snr sin120593

1cos12059311205763

(A14)



Acknowledgments



References
































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation












120588 (119891) =

1205880

10038161003816100381610038161198911003816100381610038161003816 le 1198611199082

010038161003816100381610038161198911003816100381610038161003816 gt 1198611199082

1198611199082radic3


0exp (minus2119891212059021198612

119908) 120590119861

1199082



119888 increas-








120593and 120575

120601




1and 120593

1 Moreover it


120601







119888= 9GHz and 119861

119908= 02119891

119888 then the RMS



119905= 119903119903= 119903 = 1m










1=




AntennaTarget

(a)

AntennaTarget

(b)

AntennaTarget

(c)

AntennaTarget

(d)



SARL

of e

leva

tion

(deg

)

Type (a)Type (b)

Type (c)Type (d)

10minus3

10minus4

10minus5



SARL

of a

zim

uth

(deg

)

Type (a)Type (b)

Type (c)Type (d)

10minus3

10minus4

10minus5









120601) From


119901119891119886


119891119886lead



119889



2

15

05

1

0

0

300200

100

Elevation (deg)

Azimuth (deg)

8060

4020

2

18

16

14

12

08

06

04

02

1

times10minus3

times10minus3radic

MSA

E CRL

B

(a)

radicM

SAE C

RLB

2

15

05

1

0

0

300200

100

Elevation (deg)

Azimuth (deg)

8060

4020

2

18

16

14

12

08

06

04

02

1

times10minus3

times10minus3

(b)

0

0

300200

100

Elevation (deg)

Azimuth (deg)

8060

4020

times10minus4

radicM

SAE C

RLB

5

4

3

2

1

times10minus4

5

4

3

2

1

(c)

0

0

300200

100Elevation (deg)

Azimuth (deg)

8060

4020

times10minus4

times10minus4

2

1

radicM

SAE C

RLB

2

15

1

05

(d)



119891119886 119901119889) = (001 099) For visual





119905= 119903119903= 119903 isin 1m 2m 3m



6 Conclusion



01 02 03 04 05 06 07 08 09

051

152

253

354

455

55

SARL

times10minus4

pfa = 001

pfa = 005

pfa = 01

pfa = 02

pd


SNR = 20 dB and 1205931= 1∘

Elevation (rad)

Azi

mut

h (r

ad)

0 05 1 150

05

1

15





10∘ 30∘ 60∘ and (119901119891119886 119901119889) = (001 099)


10 15 20 25 30 35 40 45 50SNR (dB)

SARL

(rad

)

10minus3

10minus4

10minus5

r = 1mr = 2mr = 3m


Appendix



120597 log119901 (y | 120585)1205971205721198772

=2

1205902119899

119872

sum119898=1

int119879

Re(119910119898 (119905) minus

2

sum119897=1

120572119897

119873

sum119899=1

119904119899(119905 minus 120591

119897

119898119899) 119890minus1198952120587119891

119888120591119897

119898119899)

times (

119873

sum119899=1

119904119899(119905 minus 120591

2

119898119899) 119890minus1198952120587119891

1198881205912

119898119899)

lowast

119889119905

(A1)120597 log119901 (y | 120585)

1205971205721198682

=2

1205902119899

119872

sum119898=1

int119879

Re(119910119898 (119905) minus

2

sum119897=1

120572119897

119873

sum119899=1

119904119899(119905 minus 120591

119897

119898119899) 119890minus1198952120587119891

119888120591119897

119898119899)

times (119895

119873

sum119899=1

119904119899(119905 minus 120591

2

119898119899) 119890minus1198952120587119891

1198881205912

119898119899)

lowast

119889119905

(A2)120597 log119901 (y | 120585)

120597120575120593

=minus2 cos120593

1

1205902119899

times

119872

sum119898=1

int119879

Re

times (119910119898 (119905) minus

2

sum119897=1

120572119897

119873

sum119899=1

119904119899(119905 minus 120591

119897

119898119899) 119890minus1198952120587119891

119888120591119897

119898119899)


times (1205722

119873

sum119899=1

119886119898119899( 119904119899(119905minus1205912

119898119899)+1198952120587119891

119888119904119899(119905minus1205912

119898119899))

times119890minus1198952120587119891

1198881205912

119898119899)

lowast

119889119905

(A3)

120597 log119901 (y | 120585)120597120575120601

=2 sin120593

1

1205902119899

times

119872

sum119898=1

int119879

Re(119910119898 (119905) minus

2

sum119897=1

120572119897

119873

sum119899=1

119904119899(119905 minus 120591

119897

119898119899) 119890minus1198952120587119891

119888120591119897

119898119899)

times (1205722

119873

sum119899=1

119887119898119899( 119904119899(119905 minus 120591

2

119898119899)

+ 1198952120587119891119888119904119899(119905 minus 120591

2

119898119899))

times 119890minus1198952120587119891

1198881205912

119898119899)

lowast

119889119905

(A4)



[J (120585)]11 = minus 119864y|120585 1205972 log119901 (y | 120585)

(1205971205721198772)2

1003816100381610038161003816100381610038161003816100381610038161003816120575120593=0 120575120601=0

= 2119872119873snr

(A5)

[J (120585)]22 = minus 119864y|120585 1205972 log119901 (y | 120585)

(1205971205721198682)2

1003816100381610038161003816100381610038161003816100381610038161003816120575120593=0 120575120601=0

= 2119872119873snr

(A6)

[J (120585)]21 = [J (120585)]12

= minus 119864y|120585 1205972 log119901 (y | 120585)

12059712057211987721205971205721198682

100381610038161003816100381610038161003816100381610038161003816120575120593=0 120575120601=0

=2

1205902119899

Re minus1198951198721198731205902

119904

= 0

(A7)

[J (120585)]31 = [J (120585)]13

= minus 119864y|120585 1205972 log119901 (y | 120585)

1205971205721198772120597120575120593

1003816100381610038161003816100381610038161003816100381610038161003816120575120593=0 120575120601=0

= 41205871198911198881205722snr cos120593

11205764

(A8)

[J (120585)]41 = [J (120585)]14

= minus 119864y|120585 1205972 log119901 (y | 120585)

1205971205721198772120597120575120593

1003816100381610038161003816100381610038161003816100381610038161003816120575120593=0 120575120601=0

= minus4120587119891119888120572119868

2snr sin120593

11205765

(A9)

[J (120585)]32 = [J (120585)]23

= minus 119864y|120585 1205972 log119901 (y | 120585)

1205971205721198682120597120575120593

1003816100381610038161003816100381610038161003816100381610038161003816120575120593=0120575120601=0

= minus4120587119891119888120572119877

2snr cos120593

11205764

(A10)

[J (120585)]42 = [J (120585)]24

= minus 119864y|120585 1205972 log119901 (y | 120585)

1205971205721198682120597120575120601

1003816100381610038161003816100381610038161003816100381610038161003816120575120593=0 120575120601=0

= 4120587119891119888120572119877

2snr sin120593

11205765

(A11)

[J (120585)]33 = minus119864y|120585

1205972 log119901 (y | 120585)

(120597120575120593)2

100381610038161003816100381610038161003816100381610038161003816100381610038161003816120575120593=0 120575120601=0

= 81205872 10038171003817100381710038171205722

10038171003817100381710038172(1198612

rms + 1198912

119888) snr cos2120593

11205761

(A12)

[J (120585)]44 = minus119864y|120585

1205972 log119901 (y | 120585)

(120597120575120601)2

100381610038161003816100381610038161003816100381610038161003816100381610038161003816120575120593=0120575120601=0

= 81205872 10038171003817100381710038171205722

10038171003817100381710038172(1198612

rms + 1198912

119888) snr sin2120593

11205762

(A13)

[J (120585)]34 = [J (120585)]34

= minus119864y|120585 1205972 log119901 (y | 120585)

120597120575120593120597120575120601

1003816100381610038161003816100381610038161003816100381610038161003816120575120593=0120575120601=0

= minus81205872 10038171003817100381710038171205722

10038171003817100381710038172(1198612

rms + 1198912

119888) snr sin120593

1cos12059311205763

(A14)



Acknowledgments



References
































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










AntennaTarget

(a)

AntennaTarget

(b)

AntennaTarget

(c)

AntennaTarget

(d)



SARL

of e

leva

tion

(deg

)

Type (a)Type (b)

Type (c)Type (d)

10minus3

10minus4

10minus5



SARL

of a

zim

uth

(deg

)

Type (a)Type (b)

Type (c)Type (d)

10minus3

10minus4

10minus5









120601) From


119901119891119886


119891119886lead



119889



2

15

05

1

0

0

300200

100

Elevation (deg)

Azimuth (deg)

8060

4020

2

18

16

14

12

08

06

04

02

1

times10minus3

times10minus3radic

MSA

E CRL

B

(a)

radicM

SAE C

RLB

2

15

05

1

0

0

300200

100

Elevation (deg)

Azimuth (deg)

8060

4020

2

18

16

14

12

08

06

04

02

1

times10minus3

times10minus3

(b)

0

0

300200

100

Elevation (deg)

Azimuth (deg)

8060

4020

times10minus4

radicM

SAE C

RLB

5

4

3

2

1

times10minus4

5

4

3

2

1

(c)

0

0

300200

100Elevation (deg)

Azimuth (deg)

8060

4020

times10minus4

times10minus4

2

1

radicM

SAE C

RLB

2

15

1

05

(d)



119891119886 119901119889) = (001 099) For visual





119905= 119903119903= 119903 isin 1m 2m 3m



6 Conclusion



01 02 03 04 05 06 07 08 09

051

152

253

354

455

55

SARL

times10minus4

pfa = 001

pfa = 005

pfa = 01

pfa = 02

pd


SNR = 20 dB and 1205931= 1∘

Elevation (rad)

Azi

mut

h (r

ad)

0 05 1 150

05

1

15





10∘ 30∘ 60∘ and (119901119891119886 119901119889) = (001 099)


10 15 20 25 30 35 40 45 50SNR (dB)

SARL

(rad

)

10minus3

10minus4

10minus5

r = 1mr = 2mr = 3m


Appendix



120597 log119901 (y | 120585)1205971205721198772

=2

1205902119899

119872

sum119898=1

int119879

Re(119910119898 (119905) minus

2

sum119897=1

120572119897

119873

sum119899=1

119904119899(119905 minus 120591

119897

119898119899) 119890minus1198952120587119891

119888120591119897

119898119899)

times (

119873

sum119899=1

119904119899(119905 minus 120591

2

119898119899) 119890minus1198952120587119891

1198881205912

119898119899)

lowast

119889119905

(A1)120597 log119901 (y | 120585)

1205971205721198682

=2

1205902119899

119872

sum119898=1

int119879

Re(119910119898 (119905) minus

2

sum119897=1

120572119897

119873

sum119899=1

119904119899(119905 minus 120591

119897

119898119899) 119890minus1198952120587119891

119888120591119897

119898119899)

times (119895

119873

sum119899=1

119904119899(119905 minus 120591

2

119898119899) 119890minus1198952120587119891

1198881205912

119898119899)

lowast

119889119905

(A2)120597 log119901 (y | 120585)

120597120575120593

=minus2 cos120593

1

1205902119899

times

119872

sum119898=1

int119879

Re

times (119910119898 (119905) minus

2

sum119897=1

120572119897

119873

sum119899=1

119904119899(119905 minus 120591

119897

119898119899) 119890minus1198952120587119891

119888120591119897

119898119899)


times (1205722

119873

sum119899=1

119886119898119899( 119904119899(119905minus1205912

119898119899)+1198952120587119891

119888119904119899(119905minus1205912

119898119899))

times119890minus1198952120587119891

1198881205912

119898119899)

lowast

119889119905

(A3)

120597 log119901 (y | 120585)120597120575120601

=2 sin120593

1

1205902119899

times

119872

sum119898=1

int119879

Re(119910119898 (119905) minus

2

sum119897=1

120572119897

119873

sum119899=1

119904119899(119905 minus 120591

119897

119898119899) 119890minus1198952120587119891

119888120591119897

119898119899)

times (1205722

119873

sum119899=1

119887119898119899( 119904119899(119905 minus 120591

2

119898119899)

+ 1198952120587119891119888119904119899(119905 minus 120591

2

119898119899))

times 119890minus1198952120587119891

1198881205912

119898119899)

lowast

119889119905

(A4)



[J (120585)]11 = minus 119864y|120585 1205972 log119901 (y | 120585)

(1205971205721198772)2

1003816100381610038161003816100381610038161003816100381610038161003816120575120593=0 120575120601=0

= 2119872119873snr

(A5)

[J (120585)]22 = minus 119864y|120585 1205972 log119901 (y | 120585)

(1205971205721198682)2

1003816100381610038161003816100381610038161003816100381610038161003816120575120593=0 120575120601=0

= 2119872119873snr

(A6)

[J (120585)]21 = [J (120585)]12

= minus 119864y|120585 1205972 log119901 (y | 120585)

12059712057211987721205971205721198682

100381610038161003816100381610038161003816100381610038161003816120575120593=0 120575120601=0

=2

1205902119899

Re minus1198951198721198731205902

119904

= 0

(A7)

[J (120585)]31 = [J (120585)]13

= minus 119864y|120585 1205972 log119901 (y | 120585)

1205971205721198772120597120575120593

1003816100381610038161003816100381610038161003816100381610038161003816120575120593=0 120575120601=0

= 41205871198911198881205722snr cos120593

11205764

(A8)

[J (120585)]41 = [J (120585)]14

= minus 119864y|120585 1205972 log119901 (y | 120585)

1205971205721198772120597120575120593

1003816100381610038161003816100381610038161003816100381610038161003816120575120593=0 120575120601=0

= minus4120587119891119888120572119868

2snr sin120593

11205765

(A9)

[J (120585)]32 = [J (120585)]23

= minus 119864y|120585 1205972 log119901 (y | 120585)

1205971205721198682120597120575120593

1003816100381610038161003816100381610038161003816100381610038161003816120575120593=0120575120601=0

= minus4120587119891119888120572119877

2snr cos120593

11205764

(A10)

[J (120585)]42 = [J (120585)]24

= minus 119864y|120585 1205972 log119901 (y | 120585)

1205971205721198682120597120575120601

1003816100381610038161003816100381610038161003816100381610038161003816120575120593=0 120575120601=0

= 4120587119891119888120572119877

2snr sin120593

11205765

(A11)

[J (120585)]33 = minus119864y|120585

1205972 log119901 (y | 120585)

(120597120575120593)2

100381610038161003816100381610038161003816100381610038161003816100381610038161003816120575120593=0 120575120601=0

= 81205872 10038171003817100381710038171205722

10038171003817100381710038172(1198612

rms + 1198912

119888) snr cos2120593

11205761

(A12)

[J (120585)]44 = minus119864y|120585

1205972 log119901 (y | 120585)

(120597120575120601)2

100381610038161003816100381610038161003816100381610038161003816100381610038161003816120575120593=0120575120601=0

= 81205872 10038171003817100381710038171205722

10038171003817100381710038172(1198612

rms + 1198912

119888) snr sin2120593

11205762

(A13)

[J (120585)]34 = [J (120585)]34

= minus119864y|120585 1205972 log119901 (y | 120585)

120597120575120593120597120575120601

1003816100381610038161003816100381610038161003816100381610038161003816120575120593=0120575120601=0

= minus81205872 10038171003817100381710038171205722

10038171003817100381710038172(1198612

rms + 1198912

119888) snr sin120593

1cos12059311205763

(A14)



Acknowledgments



References
































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










2

15

05

1

0

0

300200

100

Elevation (deg)

Azimuth (deg)

8060

4020

2

18

16

14

12

08

06

04

02

1

times10minus3

times10minus3radic

MSA

E CRL

B

(a)

radicM

SAE C

RLB

2

15

05

1

0

0

300200

100

Elevation (deg)

Azimuth (deg)

8060

4020

2

18

16

14

12

08

06

04

02

1

times10minus3

times10minus3

(b)

0

0

300200

100

Elevation (deg)

Azimuth (deg)

8060

4020

times10minus4

radicM

SAE C

RLB

5

4

3

2

1

times10minus4

5

4

3

2

1

(c)

0

0

300200

100Elevation (deg)

Azimuth (deg)

8060

4020

times10minus4

times10minus4

2

1

radicM

SAE C

RLB

2

15

1

05

(d)



119891119886 119901119889) = (001 099) For visual





119905= 119903119903= 119903 isin 1m 2m 3m



6 Conclusion



01 02 03 04 05 06 07 08 09

051

152

253

354

455

55

SARL

times10minus4

pfa = 001

pfa = 005

pfa = 01

pfa = 02

pd


SNR = 20 dB and 1205931= 1∘

Elevation (rad)

Azi

mut

h (r

ad)

0 05 1 150

05

1

15





10∘ 30∘ 60∘ and (119901119891119886 119901119889) = (001 099)


10 15 20 25 30 35 40 45 50SNR (dB)

SARL

(rad

)

10minus3

10minus4

10minus5

r = 1mr = 2mr = 3m


Appendix



120597 log119901 (y | 120585)1205971205721198772

=2

1205902119899

119872

sum119898=1

int119879

Re(119910119898 (119905) minus

2

sum119897=1

120572119897

119873

sum119899=1

119904119899(119905 minus 120591

119897

119898119899) 119890minus1198952120587119891

119888120591119897

119898119899)

times (

119873

sum119899=1

119904119899(119905 minus 120591

2

119898119899) 119890minus1198952120587119891

1198881205912

119898119899)

lowast

119889119905

(A1)120597 log119901 (y | 120585)

1205971205721198682

=2

1205902119899

119872

sum119898=1

int119879

Re(119910119898 (119905) minus

2

sum119897=1

120572119897

119873

sum119899=1

119904119899(119905 minus 120591

119897

119898119899) 119890minus1198952120587119891

119888120591119897

119898119899)

times (119895

119873

sum119899=1

119904119899(119905 minus 120591

2

119898119899) 119890minus1198952120587119891

1198881205912

119898119899)

lowast

119889119905

(A2)120597 log119901 (y | 120585)

120597120575120593

=minus2 cos120593

1

1205902119899

times

119872

sum119898=1

int119879

Re

times (119910119898 (119905) minus

2

sum119897=1

120572119897

119873

sum119899=1

119904119899(119905 minus 120591

119897

119898119899) 119890minus1198952120587119891

119888120591119897

119898119899)


times (1205722

119873

sum119899=1

119886119898119899( 119904119899(119905minus1205912

119898119899)+1198952120587119891

119888119904119899(119905minus1205912

119898119899))

times119890minus1198952120587119891

1198881205912

119898119899)

lowast

119889119905

(A3)

120597 log119901 (y | 120585)120597120575120601

=2 sin120593

1

1205902119899

times

119872

sum119898=1

int119879

Re(119910119898 (119905) minus

2

sum119897=1

120572119897

119873

sum119899=1

119904119899(119905 minus 120591

119897

119898119899) 119890minus1198952120587119891

119888120591119897

119898119899)

times (1205722

119873

sum119899=1

119887119898119899( 119904119899(119905 minus 120591

2

119898119899)

+ 1198952120587119891119888119904119899(119905 minus 120591

2

119898119899))

times 119890minus1198952120587119891

1198881205912

119898119899)

lowast

119889119905

(A4)



[J (120585)]11 = minus 119864y|120585 1205972 log119901 (y | 120585)

(1205971205721198772)2

1003816100381610038161003816100381610038161003816100381610038161003816120575120593=0 120575120601=0

= 2119872119873snr

(A5)

[J (120585)]22 = minus 119864y|120585 1205972 log119901 (y | 120585)

(1205971205721198682)2

1003816100381610038161003816100381610038161003816100381610038161003816120575120593=0 120575120601=0

= 2119872119873snr

(A6)

[J (120585)]21 = [J (120585)]12

= minus 119864y|120585 1205972 log119901 (y | 120585)

12059712057211987721205971205721198682

100381610038161003816100381610038161003816100381610038161003816120575120593=0 120575120601=0

=2

1205902119899

Re minus1198951198721198731205902

119904

= 0

(A7)

[J (120585)]31 = [J (120585)]13

= minus 119864y|120585 1205972 log119901 (y | 120585)

1205971205721198772120597120575120593

1003816100381610038161003816100381610038161003816100381610038161003816120575120593=0 120575120601=0

= 41205871198911198881205722snr cos120593

11205764

(A8)

[J (120585)]41 = [J (120585)]14

= minus 119864y|120585 1205972 log119901 (y | 120585)

1205971205721198772120597120575120593

1003816100381610038161003816100381610038161003816100381610038161003816120575120593=0 120575120601=0

= minus4120587119891119888120572119868

2snr sin120593

11205765

(A9)

[J (120585)]32 = [J (120585)]23

= minus 119864y|120585 1205972 log119901 (y | 120585)

1205971205721198682120597120575120593

1003816100381610038161003816100381610038161003816100381610038161003816120575120593=0120575120601=0

= minus4120587119891119888120572119877

2snr cos120593

11205764

(A10)

[J (120585)]42 = [J (120585)]24

= minus 119864y|120585 1205972 log119901 (y | 120585)

1205971205721198682120597120575120601

1003816100381610038161003816100381610038161003816100381610038161003816120575120593=0 120575120601=0

= 4120587119891119888120572119877

2snr sin120593

11205765

(A11)

[J (120585)]33 = minus119864y|120585

1205972 log119901 (y | 120585)

(120597120575120593)2

100381610038161003816100381610038161003816100381610038161003816100381610038161003816120575120593=0 120575120601=0

= 81205872 10038171003817100381710038171205722

10038171003817100381710038172(1198612

rms + 1198912

119888) snr cos2120593

11205761

(A12)

[J (120585)]44 = minus119864y|120585

1205972 log119901 (y | 120585)

(120597120575120601)2

100381610038161003816100381610038161003816100381610038161003816100381610038161003816120575120593=0120575120601=0

= 81205872 10038171003817100381710038171205722

10038171003817100381710038172(1198612

rms + 1198912

119888) snr sin2120593

11205762

(A13)

[J (120585)]34 = [J (120585)]34

= minus119864y|120585 1205972 log119901 (y | 120585)

120597120575120593120597120575120601

1003816100381610038161003816100381610038161003816100381610038161003816120575120593=0120575120601=0

= minus81205872 10038171003817100381710038171205722

10038171003817100381710038172(1198612

rms + 1198912

119888) snr sin120593

1cos12059311205763

(A14)



Acknowledgments



References
































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










01 02 03 04 05 06 07 08 09

051

152

253

354

455

55

SARL

times10minus4

pfa = 001

pfa = 005

pfa = 01

pfa = 02

pd


SNR = 20 dB and 1205931= 1∘

Elevation (rad)

Azi

mut

h (r

ad)

0 05 1 150

05

1

15





10∘ 30∘ 60∘ and (119901119891119886 119901119889) = (001 099)


10 15 20 25 30 35 40 45 50SNR (dB)

SARL

(rad

)

10minus3

10minus4

10minus5

r = 1mr = 2mr = 3m


Appendix



120597 log119901 (y | 120585)1205971205721198772

=2

1205902119899

119872

sum119898=1

int119879

Re(119910119898 (119905) minus

2

sum119897=1

120572119897

119873

sum119899=1

119904119899(119905 minus 120591

119897

119898119899) 119890minus1198952120587119891

119888120591119897

119898119899)

times (

119873

sum119899=1

119904119899(119905 minus 120591

2

119898119899) 119890minus1198952120587119891

1198881205912

119898119899)

lowast

119889119905

(A1)120597 log119901 (y | 120585)

1205971205721198682

=2

1205902119899

119872

sum119898=1

int119879

Re(119910119898 (119905) minus

2

sum119897=1

120572119897

119873

sum119899=1

119904119899(119905 minus 120591

119897

119898119899) 119890minus1198952120587119891

119888120591119897

119898119899)

times (119895

119873

sum119899=1

119904119899(119905 minus 120591

2

119898119899) 119890minus1198952120587119891

1198881205912

119898119899)

lowast

119889119905

(A2)120597 log119901 (y | 120585)

120597120575120593

=minus2 cos120593

1

1205902119899

times

119872

sum119898=1

int119879

Re

times (119910119898 (119905) minus

2

sum119897=1

120572119897

119873

sum119899=1

119904119899(119905 minus 120591

119897

119898119899) 119890minus1198952120587119891

119888120591119897

119898119899)


times (1205722

119873

sum119899=1

119886119898119899( 119904119899(119905minus1205912

119898119899)+1198952120587119891

119888119904119899(119905minus1205912

119898119899))

times119890minus1198952120587119891

1198881205912

119898119899)

lowast

119889119905

(A3)

120597 log119901 (y | 120585)120597120575120601

=2 sin120593

1

1205902119899

times

119872

sum119898=1

int119879

Re(119910119898 (119905) minus

2

sum119897=1

120572119897

119873

sum119899=1

119904119899(119905 minus 120591

119897

119898119899) 119890minus1198952120587119891

119888120591119897

119898119899)

times (1205722

119873

sum119899=1

119887119898119899( 119904119899(119905 minus 120591

2

119898119899)

+ 1198952120587119891119888119904119899(119905 minus 120591

2

119898119899))

times 119890minus1198952120587119891

1198881205912

119898119899)

lowast

119889119905

(A4)



[J (120585)]11 = minus 119864y|120585 1205972 log119901 (y | 120585)

(1205971205721198772)2

1003816100381610038161003816100381610038161003816100381610038161003816120575120593=0 120575120601=0

= 2119872119873snr

(A5)

[J (120585)]22 = minus 119864y|120585 1205972 log119901 (y | 120585)

(1205971205721198682)2

1003816100381610038161003816100381610038161003816100381610038161003816120575120593=0 120575120601=0

= 2119872119873snr

(A6)

[J (120585)]21 = [J (120585)]12

= minus 119864y|120585 1205972 log119901 (y | 120585)

12059712057211987721205971205721198682

100381610038161003816100381610038161003816100381610038161003816120575120593=0 120575120601=0

=2

1205902119899

Re minus1198951198721198731205902

119904

= 0

(A7)

[J (120585)]31 = [J (120585)]13

= minus 119864y|120585 1205972 log119901 (y | 120585)

1205971205721198772120597120575120593

1003816100381610038161003816100381610038161003816100381610038161003816120575120593=0 120575120601=0

= 41205871198911198881205722snr cos120593

11205764

(A8)

[J (120585)]41 = [J (120585)]14

= minus 119864y|120585 1205972 log119901 (y | 120585)

1205971205721198772120597120575120593

1003816100381610038161003816100381610038161003816100381610038161003816120575120593=0 120575120601=0

= minus4120587119891119888120572119868

2snr sin120593

11205765

(A9)

[J (120585)]32 = [J (120585)]23

= minus 119864y|120585 1205972 log119901 (y | 120585)

1205971205721198682120597120575120593

1003816100381610038161003816100381610038161003816100381610038161003816120575120593=0120575120601=0

= minus4120587119891119888120572119877

2snr cos120593

11205764

(A10)

[J (120585)]42 = [J (120585)]24

= minus 119864y|120585 1205972 log119901 (y | 120585)

1205971205721198682120597120575120601

1003816100381610038161003816100381610038161003816100381610038161003816120575120593=0 120575120601=0

= 4120587119891119888120572119877

2snr sin120593

11205765

(A11)

[J (120585)]33 = minus119864y|120585

1205972 log119901 (y | 120585)

(120597120575120593)2

100381610038161003816100381610038161003816100381610038161003816100381610038161003816120575120593=0 120575120601=0

= 81205872 10038171003817100381710038171205722

10038171003817100381710038172(1198612

rms + 1198912

119888) snr cos2120593

11205761

(A12)

[J (120585)]44 = minus119864y|120585

1205972 log119901 (y | 120585)

(120597120575120601)2

100381610038161003816100381610038161003816100381610038161003816100381610038161003816120575120593=0120575120601=0

= 81205872 10038171003817100381710038171205722

10038171003817100381710038172(1198612

rms + 1198912

119888) snr sin2120593

11205762

(A13)

[J (120585)]34 = [J (120585)]34

= minus119864y|120585 1205972 log119901 (y | 120585)

120597120575120593120597120575120601

1003816100381610038161003816100381610038161003816100381610038161003816120575120593=0120575120601=0

= minus81205872 10038171003817100381710038171205722

10038171003817100381710038172(1198612

rms + 1198912

119888) snr sin120593

1cos12059311205763

(A14)



Acknowledgments



References
































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










times (1205722

119873

sum119899=1

119886119898119899( 119904119899(119905minus1205912

119898119899)+1198952120587119891

119888119904119899(119905minus1205912

119898119899))

times119890minus1198952120587119891

1198881205912

119898119899)

lowast

119889119905

(A3)

120597 log119901 (y | 120585)120597120575120601

=2 sin120593

1

1205902119899

times

119872

sum119898=1

int119879

Re(119910119898 (119905) minus

2

sum119897=1

120572119897

119873

sum119899=1

119904119899(119905 minus 120591

119897

119898119899) 119890minus1198952120587119891

119888120591119897

119898119899)

times (1205722

119873

sum119899=1

119887119898119899( 119904119899(119905 minus 120591

2

119898119899)

+ 1198952120587119891119888119904119899(119905 minus 120591

2

119898119899))

times 119890minus1198952120587119891

1198881205912

119898119899)

lowast

119889119905

(A4)



[J (120585)]11 = minus 119864y|120585 1205972 log119901 (y | 120585)

(1205971205721198772)2

1003816100381610038161003816100381610038161003816100381610038161003816120575120593=0 120575120601=0

= 2119872119873snr

(A5)

[J (120585)]22 = minus 119864y|120585 1205972 log119901 (y | 120585)

(1205971205721198682)2

1003816100381610038161003816100381610038161003816100381610038161003816120575120593=0 120575120601=0

= 2119872119873snr

(A6)

[J (120585)]21 = [J (120585)]12

= minus 119864y|120585 1205972 log119901 (y | 120585)

12059712057211987721205971205721198682

100381610038161003816100381610038161003816100381610038161003816120575120593=0 120575120601=0

=2

1205902119899

Re minus1198951198721198731205902

119904

= 0

(A7)

[J (120585)]31 = [J (120585)]13

= minus 119864y|120585 1205972 log119901 (y | 120585)

1205971205721198772120597120575120593

1003816100381610038161003816100381610038161003816100381610038161003816120575120593=0 120575120601=0

= 41205871198911198881205722snr cos120593

11205764

(A8)

[J (120585)]41 = [J (120585)]14

= minus 119864y|120585 1205972 log119901 (y | 120585)

1205971205721198772120597120575120593

1003816100381610038161003816100381610038161003816100381610038161003816120575120593=0 120575120601=0

= minus4120587119891119888120572119868

2snr sin120593

11205765

(A9)

[J (120585)]32 = [J (120585)]23

= minus 119864y|120585 1205972 log119901 (y | 120585)

1205971205721198682120597120575120593

1003816100381610038161003816100381610038161003816100381610038161003816120575120593=0120575120601=0

= minus4120587119891119888120572119877

2snr cos120593

11205764

(A10)

[J (120585)]42 = [J (120585)]24

= minus 119864y|120585 1205972 log119901 (y | 120585)

1205971205721198682120597120575120601

1003816100381610038161003816100381610038161003816100381610038161003816120575120593=0 120575120601=0

= 4120587119891119888120572119877

2snr sin120593

11205765

(A11)

[J (120585)]33 = minus119864y|120585

1205972 log119901 (y | 120585)

(120597120575120593)2

100381610038161003816100381610038161003816100381610038161003816100381610038161003816120575120593=0 120575120601=0

= 81205872 10038171003817100381710038171205722

10038171003817100381710038172(1198612

rms + 1198912

119888) snr cos2120593

11205761

(A12)

[J (120585)]44 = minus119864y|120585

1205972 log119901 (y | 120585)

(120597120575120601)2

100381610038161003816100381610038161003816100381610038161003816100381610038161003816120575120593=0120575120601=0

= 81205872 10038171003817100381710038171205722

10038171003817100381710038172(1198612

rms + 1198912

119888) snr sin2120593

11205762

(A13)

[J (120585)]34 = [J (120585)]34

= minus119864y|120585 1205972 log119901 (y | 120585)

120597120575120593120597120575120601

1003816100381610038161003816100381610038161003816100381610038161003816120575120593=0120575120601=0

= minus81205872 10038171003817100381710038171205722

10038171003817100381710038172(1198612

rms + 1198912

119888) snr sin120593

1cos12059311205763

(A14)



Acknowledgments



References
































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










References
































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation









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