A Novel Two-Level Nested STAP Strategy for Clutter ...downloads.hindawi.com/journals/mpe/2019/2540858.pdfpotential to detect a slow-moving weak target since the MIMO radar can gain

Research ArticleA Novel Two-Level Nested STAP Strategy for ClutterSuppression in Airborne Radar

WeiWang 12 Lin Zou1 and XuegangWang1

1School of Information and Communication Engineering University of Electronic Science and Technology of ChinaChengdu 611731 China2China Aerodynamics Research and Development Center Mianyang 621000 China

Correspondence should be addressed to Wei Wang 415382448qqcom

Received 11 February 2019 Revised 14 May 2019 Accepted 23 May 2019 Published 4 June 2019

Academic Editor Kishin Sadarangani

Copyright copy 2019 Wei Wang et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

Nested arrays have been studied recently in array signal processing field because of their closed-form expressions for the sensorlocations and achievable degrees of freedom (DOFs) In this paper the concept of nesting is further extended to space-time adaptiveprocessing (STAP) Different from the traditional uniform-STAP method that calculates the clutter plus noise covariance matrix(CNCM) and performs the STAP filter direct using the data snapshots collected from the uniform linear array (ULA) and thetransmitting pulses with uniform pulse repetition interval (PRI) we present a new optimum two-level nested STAP (O2LN-STAP)strategy which employs an optimum two-level nested array (O2LNA) and an optimum two-level nested PRI (O2LN-PRI) to exploitthe enhancedDOFs embedded in the space-time O2LN structure Similar to the difference coarray perspective we first construct avirtual space-time snapshot from the direct covariancematrix of the received signalsThen a newCNCMestimation correspondingto the virtual space-time snapshot can be computed by the spatial-temporal smoothing technique for STAP filter Furthermore thecomparative simulations and analyses with the traditional uniform-STAP and the recently reported coprime-STAP are carried outto verify the effectiveness of the O2LN-STAP approach

1 Introduction

Space-time adaptive processing (STAP) is an attractive tech-nique for clutter suppression and target detection in theairbornespaceborne moving target indicator (MTI) radar[1ndash3] It is well known that the STAP performance is com-monly relevant to the effective system degrees of freedom(DOFs) For example in contrast with the phased array(PA) STAP radar composed of uniform linear array (ULA)multiple-input-multiple-output (MIMO) STAP radar ismorepotential to detect a slow-moving weak target since theMIMO radar can gain larger DOFs due to its orthogo-nal transmitting waveforms [4] However these orthogonalwaveforms are generated by a number of extra waveformcontrol modules at the expense of design and controlcomplexities

Considering the fact that the PA radar is widely used inSTAP applications it is still valuable to find some valid waysto increase its effective DOFs Intuitively the DOFs of the

PA radar with ULA and uniform pulse repetition interval(PRI) can be simply increased by adding array elements andtransmitting pulses However there may exist some issues forthis uniform configuration Firstly the angle-Doppler ambi-guities are unavoidable Secondly the electronic counter-countermeasures (ECCM) capabilities are restricted by theuniform PRI [5] Finally it may be restricted by the largenumber of microwave modules analog-to-digital converter(ADC)modules and so on for the airborne radar applicationswhere the significant constraints of equipment weight sizehardware complexity and power consumption are required[6] On the other hand the PA radar with nonuniformsparse (or thinned) array and PRI can provide an alternativeway to heighten the DOFs while overcoming these issues[5ndash9] Nevertheless the methods with nonuniform sparsePRI require sparse recovery (SR) technique that has highcomputational complexity [5 8] The STAP method in [7]does not use the SR technique but it cannot avoid thetypically high-average angle-Doppler side-lobe levels caused

HindawiMathematical Problems in EngineeringVolume 2019 Article ID 2540858 16 pageshttpsdoiorg10115520192540858

2 Mathematical Problems in Engineering

by randomly spaced sparse array and PRI [6 9] whichdeteriorates the STAP performance of clutter suppression

It is noted that the aforementioned approaches are allbased on the physical nonuniform sparse antenna arraysActually the virtual difference coarray concept with DOFssuperiority has also been introduced into the nonuniformarray signal processing domain In the past years thisconcept which is able to reduce the average angle-Dopplerside-lobe levels of the nonuniform sparse array without theusage of SR technique has been applied to the direction-of-arrival (DOA) estimation and the spatial beamformingAccording to the array geometry we classify those methodsfor exploiting the enhanced DOFs by use of the differencecoarray perspective into three categories The first one isbased on the minimum redundancy array (MRA) [10] Itis shown that the DOFs of the MRA can be improved byconstructing an augmented covariance matrix which is notpositive semidefinite for finite number of snapshots [11 12]And in [13 14] a transformation of this augmented matrixinto a suitable positive definite Toeplitz matrix is suggestedand an elaborate algorithm is presented to construct thisToeplitz matrix Nevertheless this class of approaches hastwo issues One is that for MRA there are no closed-formexpressions for the array geometry and the achievable DOFsfor a given number of sensors and the optimum design ofsuch arrays is always complicated [15 16] Another issue isthat the algorithm for searching an appropriate augmentedcovariance matrix for a large array is a complex iterativeprocedure which converges only to a local optimum [13 14]Moreover the second class of methods about the differencecoarray is related to the recently developed coprime arrayswhich have received much attention since they can be formu-lated more easily than the MRAwhile resolving more sourcesthan the number of sensors [17 18] In [19] performancesof a generalized coprime array have been evaluated by useof their difference coarray equivalence and this generalizedcoprime array has been applied to DOA estimation withboth subspace-basedmethod and sparse reconstruction tech-nique A new approach for passive beamforming and targetdetection using coprime array is also presented in [20] whichcan get the developments of peak side-lobe ratio and the inte-grated side-lobe ratio without giving rise to ldquoghostrdquo targets inthe presence of multiple interferers Another multifrequencyoperation with coprime array is presented in [21] for increas-ing the number of resolvable sources in high-resolution DOAestimation The authors employ multiple frequencies to fillthe missing coarray elements thereby enabling the coprimearray to utilize all of the offered DOFs effectively Finally thetechniques for further exploiting the increased DOFs usingnested arrays [22 23] can be regarded as the third class ofdifference coarray methods An effective nested array designscheme can yield a fourth-order difference coarray to providea higher number of DOFs for DOA estimation [24] Anotherapplication for DOA estimation of the distributed sourceswith nested arrays has been studied in [25] To use the nestedarray for wideband case authors in [26] have proposed aneffective strategy to apply the processing algorithm of nestedarray to each frequency component and the whole spectralinformation of various frequencies has been cooperated to

conduct the detection and estimation Nested array has alsobeen extended to MIMO radar for DOA estimation [27 28]

In recent years the difference coarray perspective hasbeen considered as an effective approach to increase thesystem DOFs for STAP applications To acquire more DOFsauthors in [29] have arranged the array geometry and thetemporal sample interval to make the location of jointspace-time samples satisfy minimum redundancy intervalHowever this STAP algorithm is limited to a time-consumingprocedure of computer searching Moreover a new STAPstrategy for the airborne radar with space-time coprime sam-pling structure (which is referred to as coprime-STAP in thefollowing) has been discussed in [30ndash32] It is demonstratedthat using few array elements and transmitting pulses thecoprime-STAPmethod can achieve comparable performancewith respect to the case of large ULA and great many pulseswith uniform PRI Additionally a fully STAP with nestedarray is also presented in [33] where a high resolution inangle-Doppler plane is obtained However to complete thisfull STAP the spatial and Doppler frequencies of all jammingand clutter sources need to be priorly known

Inspired by the nesting concept in the spatial dimen-sion we think that using PRI with nested time intervalmight also obtain an increase of DOFs in the temporal(or Doppler) dimension Therefore following the differencecoarray perspective we have applied an optimum two-levelnested STAP (O2LN-STAP) strategy to improve the STAPperformance in this paper Different from the traditionaluniform-STAP method that calculates the clutter plus noisecovariance matrix (CNCM) and performs the STAP filterdirect using the received data collected from the physicalULA with uniform PRI the STAP strategy shown in thispaper presents a novel processing idea with an optimumtwo-level nested array (O2LNA) and an optimum two-level nested PRI (O2LN-PRI) In our proposed methodwe first construct a virtual space-time snapshot from thedirect covariance matrix of the received signals The virtualsnapshot can be seen as the collection of the returned signalsfrom a virtual large ULA and a large number of transmittingpulses with uniform PRI Then using the spatial-temporalsmoothing approach [22] a new CNCM estimation of thevirtual space-time snapshot can be computed for STAP filterRemarkably the existing coprime-STAP method which isalso considered in the virtual difference coarray domaincan get more DOFs than the traditional uniform-STAP thatis implemented on the physical array and pulses Howeversimulation results and comparative analyses in the followinghave indicated that the O2LN-STAP method can obtainmore DOFs than both the traditional uniform-STAP and thecoprime-STAP under the constraint of the same number ofphysical array elements and transmitting pulses whichmakesa higher angle-Doppler resolution lower side-lobe levelslarger signal-to-interference-plus-noise ratio (SINR) andbetter minimum detectable velocity (MDV) performancesOn the other hand for a specified number of DOFs and SINRlevel the O2LN-STAP method can reach them with fewerphysical array elements and transmitting pulses (ie lesshardware complexity and power consumption) in contrastwith the traditional uniform-STAP and the coprime-STAP

Mathematical Problems in Engineering 3

0 1d 2d 3d 7d 11d

Level 1 Level 2

(a)-10d -8d -6d -4d -2d 0 2d 4d 6d 8d 10d

(b)

Figure 1 A two-level nested array and its difference coarray (a) A two-level nested array with 3 sensors in each level (b)The correspondingdifference coarray

The remainder of this paper is organized as followsSection 2 introduces the nesting concept of array and PRIThe formulation of the O2LN-STAP strategy is deduced inSection 3 Section 4 gives simulation results and discussionsto verify the effectiveness of theO2LN-STAPmethod Finallyconclusion is presented in Section 5

2 Nesting Concept of Array and PRI

21 Nested Array We first briefly describe the nested arraygeometry in this subsection and more details about itcan be found in [22] The nested array geometry can begenerated very easily in a systematic fashion and we canexactly predict the DOFs of its difference coarray for a givennumber of sensors Moreover using this nested structureit is indeed possible to generate 119874(1198732) DOFs from 119874(119873)physical elements The basic nesting strategy can generate atwo-level nested array (2LNA) whose difference coarray isa filled ULA [22] Though the idea of nesting is easy to beextended to higher dimensions the nested arrays beyond twolevels fail to produce an ULA difference coarray only using asecond-order statistics of the received data [22] Of course itis also possible to produce a virtual filled ULA if we considera higher-order statistics (eg the fourth-order cumulant)for the nested arrays beyond two levels [24] Howeverusing higher-order statistics usually brings hardly acceptablecomputational burden which is not appropriate for practicalapplications Considering that so many conventional arrayprocessing methods (such as beamforming algorithms) arebased on the ULA we prefer to deal with those signalprocessing issues in the ULA domain for simplicity andfeasibility Therefore we just focus on the 2LNA in this papersince it can build a filled ULA with lower cost

The 2LNA is basically a combination of two ULAs Oneis called the inner ULA which has1198731 elements with spacing119889119894119899 Another is named the outer ULA which has1198732 elementswith spacing 119889119900119906119905 such that 119889119900119906119905 = (1198731+1)119889119894119899 In other wordsit is a linear array with sensor locations given by the union ofsets

119878119894119899 = 119899119894119899119889119894119899 119899119894119899 = 0 1 1198731 minus 1 (1a)

and

119878119900119906119905 = 119899119900119906119905 (1198731 + 1) 119889119894119899 minus 119889119894119899 119899119900119906119905 = 1 1198732 (1b)

Obviously the total number of 2LNA is 119873 = 1198731 + 1198732Figure 1(a) illustrates a 2LNA with 1198731 = 1198732 = 3 and

119889 = 119889119894119899 which is similar to the union of the transmitting andthe receiving arrays of the MIMO radar and uses the samenumber of sensors

Next let us recall the definition of difference coarray[22] Considering an array ofN sensors with p119894 denoting theposition vector of the ith sensor a set is defined as

119863 = p119894 minus p1198941015840 119894 = 1 2 119873 1198941015840 = 1 2 119873 (2)

We also define a set119863119906 which consists of the distinct elementsof the set D Then the difference coarray of a given physicalarray can be defined as a virtual array which has sensorslocated at the positions given by the set 119863119906 The differencecoarray of the 2LNA presented in Figure 1(a) is shown inFigure 1(b) Note that the difference coarray of a 2LNA is afilledULAwith = 21198732(1198731+1)minus1 elementswhose positionsare provided by the set

119878119888119900 = 119899119888119900119889119894119899 119899119888119900 = minus119873119888119900 119873119888119900 119873119888119900 = 1198732 (1198731 + 1)minus 1 (3)

where 119899119888119900 is an integer Itmeans thatwe can get 21198732 (1198731+1)minus1virtual elements (ie DOFs) in the coarray domain using only1198731 + 1198732 physical elements In order to maximize the totalDOFs of the coarray authors in [22] have offered the optimalsolutions of1198731 and1198732 for the expression = 21198732(1198731+1)minus1under the constraint of fixed total number of array sensors(ie119873 = 1198731 + 1198732 is fixed) which can be given as follows

Proposition 1 When N is an even integer the optimal solu-tions are1198731 = 1198732 = 1198732 the corresponding maximum valueof (ie DOFs) is (1198732 minus 2)2 + 119873

Proposition 2 When N is an odd integer the optimal solu-tions are1198731 = (119873minus1)2 and1198732 = (119873+1)2 the correspondingmaximum value of (ie DOFs) is (1198732 minus 1)2 + 119873

In the following contexts we call the two-level nestedarray that satisfies Proposition 1 or Proposition 2 as theoptimum two-level nested array (O2LNA)

22 Nested PRI The nesting concept in array design can alsobe applied to PRI design for exploiting the enhanced DOFsin temporal (Doppler) dimension Generally speaking justusing the parameter of time interval to replace the parameterof array element spacing defined in the nested array we canstill determine and analyze the property of the nested PRI


0 1T 2T 3T 7T 11T

Level 1 Level 2

(a)-10T -8T -6T -4T -2T 0 2T 4T 6T 8T 10T

(b)

Figure 2 Transmitting pulses in a CPI with two-level nested PRI and its difference copulses (a) A two-level nested PRI with 3 pulses in eachlevel (b)The corresponding difference copulses

Specifically we just describe the two-level nested PRI (2LN-PRI) in short due to the space limitation of this article Weassume that 119872 = 1198721 + 1198722 transmitting pulses with 2LN-PRI are received in a coherent processing interval (CPI) foreach range gate where1198721 and1198722 have analogous meaningswith 1198731 and 1198732 respectively Like the construction of the2LNA the 2LN-PRI is also a combination of two uniformPRIs with constant time intervals 119879119894119899 and 119879119900119906119905 = (1198721 + 1)119879119894119899respectively Then the pulse position can be given by theunion of sets

119876119894119899 = 119898119894119899119879119894119899 119898119894119899 = 0 1 1198721 minus 1 (4a)

and

119876119900119906119905 = 119898119900119906119905 (1198721 + 1)119879119894119899 minus 119879119894119899 119898119900119906119905 = 1 1198722 (4b)

Figure 2(a) presents an example of 2LN-PRI in a CPI with1198721 = 1198722 = 3 and 119879 = 119879119894119899Similar to the definition of difference coarray we give a

definition of difference copulses herein Considering a CPIwith M pulses using q119895 denoting the timing position of thejth pulse a set is defined as

119866 = q119895 minus q1198951015840 119895 = 1 2 119872 1198951015840 = 1 2 119872 (5)

Also defining a set 119866119906 which consists of the distinct elementsof the setG the difference copulses of the physical pulses in aCPI can be defined as a number of virtual pulseswhose timingpositions are given by the set 119866119906 For instance Figure 2(b)shows the difference copulses corresponding to Figure 2(a)Note that the difference copulses of a 2LN-PRI in a CPI have = 21198722(1198721+1)minus1 pulses with uniform PRIThe positionsof those copulses are provided by the set

119876119888119900 = 119898119888119900119879119894119899 119898119888119900 = minus119872119888119900 119872119888119900 119872119888119900= 1198722 (1198721 + 1) minus 1 (6)

where 119898119888119900 is an integer It means that we can get 21198722(1198721 +1)minus1 virtual pulses (ie DOFs) in the copulses domain usingonly1198721 +1198722 physical pulses Furthermore substituting11987211198722 and M for 1198731 1198732 and N in Propositions 1 and 2 wecan also construct an optimum two-level nested PRI (O2LN-PRI) which will not be rewritten here for the limited length

3 Two-Level Nested STAP Strategy

In this section we further investigate the STAP with nestedarray and nested PRI to exploit the increasing of DOFs for

improving STAP performance We first present the STAPsignal model with 2LNA and 2LN-PRI in Section 31 Andthen in view of difference coarray and copulses perspectivesthe strategy for virtual space-time snapshot constructionis described in Section 32 Next we employ the spatial-temporal smoothing technique to estimate the virtual CNCMmatching to the virtual space-time snapshot in Section 33 Atlast STAP filter based on the virtual snapshot and the virtualCNCM estimation is illustrated in Section 34

31 STAP Signal Model with 2LNA and 2LN-PRI Considera side-looking radar that consists of a 2LNA with 119873 =1198731 + 1198732 identical antenna elements whose locations can bedetermined by the union of sets 119878119894119899 and 119878119900119906119905 But hereindenoting 120582 as the radar wavelength we can rewrite the unionas a new set 119878 = 119899119889 119899 = 0 1198991 1198992 119899119873minus1 where 119889 =119889119894119899 = 1205822 We name d the basic array spacing parameterof the 2LNA in this paper In the same way we assume that119872 = 1198721+1198722 transmitting pulses with 2LN-PRI are receivedin a CPI for each range gate where the pulse locations can bedetermined by the union of sets 119876119894119899 and 119876119900119906119905 For simplicitywe also represent the pulse positions using a new set 119876 =119898119879 119898 = 01198981 1198982 119898119872minus1 where 119879 = 119879119894119899 is named thebasic PRI parameter of the 2LN-PRI in this paper

According to this radar configuration the spatial steeringvector can be given by

a (119891119904) = [1 11989011989521205871198911199041198991 1198901198952120587119891119904119899119873minus1]119879 (7)

where [∙]119879 denotes the transpose operation and 119891119904 is thespatial frequency which is given by

119891119904 = 119889 sin (120579)120582 (8)

where 120579 is the angle of arrival (AOA) At the same time theDoppler steering vector b(119891119889) can be written as

b (119891119889) = [1 11989011989521205871198911198891198981 1198901198952120587119891119889119898119872minus1]119879 (9)

where 119891119889 is the normalized Doppler frequency given by

119891119889 = 2V sin (120579) 119879120582 = 120573119891119904 (10)

where 120573 = 2V119879119889 and v is the relative velocity between themoving radar platform and the signal source from direction


120579 119879 = 119879119894119899 denotes the basic PRI parameter of the 2LN-PRIThen the space-time steering vector k can be expressed as

k (119891119904 119891119889) = b (119891119889) otimes a (119891119904) (11)

where otimes denotes the Kronecker productSuppose that119873119888 clutter patches are uniformly distributed

among the angles from 0 degrees to 360 degrees in afixed range bin We denote 120572119894119891119894119904119888 and 119891119894119889119888 as the complexamplitude the spatial frequency and the normalized Dopplerfrequency of the ith clutter patch respectively The receivedsignal x119888 which contains clutter c plus noise n can beexpressed by

x119888 = c + n = 119873119888sum119894=1

120572119894b (119891119894119889119888) otimes a (119891119894119904119888) + n = 119873119888sum119894=1

120572119894k119888119894 + n

= 119873119888sum119894=1

c119894 + n

(12)

where k119888119894 = k(119891119894119904119888 119891119894119889119888) = b(119891119894119889119888) otimes a(119891119894119904119888) and c119894 denotethe space-time steering vector and the returned signal ofthe ith clutter patch respectively It is assumed that theclutter patches aremutually uncorrelated and then the cluttercovariance matrix (CCM) corresponding to the physicalarray and pulses can be given by [1ndash3]

R119888 =119873119888sum119894=1

119864 [c119894c119867119894 ] =119873119888sum119894=1

1205902119894 k119888119894k119867119888119894 = V119888PV119888119867 (13)

where V119888 = [k1198881 k1198882 k119888119894 k119888119873119888] is the clutter space-time steering matrix 1205902119894 = 119864[|1205722119894 |] denotes the power of theith clutter patch P = diag([12059021 12059022 1205902119873119888]119879) is a diagonalmatrix [∙]119867 is the conjugate transpose operation and 119864[∙]indicates the expectation operator Additionally the noisesare also assumed to be mutually uncorrelated and then thenoise covariance matrix R119899 is described as

R119899 = 1205902119899I119872119873 (14)

where 1205902119899 is the noise power and I119872119873 is an 119872119873 times 119872119873identity matrix Then in the assumption that the clutter isindependent of the noise the CNCM can be given by

R119906 = 119864 [x119888x119867119888 ] = V119888PV119888119867 + 1205902119899I119872119873 = R119888 + R119899 (15)

We call this CNCM as the traditional CNCM in order todistinguish it from the virtual CNCM that will be presentedin the later contents

As is well known denoting 119891119904119905 and 119891119889119905 as the spatialfrequency and the normalized Doppler frequency of aninteresting target the traditional optimum adaptive filterweights of STAP can be given by [1]

w = Rminus1119906 k119905radick119867119905 Rminus1119906 k119905

(16)

where k119905 = k(119891119904119905 119891119889119905) = b(119891119889119905)otimesa(119891119904119905) denotes the space-timesteering vector of the interesting target and Rminus1119906 denotes theinverse matrix of R119906 The term (k119867119905 Rminus1119906 k119905)12 is set to satisfythe constant false alarm rate (CFAR) property In additionthe traditional optimum output SINR can be given by [3]

119878119868119873119877 = 1205902119905 k119867119905 Rminus1119906 k119905 (17)

where 1205902119905 denotes the target powerHowever the true traditional CNCM is commonly

unknown and needs to be estimated by the MLE approachusing the training samples which is given by [3]

R119906 = 1119870X119888X

119867119888 (18)

where R119906 is the estimation of the traditional CNCM Kdenotes the number of training samples and X119888 indicatesthe training sample matrix formed with the training samplesnapshots collected from K range cells

32 Derivation of Virtual Space-Time Snapshot An approachto acquire virtual space-time snapshot was firstly reportedin [30] for coprime arrays Herein we employ a similarprocess to derive the virtual space-time snapshot for the2LNA with 2LN-PRI Note that the term k(119891119904 119891119889)k(119891119904 119891119889)119867can be rewritten as

kk119867 = [b (119891119889) otimes a (119891119904)] [b (119891119889) otimes a (119891119904)]119867= [b (119891119889) b (119891119889)119867] otimes [a (119891119904) a (119891119904)119867]

(19)

owing to the property of (119860 otimes 119861)(119862 otimes 119863) = (119860119862) otimes (119861119863)Moreover the (p q)th entry in a(119891119904)a(119891119904)119867 has the form

[a (119891119904) a (119891119904)119867]119901119902 = 1198901198952120587119891119904(119899119901minus119899119902) = 1198901198952120587119891119904119899119901119902 (20)

We can observe that (20) actually reflects the differencecoarray of a 2LNA according to the definition in (2) Thenfor the considered 2LNAwhose difference coarray is an ULAwith elements the set 119899119901119902 119901 119902 = 0 1198991 sdot sdot sdot 119899119873minus1 in (20)should contain all integers from minus119873119888119900 to119873119888119900 according to (3)And we define a new set 119899 which consists of the distinctelements of the set 119899119901119902 Then there exists an arrangementthat could convert a(119891119904)a(119891119904)119867 to a new spatial steeringvector a(119891119904) which is corresponding to a virtual ULA with elements and 119899119889 spacing in the coarray domain a(119891119904) canbe written as

a (119891119904) = [119890minus1198952120587119891119904119873119888119900 sdot sdot sdot 1 1198901198952120587119891119904119873119888119900]119879 (21)

Similarly the (k l)th entry in b(119891119889)b(119891119889)119867 takes the form[b (119891119889) b (119891119889)119867]119896119897 = 1198901198952120587119891119889(119898119896minus119898119897) = 1198901198952120587119891119889119898119896119897 (22)

where the set 119896119897 119896 119897 = 01198981 sdot sdot sdot 119898119872minus1 contains allintegers from minus119872119888119900 to119872119888119900 in the condition of 2LN-PRI Stilldefining a new set which consists of the distinct elements


of the set 119896119897 the term b(119891119889)b(119891119889)119867 can be rearranged intoa new Doppler steering vector b(119891119889) which is correspondingto virtual pulses with constant PRI 119879 in a CPI and canbe expressed as

b (119891119889) = [119890minus1198952120587119891119889119872119888119900 sdot sdot sdot 1 1198901198952120587119891119889119872119888119900]119879 (23)

Furthermore it is noted that the expression (15) of CNCMR119906 includes the term k119888119894k

119867119888119894 Thus based on the analysis shown

from (19) to (23) it is possible to derive a virtual space-timedata snapshot corresponding to a virtual ULA and virtualpulses with constant PRI from the expression (15) We canextract the specified elements in the matrix R119906 to construct anew times matrix Z such that

Z = 119873119888sum119894=1

1205902119894 a (119891119894119904119888) b (119891119894119889119888)119879 + 1205902119899e1e1198792 (24)

where e1 and e2 are times1 and times1 column vectors of all zerosexcept a one at center position respectively Thus a times 1virtual space-time snapshot of the clutter plus noise can begot through vectoring (24) which can be expressed as

z = 119873119888sum119894=1

1205902119894 b (119891119894119889119888) otimes a (119891119894119904119888) + 1205902119899e2 otimes e1 =119873119888sum119894=1

1205902119894 k119888119894 + n

= c + n

(25)

where k119888119894 = b(119891119894119889119888) otimes a(119891119894119904119888) denotes the virtual space-timesteering vector of the ith clutter patch c and n indicate thevirtual clutter vector and the virtual noise vector respectively

Comparing (25) with (12) we can find that z behaveslike a virtual clutter plus noise signal collected from a virtualULA with elements and 119899119889 spacing in a virtual CPI with uniform transmitting pulses with time intervals 119879 Butthe differences are that the amplitudes of the virtual clutterpatches are indeed the powers of the realistic clutter patchesand the virtual noise becomes a deterministic vector which isalso determined by the power of the realistic noise Moreoverit is noted that under the constraints of 119873 = 1198731 + 1198732and 119872 = 1198721 + 1198722 the inequalities that gt 119873 and gt 119872 are always maintained whatever the values of(1198731 1198732) and (11987211198722) are That is to say the virtual space-time snapshot z has an increased number of DOFs in contrastwith the physical one as shown in (12) Thus it is attractiveto implement STAP on the virtual space-time snapshot z forimproving the performance of clutter suppression and targetdetection However in view of the fact that z is just a singlesnapshot the corresponding CNCM estimation in term ofexpression (18) is a rank-one matrix which is hard to be usedto design the STAP filter [2 3] In order to build a full-rankCNCM estimation we make the spatial-temporal smoothingtechnique operate on the virtual space-time snapshot to geta virtual CNCM estimation for STAP filter The detailedprocess will be presented in Section 33

33 Derivation of Virtual CNCM Estimation For applyingthe spatial-temporal smoothing technique the data matrix

0

0

helliphellip

helliphellip

helliphellip helliphellip

Z

Z

Z00

(N minus 1)

minus(N minus 1)

(M minus 1)minus(M minus 1)

Figure 3 Partition strategy of the virtual space-time snapshot

Z should be divided into 119872119873 number of submatrixes eachwith size119873 times119872 where119873 = 1198732(1198731 + 1) and119872 = 1198722(1198721 +1) The partition strategy is illustrated in Figure 3 and thesubmatrix can be defined as

Z120588120574 =119873119888sum119894=1

119890minus119895(120588119891119894119904119888+120574119891119894119889119888)1205902119894 a (119891119894119904119888) b (119891119894119889119888)119879 + 1205902119899e1120588e1198792120574 (26)

where120588 = 0 1 119873minus1 120574 = 0 1 119872minus1 e1120588 is a subvectorconstructed from the (119873 minus 120588)th entry to the (2119873 minus 1 minus 120588)thentry of e1 e2120574 is a subvector constructed from the (119872minus120574)thentry to the (119872minus120574) th entry of e2 a(119891119894119904119888) is the spatial steeringvector corresponding to a virtual sub-ULA with 119873 elementsand basic spacing d and b(119891119894119889119888) is the Doppler steering vectorcorresponding to 119872 virtual pulses in a sub-CPI with basicPRI parameter T

Then we can define a new covariance matrix via spatial-temporal smoothing approach as follows

R119904 = 1119872119873119873minus1sum120588=0

119872minus1sum120574=0

z120588120574z119867120588120574 (27)

where z120588120574 = vec(Z120588120574) and vec(∙) means to transform amatrix into a column vector It is noted that R119904 is a Hermitianpositive semidefinite matrix of size 119872119873 times 119872119873 Followingthe same proof process of Theorem 2 in [22] or Theorem 3 in[34] the expression (27) can be reformulated as

R119904 = 1119872119873R2119906 (28)

where

R119906 = V119888PV119867

119888 + 1205902119899I119872119873 (29)

where V119888 = [k1198881 k1198882 k119888119873119888] k119888119894 = b(119891119894119889119888) otimes a(119891119894119904119888) andP is the diagonal matrix which represents the powers of allclutter patches as aforementioned in Section 31 Then likethe expression (15) R119906 can be exactly viewed as the virtual


CNCM estimation of the signals received from a virtual sub-ULA with 119873 antenna elements and 119872 virtual transmittingpulses in a sub-CPI with constant PRI k119888119894 is regarded asanother virtual space-time steering vector corresponding tothe sub-ULA and sub-CPI On the other hand it is hard tocalculate 119877119906 via (29) in practice because we can hardly getthe precise 119891119894119889119888 119891119894119904119888 and 1205902119894 of the ith clutter patch direct fromthe echoes to compute the matrixes V119888 and P Contrarilysince R119904 can be easily computed by a sum of vector outerproducts with finite snapshots as presented in (27) R119906 issuggested to be calculated by (28) which can be rewrittenas

R119906 = (119872119873R119904)12 (30)

Thus as is described in Section 34 the matrix R119906 canbe employed as the CNCM estimation for O2LN-STAPfilter

34 STAP Filter Finally we can perform the STAP filterbased on the virtual sub-ULA with 119873 antenna elements andthe119872 virtual transmitting pulses in a sub-CPI with constantPRIThen the corresponding virtual optimum adaptive filterweights can be given by

w = Rminus1119906 k119905radick119867119905 R

minus1

119906 k119905(31)

where k119905 = b(119891119889119905) otimes a(119891119904119905) denotes the virtual target space-time steering vector with respect to the sub-ULA and sub-CPI and Rminus1119906 denotes the inverse matrix of R119906 Note that theexpression (31) is actually acquired by substituting R119906 and k119905for R119906 and k119905 in (16) respectively Additionally the virtualoptimum output SINR corresponding to the sub-ULA andsub-CPI can be written as

119881119878119868119873119877 = 1205902119905 k119867119905 Rminus1119906 k119905 (32)

Significantly using the space-time two-level nested structurethe final available DOFs for STAP filter are indeed equivalentto119872119873 rather than Intuitively we rewrite the value ofDOFs again as follows

1198632minus119899119890119904119905119890119889 = 119872119873 = 1198722 (1198721 + 1) times 1198732 (1198731 + 1) (33)

Remarkably though 119873 lt and 119872 lt the inequalitiesthat 119873 ge 119873 and 119872 ge 119872 are still maintained underthe constraints of 119873 = 1198731 + 1198732 and 119872 = 1198721 + 1198722In other words the available DOFs of the STAP with thespace-time two-level nested structure are still larger thanthat of the traditional approaches Using w for STAP filteris able to get more DOFs than using w that is directlyderived from the physical antenna array and transmittingpulses

1198631199001199011199052minus119899119890119904119905119890119889

=

(119873 + 1)42 times (119872 + 1)

42 119873 119900119889119889 119872 119900119889119889 (119886)

(11987324 + 1198732 ) times (11987224 + 119872

2 ) 119873 119890V119890119899 119872 119890V119890119899 (119887)(119873 + 1)

42 times (11987224 + 119872

2 ) 119873 119900119889119889 119872 119890V119890119899 (119888)(11987324 + 119873

2 ) times (119872 + 1)42 119873 119890V119890119899 119872 119900119889119889 (119889)

(34)

In particular in the case of O2LNA and O2LN-PRI themaximum available DOFs can be given by expression (34)which is able to generate 119874(11987221198732) DOFs from 119874(119872119873)physical elementsThen the STAP strategy using O2LNAandO2LN-PRI is referred to as O2LN-STAP

4 Results and Discussion

In this section numerical experiments and results are offeredto verify the theoretical derivations and compare the per-formances of the O2LN-STAP with those of the traditionaluniform-STAP and the recently reported coprime-STAPWe assume a side-looking airborne radar composed of anO2LNA with 1198731 = 1198732 = 3 and an O2LN-PRI with 1198721 =1198722 = 3 Obviously the total numbers of array elementsand transmitting pulses in a CPI are 119873 = 6 and 119872 = 6respectively The carrier frequency is 10GHz the basic PRIis 119879 = 1600119867119911 and the basic array element spacing is119889 = 1205822 Without loss of generality we set 120573 = 1 to avoidtemporal aliasing There are 180 clutter patches uniformlydistributed in the forward area of the antenna array which areassumed to follow the zero-mean complex-valued Gaussiandistributions The noise power is normalized to 0dB and theclutter-to-noise ratio (CNR) is set to 30dB It is assumed thata target is located at angle 165 degrees and its normalizedDoppler frequency is -027

41 Smoothed Virtual CNCM Estimation In STAP appli-cations the eigenspectrum is usually utilized to measurethe similarity of different covariance matrices Herein wealso employ it to validate the derivation of virtual CNCMestimation from the spatial-temporal smoothing techniqueAccording to the simulation parameters we can get that119873 =12 and119872 = 12 We define a new matrix RV to represent thetheoretical CNCMof an ULAwith119873 elements and119872 pulseswhere the clutter space-time steering vectors and amplitudesare both assumed to be priorly known In other words thematrix RV is able to be calculated via (29) Thus we compareR119906 which is computed through (30) with RV and R119904 in termsof their eigenspectra Figure 4 depicts the comparative resultsAccording to the Brennan rule [3] the rank of RV should be119873 + 120573(119872 minus 1) = 23 And we can see from the figure that R119906andR119904 have the same rank asRV But the eigenvalues ofR119904 arevery different from the others while the eigenvalues ofR119906 andRV are nearly overlapped completely Thus we can see thatR119906 has the same eigenspectrum as RV Therefore on the basisof these eigenspectra we have demonstrated the theoreticalderivation of R119906 which is indeed matching to a virtual ULA


20

40

60

80

100

120

00 5 10 15 20 25 30

Eige

nval

ues (

dB)

Eigenvalue index

R

Rs

Ru

Figure 4 Eigenspectra of different covariance matrices

with 119873 antenna elements and 119872 virtual transmitting pulsesin a CPI with constant PRI as is expressed in (29) and (30)

42 PerformanceComparisonwithTraditional Uniform-STAPAs we know the traditional uniform-STAP is commonlybased on the ULA with uniform PRI In addition we willtake another method named as direct-nested-STAP intoconsideration too which adopts the same direct processingas the traditional uniform-STAP but acts on the physicalO2LNA and pulses with O2LN-PRI In this subsectionwe compare the performance of the O2LN-STAP with thetraditional uniform-STAP and the direct-nested-STAP in twodifferent conditions one is setting a fixed number of physicalarray elements and transmitting pulses another is setting afixed objective of DOFs

(1) Setting a Fixed Number of Physical Array Elements andPulses Firstly we implement the comparison under therestriction of fixed numbers of physical array elements N andtransmitting pulsesMWithout loss of generality we supposethat N and M are even numbers For O2LN-STAP we canget that 1198731 = 1198732 = 1198732 and 1198721 = 1198722 = 1198722 as isdescribed inProposition 1 And the final availableDOFs valuefor the O2LN-STAP is given by (34)(b) In contrast it is wellknown that the available DOFs of the traditional uniform-STAP and the direct-nested-STAP are only equivalent to119872119873which is evidently much smaller than the value of (34)(b) Forexample fixing119873 = 6 and119872 = 6 the value of (34)(b) is 144but119872119873 equals to 36

Furthermore we study the normalized STAP filter outputresponse and the optimum output SINR performance toillustrate the advantages of the O2LN-STAP in the case offixed number of array elements and transmitting pulsesFigure 5 presents the normalized STAP filter output responsefor the three different STAP strategies Specifically Figures5(a) 5(b) and 5(c) are corresponding to the O2LN-STAPthe traditional uniform-STAP and the direct-nested-STAPrespectively It is observed that all of the three strategies canform a deep notch at the clutter ridge and make a maximumpeak at the target position Nevertheless the side-lobe levelsand the angle-Doppler resolution of the O2LN-STAP are

better than those of the other two approaches owing to theincreased available DOFs of the O2LN-STAP Moreover thedirect-nested-STAP even shows much higher side-lobe levelsthan the traditional uniform-STAP obviously It is becausethat though the direct-nested-STAP seems to utilize thenested array and pulses too it actually fails to construct alarger virtual ULA and more virtual pulses with uniformPRIThedirect-nested-STAP also just directly operates on thephysical nested array and pulses rather than takes advantageof the difference coarray and copulses Therefore the sparseproperty of the nested array and pulses results in the poorside-lobe performance of the direct-nested-STAP

In addition in the condition of fixed number of arrayelements and transmitting pulses the optimum output SINRcurves against the target normalized Doppler frequency areshown in Figure 6 The SINR curve of the O2LN-STAPis derived from (32) and the others are from (17) It isclear that the SINR of the O2LN-STAP in the nonclutterregions is much higher than those of the traditional uniform-STAP and the direct-nested-STAP owing to the enhancedDOFs of O2LN-STAP Moreover the O2LN-STAP methodand the direct-nested-STAP both possess a narrower clutternotch than the traditional uniform-STAP which means animprovement of MDV performance However in view of thebad filter output response as shown in Figure 5(c) we donot suggest to use the direct-nested-STAP for improving theMDV performance in practice

Note that almost all conventional STAP methods em-ploy constant uniform PRI But the electronic counter-countermeasures (ECCM) capabilities are restricted by theuniform PRI Moreover in contrast with the uniform PRIthe two-level nested PRI approach can get a comparativeperformance with the uniform PRI approach that has alarger number of physical pulses In other words the two-level nested PRI approach is potential to reduce the radarprobability of intercept owing to the low accumulation oftransmitting energy and timeTherefore the two-level nestedPRI approach can get some advantages over the conventionalway in these aspects On the other hand the small numberof pulses ought to cause a performance decrease (eg thedecrease of Doppler resolution and high side-lobe levels intemporal dimension) as shown in Figure 5(b) However wecan obtain an increased number of virtual pulses via thetransformation described in Section 3 to offset the deficiencyof the physical pulses for performance improvements as ispresented in Figure 5(a) Additionally from the hardwareperspective the conventional uniform PRI may be easier torealize Nevertheless the two-level nested PRI approach isalso not hard to realize for the reason that the two-level nestedstructure is actually thinned from the conventional uniformone and close to the uniform structure besides it may getseveral benefits in contrast with the conventional way

With respect to the other situations that N and M arenot even integers similar analysis procedures and simulationresults can also be carried out to confirm the effectivenessof the O2LN-STAP which will not be presented here onaccount of the space limitation In summary we can drawa conclusion that (which is referred to as conclusion 1 in thefollowing) given a fixed number of physical array elements


Spatial frequency050minus05

0

minus60

minus50

minus40

minus30

minus20

minus10N

orm

aliz

ed D

oppl

er fr

eque

ncy

00102030405

minus05

minus04

minus03

minus02

minus01

(a)

(a)Spatial frequency

050minus05

0

minus60

minus50

minus40

minus30

minus20

minus10

Nor

mal

ized

Dop

pler

freq

uenc

y

00102030405

minus05

minus04

minus03

minus02

minus01

(b)


050minus05

0

minus60

minus50

minus40

minus30

minus20

minus10

Nor

mal

ized

Dop

pler

freq

uenc

y

00102030405

minus05

minus04

minus03

minus02

minus01

(c)

Figure 5 Normalized STAP filter output responses in the case of fixed total number of array elements and transmitting pulses (a) Theproposed O2LN-STAP (b)The traditional uniform-STAP (c)The direct-nested-STAP

0 01 02 03 04 05

0

10

20

Normalized Doppler frequency

SIN

R (d

B)

Proposed O2LN-STAPTraditional uniform-STAPDirect-nested-STAP

minus10

minus20

minus30

minus01minus02minus03minus04minus05

Figure 6 Optimum output SINR performances against target normalized Doppler frequency with N = 6 array elements and M = 6transmitting pulses


and transmitting pulses the O2LN-STAP can obtain moreDOFs than the traditional uniform-STAP and the direct-nested-STAP which improves the filter output responseSINR and MDV performances

(2) Setting a Fixed Objective of DOFs In most of the STAPapplications we prefer to achieve a fixed objective of DOFsusing as low cost as possible for performance improvementRemarkably the converse of conclusion 1 means that theO2LN-STAP needs fewer physical array elements and trans-mitting pulses than the other two methods to achieve acertain designated number of DOFs Considering that thedirect-nested-STAP is not suitable for the practical applica-tions due to its high side-lobe levels we mainly focus onthe comparison between O2LN-STAP and the traditionaluniform-STAP in this subsection For instance if the objec-tive of DOFs is set to 144 the O2LN-STAP can reach it with119873 = 119872 = 6 but the traditional uniform-STAP needs theconfiguration of 119873 = 119872 = 12 Thereby the O2LN-STAPmethod is potential to save the hardware resource and reducethe power consumption

Furthermore as is deduced in Section 3 we note thatthe O2LN-STAP method originates from the traditionalCNCM R119906 However the true traditional CNCM is com-monly unknown and needs to be estimated by the MLEapproach using the training samples as is described in (18)The accuracy of the estimation of the traditional CNCM isproportional to the ratio of the number of training samplesK to the product between the number of array elements andthe number of transmitting pulses (ieMN) according to theRMB rule [3] It is also well known that with the increaseof the number of training samples the estimation error ofthe traditional CNCM for small MN would converge to zeromore quickly than that for largeMNTherefore using CNCMestimation to replace the true CNCM for STAP filter wouldresult in SINR loss due to the estimation error Note thatthe O2LN-STAP could achieve an objective of DOFs withsmaller MN than the traditional uniform-STAP Hence itseems that the output SINR of the O2LN-STAPmay convergeto an identical optimum level more rapidly than that of thetraditional uniform-STAP since the O2LN-STAP employsfewer physical array elements and transmitting pulses Thatis to say it is supposed that the O2LN-STAP may achieve aspecific output SINRwith smaller number of training samplesthan the traditional uniform-STAP which is referred to assupposition 1 in the following

In order to further study this issue we carry out anexperiment to analyze the convergence performance of theoutput SINR versus the number of training samples by theway of substituting the CNCMestimation for the true CNCMR119906 to realize the deducing processing shown in Section 3 andthen get the corresponding estimations of R119904 and R119906 via (27)and (30) for the final filter processing respectively Note thatwe use the CNCM estimation in (18) that is actually acquiredby the sample matrix inverse (SMI) algorithm [1] and theCNCM estimation calculated through principal components(PC) technique [3] to replace the trueCNCMR119906 respectivelyThis experiment is applied to both the O2LN-STAP and thetraditional uniform-STAP for comparison The results are

13141516171819202122

Number of training samples

Out

put S

INR

(dB)

Objective SINRTraditional uniform-STAP using SMI filter(N=M=12)Traditional uniform-STAP using PC filter(N=M=12)Proposed O2LN-STAP using SMI filter(N=M=6)Proposed O2LN-STAP using PC filter(N=M=6)

102 103 104

Figure 7 Output SINR performances versus the number of trainingsamples

provided in Figure 7 In the following the term ldquousing SMIfilterrdquo is corresponding to the aforementioned SMI approachand ldquousing PC filterrdquo is matching to the PC approach Theobjective SINR curve in the figure is actually the theoreticalupper bound of the output SINR corresponding to thetraditional uniform-STAP with 119873 = 119872 = 12 It can beseen that the O2LN-STAP using SMI filter has not convergedto the objective SINR while the others have reached itAdditionally the approaches using PC filter present betterconvergence performance than those using SMI filter asexpected since the PC algorithm utilizes the clutter subspacetechnique to eliminate a part of cross correlation amongclutter patches and then lower the requirement for trainingsamples [3] Moreover the traditional uniform-STAP usingPC filter (or SMI filter) exhibits faster convergence propertyand higher SINR in the case of a small number of trainingsamples than the O2LN-STAP using PC filter (or SMI filter)which is not coincident with supposition 1 proposed in thepreceding paragraph This is because that for the O2LN-STAP method the estimation error of R119906 would propagatethroughout the additional procedures for the derivations ofvirtual space-time snapshot and virtual CNCM estimationand the spatial-temporal smoothing process would also bringabout a new estimation error Therefore we need moretraining samples to reduce this effect of error accumulationthat is not existent in the traditional uniform-STAP But fromanother point of view we can see from Figure 7 that onlyemploying half of the array elements and transmitting pulsesof the traditional uniform-STAP the O2LN-STAP using PCfilter can achieve the specified aim of SINR with a closeconvergence performance to the traditional uniform-STAPusing PC filter On the other hand the O2LN-STAP usingPC filter still has a superior convergence performance thanthe traditional uniform-STAP using SMI filter It is becausethat the requirement of sample numbers for PC techniqueis less than that for SMI approach [3] which makes theestimation error of R119906 for PC technique become smaller than


Nor

mal

ized

Dop

pler

freq

uenc

y

Spatial frequency

0

0

0102030405

050

minus05

minus05minus60

minus50

minus40

minus30

minus20

minus10minus04

minus03

minus02

minus01

(a)


Nor

mal

ized

Dop

pler

freq

uenc

y

00102030405

minus05

minus04

minus03

minus02

minus01

0

minus60

minus50

minus40

minus30

minus20

minus10

(b)

Figure 8 Normalized STAP filter output responses with N = 6 array elements andM = 6 transmitting pulses (a)The proposed O2LN-STAP(b)The coprime-STAP

that for SMI approach Therefore using fewer array elementsand pulses the O2LN-STAP is still potential to be used forclutter suppression with less hardware complexity and powerconsumption

43 Performance Comparison with Coprime-STAP As afore-mentioned in the introduction section the recently reportedcoprime-STAP strategy can also achieve enhanced DOFs toimprove the STAP performance with low cost in contrastwith the traditional uniform-STAP [32] In this subsectionwe will further compare the performance of the O2LN-STAPwith that of the coprime-STAP Herein the coprime-STAPmodel used for analysis and simulation is identical with thosein [30 32] It is assumed that the spatial coprime array iscomposed of two ULA with interelement spacing 11987310158401119889 and11987310158402119889 respectively where 11987310158401 and 11987310158402 are coprime integersand satisfy 11987310158401 lt 11987310158402 Meanwhile we also assume that thecoprime PRI is a combination of two uniform PRIs denotedas11987210158401119879 and11987210158402119879 where11987210158401 and11987210158402 are coprime integers andsatisfy 11987210158401 lt 11987210158402 Then there are 1198731015840 = 11987310158402 + 211987310158401 minus 1 arrayelements and M1015840 = 11987210158402 + 211987210158401 minus 1 pulses in a CPI in total forcoprime-STAP According to [30] the final available DOFs ofthe coprime-STAP can be given by

119863119888119900119901119903119894119898119890 = (1198731015840111987310158402 + 1) times (1198721015840111987210158402 + 1) (35)

We also compare the performance of the O2LN-STAPwith the coprime-STAP in the two different conditions asdescribed in Section 42

(1) Setting a Fixed Number of Physical Array Elements andPulses We first make the comparison in the case of afixed number of physical array elements and transmittingpulses We give a new proposition with respect to 119863119888119900119901119903119894119898119890and 119863119900119901119905

2minus119899119890119904119905119890119889under this situation which is described as

Proposition 3 in the following

Proposition 3 Under the constraint of the same number ofphysical antenna array elements and transmitting pulses (ie119873 = 1198731015840 119872 = 1198721015840) the final available DOFs of the

O2LN-STAP1198631199001199011199052minus119899119890119904119905119890119889 is larger than that of the coprime-STAP119863119888119900119901119903119894119898119890 (ie1198631199001199011199052minus119899119890119904119905119890119889 gt 119863119888119900119901119903119894119898119890) when119873 gt 2 and119872 gt 2

Significantly the conditions in Proposition 3 that 119873 gt2 and 119872 gt 2 are almost always met in the practicalapplications The detailed proof process can be found inappendix For example fixing 119873 = 6 array elements and119872 = 6 transmitting pulses the parameters of coprime-STAPhave to be set as11987310158401 = 11987210158401 = 2 and11987310158402 = 11987210158402 = 3 for achievinga maximum value of DOFs while the parameters of O2LN-STAP are set as 1198731 = 1198721 = 3 and 1198732 = 1198722 = 3 Then wecan get 119863119888119900119901119903119894119898119890 = 49 via (35) while 1198631199001199011199052minus119899119890119904119905119890119889 = 144 through(34)(b) Their normalized STAP filter output responses andoptimum output SINR curves are illustrated in Figures 8and 9 respectively It is clearly observed that with the samenumber of physical array elements and transmitting pulsesthe O2LN-STAP gains better angle-Doppler resolution lowerside-lobe levels higher SINR and narrower clutter notch(improved MDV performance) than the coprime-STAP

(2) Setting a Fixed Objective of DOFs We then compare theperformances in the situation of a fixed objective of DOFsSimilar to the analysis in the previous subsection it is notedthat Proposition 3 can be illustrated in another way Thatis to say given a specified objective of DOFs the O2LN-STAP can reach it with fewer physical array elements andtransmitting pulses than the coprime-STAP For examplewe first assume that there exists a virtual space-time systemconsisting of a virtual ULA with 16 elements and 16 virtualtransmitting pulses with uniform PRI in a CPI (this virtualspace-time system is denoted as V-ULA-UPRI-16x16) whoseDOFs equal to 16 times 16 = 256 Then in order to constructan identical virtual space-time structure and obtain the sameDOFs as V-ULA-UPRI-16x16 we can employ the O2LN-STAP with the configurations of 1198731 = 1198721 = 3 1198732 =1198722 = 4 and 119873 = 119872 = 7 On the other hand we haveto set 11987310158401 = 11987210158401 = 3 11987310158402 = 11987210158402 = 5 and 1198731015840 = 1198721015840 =10 to reach the DOFs goal by use of the coprime-STAPObviously the O2LN-STAP can achieve the DOFs aim withfewer physical array elements and transmitting pulses thanthe coprime-STAP In addition as is reported in the literature


0 01 02 03 04 05

0

10

20


SIN

R (d

B)

Proposed O2LN-STAPCoprime-STAP

minus10

minus20

minus30


Figure9Optimumoutput SINRperformances versus target normalizedDoppler frequencywithN=6 array elements andM=6 transmittingpulses

205

21

215

22

225

23

235

24

245


Out

put S

INR

(dB)

Objective SINR

Proposed O2LN-STAP(N=M=7)

102 103 104

Coprime-STAP(=－=10)

Figure 10 Output SINR performances versus the number of training samples

[30 32] the coprime-STAP method is also derived from thetraditional CNCM R119906 Hence there also exists SINR loss dueto the estimation error when the true traditional CNCM isreplaced by its estimation in practice Thus based on theCNCM estimation we still further analyze the output SINRperformance versus the number of training samples for theO2LN-STAP and the coprime-STAP Considering that thePC technique is more suitable for the sample-starved envi-ronment than the SMI algorithm as aforementioned beforewe only use the PC technique to implement the practicalSTAP filter for both O2LN-STAP and coprime-STAP in thecomparative simulation in this subsection The results are

presented in Figure 10 The objective curve in the figureis indeed the theoretical upper bound of the output SINRcorresponding to the V-ULA-UPRI-16x16 STAP system Wecan see that the O2LN-STAP and the coprime-STAP canconverge to the objective SINR level with different number oftraining samples Concretely the O2LN-STAP exhibits fasterconvergence property and higher SINR in the case of a smallnumber of training samples than the coprime-STAP This isbecause that the traditional CNCM for the O2LN-STAP hassmaller size than that for the coprime-STAP in the conditionsof 119873 = 119872 = 7 and 1198731015840 = 1198721015840 = 10 which makes therequirement of training samples for estimating the small-size


CNCM less than that for estimating the large-size CNCMaccording to the RMB rule Therefore in contrast with thecoprime-STAP the O2LN-STAP is more potential for clut-ter suppression with fewer training samples physical arrayelements and pulses in the sample-limited heterogeneousenvironment

Finally note that the key point of our research is focusedon how to enhance the DOFs based on the nested space-timestructure and illustrating the advantages of this structureAll the comparative simulation results are carried out bycombining different space-time structures with commonSTAP procedure including CCM estimation and angle-Doppler filter Then the computational complexity is mainlyembedded in the STAP processing related toCCMestimationor angle-Doppler filter but neither in the nested space-timestructure nor in other contrastive structures Hence by ana-lyzing the computational complexity of the algorithms withrespect to CCM estimation and angle-Doppler filter one canfurther discuss the computational complexity of the STAPmethods concerning the different space-time structures usedherein

5 Conclusion

In this paper we have considered the issue of STAP withnested array and nested PRI for clutter suppression on theairbornespaceborne MTI radar and proposed an O2LN-STAP strategy The O2LN-STAP is capable of providingenhanced available DOFs in the difference coarray andcopulses domains to further improve the STAP performanceSimulation results have validated the theoretical derivationsof the proposed O2LN-STAP and revealed the following (1)Under the constraint of the same number of physical arrayelements and transmitting pulses the O2LN-STAP can offerhigher available DOFs than the traditional uniform-STAPand the coprime-STAP which makes an improved angle-Doppler resolution lower side-lobe levels higher SINRand better MDV performance (2) For achieving a specifiednumber of DOFs to improve the STAP performance theO2LN-STAP needs fewer number of physical array elementsand transmitting pulses than the traditional uniform-STAPand the coprime-STAP which reduces the hardware com-plexity and power consumption (3) The O2LN-STAP ismore potential for clutter suppression with fewer trainingsamples physical array elements and pulses in the sample-limited heterogeneous environment than the coprime-STAPIn addition in contrast with the coprime-STAP the enhancedavailable DOFs and the improved performance of the O2LN-STAP are obtained at the sacrifice of array mutual couplingperformance Hence the issue for reducing the mutualcoupling while holding the DOFs advantage of the nestedclass of STAP method might be an interesting research workin the future

Appendix

Proof of the Proposition 3 Firstly fix the total number ofphysical array elements and transmitting pulses to N and

M respectively N and M are both positive integers For theO2LN-STAP we have

119873 = 1198731 + 1198732119872 = 1198721 +1198722 (A1)

Using the inequality of arithmetic mean and geometric mean(AM-GM inequality) we can get that

1198732 (1198731 + 1) le 11987322 + (1198731 + 1)22 (A2)

with equality if (and only if)1198732 = (1198731 + 1) Combining (A1)and the constraint that 1198731 and 1198732 should be both positiveintegers we can further obtain the maximum value of theterm1198732(1198731 + 1) which can be written as

for odd 119873 1198601 = max 1198732 (1198731 + 1) = (119873 + 1)24 (A3a)

for even 119873 1198611 = max 1198732 (1198731 + 1) = 11987324 + 119873

2 (A3b)

where1198731 and1198732 are computed by

for odd 1198731198731 = 119873 minus 1

2 1198732 = 119873 + 1

2 (A4a)

for even 119873 1198731 = 1198732 = 1198732 (A4b)

Following a similar deducing process we can also obtain themaximum value of the term1198722(1198721 + 1) given by

for odd 119872 1198602 = max 1198722 (1198721 + 1)= (119872 + 1)2

4 (A5a)

for even 119872 1198612 = max 1198722 (1198721 + 1)= 1198722

4 + 1198722 (A5b)

where1198721 and1198722 are given by

for odd 1198721198721 = 119872 minus 1

2 1198722 = 119872 + 1

2 (A6a)

for even 119872 1198721 = 1198722 = 1198722 (A6b)

On the other hand for the coprime-STAP we have

119873 = 1198731015840 = 211987310158401 + 11987310158402 minus 1119872 = 1198721015840 = 211987210158401 +11987210158402 minus 1

(A7)


0 5 10 15 200

20

40

60

80

100

120

Values of N

Func

tion

valu

es

A1B1C1

(a)

A1B1C1

1 2 3 4 50

1

2

3

4

5

6

7

8

9

Values of N

Func

tion

valu

es(b)

Figure 11 Three different function values versus N (a) Original curves (b) Close-up of the original curves

Still using the AM-GM inequality we can get that

1198731015840111987310158402 + 1 le 119873101584021 + 1198731015840222 + 1 (A8)

with equality if (and only if) 11987310158401 = 11987310158402 Now if we donot consider the constraint that 11987310158401 and 11987310158402 should be bothintegers we can get themaximumvalue of1198731015840111987310158402+1 accordingto (A7) and (A8) which is expressed as

1198621 = max 1198731015840111987310158402 + 1 = (119873 + 1)29 + 1 (A9)


11987310158401 = 11987310158402 = 119873 + 13 (A10)

Similarly we can get themaximumvalues of1198721015840111987210158402+1 whichis written by

1198622 = max 1198721015840111987210158402 + 1 = (119872 + 1)29 + 1 (A11)


11987210158401 = 11987210158402 = 119872 + 13 (A12)

Note that 1198601 1198611 and 1198621 can be seen as three differentfunction values with respect to the variable N And theirfunction values versus the variable N are presented inFigure 11 Figure 11(b) is the close-up of Figure 11(a) It is clearthat the values of1198601 and 1198611 are both larger than 1198621 when 2 lt

119873 le 20 and 1198611 is equivalent to1198621 when119873 = 2 Additionallyit is noted that the second-order derivatives of 1198601 1198611 and1198621 with respect to N are 12 12 and 29 respectively That isto say since the function values of 1198601 and 1198611 increase morerapidly with the increasing ofN than that of 1198621 the curve 1198621would no longer intersect the curves 1198601 and 1198611 after N = 20It means that the values of 1198601 and 1198611 are always larger thanthat of1198621 when119873 gt 2 which is not shown in Figure 11 due tothe space limitation Similarly we can also say that the valuesof 1198602 and 1198612 are always larger than that of 1198622 when119872 gt 2

Furthermore we rewrite the expression of 1198631199001199011199052minus119899119890119904119905119890119889

and119863119888119900119901119903119894119898119890 as follows

1198631199001199011199052minus119899119890119904119905119890119889

=

11986011198602 119873 119900119889119889 119872 119900119889119889 (119886)11986111198612 119873 119890V119890119899 119872 119890V119890119899 (119887)11986011198612 119873 119900119889119889 119872 119890V119890119899 (119888)11986111198602 119873 119890V119890119899 119872 119900119889119889 (119889)

(A13)

119863119888119900119901119903119894119898119890 lt 11986211198622 (A14)

Note that if we consider the constraint that 11987310158401 (or 11987210158401) and11987310158402 (or11987210158402) are coprime integers and11987310158401 lt 11987310158402 (or11987210158401 lt 11987210158402)the conditions of (A10) and (A12) cannot be satisfied and themaximum values of 1198621 and 1198622 cannot be even reached toowhich results in expression (A14)Therefore considering theaforementioned analysis results that 1198601 gt 1198621 1198611 gt 1198621 when119873 gt 2 and1198602 gt 1198622 1198612 gt 1198622 when119872 gt 2 we can deduce theconclusion that 119863119900119901119905

2minus119899119890119904119905119890119889gt 119863119888119900119901119903119894119898119890 when119873 gt 2 and119872 gt 2

according to (A13) and (A14) Thus Proposition 3 has beenproved


Data Availability

The datasets supporting the conclusions of this article aregenerated using MATLAB software by authors according tothe radar parameters described in the manuscript

Conflicts of Interest

The authors declare that they have no competing interests

Acknowledgments

The authors express their gratitude to the editor and anony-mous reviewers for their helpful comments and suggestionsthat improved this paper

References

[1] W LMelvin ldquoA STAP overviewrdquo IEEEAerospace and ElectronicSystems Magazine vol 19 no 1 pp 19ndash35 2004

[2] R Klemm Principles of Space-Time Adaptive Processing TheInstitution of Engineering and Technology London UK 2006

[3] J R Guerci Space-Time Adaptive Processing for Radar ArtechHouse Boston Mass USA 2003

[4] C Chen and P P Vaidyanathan ldquoMIMO radar space-timeadaptive processing using prolate spheroidal wave functionsrdquoIEEE Transactions on Signal Processing vol 56 no 2 pp 623ndash635 2008

[5] Z Liu X Wei and X Li ldquoAliasing-free moving target detectionin random pulse repetition interval radar based on compressedsensingrdquo IEEE Sensors Journal vol 13 no 7 pp 2523ndash25342013

[6] J Ward ldquoSpace-time adaptive processing with sparse antennaarraysrdquo in Proceedings of the 1998 32nd Asilomar Conferenceon Signals Systems amp Computers Part 1 (of 2) pp 1537ndash1541November 1998

[7] Z Yang G Quan Y Ma and J Huang ldquoCompressive space-time adaptive processing airborne radarwith randompulse rep-etition interval and random arraysrdquo in Proceedings of the 20164th International Workshop on Compressed Sensing Theory andits Applications to Radar Sonar and Remote Sensing (CoSeRa)pp 247ndash251 Aachen Germany September 2016

[8] D Cohen A Dikopoltsev and Y C Eldar ldquoExtensions of sub-Nyquist radar reduced time-on-target and cognitive radarrdquo inProceedings of the 3rd Int Workshop CoSeRa vol 62 pp 31ndash352015

[9] A Sarajedini ldquoAdaptive array thinning for space-Time beam-formingrdquo in Proceedings of the 33rd Asilomar Conference onSignals Systems and Computers ACSSC 1999 pp 1572ndash1576USA October 1999

[10] A Moffet ldquoMinimum-redundancy linear arraysrdquo IEEE Trans-actions on Antennas and Propagation vol 16 no 2 pp 172ndash1751968

[11] S U Pillai Y B Ness and F Haber ldquoA new approach toarray geometry for improved spatial spectrum estimationrdquoProceedings of the IEEE vol 73 no 10 pp 1522ndash1524 1985

[12] S U Pillai and F Haber ldquoStatistical analysis of a high resolutionspatial spectrum estimator utilizing an augmented covariancematrixrdquo Institute of Electrical and Electronics Engineers Transac-tions on Acoustics Speech and Signal Processing vol 35 no 11pp 1517ndash1523 1987

[13] Y I Abramovich D A Gray A Gorokhov and N K SpencerldquoPositive-definite Toeplitz completion in DOA estimation fornonuniform linear antenna arrays I Fully augmentable arraysrdquoIEEE Transactions on Signal Processing vol 46 no 9 pp 2458ndash2471 1998

[14] Y I Abramovich N K Spencer and A Y Gorokhov ldquoPositive-definite Toeplitz completion in DOA estimation for nonuni-form linear antenna arrays II Partially augmentable arraysrdquoIEEE Transactions on Signal Processing vol 47 no 6 pp 1502ndash1521 1999

[15] D H Johnson and D E Dudgeon Array Signal Processing -Concepts and Techniques Prentice-Hall Englewood Cliffs NJUSA 1993

[16] C S Ruf ldquoNumerical annealing of low-redundancy lineararraysrdquo IEEE Transactions on Antennas and Propagation vol 41no 1 pp 85ndash90 1993

[17] P P Vaidyanathan and P Pal ldquoSparse sensing with co-primesamplers and arraysrdquo IEEE Transactions on Signal Processingvol 59 no 2 pp 573ndash586 2011

[18] P Pal and P P Vaidyanathan ldquoCoprime sampling and themusicalgorithmrdquo in Proceedings of the IEEE Digital Signal ProcessingWorkshop and IEEE Signal Processing Education Workshop(DSPSPE rsquo11) pp 289ndash294 IEEE Sedona Ariz USA January2011

[19] S Qin Y D Zhang and M G Amin ldquoGeneralized coprimearray configurations for direction-of-arrival estimationrdquo IEEETransactions on Signal Processing vol 63 no 6 pp 1377ndash13902015

[20] G Di Martino and A Iodice ldquoPassive beamforming withcoprime arraysrdquo IET Radar Sonar amp Navigation vol 11 no 6pp 964ndash971 2017

[21] E BouDaher Y Jia F Ahmad and M G Amin ldquoMulti-frequency co-prime arrays for high-resolution direction-of-arrival estimationrdquo IEEE Transactions on Signal Processing vol63 no 14 pp 3797ndash3808 2015

[22] P Pal and P P Vaidyanathan ldquoNested arrays a novel approachto array processing with enhanced degrees of freedomrdquo IEEETransactions on Signal Processing vol 58 no 8 pp 4167ndash41812010

[23] P Pal and P P Vaidyanathan ldquoNested arrays in two dimensionsPart I Geometrical considerationsrdquo IEEETransactions on SignalProcessing vol 60 no 9 pp 4694ndash4705 2012

[24] A Ahmed Y D Zhang and B Himed ldquoEffective nested arraydesign for fourth-order cumulant-based DOA estimationrdquo inProceedings of the 2017 IEEE Radar Conference RadarConf 2017pp 0998ndash1002 USA May 2017

[25] K Han andANehorai ldquoNested array processing for distributedsourcesrdquo IEEE Signal Processing Letters vol 21 no 9 pp 1111ndash1114 2014

[26] K Han and A Nehorai ldquoWideband gaussian source processingusing a linear nested arrayrdquo IEEE Signal Processing Letters vol20 no 11 pp 1110ndash1113 2013

[27] W Zheng X Zhang and J Shi ldquoSparse extension arraygeometry for DOA estimation with nested MIMO radarrdquo IEEEAccess vol 5 pp 9580ndash9586 2017

[28] X Lan Y Li and E Wang ldquoA rare algorithm for 2D DOAestimation based on nested array in massive MIMO systemrdquoIEEE Access vol 4 pp 3806ndash3814 2016

[29] R Li Y Wang Z He J Li and G Sun ldquoMinimum redundancyspace-time adaptive processing utilizing reconstructed covari-ance matrixrdquo in Proceedings of the 2017 IEEE Radar ConferenceRadarConf 2017 pp 0722ndash0726 USA May 2017


[30] C-L Liu and P P Vaidyanathan ldquoCoprime arrays and samplersfor space-time adaptive processingrdquo in Proceedings of the 40thIEEE International Conference on Acoustics Speech and SignalProcessing ICASSP 2015 pp 2364ndash2368 Australia April 2014

[31] Y Ma Z Yang J Huang J Huang and L Kang ldquoSpace-time adaptive processing airborne radar with coprime pulserepetition intervalrdquo in Proceedings of the 2016 CIE InternationalConference on Radar RADAR 2016 pp 1ndash4 China October2016

[32] X Wang Z Yang H Huang J Huang and M Jiang ldquoSpace-time adaptive processing for airborne radars with space-timecoprime sampling structurerdquo IEEE Access vol 6 pp 20031ndash20046 2018

[33] P Vouras ldquoFully Adaptive Space-Time Processing on nestedarraysrdquo in Proceedings of the 2015 IEEE International RadarConference (RadarCon) pp 0858ndash0863 Arlington VA USAMay 2015

[34] P Pal and P P Vaidyanathan ldquoNested arrays in two dimensionspart II Application in two dimensional array processingrdquo IEEETransactions on Signal Processing vol 60 no 9 pp 4706ndash47182012

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by randomly spaced sparse array and PRI [6 9] whichdeteriorates the STAP performance of clutter suppression

It is noted that the aforementioned approaches are allbased on the physical nonuniform sparse antenna arraysActually the virtual difference coarray concept with DOFssuperiority has also been introduced into the nonuniformarray signal processing domain In the past years thisconcept which is able to reduce the average angle-Dopplerside-lobe levels of the nonuniform sparse array without theusage of SR technique has been applied to the direction-of-arrival (DOA) estimation and the spatial beamformingAccording to the array geometry we classify those methodsfor exploiting the enhanced DOFs by use of the differencecoarray perspective into three categories The first one isbased on the minimum redundancy array (MRA) [10] Itis shown that the DOFs of the MRA can be improved byconstructing an augmented covariance matrix which is notpositive semidefinite for finite number of snapshots [11 12]And in [13 14] a transformation of this augmented matrixinto a suitable positive definite Toeplitz matrix is suggestedand an elaborate algorithm is presented to construct thisToeplitz matrix Nevertheless this class of approaches hastwo issues One is that for MRA there are no closed-formexpressions for the array geometry and the achievable DOFsfor a given number of sensors and the optimum design ofsuch arrays is always complicated [15 16] Another issue isthat the algorithm for searching an appropriate augmentedcovariance matrix for a large array is a complex iterativeprocedure which converges only to a local optimum [13 14]Moreover the second class of methods about the differencecoarray is related to the recently developed coprime arrayswhich have received much attention since they can be formu-lated more easily than the MRAwhile resolving more sourcesthan the number of sensors [17 18] In [19] performancesof a generalized coprime array have been evaluated by useof their difference coarray equivalence and this generalizedcoprime array has been applied to DOA estimation withboth subspace-basedmethod and sparse reconstruction tech-nique A new approach for passive beamforming and targetdetection using coprime array is also presented in [20] whichcan get the developments of peak side-lobe ratio and the inte-grated side-lobe ratio without giving rise to ldquoghostrdquo targets inthe presence of multiple interferers Another multifrequencyoperation with coprime array is presented in [21] for increas-ing the number of resolvable sources in high-resolution DOAestimation The authors employ multiple frequencies to fillthe missing coarray elements thereby enabling the coprimearray to utilize all of the offered DOFs effectively Finally thetechniques for further exploiting the increased DOFs usingnested arrays [22 23] can be regarded as the third class ofdifference coarray methods An effective nested array designscheme can yield a fourth-order difference coarray to providea higher number of DOFs for DOA estimation [24] Anotherapplication for DOA estimation of the distributed sourceswith nested arrays has been studied in [25] To use the nestedarray for wideband case authors in [26] have proposed aneffective strategy to apply the processing algorithm of nestedarray to each frequency component and the whole spectralinformation of various frequencies has been cooperated to

conduct the detection and estimation Nested array has alsobeen extended to MIMO radar for DOA estimation [27 28]

In recent years the difference coarray perspective hasbeen considered as an effective approach to increase thesystem DOFs for STAP applications To acquire more DOFsauthors in [29] have arranged the array geometry and thetemporal sample interval to make the location of jointspace-time samples satisfy minimum redundancy intervalHowever this STAP algorithm is limited to a time-consumingprocedure of computer searching Moreover a new STAPstrategy for the airborne radar with space-time coprime sam-pling structure (which is referred to as coprime-STAP in thefollowing) has been discussed in [30ndash32] It is demonstratedthat using few array elements and transmitting pulses thecoprime-STAPmethod can achieve comparable performancewith respect to the case of large ULA and great many pulseswith uniform PRI Additionally a fully STAP with nestedarray is also presented in [33] where a high resolution inangle-Doppler plane is obtained However to complete thisfull STAP the spatial and Doppler frequencies of all jammingand clutter sources need to be priorly known

Inspired by the nesting concept in the spatial dimen-sion we think that using PRI with nested time intervalmight also obtain an increase of DOFs in the temporal(or Doppler) dimension Therefore following the differencecoarray perspective we have applied an optimum two-levelnested STAP (O2LN-STAP) strategy to improve the STAPperformance in this paper Different from the traditionaluniform-STAP method that calculates the clutter plus noisecovariance matrix (CNCM) and performs the STAP filterdirect using the received data collected from the physicalULA with uniform PRI the STAP strategy shown in thispaper presents a novel processing idea with an optimumtwo-level nested array (O2LNA) and an optimum two-level nested PRI (O2LN-PRI) In our proposed methodwe first construct a virtual space-time snapshot from thedirect covariance matrix of the received signals The virtualsnapshot can be seen as the collection of the returned signalsfrom a virtual large ULA and a large number of transmittingpulses with uniform PRI Then using the spatial-temporalsmoothing approach [22] a new CNCM estimation of thevirtual space-time snapshot can be computed for STAP filterRemarkably the existing coprime-STAP method which isalso considered in the virtual difference coarray domaincan get more DOFs than the traditional uniform-STAP thatis implemented on the physical array and pulses Howeversimulation results and comparative analyses in the followinghave indicated that the O2LN-STAP method can obtainmore DOFs than both the traditional uniform-STAP and thecoprime-STAP under the constraint of the same number ofphysical array elements and transmitting pulses whichmakesa higher angle-Doppler resolution lower side-lobe levelslarger signal-to-interference-plus-noise ratio (SINR) andbetter minimum detectable velocity (MDV) performancesOn the other hand for a specified number of DOFs and SINRlevel the O2LN-STAP method can reach them with fewerphysical array elements and transmitting pulses (ie lesshardware complexity and power consumption) in contrastwith the traditional uniform-STAP and the coprime-STAP


0 1d 2d 3d 7d 11d

Level 1 Level 2

(a)-10d -8d -6d -4d -2d 0 2d 4d 6d 8d 10d

(b)






119878119894119899 = 119899119894119899119889119894119899 119899119894119899 = 0 1 1198731 minus 1 (1a)

and

119878119900119906119905 = 119899119900119906119905 (1198731 + 1) 119889119894119899 minus 119889119894119899 119899119900119906119905 = 1 1198732 (1b)




119863 = p119894 minus p1198941015840 119894 = 1 2 119873 1198941015840 = 1 2 119873 (2)


119878119888119900 = 119899119888119900119889119894119899 119899119888119900 = minus119873119888119900 119873119888119900 119873119888119900 = 1198732 (1198731 + 1)minus 1 (3)







0 1T 2T 3T 7T 11T

Level 1 Level 2

(a)-10T -8T -6T -4T -2T 0 2T 4T 6T 8T 10T

(b)



119876119894119899 = 119898119894119899119879119894119899 119898119894119899 = 0 1 1198721 minus 1 (4a)

and

119876119900119906119905 = 119898119900119906119905 (1198721 + 1)119879119894119899 minus 119879119894119899 119898119900119906119905 = 1 1198722 (4b)



119866 = q119895 minus q1198951015840 119895 = 1 2 119872 1198951015840 = 1 2 119872 (5)


119876119888119900 = 119898119888119900119879119894119899 119898119888119900 = minus119872119888119900 119872119888119900 119872119888119900= 1198722 (1198721 + 1) minus 1 (6)







a (119891119904) = [1 11989011989521205871198911199041198991 1198901198952120587119891119904119899119873minus1]119879 (7)


119891119904 = 119889 sin (120579)120582 (8)


b (119891119889) = [1 11989011989521205871198911198891198981 1198901198952120587119891119889119898119872minus1]119879 (9)


119891119889 = 2V sin (120579) 119879120582 = 120573119891119904 (10)




k (119891119904 119891119889) = b (119891119889) otimes a (119891119904) (11)



x119888 = c + n = 119873119888sum119894=1

120572119894b (119891119894119889119888) otimes a (119891119894119904119888) + n = 119873119888sum119894=1

120572119894k119888119894 + n

= 119873119888sum119894=1

c119894 + n

(12)


R119888 =119873119888sum119894=1

119864 [c119894c119867119894 ] =119873119888sum119894=1

1205902119894 k119888119894k119867119888119894 = V119888PV119888119867 (13)


R119899 = 1205902119899I119872119873 (14)


R119906 = 119864 [x119888x119867119888 ] = V119888PV119888119867 + 1205902119899I119872119873 = R119888 + R119899 (15)




(16)


119878119868119873119877 = 1205902119905 k119867119905 Rminus1119906 k119905 (17)



R119906 = 1119870X119888X

119867119888 (18)




(19)


[a (119891119904) a (119891119904)119867]119901119902 = 1198901198952120587119891119904(119899119901minus119899119902) = 1198901198952120587119891119904119899119901119902 (20)











Z = 119873119888sum119894=1

1205902119894 a (119891119894119904119888) b (119891119894119889119888)119879 + 1205902119899e1e1198792 (24)


z = 119873119888sum119894=1


1205902119894 k119888119894 + n

= c + n

(25)




0

0

helliphellip

helliphellip


Z

Z

Z00

(N minus 1)

minus(N minus 1)




Z120588120574 =119873119888sum119894=1

119890minus119895(120588119891119894119904119888+120574119891119894119889119888)1205902119894 a (119891119894119904119888) b (119891119894119889119888)119879 + 1205902119899e1120588e1198792120574 (26)



R119904 = 1119872119873119873minus1sum120588=0

119872minus1sum120574=0

z120588120574z119867120588120574 (27)


R119904 = 1119872119873R2119906 (28)

where

R119906 = V119888PV119867

119888 + 1205902119899I119872119873 (29)




R119906 = (119872119873R119904)12 (30)




minus1

119906 k119905(31)


119881119878119868119873119877 = 1205902119905 k119867119905 Rminus1119906 k119905 (32)


1198632minus119899119890119904119905119890119889 = 119872119873 = 1198722 (1198721 + 1) times 1198732 (1198731 + 1) (33)


1198631199001199011199052minus119899119890119904119905119890119889

=

(119873 + 1)42 times (119872 + 1)

42 119873 119900119889119889 119872 119900119889119889 (119886)

(11987324 + 1198732 ) times (11987224 + 119872

2 ) 119873 119890V119890119899 119872 119890V119890119899 (119887)(119873 + 1)

42 times (11987224 + 119872

2 ) 119873 119900119889119889 119872 119890V119890119899 (119888)(11987324 + 119873

2 ) times (119872 + 1)42 119873 119890V119890119899 119872 119900119889119889 (119889)

(34)






20

40

60

80

100

120

00 5 10 15 20 25 30

Eige

nval

ues (

dB)

Eigenvalue index

R

Rs

Ru












0

minus60

minus50

minus40

minus30

minus20

minus10N

orm

aliz

ed D

oppl

er fr

eque

ncy

00102030405

minus05

minus04

minus03

minus02

minus01

(a)


050minus05

0

minus60

minus50

minus40

minus30

minus20

minus10

Nor

mal

ized

Dop

pler

freq

uenc

y

00102030405

minus05

minus04

minus03

minus02

minus01

(b)


050minus05

0

minus60

minus50

minus40

minus30

minus20

minus10

Nor

mal

ized

Dop

pler

freq

uenc

y

00102030405

minus05

minus04

minus03

minus02

minus01

(c)


0 01 02 03 04 05

0

10

20


SIN

R (d

B)


minus10

minus20

minus30








13141516171819202122


Out

put S

INR

(dB)


102 103 104




Nor

mal

ized

Dop

pler

freq

uenc

y

Spatial frequency

0

0

0102030405

050

minus05

minus05minus60

minus50

minus40

minus30

minus20

minus10minus04

minus03

minus02

minus01

(a)


Nor

mal

ized

Dop

pler

freq

uenc

y

00102030405

minus05

minus04

minus03

minus02

minus01

0

minus60

minus50

minus40

minus30

minus20

minus10

(b)




119863119888119900119901119903119894119898119890 = (1198731015840111987310158402 + 1) times (1198721015840111987210158402 + 1) (35)










0 01 02 03 04 05

0

10

20


SIN

R (d

B)


minus10

minus20

minus30



205

21

215

22

225

23

235

24

245


Out

put S

INR

(dB)

Objective SINR


102 103 104








5 Conclusion


Appendix



119873 = 1198731 + 1198732119872 = 1198721 +1198722 (A1)


1198732 (1198731 + 1) le 11987322 + (1198731 + 1)22 (A2)


for odd 119873 1198601 = max 1198732 (1198731 + 1) = (119873 + 1)24 (A3a)

for even 119873 1198611 = max 1198732 (1198731 + 1) = 11987324 + 119873

2 (A3b)


for odd 1198731198731 = 119873 minus 1

2 1198732 = 119873 + 1

2 (A4a)

for even 119873 1198731 = 1198732 = 1198732 (A4b)


for odd 119872 1198602 = max 1198722 (1198721 + 1)= (119872 + 1)2

4 (A5a)

for even 119872 1198612 = max 1198722 (1198721 + 1)= 1198722

4 + 1198722 (A5b)


for odd 1198721198721 = 119872 minus 1

2 1198722 = 119872 + 1

2 (A6a)

for even 119872 1198721 = 1198722 = 1198722 (A6b)


119873 = 1198731015840 = 211987310158401 + 11987310158402 minus 1119872 = 1198721015840 = 211987210158401 +11987210158402 minus 1

(A7)


0 5 10 15 200

20

40

60

80

100

120

Values of N

Func

tion

valu

es

A1B1C1

(a)

A1B1C1

1 2 3 4 50

1

2

3

4

5

6

7

8

9

Values of N

Func

tion

valu

es(b)



1198731015840111987310158402 + 1 le 119873101584021 + 1198731015840222 + 1 (A8)


1198621 = max 1198731015840111987310158402 + 1 = (119873 + 1)29 + 1 (A9)


11987310158401 = 11987310158402 = 119873 + 13 (A10)


1198622 = max 1198721015840111987210158402 + 1 = (119872 + 1)29 + 1 (A11)


11987210158401 = 11987210158402 = 119872 + 13 (A12)




and119863119888119900119901119903119894119898119890 as follows

1198631199001199011199052minus119899119890119904119905119890119889

=

11986011198602 119873 119900119889119889 119872 119900119889119889 (119886)11986111198612 119873 119890V119890119899 119872 119890V119890119899 (119887)11986011198612 119873 119900119889119889 119872 119890V119890119899 (119888)11986111198602 119873 119890V119890119899 119872 119900119889119889 (119889)

(A13)

119863119888119900119901119903119894119898119890 lt 11986211198622 (A14)





Data Availability




Acknowledgments


References











































Journal of












Journal of







Volume 2018






Volume 2018









0 1d 2d 3d 7d 11d

Level 1 Level 2

(a)-10d -8d -6d -4d -2d 0 2d 4d 6d 8d 10d

(b)






119878119894119899 = 119899119894119899119889119894119899 119899119894119899 = 0 1 1198731 minus 1 (1a)

and

119878119900119906119905 = 119899119900119906119905 (1198731 + 1) 119889119894119899 minus 119889119894119899 119899119900119906119905 = 1 1198732 (1b)




119863 = p119894 minus p1198941015840 119894 = 1 2 119873 1198941015840 = 1 2 119873 (2)


119878119888119900 = 119899119888119900119889119894119899 119899119888119900 = minus119873119888119900 119873119888119900 119873119888119900 = 1198732 (1198731 + 1)minus 1 (3)







0 1T 2T 3T 7T 11T

Level 1 Level 2

(a)-10T -8T -6T -4T -2T 0 2T 4T 6T 8T 10T

(b)



119876119894119899 = 119898119894119899119879119894119899 119898119894119899 = 0 1 1198721 minus 1 (4a)

and

119876119900119906119905 = 119898119900119906119905 (1198721 + 1)119879119894119899 minus 119879119894119899 119898119900119906119905 = 1 1198722 (4b)



119866 = q119895 minus q1198951015840 119895 = 1 2 119872 1198951015840 = 1 2 119872 (5)


119876119888119900 = 119898119888119900119879119894119899 119898119888119900 = minus119872119888119900 119872119888119900 119872119888119900= 1198722 (1198721 + 1) minus 1 (6)







a (119891119904) = [1 11989011989521205871198911199041198991 1198901198952120587119891119904119899119873minus1]119879 (7)


119891119904 = 119889 sin (120579)120582 (8)


b (119891119889) = [1 11989011989521205871198911198891198981 1198901198952120587119891119889119898119872minus1]119879 (9)


119891119889 = 2V sin (120579) 119879120582 = 120573119891119904 (10)




k (119891119904 119891119889) = b (119891119889) otimes a (119891119904) (11)



x119888 = c + n = 119873119888sum119894=1

120572119894b (119891119894119889119888) otimes a (119891119894119904119888) + n = 119873119888sum119894=1

120572119894k119888119894 + n

= 119873119888sum119894=1

c119894 + n

(12)


R119888 =119873119888sum119894=1

119864 [c119894c119867119894 ] =119873119888sum119894=1

1205902119894 k119888119894k119867119888119894 = V119888PV119888119867 (13)


R119899 = 1205902119899I119872119873 (14)


R119906 = 119864 [x119888x119867119888 ] = V119888PV119888119867 + 1205902119899I119872119873 = R119888 + R119899 (15)




(16)


119878119868119873119877 = 1205902119905 k119867119905 Rminus1119906 k119905 (17)



R119906 = 1119870X119888X

119867119888 (18)




(19)


[a (119891119904) a (119891119904)119867]119901119902 = 1198901198952120587119891119904(119899119901minus119899119902) = 1198901198952120587119891119904119899119901119902 (20)











Z = 119873119888sum119894=1

1205902119894 a (119891119894119904119888) b (119891119894119889119888)119879 + 1205902119899e1e1198792 (24)


z = 119873119888sum119894=1


1205902119894 k119888119894 + n

= c + n

(25)




0

0

helliphellip

helliphellip


Z

Z

Z00

(N minus 1)

minus(N minus 1)




Z120588120574 =119873119888sum119894=1

119890minus119895(120588119891119894119904119888+120574119891119894119889119888)1205902119894 a (119891119894119904119888) b (119891119894119889119888)119879 + 1205902119899e1120588e1198792120574 (26)



R119904 = 1119872119873119873minus1sum120588=0

119872minus1sum120574=0

z120588120574z119867120588120574 (27)


R119904 = 1119872119873R2119906 (28)

where

R119906 = V119888PV119867

119888 + 1205902119899I119872119873 (29)




R119906 = (119872119873R119904)12 (30)




minus1

119906 k119905(31)


119881119878119868119873119877 = 1205902119905 k119867119905 Rminus1119906 k119905 (32)


1198632minus119899119890119904119905119890119889 = 119872119873 = 1198722 (1198721 + 1) times 1198732 (1198731 + 1) (33)


1198631199001199011199052minus119899119890119904119905119890119889

=

(119873 + 1)42 times (119872 + 1)

42 119873 119900119889119889 119872 119900119889119889 (119886)

(11987324 + 1198732 ) times (11987224 + 119872

2 ) 119873 119890V119890119899 119872 119890V119890119899 (119887)(119873 + 1)

42 times (11987224 + 119872

2 ) 119873 119900119889119889 119872 119890V119890119899 (119888)(11987324 + 119873

2 ) times (119872 + 1)42 119873 119890V119890119899 119872 119900119889119889 (119889)

(34)






20

40

60

80

100

120

00 5 10 15 20 25 30

Eige

nval

ues (

dB)

Eigenvalue index

R

Rs

Ru












0

minus60

minus50

minus40

minus30

minus20

minus10N

orm

aliz

ed D

oppl

er fr

eque

ncy

00102030405

minus05

minus04

minus03

minus02

minus01

(a)


050minus05

0

minus60

minus50

minus40

minus30

minus20

minus10

Nor

mal

ized

Dop

pler

freq

uenc

y

00102030405

minus05

minus04

minus03

minus02

minus01

(b)


050minus05

0

minus60

minus50

minus40

minus30

minus20

minus10

Nor

mal

ized

Dop

pler

freq

uenc

y

00102030405

minus05

minus04

minus03

minus02

minus01

(c)


0 01 02 03 04 05

0

10

20


SIN

R (d

B)


minus10

minus20

minus30








13141516171819202122


Out

put S

INR

(dB)


102 103 104




Nor

mal

ized

Dop

pler

freq

uenc

y

Spatial frequency

0

0

0102030405

050

minus05

minus05minus60

minus50

minus40

minus30

minus20

minus10minus04

minus03

minus02

minus01

(a)


Nor

mal

ized

Dop

pler

freq

uenc

y

00102030405

minus05

minus04

minus03

minus02

minus01

0

minus60

minus50

minus40

minus30

minus20

minus10

(b)




119863119888119900119901119903119894119898119890 = (1198731015840111987310158402 + 1) times (1198721015840111987210158402 + 1) (35)










0 01 02 03 04 05

0

10

20


SIN

R (d

B)


minus10

minus20

minus30



205

21

215

22

225

23

235

24

245


Out

put S

INR

(dB)

Objective SINR


102 103 104








5 Conclusion


Appendix



119873 = 1198731 + 1198732119872 = 1198721 +1198722 (A1)


1198732 (1198731 + 1) le 11987322 + (1198731 + 1)22 (A2)


for odd 119873 1198601 = max 1198732 (1198731 + 1) = (119873 + 1)24 (A3a)

for even 119873 1198611 = max 1198732 (1198731 + 1) = 11987324 + 119873

2 (A3b)


for odd 1198731198731 = 119873 minus 1

2 1198732 = 119873 + 1

2 (A4a)

for even 119873 1198731 = 1198732 = 1198732 (A4b)


for odd 119872 1198602 = max 1198722 (1198721 + 1)= (119872 + 1)2

4 (A5a)

for even 119872 1198612 = max 1198722 (1198721 + 1)= 1198722

4 + 1198722 (A5b)


for odd 1198721198721 = 119872 minus 1

2 1198722 = 119872 + 1

2 (A6a)

for even 119872 1198721 = 1198722 = 1198722 (A6b)


119873 = 1198731015840 = 211987310158401 + 11987310158402 minus 1119872 = 1198721015840 = 211987210158401 +11987210158402 minus 1

(A7)


0 5 10 15 200

20

40

60

80

100

120

Values of N

Func

tion

valu

es

A1B1C1

(a)

A1B1C1

1 2 3 4 50

1

2

3

4

5

6

7

8

9

Values of N

Func

tion

valu

es(b)



1198731015840111987310158402 + 1 le 119873101584021 + 1198731015840222 + 1 (A8)


1198621 = max 1198731015840111987310158402 + 1 = (119873 + 1)29 + 1 (A9)


11987310158401 = 11987310158402 = 119873 + 13 (A10)


1198622 = max 1198721015840111987210158402 + 1 = (119872 + 1)29 + 1 (A11)


11987210158401 = 11987210158402 = 119872 + 13 (A12)




and119863119888119900119901119903119894119898119890 as follows

1198631199001199011199052minus119899119890119904119905119890119889

=

11986011198602 119873 119900119889119889 119872 119900119889119889 (119886)11986111198612 119873 119890V119890119899 119872 119890V119890119899 (119887)11986011198612 119873 119900119889119889 119872 119890V119890119899 (119888)11986111198602 119873 119890V119890119899 119872 119900119889119889 (119889)

(A13)

119863119888119900119901119903119894119898119890 lt 11986211198622 (A14)





Data Availability




Acknowledgments


References











































Journal of












Journal of







Volume 2018






Volume 2018









0 1T 2T 3T 7T 11T

Level 1 Level 2

(a)-10T -8T -6T -4T -2T 0 2T 4T 6T 8T 10T

(b)



119876119894119899 = 119898119894119899119879119894119899 119898119894119899 = 0 1 1198721 minus 1 (4a)

and

119876119900119906119905 = 119898119900119906119905 (1198721 + 1)119879119894119899 minus 119879119894119899 119898119900119906119905 = 1 1198722 (4b)



119866 = q119895 minus q1198951015840 119895 = 1 2 119872 1198951015840 = 1 2 119872 (5)


119876119888119900 = 119898119888119900119879119894119899 119898119888119900 = minus119872119888119900 119872119888119900 119872119888119900= 1198722 (1198721 + 1) minus 1 (6)







a (119891119904) = [1 11989011989521205871198911199041198991 1198901198952120587119891119904119899119873minus1]119879 (7)


119891119904 = 119889 sin (120579)120582 (8)


b (119891119889) = [1 11989011989521205871198911198891198981 1198901198952120587119891119889119898119872minus1]119879 (9)


119891119889 = 2V sin (120579) 119879120582 = 120573119891119904 (10)




k (119891119904 119891119889) = b (119891119889) otimes a (119891119904) (11)



x119888 = c + n = 119873119888sum119894=1

120572119894b (119891119894119889119888) otimes a (119891119894119904119888) + n = 119873119888sum119894=1

120572119894k119888119894 + n

= 119873119888sum119894=1

c119894 + n

(12)


R119888 =119873119888sum119894=1

119864 [c119894c119867119894 ] =119873119888sum119894=1

1205902119894 k119888119894k119867119888119894 = V119888PV119888119867 (13)


R119899 = 1205902119899I119872119873 (14)


R119906 = 119864 [x119888x119867119888 ] = V119888PV119888119867 + 1205902119899I119872119873 = R119888 + R119899 (15)




(16)


119878119868119873119877 = 1205902119905 k119867119905 Rminus1119906 k119905 (17)



R119906 = 1119870X119888X

119867119888 (18)




(19)


[a (119891119904) a (119891119904)119867]119901119902 = 1198901198952120587119891119904(119899119901minus119899119902) = 1198901198952120587119891119904119899119901119902 (20)











Z = 119873119888sum119894=1

1205902119894 a (119891119894119904119888) b (119891119894119889119888)119879 + 1205902119899e1e1198792 (24)


z = 119873119888sum119894=1


1205902119894 k119888119894 + n

= c + n

(25)




0

0

helliphellip

helliphellip


Z

Z

Z00

(N minus 1)

minus(N minus 1)




Z120588120574 =119873119888sum119894=1

119890minus119895(120588119891119894119904119888+120574119891119894119889119888)1205902119894 a (119891119894119904119888) b (119891119894119889119888)119879 + 1205902119899e1120588e1198792120574 (26)



R119904 = 1119872119873119873minus1sum120588=0

119872minus1sum120574=0

z120588120574z119867120588120574 (27)


R119904 = 1119872119873R2119906 (28)

where

R119906 = V119888PV119867

119888 + 1205902119899I119872119873 (29)




R119906 = (119872119873R119904)12 (30)




minus1

119906 k119905(31)


119881119878119868119873119877 = 1205902119905 k119867119905 Rminus1119906 k119905 (32)


1198632minus119899119890119904119905119890119889 = 119872119873 = 1198722 (1198721 + 1) times 1198732 (1198731 + 1) (33)


1198631199001199011199052minus119899119890119904119905119890119889

=

(119873 + 1)42 times (119872 + 1)

42 119873 119900119889119889 119872 119900119889119889 (119886)

(11987324 + 1198732 ) times (11987224 + 119872

2 ) 119873 119890V119890119899 119872 119890V119890119899 (119887)(119873 + 1)

42 times (11987224 + 119872

2 ) 119873 119900119889119889 119872 119890V119890119899 (119888)(11987324 + 119873

2 ) times (119872 + 1)42 119873 119890V119890119899 119872 119900119889119889 (119889)

(34)






20

40

60

80

100

120

00 5 10 15 20 25 30

Eige

nval

ues (

dB)

Eigenvalue index

R

Rs

Ru












0

minus60

minus50

minus40

minus30

minus20

minus10N

orm

aliz

ed D

oppl

er fr

eque

ncy

00102030405

minus05

minus04

minus03

minus02

minus01

(a)


050minus05

0

minus60

minus50

minus40

minus30

minus20

minus10

Nor

mal

ized

Dop

pler

freq

uenc

y

00102030405

minus05

minus04

minus03

minus02

minus01

(b)


050minus05

0

minus60

minus50

minus40

minus30

minus20

minus10

Nor

mal

ized

Dop

pler

freq

uenc

y

00102030405

minus05

minus04

minus03

minus02

minus01

(c)


0 01 02 03 04 05

0

10

20


SIN

R (d

B)


minus10

minus20

minus30








13141516171819202122


Out

put S

INR

(dB)


102 103 104




Nor

mal

ized

Dop

pler

freq

uenc

y

Spatial frequency

0

0

0102030405

050

minus05

minus05minus60

minus50

minus40

minus30

minus20

minus10minus04

minus03

minus02

minus01

(a)


Nor

mal

ized

Dop

pler

freq

uenc

y

00102030405

minus05

minus04

minus03

minus02

minus01

0

minus60

minus50

minus40

minus30

minus20

minus10

(b)




119863119888119900119901119903119894119898119890 = (1198731015840111987310158402 + 1) times (1198721015840111987210158402 + 1) (35)










0 01 02 03 04 05

0

10

20


SIN

R (d

B)


minus10

minus20

minus30



205

21

215

22

225

23

235

24

245


Out

put S

INR

(dB)

Objective SINR


102 103 104








5 Conclusion


Appendix



119873 = 1198731 + 1198732119872 = 1198721 +1198722 (A1)


1198732 (1198731 + 1) le 11987322 + (1198731 + 1)22 (A2)


for odd 119873 1198601 = max 1198732 (1198731 + 1) = (119873 + 1)24 (A3a)

for even 119873 1198611 = max 1198732 (1198731 + 1) = 11987324 + 119873

2 (A3b)


for odd 1198731198731 = 119873 minus 1

2 1198732 = 119873 + 1

2 (A4a)

for even 119873 1198731 = 1198732 = 1198732 (A4b)


for odd 119872 1198602 = max 1198722 (1198721 + 1)= (119872 + 1)2

4 (A5a)

for even 119872 1198612 = max 1198722 (1198721 + 1)= 1198722

4 + 1198722 (A5b)


for odd 1198721198721 = 119872 minus 1

2 1198722 = 119872 + 1

2 (A6a)

for even 119872 1198721 = 1198722 = 1198722 (A6b)


119873 = 1198731015840 = 211987310158401 + 11987310158402 minus 1119872 = 1198721015840 = 211987210158401 +11987210158402 minus 1

(A7)


0 5 10 15 200

20

40

60

80

100

120

Values of N

Func

tion

valu

es

A1B1C1

(a)

A1B1C1

1 2 3 4 50

1

2

3

4

5

6

7

8

9

Values of N

Func

tion

valu

es(b)



1198731015840111987310158402 + 1 le 119873101584021 + 1198731015840222 + 1 (A8)


1198621 = max 1198731015840111987310158402 + 1 = (119873 + 1)29 + 1 (A9)


11987310158401 = 11987310158402 = 119873 + 13 (A10)


1198622 = max 1198721015840111987210158402 + 1 = (119872 + 1)29 + 1 (A11)


11987210158401 = 11987210158402 = 119872 + 13 (A12)




and119863119888119900119901119903119894119898119890 as follows

1198631199001199011199052minus119899119890119904119905119890119889

=

11986011198602 119873 119900119889119889 119872 119900119889119889 (119886)11986111198612 119873 119890V119890119899 119872 119890V119890119899 (119887)11986011198612 119873 119900119889119889 119872 119890V119890119899 (119888)11986111198602 119873 119890V119890119899 119872 119900119889119889 (119889)

(A13)

119863119888119900119901119903119894119898119890 lt 11986211198622 (A14)





Data Availability




Acknowledgments


References











































Journal of












Journal of







Volume 2018






Volume 2018










k (119891119904 119891119889) = b (119891119889) otimes a (119891119904) (11)



x119888 = c + n = 119873119888sum119894=1

120572119894b (119891119894119889119888) otimes a (119891119894119904119888) + n = 119873119888sum119894=1

120572119894k119888119894 + n

= 119873119888sum119894=1

c119894 + n

(12)


R119888 =119873119888sum119894=1

119864 [c119894c119867119894 ] =119873119888sum119894=1

1205902119894 k119888119894k119867119888119894 = V119888PV119888119867 (13)


R119899 = 1205902119899I119872119873 (14)


R119906 = 119864 [x119888x119867119888 ] = V119888PV119888119867 + 1205902119899I119872119873 = R119888 + R119899 (15)




(16)


119878119868119873119877 = 1205902119905 k119867119905 Rminus1119906 k119905 (17)



R119906 = 1119870X119888X

119867119888 (18)




(19)


[a (119891119904) a (119891119904)119867]119901119902 = 1198901198952120587119891119904(119899119901minus119899119902) = 1198901198952120587119891119904119899119901119902 (20)











Z = 119873119888sum119894=1

1205902119894 a (119891119894119904119888) b (119891119894119889119888)119879 + 1205902119899e1e1198792 (24)


z = 119873119888sum119894=1


1205902119894 k119888119894 + n

= c + n

(25)




0

0

helliphellip

helliphellip


Z

Z

Z00

(N minus 1)

minus(N minus 1)




Z120588120574 =119873119888sum119894=1

119890minus119895(120588119891119894119904119888+120574119891119894119889119888)1205902119894 a (119891119894119904119888) b (119891119894119889119888)119879 + 1205902119899e1120588e1198792120574 (26)



R119904 = 1119872119873119873minus1sum120588=0

119872minus1sum120574=0

z120588120574z119867120588120574 (27)


R119904 = 1119872119873R2119906 (28)

where

R119906 = V119888PV119867

119888 + 1205902119899I119872119873 (29)




R119906 = (119872119873R119904)12 (30)




minus1

119906 k119905(31)


119881119878119868119873119877 = 1205902119905 k119867119905 Rminus1119906 k119905 (32)


1198632minus119899119890119904119905119890119889 = 119872119873 = 1198722 (1198721 + 1) times 1198732 (1198731 + 1) (33)


1198631199001199011199052minus119899119890119904119905119890119889

=

(119873 + 1)42 times (119872 + 1)

42 119873 119900119889119889 119872 119900119889119889 (119886)

(11987324 + 1198732 ) times (11987224 + 119872

2 ) 119873 119890V119890119899 119872 119890V119890119899 (119887)(119873 + 1)

42 times (11987224 + 119872

2 ) 119873 119900119889119889 119872 119890V119890119899 (119888)(11987324 + 119873

2 ) times (119872 + 1)42 119873 119890V119890119899 119872 119900119889119889 (119889)

(34)






20

40

60

80

100

120

00 5 10 15 20 25 30

Eige

nval

ues (

dB)

Eigenvalue index

R

Rs

Ru












0

minus60

minus50

minus40

minus30

minus20

minus10N

orm

aliz

ed D

oppl

er fr

eque

ncy

00102030405

minus05

minus04

minus03

minus02

minus01

(a)


050minus05

0

minus60

minus50

minus40

minus30

minus20

minus10

Nor

mal

ized

Dop

pler

freq

uenc

y

00102030405

minus05

minus04

minus03

minus02

minus01

(b)


050minus05

0

minus60

minus50

minus40

minus30

minus20

minus10

Nor

mal

ized

Dop

pler

freq

uenc

y

00102030405

minus05

minus04

minus03

minus02

minus01

(c)


0 01 02 03 04 05

0

10

20


SIN

R (d

B)


minus10

minus20

minus30








13141516171819202122


Out

put S

INR

(dB)


102 103 104




Nor

mal

ized

Dop

pler

freq

uenc

y

Spatial frequency

0

0

0102030405

050

minus05

minus05minus60

minus50

minus40

minus30

minus20

minus10minus04

minus03

minus02

minus01

(a)


Nor

mal

ized

Dop

pler

freq

uenc

y

00102030405

minus05

minus04

minus03

minus02

minus01

0

minus60

minus50

minus40

minus30

minus20

minus10

(b)




119863119888119900119901119903119894119898119890 = (1198731015840111987310158402 + 1) times (1198721015840111987210158402 + 1) (35)










0 01 02 03 04 05

0

10

20


SIN

R (d

B)


minus10

minus20

minus30



205

21

215

22

225

23

235

24

245


Out

put S

INR

(dB)

Objective SINR


102 103 104








5 Conclusion


Appendix



119873 = 1198731 + 1198732119872 = 1198721 +1198722 (A1)


1198732 (1198731 + 1) le 11987322 + (1198731 + 1)22 (A2)


for odd 119873 1198601 = max 1198732 (1198731 + 1) = (119873 + 1)24 (A3a)

for even 119873 1198611 = max 1198732 (1198731 + 1) = 11987324 + 119873

2 (A3b)


for odd 1198731198731 = 119873 minus 1

2 1198732 = 119873 + 1

2 (A4a)

for even 119873 1198731 = 1198732 = 1198732 (A4b)


for odd 119872 1198602 = max 1198722 (1198721 + 1)= (119872 + 1)2

4 (A5a)

for even 119872 1198612 = max 1198722 (1198721 + 1)= 1198722

4 + 1198722 (A5b)


for odd 1198721198721 = 119872 minus 1

2 1198722 = 119872 + 1

2 (A6a)

for even 119872 1198721 = 1198722 = 1198722 (A6b)


119873 = 1198731015840 = 211987310158401 + 11987310158402 minus 1119872 = 1198721015840 = 211987210158401 +11987210158402 minus 1

(A7)


0 5 10 15 200

20

40

60

80

100

120

Values of N

Func

tion

valu

es

A1B1C1

(a)

A1B1C1

1 2 3 4 50

1

2

3

4

5

6

7

8

9

Values of N

Func

tion

valu

es(b)



1198731015840111987310158402 + 1 le 119873101584021 + 1198731015840222 + 1 (A8)


1198621 = max 1198731015840111987310158402 + 1 = (119873 + 1)29 + 1 (A9)


11987310158401 = 11987310158402 = 119873 + 13 (A10)


1198622 = max 1198721015840111987210158402 + 1 = (119872 + 1)29 + 1 (A11)


11987210158401 = 11987210158402 = 119872 + 13 (A12)




and119863119888119900119901119903119894119898119890 as follows

1198631199001199011199052minus119899119890119904119905119890119889

=

11986011198602 119873 119900119889119889 119872 119900119889119889 (119886)11986111198612 119873 119890V119890119899 119872 119890V119890119899 (119887)11986011198612 119873 119900119889119889 119872 119890V119890119899 (119888)11986111198602 119873 119890V119890119899 119872 119900119889119889 (119889)

(A13)

119863119888119900119901119903119894119898119890 lt 11986211198622 (A14)





Data Availability




Acknowledgments


References











































Journal of












Journal of







Volume 2018






Volume 2018














Z = 119873119888sum119894=1

1205902119894 a (119891119894119904119888) b (119891119894119889119888)119879 + 1205902119899e1e1198792 (24)


z = 119873119888sum119894=1


1205902119894 k119888119894 + n

= c + n

(25)




0

0

helliphellip

helliphellip


Z

Z

Z00

(N minus 1)

minus(N minus 1)




Z120588120574 =119873119888sum119894=1

119890minus119895(120588119891119894119904119888+120574119891119894119889119888)1205902119894 a (119891119894119904119888) b (119891119894119889119888)119879 + 1205902119899e1120588e1198792120574 (26)



R119904 = 1119872119873119873minus1sum120588=0

119872minus1sum120574=0

z120588120574z119867120588120574 (27)


R119904 = 1119872119873R2119906 (28)

where

R119906 = V119888PV119867

119888 + 1205902119899I119872119873 (29)




R119906 = (119872119873R119904)12 (30)




minus1

119906 k119905(31)


119881119878119868119873119877 = 1205902119905 k119867119905 Rminus1119906 k119905 (32)


1198632minus119899119890119904119905119890119889 = 119872119873 = 1198722 (1198721 + 1) times 1198732 (1198731 + 1) (33)


1198631199001199011199052minus119899119890119904119905119890119889

=

(119873 + 1)42 times (119872 + 1)

42 119873 119900119889119889 119872 119900119889119889 (119886)

(11987324 + 1198732 ) times (11987224 + 119872

2 ) 119873 119890V119890119899 119872 119890V119890119899 (119887)(119873 + 1)

42 times (11987224 + 119872

2 ) 119873 119900119889119889 119872 119890V119890119899 (119888)(11987324 + 119873

2 ) times (119872 + 1)42 119873 119890V119890119899 119872 119900119889119889 (119889)

(34)






20

40

60

80

100

120

00 5 10 15 20 25 30

Eige

nval

ues (

dB)

Eigenvalue index

R

Rs

Ru












0

minus60

minus50

minus40

minus30

minus20

minus10N

orm

aliz

ed D

oppl

er fr

eque

ncy

00102030405

minus05

minus04

minus03

minus02

minus01

(a)


050minus05

0

minus60

minus50

minus40

minus30

minus20

minus10

Nor

mal

ized

Dop

pler

freq

uenc

y

00102030405

minus05

minus04

minus03

minus02

minus01

(b)


050minus05

0

minus60

minus50

minus40

minus30

minus20

minus10

Nor

mal

ized

Dop

pler

freq

uenc

y

00102030405

minus05

minus04

minus03

minus02

minus01

(c)


0 01 02 03 04 05

0

10

20


SIN

R (d

B)


minus10

minus20

minus30








13141516171819202122


Out

put S

INR

(dB)


102 103 104




Nor

mal

ized

Dop

pler

freq

uenc

y

Spatial frequency

0

0

0102030405

050

minus05

minus05minus60

minus50

minus40

minus30

minus20

minus10minus04

minus03

minus02

minus01

(a)


Nor

mal

ized

Dop

pler

freq

uenc

y

00102030405

minus05

minus04

minus03

minus02

minus01

0

minus60

minus50

minus40

minus30

minus20

minus10

(b)




119863119888119900119901119903119894119898119890 = (1198731015840111987310158402 + 1) times (1198721015840111987210158402 + 1) (35)










0 01 02 03 04 05

0

10

20


SIN

R (d

B)


minus10

minus20

minus30



205

21

215

22

225

23

235

24

245


Out

put S

INR

(dB)

Objective SINR


102 103 104








5 Conclusion


Appendix



119873 = 1198731 + 1198732119872 = 1198721 +1198722 (A1)


1198732 (1198731 + 1) le 11987322 + (1198731 + 1)22 (A2)


for odd 119873 1198601 = max 1198732 (1198731 + 1) = (119873 + 1)24 (A3a)

for even 119873 1198611 = max 1198732 (1198731 + 1) = 11987324 + 119873

2 (A3b)


for odd 1198731198731 = 119873 minus 1

2 1198732 = 119873 + 1

2 (A4a)

for even 119873 1198731 = 1198732 = 1198732 (A4b)


for odd 119872 1198602 = max 1198722 (1198721 + 1)= (119872 + 1)2

4 (A5a)

for even 119872 1198612 = max 1198722 (1198721 + 1)= 1198722

4 + 1198722 (A5b)


for odd 1198721198721 = 119872 minus 1

2 1198722 = 119872 + 1

2 (A6a)

for even 119872 1198721 = 1198722 = 1198722 (A6b)


119873 = 1198731015840 = 211987310158401 + 11987310158402 minus 1119872 = 1198721015840 = 211987210158401 +11987210158402 minus 1

(A7)


0 5 10 15 200

20

40

60

80

100

120

Values of N

Func

tion

valu

es

A1B1C1

(a)

A1B1C1

1 2 3 4 50

1

2

3

4

5

6

7

8

9

Values of N

Func

tion

valu

es(b)



1198731015840111987310158402 + 1 le 119873101584021 + 1198731015840222 + 1 (A8)


1198621 = max 1198731015840111987310158402 + 1 = (119873 + 1)29 + 1 (A9)


11987310158401 = 11987310158402 = 119873 + 13 (A10)


1198622 = max 1198721015840111987210158402 + 1 = (119872 + 1)29 + 1 (A11)


11987210158401 = 11987210158402 = 119872 + 13 (A12)




and119863119888119900119901119903119894119898119890 as follows

1198631199001199011199052minus119899119890119904119905119890119889

=

11986011198602 119873 119900119889119889 119872 119900119889119889 (119886)11986111198612 119873 119890V119890119899 119872 119890V119890119899 (119887)11986011198612 119873 119900119889119889 119872 119890V119890119899 (119888)11986111198602 119873 119890V119890119899 119872 119900119889119889 (119889)

(A13)

119863119888119900119901119903119894119898119890 lt 11986211198622 (A14)





Data Availability




Acknowledgments


References











































Journal of












Journal of







Volume 2018






Volume 2018










R119906 = (119872119873R119904)12 (30)




minus1

119906 k119905(31)


119881119878119868119873119877 = 1205902119905 k119867119905 Rminus1119906 k119905 (32)


1198632minus119899119890119904119905119890119889 = 119872119873 = 1198722 (1198721 + 1) times 1198732 (1198731 + 1) (33)


1198631199001199011199052minus119899119890119904119905119890119889

=

(119873 + 1)42 times (119872 + 1)

42 119873 119900119889119889 119872 119900119889119889 (119886)

(11987324 + 1198732 ) times (11987224 + 119872

2 ) 119873 119890V119890119899 119872 119890V119890119899 (119887)(119873 + 1)

42 times (11987224 + 119872

2 ) 119873 119900119889119889 119872 119890V119890119899 (119888)(11987324 + 119873

2 ) times (119872 + 1)42 119873 119890V119890119899 119872 119900119889119889 (119889)

(34)






20

40

60

80

100

120

00 5 10 15 20 25 30

Eige

nval

ues (

dB)

Eigenvalue index

R

Rs

Ru












0

minus60

minus50

minus40

minus30

minus20

minus10N

orm

aliz

ed D

oppl

er fr

eque

ncy

00102030405

minus05

minus04

minus03

minus02

minus01

(a)


050minus05

0

minus60

minus50

minus40

minus30

minus20

minus10

Nor

mal

ized

Dop

pler

freq

uenc

y

00102030405

minus05

minus04

minus03

minus02

minus01

(b)


050minus05

0

minus60

minus50

minus40

minus30

minus20

minus10

Nor

mal

ized

Dop

pler

freq

uenc

y

00102030405

minus05

minus04

minus03

minus02

minus01

(c)


0 01 02 03 04 05

0

10

20


SIN

R (d

B)


minus10

minus20

minus30








13141516171819202122


Out

put S

INR

(dB)


102 103 104




Nor

mal

ized

Dop

pler

freq

uenc

y

Spatial frequency

0

0

0102030405

050

minus05

minus05minus60

minus50

minus40

minus30

minus20

minus10minus04

minus03

minus02

minus01

(a)


Nor

mal

ized

Dop

pler

freq

uenc

y

00102030405

minus05

minus04

minus03

minus02

minus01

0

minus60

minus50

minus40

minus30

minus20

minus10

(b)




119863119888119900119901119903119894119898119890 = (1198731015840111987310158402 + 1) times (1198721015840111987210158402 + 1) (35)










0 01 02 03 04 05

0

10

20


SIN

R (d

B)


minus10

minus20

minus30



205

21

215

22

225

23

235

24

245


Out

put S

INR

(dB)

Objective SINR


102 103 104








5 Conclusion


Appendix



119873 = 1198731 + 1198732119872 = 1198721 +1198722 (A1)


1198732 (1198731 + 1) le 11987322 + (1198731 + 1)22 (A2)


for odd 119873 1198601 = max 1198732 (1198731 + 1) = (119873 + 1)24 (A3a)

for even 119873 1198611 = max 1198732 (1198731 + 1) = 11987324 + 119873

2 (A3b)


for odd 1198731198731 = 119873 minus 1

2 1198732 = 119873 + 1

2 (A4a)

for even 119873 1198731 = 1198732 = 1198732 (A4b)


for odd 119872 1198602 = max 1198722 (1198721 + 1)= (119872 + 1)2

4 (A5a)

for even 119872 1198612 = max 1198722 (1198721 + 1)= 1198722

4 + 1198722 (A5b)


for odd 1198721198721 = 119872 minus 1

2 1198722 = 119872 + 1

2 (A6a)

for even 119872 1198721 = 1198722 = 1198722 (A6b)


119873 = 1198731015840 = 211987310158401 + 11987310158402 minus 1119872 = 1198721015840 = 211987210158401 +11987210158402 minus 1

(A7)


0 5 10 15 200

20

40

60

80

100

120

Values of N

Func

tion

valu

es

A1B1C1

(a)

A1B1C1

1 2 3 4 50

1

2

3

4

5

6

7

8

9

Values of N

Func

tion

valu

es(b)



1198731015840111987310158402 + 1 le 119873101584021 + 1198731015840222 + 1 (A8)


1198621 = max 1198731015840111987310158402 + 1 = (119873 + 1)29 + 1 (A9)


11987310158401 = 11987310158402 = 119873 + 13 (A10)


1198622 = max 1198721015840111987210158402 + 1 = (119872 + 1)29 + 1 (A11)


11987210158401 = 11987210158402 = 119872 + 13 (A12)




and119863119888119900119901119903119894119898119890 as follows

1198631199001199011199052minus119899119890119904119905119890119889

=

11986011198602 119873 119900119889119889 119872 119900119889119889 (119886)11986111198612 119873 119890V119890119899 119872 119890V119890119899 (119887)11986011198612 119873 119900119889119889 119872 119890V119890119899 (119888)11986111198602 119873 119890V119890119899 119872 119900119889119889 (119889)

(A13)

119863119888119900119901119903119894119898119890 lt 11986211198622 (A14)





Data Availability




Acknowledgments


References











































Journal of












Journal of







Volume 2018






Volume 2018









20

40

60

80

100

120

00 5 10 15 20 25 30

Eige

nval

ues (

dB)

Eigenvalue index

R

Rs

Ru












0

minus60

minus50

minus40

minus30

minus20

minus10N

orm

aliz

ed D

oppl

er fr

eque

ncy

00102030405

minus05

minus04

minus03

minus02

minus01

(a)


050minus05

0

minus60

minus50

minus40

minus30

minus20

minus10

Nor

mal

ized

Dop

pler

freq

uenc

y

00102030405

minus05

minus04

minus03

minus02

minus01

(b)


050minus05

0

minus60

minus50

minus40

minus30

minus20

minus10

Nor

mal

ized

Dop

pler

freq

uenc

y

00102030405

minus05

minus04

minus03

minus02

minus01

(c)


0 01 02 03 04 05

0

10

20


SIN

R (d

B)


minus10

minus20

minus30








13141516171819202122


Out

put S

INR

(dB)


102 103 104




Nor

mal

ized

Dop

pler

freq

uenc

y

Spatial frequency

0

0

0102030405

050

minus05

minus05minus60

minus50

minus40

minus30

minus20

minus10minus04

minus03

minus02

minus01

(a)


Nor

mal

ized

Dop

pler

freq

uenc

y

00102030405

minus05

minus04

minus03

minus02

minus01

0

minus60

minus50

minus40

minus30

minus20

minus10

(b)




119863119888119900119901119903119894119898119890 = (1198731015840111987310158402 + 1) times (1198721015840111987210158402 + 1) (35)










0 01 02 03 04 05

0

10

20


SIN

R (d

B)


minus10

minus20

minus30



205

21

215

22

225

23

235

24

245


Out

put S

INR

(dB)

Objective SINR


102 103 104








5 Conclusion


Appendix



119873 = 1198731 + 1198732119872 = 1198721 +1198722 (A1)


1198732 (1198731 + 1) le 11987322 + (1198731 + 1)22 (A2)


for odd 119873 1198601 = max 1198732 (1198731 + 1) = (119873 + 1)24 (A3a)

for even 119873 1198611 = max 1198732 (1198731 + 1) = 11987324 + 119873

2 (A3b)


for odd 1198731198731 = 119873 minus 1

2 1198732 = 119873 + 1

2 (A4a)

for even 119873 1198731 = 1198732 = 1198732 (A4b)


for odd 119872 1198602 = max 1198722 (1198721 + 1)= (119872 + 1)2

4 (A5a)

for even 119872 1198612 = max 1198722 (1198721 + 1)= 1198722

4 + 1198722 (A5b)


for odd 1198721198721 = 119872 minus 1

2 1198722 = 119872 + 1

2 (A6a)

for even 119872 1198721 = 1198722 = 1198722 (A6b)


119873 = 1198731015840 = 211987310158401 + 11987310158402 minus 1119872 = 1198721015840 = 211987210158401 +11987210158402 minus 1

(A7)


0 5 10 15 200

20

40

60

80

100

120

Values of N

Func

tion

valu

es

A1B1C1

(a)

A1B1C1

1 2 3 4 50

1

2

3

4

5

6

7

8

9

Values of N

Func

tion

valu

es(b)



1198731015840111987310158402 + 1 le 119873101584021 + 1198731015840222 + 1 (A8)


1198621 = max 1198731015840111987310158402 + 1 = (119873 + 1)29 + 1 (A9)


11987310158401 = 11987310158402 = 119873 + 13 (A10)


1198622 = max 1198721015840111987210158402 + 1 = (119872 + 1)29 + 1 (A11)


11987210158401 = 11987210158402 = 119872 + 13 (A12)




and119863119888119900119901119903119894119898119890 as follows

1198631199001199011199052minus119899119890119904119905119890119889

=

11986011198602 119873 119900119889119889 119872 119900119889119889 (119886)11986111198612 119873 119890V119890119899 119872 119890V119890119899 (119887)11986011198612 119873 119900119889119889 119872 119890V119890119899 (119888)11986111198602 119873 119890V119890119899 119872 119900119889119889 (119889)

(A13)

119863119888119900119901119903119894119898119890 lt 11986211198622 (A14)





Data Availability




Acknowledgments


References











































Journal of












Journal of







Volume 2018






Volume 2018










0

minus60

minus50

minus40

minus30

minus20

minus10N

orm

aliz

ed D

oppl

er fr

eque

ncy

00102030405

minus05

minus04

minus03

minus02

minus01

(a)


050minus05

0

minus60

minus50

minus40

minus30

minus20

minus10

Nor

mal

ized

Dop

pler

freq

uenc

y

00102030405

minus05

minus04

minus03

minus02

minus01

(b)


050minus05

0

minus60

minus50

minus40

minus30

minus20

minus10

Nor

mal

ized

Dop

pler

freq

uenc

y

00102030405

minus05

minus04

minus03

minus02

minus01

(c)


0 01 02 03 04 05

0

10

20


SIN

R (d

B)


minus10

minus20

minus30








13141516171819202122


Out

put S

INR

(dB)


102 103 104




Nor

mal

ized

Dop

pler

freq

uenc

y

Spatial frequency

0

0

0102030405

050

minus05

minus05minus60

minus50

minus40

minus30

minus20

minus10minus04

minus03

minus02

minus01

(a)


Nor

mal

ized

Dop

pler

freq

uenc

y

00102030405

minus05

minus04

minus03

minus02

minus01

0

minus60

minus50

minus40

minus30

minus20

minus10

(b)




119863119888119900119901119903119894119898119890 = (1198731015840111987310158402 + 1) times (1198721015840111987210158402 + 1) (35)










0 01 02 03 04 05

0

10

20


SIN

R (d

B)


minus10

minus20

minus30



205

21

215

22

225

23

235

24

245


Out

put S

INR

(dB)

Objective SINR


102 103 104








5 Conclusion


Appendix



119873 = 1198731 + 1198732119872 = 1198721 +1198722 (A1)


1198732 (1198731 + 1) le 11987322 + (1198731 + 1)22 (A2)


for odd 119873 1198601 = max 1198732 (1198731 + 1) = (119873 + 1)24 (A3a)

for even 119873 1198611 = max 1198732 (1198731 + 1) = 11987324 + 119873

2 (A3b)


for odd 1198731198731 = 119873 minus 1

2 1198732 = 119873 + 1

2 (A4a)

for even 119873 1198731 = 1198732 = 1198732 (A4b)


for odd 119872 1198602 = max 1198722 (1198721 + 1)= (119872 + 1)2

4 (A5a)

for even 119872 1198612 = max 1198722 (1198721 + 1)= 1198722

4 + 1198722 (A5b)


for odd 1198721198721 = 119872 minus 1

2 1198722 = 119872 + 1

2 (A6a)

for even 119872 1198721 = 1198722 = 1198722 (A6b)


119873 = 1198731015840 = 211987310158401 + 11987310158402 minus 1119872 = 1198721015840 = 211987210158401 +11987210158402 minus 1

(A7)


0 5 10 15 200

20

40

60

80

100

120

Values of N

Func

tion

valu

es

A1B1C1

(a)

A1B1C1

1 2 3 4 50

1

2

3

4

5

6

7

8

9

Values of N

Func

tion

valu

es(b)



1198731015840111987310158402 + 1 le 119873101584021 + 1198731015840222 + 1 (A8)


1198621 = max 1198731015840111987310158402 + 1 = (119873 + 1)29 + 1 (A9)


11987310158401 = 11987310158402 = 119873 + 13 (A10)


1198622 = max 1198721015840111987210158402 + 1 = (119872 + 1)29 + 1 (A11)


11987210158401 = 11987210158402 = 119872 + 13 (A12)




and119863119888119900119901119903119894119898119890 as follows

1198631199001199011199052minus119899119890119904119905119890119889

=

11986011198602 119873 119900119889119889 119872 119900119889119889 (119886)11986111198612 119873 119890V119890119899 119872 119890V119890119899 (119887)11986011198612 119873 119900119889119889 119872 119890V119890119899 (119888)11986111198602 119873 119890V119890119899 119872 119900119889119889 (119889)

(A13)

119863119888119900119901119903119894119898119890 lt 11986211198622 (A14)





Data Availability




Acknowledgments


References











































Journal of












Journal of







Volume 2018






Volume 2018













13141516171819202122


Out

put S

INR

(dB)


102 103 104




Nor

mal

ized

Dop

pler

freq

uenc

y

Spatial frequency

0

0

0102030405

050

minus05

minus05minus60

minus50

minus40

minus30

minus20

minus10minus04

minus03

minus02

minus01

(a)


Nor

mal

ized

Dop

pler

freq

uenc

y

00102030405

minus05

minus04

minus03

minus02

minus01

0

minus60

minus50

minus40

minus30

minus20

minus10

(b)




119863119888119900119901119903119894119898119890 = (1198731015840111987310158402 + 1) times (1198721015840111987210158402 + 1) (35)










0 01 02 03 04 05

0

10

20


SIN

R (d

B)


minus10

minus20

minus30



205

21

215

22

225

23

235

24

245


Out

put S

INR

(dB)

Objective SINR


102 103 104








5 Conclusion


Appendix



119873 = 1198731 + 1198732119872 = 1198721 +1198722 (A1)


1198732 (1198731 + 1) le 11987322 + (1198731 + 1)22 (A2)


for odd 119873 1198601 = max 1198732 (1198731 + 1) = (119873 + 1)24 (A3a)

for even 119873 1198611 = max 1198732 (1198731 + 1) = 11987324 + 119873

2 (A3b)


for odd 1198731198731 = 119873 minus 1

2 1198732 = 119873 + 1

2 (A4a)

for even 119873 1198731 = 1198732 = 1198732 (A4b)


for odd 119872 1198602 = max 1198722 (1198721 + 1)= (119872 + 1)2

4 (A5a)

for even 119872 1198612 = max 1198722 (1198721 + 1)= 1198722

4 + 1198722 (A5b)


for odd 1198721198721 = 119872 minus 1

2 1198722 = 119872 + 1

2 (A6a)

for even 119872 1198721 = 1198722 = 1198722 (A6b)


119873 = 1198731015840 = 211987310158401 + 11987310158402 minus 1119872 = 1198721015840 = 211987210158401 +11987210158402 minus 1

(A7)


0 5 10 15 200

20

40

60

80

100

120

Values of N

Func

tion

valu

es

A1B1C1

(a)

A1B1C1

1 2 3 4 50

1

2

3

4

5

6

7

8

9

Values of N

Func

tion

valu

es(b)



1198731015840111987310158402 + 1 le 119873101584021 + 1198731015840222 + 1 (A8)


1198621 = max 1198731015840111987310158402 + 1 = (119873 + 1)29 + 1 (A9)


11987310158401 = 11987310158402 = 119873 + 13 (A10)


1198622 = max 1198721015840111987210158402 + 1 = (119872 + 1)29 + 1 (A11)


11987210158401 = 11987210158402 = 119872 + 13 (A12)




and119863119888119900119901119903119894119898119890 as follows

1198631199001199011199052minus119899119890119904119905119890119889

=

11986011198602 119873 119900119889119889 119872 119900119889119889 (119886)11986111198612 119873 119890V119890119899 119872 119890V119890119899 (119887)11986011198612 119873 119900119889119889 119872 119890V119890119899 (119888)11986111198602 119873 119890V119890119899 119872 119900119889119889 (119889)

(A13)

119863119888119900119901119903119894119898119890 lt 11986211198622 (A14)





Data Availability




Acknowledgments


References











































Journal of












Journal of







Volume 2018






Volume 2018









Nor

mal

ized

Dop

pler

freq

uenc

y

Spatial frequency

0

0

0102030405

050

minus05

minus05minus60

minus50

minus40

minus30

minus20

minus10minus04

minus03

minus02

minus01

(a)


Nor

mal

ized

Dop

pler

freq

uenc

y

00102030405

minus05

minus04

minus03

minus02

minus01

0

minus60

minus50

minus40

minus30

minus20

minus10

(b)




119863119888119900119901119903119894119898119890 = (1198731015840111987310158402 + 1) times (1198721015840111987210158402 + 1) (35)










0 01 02 03 04 05

0

10

20


SIN

R (d

B)


minus10

minus20

minus30



205

21

215

22

225

23

235

24

245


Out

put S

INR

(dB)

Objective SINR


102 103 104








5 Conclusion


Appendix



119873 = 1198731 + 1198732119872 = 1198721 +1198722 (A1)


1198732 (1198731 + 1) le 11987322 + (1198731 + 1)22 (A2)


for odd 119873 1198601 = max 1198732 (1198731 + 1) = (119873 + 1)24 (A3a)

for even 119873 1198611 = max 1198732 (1198731 + 1) = 11987324 + 119873

2 (A3b)


for odd 1198731198731 = 119873 minus 1

2 1198732 = 119873 + 1

2 (A4a)

for even 119873 1198731 = 1198732 = 1198732 (A4b)


for odd 119872 1198602 = max 1198722 (1198721 + 1)= (119872 + 1)2

4 (A5a)

for even 119872 1198612 = max 1198722 (1198721 + 1)= 1198722

4 + 1198722 (A5b)


for odd 1198721198721 = 119872 minus 1

2 1198722 = 119872 + 1

2 (A6a)

for even 119872 1198721 = 1198722 = 1198722 (A6b)


119873 = 1198731015840 = 211987310158401 + 11987310158402 minus 1119872 = 1198721015840 = 211987210158401 +11987210158402 minus 1

(A7)


0 5 10 15 200

20

40

60

80

100

120

Values of N

Func

tion

valu

es

A1B1C1

(a)

A1B1C1

1 2 3 4 50

1

2

3

4

5

6

7

8

9

Values of N

Func

tion

valu

es(b)



1198731015840111987310158402 + 1 le 119873101584021 + 1198731015840222 + 1 (A8)


1198621 = max 1198731015840111987310158402 + 1 = (119873 + 1)29 + 1 (A9)


11987310158401 = 11987310158402 = 119873 + 13 (A10)


1198622 = max 1198721015840111987210158402 + 1 = (119872 + 1)29 + 1 (A11)


11987210158401 = 11987210158402 = 119872 + 13 (A12)




and119863119888119900119901119903119894119898119890 as follows

1198631199001199011199052minus119899119890119904119905119890119889

=

11986011198602 119873 119900119889119889 119872 119900119889119889 (119886)11986111198612 119873 119890V119890119899 119872 119890V119890119899 (119887)11986011198612 119873 119900119889119889 119872 119890V119890119899 (119888)11986111198602 119873 119890V119890119899 119872 119900119889119889 (119889)

(A13)

119863119888119900119901119903119894119898119890 lt 11986211198622 (A14)





Data Availability




Acknowledgments


References











































Journal of












Journal of







Volume 2018






Volume 2018









0 01 02 03 04 05

0

10

20


SIN

R (d

B)


minus10

minus20

minus30



205

21

215

22

225

23

235

24

245


Out

put S

INR

(dB)

Objective SINR


102 103 104








5 Conclusion


Appendix



119873 = 1198731 + 1198732119872 = 1198721 +1198722 (A1)


1198732 (1198731 + 1) le 11987322 + (1198731 + 1)22 (A2)


for odd 119873 1198601 = max 1198732 (1198731 + 1) = (119873 + 1)24 (A3a)

for even 119873 1198611 = max 1198732 (1198731 + 1) = 11987324 + 119873

2 (A3b)


for odd 1198731198731 = 119873 minus 1

2 1198732 = 119873 + 1

2 (A4a)

for even 119873 1198731 = 1198732 = 1198732 (A4b)


for odd 119872 1198602 = max 1198722 (1198721 + 1)= (119872 + 1)2

4 (A5a)

for even 119872 1198612 = max 1198722 (1198721 + 1)= 1198722

4 + 1198722 (A5b)


for odd 1198721198721 = 119872 minus 1

2 1198722 = 119872 + 1

2 (A6a)

for even 119872 1198721 = 1198722 = 1198722 (A6b)


119873 = 1198731015840 = 211987310158401 + 11987310158402 minus 1119872 = 1198721015840 = 211987210158401 +11987210158402 minus 1

(A7)


0 5 10 15 200

20

40

60

80

100

120

Values of N

Func

tion

valu

es

A1B1C1

(a)

A1B1C1

1 2 3 4 50

1

2

3

4

5

6

7

8

9

Values of N

Func

tion

valu

es(b)



1198731015840111987310158402 + 1 le 119873101584021 + 1198731015840222 + 1 (A8)


1198621 = max 1198731015840111987310158402 + 1 = (119873 + 1)29 + 1 (A9)


11987310158401 = 11987310158402 = 119873 + 13 (A10)


1198622 = max 1198721015840111987210158402 + 1 = (119872 + 1)29 + 1 (A11)


11987210158401 = 11987210158402 = 119872 + 13 (A12)




and119863119888119900119901119903119894119898119890 as follows

1198631199001199011199052minus119899119890119904119905119890119889

=

11986011198602 119873 119900119889119889 119872 119900119889119889 (119886)11986111198612 119873 119890V119890119899 119872 119890V119890119899 (119887)11986011198612 119873 119900119889119889 119872 119890V119890119899 (119888)11986111198602 119873 119890V119890119899 119872 119900119889119889 (119889)

(A13)

119863119888119900119901119903119894119898119890 lt 11986211198622 (A14)





Data Availability




Acknowledgments


References











































Journal of












Journal of







Volume 2018






Volume 2018











5 Conclusion


Appendix



119873 = 1198731 + 1198732119872 = 1198721 +1198722 (A1)


1198732 (1198731 + 1) le 11987322 + (1198731 + 1)22 (A2)


for odd 119873 1198601 = max 1198732 (1198731 + 1) = (119873 + 1)24 (A3a)

for even 119873 1198611 = max 1198732 (1198731 + 1) = 11987324 + 119873

2 (A3b)


for odd 1198731198731 = 119873 minus 1

2 1198732 = 119873 + 1

2 (A4a)

for even 119873 1198731 = 1198732 = 1198732 (A4b)


for odd 119872 1198602 = max 1198722 (1198721 + 1)= (119872 + 1)2

4 (A5a)

for even 119872 1198612 = max 1198722 (1198721 + 1)= 1198722

4 + 1198722 (A5b)


for odd 1198721198721 = 119872 minus 1

2 1198722 = 119872 + 1

2 (A6a)

for even 119872 1198721 = 1198722 = 1198722 (A6b)


119873 = 1198731015840 = 211987310158401 + 11987310158402 minus 1119872 = 1198721015840 = 211987210158401 +11987210158402 minus 1

(A7)


0 5 10 15 200

20

40

60

80

100

120

Values of N

Func

tion

valu

es

A1B1C1

(a)

A1B1C1

1 2 3 4 50

1

2

3

4

5

6

7

8

9

Values of N

Func

tion

valu

es(b)



1198731015840111987310158402 + 1 le 119873101584021 + 1198731015840222 + 1 (A8)


1198621 = max 1198731015840111987310158402 + 1 = (119873 + 1)29 + 1 (A9)


11987310158401 = 11987310158402 = 119873 + 13 (A10)


1198622 = max 1198721015840111987210158402 + 1 = (119872 + 1)29 + 1 (A11)


11987210158401 = 11987210158402 = 119872 + 13 (A12)




and119863119888119900119901119903119894119898119890 as follows

1198631199001199011199052minus119899119890119904119905119890119889

=

11986011198602 119873 119900119889119889 119872 119900119889119889 (119886)11986111198612 119873 119890V119890119899 119872 119890V119890119899 (119887)11986011198612 119873 119900119889119889 119872 119890V119890119899 (119888)11986111198602 119873 119890V119890119899 119872 119900119889119889 (119889)

(A13)

119863119888119900119901119903119894119898119890 lt 11986211198622 (A14)





Data Availability




Acknowledgments


References











































Journal of












Journal of







Volume 2018






Volume 2018









0 5 10 15 200

20

40

60

80

100

120

Values of N

Func

tion

valu

es

A1B1C1

(a)

A1B1C1

1 2 3 4 50

1

2

3

4

5

6

7

8

9

Values of N

Func

tion

valu

es(b)



1198731015840111987310158402 + 1 le 119873101584021 + 1198731015840222 + 1 (A8)


1198621 = max 1198731015840111987310158402 + 1 = (119873 + 1)29 + 1 (A9)


11987310158401 = 11987310158402 = 119873 + 13 (A10)


1198622 = max 1198721015840111987210158402 + 1 = (119872 + 1)29 + 1 (A11)


11987210158401 = 11987210158402 = 119872 + 13 (A12)




and119863119888119900119901119903119894119898119890 as follows

1198631199001199011199052minus119899119890119904119905119890119889

=

11986011198602 119873 119900119889119889 119872 119900119889119889 (119886)11986111198612 119873 119890V119890119899 119872 119890V119890119899 (119887)11986011198612 119873 119900119889119889 119872 119890V119890119899 (119888)11986111198602 119873 119890V119890119899 119872 119900119889119889 (119889)

(A13)

119863119888119900119901119903119894119898119890 lt 11986211198622 (A14)





Data Availability




Acknowledgments


References











































Journal of












Journal of







Volume 2018






Volume 2018









Data Availability




Acknowledgments


References











































Journal of












Journal of







Volume 2018






Volume 2018





















Journal of












Journal of







Volume 2018






Volume 2018















Journal of












Journal of







Volume 2018






Volume 2018








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A Novel Two-Level Nested STAP Strategy for Clutter ...downloads.hindawi.com/journals/mpe/2019/2540858.pdfpotential to detect a slow-moving weak target since the MIMO radar can gain