review article aspects of the subarrayed array processing...

Review ArticleAspects of the Subarrayed Array Processing forthe Phased Array Radar

Hang Hu

School of Electronics and Information Engineering Harbin Institute of Technology Harbin 150001 China

Correspondence should be addressed to Hang Hu huhanghiteducn

Received 20 June 2014 Revised 25 November 2014 Accepted 25 November 2014

Academic Editor Michelangelo Villano

Copyright copy 2015 Hang Hu This is an open access article distributed under the Creative Commons Attribution License whichpermits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

This paper gives an overview on the research status developments and achievements of subarrayed array processing for themultifunction phased array radar We address some issues concerning subarrayed adaptive beamforming subarrayed fast-timespace-time adaptive processing subarray-based sidelobe reduction of sum and difference beam subarrayed adapted monopulsesubarrayed superresolution direction finding subarray configuration optimization in ECCM (electronic counter-countermeasure)and subarrayed array processing for MIMO-PAR In this review several viewpoints relevant to subarrayed array processing arepointed out and the achieved results are demonstrated by numerical examples

1 Introduction

PAR (phased array radar) and especially MFPAR (multifunc-tion PAR) often adopt a subarrayed antenna array SASP (sub-arrayed array signal processing) is one of the key technologiesin modern PARs and plays a key role in the MFPAR

In this paper we consider the modern MFPAR with anelement number of the order of 1000 (assume that the sub-arrays are compact and nonoverlapping) and with amplitudetapering (eg Taylor weighting) applied at the elements forlow sidelobes And the steering into the look direction isdone by phase shifting at the elements Assume the MFPARwith many modes of operation (eg search track and highresolution) and all processing to be performed digitally withthe subarray outputs

The key technologies and issues of SASP include sub-array weighting for quiescent pattern synthesis and PSL(peak sidelobe) reduction subarrayed adaptive interferencesuppression with subarrayed ABF (adaptive beamforming)or ASLB (adaptive sidelobe blanking) subarrayed adaptivedetection subarrayed parameter estimation for instanceadaptive monopulse and superresolution direction findingoptimization of subarray configuration and an expansionof the SASP that is application in MIMO- (multiple-inputmultiple-output-) PAR and so forth

Nickel is the pioneer and trailblazer in the field of theSASP Since the 1990s he has firstly systematically per-formed a series of thorough researches which theoreticallyand technically lay the solid foundation of the SASP Hisresearch achievements represent the development level ofSASP technology [1] Nickelrsquos reserches covered all kinds ofkey technologies in SASP including subarray weighting forPSL reduction [2] adaptive interference suppression (ABFand ASLB) [3ndash5] slow- and fast-time STAP (space-timeadaptive processing) [6ndash9] influence of channel errors [610] multiple BF (beam forming) with low sidelobes [11]adaptive detection [5 8] adaptive monopulse [9 12ndash18]superresolution [6 19] adaptive tracking [20ndash24] and designof optimum subarrays [2] The achievements have formed acomplete theoretical system and framework of the SASP

Similarly Farina with different coauthors has given cre-ative and pioneering contributions to the following aspectsincluding subarrayed weighting for SLC [25 26] subarrayedadaptation and superresolution [25ndash27] and subarray opti-mization [25 28] Lombardo and his co-authors contributedgreat achievements to subarrayed adaptive processing andpattern control [29 30] SASP for thinned arrays [31 32]while Massa et al contributed in the fields of subarrayedweighting synthesis of sum and difference patterns formonopulse [33ndash36] Liao et al in the field of subarrayed

Hindawi Publishing CorporationInternational Journal of Antennas and PropagationVolume 2015 Article ID 797352 21 pageshttpdxdoiorg1011552015797352

2 International Journal of Antennas and Propagation

ABF subarrayed STAP and subarray architectures [23 37 38]Wang et al in the field of subarrayed STAP and subarrayconfiguration [39ndash42] Klemm in the field of the design of thesubarray configuration for STAP [43] and so forth

SASP possesses a good application prospect for mil-limeter wave PAR seeker airborne multifunction radar andspace-borne early-warning radar A famous and outstandingexample is the tri-national X-band AMSAR project for afuture European airborne radar [7]

This paper summarizes the research achievements in thefield of SASP including the study results of the author Therelevant issues focus on subarrayed ABF subarrayed fast-time STAP subarrayed PSL reduction for sum and differencebeam PSL reduction for subarrayed beam scanning subar-rayed adapted monopulse subarrayed superresolution direc-tion finding subarray optimization for ECCM (electroniccounter-countermeasure) and the SASP for MIMO-PAR

In this review viewpoints and innovative ways are pre-sented which are relevant to the SASP For the proposedalgorithms we give examples

2 Subarrayed ABF

In this section we investigate subarrayed ABF and adaptiveinterference suppression Actually the adaptive weighting isimplemented at subarray levelThe subarrayed ABF has threeconfigurations that is DSW (direct subarray weighting) SLC(sidelobe canceller) and GSLC (generalized SLC) types [16]The three concepts are shown in Figure 1

The limitation of DSW configuration is that it requires anestimation of the interference-plus-noise covariance matrixThematrix is necessary for the estimation of adaptiveweightsThe limitation of the subarrayed SLC configuration is thatequipment with auxiliary antennas and auxiliary channelsis required as indicated in Figure 1(b) The advantage ofsubarrayed GSLC is less equipment complexity comparedwith subarrayed SLC because the auxiliary channels neednot single elements but may be generated from the arrayitself by subarrays as indicated in Figure 1(c) Furthermorecompared with the DSW the SLC and the GSLC do not needan estimation of interference plus noise covariance matrix(since the jammer signal can be subtracted from the mainchannel by the weighted sum of auxiliary channels outputsFigures 1(b) and 1(c))

This section is devoted to the DSW and GSLC configura-tions

21 DSW Type Subarrayed ABF211 Subarrayed Optimum ABF First consider the subar-rayed optimum BF (optimum adaptive filter [30]) in whichweight vector w is obtained by direct weighting of thesubarray outputs From the likelihood ratio test criterionthe probability of detection is maximized if the weightvector maximizes the SINR for a desired signal [1] As aconsequencew = 120583

Rminus1a(1205790

1205930

) whereR is an estimate of theinterference-plus-noise covariance matrix at subarray levela(1205790

1205930

) denotes the steering vector at look direction and120583 is a nonzero constant The maximum likelihood estimate

of covariance matrix is RSMI that is taking the average overthe subarray output data [1] This method is also called thesubarrayed SMI (sample matrix inverse)

212 ImprovedVersions of Subarrayed SMI Theshortcomingof the subarrayed SMI is that the sidelobe level of adaptedpattern is increased remarkably compared with quiescentpattern with tapering weighting due to subarrayed weighting(assuming the subarray transformation matrix T is notnormalized) especially in the case of the absence of thejamming

Suppose that we have an array with 119873 elements dividedinto 119871 subarrays T can be written by

T = ΦWT0

(1)

with Φ = diag[120593119899

(1205790

1205930

)]119899=0119873minus1

hereinto 120593119899

(1205790

1205930

) rep-resents the effect of the phase shifting of the 119899th elementW =

diag(119908119899

)119899=0119873minus1

where 119908119899

is the weight of the 119899th elementused to suppress sidelobe level of sumpatternT

0

is the119873times119871-dimensional subarray forming matrix in all the elements ofthe 119897th (119897 = 0 119871 minus 1) column only if the element belongsto 119897th subarray the element value is 1 otherwise it is 0

On the other hand the subarrayed SMI has only asymp-totically a good performance In practice we have to drawon a variety of techniques to provide robust performancein the case of small sample size The improved versionsof subarrayed SMI comprise the subarrayed LSMI (loadsample matrix inversion) the subarrayed LMI (lean matrixinversion) and so forth The subarrayed LSMI and thesubarrayed LMI are the extensions of LSMI [44] and LMI[45] respectively

213 Quiescent Pattern Control Approaches Quiescent pat-tern control approaches are another kind of the improvedversions of subarrayed optimum ABF The approaches pre-serve the desired quiescent pattern in the absence of jammerTherefore they make system automatically converge to thenonadapting mode Thus the conversion problem betweenthe two working modes of the subarrayed ABF is resolved[30] Furthermore quiescent pattern control approachesimprove the PSL remarkably compared with subarrayedoptimum ABF in the presence of jammers [46]

The pattern control approaches include normalizationmethod MOD (mismatched optimum detector) and SSP(subspace projection) Therein the starting point of thenormalization method [2 25 29 30] is the amplitude tapercombined with the different number of elements for eachsubarray which implies the receiver noise level is differentat the subarray output The subarrayed ABF tries to find aweight giving equal noise power in all subarrays and cancelsthe low sidelobe taper that is the adapted weight tends tobe the vector with uniform tapering Then we can adoptnormalization method to overcome this effect that is tonormalize the outputs of subarrays before adaptation so thatthe noise powers at the subarray outputs are equal

The MOD [29] introduces a mismatched steering vectorso that the adaptive beamformer detects such a mismatchedtarget and yields quiescent pattern in the absence of jammers

International Journal of Antennas and Propagation 3

Subarr-aying

ADC

Adaptive weighting

Σ Σ Σ

Σ

Σ

wopt1 wopt2 wopt3 woptL

Rminus1sub

middot middot middot


(a) DSW

ADC Adaptive weighting

Σ Σ Σ Σ

Σ


minus

(b) Subarrayed SLC

Subarraying

ADC

Main channel

Auxiliarychannel

Adaptive weighting

Σ


Blocking matrix M

wAdap

wQuie

yMain (t)

yAuxi (t)+

+minus

y(t)

(c) Subarrayed GSLC

Figure 1 Configurations of subarrayed ABF

The mismatched steering vector is written by a1015840(1205790

1205930

) =

RSMI(0)q the superscript (0) indicates the absence of jammer

Assume that q denotes the quiescent control vector namelythe 119871times 1 column vector in which all the elements are equally1 As a consequence in the case of absence of jammer theadaptive weights are w(0) = 120583q Thereby the quiescent pat-tern control is achieved Compared with the normalizationmethod the MOD does not need preprocessing and a recov-ery operator therefore the computational cost is reducedremarkably

The SSP [29] consists of the following steps (1) obtainingthe first subspace and getting the quiescent patterns by usingq to combine subarray outputs (2) finding the subspacewhich distorts the quiescent pattern and discarding it and(3) making the optimum interference suppression in theremaining 119871 minus 2 dimensional space which does not impactthe quiescent pattern

Table 1 presents the contrast of the performance for thequiescent pattern control approaches obviously in which theMOD is suitable for the overlapped subarrays and has smallcomputational burden compared with other models

214 The Approach Combining the Subarrayed OptimumABF with the Quiescent Pattern Control [46] Followingwe present an improvement of quiescent pattern controlapproaches For the pattern control the PSL reduction capa-bility is at the cost of degradation of adaptationThis results inan output SINR loss On the other hand for the subarrayedoptimum ABF the anti-jamming capability is optimum Inorder to compromise PSL and adaptation capability we putforward the methods combining subarrayed optimum ABFand quiescent pattern control The adaptive weight is

w(com) = 119870OptwOpt + 119870QPCwQPC (2)

where wOpt and wQPC denote the weights of the subar-rayed optimumABF and quiescent pattern control approachrespectively And119870Opt + 119870QPC = 10

The combined method unifies subarrayed optimum ABFand quiescent pattern control both of which are two extremecases of the combined method The combined methodimproves the flexibility of the PSL reduction for adaptivepattern and makes the trade-off between PSL and SINRaccording to real requirements When we want to obtain


Table 1 Performance contrast of quiescent pattern control approaches

Suitability for overlapped subarrays Computational burdenMOD Suitable Very smallNormalization method Nonsuitable LargeSSP Suitable Very heavy

minus80

minus60

minus40

minus20

0

Gai

n (d

B)

Jammerdirection

Subarrayed GSLCDSW

minus40 minus20 0 20 40

120579 (∘)

(a) Amplitude errors

minus40 minus20 0 20 40minus80

minus70

minus60

minus50

minus40

minus30

minus20

minus10

0

Gai

n (d

B)

JammerdirectionSubarrayed GSLC

DSW

120579 (∘)

(b) Phase errors

Figure 2 Adaptive patterns with IQ errors

better PSL reduction capability we can choose higher value of119870QPC otherwise we can set a higher value of 119870Opt to reduceSINR loss Through choosing appropriate parameters com-bined method can reduce PSL effectively and the achievedSINR is closely approximate to the SINR of subarrayedoptimum ABF as well

The above-mentioned quiescent pattern control methodand the combining method can be extended into subarrayedfast-time STAP as will be shown in Section 3

22 GSLC-Based Subarrayed ABF In the subarrayed GSLCthe blocking matrix in auxiliary channel (Figure 1(c)) is usedfor removing the desired signal component from this channelin which simplest form is only difference operation of adja-cent subarray outputs [47] However its blocking capabilityis poor Consequently residual signal component would becancelled which leads to the degradation of output SINRFor this reason we introduce the Householder transform todetermine the blocking matrix [48]

Let

e = [1 0 0]

119879

119871times1

p =

a (1205790

1205930

) minus

1003817100381710038171003817

a (1205790

1205930

)

10038171003817100381710038172

e1003817100381710038171003817

a (1205790

1205930

) minus

1003817100381710038171003817

a (1205790

1205930

)

10038171003817100381710038172

e10038171003817100381710038172

(3)

where sdot 2

is 2-norm Then the Householder matrix is

H = I minus 2pp119879 (4)

of which the first column is a(1205790

1205930

) and the remaining 119871 minus 1

ones are orthogonal to a(1205790

1205930

) respectively AccordinglyMin Figure 1(c) is

M = H[

[

[

[

[

[

[

0 sdot sdot sdot 0 0


d



]

]

]

]

]

]

]119871times(119871minus1)

(5)

The adaptive patterns and output SINR obtained byHouseholder transform-based GSLC are the same as that ofthe optimum ABF-based DSW [48] On the other hand therobustness of the subarrayed GSLC is much better than DSWWe assume a ULA (uniform linear array) comprising 92 ele-ments is partitioned into 10 nonoverlapped subarrays Thenminus40 dB Taylor weighting is applied The jammer direction 120579

119895

is 12∘ JNR (jammer-to-noise ratio) is 35 dB signal power119901119904

=

1 and the noise at element is AGWN (additional Gaussianwhite noise) Following we consider the errors at subarraylevel produced by the receivers which create the I- and Q-channels Assume the amplitude errors obey the distributionof 119873(0 01) with the ideal error-free amplitude gain of theI- and Q-channel being 1 and a normalized parameter being01 while phase errors obey the distribution of 119873(0 1) withthe unit of degree Figure 2 shows the adaptive patterns of thesubarrayed GSLC with Householder transform It is obviousthat all respects such as the look direction the beam shape


0 20 40 60 80minus80

minus70

minus60

minus50

minus40

minus30

minus20

minus10

0

Gai

n (d

B)

Direction ofbroadband jammer

SABF

120579 (∘)

Broadband steering vector method

Figure 3 Adapted pattern obtained with subarrayed broadbandABF

the direction and depth of jammer notch and the PSL aremuch better than those of optimum ABF-based DSW

The subarrayed GSLC with Householder transform isvalid no matter it is with amplitude errors or phase errorswhich is insensitive to phase errors even

23 The Subarrayed Broadband ABF For the broadbandarray additional time delays at subarray outputs are adoptedin order to compensate phase difference of internal subarrayof each frequency component The look direction is con-trolled by subarray time-delay and phase shifter together forwhole working bandwidth

The adaptive weights of subarrayed broadband ABF canbe obtained through broadband steering vector methodThismethod controls the look direction for each frequency ofworking bandwidth and realizes gain condition constraintsthat is makes the gain for each frequency to be equal Assumebandwidth of all jammers is 119861 and the power spectrumdensity obeys the uniform distribution in [119891

0

minus1198612 1198910

+1198612]then the subarrayed steering vector is

a (120579 120593) = int

1198910+1198612

1198910minus1198612

T119863

119867aele (119891 120579 120593) d119891 (6)

where T119863

represents subarray transformation matrixwith subarray time-delay effect aele(119891 120579 120593) =

[1 119890

minus1198952120587119891120591119899(120579120593)

119890

minus1198952120587119891120591119873minus1(120579120593)

]

119879 hereinto 120591119899

(120579 120593)

is the time difference relative to reference elementAssume that a ULA consisting of 99 elements

is partitioned into 25 nonuniform subarrays therelative bandwidth is 119861119891

0

= 10 element spacing is1205820

2 (1205820

is wavelength at 1198910

) And minus40 dB Taylor weightingis applied Let 120579

0

= minus60∘ and 120579119895

= minus20∘ Figure 3 illustrates aresulting adapted pattern obtained with broadband steeringvector method-based subarrayed broadband ABF [49] Itis drawn that the look direction and mainlobe shape of theadapted pattern is maintained whereas the subarrayed ABF(SABF for short) has an obvious mainlobe broadening


minus80

minus60

minus40

minus20

0

Gai

n (d

B)

Direction ofjammer

SABF

Elevation (∘)

Conventional SFT STAP

Figure 4 Adaptive patterns in the case of broadband jammer (cutpatterns in azimuth plane)

3 Subarrayed Fast-Time STAP

In the recent twenty years extensive and thorough researchon slow-time STAP used for clutter mitigation in airborneradar has been carried out [40 43] This paper only focuseson the fast-time STAP and this section is concerned with thesubarrayed fast-time STAP

In the presence of broadband jammers subarrayed ABFwould form broad adaptive nulls which consumes spatialdegrees of freedom and distorts the beam shape as indicatedin Figure 3 For this reason we should adopt subarrayed fast-time STAP [6]

The subarrayed ABF methods presented in Section 2can be applied analogously in the space-time domain Thesubarrayed fast-time STAP includes three kinds of structuresnamelyDSW SLC andGSLC type For example for theDSWtype subarrayed fast-time STAP delay-weighting networkis used at each subarray output (the number of delays isequal for each subarray) The delays aim to provide phasecompensation for each frequency component in bandwidthMeanwhile adaptive weights are applied at all delay outputsof each subarray to achieve broadband interference suppres-sion

To give an example suppose a UPA (uniform planararray) with 32 times 34 elements and in both 119909 and 119910 directionsthe element spacing is 120582

0

2 A minus40 dB Taylor weighting isapplied in both 119909 and 119910 directions Array is partitioned into6 times 6 nonuniform subarrays each of which is a rectangleplanar array Suppose 119861119891

0

= 10 for the jammer We setthe number of delay 119870 = 4 Figure 4 plots patterns in thepresence of SLJ (sidelobe jamming) where SFT STAP is anabbreviation of subarrayed fast-time STAP By comparing twoapproaches we see a narrow null of subarrayed fast-timeSTAP while subarrayed ABF results in rather wide null

On the other hand for broad array similar to subarrayedbroadband ABF the subarrayed fast-time STAP should adopttime delays at subarray outputs to compensate phase dif-ference This is subarrayed fast-time STAP with broadband


minus80 minus60 minus40 minus20 0minus80

minus70

minus60

minus50

minus40

minus30

minus20

minus10

0

Gai

n (d

B)


120579 (∘)

Broadband steering vector methodConventional SFT STAP

Figure 5 Adaptive pattern by broadband steering vector-basedsubarrayed fast-time STAP

steering vector method [50] Here broadband steering vec-tor method modifies constraint condition of conventionalLCMV criterion so that the gain of the pattern in desiredsignal direction is a constant within the whole receivingbandwidth

We give the simulation results in which the parametersof array are set the same as Figure 3 The central frequencyof the bandwidth of array is the same as that of the band-width of jammer Adaptive pattern obtained with broadbandsteering vector method-based subarrayed fast-time STAP isillustrated in Figure 5 [50] It is seen that the look directionand mainlobe shape are preserved which is impossible forthe conventional subarrayed fast-time STAP (conventionalSFT STAP for short)

The optimum adaptive weights of subarrayed fast-timeSTAP can be determined by the LCMV criterion [51] How-ever the PSL is very high which is similar to optimum sub-arrayed ABF Therefore we adopt the approach combiningthe optimum subarrayed fast-time STAP and subarrayed fast-time STAP with the quiescent pattern control capability inorder tomake a trade-off between the PSL and the broadbandjammer suppression capability

The adaptive pattern in the case of broadband SLJ isgiven in Figure 6 [50] meanwhile 119870Opt is chosen to be06 for the combined subarrayed fast-time STAP It is seenthat the PSL of combined subarrayed fast-time STAP isimproved remarkably namely by 132 dB in comparison withthe optimum one

4 The Subarray Weighting-Based SidelobeReduction of Sum and Difference Beam

Thesidelobe reduction of patterns is the basic task for the PARsystems In this section we deal with subarray weighting-based algorithms for the PSL reduction of sum and differencebeam The goal is to obtain a trade-off between hardwarecomplexity and achievable sidelobe level

minus40 minus30 minus20 minus10 0 10 20 30 40minus90

minus80

minus70

minus60

minus50

minus40

minus30

minus20

minus10

0

Elevation (deg)

Gai

n (d

B)

SLJ

Conventional SFT STAPCombined SFT STAP

Figure 6 Comparison of combined subarrayed fast-time STAPwiththe optimum one

One function of subarray weighting is to reduce the PSLof both the sum and the difference beam simultaneously for amonopulse PAR in which digital subarray weighting substi-tutes the analog element amplitude weighting Accordinglyhardware cost and complexity are greatly reduced

In this section two techniques are presented as follows

41 Full Digital Weighting Scheme This approach completelysubstitutes the element weighting adopting subarray weight-ing (for instance a Taylor and Bayliss weighting for sumand difference beam resp) We assume that no amplitudeweighting at element level is possible The two schemes fordetermining subarray weights are as follows

411 The Analytical Approach The analytical approachincludes twoways namely weight approximation and patternapproximation The former makes the subarray weight toapproximate the Taylor or Bayliss weight in LMS (least meansquare) sense while the latter makes the patterns obtainedwith the subarray weights approximate to that obtained bythe Taylor or Bayliss weight in LMS sense [2]

The weight approximation-based subarray weights aredetermined by

Sum beam

wsum WA(opt)

= arg minwsum WA

10038171003817100381710038171003817

Twsum WA minus wTay

10038171003817100381710038171003817

2

2

(7)

Difference beam

wΔ WA(opt)

= arg minwΔ WA

10038171003817100381710038171003817

TwΔ WA minus wBay

10038171003817100381710038171003817

2

2

a119867 (1205790

1205930

) (TwΔ WA(opt)

) = 0(8)

where wTay wBay denote Taylor weigh vector Bayliss weighvector respectively


minus50 minus40 minus30 minus20 minus10 0 10 20 30 40 50 60minus50

minus40

minus30

minus20

minus10

0Po

wer

(dB)

Subarray weightingWithout weighting

Angle (deg)

(a) Sum beam



minus40

minus30

minus20

minus10

0

Pow

er (d

B)

Angle (deg)

(b) Difference beam

Figure 7 Subarrayed weighting-based quiescent patterns

Take the sum beam for example the pattern approxima-tion-based subarray weights are determined by

wsum PA(opt)

= arg minwsum PAint

1205872

minus1205872

int

1205872

minus1205872

10038161003816100381610038161003816

w119867sum PAT

119867a (120579 120593) minus w119867Taya (120579 120593)10038161003816100381610038161003816

2

times 119875 (120579 120593) d120579 d120593(9)

where 119875(120579 120593) is the directional weighted function It couldadjust the approximation of pattern obtained by the patternapproximation method and the pattern obtained by theTaylorweights in a certain direction areawhich leads to bettersidelobe reduction capability

Assume a ULA with 30 omnidirectional elements with1205822 of element spacing The look direction is 10∘ and thearray is divided into 11 nonuniform subarrays Figure 7 showspatterns obtained by pattern approximation method whichare used for approximating the patterns obtained by minus30 dBTaylor and minus30 dB Bayliss weighting respectively The PSL isminus2518 dB for the sum beam (Figure 7(a)) which is improvedby 1195 dB compared with that without weighting while PSLis minus2022 dB for the difference beam (Figure 7(b)) which isimproved by 978 dB compared with that without weightingIt is seen that the PSL reduction effect of difference beam isinferior to the one of the sumbeam for the subarrayweightingmethod since the constraint for the null restricts a furtherimprovement of PSL

Note the subarray weighting-based PSL reduction capa-bility cannot be the same as that with element weightingwhich is a price required for reducing the hardware costThisapproach is more suitable for the difference beam which isused for target tracking

412 GA (Genetic Algorithm) This approach optimizes thesubarray weights using GA The GA could achieve global

optimum solution which is unconstrained by issues relatedsolution

Here fitness function has a great influence to PSL reduc-tion capability We choose two kinds of fitness functions(1) the weight approximation-based fitness function that is119891WA = Twsub minus wele

2

2

where wsub and wele are the subarrayweights and element weights respectively (2) the PA-basedfitness function namely making PSL of pattern obtainedby subarray weighting as low as possible which turns tobe more reasonable Meanwhile main beam broadeningshould be limited in a certain range Furthermore it shouldform a notch with enough null depth in a certain directionfor the difference beam For example for difference beamthe fitness function is designed as 119891patt = 119896SLL(Δ SLL)

2

+

119896BW(Δ BW)

2

+ 119896ND(ΔND)2 where Δ PSL Δ BW and ΔND are

difference between expected value and practice value of PSLbeamwidth and null depth respectively 119896SLL 119896BW and 119896NDare used for adjusting the weighing coefficients of Δ SLL Δ BWand ΔND respectively

Adopting 119891weight as fitness function brings high operat-ing efficiency however the PSL reduction performance isunsatisfactory because only the weight vector is optimizedWhile adopting 119891patt as the fitness function can improve theperformance because the PSL itself is optimized in this caseHowever its drawbacks are that the genetic operation is easilyconstrained on local optimal solution and the computingefficiency is poor

In order to overcome the limitation of two approachesmentioned above we present an improved GA that ispartition genetic process into two stages whose process is asbelowThe first stage is used for the preliminary optimizationusing 119891weight as fitness function Then we turn into stage2 which is used for improving PSL reduction capabilityfurther namely according to descending order of fitnessvalues of the first stage select a certain scale of excellentfilial generations among optimizing results to compose new


Table 2 The PLSs of sum beam with several approaches

Without weighing Conventional GA(with fitness function 119891WA)

Conventional GA(with fitness function 119891patt)

Improved GA

PSL (dB) minus1343 minus2618 minus2770 minus2777


minus40

minus30

minus20

minus10

0

Pow

er (d

B)

Without weightingImproved GA

v

Figure 8 Sum patterns improved GA

population Afterwards continue to optimize by using119891patt asfitness function

There is a UPA containing 30 times 30 omnidirectionalelements with the element spacing 119889 = 1205822 in the directionof 119909 and 119910 The beam direction is (0∘ 0∘)The array is dividedinto 10 times 10 nonuniform subarrays The improved GA-basedsubarray weights are used to approximate minus30 dB Taylorweights in which number of generations is 1300 in first stageand 20 in second one respectively Figure 8 reports the sumpattern it is seen that obtained PSL is close to the expectedminus30 dB

Table 2 shows the PSLs with several methods in whichPSL reduction capability with 119891patt fitness function is betterthan that with 119891WA (it is improved by 158 dB) for conven-tional GA And PSL reduction performance of improvedGA quite is close to the one of conventional GA with 119891patt(meanwhile improved GA enhances calculation efficiencygreatly) which is degraded only by 223 dB compared withexpected Taylor weight

42 Combining of Element and Subarray Weighting Thescheme generates sum and difference channels simultane-ously only using an analog weighting In order to deter-mine the analog weights it is assumed that the supposedinterferences locate within the sidelobe area of sum anddifference beams [25] Then the supposed interferences areadaptively suppressed (eg based on LCMV criterion) Andthe obtained adaptive weight is regarded as the desired analogweight Furthermore the subarray weighting is adopted thatis analog weight is combined with the digital weights inorder to improve the PSL reduction capability The subarrayweighting has two forms which are used for sum and dif-ference beam respectively Then the subarray weights could

1st

quadrant

3rd

quadrant

2nd

quadrant

4th

quadrant

Figure 9 The division of planar array into four quadrants

be determined by the weight approximation or the patternapproximation approach as mentioned in Section 41

The literature [25] presented above-mentioned approachfor the ULA Next we focus on planar array [52] Consider aUPA with 119873

119909

times 119873119910

= 119873 elements First the array is dividedinto four equal quadrants as shown in Figure 9 And then wenumber the elements as 1 le 119899 le 1198734 in the first quadrant1198734 + 1 le 119899 le 1198732 for the elements in the second quadrant1198732 + 1 le 119899 le 31198734 for the elements in the third quadrantand 31198734+1 le 119899 le 119873 for the elements in the fourth quadrantThe sum and difference channels are shown in Figure 10

Assume coordinate of 119899th (119899 = 1 119873) elementis (119909119899

119910119899

) The analog weight is wele = 120583Rminus1a(1205790

1205930

)where 120583 is a nonzero constant a(120579

0

1205930

) is the steeringvector into the look direction R = Rxsum + J

Δ

RxΔJΔ withJΔ

= diag( q p sdot sdot sdot q p⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

1198731199102

p q sdot sdot sdot p q⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

1198731199102

) and q is 119873119909

2-

dimensional unit row vector while p is 119873119910

2-dimensionalrow vector in which every element is minus1 Assuming theinterferences are with uniform distribution let Rxsum =

(119903119894119896 sum

)119894119896=12119873

then we have

119903119894119896 sum

=

120587 (1 minus 119877sum

2

) 120590

2

119868 sum

+ 120590

2

119899

119894 = 119896

2120587119890

119895120587[(119909119894minus119909119896)1199060+(119910119894minus119910119896)V0](1198970minus119877sum 119897sum )1205902

119868 sum 119894 = 119896

(10)

with 1198970

= 1198691

(119888119894119896

)119888119894119896

119897sum

= 1198691

(119888119894119896

119877sum

)119888119894119896

where 1198691

(sdot)

is first order Bessel function of the first kind and 119888119894119896

=

2120587radic(119909119894

minus 119909119896

)

2

+ (119910119894

minus 119910119896

)

2

120582 119877sum

is mainlobe radius of sumbeam 1205902

119868 sum

is power of interference imposing on sum beam120590

2

119899

is thermal noise power Similarly we can obtain 119903119894119896 Δ

withRxΔ = (119903

119894119896 Δ

)119894119896=12119873

It should be pointed out that the PSL reduction effect with

only analog weighting approach is determined by the selecteddesign parameters such as JNR and spatial distribution ofsupposed interferences


1

Elements of 1st quadrant

Elements of 2nd quadrant

Elements of 3rd quadrant

Elements of 4th quadrant

⨁

N4 N4 + 1 N2 N2 + 1 3N4 3N4 + 1 N

w1 wN4 wN4+1 wN2 wN2+1 w3N4 w3N4+1 wN

+

+ + + +

+ + + + + +

+

minus minus

rarrotimes rarrotimes rarrotimes rarrotimes rarrotimes rarrotimes rarrotimes rarrotimes

yΣ yΔ

y

middot middot middotmiddot middot middotmiddot middot middotmiddot middot middot

Figure 10 Construction of sum and difference channels

0

minus05

0

05

minus30

minus20

minus10

0

Pow

er (d

B)

u

minus05

05

(a) Sum beam

minus30

minus20

minus10

0

Pow

er (d

B)

0u minus05

0

05

minus05

05

(b) Double difference beam

Figure 11 Patterns obtained by only analog weighting

minus03minus02

minus010

0102

03 minus03

minus02

minus01

0

01

0203

minus30

minus20

minus10

0

Pow

er (d

B)

u

(a) Sum beam

minus03minus02

minus010

0102

03 minus03

minus02

minus010

0102

03

minus30

minus20

minus10

0

Pow

er (d

B)

u


Figure 12 Patterns obtained by analog and digital weighting (with weight approximation approach)


Assume that a UPA consists of 30 times 30 elements on arectangular grid at half wavelength spacing configured into10 times 10 uniform subarrays The look direction is (0∘ 0∘)For sum and double difference beam (double difference beamis useful for main beam ECCM application for instancethe mainlobe canceller) the JNRs of supposed interferencesare both of 30 dB Figure 11 shows patterns generated byonly analog weighting The PSLs of the sum and differencepattern are minus1451 dB and minus1077 dB respectively Figure 12shows the results of adopting both element weighting andweight approximation-based subarray weighting in whichthe PSLs of sum and double difference patterns are improvedby 834 dB and 527 dB respectively compared with Figure 11

In fact the PSL reduction capabilities of weight approxi-mation- and pattern approximation-based subarray weight-ing are quite close

5 The Sidelobe Reduction forSubarrayed Beam Scanning

The look direction is usually controlled by phase shiftersfor PAR However further subarrayed digital beam scanningmay be required for forming multiple beams and many otherapplications which are used for a limited sector of lookdirections around the presteered direction

Typically with subarrayed scanning the PSL will increaserapidly with the scanning direction departing from theoriginal look direction The larger the scan angle is thegreater the PSL increase is [11]Therefore the requirement forsidelobe reduction arises

In this section we discuss the PSL reduction approachesfor subarrayed beam scanning namely beam clusters createdat subarray level

With subarrayed scanning the resulting pattern is com-posed of the pattern of each subarray Each subarray patternhas a high PSL which contributes to a high PSL of the arraypattern If pattern of each subarray would be superimposedproperly in the main beam and has a low PSL the PSL of thearray pattern could be reduced effectively

Therefore one can post-process the subarray outputsusing a weighting network consequently creating new sub-arrays with the patterns with similar shape within the mainbeam and with PSLs as low as possible The starting point istomake the new subarray patterns to approximate the desiredone

The natural form of desired subarray pattern is theISP (ideal subarray pattern) which is constant within themainlobe and zero else [11] It has two kinds of forms (1)projection of mainlobe in array plane is a rectangular areaand (2) projection of mainlobe is a circular area

As an example assume that a UPA consists of 32 times

34 omnidirectional elements on a rectangular grid at halfwavelength spacing A minus40 dB Taylor weighting is appliedin both 119909 and 119910 directions The array is divided into 6 times 6

nonuniform subarrays and each of which is a rectangulararray Assume (119906

0

V0

) = (minus05 0)By using the ISP based on rectangular projection we

give an example for reducing the sidelobes in Figure 13(a)

Obviously the shape and beamwidth of the main beam ofall subarrays are very similar but the gain of the differentsubarrays makes a great difference in the results Figure 13(b)shows patterns obtained by ISP based circular projectionObviously the shape the beamwidth and the gains of themain beam of all subarrays are very similar which is superiorto rectangular projection method

Figure 14 shows array patterns after scanning carriedout in 119906 direction Subplots (a) (b) and (c) show thepatterns obtained without the weighting network processingwith ISP based rectangular projection and ISP based oncircular projection respectively The different curves aregiven for nonscanning and scan angles of 05 Bw and 10 Bwrespectively where Bw is the 3 dB beam width Comparedto the case without-weighting network for ISP based onrectangular projection the PSL is reduced by 134 dB for05 Bw scanning and 172 dB for the 1 Bw while for ISP basedcircular projection the PSL is reduced by 199 dB and 239 dBrespectively

It is seen from above-mentioned examples that the PSLreduction capability with ISP is not satisfied For the reasonwe adopt GSP- (Gaussian subarray pattern-) based approach[53] Therein desired pattern is the Gaussian pattern whichis more smooth than the ISP [11]

The result obtained with the GSP approach is shownin Figure 15 Comparing Figure 15 with Figure 14(b) we seethat the PSL by GSP is improved obviously compared withISP based rectangular projection namely PSL is reduced by149 dB for the 05 Bw scanning and 205 dB for the 1 Bwrespectively Consequently the GSP improves remarkablyISP as PSL of the latter is only reduced by more than1 dB compared with non-weighting network meanwhile thecomputational burden is equivalent to the latter [53]

6 The Subarrayed Adaptive Monopulse

In order to ensure the accuracy of angle estimation ofmonopulse PAR in jammer environments we should adoptsubarrayed adaptive monopulse technique The subarrayedABF can improve SINR and detection capability Howeverunder circumstances of MLJ (mainlobe jamming) the adap-tive null will distort the main beam of pattern greatly thusleading to serious deviation of the angle estimation Therebythe subarrayed adaptive monopulse should be applied

Existing approaches usually take the adaptive patternsinto account to avoid the degradation of monopulse perfor-mance near the null For example one can use LMS-basedtarget direction search to determine adaptively the weights ofoptimum difference beam [54] This kind of methods needsto adopt the outputs of the sum and difference beam toachieve the corrected monopulse ratio and it needs highercomputational cost Subarrayed linearly constrained adap-tive monopulse is generalized from the linearly constrainedadaptive monopulse [55] But it reduces degree of freedom byadopting adaptive difference beam constraints

The reined subarrayed adaptivemonopulsemethods withhigher monopulse property have been suggested for exam-ple the approximativemaximum-likelihood angle estimators



minus50

minus40

minus30

minus20

minus10

0

Pow

er (d

B)


minus50

minus40

minus30

minus20

minus10

0Po

wer

(dB)

u v

Azimuth cut Elevation cut

(a) Rectangular projection method


minus50

minus40

minus30

minus20

minus10

0

Pow

er (d

B)


minus50

minus40

minus30

minus20

minus10

0

Pow

er (d

B)

u


(b) Circular projection method

Figure 13 Subarray patterns obtained by ISP

[12 13 15 16] proposed by Nickel In this section we focus ontwo-stage subarrayed adaptive monopulse Here we improvethe conventional two-stage subarrayed adaptive monopulsewhich is an extension of the two-stage adaptive monopulsetechnique [56]

The realization process of the two-stage subarrayed adap-tive monopulse is as follows firstly the subarrayed ABF isused to suppress the SLJ while maintaining the mainlobeshape secondly theMLJ is suppressed whilemaintaining themonopulse performance that is the jammers are canceledwith nulls along one direction (elevation or azimuth) andundistorted monopulse ratio along the orthogonal direction(azimuth or elevation) is preservedThe two-stage subarrayedadaptive monopulse requires four channels in which thedelta-delta channel is used as an auxiliary channel for themainlobe cancellation

However the method possesses two limitations (1) afterthe first-stage processing the PSLs of patterns increasegreatly and (2) the monopulse performance is undesirablefor there exists serious distortion in deviation of the lookdirection for adaptive monopulse ratio the reason is that theMLM(mainlobemaintaining) effect is not idealTherefore inthe first-stage processing subarrayed optimum ABF is sub-stituted by MOD consequently the monopulse performancewould be improved greatly [57]

Suppose a rectangular UPAwith 56times42 elements and theelement spacing is 1205822 A minus40 dB Taylor weighting is appliedin 119909 and a minus30 dB in the 119910 direction Array is partitionedinto 6 times 6 nonuniform subarrays and each subarray is arectangular array (120579

0

1205930

) = (0

∘

0

∘

) Assuming that MLJ andSLJ are located at (1∘ 2∘) and (10

∘

15

∘

) respectively then allthe JNRS are 30 dB



minus40

minus30

minus20

minus10

0Po

wer

(dB)

Without scanningScanning in 05BWScanning in 10BW

u

(a) Non-weighing network


minus40

minus30

minus20

minus10

0

Pow

er (d

B)


u

(b) Rectangular projection method


minus40

minus30

minus20

minus10

0

Pow

er (d

B)


u

(c) Circular projection method

Figure 14 Array patterns with beam scanning obtained by ISP method (azimuth cut)

u


minus40

minus30

minus20

minus10

0

Pow

er (d

B)


Figure 15 Array patterns with beam scanning obtained by GSP (azimuth cut)


minus15 minus10 minus5 0 5 10 15 20

minus80

minus70

minus60

minus50

minus40

minus30

minus20

minus10

0

Pow

er (d

B)

SLJ

Quiescent patternConventional TSSAMImproved TSSAM

120579 (∘)

Figure 16 Adaptive patterns of elevation sum beam obtained byfirst-stage adaptive processing

Next we illustrate the simulation results taking theelevation direction as an example Figure 16 is the cut ofthe adaptive patterns of elevation sum beam applying first-stage processing It is obvious that the conventional two-stage subarrayed adaptive monopulse (TSSAM for short)namely the subarrayed optimum ABF combined with MLMobviously increases the PSL compared with the quiescentpattern and the mainlobe width shows apparent reductionwhich leads to the distortion of adaptive monopulse ratioaround the look direction The PSL is improved greatlyby using the MOD and more important the remarkableimprovement of the MLM effect makes the mainlobe shapeclose to that of the quiescent pattern therefore a greaterimprovement of adaptive monopulse ratio can be obtained

Figure 17 presents an elevation monopulse ratio obtainedby the two-stage subarrayed adaptive monopulse Let 119906 =

cos 120579 sin120593 and V = sin 120579 where 120593 and 120579 are the azimuth andelevation angle correspondingly The monopulse ratio alongelevation direction is 119870

119864

(119906 V) = 119865Δ119864

(119906 V)119865sum

(119906 V) where119865sum

(119906 V) and 119865Δ119864

(119906 V) denote the pattern of sum beam andelevation difference beam respectivelyThe two-dimensionalantenna patterns are separable therefore 119870

119864

(119906 V) = 119865sumA

(119906)

119865Δ119864

(V)[119865sumA

(119906)119865sum

119864

(V)] = 119865Δ119864

(V)119865sum

119864

(V) [56]Since the subarrayed optimum ABF combined with

MLM is used in the first-stage processing the monopulseratio deviated from the look direction shows serious dis-tortion Undoubtedly the MOD combined with MLM is anappropriate choice to enhance adaptive monopulse ratio forits similarity to quiescent monopulse ratio and the smalldistortion when deviated from the look direction And therelative error of adaptive monopulse ratio with improvedtwo-stage monopulse (MOD+MLM for first-stage process-ing) is only 349 compared with quiescent monopulse ratiowhen the elevation is minus27∘ while the conventional two-stage monopulse (subarrayed optimum ABF+MLM for thefirst-stage processing) is as much as 6970 So improved

minus3 minus2 minus1 0 1 2 3

minus2

minus1

0

1

2

Mon

opul

se ra

tio

Quiescent monopulse ratioConventional TSSAM

Subarrayed optimum ABFImproved TSSAM

120579 (∘)

Figure 17 Elevation monopulse ratio obtained by two-stage adap-tive processing

method monopulse reduces error of monopulse ratio greatlycompared with conventional method The numerical resultsdemonstrate that in the range of 3 dB bandwidth of patternsfor the improved method monopulse the relative error ofadaptive monopulse ratio is less 1 and even reaches 001compared with quiescent monopulse ratio

Two-stage subarrayed adaptivemonopulse integrates sev-eral techniques including subarrayed ABF MLM quiescentpattern control and four-channel MJC The SLJ and MLJ aresuppressed respectively so it is unnecessary to design thesophisticated monopulse techniques to match the mainlobeof patterns

For the wideband MLJs the SLC type STAP can beused to form both sum and difference beam If auxiliaryarray is separated from the main array by distances thatare sufficiently large the array can place narrow nulls onthe MLJ while maintaining peak gain on a closely spacedtarget [58] Consequently it can suppress MLJs effectivelywhile preserving superior monopulse capability

7 The Subarrayed Superresolution

To address angle superresolution issues a variety of methodshave been developed This paper only focuses on subar-rayed superresolution In this section two types of theapproaches are described the first refers to the narrowbandsuperresolution the second approach presents the broadbandsuperresolution [6]

71 The Subarrayed Superresolution The subarrayed super-resolution direction finding could be achieved by extendingthe conventional superresolution algorithms into the subar-ray level However this kind of algorithms has to calibratethe whole array manifold But for the active calibrationtechnique the realization is very complicated and costlywhile for the self-calibrate solutions there are limitations forexample difficulty of implementation high computationalburden (some algorithms are many times more than original


0 05 1 150

02

04

06

08

1 P

roba

bilit

y of

reso

lutio

n

SCSSTSISSMSWAVES

Azimuth difference of two targets (∘)

(a) Probability of resolution versus azimuth difference of two targets

Pro

babi

lity

of re

solu

tion

SCSSTSISSMSWAVES

1

08

06

04

02

0minus20 minus10 0 10 20

SNR (dB)

(b) Probability of resolution versus SNRs

SCSST azimuthSCSST elevation

SWAVES azimuthSWAVES elevation

02

018

016

014

012

01

008

006

004

002

00 5 10 15 20minus20 minus15 minus10 minus5

SNR (dB)

RSM

E(∘)

(c) The std of direction finding

Figure 18 Contrast of performances of three broadband subarrayed superresolution algorithms

superresolution algorithms even) the convergence problemsfor some iterative algorithms the performances are limitedin the case of a lot of array position errors some are onlyeffective for specific errors of one or two kinds only [59]

Thus simplified arraymanifolds can be usedThe essenceof these methods is to treat the whole array as a super-array and each subarray as a superelement The idea isto approximate the patterns at subarray outputs by thoseof superelements located at the subarray centres [19] Thesimplified array manifolds are only determined by phasecenters and gain of subarrays [19] Because the calibrationneeds to be implemented only inside the subarrays thecalibration cost and complexity are reduced greatly

These methods can eliminate uncertain information insidelobe area The available region of direction finding ofthem is within 3 dB beamwidth of patterns around the

center of the look direction However combined with beamscanning by element phase shifting superresolution canbe achieved in any possible direction that is make thesuperresolution be carried out repeatedly by changing lookdirection The advantage is suppressing sidelobe sourcesin complicated situation such as multipathway reflectionconsequently number of sources anddimension of parameterestimation are reduced and the process of superresolution issimplified [19] Meanwhile we can use beam scanning to findthe area with no signal source and wipe it off in advancetherefore 2-D peak searching cost is reduced greatly

The available direction finding area of direct simplifiedarray manifold-based subarrayed superresolution is fixedwhich cannot be adjusted and the sidelobe sources cannotbe suppressed completely Changing subarray patterns canovercome the limitations but it is unable to be achieved by


restructuring the subarray structure The subarray structureis an optimized result based on various factors and hardwareis fixed Therefore we post-process the subarray outputs byintroducing a weighting network which constructs requirednew subarray patterns This kind of methods improved theflexibility of array processing greatly and the optimization ofsubarrays for different purpose and distinct processing taskscan be achieved based the same hardware [19]

Compared with the method based direct simplified arraymanifold the ISP and GSP methods can adjust the availableregion of direction estimation and suppress sidelobe sourcesbetter [19 60 61] On the other hand compared with theISP and GSPmethods the approximate ISP and approximateGSPmethods reduce the computational cost greatly while theprecision of direction finding and resolution probability arevery close to ISP and GSP [62 63] respectively

72 The Broadband Subarrayed Superresolution The subar-rayed superresolution approaches presented in the Section 71are based on the scenario of narrowband signals Next letus consider the broadband subarrayed superresolutionwhichincludes three kinds of approaches namely subband pro-cessing space-time processing and subarray time-delayingprocessing [6]

The subarrayed ISSM (incoherent signal subspace me-thod) subarrayed CSST (coherent signal subspace trans-form) and subarrayed WAVES (weighted average of signalsubspace) translate broadband problems into a subband pro-cessing All these algorithms need to make delay-weighingfor outputs of subbands The subarrayed ISSM only averagesthe direction finding results of each subband obtained bysubarrayed MUSIC The subarrayed CSST and subarrayedWAVES should choose the suitable focusing transformationmatrix and therefore need to preestimate source direction Ifthe preestimated direction is erroneous the direction findingperformancewill be degradedThe subarrayedWAVES deter-mines a joint space from all subspaces of subband covariancematrix which is only one representative optimum signalsubspace [6] The mentioned-above subband processingmethods can be combined with simplified array manifoldsConsequently the broadband subarrayed superresolutioncan be achieved

The broadband superresolution based on subarray timedelay does not need any subband processing Accordingly ithas less calculationwhich is easy to be implemented but needsa focusing transform For the simplified arraymanifold-basedapproaches they have better performance if the preestimatedtarget direction is near the look direction otherwise theperformance would be degraded rapidly

The narrowband subarrayed superresolution can beextended to the space-time superresolution (such as thesubarrayed space-time MUSIC) Through compensating thephase using a subarray delay network this kind of algorithmsreduces the frequency-angle ambiguitiesThen use the space-time steering vector to calculate the space-time covariancematrix and extract the dominant signal subspace [6] Thespace-time broadband subarrayed superresolution does notneed any focusing transform and direction preestimation butthe computation is costly

To give an example suppose a UPA with 19 times 43 = 817

elements and the element spacing is 1205820

2 A minus40 dB Taylorweighting is applied in 119909 and a minus30 dB in the 119910 directionArray is partitioned into 7 times 11 = 77 nonuniform subarraysand each subarray is a rectangular array Assume lookdirection is (5∘ 45∘) The sources are incoherent broadbandwith 119861 = 02119891

0

and with equal power Suppose there aretwo targets in both the elevation is 430∘ and the azimuthdifference varies from 50∘ to 0∘ and SNR (signal-to-noiseratio) is 0 dB

Two targets are deemed to be separable if their peakvalues of spatial spectrum both are greater than the value inthe central direction of targets Namely targets are resolvableif the following two conditions are both met 119875(120579

1

) gt 119875((1205791

+

1205792

)2) and 119875(1205792

) gt 119875((1205791

+ 1205792

)2) where 1205791

and 1205792

are direc-tions of spectrum peaks of the targets respectively And theprobability of resolution is the probability that the two targetsare successfully resolved

Figure 18(a) plots the probability of resolution curvesobtained by several algorithms [64] It can be seen that theangle resolution capability of subarrayed WAVES (SWAVESfor short) is the best while subarrayed ISSM (SISSM forshort) is the worst Assuming two targets located at (40∘450∘) and (5∘ 430∘) Figure 18(b) shows the curves of theprobability of resolution versus SNRs From the figure itcan be found that probability of resolution of subarrayedWAVES is the best followed by subarrayed CSST (SCSST forshort) and subarrayed ISSM is the worst Figure 18(c) givesthe curves of RMSE of direction estimation versus SNRs inwhich RMSEs are the average value of the estimations of thetwo targets [64]

8 Subarray ConfigurationOptimization for ECCM

For the PAR equippedwith ABF the optimization of subarrayconfiguration is a key problem and challenge in the field of theSASP

For its obvious influence on the performance of PARthe optimization of subarray configuration can bring muchimprovement of the system performances such as sidelobelevel detection capability accuracy of angle estimationECCM capability (anti-MLJ) and so forth The subarraydivision is a system design problem the optimized resulthas to take into account various factors However differentcapabilities may be contradictory mutually

There are different solutions to this contradiction Inthis section we briefly overview the MOEA (multiobjectiveevolutionary algorithm) procedure working on this issueMOEA might make an optimal trade-off between above-mentioned capabilities

The objective functions in the MOEA could be themean 119875

119889

for different positions of the target the meanCRB (Cramer-Rao Bound) of the target azimuthelevationestimation the mean PSLs of sum adapted pattern inthe azimuthelevation plane and so forth [28] But theset of optimum solutions is obtained after the subarrayoptimization through MOEA Therefore we need to deter-mine posteriorly an optimized array through optimizing the


specific capability or introducing constraints (such as thegeometric structures or the realization cost and complexity ofarray)

On the other hand several issues should be consideredon engineering realization for example all the subarraysare nonoverlapped and array is the fully filled subarrayconfiguration is relatively regular and all elements insidea subarray are relatively concentrative (subarrayrsquos shapesshould not be disjointed) Therefore multiple constraintsshould be set during the process of genetic optimization

The process of optimizing subarray is quite complicatedfor the great amount of elements in a PAR and the largesearching spaceThuswe can adopt the improvedGA tomakeoptimizing with a characteristic of adaptive crossover com-bination which is based on the adaptive crossover operatorthis will result in remarkable enhancement on convergencespeed of optimization and calculation efficiency [65]

Suppose a UPA with 30 times 32 = 960 omnidirectionalelements and element spacing is 1205822 A minus40 dB Taylorweighting is applied in 119909 and in the 119910 directionThe subarraynumber is 64 (120579

0

1205930

) = (0

∘

0

∘

) The original subarraystructure is selected randomly Suppose that the direction ofthe MLJ is (1∘ 1∘) and JNR = 35 dB

We optimize the five objective functions simultaneouslybased on the MOEA Hereinto the first objective func-tion is the PSL of adapted sum pattern in 119906 = 0 cutFigure 19(a) shows the adapted sum pattern obtained bythe optimized subarray configuration and the original onerespectively Here the original and optimized PSL are minus1236and minus2237 dB respectively namely the PSL is improvedby 1001 dB The second objective function is the PSL ofadapted sum pattern in V = 0 cut Figure 19(b) shows theadapted sum pattern cut obtained by the optimized subarrayconfiguration and the original one respectively in whichthe original and optimized PSL are minus893 dB and minus1598 dBrespectively therefore the PSL is improved by 705 dB Notehere the PSL is the third high sidelobe in the patterns becausethe second one is caused by adaptive null

On the other hand for the PAR with subarrayed ABF theideal optimum subarray configuration should keep systemrsquosperformances be optimal for various interference environ-ments interference environments But this is impossible torealize in principle

9 The Subarrayed Processing for MIMO-PAR

Historically the SASP techniques aremainly used for the PARsystems In this section we discuss the concept of the SASPsuitable forMIMO-PAR system Creating this technique is animpetus following from previous sections

The MIMO structure is impracticable when the arrayis composed of hundreds or thousands of elements dueto the huge quantity of independent transmitting signalsand transmitting and receiving channels Hardware cost andalgorithmic complexity will exceed the acceptable level

Therefore we present the subarrayed MIMO-PAR It isan extension of the subarrayed PAR The MIMO-PAR is thecombination of MIMO radar and PAR the array is dividedinto several subarrays inside each subarray a coherent signal

is transmitted working as PAR mode orthogonal signalsare transmitted between subarrays to form a MIMO systemThe MIMO-PAR presented in this paper adopts TxRx arraymodules and transmitting and receiving arrays have the samesubarray configuration while both at the transmitting andreceiving ends the SASP techniques are be applied

91 The Characteristics of MIMO-PAR The features of MI-MO-PAR are listed as follows

(1) It maintains advantages of MIMO radar as well asthe characteristics of PAR (such as the coherentprocessing gain)

(2) Compared with MIMO radar the MIMO-PARreduces the cost and complexity of the hardware(the number of the transmitting and receivingchannels) and the computational burden greatly Forexample the dimension of optimization algorithmsfor designing transmitting signals is reduced tosubarray number while it is the same as the elementnumber in MIMO Typically element number has anorder of magnitude from several hundreds to severalthousands and subarray number is only a few dozensof magnitude

(3) Hardware cost and algorithm complexity can beflexibly controlled by adjusting the number of thesubarrays The ability of coherent processing (SNR)and the characteristics of waveform diversity (angleresolution) can be compromised by adjusting thesubarray structure and subarray number

(4) ComparedwithMIMOradar the transmitting signalsof different subarrays can be combined with indepen-dent beam look directions to improve the capabilityof beamcontrol producingmore flexible transmittingbeam patterns thus improving the flexibility of objecttracking

(5) The characteristics with further advantages overMIMO radar such as lower PSL of total pattern canbe achieved by design of the weighting of subarrayedtransmittingreceiving BF Moreover MIMO-PARcan enhance the anti-jamming capability and gener-ate higher output SINRunder strong interference andrealize lowPSLwithout expense ofmainlobe gain loss

(6) The performance of the system can be enhancedby optimizing the structure of subarray such astransmitting diversity SNR PSL of total pattern theperformance of transmitting beam and the ECCMcapability

(7) The exiting research achievements related to receivingSASP for PAR can be generalized to the MIMO-PARThis is a promising direction of further studies

(8) The configuration of MIMO-PAR with TxRx mod-ules can be easily made compatible with existingsubarrayed PAR

92 The Research Topics of SASP for MIMO-PAR Theresearch topics are as follows



minus50

minus40

minus30

minus20

minus10

0Po

wer

(dB)

Original patternOptimized pattern

v

(a) First objective function sum adapted pattern in V = 0 cut



minus40

minus30

minus20

minus10

0

Pow

er (d

B)

u

(b) Second objective function sum adapted pattern in 119906 = 0 cut

Figure 19 Optimized results based on MOEA

0

minus20

minus40

minus60

minus80minus1 minus05 0 05 1

(dB)

MIMO-PARPARMIMO

(a) 119906 direction

(dB)

0

minus10

minus20

minus30

minus40

minus50

minus60

minus70

minus80minus04 minus02 0 02 04 06

MIMO-PARPARMIMO

(b) v direction

Figure 20 Total patterns of PAR MIMO and MIMO-PAR

921 Subarrayed TransmittingReceiving BF

(1) Subarrayed Transmitting BF The use of robust subarrayedtransmitting BF can minimize the transmitted power Thestarting point is minimizing the norm of the transmitting BFweight vector while restricting the upper bound of sidelobes[66]

(2) Subarrayed Receiving BF The subarrayed ABF for PARis described in Section 2 can be employed for MIMO-PARFurthermore the subarrayed LSMI and subarrayed CAPS(constrained adaptive pattern synthesis) and so forth [1] canalso be applied to MIMO-PAR

(3) The Characteristic of the TransmittingWaveform Diver-sityReceiving Total Pattern The virtual array steering vectorof the MIMO-PAR is decided by the coherent processinggain vector the waveform diversity vector and the steeringvector of the receiving array And the total patterns are theproduction of transmitting pattern wave diversity patternsand receiving patterns

Assume a UPA of TxRx community with 960 omnidi-rectional elements on a rectangular grid at half wavelengthspacing The array is divided into 8 times 8 = 64 uniformsubarrays each of which is a rectangular array Figure 20illustrates the total pattern for the three radar models [67]Table 3 shows the results of the comparison It is seen that


12

10

8

6

4

2

01

105050

0minus05minus05

minus1 minus1

times109 times109

u

105

0minus05

minus1u

105

0minus05

minus1 u

105

0minus05

minus1 u

1

05

0

minus05minus1

1

050

minus05minus1

1

05

0minus05

minus1

2

15

1

05

0

2

15

1

05

0

3

25

2

15

1

times107times108

(a) 120588 = 0001 (b) 120588 = 0125

(c) 120588 = 0325 (d) 120588 = 055

Figure 21 Transmit beam patterns based on synthesis of subarray transmitted signal

Table 3 PSLs of total pattern for different radar models (dB)

Radar model PAR MIMO MIMO-PARPSLs in u direction minus1500 minus2466 minus2656PSLs in v direction minus1516 minus2526 minus2695

MIMO-PAR is superior to the PAR and the MIMO in termsof the PSL

922 Synthesis of Transmitting Beam Pattern The rectan-gle transmitting beam pattern can be used to radiate themaximum transmitting energy to interesting areas so asto improve the exploring ability As a result a rectangletransmitting beam pattern can be synthesized from cross-correlation matrix of transmit signals [68]

The transmitting beam control includes two scenarios (1)Each subarray has different beam directions to form a broadtransmitting beam (for radar searchmode) (2) Each subarrayhas the same beam direction to focus on transmitting beam(for tracking mode)

Aiming at a UPA Figure 21 shows the transmitting beampatterns with different signal correlation coefficient 120588 [69]We observe that the beam width can be adjusted

The performance evaluation of transmitted signalincludes (1) orthogonality and (2) range resolution andmultiple-target resolution (pulse compressing performance)On the other hand the orthogonality degrades with increaseof the subarray number

923 Optimization of the Subarray Configuration In theMI-MO-PAR when the element number and subarray numberare fixed the subarray structure has a significant impact onthe system performance Then the subarray structure has tobe optimized While in the MOEA method the followingconstraints can be chosen as objective functions (1) thewaveform diversity capability (2) the coherent processinggain (3) the PSL of the total pattern in elevation direction (4)the PSL of the total pattern in azimuth direction and (5) theRMSE of the transmitting beam pattern and the rectangularpattern

10 Conclusions and Remarks

In this paper we describe some aspects of the SASP Fromthese investigations we draw the concluding remarks asfollows


(1) For the SASP the achievable capabilities in applicationare constrained by some hardware factors for exam-ple channel errors

(2) The amalgamation as well as integration of multipleSASP techniques is a trend such as the combina-tion of subarrayed ABF adaptive monopulse andsuperresolution Consequently the performance ofthe SASP could be improved

(3) The subarray optimization is still a complicated andhard problem compared with the algorithms

(4) The extension of the SASP into MIMO-PAR couldpromote and deepen development of the SASP

Furthermore we point out the problems to be dealt withThe challenging works are in the following areas

(1) The more thorough study of subarray optimizationshould be carried out It is important for improvingcapabilities of system (including ECCM)

(2) The SASP for anti-MLJs is still a hard research topic

(3) The SASP techniques for the thinned arrays should befurther developed

(4) At present the research focuses mainly on the planararrays which are only suitable for small angle of view(for instance plusmn45∘) The SASP should be extended tothe conformal arrays (eg seekers)

List of Acronyms

ABF Adaptive beamformingADC Analogue-to-digital conversionASLB Adaptive sidelobe blankingBF Beam formingCAPS Constrained adaptive pattern synthesisCSST Coherent signal subspace transformCRB Cramer-Rao BoundDSW Direct subarray weightingECCM Electronic counter-countermeasureGA Genetic algorithmGSLC Generalized sidelobe cancellerGSP Gaussian subarray patternISP Ideal subarray patternISSM Incoherent signal subspace methodJNR Jammer-to-noise ratioLCMV Linearly constrained minimum varianceLMI Lean matrix inversionLMS Least mean squareLSMI Load sample matrix inversionMFPAR Multifunction phased array radarMIMO Multiple-input multiple-outputMLJ Mainlobe jammingMLM Mainlobe maintainingMOD Mismatched optimum detectorMOEA Multiobjective evolutionary algorithmPAR Phased array radarPSL Peak sidelobe level

SASP Subarrayed array signal processingSINR Signal-to-interference-plus-noise ratioSLC Sidelobe cancellerSLJ Sidelobe jammingSMI Sample matrix inverseSNR Signal-to-noise ratioSSP Subspace projectionSTAP Space-time adaptive processingULA Uniform linear arrayUPA Uniform planar arrayWAVES Weighted average of signal subspace

Conflict of Interests

The author declares that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The author would like to thank Dr Ulrich Nickel of Fraun-hofer FKIE Germany for his insightful comments and usefuldiscussions This work has been partially supported by ldquotheFundamental Research Funds for the Central Universitiesrdquo(Grant no HITNSRIF201152) the ASFC (Aeronautical Sci-ence Foundation of China no 20132077016) and the SASTFoundation (no SAST201339)

References

[1] U Nickel ldquoArray processing for radar achievements andchallengesrdquo International Journal of Antennas and Propagationvol 2013 Article ID 261230 21 pages 2013

[2] U R O Nickel ldquoSubarray configurations for digital beamform-ing with low sidelobes and adaptive interference suppressionrdquoin Proceedings of the IEEE International Radar Conference pp714ndash719 IEEE Alexandria VA USA May 1995

[3] U Nickel ldquoAdaptive beamforming for phased array radarsrdquo inProceedings of the International Radar Symposium pp 897ndash906Munich Germany September 1998

[4] UNickel ldquoSubarray configurations for interference suppressionwith phased array radarrdquo in Proceedings of the InternationalConference on Radar pp 82ndash86 Paris France 1989

[5] U Nickel ldquoDetection with adaptive arrays with irregular digitalsubarraysrdquo in Proceedings of the IEEE Radar Conference pp635ndash640 Waltham Mass USA April 2007

[6] U Nickel ldquoSuper-resolution and jammer suppression withbroadband arrays for multi-function radarrdquo in Applications ofSpace-Time Adaptive Processing R Klemm Ed chapter 16 pp543ndash599 IEE 2004

[7] U Nickel P G Richardson J C Medley and E Briemle ldquoArraysignal processing using digital subarraysrdquo in Proceedings of theIET Conference on Radar Systems Edinburgh UK October2007

[8] W Burger and U Nickel ldquoSpace-time adaptive detection forairborne multifunction radarrdquo in Proceedings of the IEEE RadarConference (RADAR rsquo08) pp 1ndash5 Rome Italy May 2008

[9] R Klemm and U Nickel ldquoAdaptive monopulse with STAPrdquo inProceedings of the CIE International Conference on Radar (CIEICR rsquo06) Shanghai China October 2006


[10] U Nickel ldquoOn the influence of channel errors on array signalprocessing methodsrdquo International Journal of Electronics andCommunication vol 47 no 4 pp 209ndash219 1993

[11] U R O Nickel ldquoProperties of digital beamforming withsubarraysrdquo in Proceedings of the International Conference onRadar (ICR rsquo06) Shanghai China October 2006

[12] U Nickel ldquoMonopulse estimation with adaptive arraysrdquo IEEProceedings Part F Radar and Signal Processing vol 140 no5 pp 303ndash308 1993

[13] U Nickel ldquoMonopulse estimation with sub-array adaptivearrays and arbitrary sum- and difference beamsrdquo IEE Pro-ceedingsmdashRadar Sonar and Navigation vol 143 no 4 pp 232ndash238 1996

[14] U Nickel ldquoPerformance of corrected adaptive monopulseestimationrdquo IEE Proceedings Radar Sonar and Navigation vol146 no 1 pp 17ndash24 1999

[15] U Nickel ldquoPerformance analysis of space-time-adaptivemonopulserdquo Signal Processing vol 84 Special section on newtrends and findings in antenna array processing for radar no9 pp 1561ndash1579 2004

[16] U Nickel ldquoOverview of generalized monopulse estimationrdquoIEEE Aerospace and Electronic Systems Magazine vol 21 no 6pp 27ndash55 2006

[17] U R O Nickel E Chaumette and P Larzabal ldquoStatisticalperformance prediction of generalized monopulse estimationrdquoIEEE Transactions on Aerospace and Electronic Systems vol 47no 1 pp 381ndash404 2011

[18] E Chaumette U Nickel and P Larzabal ldquoDetection andparameter estimation of extended targets using the generalizedmonopulse estimatorrdquo IEEE Transactions on Aerospace andElectronic Systems vol 48 no 4 pp 3389ndash3417 2012

[19] U Nickel ldquoSpotlight MUSIC super-resolution with subarrayswith low calibration effortrdquo IEE Proceedings Radar Sonar andNavigation vol 149 no 4 pp 166ndash173 2002

[20] R Klemm W Koch and U Nickel ldquoGround target trackingwith adaptive monopulse radarrdquo in Proceedings of the EuropeanMicrowave Association vol 3 pp 47ndash56 2007

[21] W R Blanding W Koch and U Nickel ldquoAdaptive phased-array tracking in ECM using negative informationrdquo IEEETransactions on Aerospace and Electronic Systems vol 45 no1 pp 152ndash166 2009

[22] M Feldmann and U Nickel ldquoTarget parameter estimationand tracking with adaptive beamformingrdquo in Proceedings ofthe International Radar Symposium (IRS rsquo11) pp 585ndash590September 2011

[23] G S Liao Z Bao and Y H Zhang ldquoOn clutter DOF for phasedarray airborne early warning radarsrdquo Journal of Electronics vol10 no 4 pp 307ndash314 1993

[24] W Blanding W Koch and U Nickel ldquoOn adaptive phased-array tracking in the presence of main-lobe jammer suppres-sionrdquo in Signal and Data Processing of Small Targets vol 6236of Proceedings of SPIE Orlando Fla USA April 2006

[25] A Farina G Golino S Immediata L Ortenzi and L Tim-moneri ldquoTechniques to design sub-arrays for radar antennasrdquoin Proceedings of the 12th International Conference on Antennasand Propagation (ICAP rsquo03) vol 1 pp 17ndash23 2003

[26] A Farina ldquoElectronic counter-countermeasuresrdquo in RadarHandbook M I Skolnik Ed chapter 24 McGraw-Hill 3rdedition 2008

[27] A Farina G Golino and L Timmoneri ldquoMaximum likelihoodapproach to the estimate of target angular co-ordinates under

a main beam interference conditionrdquo in Proceedings of theInternational Conference on Radar pp 834ndash838 Beijing China2001

[28] G Golino ldquoImproved genetic algorithm for the design ofthe optimal antenna division in sub-arrays a multi-objectivegenetic algorithmrdquo in Proceedings of the IEEE InternationalRadar Conference pp 629ndash634 Washington DC USA 2005

[29] P Lombardo and D Pastina ldquoPattern control for adaptiveantenna processing with overlapped sub-arraysrdquo in Proceedingsof the International Conference on Radar pp 188ndash193 AdelaideAustralia September 2003

[30] P Lombardo and D Pastina ldquoQuiescent pattern control inadaptive antenna processing at sub-array levelrdquo in Proceedingsof the IEEE International Symposium on Phased Array Systemsand Technology pp 176ndash181 Boston Mass USA October 2003

[31] P Lombardo R Cardinali M Bucciarelli D Pastina andA Farina ldquoPlanar thinned arrays optimization and subarraybased adaptive processingrdquo International Journal of Antennasand Propagation vol 2013 Article ID 206173 13 pages 2013

[32] P Lombardo R Cardinali D Pastina M Bucciarelli and AFarina ldquoArray optimization and adaptive processing for sub-array based thinned arraysrdquo in Proceedings of the InternationalConference on Radar (Radar rsquo08) pp 197ndash202 Adelaide Aus-tralia September 2008

[33] A Massa M Pastorino and A Randazzo ldquoOptimization of thedirectivity of a monopulse antenna with a subarray weightingby a hybrid differential evolution methodrdquo IEEE Antennas andWireless Propagation Letters vol 5 no 1 pp 155ndash158 2006

[34] S Caorsi AMassaM Pastorino and A Randazzo ldquoOptimiza-tion of the difference patterns for monopulse antennas by ahybrid realinteger-coded differential evolution methodrdquo IEEETransactions on Antennas and Propagation vol 53 no 1 pp372ndash376 2005

[35] L Manica P Rocca and A Massa ldquoDesign of subarrayedlinear and planar array antennas with SLL control based on anexcitation matching approachrdquo IEEE Transactions on Antennasand Propagation vol 57 no 6 pp 1684ndash1691 2009

[36] L Manica P Rocca and A Massa ldquoExcitation matchingprocedure for sub-arrayed monopulse arrays with maximumdirectivityrdquo IET Radar Sonar and Navigation vol 3 no 1 pp42ndash48 2009

[37] Y F Kong C Zeng G S Liao and H H Tao ldquoA newreduced-dimension GSC for target tracking and interferencesuppressionrdquo in Proceedings of the 6th International Conferenceon Radar (RADAR rsquo11) pp 785ndash788 Chengdu China October2011

[38] Z Y Xu Z Bao andG S Liao ldquoAmethod of designing irregularsubarray architectures for partially adaptive processingrdquo inProceedings of the CIE International Conference on Radar (ICRrsquo96) pp 461ndash464 Beijing China October 1996

[39] Y Wang K Duan and W Xie ldquoCross beam STAP for non-stationary clutter suppression in airborne radarrdquo InternationalJournal of Antennas and Propagation vol 2013 Article ID276310 5 pages 2013

[40] W-C Xie and Y-L Wang ldquoOptimization of subarray partitionbased on genetic algorithmrdquo in Proceedings of the 8th Inter-national Conference on Signal Processing (ICSP rsquo06) BeijingChina November 2006

[41] Z H Zhang J B Zhu and Y L Wang ldquoLocal degrees offreedom of clutter for airborne space-time adaptive processingradar with subarraysrdquo IET Radar Sonar and Navigation vol 6no 3 pp 130ndash136 2012


[42] L Y Dai R F Li and C Rao ldquoCombining sum-difference andauxiliary beams for adaptivemonopulse in jammingrdquo Journal ofSystems Engineering and Electronics vol 24 no 3 pp 372ndash3812013

[43] R Klemm Principles of Space-Time Adaptive Processing IEE3rd edition 2006

[44] B D Carlson ldquoCovariance matrix estimation errors and diago-nal loading in adaptive arraysrdquo IEEE Transactions on Aerospaceand Electronic Systems vol 24 no 4 pp 397ndash401 1988

[45] C H Gierull ldquoFast and effective method for low-rank inter-ference suppression in presence of channel errorsrdquo ElectronicsLetters vol 34 no 6 pp 518ndash520 1998

[46] H Hu and X H Deng ldquoAn effective ADBF method atsubarray level for plane phased arrayrdquo in Proceedings of the 8thInternational Conference on Signal Processing (ICSP rsquo06) vol 4Beijing China November 2006

[47] H Hu E Liu and Y Xiao ldquoADBF at subarray level using ageneralized sidelobe cancellerrdquo in Proceedings of the 3rd IEEEInternational Symposium on Microwave Antenna Propagationand EMC Technologies for Wireless Communications pp 697ndash700 Beijing China October 2009

[48] H Hu and E X Liu ldquoAn improved GSLC at subarray levelrdquoSubmitted

[49] H Hu and X Qu ldquoAn improved subarrayed broadband ABFrdquoSubmitted

[50] H Hu and X Qu ldquoAn improved subarrayed fast-time STAPalgorithmrdquo Submitted

[51] D Wang H Hu and X Qu ldquoSpace-time adaptive processingmethod at subarray level for broadband jammer suppressionrdquoin Proceedings of the 2nd IEEE International Conference onMicrowave Technology and Computational Electromagnetics(ICMTCE rsquo11) pp 281ndash284 Beijing China May 2011

[52] H Hu and W H Liu ldquoA novel analog and digital weightingapproach for sum and difference beam of planar array atsubarray levelrdquo Chinese Journal of Electronics vol 19 no 3 pp468ndash472 2010

[53] H Hu W-H Liu and Q Wu ldquoA kind of effective side lobesuppression approach for beam scanning at sub-array levelrdquoJournal of Electronics amp Information Technology vol 31 no 8pp 1867ndash1871 2009 (Chinese)

[54] A S Paine ldquoMinimum variance monopulse technique for anadaptive phased array radarrdquo IEE ProceedingsmdashRadar Sonarand Navigation vol 145 no 6 pp 374ndash380 1998

[55] R L Fante ldquoSynthesis of adaptive monopulse patternsrdquo IEEETransactions on Antennas and Propagation vol 47 no 5 pp773ndash774 1999

[56] K B Yu and D J Murrow ldquoAdaptive digital beamforming forangle estimation in jammingrdquo IEEE Transactions on Aerospaceand Electronic Systems vol 37 no 2 pp 508ndash523 2001

[57] H Hu and H Zhang ldquoAn improved two-stage processingapproach of adaptive monopulse at subarray levelrdquo Acta Elec-tronica Sinica vol 37 no 9 pp 1996ndash2003 2009 (Chinese)

[58] R L Fante R M Davis and T P Guella ldquoWideband cancel-lation of multiple mainbeam jammersrdquo IEEE Transactions onAntennas and Propagation vol 44 no 10 pp 1402ndash1413 1996

[59] CM See andA B Gershman ldquoDirection-of-arrival estimationin partly calibrated subarray-based sensor arraysrdquo IEEE Trans-actions on Signal Processing vol 52 no 2 pp 329ndash337 2004

[60] H Hu and XW Jing ldquo2-DWSFmethod at subarray level basedon ideal patternsrdquo in Proceedings of the International Conferenceon Radar pp 1161ndash1164 Beijing China 2006

[61] H Hu and X W Jing ldquo2-D spatial spectrum estimationmethods at subarray level for phased arrayrdquo Acta ElectronicaSinica vol 35 pp 415ndash419 2007 (Chinese)

[62] H Hu and X W Jing ldquoAn improved super-resolution directionfinding method at subarray level for coherent sourcesrdquo inProceedings of the CIE International Conference on Radar (ICRrsquo06) pp 525ndash528 Shanghai China October 2006

[63] H Hu and X-W Jing ldquoA new super-resolution method atsubarray level in planar phased array for coherent sourcerdquo ActaElectronica Sinica vol 36 no 6 pp 1052ndash1057 2008 (Chinese)

[64] H Hu X Liu and L J Liang ldquoStudy on the broadbandsuperresolution direction finding at subarray levelrdquo Submitted

[65] H Hu W C Qin and S Feng ldquoOptimizing the architectureof planar phased array by improved genetic algorithmrdquo inProceedings of the IEEE International Symposium onMicrowaveAntenna Propagation and EMC Technologies for WirelessCommunications (MAPE rsquo07) pp 676ndash679 Hangzhou ChinaAugust 2007

[66] A Hassanien and S A Vorobyov ldquoPhased-MIMO radar atradeoff between phased-array andMIMO radarsrdquo IEEE Trans-actions on Signal Processing vol 58 no 6 pp 3137ndash3151 2010

[67] X P Luan H Hu and X Hu ldquoAnalysis on pattern features ofMIMO-phased array hybrid radarrdquo Submitted

[68] D R Fuhrmann J P Browning and M Rangaswamy ldquoSignal-ing strategies for the hybrid MIMO phased-array radarrdquo IEEEJournal on Selected Topics in Signal Processing vol 4 no 1 pp66ndash78 2010

[69] X P Luan H Hu and N Xiao ldquoA synthesis algorithm oftransmitting beam pattern for MIMO-PARrdquo Submitted

International Journal of

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Active and Passive Electronic Components

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

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Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Chemical EngineeringInternational Journal of Antennas and

Propagation




Navigation and Observation



DistributedSensor Networks



ABF subarrayed STAP and subarray architectures [23 37 38]Wang et al in the field of subarrayed STAP and subarrayconfiguration [39ndash42] Klemm in the field of the design of thesubarray configuration for STAP [43] and so forth

SASP possesses a good application prospect for mil-limeter wave PAR seeker airborne multifunction radar andspace-borne early-warning radar A famous and outstandingexample is the tri-national X-band AMSAR project for afuture European airborne radar [7]

This paper summarizes the research achievements in thefield of SASP including the study results of the author Therelevant issues focus on subarrayed ABF subarrayed fast-time STAP subarrayed PSL reduction for sum and differencebeam PSL reduction for subarrayed beam scanning subar-rayed adapted monopulse subarrayed superresolution direc-tion finding subarray optimization for ECCM (electroniccounter-countermeasure) and the SASP for MIMO-PAR

In this review viewpoints and innovative ways are pre-sented which are relevant to the SASP For the proposedalgorithms we give examples

2 Subarrayed ABF

In this section we investigate subarrayed ABF and adaptiveinterference suppression Actually the adaptive weighting isimplemented at subarray levelThe subarrayed ABF has threeconfigurations that is DSW (direct subarray weighting) SLC(sidelobe canceller) and GSLC (generalized SLC) types [16]The three concepts are shown in Figure 1

The limitation of DSW configuration is that it requires anestimation of the interference-plus-noise covariance matrixThematrix is necessary for the estimation of adaptiveweightsThe limitation of the subarrayed SLC configuration is thatequipment with auxiliary antennas and auxiliary channelsis required as indicated in Figure 1(b) The advantage ofsubarrayed GSLC is less equipment complexity comparedwith subarrayed SLC because the auxiliary channels neednot single elements but may be generated from the arrayitself by subarrays as indicated in Figure 1(c) Furthermorecompared with the DSW the SLC and the GSLC do not needan estimation of interference plus noise covariance matrix(since the jammer signal can be subtracted from the mainchannel by the weighted sum of auxiliary channels outputsFigures 1(b) and 1(c))

This section is devoted to the DSW and GSLC configura-tions

21 DSW Type Subarrayed ABF211 Subarrayed Optimum ABF First consider the subar-rayed optimum BF (optimum adaptive filter [30]) in whichweight vector w is obtained by direct weighting of thesubarray outputs From the likelihood ratio test criterionthe probability of detection is maximized if the weightvector maximizes the SINR for a desired signal [1] As aconsequencew = 120583

Rminus1a(1205790

1205930

) whereR is an estimate of theinterference-plus-noise covariance matrix at subarray levela(1205790

1205930

) denotes the steering vector at look direction and120583 is a nonzero constant The maximum likelihood estimate

of covariance matrix is RSMI that is taking the average overthe subarray output data [1] This method is also called thesubarrayed SMI (sample matrix inverse)

212 ImprovedVersions of Subarrayed SMI Theshortcomingof the subarrayed SMI is that the sidelobe level of adaptedpattern is increased remarkably compared with quiescentpattern with tapering weighting due to subarrayed weighting(assuming the subarray transformation matrix T is notnormalized) especially in the case of the absence of thejamming

Suppose that we have an array with 119873 elements dividedinto 119871 subarrays T can be written by

T = ΦWT0

(1)

with Φ = diag[120593119899

(1205790

1205930

)]119899=0119873minus1

hereinto 120593119899

(1205790

1205930

) rep-resents the effect of the phase shifting of the 119899th elementW =

diag(119908119899

)119899=0119873minus1

where 119908119899

is the weight of the 119899th elementused to suppress sidelobe level of sumpatternT

0

is the119873times119871-dimensional subarray forming matrix in all the elements ofthe 119897th (119897 = 0 119871 minus 1) column only if the element belongsto 119897th subarray the element value is 1 otherwise it is 0

On the other hand the subarrayed SMI has only asymp-totically a good performance In practice we have to drawon a variety of techniques to provide robust performancein the case of small sample size The improved versionsof subarrayed SMI comprise the subarrayed LSMI (loadsample matrix inversion) the subarrayed LMI (lean matrixinversion) and so forth The subarrayed LSMI and thesubarrayed LMI are the extensions of LSMI [44] and LMI[45] respectively

213 Quiescent Pattern Control Approaches Quiescent pat-tern control approaches are another kind of the improvedversions of subarrayed optimum ABF The approaches pre-serve the desired quiescent pattern in the absence of jammerTherefore they make system automatically converge to thenonadapting mode Thus the conversion problem betweenthe two working modes of the subarrayed ABF is resolved[30] Furthermore quiescent pattern control approachesimprove the PSL remarkably compared with subarrayedoptimum ABF in the presence of jammers [46]

The pattern control approaches include normalizationmethod MOD (mismatched optimum detector) and SSP(subspace projection) Therein the starting point of thenormalization method [2 25 29 30] is the amplitude tapercombined with the different number of elements for eachsubarray which implies the receiver noise level is differentat the subarray output The subarrayed ABF tries to find aweight giving equal noise power in all subarrays and cancelsthe low sidelobe taper that is the adapted weight tends tobe the vector with uniform tapering Then we can adoptnormalization method to overcome this effect that is tonormalize the outputs of subarrays before adaptation so thatthe noise powers at the subarray outputs are equal

The MOD [29] introduces a mismatched steering vectorso that the adaptive beamformer detects such a mismatchedtarget and yields quiescent pattern in the absence of jammers


Subarr-aying

ADC

Adaptive weighting

Σ Σ Σ

Σ

Σ


Rminus1sub



(a) DSW


Σ Σ Σ Σ

Σ


minus

(b) Subarrayed SLC

Subarraying

ADC

Main channel

Auxiliarychannel

Adaptive weighting

Σ


Blocking matrix M

wAdap

wQuie

yMain (t)

yAuxi (t)+

+minus

y(t)

(c) Subarrayed GSLC



1205930

) =












minus80

minus60

minus40

minus20

0

Gai

n (d

B)

Jammerdirection

Subarrayed GSLCDSW


120579 (∘)



minus70

minus60

minus50

minus40

minus30

minus20

minus10

0

Gai

n (d

B)


DSW

120579 (∘)

(b) Phase errors





Let

e = [1 0 0]

119879

119871times1

p =

a (1205790

1205930

) minus

1003817100381710038171003817

a (1205790

1205930

)

10038171003817100381710038172

e1003817100381710038171003817

a (1205790

1205930

) minus

1003817100381710038171003817

a (1205790

1205930

)

10038171003817100381710038172

e10038171003817100381710038172

(3)

where sdot 2


H = I minus 2pp119879 (4)


1205930



1205930


M = H[

[

[

[

[

[

[



d



]

]

]

]

]

]

]119871times(119871minus1)

(5)


119895


=



0 20 40 60 80minus80

minus70

minus60

minus50

minus40

minus30

minus20

minus10

0

Gai

n (d

B)


SABF

120579 (∘)







0

minus1198612 1198910


a (120579 120593) = int

1198910+1198612

1198910minus1198612

T119863

119867aele (119891 120579 120593) d119891 (6)

where T119863


[1 119890

minus1198952120587119891120591119899(120579120593)

119890

minus1198952120587119891120591119873minus1(120579120593)

]

119879 hereinto 120591119899

(120579 120593)



0


2 (1205820



0

= minus60∘ and 120579119895



minus80

minus60

minus40

minus20

0

Gai

n (d

B)

Direction ofjammer

SABF

Elevation (∘)








0


0





minus70

minus60

minus50

minus40

minus30

minus20

minus10

0

Gai

n (d

B)


120579 (∘)










minus80

minus70

minus60

minus50

minus40

minus30

minus20

minus10

0

Elevation (deg)

Gai

n (d

B)

SLJ








Sum beam

wsum WA(opt)

= arg minwsum WA

10038171003817100381710038171003817

Twsum WA minus wTay

10038171003817100381710038171003817

2

2

(7)

Difference beam

wΔ WA(opt)

= arg minwΔ WA

10038171003817100381710038171003817

TwΔ WA minus wBay

10038171003817100381710038171003817

2

2

a119867 (1205790

1205930

) (TwΔ WA(opt)

) = 0(8)




minus40

minus30

minus20

minus10

0Po

wer

(dB)


Angle (deg)

(a) Sum beam



minus40

minus30

minus20

minus10

0

Pow

er (d

B)

Angle (deg)

(b) Difference beam



wsum PA(opt)

= arg minwsum PAint

1205872

minus1205872

int

1205872

minus1205872

10038161003816100381610038161003816

w119867sum PAT

119867a (120579 120593) minus w119867Taya (120579 120593)10038161003816100381610038161003816

2

times 119875 (120579 120593) d120579 d120593(9)







2

2


2

+

119896BW(Δ BW)

2









Improved GA



minus40

minus30

minus20

minus10

0

Pow

er (d

B)


v






1st

quadrant

3rd

quadrant

2nd

quadrant

4th

quadrant




119909

times 119873119910



119910119899


1205930


0

1205930


Δ

RxΔJΔ withJΔ


1198731199102


1198731199102

) and q is 119873119909

2-



(119903119894119896 sum

)119894119896=12119873

then we have

119903119894119896 sum

=

120587 (1 minus 119877sum

2

) 120590

2

119868 sum

+ 120590

2

119899

119894 = 119896

2120587119890


119868 sum 119894 = 119896

(10)

with 1198970

= 1198691

(119888119894119896

)119888119894119896

119897sum

= 1198691

(119888119894119896

119877sum

)119888119894119896

where 1198691

(sdot)


=

2120587radic(119909119894

minus 119909119896

)

2

+ (119910119894

minus 119910119896

)

2

120582 119877sum


119868 sum


2

119899


withRxΔ = (119903

119894119896 Δ

)119894119896=12119873




1





⨁

N4 N4 + 1 N2 N2 + 1 3N4 3N4 + 1 N


+

+ + + +

+ + + + + +

+

minus minus


yΣ yΔ

y



0

minus05

0

05

minus30

minus20

minus10

0

Pow

er (d

B)

u

minus05

05

(a) Sum beam

minus30

minus20

minus10

0

Pow

er (d

B)

0u minus05

0

05

minus05

05



minus03minus02

minus010

0102

03 minus03

minus02

minus01

0

01

0203

minus30

minus20

minus10

0

Pow

er (d

B)

u

(a) Sum beam

minus03minus02

minus010

0102

03 minus03

minus02

minus010

0102

03

minus30

minus20

minus10

0

Pow

er (d

B)

u
















0

V0













minus50

minus40

minus30

minus20

minus10

0

Pow

er (d

B)


minus50

minus40

minus30

minus20

minus10

0Po

wer

(dB)

u v




minus50

minus40

minus30

minus20

minus10

0

Pow

er (d

B)


minus50

minus40

minus30

minus20

minus10

0

Pow

er (d

B)

u








0

1205930

) = (0

∘

0

∘


∘

15

∘




minus40

minus30

minus20

minus10

0Po

wer

(dB)


u



minus40

minus30

minus20

minus10

0

Pow

er (d

B)


u



minus40

minus30

minus20

minus10

0

Pow

er (d

B)


u



u


minus40

minus30

minus20

minus10

0

Pow

er (d

B)





minus80

minus70

minus60

minus50

minus40

minus30

minus20

minus10

0

Pow

er (d

B)

SLJ


120579 (∘)





119864

(119906 V) = 119865Δ119864

(119906 V)119865sum

(119906 V) where119865sum

(119906 V) and 119865Δ119864


119864

(119906 V) = 119865sumA

(119906)

119865Δ119864

(V)[119865sumA

(119906)119865sum

119864

(V)] = 119865Δ119864

(V)119865sum

119864




minus2

minus1

0

1

2

Mon

opul

se ra

tio



120579 (∘)









0 05 1 150

02

04

06

08

1 P

roba

bilit

y of

reso

lutio

n

SCSSTSISSMSWAVES



Pro

babi

lity

of re

solu

tion

SCSSTSISSMSWAVES

1

08

06

04

02


SNR (dB)




02

018

016

014

012

01

008

006

004

002


SNR (dB)

RSM

E(∘)


















0



1

) gt 119875((1205791

+

1205792

)2) and 119875(1205792

) gt 119875((1205791

+ 1205792

)2) where 1205791

and 1205792








119889







0

1205930

) = (0

∘

0

∘





















minus50

minus40

minus30

minus20

minus10

0Po

wer

(dB)


v




minus40

minus30

minus20

minus10

0

Pow

er (d

B)

u



0

minus20

minus40

minus60


(dB)

MIMO-PARPARMIMO


(dB)

0

minus10

minus20

minus30

minus40

minus50

minus60

minus70


MIMO-PARPARMIMO

(b) v direction








12

10

8

6

4

2

01

105050

0minus05minus05

minus1 minus1

times109 times109

u

105

0minus05

minus1u

105

0minus05

minus1 u

105

0minus05

minus1 u

1

05

0

minus05minus1

1

050

minus05minus1

1

05

0minus05

minus1

2

15

1

05

0

2

15

1

05

0

3

25

2

15

1

times107times108

(a) 120588 = 0001 (b) 120588 = 0125

(c) 120588 = 0325 (d) 120588 = 055






















List of Acronyms





Acknowledgments


References











































































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










Subarr-aying

ADC

Adaptive weighting

Σ Σ Σ

Σ

Σ


Rminus1sub



(a) DSW


Σ Σ Σ Σ

Σ


minus

(b) Subarrayed SLC

Subarraying

ADC

Main channel

Auxiliarychannel

Adaptive weighting

Σ


Blocking matrix M

wAdap

wQuie

yMain (t)

yAuxi (t)+

+minus

y(t)

(c) Subarrayed GSLC



1205930

) =












minus80

minus60

minus40

minus20

0

Gai

n (d

B)

Jammerdirection

Subarrayed GSLCDSW


120579 (∘)



minus70

minus60

minus50

minus40

minus30

minus20

minus10

0

Gai

n (d

B)


DSW

120579 (∘)

(b) Phase errors





Let

e = [1 0 0]

119879

119871times1

p =

a (1205790

1205930

) minus

1003817100381710038171003817

a (1205790

1205930

)

10038171003817100381710038172

e1003817100381710038171003817

a (1205790

1205930

) minus

1003817100381710038171003817

a (1205790

1205930

)

10038171003817100381710038172

e10038171003817100381710038172

(3)

where sdot 2


H = I minus 2pp119879 (4)


1205930



1205930


M = H[

[

[

[

[

[

[



d



]

]

]

]

]

]

]119871times(119871minus1)

(5)


119895


=



0 20 40 60 80minus80

minus70

minus60

minus50

minus40

minus30

minus20

minus10

0

Gai

n (d

B)


SABF

120579 (∘)







0

minus1198612 1198910


a (120579 120593) = int

1198910+1198612

1198910minus1198612

T119863

119867aele (119891 120579 120593) d119891 (6)

where T119863


[1 119890

minus1198952120587119891120591119899(120579120593)

119890

minus1198952120587119891120591119873minus1(120579120593)

]

119879 hereinto 120591119899

(120579 120593)



0


2 (1205820



0

= minus60∘ and 120579119895



minus80

minus60

minus40

minus20

0

Gai

n (d

B)

Direction ofjammer

SABF

Elevation (∘)








0


0





minus70

minus60

minus50

minus40

minus30

minus20

minus10

0

Gai

n (d

B)


120579 (∘)










minus80

minus70

minus60

minus50

minus40

minus30

minus20

minus10

0

Elevation (deg)

Gai

n (d

B)

SLJ








Sum beam

wsum WA(opt)

= arg minwsum WA

10038171003817100381710038171003817

Twsum WA minus wTay

10038171003817100381710038171003817

2

2

(7)

Difference beam

wΔ WA(opt)

= arg minwΔ WA

10038171003817100381710038171003817

TwΔ WA minus wBay

10038171003817100381710038171003817

2

2

a119867 (1205790

1205930

) (TwΔ WA(opt)

) = 0(8)




minus40

minus30

minus20

minus10

0Po

wer

(dB)


Angle (deg)

(a) Sum beam



minus40

minus30

minus20

minus10

0

Pow

er (d

B)

Angle (deg)

(b) Difference beam



wsum PA(opt)

= arg minwsum PAint

1205872

minus1205872

int

1205872

minus1205872

10038161003816100381610038161003816

w119867sum PAT

119867a (120579 120593) minus w119867Taya (120579 120593)10038161003816100381610038161003816

2

times 119875 (120579 120593) d120579 d120593(9)







2

2


2

+

119896BW(Δ BW)

2









Improved GA



minus40

minus30

minus20

minus10

0

Pow

er (d

B)


v






1st

quadrant

3rd

quadrant

2nd

quadrant

4th

quadrant




119909

times 119873119910



119910119899


1205930


0

1205930


Δ

RxΔJΔ withJΔ


1198731199102


1198731199102

) and q is 119873119909

2-



(119903119894119896 sum

)119894119896=12119873

then we have

119903119894119896 sum

=

120587 (1 minus 119877sum

2

) 120590

2

119868 sum

+ 120590

2

119899

119894 = 119896

2120587119890


119868 sum 119894 = 119896

(10)

with 1198970

= 1198691

(119888119894119896

)119888119894119896

119897sum

= 1198691

(119888119894119896

119877sum

)119888119894119896

where 1198691

(sdot)


=

2120587radic(119909119894

minus 119909119896

)

2

+ (119910119894

minus 119910119896

)

2

120582 119877sum


119868 sum


2

119899


withRxΔ = (119903

119894119896 Δ

)119894119896=12119873




1





⨁

N4 N4 + 1 N2 N2 + 1 3N4 3N4 + 1 N


+

+ + + +

+ + + + + +

+

minus minus


yΣ yΔ

y



0

minus05

0

05

minus30

minus20

minus10

0

Pow

er (d

B)

u

minus05

05

(a) Sum beam

minus30

minus20

minus10

0

Pow

er (d

B)

0u minus05

0

05

minus05

05



minus03minus02

minus010

0102

03 minus03

minus02

minus01

0

01

0203

minus30

minus20

minus10

0

Pow

er (d

B)

u

(a) Sum beam

minus03minus02

minus010

0102

03 minus03

minus02

minus010

0102

03

minus30

minus20

minus10

0

Pow

er (d

B)

u
















0

V0













minus50

minus40

minus30

minus20

minus10

0

Pow

er (d

B)


minus50

minus40

minus30

minus20

minus10

0Po

wer

(dB)

u v




minus50

minus40

minus30

minus20

minus10

0

Pow

er (d

B)


minus50

minus40

minus30

minus20

minus10

0

Pow

er (d

B)

u








0

1205930

) = (0

∘

0

∘


∘

15

∘




minus40

minus30

minus20

minus10

0Po

wer

(dB)


u



minus40

minus30

minus20

minus10

0

Pow

er (d

B)


u



minus40

minus30

minus20

minus10

0

Pow

er (d

B)


u



u


minus40

minus30

minus20

minus10

0

Pow

er (d

B)





minus80

minus70

minus60

minus50

minus40

minus30

minus20

minus10

0

Pow

er (d

B)

SLJ


120579 (∘)





119864

(119906 V) = 119865Δ119864

(119906 V)119865sum

(119906 V) where119865sum

(119906 V) and 119865Δ119864


119864

(119906 V) = 119865sumA

(119906)

119865Δ119864

(V)[119865sumA

(119906)119865sum

119864

(V)] = 119865Δ119864

(V)119865sum

119864




minus2

minus1

0

1

2

Mon

opul

se ra

tio



120579 (∘)









0 05 1 150

02

04

06

08

1 P

roba

bilit

y of

reso

lutio

n

SCSSTSISSMSWAVES



Pro

babi

lity

of re

solu

tion

SCSSTSISSMSWAVES

1

08

06

04

02


SNR (dB)




02

018

016

014

012

01

008

006

004

002


SNR (dB)

RSM

E(∘)


















0



1

) gt 119875((1205791

+

1205792

)2) and 119875(1205792

) gt 119875((1205791

+ 1205792

)2) where 1205791

and 1205792








119889







0

1205930

) = (0

∘

0

∘





















minus50

minus40

minus30

minus20

minus10

0Po

wer

(dB)


v




minus40

minus30

minus20

minus10

0

Pow

er (d

B)

u



0

minus20

minus40

minus60


(dB)

MIMO-PARPARMIMO


(dB)

0

minus10

minus20

minus30

minus40

minus50

minus60

minus70


MIMO-PARPARMIMO

(b) v direction








12

10

8

6

4

2

01

105050

0minus05minus05

minus1 minus1

times109 times109

u

105

0minus05

minus1u

105

0minus05

minus1 u

105

0minus05

minus1 u

1

05

0

minus05minus1

1

050

minus05minus1

1

05

0minus05

minus1

2

15

1

05

0

2

15

1

05

0

3

25

2

15

1

times107times108

(a) 120588 = 0001 (b) 120588 = 0125

(c) 120588 = 0325 (d) 120588 = 055






















List of Acronyms





Acknowledgments


References











































































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation












minus80

minus60

minus40

minus20

0

Gai

n (d

B)

Jammerdirection

Subarrayed GSLCDSW


120579 (∘)



minus70

minus60

minus50

minus40

minus30

minus20

minus10

0

Gai

n (d

B)


DSW

120579 (∘)

(b) Phase errors





Let

e = [1 0 0]

119879

119871times1

p =

a (1205790

1205930

) minus

1003817100381710038171003817

a (1205790

1205930

)

10038171003817100381710038172

e1003817100381710038171003817

a (1205790

1205930

) minus

1003817100381710038171003817

a (1205790

1205930

)

10038171003817100381710038172

e10038171003817100381710038172

(3)

where sdot 2


H = I minus 2pp119879 (4)


1205930



1205930


M = H[

[

[

[

[

[

[



d



]

]

]

]

]

]

]119871times(119871minus1)

(5)


119895


=



0 20 40 60 80minus80

minus70

minus60

minus50

minus40

minus30

minus20

minus10

0

Gai

n (d

B)


SABF

120579 (∘)







0

minus1198612 1198910


a (120579 120593) = int

1198910+1198612

1198910minus1198612

T119863

119867aele (119891 120579 120593) d119891 (6)

where T119863


[1 119890

minus1198952120587119891120591119899(120579120593)

119890

minus1198952120587119891120591119873minus1(120579120593)

]

119879 hereinto 120591119899

(120579 120593)



0


2 (1205820



0

= minus60∘ and 120579119895



minus80

minus60

minus40

minus20

0

Gai

n (d

B)

Direction ofjammer

SABF

Elevation (∘)








0


0





minus70

minus60

minus50

minus40

minus30

minus20

minus10

0

Gai

n (d

B)


120579 (∘)










minus80

minus70

minus60

minus50

minus40

minus30

minus20

minus10

0

Elevation (deg)

Gai

n (d

B)

SLJ








Sum beam

wsum WA(opt)

= arg minwsum WA

10038171003817100381710038171003817

Twsum WA minus wTay

10038171003817100381710038171003817

2

2

(7)

Difference beam

wΔ WA(opt)

= arg minwΔ WA

10038171003817100381710038171003817

TwΔ WA minus wBay

10038171003817100381710038171003817

2

2

a119867 (1205790

1205930

) (TwΔ WA(opt)

) = 0(8)




minus40

minus30

minus20

minus10

0Po

wer

(dB)


Angle (deg)

(a) Sum beam



minus40

minus30

minus20

minus10

0

Pow

er (d

B)

Angle (deg)

(b) Difference beam



wsum PA(opt)

= arg minwsum PAint

1205872

minus1205872

int

1205872

minus1205872

10038161003816100381610038161003816

w119867sum PAT

119867a (120579 120593) minus w119867Taya (120579 120593)10038161003816100381610038161003816

2

times 119875 (120579 120593) d120579 d120593(9)







2

2


2

+

119896BW(Δ BW)

2









Improved GA



minus40

minus30

minus20

minus10

0

Pow

er (d

B)


v






1st

quadrant

3rd

quadrant

2nd

quadrant

4th

quadrant




119909

times 119873119910



119910119899


1205930


0

1205930


Δ

RxΔJΔ withJΔ


1198731199102


1198731199102

) and q is 119873119909

2-



(119903119894119896 sum

)119894119896=12119873

then we have

119903119894119896 sum

=

120587 (1 minus 119877sum

2

) 120590

2

119868 sum

+ 120590

2

119899

119894 = 119896

2120587119890


119868 sum 119894 = 119896

(10)

with 1198970

= 1198691

(119888119894119896

)119888119894119896

119897sum

= 1198691

(119888119894119896

119877sum

)119888119894119896

where 1198691

(sdot)


=

2120587radic(119909119894

minus 119909119896

)

2

+ (119910119894

minus 119910119896

)

2

120582 119877sum


119868 sum


2

119899


withRxΔ = (119903

119894119896 Δ

)119894119896=12119873




1





⨁

N4 N4 + 1 N2 N2 + 1 3N4 3N4 + 1 N


+

+ + + +

+ + + + + +

+

minus minus


yΣ yΔ

y



0

minus05

0

05

minus30

minus20

minus10

0

Pow

er (d

B)

u

minus05

05

(a) Sum beam

minus30

minus20

minus10

0

Pow

er (d

B)

0u minus05

0

05

minus05

05



minus03minus02

minus010

0102

03 minus03

minus02

minus01

0

01

0203

minus30

minus20

minus10

0

Pow

er (d

B)

u

(a) Sum beam

minus03minus02

minus010

0102

03 minus03

minus02

minus010

0102

03

minus30

minus20

minus10

0

Pow

er (d

B)

u
















0

V0













minus50

minus40

minus30

minus20

minus10

0

Pow

er (d

B)


minus50

minus40

minus30

minus20

minus10

0Po

wer

(dB)

u v




minus50

minus40

minus30

minus20

minus10

0

Pow

er (d

B)


minus50

minus40

minus30

minus20

minus10

0

Pow

er (d

B)

u








0

1205930

) = (0

∘

0

∘


∘

15

∘




minus40

minus30

minus20

minus10

0Po

wer

(dB)


u



minus40

minus30

minus20

minus10

0

Pow

er (d

B)


u



minus40

minus30

minus20

minus10

0

Pow

er (d

B)


u



u


minus40

minus30

minus20

minus10

0

Pow

er (d

B)





minus80

minus70

minus60

minus50

minus40

minus30

minus20

minus10

0

Pow

er (d

B)

SLJ


120579 (∘)





119864

(119906 V) = 119865Δ119864

(119906 V)119865sum

(119906 V) where119865sum

(119906 V) and 119865Δ119864


119864

(119906 V) = 119865sumA

(119906)

119865Δ119864

(V)[119865sumA

(119906)119865sum

119864

(V)] = 119865Δ119864

(V)119865sum

119864




minus2

minus1

0

1

2

Mon

opul

se ra

tio



120579 (∘)









0 05 1 150

02

04

06

08

1 P

roba

bilit

y of

reso

lutio

n

SCSSTSISSMSWAVES



Pro

babi

lity

of re

solu

tion

SCSSTSISSMSWAVES

1

08

06

04

02


SNR (dB)




02

018

016

014

012

01

008

006

004

002


SNR (dB)

RSM

E(∘)


















0



1

) gt 119875((1205791

+

1205792

)2) and 119875(1205792

) gt 119875((1205791

+ 1205792

)2) where 1205791

and 1205792








119889







0

1205930

) = (0

∘

0

∘





















minus50

minus40

minus30

minus20

minus10

0Po

wer

(dB)


v




minus40

minus30

minus20

minus10

0

Pow

er (d

B)

u



0

minus20

minus40

minus60


(dB)

MIMO-PARPARMIMO


(dB)

0

minus10

minus20

minus30

minus40

minus50

minus60

minus70


MIMO-PARPARMIMO

(b) v direction








12

10

8

6

4

2

01

105050

0minus05minus05

minus1 minus1

times109 times109

u

105

0minus05

minus1u

105

0minus05

minus1 u

105

0minus05

minus1 u

1

05

0

minus05minus1

1

050

minus05minus1

1

05

0minus05

minus1

2

15

1

05

0

2

15

1

05

0

3

25

2

15

1

times107times108

(a) 120588 = 0001 (b) 120588 = 0125

(c) 120588 = 0325 (d) 120588 = 055






















List of Acronyms





Acknowledgments


References











































































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










0 20 40 60 80minus80

minus70

minus60

minus50

minus40

minus30

minus20

minus10

0

Gai

n (d

B)


SABF

120579 (∘)







0

minus1198612 1198910


a (120579 120593) = int

1198910+1198612

1198910minus1198612

T119863

119867aele (119891 120579 120593) d119891 (6)

where T119863


[1 119890

minus1198952120587119891120591119899(120579120593)

119890

minus1198952120587119891120591119873minus1(120579120593)

]

119879 hereinto 120591119899

(120579 120593)



0


2 (1205820



0

= minus60∘ and 120579119895



minus80

minus60

minus40

minus20

0

Gai

n (d

B)

Direction ofjammer

SABF

Elevation (∘)








0


0





minus70

minus60

minus50

minus40

minus30

minus20

minus10

0

Gai

n (d

B)


120579 (∘)










minus80

minus70

minus60

minus50

minus40

minus30

minus20

minus10

0

Elevation (deg)

Gai

n (d

B)

SLJ








Sum beam

wsum WA(opt)

= arg minwsum WA

10038171003817100381710038171003817

Twsum WA minus wTay

10038171003817100381710038171003817

2

2

(7)

Difference beam

wΔ WA(opt)

= arg minwΔ WA

10038171003817100381710038171003817

TwΔ WA minus wBay

10038171003817100381710038171003817

2

2

a119867 (1205790

1205930

) (TwΔ WA(opt)

) = 0(8)




minus40

minus30

minus20

minus10

0Po

wer

(dB)


Angle (deg)

(a) Sum beam



minus40

minus30

minus20

minus10

0

Pow

er (d

B)

Angle (deg)

(b) Difference beam



wsum PA(opt)

= arg minwsum PAint

1205872

minus1205872

int

1205872

minus1205872

10038161003816100381610038161003816

w119867sum PAT

119867a (120579 120593) minus w119867Taya (120579 120593)10038161003816100381610038161003816

2

times 119875 (120579 120593) d120579 d120593(9)







2

2


2

+

119896BW(Δ BW)

2









Improved GA



minus40

minus30

minus20

minus10

0

Pow

er (d

B)


v






1st

quadrant

3rd

quadrant

2nd

quadrant

4th

quadrant




119909

times 119873119910



119910119899


1205930


0

1205930


Δ

RxΔJΔ withJΔ


1198731199102


1198731199102

) and q is 119873119909

2-



(119903119894119896 sum

)119894119896=12119873

then we have

119903119894119896 sum

=

120587 (1 minus 119877sum

2

) 120590

2

119868 sum

+ 120590

2

119899

119894 = 119896

2120587119890


119868 sum 119894 = 119896

(10)

with 1198970

= 1198691

(119888119894119896

)119888119894119896

119897sum

= 1198691

(119888119894119896

119877sum

)119888119894119896

where 1198691

(sdot)


=

2120587radic(119909119894

minus 119909119896

)

2

+ (119910119894

minus 119910119896

)

2

120582 119877sum


119868 sum


2

119899


withRxΔ = (119903

119894119896 Δ

)119894119896=12119873




1





⨁

N4 N4 + 1 N2 N2 + 1 3N4 3N4 + 1 N


+

+ + + +

+ + + + + +

+

minus minus


yΣ yΔ

y



0

minus05

0

05

minus30

minus20

minus10

0

Pow

er (d

B)

u

minus05

05

(a) Sum beam

minus30

minus20

minus10

0

Pow

er (d

B)

0u minus05

0

05

minus05

05



minus03minus02

minus010

0102

03 minus03

minus02

minus01

0

01

0203

minus30

minus20

minus10

0

Pow

er (d

B)

u

(a) Sum beam

minus03minus02

minus010

0102

03 minus03

minus02

minus010

0102

03

minus30

minus20

minus10

0

Pow

er (d

B)

u
















0

V0













minus50

minus40

minus30

minus20

minus10

0

Pow

er (d

B)


minus50

minus40

minus30

minus20

minus10

0Po

wer

(dB)

u v




minus50

minus40

minus30

minus20

minus10

0

Pow

er (d

B)


minus50

minus40

minus30

minus20

minus10

0

Pow

er (d

B)

u








0

1205930

) = (0

∘

0

∘


∘

15

∘




minus40

minus30

minus20

minus10

0Po

wer

(dB)


u



minus40

minus30

minus20

minus10

0

Pow

er (d

B)


u



minus40

minus30

minus20

minus10

0

Pow

er (d

B)


u



u


minus40

minus30

minus20

minus10

0

Pow

er (d

B)





minus80

minus70

minus60

minus50

minus40

minus30

minus20

minus10

0

Pow

er (d

B)

SLJ


120579 (∘)





119864

(119906 V) = 119865Δ119864

(119906 V)119865sum

(119906 V) where119865sum

(119906 V) and 119865Δ119864


119864

(119906 V) = 119865sumA

(119906)

119865Δ119864

(V)[119865sumA

(119906)119865sum

119864

(V)] = 119865Δ119864

(V)119865sum

119864




minus2

minus1

0

1

2

Mon

opul

se ra

tio



120579 (∘)









0 05 1 150

02

04

06

08

1 P

roba

bilit

y of

reso

lutio

n

SCSSTSISSMSWAVES



Pro

babi

lity

of re

solu

tion

SCSSTSISSMSWAVES

1

08

06

04

02


SNR (dB)




02

018

016

014

012

01

008

006

004

002


SNR (dB)

RSM

E(∘)


















0



1

) gt 119875((1205791

+

1205792

)2) and 119875(1205792

) gt 119875((1205791

+ 1205792

)2) where 1205791

and 1205792








119889







0

1205930

) = (0

∘

0

∘





















minus50

minus40

minus30

minus20

minus10

0Po

wer

(dB)


v




minus40

minus30

minus20

minus10

0

Pow

er (d

B)

u



0

minus20

minus40

minus60


(dB)

MIMO-PARPARMIMO


(dB)

0

minus10

minus20

minus30

minus40

minus50

minus60

minus70


MIMO-PARPARMIMO

(b) v direction








12

10

8

6

4

2

01

105050

0minus05minus05

minus1 minus1

times109 times109

u

105

0minus05

minus1u

105

0minus05

minus1 u

105

0minus05

minus1 u

1

05

0

minus05minus1

1

050

minus05minus1

1

05

0minus05

minus1

2

15

1

05

0

2

15

1

05

0

3

25

2

15

1

times107times108

(a) 120588 = 0001 (b) 120588 = 0125

(c) 120588 = 0325 (d) 120588 = 055






















List of Acronyms





Acknowledgments


References











































































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











minus70

minus60

minus50

minus40

minus30

minus20

minus10

0

Gai

n (d

B)


120579 (∘)










minus80

minus70

minus60

minus50

minus40

minus30

minus20

minus10

0

Elevation (deg)

Gai

n (d

B)

SLJ








Sum beam

wsum WA(opt)

= arg minwsum WA

10038171003817100381710038171003817

Twsum WA minus wTay

10038171003817100381710038171003817

2

2

(7)

Difference beam

wΔ WA(opt)

= arg minwΔ WA

10038171003817100381710038171003817

TwΔ WA minus wBay

10038171003817100381710038171003817

2

2

a119867 (1205790

1205930

) (TwΔ WA(opt)

) = 0(8)




minus40

minus30

minus20

minus10

0Po

wer

(dB)


Angle (deg)

(a) Sum beam



minus40

minus30

minus20

minus10

0

Pow

er (d

B)

Angle (deg)

(b) Difference beam



wsum PA(opt)

= arg minwsum PAint

1205872

minus1205872

int

1205872

minus1205872

10038161003816100381610038161003816

w119867sum PAT

119867a (120579 120593) minus w119867Taya (120579 120593)10038161003816100381610038161003816

2

times 119875 (120579 120593) d120579 d120593(9)







2

2


2

+

119896BW(Δ BW)

2









Improved GA



minus40

minus30

minus20

minus10

0

Pow

er (d

B)


v






1st

quadrant

3rd

quadrant

2nd

quadrant

4th

quadrant




119909

times 119873119910



119910119899


1205930


0

1205930


Δ

RxΔJΔ withJΔ


1198731199102


1198731199102

) and q is 119873119909

2-



(119903119894119896 sum

)119894119896=12119873

then we have

119903119894119896 sum

=

120587 (1 minus 119877sum

2

) 120590

2

119868 sum

+ 120590

2

119899

119894 = 119896

2120587119890


119868 sum 119894 = 119896

(10)

with 1198970

= 1198691

(119888119894119896

)119888119894119896

119897sum

= 1198691

(119888119894119896

119877sum

)119888119894119896

where 1198691

(sdot)


=

2120587radic(119909119894

minus 119909119896

)

2

+ (119910119894

minus 119910119896

)

2

120582 119877sum


119868 sum


2

119899


withRxΔ = (119903

119894119896 Δ

)119894119896=12119873




1





⨁

N4 N4 + 1 N2 N2 + 1 3N4 3N4 + 1 N


+

+ + + +

+ + + + + +

+

minus minus


yΣ yΔ

y



0

minus05

0

05

minus30

minus20

minus10

0

Pow

er (d

B)

u

minus05

05

(a) Sum beam

minus30

minus20

minus10

0

Pow

er (d

B)

0u minus05

0

05

minus05

05



minus03minus02

minus010

0102

03 minus03

minus02

minus01

0

01

0203

minus30

minus20

minus10

0

Pow

er (d

B)

u

(a) Sum beam

minus03minus02

minus010

0102

03 minus03

minus02

minus010

0102

03

minus30

minus20

minus10

0

Pow

er (d

B)

u
















0

V0













minus50

minus40

minus30

minus20

minus10

0

Pow

er (d

B)


minus50

minus40

minus30

minus20

minus10

0Po

wer

(dB)

u v




minus50

minus40

minus30

minus20

minus10

0

Pow

er (d

B)


minus50

minus40

minus30

minus20

minus10

0

Pow

er (d

B)

u








0

1205930

) = (0

∘

0

∘


∘

15

∘




minus40

minus30

minus20

minus10

0Po

wer

(dB)


u



minus40

minus30

minus20

minus10

0

Pow

er (d

B)


u



minus40

minus30

minus20

minus10

0

Pow

er (d

B)


u



u


minus40

minus30

minus20

minus10

0

Pow

er (d

B)





minus80

minus70

minus60

minus50

minus40

minus30

minus20

minus10

0

Pow

er (d

B)

SLJ


120579 (∘)





119864

(119906 V) = 119865Δ119864

(119906 V)119865sum

(119906 V) where119865sum

(119906 V) and 119865Δ119864


119864

(119906 V) = 119865sumA

(119906)

119865Δ119864

(V)[119865sumA

(119906)119865sum

119864

(V)] = 119865Δ119864

(V)119865sum

119864




minus2

minus1

0

1

2

Mon

opul

se ra

tio



120579 (∘)









0 05 1 150

02

04

06

08

1 P

roba

bilit

y of

reso

lutio

n

SCSSTSISSMSWAVES



Pro

babi

lity

of re

solu

tion

SCSSTSISSMSWAVES

1

08

06

04

02


SNR (dB)




02

018

016

014

012

01

008

006

004

002


SNR (dB)

RSM

E(∘)


















0



1

) gt 119875((1205791

+

1205792

)2) and 119875(1205792

) gt 119875((1205791

+ 1205792

)2) where 1205791

and 1205792








119889







0

1205930

) = (0

∘

0

∘





















minus50

minus40

minus30

minus20

minus10

0Po

wer

(dB)


v




minus40

minus30

minus20

minus10

0

Pow

er (d

B)

u



0

minus20

minus40

minus60


(dB)

MIMO-PARPARMIMO


(dB)

0

minus10

minus20

minus30

minus40

minus50

minus60

minus70


MIMO-PARPARMIMO

(b) v direction








12

10

8

6

4

2

01

105050

0minus05minus05

minus1 minus1

times109 times109

u

105

0minus05

minus1u

105

0minus05

minus1 u

105

0minus05

minus1 u

1

05

0

minus05minus1

1

050

minus05minus1

1

05

0minus05

minus1

2

15

1

05

0

2

15

1

05

0

3

25

2

15

1

times107times108

(a) 120588 = 0001 (b) 120588 = 0125

(c) 120588 = 0325 (d) 120588 = 055






















List of Acronyms





Acknowledgments


References











































































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











minus40

minus30

minus20

minus10

0Po

wer

(dB)


Angle (deg)

(a) Sum beam



minus40

minus30

minus20

minus10

0

Pow

er (d

B)

Angle (deg)

(b) Difference beam



wsum PA(opt)

= arg minwsum PAint

1205872

minus1205872

int

1205872

minus1205872

10038161003816100381610038161003816

w119867sum PAT

119867a (120579 120593) minus w119867Taya (120579 120593)10038161003816100381610038161003816

2

times 119875 (120579 120593) d120579 d120593(9)







2

2


2

+

119896BW(Δ BW)

2









Improved GA



minus40

minus30

minus20

minus10

0

Pow

er (d

B)


v






1st

quadrant

3rd

quadrant

2nd

quadrant

4th

quadrant




119909

times 119873119910



119910119899


1205930


0

1205930


Δ

RxΔJΔ withJΔ


1198731199102


1198731199102

) and q is 119873119909

2-



(119903119894119896 sum

)119894119896=12119873

then we have

119903119894119896 sum

=

120587 (1 minus 119877sum

2

) 120590

2

119868 sum

+ 120590

2

119899

119894 = 119896

2120587119890


119868 sum 119894 = 119896

(10)

with 1198970

= 1198691

(119888119894119896

)119888119894119896

119897sum

= 1198691

(119888119894119896

119877sum

)119888119894119896

where 1198691

(sdot)


=

2120587radic(119909119894

minus 119909119896

)

2

+ (119910119894

minus 119910119896

)

2

120582 119877sum


119868 sum


2

119899


withRxΔ = (119903

119894119896 Δ

)119894119896=12119873




1





⨁

N4 N4 + 1 N2 N2 + 1 3N4 3N4 + 1 N


+

+ + + +

+ + + + + +

+

minus minus


yΣ yΔ

y



0

minus05

0

05

minus30

minus20

minus10

0

Pow

er (d

B)

u

minus05

05

(a) Sum beam

minus30

minus20

minus10

0

Pow

er (d

B)

0u minus05

0

05

minus05

05



minus03minus02

minus010

0102

03 minus03

minus02

minus01

0

01

0203

minus30

minus20

minus10

0

Pow

er (d

B)

u

(a) Sum beam

minus03minus02

minus010

0102

03 minus03

minus02

minus010

0102

03

minus30

minus20

minus10

0

Pow

er (d

B)

u
















0

V0













minus50

minus40

minus30

minus20

minus10

0

Pow

er (d

B)


minus50

minus40

minus30

minus20

minus10

0Po

wer

(dB)

u v




minus50

minus40

minus30

minus20

minus10

0

Pow

er (d

B)


minus50

minus40

minus30

minus20

minus10

0

Pow

er (d

B)

u








0

1205930

) = (0

∘

0

∘


∘

15

∘




minus40

minus30

minus20

minus10

0Po

wer

(dB)


u



minus40

minus30

minus20

minus10

0

Pow

er (d

B)


u



minus40

minus30

minus20

minus10

0

Pow

er (d

B)


u



u


minus40

minus30

minus20

minus10

0

Pow

er (d

B)





minus80

minus70

minus60

minus50

minus40

minus30

minus20

minus10

0

Pow

er (d

B)

SLJ


120579 (∘)





119864

(119906 V) = 119865Δ119864

(119906 V)119865sum

(119906 V) where119865sum

(119906 V) and 119865Δ119864


119864

(119906 V) = 119865sumA

(119906)

119865Δ119864

(V)[119865sumA

(119906)119865sum

119864

(V)] = 119865Δ119864

(V)119865sum

119864




minus2

minus1

0

1

2

Mon

opul

se ra

tio



120579 (∘)









0 05 1 150

02

04

06

08

1 P

roba

bilit

y of

reso

lutio

n

SCSSTSISSMSWAVES



Pro

babi

lity

of re

solu

tion

SCSSTSISSMSWAVES

1

08

06

04

02


SNR (dB)




02

018

016

014

012

01

008

006

004

002


SNR (dB)

RSM

E(∘)


















0



1

) gt 119875((1205791

+

1205792

)2) and 119875(1205792

) gt 119875((1205791

+ 1205792

)2) where 1205791

and 1205792








119889







0

1205930

) = (0

∘

0

∘





















minus50

minus40

minus30

minus20

minus10

0Po

wer

(dB)


v




minus40

minus30

minus20

minus10

0

Pow

er (d

B)

u



0

minus20

minus40

minus60


(dB)

MIMO-PARPARMIMO


(dB)

0

minus10

minus20

minus30

minus40

minus50

minus60

minus70


MIMO-PARPARMIMO

(b) v direction








12

10

8

6

4

2

01

105050

0minus05minus05

minus1 minus1

times109 times109

u

105

0minus05

minus1u

105

0minus05

minus1 u

105

0minus05

minus1 u

1

05

0

minus05minus1

1

050

minus05minus1

1

05

0minus05

minus1

2

15

1

05

0

2

15

1

05

0

3

25

2

15

1

times107times108

(a) 120588 = 0001 (b) 120588 = 0125

(c) 120588 = 0325 (d) 120588 = 055






















List of Acronyms





Acknowledgments


References











































































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation













Improved GA



minus40

minus30

minus20

minus10

0

Pow

er (d

B)


v






1st

quadrant

3rd

quadrant

2nd

quadrant

4th

quadrant




119909

times 119873119910



119910119899


1205930


0

1205930


Δ

RxΔJΔ withJΔ


1198731199102


1198731199102

) and q is 119873119909

2-



(119903119894119896 sum

)119894119896=12119873

then we have

119903119894119896 sum

=

120587 (1 minus 119877sum

2

) 120590

2

119868 sum

+ 120590

2

119899

119894 = 119896

2120587119890


119868 sum 119894 = 119896

(10)

with 1198970

= 1198691

(119888119894119896

)119888119894119896

119897sum

= 1198691

(119888119894119896

119877sum

)119888119894119896

where 1198691

(sdot)


=

2120587radic(119909119894

minus 119909119896

)

2

+ (119910119894

minus 119910119896

)

2

120582 119877sum


119868 sum


2

119899


withRxΔ = (119903

119894119896 Δ

)119894119896=12119873




1





⨁

N4 N4 + 1 N2 N2 + 1 3N4 3N4 + 1 N


+

+ + + +

+ + + + + +

+

minus minus


yΣ yΔ

y



0

minus05

0

05

minus30

minus20

minus10

0

Pow

er (d

B)

u

minus05

05

(a) Sum beam

minus30

minus20

minus10

0

Pow

er (d

B)

0u minus05

0

05

minus05

05



minus03minus02

minus010

0102

03 minus03

minus02

minus01

0

01

0203

minus30

minus20

minus10

0

Pow

er (d

B)

u

(a) Sum beam

minus03minus02

minus010

0102

03 minus03

minus02

minus010

0102

03

minus30

minus20

minus10

0

Pow

er (d

B)

u
















0

V0













minus50

minus40

minus30

minus20

minus10

0

Pow

er (d

B)


minus50

minus40

minus30

minus20

minus10

0Po

wer

(dB)

u v




minus50

minus40

minus30

minus20

minus10

0

Pow

er (d

B)


minus50

minus40

minus30

minus20

minus10

0

Pow

er (d

B)

u








0

1205930

) = (0

∘

0

∘


∘

15

∘




minus40

minus30

minus20

minus10

0Po

wer

(dB)


u



minus40

minus30

minus20

minus10

0

Pow

er (d

B)


u



minus40

minus30

minus20

minus10

0

Pow

er (d

B)


u



u


minus40

minus30

minus20

minus10

0

Pow

er (d

B)





minus80

minus70

minus60

minus50

minus40

minus30

minus20

minus10

0

Pow

er (d

B)

SLJ


120579 (∘)





119864

(119906 V) = 119865Δ119864

(119906 V)119865sum

(119906 V) where119865sum

(119906 V) and 119865Δ119864


119864

(119906 V) = 119865sumA

(119906)

119865Δ119864

(V)[119865sumA

(119906)119865sum

119864

(V)] = 119865Δ119864

(V)119865sum

119864




minus2

minus1

0

1

2

Mon

opul

se ra

tio



120579 (∘)









0 05 1 150

02

04

06

08

1 P

roba

bilit

y of

reso

lutio

n

SCSSTSISSMSWAVES



Pro

babi

lity

of re

solu

tion

SCSSTSISSMSWAVES

1

08

06

04

02


SNR (dB)




02

018

016

014

012

01

008

006

004

002


SNR (dB)

RSM

E(∘)


















0



1

) gt 119875((1205791

+

1205792

)2) and 119875(1205792

) gt 119875((1205791

+ 1205792

)2) where 1205791

and 1205792








119889







0

1205930

) = (0

∘

0

∘





















minus50

minus40

minus30

minus20

minus10

0Po

wer

(dB)


v




minus40

minus30

minus20

minus10

0

Pow

er (d

B)

u



0

minus20

minus40

minus60


(dB)

MIMO-PARPARMIMO


(dB)

0

minus10

minus20

minus30

minus40

minus50

minus60

minus70


MIMO-PARPARMIMO

(b) v direction








12

10

8

6

4

2

01

105050

0minus05minus05

minus1 minus1

times109 times109

u

105

0minus05

minus1u

105

0minus05

minus1 u

105

0minus05

minus1 u

1

05

0

minus05minus1

1

050

minus05minus1

1

05

0minus05

minus1

2

15

1

05

0

2

15

1

05

0

3

25

2

15

1

times107times108

(a) 120588 = 0001 (b) 120588 = 0125

(c) 120588 = 0325 (d) 120588 = 055






















List of Acronyms





Acknowledgments


References











































































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










1





⨁

N4 N4 + 1 N2 N2 + 1 3N4 3N4 + 1 N


+

+ + + +

+ + + + + +

+

minus minus


yΣ yΔ

y



0

minus05

0

05

minus30

minus20

minus10

0

Pow

er (d

B)

u

minus05

05

(a) Sum beam

minus30

minus20

minus10

0

Pow

er (d

B)

0u minus05

0

05

minus05

05



minus03minus02

minus010

0102

03 minus03

minus02

minus01

0

01

0203

minus30

minus20

minus10

0

Pow

er (d

B)

u

(a) Sum beam

minus03minus02

minus010

0102

03 minus03

minus02

minus010

0102

03

minus30

minus20

minus10

0

Pow

er (d

B)

u
















0

V0













minus50

minus40

minus30

minus20

minus10

0

Pow

er (d

B)


minus50

minus40

minus30

minus20

minus10

0Po

wer

(dB)

u v




minus50

minus40

minus30

minus20

minus10

0

Pow

er (d

B)


minus50

minus40

minus30

minus20

minus10

0

Pow

er (d

B)

u








0

1205930

) = (0

∘

0

∘


∘

15

∘




minus40

minus30

minus20

minus10

0Po

wer

(dB)


u



minus40

minus30

minus20

minus10

0

Pow

er (d

B)


u



minus40

minus30

minus20

minus10

0

Pow

er (d

B)


u



u


minus40

minus30

minus20

minus10

0

Pow

er (d

B)





minus80

minus70

minus60

minus50

minus40

minus30

minus20

minus10

0

Pow

er (d

B)

SLJ


120579 (∘)





119864

(119906 V) = 119865Δ119864

(119906 V)119865sum

(119906 V) where119865sum

(119906 V) and 119865Δ119864


119864

(119906 V) = 119865sumA

(119906)

119865Δ119864

(V)[119865sumA

(119906)119865sum

119864

(V)] = 119865Δ119864

(V)119865sum

119864




minus2

minus1

0

1

2

Mon

opul

se ra

tio



120579 (∘)









0 05 1 150

02

04

06

08

1 P

roba

bilit

y of

reso

lutio

n

SCSSTSISSMSWAVES



Pro

babi

lity

of re

solu

tion

SCSSTSISSMSWAVES

1

08

06

04

02


SNR (dB)




02

018

016

014

012

01

008

006

004

002


SNR (dB)

RSM

E(∘)


















0



1

) gt 119875((1205791

+

1205792

)2) and 119875(1205792

) gt 119875((1205791

+ 1205792

)2) where 1205791

and 1205792








119889







0

1205930

) = (0

∘

0

∘





















minus50

minus40

minus30

minus20

minus10

0Po

wer

(dB)


v




minus40

minus30

minus20

minus10

0

Pow

er (d

B)

u



0

minus20

minus40

minus60


(dB)

MIMO-PARPARMIMO


(dB)

0

minus10

minus20

minus30

minus40

minus50

minus60

minus70


MIMO-PARPARMIMO

(b) v direction








12

10

8

6

4

2

01

105050

0minus05minus05

minus1 minus1

times109 times109

u

105

0minus05

minus1u

105

0minus05

minus1 u

105

0minus05

minus1 u

1

05

0

minus05minus1

1

050

minus05minus1

1

05

0minus05

minus1

2

15

1

05

0

2

15

1

05

0

3

25

2

15

1

times107times108

(a) 120588 = 0001 (b) 120588 = 0125

(c) 120588 = 0325 (d) 120588 = 055






















List of Acronyms





Acknowledgments


References











































































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation






















0

V0













minus50

minus40

minus30

minus20

minus10

0

Pow

er (d

B)


minus50

minus40

minus30

minus20

minus10

0Po

wer

(dB)

u v




minus50

minus40

minus30

minus20

minus10

0

Pow

er (d

B)


minus50

minus40

minus30

minus20

minus10

0

Pow

er (d

B)

u








0

1205930

) = (0

∘

0

∘


∘

15

∘




minus40

minus30

minus20

minus10

0Po

wer

(dB)


u



minus40

minus30

minus20

minus10

0

Pow

er (d

B)


u



minus40

minus30

minus20

minus10

0

Pow

er (d

B)


u



u


minus40

minus30

minus20

minus10

0

Pow

er (d

B)





minus80

minus70

minus60

minus50

minus40

minus30

minus20

minus10

0

Pow

er (d

B)

SLJ


120579 (∘)





119864

(119906 V) = 119865Δ119864

(119906 V)119865sum

(119906 V) where119865sum

(119906 V) and 119865Δ119864


119864

(119906 V) = 119865sumA

(119906)

119865Δ119864

(V)[119865sumA

(119906)119865sum

119864

(V)] = 119865Δ119864

(V)119865sum

119864




minus2

minus1

0

1

2

Mon

opul

se ra

tio



120579 (∘)









0 05 1 150

02

04

06

08

1 P

roba

bilit

y of

reso

lutio

n

SCSSTSISSMSWAVES



Pro

babi

lity

of re

solu

tion

SCSSTSISSMSWAVES

1

08

06

04

02


SNR (dB)




02

018

016

014

012

01

008

006

004

002


SNR (dB)

RSM

E(∘)


















0



1

) gt 119875((1205791

+

1205792

)2) and 119875(1205792

) gt 119875((1205791

+ 1205792

)2) where 1205791

and 1205792








119889







0

1205930

) = (0

∘

0

∘





















minus50

minus40

minus30

minus20

minus10

0Po

wer

(dB)


v




minus40

minus30

minus20

minus10

0

Pow

er (d

B)

u



0

minus20

minus40

minus60


(dB)

MIMO-PARPARMIMO


(dB)

0

minus10

minus20

minus30

minus40

minus50

minus60

minus70


MIMO-PARPARMIMO

(b) v direction








12

10

8

6

4

2

01

105050

0minus05minus05

minus1 minus1

times109 times109

u

105

0minus05

minus1u

105

0minus05

minus1 u

105

0minus05

minus1 u

1

05

0

minus05minus1

1

050

minus05minus1

1

05

0minus05

minus1

2

15

1

05

0

2

15

1

05

0

3

25

2

15

1

times107times108

(a) 120588 = 0001 (b) 120588 = 0125

(c) 120588 = 0325 (d) 120588 = 055






















List of Acronyms





Acknowledgments


References











































































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











minus50

minus40

minus30

minus20

minus10

0

Pow

er (d

B)


minus50

minus40

minus30

minus20

minus10

0Po

wer

(dB)

u v




minus50

minus40

minus30

minus20

minus10

0

Pow

er (d

B)


minus50

minus40

minus30

minus20

minus10

0

Pow

er (d

B)

u








0

1205930

) = (0

∘

0

∘


∘

15

∘




minus40

minus30

minus20

minus10

0Po

wer

(dB)


u



minus40

minus30

minus20

minus10

0

Pow

er (d

B)


u



minus40

minus30

minus20

minus10

0

Pow

er (d

B)


u



u


minus40

minus30

minus20

minus10

0

Pow

er (d

B)





minus80

minus70

minus60

minus50

minus40

minus30

minus20

minus10

0

Pow

er (d

B)

SLJ


120579 (∘)





119864

(119906 V) = 119865Δ119864

(119906 V)119865sum

(119906 V) where119865sum

(119906 V) and 119865Δ119864


119864

(119906 V) = 119865sumA

(119906)

119865Δ119864

(V)[119865sumA

(119906)119865sum

119864

(V)] = 119865Δ119864

(V)119865sum

119864




minus2

minus1

0

1

2

Mon

opul

se ra

tio



120579 (∘)









0 05 1 150

02

04

06

08

1 P

roba

bilit

y of

reso

lutio

n

SCSSTSISSMSWAVES



Pro

babi

lity

of re

solu

tion

SCSSTSISSMSWAVES

1

08

06

04

02


SNR (dB)




02

018

016

014

012

01

008

006

004

002


SNR (dB)

RSM

E(∘)


















0



1

) gt 119875((1205791

+

1205792

)2) and 119875(1205792

) gt 119875((1205791

+ 1205792

)2) where 1205791

and 1205792








119889







0

1205930

) = (0

∘

0

∘





















minus50

minus40

minus30

minus20

minus10

0Po

wer

(dB)


v




minus40

minus30

minus20

minus10

0

Pow

er (d

B)

u



0

minus20

minus40

minus60


(dB)

MIMO-PARPARMIMO


(dB)

0

minus10

minus20

minus30

minus40

minus50

minus60

minus70


MIMO-PARPARMIMO

(b) v direction








12

10

8

6

4

2

01

105050

0minus05minus05

minus1 minus1

times109 times109

u

105

0minus05

minus1u

105

0minus05

minus1 u

105

0minus05

minus1 u

1

05

0

minus05minus1

1

050

minus05minus1

1

05

0minus05

minus1

2

15

1

05

0

2

15

1

05

0

3

25

2

15

1

times107times108

(a) 120588 = 0001 (b) 120588 = 0125

(c) 120588 = 0325 (d) 120588 = 055






















List of Acronyms





Acknowledgments


References











































































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











minus40

minus30

minus20

minus10

0Po

wer

(dB)


u



minus40

minus30

minus20

minus10

0

Pow

er (d

B)


u



minus40

minus30

minus20

minus10

0

Pow

er (d

B)


u



u


minus40

minus30

minus20

minus10

0

Pow

er (d

B)





minus80

minus70

minus60

minus50

minus40

minus30

minus20

minus10

0

Pow

er (d

B)

SLJ


120579 (∘)





119864

(119906 V) = 119865Δ119864

(119906 V)119865sum

(119906 V) where119865sum

(119906 V) and 119865Δ119864


119864

(119906 V) = 119865sumA

(119906)

119865Δ119864

(V)[119865sumA

(119906)119865sum

119864

(V)] = 119865Δ119864

(V)119865sum

119864




minus2

minus1

0

1

2

Mon

opul

se ra

tio



120579 (∘)









0 05 1 150

02

04

06

08

1 P

roba

bilit

y of

reso

lutio

n

SCSSTSISSMSWAVES



Pro

babi

lity

of re

solu

tion

SCSSTSISSMSWAVES

1

08

06

04

02


SNR (dB)




02

018

016

014

012

01

008

006

004

002


SNR (dB)

RSM

E(∘)


















0



1

) gt 119875((1205791

+

1205792

)2) and 119875(1205792

) gt 119875((1205791

+ 1205792

)2) where 1205791

and 1205792








119889







0

1205930

) = (0

∘

0

∘





















minus50

minus40

minus30

minus20

minus10

0Po

wer

(dB)


v




minus40

minus30

minus20

minus10

0

Pow

er (d

B)

u



0

minus20

minus40

minus60


(dB)

MIMO-PARPARMIMO


(dB)

0

minus10

minus20

minus30

minus40

minus50

minus60

minus70


MIMO-PARPARMIMO

(b) v direction








12

10

8

6

4

2

01

105050

0minus05minus05

minus1 minus1

times109 times109

u

105

0minus05

minus1u

105

0minus05

minus1 u

105

0minus05

minus1 u

1

05

0

minus05minus1

1

050

minus05minus1

1

05

0minus05

minus1

2

15

1

05

0

2

15

1

05

0

3

25

2

15

1

times107times108

(a) 120588 = 0001 (b) 120588 = 0125

(c) 120588 = 0325 (d) 120588 = 055






















List of Acronyms





Acknowledgments


References











































































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











minus80

minus70

minus60

minus50

minus40

minus30

minus20

minus10

0

Pow

er (d

B)

SLJ


120579 (∘)





119864

(119906 V) = 119865Δ119864

(119906 V)119865sum

(119906 V) where119865sum

(119906 V) and 119865Δ119864


119864

(119906 V) = 119865sumA

(119906)

119865Δ119864

(V)[119865sumA

(119906)119865sum

119864

(V)] = 119865Δ119864

(V)119865sum

119864




minus2

minus1

0

1

2

Mon

opul

se ra

tio



120579 (∘)









0 05 1 150

02

04

06

08

1 P

roba

bilit

y of

reso

lutio

n

SCSSTSISSMSWAVES



Pro

babi

lity

of re

solu

tion

SCSSTSISSMSWAVES

1

08

06

04

02


SNR (dB)




02

018

016

014

012

01

008

006

004

002


SNR (dB)

RSM

E(∘)


















0



1

) gt 119875((1205791

+

1205792

)2) and 119875(1205792

) gt 119875((1205791

+ 1205792

)2) where 1205791

and 1205792








119889







0

1205930

) = (0

∘

0

∘





















minus50

minus40

minus30

minus20

minus10

0Po

wer

(dB)


v




minus40

minus30

minus20

minus10

0

Pow

er (d

B)

u



0

minus20

minus40

minus60


(dB)

MIMO-PARPARMIMO


(dB)

0

minus10

minus20

minus30

minus40

minus50

minus60

minus70


MIMO-PARPARMIMO

(b) v direction








12

10

8

6

4

2

01

105050

0minus05minus05

minus1 minus1

times109 times109

u

105

0minus05

minus1u

105

0minus05

minus1 u

105

0minus05

minus1 u

1

05

0

minus05minus1

1

050

minus05minus1

1

05

0minus05

minus1

2

15

1

05

0

2

15

1

05

0

3

25

2

15

1

times107times108

(a) 120588 = 0001 (b) 120588 = 0125

(c) 120588 = 0325 (d) 120588 = 055






















List of Acronyms





Acknowledgments


References











































































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










0 05 1 150

02

04

06

08

1 P

roba

bilit

y of

reso

lutio

n

SCSSTSISSMSWAVES



Pro

babi

lity

of re

solu

tion

SCSSTSISSMSWAVES

1

08

06

04

02


SNR (dB)




02

018

016

014

012

01

008

006

004

002


SNR (dB)

RSM

E(∘)


















0



1

) gt 119875((1205791

+

1205792

)2) and 119875(1205792

) gt 119875((1205791

+ 1205792

)2) where 1205791

and 1205792








119889







0

1205930

) = (0

∘

0

∘





















minus50

minus40

minus30

minus20

minus10

0Po

wer

(dB)


v




minus40

minus30

minus20

minus10

0

Pow

er (d

B)

u



0

minus20

minus40

minus60


(dB)

MIMO-PARPARMIMO


(dB)

0

minus10

minus20

minus30

minus40

minus50

minus60

minus70


MIMO-PARPARMIMO

(b) v direction








12

10

8

6

4

2

01

105050

0minus05minus05

minus1 minus1

times109 times109

u

105

0minus05

minus1u

105

0minus05

minus1 u

105

0minus05

minus1 u

1

05

0

minus05minus1

1

050

minus05minus1

1

05

0minus05

minus1

2

15

1

05

0

2

15

1

05

0

3

25

2

15

1

times107times108

(a) 120588 = 0001 (b) 120588 = 0125

(c) 120588 = 0325 (d) 120588 = 055






















List of Acronyms





Acknowledgments


References











































































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation



















0



1

) gt 119875((1205791

+

1205792

)2) and 119875(1205792

) gt 119875((1205791

+ 1205792

)2) where 1205791

and 1205792








119889







0

1205930

) = (0

∘

0

∘





















minus50

minus40

minus30

minus20

minus10

0Po

wer

(dB)


v




minus40

minus30

minus20

minus10

0

Pow

er (d

B)

u



0

minus20

minus40

minus60


(dB)

MIMO-PARPARMIMO


(dB)

0

minus10

minus20

minus30

minus40

minus50

minus60

minus70


MIMO-PARPARMIMO

(b) v direction








12

10

8

6

4

2

01

105050

0minus05minus05

minus1 minus1

times109 times109

u

105

0minus05

minus1u

105

0minus05

minus1 u

105

0minus05

minus1 u

1

05

0

minus05minus1

1

050

minus05minus1

1

05

0minus05

minus1

2

15

1

05

0

2

15

1

05

0

3

25

2

15

1

times107times108

(a) 120588 = 0001 (b) 120588 = 0125

(c) 120588 = 0325 (d) 120588 = 055






















List of Acronyms





Acknowledgments


References











































































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation














0

1205930

) = (0

∘

0

∘





















minus50

minus40

minus30

minus20

minus10

0Po

wer

(dB)


v




minus40

minus30

minus20

minus10

0

Pow

er (d

B)

u



0

minus20

minus40

minus60


(dB)

MIMO-PARPARMIMO


(dB)

0

minus10

minus20

minus30

minus40

minus50

minus60

minus70


MIMO-PARPARMIMO

(b) v direction








12

10

8

6

4

2

01

105050

0minus05minus05

minus1 minus1

times109 times109

u

105

0minus05

minus1u

105

0minus05

minus1 u

105

0minus05

minus1 u

1

05

0

minus05minus1

1

050

minus05minus1

1

05

0minus05

minus1

2

15

1

05

0

2

15

1

05

0

3

25

2

15

1

times107times108

(a) 120588 = 0001 (b) 120588 = 0125

(c) 120588 = 0325 (d) 120588 = 055






















List of Acronyms





Acknowledgments


References











































































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











minus50

minus40

minus30

minus20

minus10

0Po

wer

(dB)


v




minus40

minus30

minus20

minus10

0

Pow

er (d

B)

u



0

minus20

minus40

minus60


(dB)

MIMO-PARPARMIMO


(dB)

0

minus10

minus20

minus30

minus40

minus50

minus60

minus70


MIMO-PARPARMIMO

(b) v direction








12

10

8

6

4

2

01

105050

0minus05minus05

minus1 minus1

times109 times109

u

105

0minus05

minus1u

105

0minus05

minus1 u

105

0minus05

minus1 u

1

05

0

minus05minus1

1

050

minus05minus1

1

05

0minus05

minus1

2

15

1

05

0

2

15

1

05

0

3

25

2

15

1

times107times108

(a) 120588 = 0001 (b) 120588 = 0125

(c) 120588 = 0325 (d) 120588 = 055






















List of Acronyms





Acknowledgments


References











































































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










12

10

8

6

4

2

01

105050

0minus05minus05

minus1 minus1

times109 times109

u

105

0minus05

minus1u

105

0minus05

minus1 u

105

0minus05

minus1 u

1

05

0

minus05minus1

1

050

minus05minus1

1

05

0minus05

minus1

2

15

1

05

0

2

15

1

05

0

3

25

2

15

1

times107times108

(a) 120588 = 0001 (b) 120588 = 0125

(c) 120588 = 0325 (d) 120588 = 055






















List of Acronyms





Acknowledgments


References











































































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation



















List of Acronyms





Acknowledgments


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









review article aspects of the subarrayed array processing...

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